FOR THE PEOPLE FOR EDVCATION FOR SCIENCE LIBRARY OF THE AMERICAN MUSEUM OF NATURAL HISTORY wah ; REP ; we aoe sd nt LE J sy Kent ey, it on aah 4 ee? Me A if f wt ‘Ab 5 ata 6 vr é hit vl ' ne Nea oe wy beft r LM IN Were bo) hi ae fens k KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:- — SO, Ob(UQ ts Mi ke PROGEEDINGS OFTHE eee LION MOE SCIENCES VOLUME XXI fee Sey \ RA. JOHANNES MULLER :—: AMSTERDAM CT OBER 1919... KONINKLIJKE AKADEMIE VAN _ WETENSCHAPPEN - TE AMSTERDAM -:- PROCEEDINGS OF THE SEE TIONGOF SCIENCES VOLUME XXI = io" BART) ES as. ae ON: pee JOHANNES MULLER :—: AMSTERDAM : APRIL 1919 Bs: Las cn vn Ge | Ee, ‘ Translated from : Veqiates van Gewone Vexeee n ly Proceedings N°. Ne N°. N° CONTENTS. Lac rib kes 467 ernaTnos KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS VOLUME XXI NS. tand 2. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vol. XXVI and XXVII). CONTENTS. J. J. VAN LAAR: “On the Course of the Values of a and b for Hydrogen at Different Temperatures and Volumes” III] and IV (Continued). (Communicated by Prof. H: A. LORENTZ), p. 2 and p. 16. aC: BEND “On the Peripheral Sensitive Nervous System”. (Communicated by Prof. J. BOEKE), p. 26. W. J. A. SCHOUTEN: “On the Parallax of some Stellar Clusters”. (Second communication). (Commu- nicated by Prof. J. C. KAPTEYN), p. 36. C. EIJKMAN and D. J. HULSHOFF POL: “Experiments with Animals on the Nutritive Value of Standard Brown-Bread and White-Bread’’, p, 48. T. EHRENFEST-AFANASSJEWA: “An indeterminateness in the interpretation of the entropy as log W”. (Communicated by Prof. J. P. KUENEN), p. 53. J. F. VAN BEMMELEN: “On the primary character of the markings in Lepidopterous pupae”, p. 58. GUNNAR NORDSTROM: “Calculation of some special cases, in EINSTEIN’s theory of gravitation”. (Communicated by Prof. H. A. LORENTZ), p. 68. J. BOESEKEN EK, VAN LOON: “Determination of the Configuration of cis-trans isomeric sub- stances”, p. 80. A. F. HOLLEMAN and B. F. H.J. MATTHES: “The Addition of Hydrogenbromide to Allylbromide”, p. 90. L. S. ORNSTEIN: “The variability with time of the distributions of Emulsion-particles”. (Communi- cated by Prof. H. A. LORENTZ), p 92. L. S. ORNSTEIN: “On the Brownian Motion”. (Communicated by Prof. H. A. LORENTZ), p. 96. L. S. ORNSTEIN and F. ZERNIKE: “The Theory of the Brownian Motion and Statistical Mechanics”. (Communicated by Prof. H. A. LORENTZ), p. 109. L. S. ORNSTEIN and F. ZERNIKE: “The Scattering of Light by Irregular Refraction in the Sun”, (Com- municated by Prof. W. H. JULIUS), p. 115. S. W. VISSER: “On the diffraction of the light in the formation of halos. II A research of the colours observed in halo-phenomena”. (Communicated by Dr. J. P. VAN DER STOK), p. 119. ARNAUD DENJOY: “Nouvelle, démonstration du théorème de JORDAN sur les courbes planes”. (Communicated by Prof. L. E. J. BROUWER), p. 125. H. ZWAARDEMAKER and F. HOGEWIND: “On the spontaneous transformation to a colloidal state of solutions of odorous substances by exposure to ultra-violet light”, p. 131. A. H. W. ATEN: “The Passivity of Chromium”. (Third Communication). (Communicated by Prof. A. F. HOLLEMAN), p. 138. S. DE BOER: “On the influence of the increase of the osmotic pressure of the fluids of the body on different cell-substrata”. (Communicated by Prof. G. VAN RIJNBERK), p. 151. A. SMITS: “On the Electrochemical Behaviour of Metals”. (Communicated by Prof. P. ZEEMAN), p. 158. J. W. VAN WIJHE: “On the Nervus Terminalis from man to Amphioxus”, p. 172. (With one Plate). M. ve Ee: “The significance of the tubercle bacteria of the Papilionaceae for the host plant”, p. 183. D. COSTER: “On the rotational oscillations of a cylinder in an infinite incompressible liquid”. (Com- municated by Prof. J. P. KUENEN), p. 193. F. M. JAEGER: “Investigations on PASTEUR’s Principle concerning the Relation between Molecular and Crystallonomical Dissymmetry. V. Optically active complex-salts of Iridium-Trioxalic Acid.” p. 203. F. M. JAEGER and WILLIAM THOMAS: Idem VI. “On the Fission of Potassium-Rhodium-Malonate into Its Optically-active Components”, p. 215. F. M. JAEGER and WILLLIAM THOMAS: ,Idem VII. “On optically active Salts of the Tri- ethylenediamine-Chromi-series”, p. 225. A. B. DROOGLEEVER FORTUYN: “The Involution of the Placenta in the Mouse after the Death of the Embryo”. (Communicated by Prof. J. BOEKE), p. 231. 1. K. A. WERTHEIM SALOMONSON: “The Limit of Sensitiveness in the String galvanometer”, p. 235. H. C. DELSMAN: “The egg-cleavage of Volvox globator and its relation to the movement of the adult form and to the cleavage types of Metazoa.” (Communicated by Prof. J. BOEKE), p. 243. 4 Proceedings Royal Acad. Amsterdam. Vol. X XI. Physics. — “On the Course of the Values of aand b for Hydrogen at Different Temperatures and Volumes”. 111. By Dr. J. J. van Laar. (Communicated by Prof. H. A. Lorenz). (Communicated in the meeting of Febr. 23, 1918). Continuation of § XVI. The factor by which the double integrals (7) are multiplied, now becomes, with n= N:v: a i gana? X MN =} X HNE X MN XX, Ss U 1 Gs, a5. 8 Ns? = dm bk: is ai | a? 1 LX de X@X5X-=0x 2x! s® v s? With omission of 1:v we get, therefore, for the constant of attraction a: 90 a by a 2a* els sin n Odd ak Gees) ae | bo rn ? Ace sin? tp (2-1); (a ) when also for F(r) and — F’(r) their values according to (8) and (8a) are substituted. When to abbreviate we write 4° for s® : (a*—s*), the above becomes: Oy a BEN sin dd 4 eye ee - ‘aL Avs = = + fina. je HD ae a: -s?) — a' sin’ 204 k°p(a*—r®) VES do Tm in which, therefore, w = 4 X (by), Xa. Let us first discuss the first integral 1 referring to all the molecules that pass the molecule which is supposed not to move, without coming in collision with it. We may write for it: 90 a 1. zE dr x sin 0de GE ra cos? (a*— (Lr) 5 Pm As was already remarked above, the above calculations only hold for temperatures above a certain limiting temperature 7, defined 3 by 0, = 90°, sin@,—=1. This is namely the lowest temperature at which a value for @, is still possible. From (6) follows namely ; a 8 : sim? 0, = = (1+ ¢), so that — (1 + p) can never become greater than a a 1, hence p never greater than (a?—s?): s? = 1: k’. When we represent this limiting value of p= M: Suu,’ by p,, we get therefore SO ea he ater | =e nara em Ta when we put the ratio s:a —=n. Accordingly, as long as p remains <,(T > T,), the quantity 1—k*p also remains > 0 in the above integral. 0, is=90° in the limiting case y=gy,; then al/ the entering molecules collide, also those that strike at an angle 6 = 90°, which just reach the rim of the sphere »=s, and will yield there a minimum value for 7 for the last time. But as soon as the temperature becomes still lower, and becomes >> gp, all the entering molecules collide without previous minimum, i.e. they all strike at angles < 90° with the normal. For these values of p we shall therefore have to execute a separate integration later on, i. e. for all the values from p > ¢, tog=a (1'=0). Now the integration with respect to 7 yields: wf: : ba EA ’ V p?r?—a? (p? — cos 6) a V p?— cos? 6 a? (p*—cos’ 0) r m Tm : s' when we put 1—/*y =p’. As sin?0, = = (1 + g), cos’0, is therefore “help EELT ee a i a nn s? De 1 ~~ ) =—.— p’. Hence the quan- a’*—s a 2 tity p? is also = , ¢0s78,, so that p*—cos*6 always remains a —s positive. For cos?6 is at most —cos?@, in J,. At the limit 7, the quantity under the rootsign, viz. p?r? —a? (p? —cos?@) is always — 0, because then dr: dt =O (compare (39). Hence we have after introduction of the ee 90 sin Ó dO cos Ô x aah — Ba tg oa ge — Vp? — cos? 6 V p?-cos?O ee Vp? — a when we serie — deosd for sin?dd, me x for cos@, so that cos 4, is represented by «,. Now dBgty = dv:V p*—a?, so that we find: + vik : Bg? t Do OE Bg? t : ia 2a 9 J Vp?—-x,' Te 2a 9 d ra | a’ s? 88 p= x,” (see above), hence p?—z,? = “ey (eee a?—s* a? — 3? 2a‘ Maltiplying by the factor w X (e =) we have therefore for the , SS first part of a: Se a? Eee 1 2 1 ae V1—n? 10 CE g—=w —-- a : s (a?—-s’) oe k n (1—n’) ey, n CE Hence we find for this a value which no longer contains p (hence 7’), so that the part of the constant of attraction which refers to the passing molecules, appears to be independent of the temperature. This seems somewhat strange, because near the limiting temperature, given by g,, 9, gets near 90°, so that then the limits of /, with respect to @ get nearer and nearer to each other, and finally coincide at 6, — 90° (¢ =p). It would therefore be expected that a, would become smaller aud smaller according as 7’ decreases, and that it would disappear at the limiting temperature. However, this is not the case according to (10). The explanation may be found by an examination of the paths of the molecules, which shows that with the diminution of the velocity wu, they occupy an ever larger portion of the path within the sphere of attraction ; to which the circumstance is added that the frequency for the angle, which is proportional to sin0, reaches its maximum exactly in the neighbourhood of 4 = 90°. When ” is near 1, i.e. a near s (very thin sphere of attraction), 2 —h Bg*tg approaches ——, so that then a, approaches w:n’ —=w. As « t n ee 2 : Ss suv is = se (i + 9), cos*0, = 2,7 = 1 —— (1 + @); 50 that z,* will a a dg lie between = 1—n? = + O at high temperatures (@ = 0), and a (O) at lower temperatures (p =p). 9, lies, therefore, in both cases in the neighbourhood of 90°, hence the limits of integration of J, will almost coincide, viz. / between + 90° and 90° at high temperatures, resp. (90°) and 90° at lower temperatures. In the case n=1 the limiting value g, = (1—n’):n’? will lie near 0, ie. T, near ow, so that the available range of temperature is exceedingly small. If, however, ” is near 0, ie. « very much larger than s (very 2 5 large sphere of attraction), then Bg*ly approaches *)'/,1°— (47 tg el st err a VE k dz as d Bgtg is again = Fae : But the second integral cannot so easily be integrated. As then wv de dBgtg is = ‚ the said integral becomes: 8 Vera V p?—2? | VE nn p?—x,? d Bg tg = fox i Vp? y a — Bgtgy X dBgtg y, To when we a (w*—w,*) : (pe Bac which causes z° to become (py? 42): Hy"), and «w—a,? to become y?(p?—w,?):(1+y’). With Bn the last integral passes into: V eae, Ig 7 Yor pa ta wd 8 sin W - p V ty? wt (we 2: p’) al ‘< a DE ERP Te 0 ; 0 sir + = cos a Ig 7 ; sin =k en Wd, V1 4-4? sin? w eee 3 as Vp?—a,*:p in consequence of p? = ae 2,7, hence p?—a,? = 3? s = 7 v,’, can be replaced by a and w,?:p? by (a?—s’): a’, while further a e ar —s? ie Ok) told a*?— s* 8 ; sin? w + = ete = sin? Ww = = 1+ EE sin? W and s? :(a? —s°)=—= k*. The last transcendental, quasi-elliptical integral can now easily (see appendix) be developed into a series, and then be approximated. Previously we may observe that z,, hence p (and therefore also 7’), no longer occur in it, so that the result — like that of the first part of (1), — will not be dependent on the temperature, as little as this was the case with /, (see $ XVI). It is further easy to see that the said integral approaches Yar k sin W VIE sin? Tp wdy = Gy i 4x42’ in the limiting case n = 1 7 (a == s), hence 4 = op; and in the opposite limiting case n = s (a: s= 00), Var hence £0, approaches ef snp X wp dp =i(— W cos p + 0 + [ eos wy a ) = k (—w cos W + sin w), which yields the value 4 between O and */,zr. Hence the integral in question lies between '/,2? andk=s:Va?—s? = —=s:a—=n (as in the latter case s is infinitely small with respect to a), so that we can represent it by SO ar, in which & will lie between 1 (when n= 1) and 8:27=0,811 (when n=O). Accordingly the factor e is little variable. It appears from the expansion into series (see Appendix A), that ¢ becomes oa. for 7 = 0,6 (1 e: s—=0;6a). We now have: 1 1 (Z), = —| | 42° — By tg — | — en X ta’ |, 2a ; k so that taking the factor w X (2a‘* : s(a’—s’)) into account, the fol- lowing equation is found: I V1 nt? (a), = © x | bat = em) Br | (LE) —n?) Ifnisnear1 (a —s), this approaches w X [47?(1-n)-(1-n?)]= Ln =w X (ta’—1) = 0,234 a. The limits of integration p and 2, are en determined by «,? =1 ORE (L4-g) = mf nf = 0) at high a a temperatures (p — 0), resp. (0) at lower temperatures (¢ = g,), and a’? p= zt = (1), resp. (0); so that @ lies between (0°) and + 90° a Ss at high temperatures, and (90°) and (90°) at lower temperatures. And if n is near 0 (a great with respect to s), then (a,), approaches to 1 Bie == Sc [4.22 2n)--(4 172) | DK (rt -—2) =S 1,14 w. Then we a a >i ie. ipa es have as limits of integration +1 for zr, (p = 0), resp. (0) at p= p‚, and (1), resp. (0) for p; so that 4 lies between (0°)and + O° at high temperatures, and (90°) and (90°) at low temperatures. When (a), is added to a,, we find for the part of the constant of attraction a that is wdependent of the temperature (coming from 8 the passing molecules and from the (not central) colliding molecules): a. =a) la we : bie eee 12 == - = wo = OF Onn SO o a, 3/1 n(1 n°) * JT ( ) According to the above this part comprises the almost totality of the angles of incidence, from 90° to near to 0°, at high temperatures ; and only a very small part, from 90° to near to 90° at Low temperatures; i. e. in the limiting cases n — 1 and n — 0. But also in intermediate cases this continues to hold, because at high temperatures p? always lies in a Op the neigbourhood of — — « = 1, and at low temperatures a; —Ss a 2 . . a always in the neighbourhood of — — x 0= 0. 8 Hence the region left for the part of a that is dependent on the temperature, is the greater as the temperature becomes smaller. : ln Now the quantity a, in (12) lies between w X ———_12 2 n(l—n*) 1 { 1 OK PX forn == 1and mX SK En Sn n(1 +») mnd n § XVIII. Calculation of (a,),. We finally come to the calculation of the part that is dependent on the temperature, and corresponds with the more central collisions of the second stage of /,. We now have for the integration with respect to 7 (cf $ sees f Vp Bn. in which cos? 0 — p* remains positwe between the limits 6 = 0° and 6 = Bg cos p. The integral yields: 1 (1 V pr? + a?(w? — p*)—a 2 Tj og = aV 4? —p* s r =S ar Wree log” (Wer PE VE when cos is put again == #, and cos 0, = #,, x," being == ik (1-5) (Cf. § XVI). Hence we have a dr da eVa hark da VEV (1) log Er ng a . a V x? —p? Vat ‘la P p We have written — dw for sin py Rees — deos 0. The minus sign has again been removed by reversing the limits of integration. Besides — for the sake of homogeneity — a factor p has still been introduced under both dog. For s/ap we may also write Vp” The first integral can again be easily integrated. d log is namely de = so that we find for it: ie ee less —p?\P a Pp 1 lo € oe = 4 log? ————-——_. J p 2 p en eRe { eeens for which with a view to log? also — } log* ————— may be P written. The second presents again the same difficulties as the correspond- ing Bgtg in § XVII. This becomes namely, d log now being = a da Bets 8 Va'—p’ hes ze Ees = Bid a, we may namely write (a°—s*): a’, and (p*—w,’): p' 10 a ; 1 is == s* ..a7. The limits “for-.¢ are {wand p, hence EN p 2 — U, ]|—,; and 0 for y, i. e. for tg hw. Thus [= P —1: (Agh), or tg O,: k and1; p?—«,? being = £*x,?, and “7 TE En V1—>p? js Vp*—a,' = „and 1 for cosh p= —Z2, being cos’? 0. Evidently the limits for w are HL tg 0, tg? 0, = by (%, [44 — Lando, as VE NVE Vp’ 5 In this g0,:kis>>1, because now p <1. Thus we obtain an integral of quite the same form as that of § XVII; with only this difference, that now hyperbolical cosinus is put instead of the former sinus. When again we expand into a series (see Appendix B), it appears that both at high temperatures (p = 0) and at low temperature (py = y, = 1:h") all the terms with higher powers of log with respect to the first term disappear, so that with close approximation we may write: Vl dp WVA) gp VII p 2 gier in which p is determined by the relation — sin? 6, = 1 + p (cf. s Va'—e, Va —p' Vp’ log Bgtghw= - log — 4nV1 + log? equation (6) of the previous paper), in consequence of which ty? 0, : k* becomes = (1 + p): (1—A?q). (n has again been written for sa=k: VI + k?). ‘ 1 + V1—p? When we now add the found integral to the first, viz. } lag? ————— then (p*? being = ae #,* = (1-++4*)2,*, and z,° being = 1 — sin?d, CxS „3 We (1 Hp), so that p? becomes = 1—k’ p) we get: === be 1+k at V1 V(l+F)@ U), =— be log? ed aR +n V1 Jp log? fe Tae ) “| 2a V1—k? » Vl p so that taking into account the factor w X(2 a‘: s(a’—s’)), we get the following form: Hi V1 Hp + VA) Pp a: ; VY Up a (a). NA x n atl =|" 1 rp log VI p 1+k Gand = itp en hen 1 og? Wares (13) V1i+9+Vl+h) gp | eae <7. (+ Fp les ; aos Se 1+ 1h) 4h Gekte be eee EL alee lp Ik Vp Tk /p 3 l oo eae kVp ) etc., og en OTN Vp 5( += (a,), will evidently at high temperature (p near 0) approach to 1 (1+4*) p ee dg kp |, RMT é Te eee | i.e. with 47 =n’ : (1—n’) to 1 n p n? | = GS — q de Ge (Arp, ole or 1 n = D= Mt Sat eo tye 18 when p is simply written for p: V1 + p. This becomes therefore properly =O for 7’=o. Then the limits of the original integral (L,),, viz. p and 1, are equal, viz. = 1, which causes the limits ot the angle of incidence @ to lie between (0°) and O° (see also the end of § XVII). For low temperatures (p near gy, = 1:47) we shall have: ( < 1 [ . ( 1 2 ) loa 2 | ARE we Sige ee n(1—n’) Z neh Pp 4 Vl p because then nVi+tg is =1, and WAH) p=Vidp=i:n. or 1 2 1 2 And as log (77) = log a + log Vv’ we may finally write with 1 omission of /og?— in comparison with the infinitely large terms: nr (a,), =o X log — X log — 1 2 x y=», = 1:#) . (18d as en rar Po = ) + (13%) 12 This gets near to logarithmically infinite. Now the limits p and 1 are evidently —O and 1, so that @ lies between (90°) and 0°, hence comprises the whole region. When n=1 (a= s), (a), does not become =o in 13%. For as p can never become greater than 1 :k? = (1—n’):n?, (a,), remains | evidently smaller than w X —W——., Le. < (/og-infinite)? in (13%, i.e. will greatly increase, when the temperature becomes lower. § XIX. Calculation of a. When we finally add the part of a that is independent of the temperature, viz. a, — a, +(a,), according to (12), to the part that is dependent on the rome es according to (13%, then we get at high temperature, taking w = } X (by), @ into account (compare § XVI) n = 1—en)1 a? = on a [ an ltn : 1 n eae Tay © Deo @ oe nne v|. n ' (1—en) (l+n)'/,2’ or also (p = 0) a=a,| in which therefore eet 1/, 1? (1—en) ete eee n a 2n(1—n*) 2 (1—en) (14) */, 77 We remind the reader of the fact that the coefficient e (see § XVII) has the value 1 for n—=1, the value 8:27 = 0,811 for n=O, and — MN, in which M is the maximum value of the function of force f(r) at contact of the molecules, and N the total number of molecules in the volume v. At low temperatures (p =p, —=1:k?) we get according to (13°): log 1/a 2, = 4 a=a,| 1+ log —— | fo 32" ee eee | er (len) ° VIB p v| =a, (1 +79) ,(14@ 13 That for p= gp, the value of a becomes. logarithmically infinite, and does not get near exponentially infinite, as is the case on assumption of _BOr.TZMANN’s temperature-distribution factor (for yd) — (e [RI —1):*/pr becomes of the order e” for 7'== 0), is already to be esteemed an advantage. But the above found logarithmic- ally infinite will lead to an ordinary /inite maximum, when we consider that only the very definite velocity u,, which causes p to be = M:tuu,*=1:#', leads to this log oo. When we take Maxwerr’s Jaw of the distribution of velocities into account, the adjacent velocities will not lead to log oo, and this will accerdingly pass into a finite maximum. We shall come back to this later on. We will, however, point out already here that the logarithmic infinity for p = g, is not bound to our special assumption (8) concerning I(r). We shall see that this log-infinite value of a for y= ~y, is found on any supposition concerning Fr). But the numerical values of the quantities a, and y in (149) e.g. will of course be dependent on the said supposition. We possess a kind of control for the case y= 0,n = 1. According to (14°) a, then becomes = '/,, 1° X (6,),,¢, because (1—en) then becomes — | —n, hence (1—-en) : n(i—n?) =1:n(-+ n)='/,. But. according to the ordinary (statical) theory, the attractive virial (see § 1X) must be a dE AL nf» ay dr. When a=s, r*=s'* can be brought before ar s the integral sign, and we have?/,7 Nns* ra. =) x Nuns? 0 (—M)j= = la Ns* X MN: v(asn = N: 1). Hence we find with MN — a for a the value (Db), X «, so that the factor by which we have to multiply, would have to be = 1, and not = '/,,2? = 0,617, as we have found. In my opinion this conclusion can only be drawn from it, that even in the limiting case 7’— (py —O) the factor of distribution at the molecule surface (the sphere of attraction is infinitely thin on the assumption @=s) is not = 1, as we assumed above in the application of the statical method, but slightly less in consequence of the influence of the passing molecules, which does not disappear even for n=1, which is the cause that the full maximum value M of the function of force is not reached. And the difference will depend on the nature of the function of force used. For n= 0,6 the factor of (6,),,@ will get the value 2,467 « 0,483 A 1,192 ——____—_—____ -- ____ — 4 §5, which comes to this, that the attraction 1,2 « 0,64 0,768 , 14 might be thought concentrated at a distance sP~ 1,55 — 1,16 s from the centre of the considered molecule (the sphere of attraction extends between s and 1,67 s for n = 0,6). We saw already that y represents the quantity M:*/, uu, *. In this wu, represents the mean relative velocity with which the mole- cules penetrate the sphere of attraction. But this velocity is augmented by a certain amount within the sphere of attraction, so that w, will not be in direct relation with the temperature. For very large volumes we may, however, entirely neglect this slight modification in the velocity -in comparison with the much larger part of the path passed over with the velocity w,. Only for small volumes this is no longer allowed, and in consequence of this new complications will make their appearance. We may now write: IT a a PRE: au? Nee SSCIERT | RT, ff == because the mean square of the relative velocity is twice that of the square of velocity JU,’ itself, and */, A7’ may be written for u N U,*. According to all that was developed above, en 1 : Je EO a KTA |p | oe Je oe may therefore be written for a, according to (14°) — at least for not too low temperatures, and when also higher powers of p are taken into consideration; whereas for low temperatures (p near y, =1:hk*) an expression of the form Re ac = Ee ie es ba Tr AE a a.( og Te a) ( ) will better answer the purpose, according to (14°).In this x = hk? & '/,a= 2 n = EE > '/,e, in which it should be borne in mind that the log Sr is now negative, so that the minus sign before A becomes positive again. , We have already pointed out before that the supposition of an exceedingly thin sphere of attraction, as is sometimes assumed, must be entirely excluded for several reasons *). To this comes the circum- stance that form — 1 the limiting temperature 7,, in which a will become logarithmically infinite (or at least maximum), is given by g, = 1:4? = = (1-—n’?):n’?, which for n=1 would give the value O for g,, i.e. 7 =o. And as it has been experimentally found that the said 1) Cf. our first paper. 15 maximum lies at very low temperatures (a continues namely to increase, for H, for instance, up to at least '/, 7%), the assumption n — 1 must be quite rejected. As the value 0,08 (about) is found for '/,@ with H,, the value 0,36 ‘Lege of RT, ='/, a: p, would become “Gd x 0,08 = 0,045 withn = 0,6 ? Gre: Soa or a= 1"/; §); 1. 0:77, about 12° vabsolute:* This -is very well possible, as we have seen that for H, the value of a is still increasing up to 16° abs. (from a, = 370 x 10-6 to aygo= = 740 X 10~® about). What is very remarkable, is the fact that the limiting temperature seems to lie so close to the triple point of H, ‘viz. 14° abs.). Fontanivent, January 1918. (To be continued). Physics. — “On the Course of the Values of a and b for Hydrogen at Different Temperatures and Volumes’. IV. (Continued). By Dr. J. J. van Laar. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of March 23, 1918). § XX. The value of a below the limiting temperature. In this case the integrations need no longer take place in different stages, since a minimum distance 7,, which is dependent on 6, need no longer be reckoned with, so that first the integration with respect to 9 can be carried out, and then with respect tor. All the entering molecules, from @ =0 to d= 90°, will now come in collision; for the limiting temperature 7’, the moleeules that strike under an angle 6 —= 90° will just pass the rim of the molecule that is supposed not to move. We have, therefore, now to integrate (see $ XVI): u (b,) . 2a' J - dr X sin 1G dé a= li (be —= a ‘s(a =) Va? cos’ 9 4 Te —? IE pie in which 4? is now always > 1, and in the limiting case p= gq, =1: A" assumes the value 1. When we put (a’—7") (k*p—1) = q’, we get therefore : . - a 0 Sen 2a° Se Sv oie 0) ae Dg EEn ay s (as?) a q gE cos” aC in which we may write for the second nel a+ Veto q log (a cos 8 + Vg at cos? 6) = == log so that we have still to integrate: a ees 1 b Ne Bat “dr / a zr ea we a’ ( 6) =— 7 eo = = UO am : a ve x ( Jo « / s el r g q q If in the first place p is near y,, then q approaches-0, and the integral approaches to £7 a a dr_ 2a “dr [ 2 UF nia f= log —= | — | log ————— — log- — |, r q r Aer Pl a S S because q is = Vol < Va?—r. Hence we have for the integral : F 2 I fi 7 Vai — y* dr og A Di og — — [log - = Vie ey 5 We have for the last integral het r:a=a,s:a=n: : ; dx a? x‘ a? 1 4 —4t Fagan 3 = i i 4 5 | Bs, TEEN) n in which e«' —1 for n=1, and 6:2? = mo for n=0. For 1 1 1 er ae n‘ IE LEK Poke ee Sane ene aoe ae a (ED) 6 1 f : = T a tor n= landen torn 0. (For n =—0,6 €! — 0,674). Hence we get finally: = 1 Sg ee = 2 Psp) a= Bell ay” De dl (l—e'n*) + log — log. Vi — |¢ 7) When we compare this with (14’), where we found for values of g in the neighbourhood of g, (but

p, in 16): Wz) el par ) + log? tog a | Pew a C a Sele og” = og og ; ae. 2n (ln?) > 4 Vi keg we observe with regard to the member that is independent of 7, a discontinuity appearing at p = ¢,. [We have added, for a comparison, to 1 : the first (finite) term the term /og?—, which was cancelled in § 18 in n form. (135) by the side of the infinitely large logarithmic term |. For n=1 we find (with the factor ; from the factor before the — n fintegrati the first ed d egration) in the first case — 2? —_— — — a?, in the second case sign of integ i io eee Ris Pe re 1 | : 1 ria ee Sie ak Zz i coe And for n=O we finc TE ik resp. 1 1 1 sea 0 n° + yeh at x* + 7. This difference can be partly accounted n for by the sudden disappearance at p= f, of the terms which refer 2 Proceedings Royal Acad. Amsterdam, Vol. XXI. 18 to the passing molecules, and which, therefore, do not occur any more in (17). But in any case the difference is of no importance, as these terms, which are independent of ~, remain finite with respect to the term that depends on p, and logarithmically approaches infinity. (In the case 2 —0, where — for infinitely large spheres of attraction — the entire quantity a would become infinite, and accordingly our ; l derivation is no longer valid, the fact that /og* — becomes infinite, n is of no importance at all). We will still point out that for p — p‚ a does not only become logarithmically infinite with the form of f(r) assumed by us, but with any arbitrary assumption about this. Compare for this Appendix C. We suppose in the second place in (16) p near oo (i.e. 7 near 0). For the integral in (16) we may then write, as q becomes very large: a a fe (2 Hees: =i) {= a fe —log{ — +4—|>/—x-=— r q q r q SVE Var? 1.e. 1 a— Va?—r?\a 1 a— V a?—s? ——| log —— So) SS SS log 1 — log —— a V k?y—1 r s Fol 1 8 1 a+ Va?—s? = — log : V i?p—1 s When the factor before the sign of integration is taken into account we get therefore: oe 1 1 1 ENT Pp ke ern DL (by), « x — —- log as, : , (18) T— 0 n (1—n’) ee Vieg—l n This approaches 0 therefore, when yg approaches oo (7’ approaches 0). We may write for 4° g—1, after substitution of the value for p, dE ee ek oh when 7’ is beni 0. 1—n? RT 1—n? RT Hence after the maximum for a at p= p, the attraction steadily decreases, and disappears at 0° abs. This result was to be foreseen. In the original integral of the virial of attraction the radical quantity in the denominator becomes namely —= oo at 0° abs., when y becomes =o. This radical quantity expresses the relative increase of velocity in the sphere of attraction, and as this increase remains finite with respect to u, — 0, the relative increase will become infinitely great. the expression 19 And this relative increase of velocity entirely determines the density in the sphere of attraction, which is in inverse ratio to it. We observe once more here, that the earlier Bonrzmann theory would give an exponentially infinite value for a at O° abs., whereas in reality itis —-0. Tai 12,3 Prats = : ee 1 For n=1 (a=s) the limiting value of io? will be = log en 5 ; 2d 1 n° = i in Vk*p—l ( f Pi With Poe in V k*p—1 (see above) this becomes ' faiyd by 1:n, so that then a will approach (b,), a x ee . 1 a : 3 For n =O (a great with respect to s) the absolute zero coincides with the limiting temperature, given by gy,=1: 4° =(1—n’):n?. 2 wil 1 For then p‚=o (7,=0). In (18) Lim — log becomes further = —log —, n n n f i 2 so that then a will approach (b,), a < — log —X — ‚ which n n va 1 n' LES AL again becomes = 0 for 7’=0, so long as 7 is not absolutely = 0, which of course would be practically impossible. Summarising we can therefore state, in agreement with the above developed exact theory concerning the quantity a for very large volume, that a, from a limiting value at 7’=o, steadily increases to a maximum value at T= T,, after which it decreases again, till a has become — 0 at the absolute zero. The mentioned limiting temperature 7, is then determined by R7,='/, a: -p,, in which ff, = (1—n’):n?. (n=s:a, in which s represents the diameter of a molecule, and a the radius of the sphere of attraction). For H, T, is about — 47%, the ratio of the values of a,, ap, and a, being Pid TPN In the next paper we shall briefly discuss the influence of Maxwerr’s distribution of velocity, and then treat the course of the quantity 6 from T=o to 7’=0, likewise at large volume. Then the values of a and b for small volumes will be considered, so as to make a complete theoretical insight possible concerning the whole course of a and b along the boundary line, both along the vapour branch and along the liquid branch. Fontanivent, January 1918. (To be continued). 2* 20 APPENDIX. Yor sin yp A. The integral /: ff — vip. (addition to § XVII VLEK ein ie ’ ) When we expand this into a series through repeated partial integration, we get: 7 . : 2 dP 4 2 5 1°P ae —ydwyw =P pio page } a ee x +4. VI ht sin? wp 6 dp 224dw? 120dy' in which (through y) all the terms at the lower limit 0 disappear. And for the upper limit all the odd differential quotients of P will disappear, because in this cos wp occurs as factor. Indeed, when we : dw . put 1+? sin p= w, so that = becomes = 2 4? sin woos w, we have: aw dP Rani ops + cos Ww =k? sin? wp 1 IT C OS — =— COS W - - mm ae = sin Y cos tf) oils cos ¥ a . oth 1—w 1 cos W ae en % =) oh. ar cos Wp sin W 3k? cos? wp 1 et ee (Mem oere “aT ji = de oh LNE wl: v( wl E wl: 30+) 2 = — sin wp = — ’ wl DE because 4* cosp = h?—k?sin®? p=k"(w—1)=(lHk")—w. We have further: 3 Tak? 6 oe 2 Elder as wp ( sh ae Je ke? cin con) — cos ( nn pS ) dp" ulle wl w?/s 15 (1 k*) 6 Snas 2 = — cos) * on — }(1—w) + 8 ( 5 Jee se wle wl: wl: oa ae AS +O 4 wle wl: And also: d* 105 (1 key 1+? Sd. 12 = = — cos | DEE pa Be + —_ } (—k? sun wp cos wp) -- dyy* wil wl: BEER) 12) pO. A ee = fa a +) 21 (105 (144? 60 (1 4-2°) + 30 12 =p ( Et) + a (= (lk) 12 (1 +A") Bio 4 )| w°/ wils zE J- = wil: w'/ P1051 +H)? GOL +4)? 4 1201144) 60144) 424 8 RADE TBE ar ne |: Etc. Etc. As has been said, all the odd differential quotients dis- appear for w='/,7, and as w becomes =1 Jk for W="!/,nr, we keep: LP ee 1 ig) ron USF Beer sla 6° 4 9 8 18’ dup’ ya oh wh wih oth TE fe (Le) For the sake of brevity we have only taken the part with sew into account in the last calculation of the two differential quotients : 4 : LR ad sin W that with cosw is namely = 0. Te. of — only the part ——_, dw w*/, a) „ Only the part with snap in the first of the three lines d and of d belonging to this. The other parts have every time been necessary for the determination of the next higher differential quotient. Proceeding, we should have found: EN 225 360 156 1— 88k? + 136k* de ennen eer The coefficients of the highest powers of 1 + 4? are in all these Beals reps Mee (AK B 5), ete. The sum of) the coefficients is always = 1. (9—8= die Db 360 411186 1). Hence we get now, taking into consideration that 4: V/d +4”) = 8 a s nn Sp, and-(P)ys md: VAAK Vars" Wars 4 | Var sin wy IUD 1 Um k w iy =n) ; “Tp ae + V1+ kh? sin? wp 0 Sh .C/ex)e 1— 88k? + 136k* ('/,2)° El Ee bid zom Us ED (LF4°)? 720 (14-29) 40320 in which we may also write 1 —n? for 1: (1 + 4’) = (a’—s’): a’. The above series is convergent, as is easily seen from the structure of the factors (1—847): (1 + hPa = 9: (4 + kh = 8 (1 +k"), ete. 22 For large values of k(a=s, i.e. nm = 1) it converges very greatly, and rapidly approaches the first term, i.e. n X ‘/,%’. For small values of & (near 0, i.e. a large with respect to's, 2 0) the series becomes: nat — dy (1D)! + rho (at — ete] =n (1 — cos ban. For the two limiting cases n =1 and n=O we, therefore, find back the same values as we had already found by direct integration in the text of § 17. When n=0,6, we get 1—8/? = 1—4,5 => — 3,5, 1— 884? + +. 13644 = 1—49,5 + 43,0 == —5,5, 1:(1 +k?) = 0,64, so that with '/,2? = 2, 4674, the integral being put = en X '/, a (cf. the text of § 17), we find from 1 ('/,0)? 1—8k? (!/,m)' 1—884? 136K (!/,2)° Eil : ize 12 ' (+) 360 (ley 201ee for « the value . 4—-0,1316—0,02425 + 0,00107 7 == 0 845 se =O eee cosh w B. The integral if : —- pdw (addition to § XVIII). V1 +h costh y tgdo:k In entirely the same way as for the above treated integral we find through repeated partial integration : 1 k re Ef k cos h w tgO,, log* k? log? ap dip == — Sa ee aoe ee Vi en cos Ti ww Sec 0, 2 sec CG. 6 7%: k 3(1+ 4?) Zs) og gk Renee aceaner = sec? , sec? 0, / 24 ty?O, 15+) 12042)+46 4 log’ En ya go, yee kt ( sec’ 0, sec* 0, bs sec? @ abe 120 deel: | TE in which log represents log (0 Bp ae ) In this it has been taken into account that d cosh y= sinh w and d sinh w == cosh w, and that further — i? cos°h w can again be replaced’ by 1—w (when namely 1 + 4? cos*h py is put =) and — h° sin*h yp by — k? cos*h w + k? = (1 + k?) — w. Now the terms with odd powers of y do not disappear, because at the lower limit the factor sinh w, which occurs for these powers, does not disappear (as for the above 23 ioe 7e aay healen integral cos wp at the upper limit), but becomes = WAE because coshp then is —{gd,:k. At the upper limit Soe il disappears, because then w — 0. (Besides, the terms with odd powers of yw still contain the factor sini yp, which now likewise becomes = 0, because cosh y becomes = 1 at the upper limit. (Cf. further the text of $ 18)). We may, therefore, write: 7 l a(t +i 2 q' ef = — | sind, | og” a (a ~ La —— log + etc. 2 (14-490)? . 1+t9?6,) 4 BEE! Lay (i B(L+e) 121444) 46 1+t9?9, |1-+t9?9, 6 (4490)? = (L+-tg°0,)? As 4 log’ ‘ alan Agta | Let us now introduce the quantity p, determined by equation (6) of the last paper but one, viz. a? M ree Orsel eee hE TATIE in which, therefore, p depends on the temperature (determined by DENTS ‘/,4uu,"). For 1-+ t9?4, we may write oe because 4? (1+ ¢): :(1—A*p) may be substituted for 49°, = cs (Ady): (1 — ds (+) a a 22 with — —=h*. For tg?0,—K* we find £?(1-+-47)@ : (1—£?¢p), so that a’—s we get k low ED (l— ( le VE Er. eel — V1+h 1---k 24 1—k#@—p loc 1—k Y — ( B) 144 6 (14-42) (1 —k’p) (8S—12k? wy) log’ — —-— ]__. + etc. 144? ) 120 | in which tg tO. VI + Vv (dee: k?) log = log G& + Be = ) = log — ae = +r yp Let us now examine, what are the limiting values to which the found integral approaches at high temperatures, and at low tempe- ratures (p near ~, = 1: k’). At high temperatures (pg =0) log draws near to log 1 —= 0, so that all the terms with high powers of log are cancelled by the 24 side of the first term, and besides the whole part with kV@ disap- pears. That in this case only the first term with /og* remains, follows also from this that ty 0, =? (1+¢):(1—A*g) approaches & for ~ = 0, so that in case of equality of the limits of the original integral the factor 4 cos hp: MAHA cos*h p=k:V1+Kk does not change between them (with respect to the log that becomes O at both the limits), and can accordingly be brought outside the integral sign. At dow temperatures (but higher than the limiting temperature T,, determined by p,=1:k?) the whole second part of ef will again disappear in consequence of the factor 1 —k*p, which approaches 0, whereas of the first part again only the first term with log” remains. In’ this’ “case cos hap =O e= oo at, the lower limit, and the ‘factor of pdw in the integral can again be placed outside the integral sign at this limit, which now prevails since the log becomes infinite there. At the other limit the /og is namely = 0. With close approximation we may, therefore, write (7 has been written for £:V1+i?=s:a): i UDO Vargo EREN V1—Kp with neglect of all the terms with higher powers of log. Only at intermediary temperatures the omitted part can have any influence — but the difference brought about by this might possibly be made to disappear entirely on a somewhat modified assumption concerning f(r) between a and s (see § XVI). C. The quantity a for p=, —1:k’. (addition to § XX). The original integral was (cf § 16): — f'(r))dr X sin0d0 —s° nih Va we may also write for the integral: )) dr d (a cos 0) LR 5 a ee. es zi jee part (7 + 14 5). Vr so En ET cos a q q s Ver when 7*y f(r) —(a*—r*)) = q? is put. When /(r) is generally a Le SG fei st =-—, so that this duly becomes — 1. for; Ss thene a) (7). 5 — and g? = —— — (a°—r’). Hence we now have: r Pe 25 ta? si—1 dr a a? a= (b,),,a@ X — logl — + | el ae Ne 5 ais rt—1 q q s in which the quantity q for the lower limit passes into ps? — (a’— s°), 2 3 2 which becomes = 0 for p = E me Bune = : as before. The a n° ke value of a will, therefore, again approach to logarithmically infinite for p=g,=1:h". This is, accordingly, entirely independent of the exponent ¢ in the assumed law of force f (sr) … 7. Physiology. — “On the Peripheral Sensitive Nervous System.” By Dr. G. C. HerINGA. (Communicated by Prof. J. Boeke). (Communicated in the meeting of February 23, 1918). When we endeavour to summarize our knowledge of the peripheral sensitive nervous system, which is a time-consuming experience as it involves the perusal of an enormous number of periodicals, we shall find amidst a mass of controversial matter a number of facts received by various controversialists, which, when put together, make up a gratifying whole. In the neurological clinic the doctrine of neurons in still all but paramount, but in the neuro-anatomic literature it is quite a different thing. There, in spite of this same doctrine of neurons, experiences come to the front pointing to the existence of a very extensive continuous retiform structure of sensory nerves close to the periphery. As has been insisted upon by Aparuy there exists a highly delicate texture of anastomotic nerve-fibers close under the surface of the body of invertebrates. This view has hardly been disqualified. It is now getting more and more evident that such a network is also to be found in vertebrates. Many data regarding the “rete amielinica subpapillare”” we owe especially to Rurrint and his school, who based upon them his theory of the “circuito chuiso delle neurofibrille.” According to the descriptions given by Rorrinr himself, the fibers of this network spring from different sources : 1. end-branches of the ordinary medullated fibers ; 2. ultraterminals of endorgans ; 3. sympathetic fibers ; 4. ultraterminals of fibers belonging to the Timorrew-system. From all sides (Borezat, LEONTOWITCH, PRENTISS, SFAMENI, DOGIeL) much evidential matter tending in the same direction, has been brought forward, so that no room is left for any doubt as to the principal facts, though there remains some difference of opinion regarding the components of the network, and though several inquirers will not go the length of subseribing to all the inferences of Rurrinv’s “teoria unitaria.” Two recent publications from the Italian school seem to me to be 27 interesting in this connection. SrepHANELLI') describes an extensive network of nerve-fibers, which he found in the skin of reptiles. This network built up of non-medullated fibers is easily distinguish- able from the familiar subepithelial plevus, which lies deeper and in which only an interlacement of nerve-fibers, for the greater part still medullated, takes place. The relations of the non-medullated network to the subepithelial plexus are also described minutely by him. In the former, which spreads diffusely as a true network of nerves in the skin, he describes by the side of very few other endings an “organo di senso in stato diffuso,” a conception which is the more plausible since the network is immediately connected with an intrapapillary extension of the same nature. Here lies the link that joins SrePHANELLI’s publication to that of VITALI. ”) Viratt examined the skin of the nail-bed also after Rurrint’s gold- chloride method. His results correspond completely with those of similar researches by Rurrint and others. In succession he describes the presence of many free endings easy to differentiate by the very melodious Italian names: gomitoli, alberelli, espansioni ad anse avoiticciati, fiochetti papillari, grappoli, and also of Rurrinr’s, Mrissner’s and Varer-Pacini’s corpuscles. The principal interest now hinges about the fact that he lays particular stress upon the occurrence of anastomoses between the terminals reciprocally and upon their contact, as a whole, with the rete amielinica subpapil- lare, therewith emphasizing the importance attached by Rvrrini long since to the ultraterminals as expounded in his teoria unitaria previously mentioned. Finally Virani comes to the conclusion that all those terminals together with the rete subpapillare form one connected amyelinic meshwork. When following up the Italian school a little further, we shall see that this meshwork must be placed on a level with Srepranerrrs diffuse network. Then also the various endorgans of the higher vertebrates will be found to be points of differentiation amidst less developed surroundings. “Eche cos’altra sono,’ as Simonelli puts it rhetorically, “quello che noi denomigniano espansioni, se non il condensarsi in punti limitati di nn simile reticolo diffuso periferico: in altri termini se non punti nodosi e 1) Augusto STEPHANELLI. Nuovo contributo alla cognoscenza della espansioni sensitivi dei Rettili e considerazioni sulla tessitura del sistemo nervoso periferico. Intern. Monatschrift. f. Anat. u. Phys. XXXII 1916. — Sui dispositivi micros copici della sensibilita cutanea a nella mucosa orale dei Rettili. (Ibid. XXXII 1916). 2) G. Viraut. Contributo allo studio istologico dell unghia. Le expansioni nervose del derma sotto ungeaie dell’ uomo. (Ibid. XXXII 1915). 28 maglie piu serrati di una rete generale, che intimamente involge e compenetra i tessuti, per meglio localizzare e precusare gli stimoli periferici ?” Thus, according to this view an unbroken series of anastomoses must be traceable in numerous varieties of free endings from the rete amielinica on the one side to the tactile corpuscles inserted in a rete intrapapillare on the other. It would perhaps be premature to consider this highly pregnant hypothesis as proven. Still, undoubtedly it is equally true that anyone who will take the trouble to look into the literature, will find attestations from other authors also pointing unmistakably in the same direction. It is evident that the border-lines demarcating the various forms of end-organs, classified into various groups, are by no means established. Nearly coeval with the study of the end- organs itself are the efforts to establish a phylogenetic pedigree of the various end-organs, in which the intricate forms are reduced to more primitive types (Merker, Krause, and others). Certain it is also that the more forms are brought to light by modern researchers, the more the border-lines between the various groups are fading out. With this we are impressed forthwith when looking at the illus- trations accompanying the several publications (see e.g. CECCHERELLI *) v. D. Verpe).*) The leading modern authors (Borrzat, DociEL, SFAMENI and followers of Rurrint) endeavour to demonstrate anasto- moses between the various endings. DocieL*) says in his article about nerve-endings in the external genitalia: “Wenn wir die Be- schreibung der Nervenendigungen in den verschiedenen Nerven- apparaten, den Genitalkérperchen, den Endkolben und den Meissner- schen Körperchen, welche in der Haut der äusseren Genitalorgane gelegen sind; vergleichen, und zugleich die beigegebenen Zeichnungen betrachten, so müssen wir zu dem Schluss kommen, dasz zwischen ihnen kein wesentlicher Unterschied: besteht”. SFAMENI *) also describes the relationship between the genital cor- puscles and Krause’s end-bulbs, Goxci-Mazzoni’s corpuscles and Vater-Pacini’s corpuscles on the one side and Rurrinr's corpuscles on the other. Botrzat*) has written a long and comprehensive paper on the system and the interrelationship of the nerve-endorgans. 1) Intern. Monatschr. XXV 1908. 2) Intern. Mon. XXVI 1909. 8) Arch. Micr. Anat. XLI. 4) Arch. di fisiol. I 1904. 5) Zeitsch. Wiss. Zool. LXXXIV. 1906. 29 But what seems to me to be more important than all this, as it falls in with Rurrmi’s views, is that also the border-lines be- tween the corpuscles and the “free” endings are gradually falling away. Here the only differential diagnostic is whether or not a capsule is present. The same characteristics of the nerve-fibers, of the supporting tissue, “‘tactile-cells” or whatever name may be given to the cells found in the endorgans are equally peculiar to either group of end-organs. This may be gathered from the illus- trations and the descriptions in all papers. Borrzar makes particular mention of this, adding that a capsule round a nerve-ending is not a question of vital importance for it, either functionally or morpho- logically. On the contrary Bornzar very often finds by the side of a capsuled ending its fellow deprived of a capsule. Thus the free “Knäuel” are found side by side with the capsuled “Knäuel’” and the bulbs of Krause; side by side with Mrrker’s cells GRANDRY’s and MerssNer’s corpuscles etc. Moreover Borrzar distinguishes all sorts of gradations between the free and the capsuled endings. In other authors we find the same again. Rvurrini’s corpuscles are according to Vitatr') nothing else but capsuled ‘‘alberelli’. Doei *) also speaks of non-capsuled corpuscles of Rurrin1. SFAMENI ®) asserts that non-capsuled varieties occur of the same Genital corpuscles, which, as has been observed, are allied to all sorts of tactile-corpuscles. Of Mrissner’s corpuscles there seems to exist a large variety of simple modifications. SFAMENI describes intermediate forms between Mrtssngk’s corpuscles and ‘‘fiochetti papillare” i.e. free endings. Dogirr’s modifications of Meissner’s corpuscles (Rurrim calls them Doginr’s corpuscles) are non-capsuled at the upper-pole from which the axis-cylinders are branching off into free endings. They are types of Rurrini’s “espan- sioni misti’. Other modifications again of Mrissner’s corpuscles (DogieL, v. D. VeLpr) are characterised by their having a slightly developed capsule and a simplified nervecourse. Doain1’s ‘“einge- kapselte Kniéiuel’ described by him in 1903 as modified Muissner’s corpuscles must therefore be closely allied to the free endings, perhaps identical with them (see supra). It seems, then, that Mrissnrr’s corpuscles are, in a higher degree than many other forms, closely allied to free nerve-endings. So when observing the several findings concerning the capsule of these corpuscles, we shall see that Lan- 1) Int. Mon. XXXI. 1915. 2) Arch. f. Mier. Anat. 1903. 8) Le. 30 GERHANS *) absolutely disproves its existence. He says: “Es besitzt der Zellhaufen*) den man Tastkörper nennt, nicht einmal eine eigene umschliessende Membran. Ueberal stossen die peripheren Zellen direct an das umgebende Bindegewebe, und nur nach längerer Einwirkung eines Reagenzes kann es vorkommen, dasz das starre Aussehen der Bindegewebsschichten eine eigene Membran vortäuscht”. Likewise Roveer, Tarani, IZQVERDO, HOGGAN, LeontowrtcH absolu- tely deny the existence of a capsule. Meissner, RENAUT, KRAUSE, Worrr, KOLrMAN and LrreBuRrE consider it as a single endothelial membrane. LEFEBURE °): “une simple lume conjoncture doublée sur une face profonde par un feuillet endothelial’. From all this it follows that the hypothesis brought forward by Docter, Rurrinr, Tromsa and Korriker that the corpuscles are provided with a true lamella-capsule, is hardly tenable. The very gradations (and they are many) between MeissNer’s corpuscles and the free endings go far to substantiate a priori the opinion of LANGERHANS, who appears to bave studied the organs under consideration thoroughly. They also support Borrzat’s view when he puts MeissNeR’s corpuscles on a level with the complicate, non-capsuled Merrkeu’s corpuscles. In virtue of my personal inquiry I incline to LANGERHANS’s view, as will appear lower down. Finally let us bestow consideration upon the problem of the genetic connections between the free endings and the tactile bodies with the subpapillary network. If we confine ourselves to the more modern authors, we mention the names of Berne, Prentiss, BorezaT, LEONTOWITCH, SFAMENI and Dogri. ©), who have, all of them, discussed more or less minutely the subepithelial network and its connections with the nerve-endorgans. Borrzar differs from the other investigators in that he considers the network to be independent of tactile corpuscles. This follows from his opinion that the rete amielinica, is built up of fibers of the so-called 2d sort *). But for the rest, he sides with the Italian School, 1) Arch. f. Mier. Anat. IX 1873. 2) The italics are mine. 3) Revue gen. d’histol. 1909. 4) Berge. Allgemeine Anat. und Phys. des Nervensystems. Leipzig 1903 PRENTISS. Journ. of Comp. neur. XIV 1904. BoTEzaT l.c. LeontowitcH Int. Mon. XVIII 1901. SFAMENI, DOGIEL l.c. 5) Medullated fibers losing their myelin already in the nerve-trunk. It seems doubtful whether these fibers are still to be considered as a separate group. 31 our starting point, when in speaking about certain free endings, he says that through anostomoses they form a widely spread end- structure, ‘““welcher in der Form eines im allgemeinen weitmaschigen varikösen Netzes von weithin ausgebreiter Ausdehnung erscheint’’, which continues into the papillae, and there adheres to ordinary medullated fibers. He looks upon this nerve-complex as a “fiir sich _bestehender sensibeler Apparat der Lederhaut”. He finds it again in fishes and amphibia, so it is beyond doubt that he describes the very network which STEPHANELIL discusses in his publication. Doe, an authority on end-organs, concurs with Rurrini that the lateral branches of the free papillary endings blend with the rete amielinica: ‘‘Wie aus dem mitgeteilten hervorgeht, so hat das aus Marklosen Aestehen und Faden zusammengesetzte subpapillaire Ner- vengeflecht, die umeingekappselte Nervenknäuel sowie die Schleifen- formig gebogene Biindel und das intrapapillaire Fädennetz einen und denselben Ursprung”. Also the Timorerw fibres of the MrissNER- corpuscles, which Doeirr, reckons among the sensory system, go to make up according to him, the intrapapillary nerve-complex by means of their ultraterminals. SFAMENI, though far from adhering to the teoria unitaria gives a description of the subepithelial plexus and of its connection with tactile corpuscles and free endings, that accords fairly with Rurrint's. Nor is it on the whole contradicted by Prentiss and LeEONTOWITCH in their publications respectively of Rana and the human skin. It surely will not do to ignore the many differences between the various authors, differences in theoretical conception, in appreciation and in interpretation of their observations. Opposed to Docter, who still holds that interlacement of the fibers is the fundamental principle governing the structure of the network, are Borrzat, Berne, RuFFINI, LrontowitcH, and SrAMENI, who are convinced of the fusion of the fibers. Prentiss wavers. It is a fact that the network is built up of sensitive fibers. However, the question whether also sympathetic elements are fused with it, is as yet unsettled. This depends in some degree on the doubtful character of the Timoreew fibers. Still, though the origin of the sensory part of the network is still uncertain, there is no denying that, also in this respect, observers concur more and more. As we observed before Borrzar considers the whole network to be made up of anastomotic free nerve-endings. Doerr, also looks upon them as the principal components, but according to him also ultraterminals of the Trmorrrw system of the tactile corpuscles unite with it. SrAMENr believes there is also some connection with the genital corpuscles; Lronrowircn, Berar, and Prentiss assume an 32 immediate connection of the network with the free endings as well as with corpuscles. All these authors, though theoretically far removed from Rurrini’s neurogenetic conceptions, have brought forward a number of facts corresponding satisfactorily with those insisted upon most emphatically by the Italian school. In short there is in the literature about the subject a tendency towards the hypothesis that there is, generally speaking, intercon- nection and coherence in the whole peripheral sensory nervous system. It is these facts, derived from the literature, that enhance the significance of recent personal studies made by the BirrscnowsKyY method on the sensory nerve-endings. The BretscHowsky method differs from the methylene blue- and the gold-chloride method in that it affords another view of the problems. It does not present those typical appearances, which, when comparatively slight magnifications of rather thick sections are examined, yield a clear survey of the relations. Its efficiency lies in the fact that when preparations counterstained in haem. eosin, are examined under a microscope of the highest power, it brings out in strong relief the relations between the fibrils and their surroundings. Along this totally different path I arrived at conclusions which, as I hope, will contribute to lend support to the hypothesis that the Meissner corpuscles are more related to the free endings than is commonly believed. In a paper read at last year’s Congress for Physics und Medicine at The Hague (1917) (see also: Verslagen Kon. Ak. v. Wetensch. 27 April 1917) I recorded some morphological data, hitherto unknown, concerning the structure of the axis-cylinder. In that paper I set forth that, when tracing an ordinary nerve-fiber from centre to periphery, the following changes in the structure are to be observed in a transverse section. First we find in the medullary sheath the axoplasm, which (in a transverse section) seems to be vacuolar in structure and embraces the neurofibrils in the protoplasmatic septa between the vacuoles. As known, the medullary sheath is surrounded by the protoplasmatic sheath of Scawann with its nucieus. More towards the periphery the medullary sheath splits up into several tubes. The always vacuolar axoplasma material with its fibrils spreads over the daughter medullary sheaths. Together they remain embedded in one undivided protoplasmatic mass, which must be considered as a continuation of the sheath of Scawann. Still further towards the terminus of the course of the nerve the medullary sheaths disappear from the section, so that the neurofibrils lie free in the proto- plasmatie envelopment which, now being of vacuolar structure like 33 the primitive axis-cylinder, must be assimilated to the sheath of SCHWANN blended with the axoplasm. These formations are seen to get thinner and thinner and their meshes to get ever wider according as they approach the terminus of the nerve. To all appearance they ultimately blend or unite with the connective tissue plasmoderms in which we find the neurofibrils in the ultimate tract of their course’). At first I was disposed to think that the described vacuolar dissolution of the axis-cylinder was characteristic of the so-called free nerve-endings, because I saw the medullated nerves force their way into the Meissner corpuscles without having undergone any modification. I can go a step farther this time, and assert on the basis of a profound investigation of MerssNer’s corpuscles that the axis-cylinders inside these corpuscles pass through precisely the same disintegration process, previously described by me for the so-called free nerve- endings, and just now designated as a vacuolar dissolution. Whereas nowadays it is maintained by many inquirers that the -axis-cylinder loses its medullary sheath, before it enters into the corpuscles, I side with ENGELMANN*), LANGERHANs, Fiscuer®) Kry— Rerzits*) and Lerrepure*), having been able to ascertain, in prepa- rations treated with Osmie acid, that the medullary sheath, just as the sheath of ScHWANN, is prolonged into the intracorpuscular course of the nerves. Moreover my preparations also proved distinctly that those medullary sheaths split up inside the sheath of ScHwANN exactly as has been indicated above. I hold with LerrBure that most likely the fact that the Osmium method has been abandoned for the modern fibril staining methods, is responsible for the erroneous opinions about the presence or the absence of medullary sheaths, prevailing in the neurological literature. As to the sheath of Scuwann, it goes without saying that [ must contest the hypothesis that it passes into the formation of the capsule, since to me it is an intrinsic part of the lemmoblastic sheath. (Dogirt. and others*)). My preparations, which are well impregnated and of good fixation also enable me to ascertain the fate of the axiseylinders 1) Cf. J. Boeke. Studien zur Nervenregeneration I, Verh. Kon. Ak. v. Wet. A'dam 2e Sectie Deel, XVIII n°. 6. 2) Zeitschr. Wiss. Zool. XII 1863. 3) Arch. f. Mikr. Anat. XII. 4) Arch. f. Mikr. Anat. IX 1873. 5) Revue génér. d’histologie 1909. 6) With more justice LANGERHAUS, KRAUSE and others assert that the sheath of ScHWANN passes into the inner capsule of the corpuscles. 3 Proceedings Royal Acad. Amsterdam. Vol. XXI. 34 inside the MuissNer corpuscles. For among the cells filling up the core of the Muissner-corpuscles we find many of the same vacuolar non-medullated nerve-sections, which we have described, with the fibrils, scattered over the spongy protoplasm. Now it was but another step to establish in well-chosen objects that those vacuolar axis-cylinders maintain their course in the cells of the core itself. In tangential sections we were in a position to observe with absolute certainty that from the axis-cylinder the fibrils pass tto the protoplasm of those cells, where they may aid in making up a regular network of the fine fibrils, and where, as a continuation of the vacuolar structure of the axis-cylinder in trans- verse section, a reticular protoplasm serves as a substratum to the neurofibrils. Just as 1 observed previously in the corpuscles of GRANDRY, 1 saw also here a similar diffuse expansion of the net- work over the cell-protoplasm, as well as the mechanical traction phenomena between protoplasm and fibril-system, so that my inter- pretation leaves hardly any room for doubt. It is beyond all question that the core cells are indeed parts of the nerve-course itself; consequently it fits in with my view *) to term them lemmo- blasts together with the other elements, building up the course of the nerve. The fibrillar networks described, are by no means terminal. As a rule the fibrils are seen to unite again and pursue their way as a new axis-cylinder. This is an additional argument for classing those cells among the structural elements of the nerve-course itself. In this way I came to the conclusion that the entire Meissner cor- puscle is built up of compact lemmoblast cords in structure completely similar to the free nerve-endings. Now this appears to me to be an “important conclusion, the more so when correlated with the above data regarding the connection between the tactile corpuscles and the free endings, as discussed in the literature. In conclusion I will. impart that in the Meissner corpuscles I found hardly anything that reminded me of a capsule, certainly not a fine fibrillary texture proper, still less a lamellar system. The enveloping connective tissue is rather of a loose spongy structure. I found in it vacuolar nerve-sections as well as “free” fibrils in- vested in the plasmoderms. I often descried that the contours of MerssNer-corpuscles are very indistinct. Especially in the tactile balls of the cat’s paw I rarely found typical Mrissner corpuscles; often, however, in the papillary connective tissue I found detached groups 1) Cf. G. GC. Herinea. Le développement des corpuscules de Granpry et de Herssr (Arch. néerl. des Sc. Exactes et nat. Serie II] B. tome III 1917). 35 of nerve-sections of the familiar appearance in various sizes. Together they presented precisely the appearance of a transverse section of a Muissner corpuscle. Only by studying serial sections it can be ascer- tained whether we have to do with a Meissner corpuscle or rather with some detached axis-cylinders of free endings. Such forms, which must no doubt be classed as modified Meissner corpuscles, are in my judgment, as many proofs of the close relationship there is indeed between tactile corpuscles and free endings. My conclusions, therefore, are the following: 1. the cells found by all inquirers') except Dogirn in the MeIssNNR corpuscles are elements of the nerve-course itself, lemmoblasts, as I have endeavoured to demonstrate for GRANDRY-corpuscles. 2. As to structure and behaviour, the nerves in the MrIssNER- corpuscles correspond exactly with those of the so-called free-endings. 3. so that it is very likely that the terminal branches of the MeissNER corpuscles (ultraterminals) form one connected whole with the free papillary endings. 1) THomsa. LANGERHANS, RANVIER, MERKEL, KRAUSE, LEONTOWITCH, SFAMENI, RUFFINE, LEFEBURE, VAN DE VELDE and others. 3* Astronomy. — “On the Parallax of some Stellar Clusters” (Second communication). By Dr. W. J. A. Scnournn. (Communicated by Prof. J. C. Kapreyn). (Communicated in the meeting of February 23, 1918). In a former communication it was shown, how it is possible to determine the parallaxes of stellar clusters from the numbers of stars of determined magnitude in the clusters by means of the luminosity curve of Kaprnyn. The calculation was performed for Messier 3 and A and y Persei. Now the same method is used in order to determine the parallax of some other clusters. | The Small Magellanic Cloud. H. S. Leavirr. 1777 Variables in the Magellanic Clouds. Annals Harvard Observ. Vol. 60, N°. 4. A preliminary catalogue containing 992 stars of the Small Cloud and 885 of the Great Magellanic Cloud. The places of 28 stars of catalogues in the neighbourhood of the Small Cloud are also given. We counted a number of stars and estimated their diameter on a photographic plate, taken at the Harvard Observatory. For orien- tation we used the catalogue-stars the position of which Miss Leavitt communicates. In order to reduce the estimates of diameters to magnitudes, we 1stly counted an area of 1000 ©’ without the Cloud, and determined from the numbers of stars of every magnitude the magnitude corre- sponding to every diameter by means of Publ. Gron. N°. 27, Table IV, 2ndly we estimated the diameters of 142 variable stars, the magni- tudes of which occur in Lravitr’s catalogue and which are equally distributed over the Cloud, and we have compared these with the mean magnitude, i.e. the average of maximum and minimum, given by Miss Leavitt, 3"ly we have estimated the diameters of the catalogue-stars mentioned above and compared these with the magnitudes in the C. P. D. and fre HAL GC Finally the magnitude corresponding to each diameter was determined from all these data by graphical smoothing. We counted an area of 240 DO! in the Cloud. The results are given in the table below. In it NV, represents the number of stars the magnitude under consideration. from the brightest star to Diameter | Magn. Nm 25 10.1 | ie 22 10.4 2 20 LO lish) (4 id fies 16 11.3 6 15 eg es (sf 14 ie aa 13 12.0 26 12 12.2 39 11 EEND 10 128i] «81 9 13.1 | 122 8 oe tte | 1 fate 20 | 6 (4.0, | 3282) | 5 ee cand 146 | 467 3 149 | 568 2 15.2 810 | 156 | 1104 0 16.0 | Magn. 10.0 10.5 11.0 11.5 12.0 gate, 13.0 13.5 14.0 14.5 Nin 16 33 60 122 202 305 438 Am 11 21 133 | Normal 11 | Cluster 16 24 122 The normal number of stars is calculated for the galactic latitude b=10°. As we always use the luminosity curve for whole numbers as values of the argument m and have counted here by half magnitudes, we may deduce from the above table the following two tables: 38 | | | | . m | Am | An+1 ee | m Am | nen ON fos a) ce 3.24 i. A425 18 | 3.39 13.0 107 1.81 12.5 61 2.48 14.0 104 | 2.35 13.5 151 1.79 | | | 15.0 455 14.5 270 | | Am : ae ge partly to be ex- LLM plained from our counting only a small part of the cluster. These numbers give the following values for the parallax: The irregular progress of the quotients ] x = 0".0004 Il 4 HI | IV 13 Vv 11 VI 4 Mean = ()",.0007- a2 00002 From 142 cluster variables that are equally distributed over the cluster and occur in Miss Leavirt’s catalogue, we find for the mean apparent magnitude of these stars m—= 14.67 and 5log.2 = — 15.77, so that the mean absolute magnitude of these d Cephei variable stars with a short period is M = 3.9 according to our determination of the parallax. From some d Cephei variable stars with a long period HERTZSPRUNG found for the parallax of the Small Magellanic Cloud «2 = 0".0001. Praesepe. ai == 834" 39°; die = | 201", b= Hr =d Dr. P. J. van Ruin. The proper motions of the stars in and near the Praesepe cluster, Publ. Groningen, N°. 26, 1916. The measurement of 2 sets of plates, taken at Potsdam. The catalogue contains 531 stars. The diameters were reduced to photo- graphic magnitudes by means of standard magnitudes, determined by Hurrzsprunc. The probable error of a magnitude is + 0”.12. We have derived the visual magnitudes from the photographic ones in the same way as Van Ratn did on page 10 of his publi- cation. The correction was determined from the value of the colour 39 index for each apparent magnitude that is based on Parkuursr and SEARES’ researches. To this objections may be raised, as for the cluster stars we have to deal with absolete magnitudes. As, however, the relation between colour index and luminosity is only inaccurately known as yet and as moreover, it cannot be decided whether a given star belongs to the cluster or not, Van Ruatn’s method is the only one possible. Van Ruwn found that the photographic magnitudes (international scale) between m == 7.5 and m = 14.5 wanted a constant correction — 0”.5 for reduction to the visual Potsdam scale. There- fore by a correction — 0”.7 they are reduced to the Harvard scale. The number of cluster stars of each magnitude we find by dimi- nishing the numbers counted by the normal number, which was determined for this cluster from Publ. Gron. N°. 27, Table V. It appears at once that the Praesepe stars have faint luminosities. The declivities that we observe in the frequency curve of the mag- nitudes are partly smaller than the smallest declivity occurring in Kaprteyn’s luminosity curve. That is why we could make only four determinations of the parallax notwithstanding the great interval of magnitudes. These give mn = 0".024 + 0".004. This parallax is considerably greater than the one which we found for other stellar clusters. Messier 52. Pear pOd — 2098, did 018, b= 4-1"; (= 81°: -class- .D 3. F. Pinesporr. Der Sternhaufen in der Cassiopeia. Diss. Bonn. 1909. Measurements of three plates, taken by Kistner. The catalogue contains 132 stars up to 15”.0. The standard magnitudes have been determined by visual observations by means of gauzes of 25 stars by ZURHELLEN. We find from 4 determinations: x = 0".002 + 0".0003. Messier 46. Bren O.2487 ie ag Th 808 Adi 4E De 6, == 2002 class: 11. W. ZurnerveN. Der Sternhaufen Messier 46. Veröffentl. Kgl. Stern- warte zu Bonn, N°. 11, 1909. Measurements of three plates, taken by Ktsrner. The catalogue 40 contains 529 stars. For standard magnitudes 47 stars were used, the brightness of which was estimated by Kisrner or determined by means of gauzes by ZURHELLEN. We find from 4 determinations: x = 0".002 + 0".0001. Messier 37. Ne G. 020995 wr ¢ = '5"45".8, de, = 4: 32°31 |, GSE 1 — 145°: class: D1. J. O. Norpiunp. Photographische Ausmessung des Sternhaufens Messier 37. Inaug. Diss. Upsala 1909, Arkiv för Matematik, Astro- nomie och Fysik, Band 5, N°. 17. Dr. H. Giegerer. Der Sternhaufen Messier 37. Veröffentl. Kgl. Sternwarte zu Bonn, N°. 12, 1914. NorDLUND measures 4 plates and gives the places and magnitudes of 842 stars. The magnitudes are derived from the diameters according to the formula of CrarLieR by means of 214 standard magnitudes that have been determined photometrically by Von ZripeL. Many of the bright stars of the cluster are red (colour index > 0”.7), e.g. some 50 or 70°/, of the stars of the 10' magnitude. : GieBeELER discusses 2 plates taken by Kistner and measured by SrroELE. The catalogue contains 1231 objects. The magnitudes have been joined with Norpiunp’s scale by comparing those of 450 stars. For the red stars too the photographic magnitude is given. For our purpose it is a drawback that for the red stars the photographie magnitude is mentioned. This is why the brightest stars, among which many red ones occur, could not be used by us. Exeluding these we find from 4 determinations: a = 0".002* + 0".0004. Messier 36. — 5h 29m 5, JS == Je 34° 4’, b= i 2% 1900 NG. 6: 1960; ( — 142°; class: D2. Dr. S. Oppennem. Ausmessung des Sternhaufens G. C. N°. 1166. Publ. der v. Kuffner’schen Sternwarte in Wien, Bd. III, pag. 271-307, 1894. Measurements of three photographic plates. The catalogue contains 200 stars. The magnitudes were derived from the diameters, measured in connection with estimates of visual magnitudes found by Dr. Patisa for the greater part of the stars. The interval of magnitudes is small. We find from 3 determinations : a = 0".005 + 0".001. t “1900 41 20 Vulpeculae. Pe BOO) 200 or 2077.6. OO NO ae ee EO H. Senurrz. Micrometrisk bestämning af 104 stjernor inom teleskopiska stjerngruppen 20 Vulpeculae. Kong]. Svenska Vetenskaps- Akademiens Handlingar, Bandet 11, N°. 3, 1873. The magnitudes have been determined by a photometer in accordance with ARGELANDER’S scale. A. Donner und QO. Backiunp. Positionen von 140 Sternen des Sternhaufens 20 Vulpeculae nach Ausmessungen photographischer Platten. Bulletin de l’Acad. Imp. des Sciences de St. Pétersbourg, Série V, Volume II, pag. 77-92, 1895. Measurements of 2 plates taken by Donner at Helsingfors. The magnitudes were taken from SHILOW. : M. Suitow. Grössenbestimmung der Sterne im Sternhaufen 20 Vulpeculae. Bulletin ete. ut supra, pp. 243-251. The magnitudes of the 140 stars, the position of which was deter- mined by Donner and BacKLUND, were found by measuring the diameters of the images. As standards those 100 magnitudes were used that Scnuurz had determined already. SHitow uses CHARLIER’s formula m = # — y log D— zD. The probable error of a difference M—Mscyuutz 1S + O™.25. We have not reduced the magnitudes based on ARGELANDER’s scale, to the Harvarp scale, because SHILOW’s magnitudes differ considerably from those of Scuuitz. We find for the parallax from 7 determinations: = 0.005: 2.9001: Messier 5. N.G.C. 5904; « class: C3. M. Smirow. Positionen von 1041 Sternen des Sternhaufens 5 Messier, aus photographischen Aufnahmen abgeleitet. Bulletin de Acad. Imp. des Sciences de St. Pétersbourg, Série V, Vol. VIII, pag. 253-312, 1898. Measurements of 2 plates, taken resp. by BrLOPOLSKY and KostTinsky. The magnitudes have been determined ina rather inaccurate manner, viz. by comparing the diameters with the images of stars of 20 Vulpeculae, the magnitudes of which are known. S. I. Bairey. Variable Stars in the Cluster Messier 5, Annals Harvard Observ., Vol. 78, Part. II, 1917. Ninety-two stars are dealt with. For 72 the period is mentioned. =15413".5, d,,,,=+2°27', b= 45°, /=333°; 1900 42 Among these 3 have long periods. Moreover the magnitudes are given for 25 comparison-stars. In Smmow’s catalogue the magnitudes of 1006 stars are mentioned. The interval of magnitudes is small and the magnitudes are inaccurate. Nor did we succeed in reducing them to a more exact scale by means of Bainuy’s magnitudes. We find the results 7 = 0".0002 and nm = 0".0009; consequently as average value: == 0" 0005" +. 00002. According to Snaruey the average photogr. magnitude of the variable stars is 15”.25 and we found 5 log. 7—=—16.3; therefore M=15".25—11".3—4".0. So we get for the mean absolute magnitude of the variable cluster stars 4.0. If we determine the parallax from the variable stars with a known period, we find, when making use of HeRTZSPRUNG’s numbers: x = 0".0002. Messier 13. N. G:C) 6205; @:,,.==16"'38".1, ¢,,,,=36°39) = LOE ae class: C3. J. Scuuiner. Der grosze Sternhaufen im Hercules Messier 13, Abhandl. Kgl. Akad. Berlin 1892. The catalogue contains 823 stars. The magnitudes are uncertain. H. Lupenporrr. Der grosze Sternhaufen im Hercules Messier 13. Publ. Astroph. Observ. Potsdam, Bd. XV, N°. 50, 1905. This catalogue contains 1118 stars. The brightness is not expressed in magnitudes; but the diameters are estimated in 16 “Helligkeitsstufen”. H. Suapiny. Studies ete. Second Paper: Thirteen hundred stars in the Hercules Cluster (Messier 13). Contrib. Mr. Wairson Observ. N° 416, 1925. The photogr. and photovis. magnitudes of 1300 stars have been determined; but of only 650 stars they have been published. For the statistical investigation 1049 magnitudes and colour indices were used. We make use of LuDeENDOrFF’s catalogue and we availed ourselves of SHAPLEY’s results in reducing the “Helligkeitsstufen” to magnitudes. First we can express the “Stufen” in photographic magnitudes by means of a table in Suaptey’s work (p. 25, Table VIII) and these may be reduced to photovisual ones by means of the Tables XIV and XVI. No correction is wanted for the difference between the scales of Harvard and Mounr Witson, because the visual Harvard 43 scale is continued only up to 12”.0 and for this magnitude agrees with the Mr. Witson scale. ; Now we determine the numbers A,,. For the brightest magnitudes we find then a declivity, which surpasses by far the greatest decli- vity, found in KapreYN’s curve. This value, great as it is, may perbaps be explained from the manner, in which the diameters have been reduced to magnitudes. Excluding of these values being unde- sirable a priort and not possible on account of the small interval of magnitudes, we have smoothed the numbers observed by a con- tinuous curve. Then we find from 4 determinations: m0 00075 = 0.00006: From Swapiey’s research (le. p. 79) we derive for the mean photographic magnitude of the variable cluster stars which are probably d Cepheids, m= 15.2 and we found 5 log. 7 = — 15.4, so that according to our determination of the parallax their mean absolute magnitude = 4.8 '). From 2 variable stars with known period SuHapizy (l.c. p. 82) found for the parallax the value: x = 0".00008. Messier 67. N.G.C. 2682; a,,,, —8'45".8, .d,,,,.==12°11', 6-=-++ 31°, —183>; class: D 2. E. Faceruotm. Ueber den Sternhaufen Messier 67. Inaug. Diss. Upsala, 1906. The catalogue contains 295 stars. The magnitudes were derived from the diameters by means of CHaARLIER’s interpolation-formula, after the visual magnitudes of 15 stars had been determined photo- metrically. H. Swapney. Studies ete. III. A catalogue of 311 Stars in Messier 67, Contrib. Mr. Witson Observ. N°. 117, 1916. For all stars the photogr. magnitudes have been determined and also the photovisual ones for all stars within 12’ of the centre. In this way 232 colour indices were found. Suarrey finds a much greater number of back-ground stars than would be expected. OrssoN’s catalogue cannot be used on account of the inaccuracy of the magnitudes. We first make use of FAGERHOLM’s catalogue. The magnitudes that are expressed in the P. D. scale, are reduced to the Harvard scale by adding a correction —O”.2. 1) The values of the parallax and the mean absolute magnitude given here, are to be preferred to the preliminary results published in the first communication. 44 Now we derive from 2 determinations (the interval being only 2 magnitudes) : x0 OON OCT According to SHaPLeY (Le. p. 10) the difference Fac.-Mr. W. is constant — + 0.24 and as Harv. = Mr. Witson photovis., we have also: Fac.-Harv. = + 0.24. We have taken Faa.-Harv. = + 0”.2, so that the magnitudes used should be correct. Upon closer inquiry, however, the difference Fac.-SHaPpLEY appears not to be constant, but to vary with the magnitude. We have determined the errors of FaGERHOLM’s scale by comparing the magnitudes of 156 stars, and afterwards we have calculated the numbers A,, for the corrected magnitudes. Now we derive for the parallax from only one deter- mination that can be used: x= 01".002. By telling off SHapLey’s catalogue we find for the parallax the values a == 0.001 and a = 0".002. Summing up, we may assume for the parallax of this cluster: 1 — 0" 002. For this cluster SHapLey determined the colour indices of all the stars, perceptible on the plate within a circle with a radius of 12’. But here, too, no great value can be attached to a comparison of the distribution of colours, found by SHapiey for every J/, with ScHWARZSCHILD’s table. For it is not certain that all stars up to 13”.0 are visible on the plate, and just here the separation of cluster stars and back-ground stars offers great difficulties. According to SHAPLEY the distribution of colours, expressed in percentages of the numbers of stars of determined absolute magnitude, is as follows: sul: bolides 50 B A FE 38 30 G 51 20 K M 45 Messier 11. Reet OD 2,,,, <= 18% ADR Oo BAS hoe de f= 355". class: (3). W. STRATONOFF. Amas stellaire de l’écu de Sobieski (Messier 11), Publ. de ’Observ. de Tachkent N°. 1, 1899. The catalogue contains 861 stars. From the estimates and measu- rements of diameters the magnitudes have been derived by means of the Southern B. D. H. Sraprey, Studies ete. IV. The galactic cluster Messier 11, Contrib. Mr. Witson Observ. N°. 126, 1916 (A. P.J. Vol. 45, 1917). For 458 stars the photogr. and photovis. magnitudes have been determined. For statistical research 364 stars were available, after the uncertain magnitudes and the stars upon which the EBrRHARD- effect may be of influence had been excluded. _We tell off Srratronorr’s catalogue and we determine the quotients hic cannot be used. Now we reduce STRATONOFF’s magnitudes to SHaPLEY’s scale. In order to do so we compare the magnitudes of 293 stars. The results are given in the table subjoined. It then appears that the magnitudes are too inaccurate and | ™Suaptey | Sh—Strat. | Len gee 10.0 + 1.53 30 5 1.94 44 ar | 1.68 38 5 1.39 26 12.0 1.45 1! 5 1.21 14 13.0 0.91 - 50 | 5 0.80 27 14.0 | 0.70 53 Afterwards we determine by interpolation A,, for the corrected magnitudes. In this way we find for the parallax from 2 determinations : a — 0".00055 + 0".000038 1) SHapLey reckons Messier 11 among the open clusters. 46 The mean parallax of the globular clusters is 0".0006 and that of the open clusters (Praesepe excluded) is 0”.003. The number of parallaxes, determined at present, is still too small to derive conclusions from them as regards the distribution of clusters in space. Perhaps this will be possible, when we shall have extended 63 O8 5 Ss OS 6h OF SE oF GL OL 59 09 47 ° our research to more clusters. It will then also be possible to investigate, how far our results give support to the well-known theory of giant and dwarf stars. From the figure subjoined it is evident that the luminosity curves of the various clusters greatly resemble that found by Prof. Kaprnyn for the stars in the neighbourhood of the sun. And so this method of determining the parallax, proposed by Prof. Kapreryn, is justified. In the graphical representation Ny; means the number of stars from the brightest star to the absolute magnitude under consideration. As it is only our purpose to compare the relative frequencies of the various absolute magnitudes, we added in each curve a constant amount to log. Ny. Amsterdam, December 1917. Physiology. — “/rperiments with Animals on the Nutritive Value of Standard Brown-Bread and White-Bread.” By Prof. C. EvwkKMAN and Dr. D. J. Hursnorr Pot. (Communicated in the meeting of April 26, 1918). Owing to the scarcity of food the old problem has latterly cropped up again whether, instead of baking white-bread, it would not be more practical to make bread of unboltered meal, since through the process of boltering the grain loses 20—30°/, of its nutritive value, according to the degree of milling. The modern technique of grinding enables the miller to separate the flour, which contains the constituents of the endosperm or starchy part, nearly entirely from the bran and the germs of the grain. The current opinion among people that brown-bread is more nourishing than white, is founded chiefly on the belief that brown bread is more satiating and appeases the appetite for a longer period than white bread does. Though this property must not be underrated, it scarcely needs to be pointed out, that it cannot be an index for the content of nutritious matter. The bran (inclusive of the germs) differs from the flour by a smaller amount of starch and more nutritive salts, fat and protein. However if also contains more cellulose, which is all but indigestible for man, and which also renders it difficult for the alimentary canal to utilize the foodstuffs contained in the bran, since they are for the greater part shut up within thick walls of cellulose. This is why many consider the bran to be useless for man, even noxious, and deem it better that only flour should be baked into bread and the bran should be given to the cattle, which can digest cellulose well and returu to us the foodstuffs of the bran in the form of flesh and dairy-products. On the other hand it has been argued that this round-about way via the cow, is also attended with great loss, and that, in striking a balance, it will turn out that man gets more food from wheat in the form of brown bread in spite of less digestibility, than from an equal amount of wheat in white bread. However, it now appears that the problem requires re-consideration, since it has been proved that, besides the foodstuffs alluded to, the bran also contains peculiar constituents, altogether lacking in flour, 49 that are highly conducive to the building up of the animal body, nay are even indispensable for its health and growth, viz. the so- called accessory foodstuffs or vitamins. Here l refer to a paper read by me (E.) some 20 years ago on a fowl’s disease (polyneuritis gallinarum) attended with degeneration of the peripheral nerves and motory disturbances arising from a polished-rice diet, and resulting in death within a few days, unless another diet was had recourse to. When the fowl was fed on un- polished rice, or when polishings were added again to the peeled rice, the disease could be prevented, or, if it had already broken out, it could be cured. It appeared namely, that the rice-polishings contained ingredients which, being diffusible, could be readily extracted with water and possessed the same prophylactic and remedial property as the polishings themselves. The fowl’s disease, which can also be produced in other birds (pigeons, rice-birds) in the manner described, shows in many respects a close resemblance to beri-beri, and the researches by VoRrDERMAN, and many others after him, demonstrated that much of what was brought forward for the one was also applicable to the other. It must be especially remembered that what has been said regarding rice, also holds for other kinds of grain. Fowls develop the disease, when fed on boltered meal, but not or exceptionally only when given the whole grain or unboltered meal. In keeping with this is the fact that beri-beri does not only manifest itself where polished rice constitutes the staple diet, but is also observed among a population living chiefly on white-bread (LrrrLr). Also in Holland the tropical beri-beri can break out, as has been proved by the cases that lately occurred among native sailors of the Rotterdam Lloyd, described by Kooremans Brynen. It is well-known, moreover, that the so-called Ship beri-beri, a comparatively mild form of the disease, which has been seen from time to time especially on Norwegian ships, is also attributed, on reasonable grounds, to too one-sided and too vitamin-poor a nourishment. Nor is it at all improbable that cases of polyneuritis among men, which do occur every now and then, are in some degree allied to beri-beri. Fortunately the accessory foodstuffs, playing a part here, occur in many other articles of food, such as peas, beans, potatoes, meat, egg-yolk etc. There need be no fear, therefore, for the immediate appearance of beri-beri, at all events not when foods such as white rice and white-bread are not the principal dish. However, if we bear in mind that, as has been seen from what we said about meat, the relative vitamins form a normal constituent of the animal 4 Proceedings Royal Acad. Amsterdam. Vol. XXI. 50 body, (not evolved in it but derived from the food), it is but natural that, especially in times of scarcity, a vitamin-poor food should be deleterious to the body, even though not causing actual illness. — Comparative experiments on the nutritive value of brown- and white-bread have repeatedly been undertaken, also when vitamins were not thought of. As early as about seven decades back MAGENDIE observed how a dog, fed exelusively on white-bread, lost flesh, got weaker, and weaker, and succumbed after 40 days; another dog, fed on bread made from the whole wheat, kept in good health. Similar results were latterly achieved in Hormeister’s laboratory with mice. The evidence from such experiments may be disqualified by contending that the laboratory animals actually starve, because they refuse to eat white-bread much sooner than brown-bread. Those nevertheless who believe in animal instinet will not wholly repudiate the significance of this phenomenon. We preferred to experiment with fowls, first of all because they react most indubitably upon vitamin-poor food with the typical aspect of polyneuritis and do not sueeumb under equivocal symptoms ; and secondly because when the appetite lessens, they readily submit to forced feeding. Forcible feeding is a method also employed in poultry-yards. Intense inanition may in this way be prevented up to the first indication of the disease, viz. atony of the muscle layer of the crop. This causes a more tardy discharge of the crop, so that the ordinary daily allowance cannot be gone through. The typical weakness in the leg-muscles, reminding so forcibly of a similar disturbance attending beri-beri, generally ensues only after some days, sometimes weeks. Here we also wish to observe that fowls are no more able to digest the cellulose of the bran than man is. The thick walls of the cells of the so-called alenrone-layer, in which chiefly protein and fat are contained, are left intact in their digestive canal. The vitamins, however, as said above, are easily isolated from the bran. The meal, from which the Standard bread was baked, was composed according to the governmental prescription for the white-bread of 60°/, inland wheat- and (or) rye-flour, 10°/, American flour and 30 °/, potato-meal; for brown-bread of 70°/, unboltered wheat- and (or) rye-meal, 25 °/, potato-meal and 5°/, grits and (or) pollard. Potato- meal is too pure and, therefore, too one-sided a food. The other nutritive constituents of the potato — protein, salts and also vitamins — get lost in the preparation. They putrefy our public waters. It would have been much more reasonable indeed, to eke bread-meal out with powder from dried potatoes, instead of potato-meal. On the 51 other hand yeast raises the vitamin-content; it has a protective and curative effect with respect to polyneuritis. An accidental advantage is that during the baking the internal temperature of the dough hardly rises above 100° C. As has been shown by Grins for rice and has been corroborated also by myself for other cereals, vitamins are destroyed by moist heat only at much higher temperatures. In the writer's laboratory two sets of three fowls have been subjected by Dr. Huursnorr Por, to feeding-experiments on brown- and white-bread; they were young, strong animals of about the same age (+ 2 years) and weight. The best fed animals were taken for the -white-bread experiment; their body-weight averaged ca. 1550 grms; that of the brown-bread fowls was ca. 1400 grms. The bread-ration was ca. 100 grms. The results of the experiments are given in the graphics. S denotes the moment when forced feeding commenced. P that when the typical symptoms of polyneuritis (disturbances in the gait) made their appearance. For purposes of accurate comparison the changes in the body-weight are not expressed in absolute measure, but in percentages of the initial body-weight. When first studying the whitebread experiments, we shall notice a fall in the body-weight almost immediately, in spite of normal appetite, which fall continued also after we proceeded to forced feeding. At the close of the 11th week the first fowl! (III) devel- oped polyneuritis and succumbed after a few days. A second (II) followed a week later. Henceforth it was fed on brown-bread, just as N°. I, which had lost flesh, indeed, but was not yet actually ill. With this diet the diseased animal recuperated and the fall in body-weight was arrested in either of them. Whereas with a polished-rice diet the fowls develop polyneuritis most often inside of five weeks, not unfrequently even as early as at the end of the 3d week, this outbreak was considerably retard- ed in the case of fowls on white-bread. It seems probable that this is due to a protective action of the baker's yeast. Much more favourable were the results of the brown-bread ex- periments. N°. IV and V remained perfectly healthy and vigorous up to the conclusion of the experiment, which lasted 20 weeks. They increased in body-weight, N°. V even considerably, so that there was no occassion for forced feeding, although a slight inap- petence ensued, as is always the case with a uniform diet. N°. VI fared worse. For the first fortnight it maintained its original weight, but after this time it lost weight constantly ; forced 4* 52 feeding was of no avail. The animal got anaemic, showed the typical aspect of polyneuritis in the 17% week and died a few days later. Change in body-weight in percentages _ Cases in which the same diet of the initial weight. is wholesome for the one and Eee injurious to the other animal are not without parallel. Every bio- TPR LES logist has to take account of i a individual differences. These dif- ferences also hold for the need of vitamins. GRYNs has even known the disease to break out after a prolonged diet of unpolished rice, though animals that have already been attacked, may most often be cured with the same diet. At- ‘tendant circumstances, such as intercurrent diseases, weakening influences may also come into play in such cases. Many even regard an infection as a neces- sary condition for beri beri to break out. Wittebrood = White-bread. Bruinbrood = Brown-bread. In resuming it may be allow- able to state that brownbread yields undoubtedly more satisfactory results than whitebread. In connection with what we said at the beginning, we believe the same to hold good also for human nourishment. The drawback of partial indigestibility must not be overestimated. Besides, by improvements in the mode of grinding the miller is able to neutralize this drawback by a finer distribution of the bran along the dry or the wet path, or by removing the coarsest and least nutritive outer layers of the grain. This method should henceforth be more generally applied. Nature, as it were, has destined the bran to eke out the flour; it seems unreasonable, therefore, to separate the two and to replace the bran by potatomeal, which last should be admixed only in the second place and preferably in the form of potato-powder. The use of white-bread should be restricted as much as possible. Foodstuffs that are fit for man, nay that are preferable for human sustenance, must in times of searcity not be given to the cattle. The Hygienic Institute of the Utrecht University. Physics. — “An indeterminateness in the interpretation of the entropy as log W”. By Mrs. T. Enrenvest-Aranasssewa. (Communicated by Prof. J. P. Kuenen). (Communicated in the meeting of March 23, 1918). I. A certain quantity of a gas may be given, so large that it may be divided into a great number of portions — great enough for the purpose we are about to discuss — without the usual statistical treatment of the parts losing its value. Regarding the matter from a thermodynamic point of view we assume : 1. that the entropy of every system strives to attain its maximum. 2. that the entropy of the total mass of gas is equal to the sum of the entropies of the parts. If in accordance with the kinetic theory, we take the entropy to be the logarithm of the probability of the state of the system, we get the following theses as the analogues of those just given: 1. The state of every system endeavours to approach the greatest probability ; 2. The logarithm of the probability of the state of the total mass of gas is equal to the sum of the logarithms of the probability of the states of its parts; or in other words: the probability of the state of the whole is equal to the product of the probability of the states of its parts. | At the same time it may easily be seen that the latter theses are only correct provided the combinations with which we reckon in the determination of the probability of the state of the whole are submitted to certain limitations, which are quite arbitrary from the combinationary point of view. II. We will illustrate this by a simple example, which depends only on the caleulus of combinations. Let us suppose 27 tables, each provided with three holes. In each of the holes a red or a black ball must come to lie. The colour of the ball may be decided by a lottery, in which the chance of drawing a red ball is °/,, and of a black ball */,. 54 In this case for each table separately — if we still distinguish between the three different holes *) — the most probable division of the balls is: two red ones and one black one. For this the probability is?) 2 1 3/12 3°63 Say ar 2 3 X We must now ask: what is the most probable distribution of the combinations over all the 27 tables? We can here still distinguish between the tables. As the most probable distribution we get that in which on only twelve tables two red balls and one black ball lie, on eight of the others three red ones, on six 2 black ones and one red one, and on the last one three black balls. For this distribution the probability is expressed by W LANE 8 NE 6" Ly 21! =(5 (27): A27 nn On the other hand, the chance that on each of the 27 tables uniformly two red balls and one black ball should come to lie is given by 12\27 SS Ss The ratio between the two is W,, 1226/87 127 Wat) BEBE Ty. which is very much smaller than 1 *). Let us now suppose the number of balls that can lie on a table, and also the number of tables to be greater; the number of different typical possibilities of division on each table separately (varying from all red to all black) then rises, as also the number of ways in which we can find these types of division spread over the collective tables. The chance of the most probable division for one particular table becomes smaller. The probability W,, that just this division will be found repeated on every table, becomes therefore represented by a high power of a very small fraction. 1) That is to say, if for a particular combination (e.g. 1 red, 2 black) we count as separate possibilities the cases in which differently coloured balls lie in a given hole. *) The chance of all three being red is ,8,, of one red and two black £, of all three black >. 68 U A2 ere! NBAT LRZ 12e E 3) For 66 ge: a7) Tage BBS ar in which further 6/ disign 1 2 ' 12 Cees GF ot en > 400 \ Tar dee ee 55 On the other hand, the chance Wn for the realisation of that case, in. which the different types are found represented amongst the collective tables in proportion to their probability, will contain a large permutation-factor, and consequently — with a suffi- ciently large number of tables the ratio W,/W,, may reach any degree of smallness. It makes a great difference, therefore, — and of course not only to the calculation of the maximum — whether we take the tables collectively as an object of higher order in the calculation of combinations or whether we determine the probability ‚for each table separately and calculate that of the whole as product of the separate probabilities. III. Suppose that the number of tables and holes for each table are not yet given, but only the total number of hollows in all the tables together, and that it was left to our choice to divide them amongst the tables, then an opinion as to what was the most probable division would be even more arbitrary. IV. It is obvious, that the above considerations may be applied to the gas, taking into consideration, where necessary, additional conditions. | If we introduce the restriction that in the parts only we attend to all the possible permutations, in defining the most probable division, and that in the system as a whole we do not take into consideration any further permutations between these parts, only then does the probability for the state of the whole appear as the product of the probability of the states of the parts. If on the other hand the total system is regarded as anew object for combinations, an object of a higher order, the probability of the distribution of a special state in the whole is not equal to the product of the probabilities of the parts corresponding to this state. The latter must be corrected by a certain permutation-factor, the magnitude of which is dependent upon the number of the parts, that is either upon the fineness of the division to be chosen at will, or — with a permanently fixed fineness of division — upon the magnitude of the total system. The question arises: with which /oy W should the entropy be identified ? Only when the said permutation-factor is neglected can it be said that the tending of the parts towards the maximum of their entropy brings with it a striving towards a, maximum of the entropy of the whole. 56 If we adopt the latter view, in other words if we say that the log W of a system is almost the same as the sum of log w of its parts, at the most a sign of inequality is changed into a sign of equality. It is not justifiable, however, to reverse the sign of inequality. But this is just what happens when, for instance, the uniform distribution of density in a gas is regarded as the most probable state, and in order to calculate the probability of a distribution slightly deviating from this the relation log W = = log w, is taken as the basis, for in this way each deviating distribution - appears as a less probable one’). V. The above analysis is by no means intended to call into question the validity of calculations similar to those indicated in the preceding paragraph, as these rest on the thesis that the entropy of the whole is equal to the sum of the entropies of the parts, a thesis that probably is physically better justified than the combinatory reasonings, at least-in the circumstances in which they are applied. The analysis is merely intended to make clear that the decision of the question whether the probability of the state of a system has reached its maximum or not, depends upon the point of view of the investigator, and that the ideas formed from purely combinatory reasonings do not form a satisfactory or conclusive foundation to direct our choice amongst many different standpoints to. any one in particular; further that the choice of our standpoint is made on the ground of various physical intuitions, which are outside the pale of the combination-calculus as such. That is to say, that the combinational reasonings in question cannot be deduced from a higher principle which may be said to rule nature. VI. We can show this more particularly in the case of a gas. Let us bring together two cubie centimetres of gas at different temperatures. If it should depend upon the “probability principle” which is to happen, it would be quite indefinite whether an equalisa- tion of temperature would take place or not. It would depend upon the question of which is more important in nature; one cubic centimetre or trillions of cubic centimetres. In the latter case our two cubic centimetres might just be those members of our trillion 1) R. Fürrn. Ueber die Entropie eines realen Gases als Funktion der mittleren radumlichen Temperatur- und Dichteverteilung. Phys. Zschr. 18, p. 395—400, 1917, 57 system, which onght to have different temperatures in order that the whole may get the most probable division of temperature over its parts (trillion tables, and upon each of them million balls). If it is advanced against this that an inequality of this kind must continually appear in precisely the same cubic centimetres, so that our two portions of gas may still equalize their temperature, it must not be forgotten that this demands that at the same moment another arbitrary pair of cubic centimetres would be obliged to change temperature in just the opposite direction. Further it must be remembered that in the case when the subdi- vision is continued as far as the single molecules we do actually take up the latter standpoint: the momentary kinetic energy accorded to each separate molecule is in itself not the most probable; over a sufficiently large number of molecules, however, the velocities are divided in such a manner that we can only talk of the most probable distribution for the whole of these molecules (quadrillion tables with one ball on each, or, what comes to the same, one table with quadrillion balls). Zoology. — “On the primary character of the markings in Lepi- dopterous pupae’. By Prof. J. F. van BEMMELEN. (Communicated in the meeting of April 26, 1918.) On p. 136 of his paper: Zur Zeiehnung des Insekten-, im be- sonderen des Dipteren- und Lepidopterenfliigels (Tijdschrift voor Entomologie, vol. LIX, 1915) pe Merrre raises objections against the comparison of the pupal stage in Lepidoptera with the subima- ginal instar of Agnatha; a comparison, which as far as I know, was first made by PovLton’), and to which I have expressed my adhe- sion in my paper on the pupae of Rhopalocera’). He says (translated by me): “It is well known that many investi- gators believe the pupa to have evolved from a flying imagolike form, the limitation of the wings to the last instar having been acquired later on. In these views | cannot agree with my colleague” (viz. vAN BEMMELEN). “In what way one may imagine the initial evolution of the pupal stage to have taken place, either from a dormant subimago, or from a dormant larva (the latter alternative according to my view being the more probable), in any case I think to be justified in supposing that the Trichoptera, Panorpata, Diptera and Lepidoptera have differentiated out of Neuroptera, after the latter had acquired the Holometabolic metamorphosis they possess to-day. Now the Neuroptera generally have a faintly coloured pupa, which leads a hidden life, concealed in the earth or in a cocoon, and usually has a thin chitinous skin. Such also is the condition with Panorpata, Diptera, and likewise with a number of lower Le- pidoptera, as Micropteryx, Lymacodides and many others. When therefore we meet with special colour-markings exactly in the freelwing pupae of diurnal butterflies, 1 am inclined to regard this as a wholly secondary feature.... (The italics are mine). This statement leads me to the following remarks: 1!) E.B. Pouuton, The external morphology of the Lepidopterous Pupa, its relation to that of other stages and to the origin and history of metamorphosis ; Transactions Linnean Society 1890—91. 2) J. F. van BEMMELEN, Die phylogenetische Bedeutung der Puppenzeichnung bei den Rhopaloceren und ihre Beziehungen zu derjenigen der Raupen und Imagines» Verh. d. Deutschen Zool. Ges. 23 Versamml. 1913. 59 Against the use of the expression ‘“subimago” in itself, for the pupal stage of Lepidoptera and other Holometabola, pr Merere does not seem to have fundamental objections, for as is seen from his own words, he declares that the pupa might be considered as an “inactive subimago,” though he himself would prefer the name “in- active larva.” In this preference I cannot agree with him. The conception “larva” implies the presence of provisional organs, as well as the manifestation of a metamorphosis, the moment of which fixes the final point of larval life. Now it is clear, that this point lies at the passage from caterpillar to pupa. Therefore the latter cannot be called an “inactive larva’, but only an “inactive subimago”. It might even be asserted to represent an “inactive imago’’, for the provi- sional larval organs have disappeared, the imaginal organs on the > contrary being all present, though still unable to functionate. But it is especially against the inference, that this subimaginal stage should have been provided with a sufficient mobility to enable it to fly about, after the fashion of the caddisflies when they leave the water, that pz Meyere raises objection. According to his view, it is much more probable that in none of their phylogenetic stages the Lepidoptera or any of their kin: Panorpata, Diptera, or Neuroptera, were ever on the wing before the very last moult, so before they fully deserved the designation “imago’’. Now I must admit, that this supposition of the occurrence of a flying subimaginal instar among the ancestors of these groups of Insects is merely a hypothesis, which can only be supported by argu- ments of probability, while most assuredly important objections can be opposed against it. One of these difficulties I will indicate my- self: Holometabolic Insects may indeed be compared still to other Hemimetabola than precisely the Agnatha, and moreover to Ame- tabola also, and this comparison may lead to raising the question, if the pupal stage might not best be compared to the last instar but one of these groups, to which belong insects, whose different instars are much more similar to each other than those of Holome- tabola, because all of them differ less from the imaginal condition, or, what means the same, because they have all deviated in a minor degree from the original Insect-type. In them we see the wings protrude at an early stage as lateral outgrowths of the dorsal body-wall and increase in size at each following ecdysis, though entering into function at the last one only. Why should this course of development be less primitive than that of caddisflies? Might not the curious phenomenon, that 60 the subimaginal instar of the Ephemeridae, after moulting at the surface of the water, flies about for a few moments, then to moult again and immediately afterwards to proceed to copulation, rather be taken as a speciality of the Agnathous life history, without any deeper significance, and therefore of no importance for the explana- tion of Holometaboly with its dormant pupal stage. On this point I dare not pronounce a definite opinion, but should like to point out, that in trying to find an answer to the above stated question, we must take into account various general consi- derations, in the first place that of the development of wings in its totality, viz. the question how Insects (at least Pterygogenea) acquired their wings. For this decides about the question whether we are to suppose that the ancestors of modern Pterygote Insects never passed through a period, in which they moved about on the wing before attaining sexual maturity, or that the beginning of the functional activity of the wings (howsoever acquired) became more and more postponed to the last instar. If we are right in accepting the second alternative, and therefore in believing that the oldest winged insects could already make use of their wings shortly after their birth, the Agnatha may have retained a last trace of ‘this ancient condition. The apparently absurd fact, that these animals fly about in their subimaginal coat for a few moments only, might then be explained by the assumption, that they gradually postponed the start on the wing to later instars, under the ever increasing influence of their secondary adaptation to life in the water. Then the difference between them and other Hemimetabola would not consist in a greater originality of the latter, but in a different mode of deviation from the primitive condition, viz. by the complete removal of the initiation of real flying to the imaginal instar. The supposition of such a retardation in the transition to flying life-habits is diametrically opposed to the explanation assumed for many other phenomena in metamorphosis, viz. that the manifesta- tion of new characteristics is gradually removed to ever younger instars. In my opinion the former supposition is as well justified as the latter. When for instance Weismann (rightly I think) assumes that changes in colour-markings of certain caterpillars, becoming visible at their last ecdysis only, have been transferred to younger stages in species near akin by a process of precession of development, the opposite course of events may also be consi- dered possible, viz. that a colour-pattern of the wings, which origi- nally came into existence together with the wings themselves, now 61 only appears a long time after the stage in which the rudiments of the wings first become visible. Now what is true for the colour-pattern, may as well be applied to the wings themselves. I do not intend to enter into these considerations more profoundly,as it is irrelevant for the solution of the question, whether or no the colour- pattern on the wing-sheaths of Rhopaloceran pupae possesses phylo- genetic significance. On the contrary it seems to me that in this way the question is made unnecessarily intricate. For the diffe- rence between the Lepidopterous pupa and the imago emerging from it, as well as between this pupa and the last instar but one in He- mimetabola, only consists in the limited mobility and the temporary suspension of food-supply and excretion in the pupa. In my opi- nion there can be no doubt that it has lost these functions, and that this loss happened gradually. For we are justified in considering the sculptured and movable pupae of primitive Lepidoptera as more original forms than the mummie-pupae, which are hardly mobile. Why then should not absence of colour and of markings be the con- sequence of a gradual regression of these characteristics? Of course this explanation may be as well applied to Neuroptera as to Lepidoptera; DE Meyere himself concedes that the pupae of Neuroptera “mostly live hidden in the earth or in cocoons, and that their chitinous envelope is thin and only poorly coloured”. (The italics are mine). The causes for the regression of existing colour: patterns — viz. ‘darkness and absence of sharpsighted enemies — which obtain all over the animal kingdom — may therefore have exerted their influence on Neuroptera. But this need not involve that the primitive Neuropterous ancestors of recent Lepidoptera already had concealed and immovable pupae. In any case those ancestors had to pass through a long range of thorough transformations, during which especially the youngest larval instars deviated ever more from the original type of the Insect, and in so doing came to differ from the last instar as well as from the last but one. Those two stages on the contrary remained alike in all important points, though they came to differ from each other in minor accessory characters, which for the pupae chiefly consisted in the loss of mobility, with all its consequences. But apart from this immobilisation it retained the old primordial characters without or with only small modifications, and where a change still occurred, this depended more on katabolie phenomena, e.g. partial or total extinction of colour- markings, than on progressive alterations. 62 Therefore I think that we need no more ascribe a secondary character to the pupal stage of Lepidoptera, than we should be inclined to do so to the larval or nymphal instar of Hemi-or Ameta- bola. A grasshopper during the succession of its moults, passes through a series of successive stages of colour-pattern as well as a moth. The idea that the last stage but one of this series bears a different character from the preceding instars or the following ultimate stage, would never occur to us. Neither is this supposition necessary or useful for the understanding of the Lepidopterous design. That the latter is secondarily modified, is beyond doubt, it has been changed in all stages, but precisely in the pupal stage less so than in the preceding larval instar or the succeeding imaginal state, as SCHIERBEEK has shown by comparing the pupal design with that of the caterpillar in its first instar. As to the colour-pattern of the pupa, the same considerations can be applied to it as to so many of its further properties. PouLTON eg. has pointed out, that in the pupae of those butterflies, whose forewings show a denticulated outer margin, the wing sheaths do not stop at that broken line, yet clearly marked out on its surface, but continue for a short bit and then end in an unbroken front line. He rightly takes this feature as an indication, that the ancestors of those butterflies at one time possessed normally rounded wings. In the same way he was able to show, that in those moths whose females have only vestigial wing-rudiments (the wings of the male sex being well developed) the female pupae differ much less from the male ones, because their wing-sheaths are only a little bit shorter than those of the males. Likewise the difference between the sheaths for harbouring the filiform antennae of the females and those for the pectinate ones of the males was found to be smaller than that between these antennae themselves. Would not all these features be caused by a recapitulation of their phylogeny, by the preservation during the subimaginal stage of former conditions which have lost their original meaning. On this topic pr Mryrre makes the following remark: “It is difficult to explain the presence of this line” (viz. Pounton’s mark) “already on the young pupal wing, otherwise than by anticipation of hereditary tendencies. Anyhow a sufficient number of instances can be adduced of cases in which features of different stages are transferred to the pupa in both directions, as well from the imago as from the larva”. . 2. To this same influence of precocious entrance into activity might also be ascribed the fact, that certain 63 markings of the imaginal wing are already visible on the pupa, e.g. the submarginal spots of Vanessidae. Especially when, as van BEMMELEN has pointed out, the imaginal wing-pattern, during the beginning of its ontogenetic development, at first shows reminiscences of older more generalised types, we can understand, that the pattern of the wing-sheaths precisely reproduces these stages, without our being obliged to assume that the imago received its colour-markings from the pupa, and that the latter onee moved about on wings ornamented in the same style”. Referring to these considerations of pr Mryrre | should like to remark, that I do not in the least suppose the imago to have drawn on the pupa for its colour-pattern, as may clearly be seen from the inferences on p. 358 of my paper: On the phylogenetic signifi- cance of the wing-markings of Rhopalocera, (Transact. 24 Entom. Congress, Oxford 1912), in which I point out the facts, that: 1. only the external surface of the wing-sheaths, harbouring the developing primaries, wear colour-markings, in contrast to that of ‚the secondaries hidden beneath it, while of course both pairs of the imaginal wings develop a colour-pattern on both their surfaces ; and 2. that the primordial or vanishing pattern on these imaginal wings is still more primitive and therefore phylogenetically older than the colour-pattern on the pupal sheath, so that there is as little reason to suppose that the latter received its pattern from the young imaginal wing hidden in its interior, as to make the opposite supposition. The transference of imaginal features to younger instars seems probable to me also, as may be seen from the foregoing remarks. When however pe Meyers calls this transference anticipated entrance into activity, he must have in view the activation of latent hereditary factors, and so must admit the presence of those factors in the genetics of the species. They therefore are connected with former periods of phylogenetic development, or in other words: the colour- pattern of the pupal sheaths must once have ornamented the wings of an insect flying about (or at least walking about) with them. Whether this insect was the imago or the subimago, is a question for itself, but in any case pr MeyerE's expression about ‘anticipated activation” includes the inference, that he also considers the pupal colour-markings as a recapitulation of a phylogenetically older stage. Trying to enter into his ideas, I suppose them to have taken the following course: The imaginal instar of Lepidoptera was of old preceded by an uncoloured pupal stage. In the ancestry of the recent butterflies the peculiar habit was acquired, that their pupae no longer lived in 64 hidden “localities, and therefore came in need of protection by mimicking- or by warning-colours. They provided for this need by means of anticipated activation, viz. by transferring the then existing pattern of their forewings to the external surface of the pupal wing- sheaths. This pattern persisted on the pupa, even after the wings of the imago had acquired the new pattern, such as is found on them to day, by the further modification of the old one. Even if this view of the course of phylogenetic development should prove right, which | consider rather improbable, it would not diminish in any way the phylogenetic significance of the pupal pattern, and so there would be no need to consider this pattern as wholly secondary and therefore destitute of all importance for the phylogeny of Lepidoptera. For this it would seem, is what pr Mererr means by his words mentioned in the beginning of this paper: which fully cited run as follows: “When precisely in the free-living pupae of the butterflies we find special colour-markings, I would consider this as a wholly secondary feature, the body having first acquired certain pigment-spots, to which sympathetic markings of the wingsheaths afterwards were added. That the latter show a certain connection with the veinal system, cannot astonish us, when we take into consideration the special importance of the veins as respiratory and circulatory vessels”. Against this view I wish fully to maintain my own, viz. that the colour-markings of the butterfly-pupae — those on the body as well as those on the wing-sheaths — should be considered as an original pattern, the whole-colour of white, yellow, brown or black pupae of most moths resulting from the loss of this primitive design. Regarding in particular the harmony between abdomen and wings, in colour-hues as well as in design, we may remark that such a similarity is a generally occurring feature, not only with pupae but even and in a higher degree with imagines. Without doubt this harmony will often root in a secondary modification of shades and markings, of the abdomen as well as the wings, which we may ascribe to sympathetic correlation, but this need not oblige us to doubt that both patterns result from a primitive one, or to abstain from searching after the vestiges of this primitive pattern on both those regions of the body. What is true for the imagines, is certainly right for the pupae, even in a higher degree; remnants of the original design may be more probably expected on them and be found there in a more complete state, because the imagines are exposed to greater versabi- 65 lity of life-conditions and external influences, even more so than the caterpillars, their habits of moving about and resting, of nourishing and propagating being more varied. Both caterpillars and imagines in these respects surpass the nearly immovable and lethargic pupae. Dr Mryerr’s views on this topic seem to be the cause, that while attaching great importance to the differences between the pattern on the pupal wing-sheaths of nearly related forms, such as Huchloe cardamines, Pieris brassicae, Aporia crataegi, he only pays very slight attention to the facts pointed out by me, viz. the great similarity between the pupal designs in several families of Rhopa- locera e. g. Papilionids, Pierids and Nymphalids, a similarity not only far exceeding the resemblance between the wing-patterns of the imagines that emerge from those pupae, but also rooting in the nearer connections of this pupal pattern with the primordial and ephemeric design, which appears on the developing wings during the course of the pupal life, and only gives place to the conclusive imaginal pattern in the very last days before the emergence of the imago. These vestigial markings on the rudiments of the wings hidden in the pupal sheaths, moreover prove to us that a primordial pattern may easily continue its existence in concealment; therefore such notions as “sympathetic colouration” or “influence of illumination and surroundings’ need not be invoked in order to explain the manifestation of such a pattern. Though the absence of markings may, in all probability, be con- nected with concealed life-habits and with absence of light, it would not do to consider these influences as the direct and unavoidable causes of the deterioration of the pattern. For the pattern is evidently able also to persist hidden under the pupal sheath, though in some forms it is retained much clearer and more complete than in others, without our being able to find an explanation for this difference. Now what holds good for the wings inside the pupal sheaths, will probably also apply to those sheaths themselves. Taking this inference for granted, we might expect, that also in some of those Lepidoptera, whose pupae conceal themselves in hidden spots, the original colour pattern, on the body as well as on the wings, might have been more or less preserved. This turns out to be really the case, as I found when studying the pupae of Chaerocampinae amongst Sphingidae, and of several genera of Geometridae. In contrast with the majority of the genera belonging to these families, whose pupae are black, brown, yellow 5 Proceedings Royal Acad. Amsterdam. Vol. XXI, 66 or white all over, the genera in question show a well marked and regular design of black markings on a light background. Yet the majority of these pupae certainly live under nearly similar circum- stances as those of their relations, i.e. concealed in the earth, in cocoons or between leaves. It is worth remarking that precisely the Chaerocampinae do not hide in the earth for the object of pupation, as many other Sphingidae do, but remain on the surface and there construct a coarse cocoon of small lumps of earth glued together with threads. In the same way many Geometridae do not pupate inside the earth, but above it; their tissue often being so loose, that the pupa may be seen inside. | suppose that this may be the cause of the colour-markings on these pupae persisting, whereas those on their near allies have disappeared by obliteration in consequence of total darkness. Yet the Chaerocampa-pupae in so far undoubtedly show the influence of their concealed habitat, as their markings not only are variable in the highest degree, but also show a marked tendency | to obliteration. In this respect they agree with the primordial design on the imaginal wings inside the pupal sheath, and also with the maculated pattern of those butterfly-pupae, in which the original colour-mosaic is replaced by a sympathetic general hue, e.g. the uniformly green pupae of Pieris napi, on which the identical spots as on P. brassicae, may easily be detected though much smaller and less sharp than on the latter (comp. vAN BEMMELEN, Phylogenetische Bedentung der Puppen-Zeichnung, and SCHIERBEEK : The significance of the setal pattern in caterpillars and its phylogeny). Therefore though the colour-design of the Chaerocampa-pupae shows deep traces of obliteration, it nevertheless is clear, that this design is founded on the same groundplan as that of butterflies. In my just- mentioned paper I have proposed a system of names (comp. fig. 6 on p. 115), according to which seven chief ranges of spots might be distinguished, called by me the dorsal, dorsolateral, epistigmal, stigmal, hypostigmal, ventrolateral and ventral rows of spots. In his essay Dr. SCHIERBEEK has pointed out, thdt the names of W. MOLLER and WersMANN, who use the expressions supra- and infrastigmal, have priority. i These rows of spots may all be met again on the pupae of sundry species of Chaerocampa as well as on those of Dedlephila (e. g. euphorbia and elpenor) in various degrees of clearness and completeness. No less striking than this correspondence in colour-design between 67 Sphingidial and Rhopaloceran pupae, is the connection between the markings on the pupae of the Sphinges and on their caterpillars and imagines respectively. Among the material at my disposal 1 found this similarity most distinetly marked in Deilephila celerio, as far as general completeness goes, though for certain details or on special parts of the body, other related forms sometimes showed the similarity still better and more complete, or in a more original form, as [ hope to point out in a following communication. Though 1 still lacked the occasion to extend my investigations to living caterpillars in their different instars, or to the development of the pupal skin beneath the last larval coat, or the imaginal epidermis inside the pupa, I do not doubt a moment but these transgressive stages will strengthen my conclusions as to the compara- bility of larval, nymphal and imaginal colour-design, viz. that all three are simply moditications of one and the same ground-plan, which manifests itself clearest in the pupa. Groningen, April 1918. Physics. — “Calculation of some special cases, in Einstrin’s theory of gravitation”. By Dr. Gunnar Norpstrém. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of April 26, 1918). As an application of the theorems deduced in two preceding papers for EINSTEINs theory *) of gravitation, we shall now calculate the gravitation field and the stresses for some special stationary systems with spherical symmetry. First the state at a surface of discontinuity will be investigated. $ 1. Introductory formulae. In a field with spherical symmetry a surface of discontinuity necessarily is a sphere. This surface will be considered as the limiting case of a layer of finite depth, and we shall only have to pay attention to such surfaces in which in the limit some component of the material stress-energy-tensor increases above every arbitrary limit so that the line-integral across the layer remains finite. In general at such a surface of discontinuity there evidently works a surface-tension P: 19 P= tin fr. EE Yg—7,—0 Tr) where 7, denotes the inner radius of the layer, and 7, the outer one. The radical component of the stress-tensor $ on the contrary we shall suppose never to pass every arbitrary limit; in other words we assume that: a lim far=0. Leeks Ee Ad ON . @) rg—r,=0 Ti First we shall consider a general surface of discontinuity and only afterwards we shall introduce special assumptions. We start from the first and third formulae (38) I and from (39) 1. (From these three formulae the second formula (38) I may also be derived, but 1) G. Norpstrém, On the mass of a material system according to the gravitation theory of E:stein. These Proceedings XX, 1917, p. 1076 (cited further on as 1) and: On the energy of the gravitation field in Einstein's theory. These Proceedings XX, 1918 p. 1238 (cited further on as Il). 69 we do not need this). The system of coördinates will be fixed by the conditions : yer, VIZ. “OSS UE on (8) Putting further: $=V—gT=uwT, Loa 2 “Eira ee (4) and applying a simple transformation, we can write for the mentioned starting formulae: 1 w' a ie Rid ase (5) u w d 1 a rma De r 1 — at ee 9 OO F, ’ . . . « . Me (6) 9 = w' 4 r dT, — he — fis = a TT fi — pete . . bd r ( i yar w er ) dr gy (7) These formulae hold for each stationary gravitation field with spherical symmetry; the system of coordinates only is determined -by the condition (3). The quantities w and w determine (when p=1) all components g,, of the fundamental tensor according to the formulae (25) I. When 7,* is given, the equation (6) determines w as a function of r. By integration across a layer which afterwards by a passage to the limit is changed into a surface of discontinuity with radius 7, =7r, = R and after division by R we obtain ra 1 1 ? he lim [tu + Io erde en en UG) u, u, rg—17;=0 ry This formula shows that w changes discontinuously at a surface of discontinuity where differs from zero. Such a surface which moreover satisfies the condition (2) will be called a material surface. The system of coördinates might be chosen in such a way that at the surface u changes continuously, but then p would change discontinuously. In general at least one of the space-components of the fundamental tensor changes discontinuously at a material surface. With the aid of formula (5) we shall now prove, that 2 on the contrary changes continuously at our. material surface, when only the condition (2) is satisfied. Equation (5) gives wu? ie 1 (tre) Gr ete nt cen ey (9) 70 and by integration across the layer we obtain rs log w‚* — log w,? ={ |= (1-- r* x 7) == | rela (10) r) We shall only consider gravitation fields in which wis every where | finite and when in the limit we pass to an infinitely thin layer the limiting value of the integral on the right-hand side becomes zero according to the assumption (2). Now we shall apply formula (7) and substitute in it the expression (9) for — and the expression (6) for 7,*. Multiplying further by w urdr ae find — Wp “ih nat l d 1 ik u (T,—T,)dr+-4{u?(1—r?%7,)—u} { — Lee ur ur dT, dr= ze dr u’ This equation must be integrated over a layer and Be, we must pass to the case of an infinitesimal depth. In order to obtain as a first term on the left-hand side the surface tension P as defined by equation (1) we must moreover multiply by w. We shall however not continue our general investigation, but rather consider two more special cases. § 2. Investigation of the state at a material surface. First we investigate the case that at the limit 7,* surpasses any value, so that the right-hand side does not become zero, but that dT; : 8 : ae remains finite, so that on both sides of the surface of discontinuity if T’ has the same value. In (11) we first consider the part of the left-hand side which after integration gives Ta d I T=t [lu d—" AT) dieu ie Ure i 1 =. — [lut (lr 7) - uid! r{ l1——]}. Ax) r° ( u We have to calculate the value of this expression for the limit r‚—r,=0. In this limiting case r constant =7, = 7, = A, so that we have 1 du pi = a ee u’ u' ad We thus obtain u ug r 2 1—R*xT, (° : ad — im I= Jeez n= En sil eG ed La rg-—1,=0 2xR 2xR u' 2xk uu, u uy Now we have treated one part of the left-hand side of (11) by integration and by passage to the limit. Of the remaining parts of this left-hand side those containing 7’ remain zero at the passage to the limit according to our assumption (2), u remaining moreover finite. The part containing 7” on the contrary does not become zero. The right-hand side has the value zero at the limit, as we have assumed 7", to change continuously at the surface of discon- tinuity. Multiplying our equation still by w, which quantity we have proved to change continuously at the surface, so that at the limit it may be considered as constant, we obtain: W Le oR (w,— u) (: = —B xT; ) ries my ed Together with (8) this formula expresses the laws for a surface of discontinuity of the kind we now consider. These formulae will be applied to the special case that all matter that is present is uus situated in the material surface. 7” being continuous, we have in this case 7” — 0. Further we have according to (6) both inside and outside the surface 1 „(1 5) = const nen ete CE u When 7 == 0, w cannot be zero, so that the value of the constant within the surface must be zero. We thus find for r << A, u=1 and therefore also SSPE VINK SOD EE Within the spherical material surface we thus have a euclidic space. (This is of course true for every hollow sphere; the distri- bution of mass and stress on the outside only has spherical symmetry). Outside the material surface the constant in equation (13) has not the value zero, but a value, proportional to the mass of the system which is given by formula (15) 11: Ana See ee ee EEO) x We thus have for w,: US ee ee ee oe For w we have at our surface: [41 88 16 we Dt . Reen er el” ( ) 2 This may be proved e.g. by putting e=0 in formula (12) II which holds outside our surface. Also by putting r== R we obtain the value (16) at the surface, and formula (9) shows afterwards (as within the surface u =—=1 and 7’. =O), that this constant value of w holds also everywhere inside the material surface. Introducing the expressions w,, u,, and w, we find for the surface This formula expresses the relation between the surface-tension, the mass and the radius. Expressed in the usual units, the surface- tension is cP (comp.I p. 1079). The constant of mass « is also con- nected with the right-hand side of equation (8). After introduction of the values of w, and u, this equation gives 3 a2 RR" hm fri est La) he ES rg—1,;—0 7) In the euclidic space inside the material surface we have not the same velocity of light as at an infinite distance from our system, but a smaller velocity a c [pA R We thus have a representation of EINSTEIN’s idea on the influence of distant masses on the velocity of light in our part of the world. Expanding the expression (17) for P in powers of a/R we obtain: p— c ee ey i TRR 4 ps B ps baat ° . . . ( a) NewroN’s theory gives for cP: km? 162 R® where k is the NeEwroniaNn gravitation constant: cP= (178) c°x > ae 73 Introducing in (175) the expressions for k and m, we find for P an expression, corresponding to the first term of (17a). As to the terms of lower order the theory of EisrriN agrees therefore with that of Newron. §. 3. Second example of a surface of discontinuity. Now we shall consider another kind of surface of discontinuity viz. one in which Yq lim [tiaso KEA eee rg-—r;=0 71 but where 7’. changes discontinuously. Such a surface of discon- tinuity we havee. g. when an electric charge is spread over the surface. Formula (8) shows that in the case in question w changes conti- nuously at the surface: (20) Above we showed already by formula (10) that w changes conti- nuously. | _ This time too we must multiply formula (11) by 2, integrate a layer and pass to the limit of an infinitesimal thickness. As in the last part of the left-hand side all quantities remain finite at the limit, this part gives the limiting value zero. As further w and w change continuously, we obtain Rk rv rp P= 5 uw (T,,— T;,), or, introducing the components of the volume-tensor © Belen a 1 — 9 Es 3 Lr). . . ee . 5 . . (21) The meaning of this equation is trivial. It expresses the equilibrium between the surface-tension P at the spherical surface and the normal force perpendicular to that surface, the magnitude of which is 3E, per unit of surface. The gravitation has evidently no influence. When on the surface we have an electric charge e and inside the surface no matter, we find (II, note p. 1240) 3 —0 ene eee ee (22) Now we shall assume that neither outside the surface there is any matter except the electric field, and “we shall calculate the mass 74 of the electric sphere. As was proved in II § 1 we have outside the sphere | uw (rn Be egg NS LT KA VEA hee ee LN 8a r' As inside the sphere and at its surface w= 1, we find from (6) by integration up to an upper limit r > kt 7 1 oe ie xe R r —_—;|J= =— - , ; u’ 5 a eae! Sar 7 82 R C7 Ss R 1 2 3 EM xe xe (25) u’ 8a:Rr 8x7? A comparison with equation (11) I] shows, that we must have: xe a= ——., 8a R and (15a) gives for the mass m e =— 26 EE (26) The charge e being expressed in electro-magnetic units (see IT p. 1202) this expression for m is equal to the electro-static energy divided by c°. Besides the electro-static energy no energy occurs in our system. That outside the electric body no gravitation energy, is present has been proved already in II $ 2. The last result says therefore that neither in the electric surface any gravitation energy is accumulated. § 4. A sphere of an incompressible jflurd. This problem has been treated already by ScHwarzscHiLD, *) but as the formulae (5), (6), and (7) lead us by another way quickly to the same result, it may be allowed to develop these calculations as shortly as possible. That the medium is incompressible means that when at rest / ay SEE BE is a constant characteristic for the medium. The fluid character of the medium demands further that no tangential stresses can occur, so that we have ES Ti Se ee ee ees Flüssigkeit nach der Einsteinschen Theorie Berl. Ber. 1916 p. 424. 75 where the pressure scalar p') meanwhile is a function of the place viz. of 7. The radius A of the sphere and the mass m and @ are related by an equation which is found by integrating (6) from 7 = 0 to r= R. As for r=0 u is not zero, while for r= R it has 1 ‚the value ——__—- (see II equation (11)), we find i= R 1 “OR! 3 and therefore Pi WEE Pe aerent apse eee This shows that @ plays the part of density. Integrated from » =O to an arbitrary upper limit r << A (6) gives further w as a function of r. We obtain: 1 (1 — = )= oe u? 3 IN WE omt jar gt 650) Now w and p have still to be determined as functions of 7. The quantities w and p are connected by equation (7). This gives w' dp (0 HP en (31) w dr so that dw — (9 + p) = — dp. w This must be integrated. The integration constant is determined by the fact that at the spherical surface p — 0 and wel 1 7 it pes — es R? (see II equation (12)). We thus obtain the asked connection between w and p: e+ h=ef41—-2R . er ea Now p will be calculated as a function of 7. Introducing in (5) the expression (30) for w and simplifying the equation we obtain w' x x Belet ende Wk. nav. rel (68) 3 3 ') We need not be afraid that this p will be confused with the quantity p which in § 1 has been put equal to 1. 76 w' : We eliminate — between this equation and (31). In this way WwW N we find 2 dp x rdr (e +9 p) (e +p 3 Hit es 3 =O te ae The integration gives 3 log eed — log ba Biles fes const. OTP 3 The integration constant has to be determined with the aid of the condition that for r= Rk p=0. We therefore find hee ee o+3p_ Fe bar ve | 0 ze Pp EF | EE En hal . . . . . . (35) Thus the pressure-scalar p is determined as a function of r. Eliminating p between this equation and (82) we obtain for was a function of r the expression : Ake lu eme 1s el. ae 2 B 3 In this way we have perfectly determined the gravitation field and the pressure distribution inside our sphere. The formulae we obtained become identical with those of ScnwarzscHiLD when for 7 we substitute B = — sin YX. ne $ 5. On the gravitation field as it may be imagined to etist in the inside of an atom. In the theory of atomic structure of RurHerForD-Bonr we meet with difficulties arising from the assumption that in an atomic nucleus of very small dimensions there exist units of charge which — at least when they are liberated in the form of electrons — have a greater diameter than the atomic nucleus. As now EINSTEIN’s gravi- tation theory states that the space in a gravitation field when expressed in natural units is non-euclidic, the question arises whether this theory leaves the possibility of the assumption that the atomic nucleus fills a greater space with a narrow neck or perhaps a space which crosses itself at a certain point. This question will be investigated here. at We consider again a stationary system with spherical symmetry. In the same way as above we may define the distance 7 from the centre of ‚symmetry by putting p=41 viz. by demanding that the periphery of a circle with its centre at the centre of symmetry is 227, when expressed in natural units. If we do so in the case in. question, the state in the field is not a single-valued but within a certain interval at least a more-valued funetion of 7. It is therefore useful to introduce a new radial space-coordinate of which the quantities in the field are single-valued functions. As such a coordinate the distance s from the centre of symmetry expressed in natural units suggests itself. In order to specialize our discussion we can prescribe a relation between the radius defined by the condition p=1 and s and investigate afterwards whether this is in agreement with a possible distribution of the components 7 of the stress- energy-tensor. s? ni 2) An Tr Mn, Ft (37) Sa’ As a trial we put where a is a constant, and we choose the sign thus that a positive value of r corresponds to a positive value of s. For small values of s r and s are proportional and the three-dimensional space is dilated when we come farther away from the centre (viz. from the point s=0). For s=a r reaches however a maximum and when s increases still further the space is contracted and crosses itself at a point in the neighbourhood of s=V 3a. For still higher values of s the space is again dilated. Before proceeding we still remark that in fact the sign of r does not play a role. Inversing the sign of 7 in our fundamental formulae (5), (6) and (7) and interchanging also the signs of dr and w’ we find from the formulae the same values as above for all remaining quantities. For this reason we take in (37) everywhere the + sign, so that r is taken negative in the intervalOV38a. In the limiting surface s = S we have according to (40) u <1. In order that in that surface w may pass continuously into the value it has in the field on the outside, « must also in the outer field be smaller than 1 for s= S. This follows also from formula (11) I], when the system has only a sufficient great electric charge. Further it does not matter that w would change discontinuously at the boundary, if only this is a material plane as considered in § 2. 4 Formula (41) shows that in the interval Vi8a | H—C— COOH H C—COO Wer dd Hg In the same way a-crotonie acid remained in solution in the form of a complex mercury salt, which could be precipitated with alcohol. From this salt 8-hydroxy-butyric acid was obtained; in consequence one deduces from this too the cis-configuration for ordinary crotonic acid with the higher melting point. In order to corroborate this result we have subjected #so-crotonic acid to the same operation and obtained an insoluble basic mercury salt, which, after decomposition by means of H‚S furnished a mixture of zso- and a-crotonic acid. It may be mentioned in passing, that x-crotonie acid must have been formed during the elimination of mercury, for this acid — so far as it originally was present in the iso-crotonic acid — was kept in solution as a complex salt and because H,S does not, or at least extremely slowly, change free iso-crotonic acid into a-crotonic acid. The coincidence of the conclusions from the dissociation constants and from the researches of BiuLMANN gives some certainty to the configuration of the erotonie acids. It follows that the formula of these complex compounds is probably different from the one that has been proposed by Brumann. However, this method of discernment is valid exclusively for a@-non-saturated acids; other ethylene deri- vatives, among which are the esters of isomeric acids, cannot be 83 distinguished in this way, because they all seem to form complex compounds with basic mercury salts’). From the dissociation constants of angelic and tiglic acid we can at the very best suspect that in the first acid a hydrogen atom is situated on the side of the COOH group, in the other one of the methyl groups. ke The configurations adopted here are supported by the consideration that the most stable acid will be the one in which the relatively positive group is situated as near as possible to the COOH group. H—C—COOH HOOC—C—H CH,— CH ae CH, | CH3;—C—H H—C—CH; : HC COOH H— COOH H— C_coon H—C_COoH CH3- C_coon CH,—C—COoH _ maleic acid fumaric acid iso crotonic acid _v-crotonic acid | angelic acid tiglic acid forms anhydride no anhydride = as | = = m2 < 10-7 93 10-4 3,6 Xx 10-5 205<10=5 "| 505 1 10-5 complex Hg. salt + 0 0 + — — stable stable stable About oleie and elaidie acid there is utter uncertainty, because the dissociation constants are not known; it can only be suspected that in the more stable elaidic acid the relatively positive carbon chain is likely to lie on the side of the carboxy! group. With cyclic cis-trans isomers the importance of cis-trans situated radicles in relation to the ring becomes less as the last widens. (The conception of von Baryer that the angle between the direc- tions of the affinities of trans-situated radicles decreases as the ring widens, is not incorrect; only von Banyer deduces this decrease from sterical considerations and then it cannot be so very important]; this consideration lessens the certainty of our conclusions about the configuration still more. But now here we meet with the very happy circumstance, that the trans-compounds frequently are asymmetrical and therefore can be resolved into optical antipodes. If this argument is annulled, as in the case of the hexahydro- terephtalie acids, which are both symmetrical, or if a resolution into optical antipodes has not been tried, there is no certainty at all. This may be backed by the following table: (See following page). We see that the formation of anhydrides, the most important argument with maleic acid, has all but lost its significance in the: case of the cyclohexane derivatives, as both 1-2-dicarboxylie acids and neither of the 1-4-acids form an anhydride. 1) B. 38 1340, 1641, 2692, (1900); 34 1385, 2906 (1901); 35 2571 (1902); 43 568 (1910). 6* 84 k Anhydride | Resolvable cis-cyclopropane-dicarboxylic acid 1.2|4 10-4 ++ | — trans 4 42124 SOA Ee OOS Nie 8 cis-cyclobutane-dicarboxylic acid 1.2 | 6.6 105 + | leunen trans 3 oe ole eel pat | N gated cis-cyclopentane-dicarboxylic acid 1.2 | 1.58 X 10—5 ote | ) trans 2 ee bees sa alee cis ” » 13/54 >< 10-5) ae | trans 5 » 13/50 10-5) Zi i cis-cyclohexane-dicarboxylic acid 1.2 4.4 X 10-5 | -+ trans ‘3 „ 12 6.2 10-5 a | ai eis 4 nek / not | st | Ònot investi- trans 4 pap bs: \ determined en \ gated ? cis = ok, yl Legh — sym- ? trans ” ne LANA >< 10-5 | en metrical This is the more true of the dissociation constants, whereof the differences in the case of the cyclopentane-dicarboxylic acids are small already, but leastways such that from the acid with the greater constant an anhydride is known. About the eyclohexane-diearboxylie acids in this respect we grope in the dark. The 1-2-acid, which has been resolved into optical antipodes and accordingly is undoubtedly the trans acid, is stronger than the eis acid; both acids easily form an anhydride. If in this case it should have been unknown which acid is resolvable, we should probably have come to a wrong conclusion. With the 1-4-diearboxylie acids, the classical case of eyclic cis- trans isomerism, there is no certainty at all; the one with the highest melting point, which von Barrer has denominated trans, has the highest dissociation constant and therefore one should perhaps call it the cis acid. As it as little forms an anhydride as the isomer and neither can be resolved into optical antipodes, the only remaining argument in favour of the current conception is the greater stability; an argument that should be termed weak, considering the slight solubility and the high melting point. Still the case is not entirely hopeless; after having discussed the 85 chemical methods that can serve to determine the configuration, we will demonstrate that here too there is a way out. It is evident that, if less characteristic radicles are bound to the nucleus, as in the cyclohexanediols 1-4 or the hexahydro-toluilic acids, with which no discernment by resolution into optical antipodes is feasible, it seems impossible to determine the configuration. Here the difference which appears in the formation of complex compounds, for instance of the diols with boric acid, has proved promising; this has been evidenced by the configuration determination of some sugars, but on that point we will not expatiate here. 3. Secondly the configuration may be deduced from what happens if the double bond is saturated; the so formed compounds are diffe- rent as they originate from the cis or from the trans isomer. The configuration may also be inferred from the way of formation, either from saturated compounds by elimination of parts of the molecule, or from acetylene derivatives by partial saturation, or by substitution of groups in compounds, of which the configuration is known. The last mentioned modes of determination, by which the bonds between the atoms are vigorously attacked, have often caused con- fusion, by which their trustworthiness has been impaired. When applied to fumaric and maleic acid, they at first seemed to answer excellently; we can still assert with satisfaction that fumarie acid is changed by KMnQ, into racemic acid and maleic acid into meso- tartaric acid. | Only, the brilliant researches of Wr1sriceNus about the bromination of both acids, followed by elimination of one molecule of HBr, by which fumaric acid furnishes first racemic dibromo-succinic acid and then bromo-maleic acid, and maleic acid first meso (#s0-)dibromo- succinic acid and then bromo-fumaric acid, have turned out to be correct only as far as the final products are concerned. McKenzir') and Bror HormBerG®) namely have demonstrated that zso-dibromo-succinic acid with the lower m.p. can be resolved into optical antipodes and this entirely overthrows the deduction. As well at the addition of bromine to both acids, as at the elimi- nation of HBr, exactly the reverse occurs from what we could expect, and this inversion appears to be rather common. By the action of PCI, on aceto-acetic acid two isomeric g-chloro- 1) Proc. Chem. Soc. 1911, 150. 2) Journ. pr. ch. 84 145 (1911). 86 erotonie acids are formed, one of which is volatile with steam. This one has a dissociation constant = 9,5.10—-°; the other acid has k = 14,4.10—-5. From this it may be concluded with some certainty, that in the first mentioned acid the chlorine atom lies farther from the COOH group than in the other. Now the relatively weaker acid on reduction furnishes the relatively stronger zso-crotonic acid; on the other hand tbe relatively stronger g-chloro-erotonic acid gives rise to the relatively weaker a@-crotonic acid; in both cases an inversion must have occurred and we come to the conclu- sion, as well as in the series maleic acid — zso-dibromo-succinice acid — bromo-fumarie acid, that an inversion has taken place at the attack of the valency, which governs the configuration. By catalytic hydrogenation only of phenylpropiolic acid in the presence of colloidal platinum 80°/, of the theoretically possible amount of allo-cinnamic acid was formed; on the other hand, by the action of zine dust and acetic acid, resp. alcohol, ordinary cinnamic acid was almost exclusively obtained *). As the catalytic hydrogenation of acetylene compounds appeared to warrant some certainty, we applied it to tetrolic acid, which ought to give chiefly a-crotonic acid. However during a microchemical investigation, which was executed some years ago with the collaboration of Miss O. B. van per Weir, a-crotonic acid could not be found among the reduction products of the sodium salt of tetrolie acid. By hydrogenation of the free acid under the influence of palladium- sol, crotonic and 7so-crotonic acid are formed in the proportion of 2: 1. We see, therefore, that no more than with the reduction of phenyl- propiolic acid, this chemical method is capable of giving us sufficient certainty about the configuration. +. To cyclic cis-trans diols also this unsafe mode of determining the configuration is generally not applicable, because the correspondent saturated diols cannot be obtained. The hydro-aromatic glycols form an exception, as they can be obtained from aromatic diphenols, which may be considered as cis diols. Of course this case is not quite to be compared to the hydrogen- ation of acetylene compounds; it is known with rather great certainty that the OH groups of the phenols are situated in the plane of the benzene nucleus; on the other hand it is to be supposed, considering 1) Hotteman and Aronstein B. 22 1181 (1889): LrEBERMANN and Trucusiiss B. 42 4674 (1909); E. Fiscuer, Ann. 386 385 (1912). 87 the number of isomers, that in acetylene derivatives the substituents lie in a line with the carbon atoms of the acetylene skeleton. The researches in this field, viz. the catalytic hydrogenation of diphenols with the aid of nickel, show, that a mirture of the cyclo- hexanediols is formed; a reduction under the influence of platinum or palladium at a low temperature apparently has not been executed yet. Now there are general syntheses of these diols, viz. from the non- saturated cyclic hydrocarbons, either by direct oxidation by KMnO, or via the oxides; we have made use of them to prepare the hydrindene- diols. According to current ideas the cis diol should be formed exclusively by this reaction‘): OH OH 0 me a H Pies br Indene oxide was hydrated as mildly as possible, that is to say at the ordinary temperature in aqueous solution with a very little acetic acid, and still we could isolate a considerable proportion of the trans isomer too. At this hydration likewise a valency of one of the carbon atoms that determine the configuration, is violated and a partial inversion takes place. In judging the cis-trans isomerism in this case, the determination of the acidity is left out for the present, as the methods of investi- gation are not sensitive enough. The forming of an anhydride too cannot yield a good result here, because the bearers of the stereo- isomerism are brought into play, which is not the case with the formation of anhydrides of acids, as of maleic or cumaric acid. Here are only left 1. the resolution into optical antipodes, but this will not be easy and has never been successfully accomplished ; 2. the formation of complex compounds, which has proved effectual with the sugars, as we came to know by it the configurations of a- and B-glucose, of «- and g-fructose and of a- and #-galactose. In the case under consideration too the last method has been to the purpose; the isomer melting at the lower temperature namely, increased the conductivity of borie acid, on the other hand the isomer with the higher melting point diminished it in some degree, 1) Versl. Kon. Akad. v. Wet 26, 1272 (1918). 88 and from this the cis configuration could be deduced for the first mentioned diol. If this method had not come to the rescue, the case would have been almost hopeless, because a resolution into optical antipodes cannot enlighten us: OH OH aid OH For it is evident that both isomers are asymmetrical and therefore can be resolved into optical antipodes. ' It has been our intention to draw attention to the fact, that as soon as valencies are attacked of atoms which determine the stereo- isomerism, the arrangement of the groups runs a risk of being changed. Of course the phenomenon going by the name of WALDEN's inversion ought also to be included here. In many cases the possible isomers are both formed and very often principally the one, which we should not expect. This will not occur only with reactions as have been mentioned, by which stereoisomers are formed; in consequence of the formation of stereoisomeric substances, which can distinetly be discerned, the phenomenon was observed here as well as with the inversion of WarpeN. But it stands to reason tbat it is of a general character and that we may compose the rule: During a chemical reaction, by which atoms are added, eliminated or substi- tuted, there is always a chance that the arrangement is changed of the valencies of the atom or of the atoms, at which the reaction takes place. It deserves further consideration to establish whether the arrangement around adjacent atoms or around remote atoms is disturbed, when such a change occurs with the valencies of some atom. This is not probable fortunately, as it would highly aggravate our task to determine the configuration of compounds, for every relation between optically active substances would fail. Besides the formation of anhydrides of maleic or citraconic acid would be worthless and the differences, that may be observed in the influence of compounds 89 on the conductivity of boric acid, would lose all significance for the determination of the configuration. Therefore, if this improbability be excluded and if we assume that an attack of the valencies somewhere in the molecule leaves unaltered the arrangement around the atoms, which are not immediately concerned, then we shall be able to obtain a solution in some apparently hopeless cases. We have seen that with the two isomeric hexahydro-terephtalic acids a comparison of the dissociation constants does not answer the purpose; neither is resolvable into optical antipodes and besides from neither an anhydride could be obtained. Of the A?-tetrahydro-terephtalic acids (see accompanying diagrams) H COOH COOH COOH COOH H H Hf Fig. 2. the trans acid should be resolvable into optical isomerides and it is there- by to be distinguished from the cis acid, which cannot be resolved. Now it should oe possible to change these acids into the corre- spondent hexahydro-terephtalic acids by catalytic reduction, without altering the arrangement of the carboxyl groups and therefore the configuration of the last mentioned acids may be definitively established. A case bearing an essential relation to this one, is: Benzoquinone furnishes maleic acid by careful oxidation; from this we may conclude with rather great certainty, that this acid has the cis configuration, as this arrangement is contained. in the quinone molecule and because at the elimination of the —CH = CH- group, the bonds that bear the isomerism, are not interfered with; if in the case of maleic and fumaric acid we were as badly equipped as with the hexahydro-terephtalic acids, then this mode of formation would have been of preponderant importance for the determination of the configuration. Chemistry. — “The Addition of Hydrogenbromide to Allylbromide”. By Prof. A. F. Hotieman and B. F. H. J. Marrars. (Communicated in the meeting of May 25, 1918). In the many cases that in my laboratory I had trinrethylene- bromide prepared by the introduction of HBr gas into allyl-bromide, I was struck with the fact that now an almost quantitative yield was obtained, now a much smaller yield, without our being able to indicate the cause of this varying yield. When now my assistant, Mr. pen HorranNper, had obtained almost exclusively trimethylene- bromide in this addition in a very brightly lighted room, whereas a few years ago Mr. Wuitr observed by the side of it considerable quantities of a product that boiled at a lower temperature (propylene bromide) in the ordinary work-room, the supposition suggested itself that daylight exerts an influence on this. Mr. Marrures undertook to inquire more closely into this matter. For this purpose a quantity of allyl-bromide was divided into two equal parts; one part was poured into an ordinary bottle, the other in a bottle that had been perfectly blackened on the outside with lacquer. The liquid in the ordinary bottle was exposed as much as possible to the sunlight during and after the introduction of HBr. Every time that no HBr was absorbed any more, it was closed, and left to itself till the next day. After some days no further HBr was absorbed. The blackened bottle was treated in the same way. The absorption of HBr took place a great deal more slowly here, so that the process had to be continued for some weeks, before complete saturation had been attained. When the contents of the two bottles was afterwards subjected to distillation, the preparation from the ordinary bottle almost entirely went over at constant temperature and at the boiling point of trimethvlene bromide. After distillation in vacuum its boiling point amounted to 167°.1 for 760 mm. The contents of the other bottle, on the other hand, presented a very considerable boiling range, viz. from 100—190°. On fractionated distillation a fraction of about 7 gr., going over between 140°—150°, was obtained, while between 155° and 165° a fraction of 22 gr. went over. The former had about the specific gravity of propylene 91 bromide, viz. 1.9259 at 23°.2; the latter had the spec. gr. of trimethylene bromide, viz. 1.9801 at 23°.2. Between 100° and 105° a few drops had also been distilled, which were still unchanged allyl-bromide, as appeared from this boiling point. Hence the conclusion is that on addition of HBr to allyl-bromide in bright daylight trimethylene bromide is almost exclusively formed; in the dark, beside this compound as chief product, also pretty much propylene bromide. Amsterdam, May 1918. Org. Chem. Lab. of the University. Physics. — “The variability with time of the distributions of Emulsion- particles’. By Prof. L. S. Ornstein. (Communicated by Prof. H. A. LORENTZ). (Communicated in the meeting of March 31, 1917). SMOLUCHOWSKI discussed this problem in different papers and gave a complete survey of his work in three lectures ad Göttingen. *) He deduced a formula for the average change of the number of particles in an element, which at the moment zero contains » particles. This formula is: Ar En) Pow cee ee ae TL) where P is the probability that a particle which lies in the element at the time zero, may have come outside in the moment ¢; whilst p is the number of particles which at a homogeneous distribution over the whole volume would come to lie in the element in con- sideration. Also for the average square with a given number of particles n at the time zero SMOoLUCHOWSKI gives a formula, viz. An =| —»v) nl Pe) ee es (2) from which follows — if the average also is determined according ton —— A? = 2p P. These relations are deduced by Smo.ucnowski with the help of calculations of probability, which ‘nach Ausführung recht kom- plizierter Summationen (yield) merkwürdigerweise das einfache Resultat”’. It goes without saying, that it must be possible to attain such a simple result also by a less complicated method. That this is indeed the case I want to demonstrate in this paper. At the same time it will be possible to give some extension to the result. 1. Let us think the space divided into a great number of equal elements, which we shall mark by the indices 1 ..x..4. Let there be at a given moment {== 0 n, .. m, . . nx particles in I Gf. Phys. Zeitschr. 1916, p. 557 and also Phys. Zeitschrift XVI. 1915. p. 323. 95 these elements. After a time ¢ has passed these numbers have become changed. Let jy, then represent the chance that a particle which at the time ¢=0O is in the element 1, is found at the time ¢ in the element x, and let p‚, represent the probability of the reversed transition. Then, if there is no predilection for any direction in the movement of the particles, it goes without saying that pi, = pa. jk Further = p,, = P if the sum is taken according to all values 2 mil except x =A, for the sum represents the probability that the particle has come after the time ¢ in one of the £—1 other elements, i.e. outside the element x. If an element 4 contains n; particles the number of particles having passed from 4 to x in a given case will be A, I shall now calculate first the average values of As, A*, and A;, A,,. The number of cases where A;, has the value s and thus 2,—s particles have remained in the element, amounts to: J N ———— py; (l—py) 9 . . . « « « (3) as is easily seen; to determine the three average-values this expres- sion must be multiplied by s resp. s? and summed from zero to 7). Then after quite an elementary calculation of these finite sums, we find P= ig A: Me Dye Ay sy tak ye ee and A», == Piz Nd), 5 0 . 5 . s 5 (4) To determine the average of a double product we need only replace (5) A by w and s by ¢ (where ¢ represent the number of emitted particles in a definite case). If the result obtained in this way is multiplied by (8) and summed with respect to » from O to n, and with respect to ¢ from 0 to7,, we find TAGE EX ac Pix Olt ve a Pee ee With the help of the relations (4), (5) and (6) SmoLucHowskt’s formulae can now immediately be deduced. The change ,4,, i.e. the total change of the number of particles in the element * may be represented by AN p=, Me Aare Lee ae es ie tas oT) Now we can write A, for Ar +... Ap, ie. the total number of particles that leaves the element in the time ¢. Then we must determine the average of (7) witb constant n,, 94 while all possible values must be given to the number n,...n, in the other elements. If now we first take the n,...m, constant and determine the average, we find nds = pie LD If then we proceed to determine the average according to n,...n, and keep in mind that n, .. =n, =v, we find JA =P (pix Teer Pie) — Pp In order to find „A, we proceed in quite an analogous way, we bring (7) into the square. Then we find Li? sae rei ee SDL ads led Are oe Ap) If now we apply (5) and (6) and determine the average with given n,...”, and n,, we find „As = (mi? — m1) Pax’ + pix +... +P? (nt — n,) + 2, P + 21 ne pir Par +... —2nP(piz m1 + -- Pre Nk) - Here the average must be determined keeping constant 7, with respect ton, ete. And we must bear in mind that n,? =7,?=...n,?= vr? Hv), that further n, =v and n,n, =v’. Consequently we find At eee + 2 vp? (pix pox +... ) — Pp (pix” - Fy J ) —2nvP? + P(r’? —n)t+nP. The three first terms together yield /* »*. The result becomes thus - pA? „=| (n—v)? P?—-n? P?} + (n + v) Py from which by determining the average eis to » the relation A?=2vrP arises. 2. The extension of the given formulae may be obtained to the case that the deviation of density in the various elements of volume are not independent, where however concerning the emission of the particles we must still presuppose independence of the events. In order to introduce the correlation of the densities I make use of the function g, which was defined by Dr. ZERNIKE and myself. *) 1) We have m =v+3, m?=v+2yd+382= 12+, nn =H) = v8 ty EH) + dp = *) Chance deviations in density in the critical point of a simple matter. These Proc. XVII, 1914. p. 582. 95 If d, is the deviation in density in a point e=0, y=0, z=0, then we get for the deviation of density d in a point z, y, z: d= 9 (a, y, Dirt beent A (8) where dv is the element of volume. Further dd, if ey yy) Ò du TG REU ee ae eee oe BD) where gp is the number of particles per unit of volume. We now have nr Piz Hompe 0 P. Now n,=v Hd, dv, if then we introduce (8) and consider 3 ‘ f Vhs pix as function of ryz, bearing in mind that d, = = we find av i = (vp—n){ P+ | gx par do} The influence of the second part may become ee with a strong correlation. Also in determining ,A?’, the correlation can be taken. into con- sideration. Then in the first place | we get the old terms, but moreover (9) yields still new terms in 7, 2, and 7; nv, Ny Nu. ny. These terms are: 2 v (v—n) [ow Gx dv ee wn | pe Yi» dv — 2n P(v—n) fe» dv + 2p i Pix Pox Jip AY) d vy. If then A?, is determined, only the last term remains and a part of the term before last, so that we get A?=2 (P+ | Pix Pux Pin AVY, AY; ze ef Px Ox dv). These considerations may also be applied, as least approximately, to the changes, which accidental derivations in density undergo in result of diffusion. Our formulae show then that close to a critical point the deviations in density as a result of their correlation, are not only stronger on the average, but also more strongly changeable. Utrecht, March 1917. Institute for Theoretical Physics. Physics. - “On the Brownian Motion”. By Prof. L. S. ORNSTEIN. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of December 29, 1917.) Von Smonucnowski') observed that the function which gives the probability, that in the Brownian Motion a particle accomplishes a definite way in a given time is a solution of the equation of diffusion. For cases in which an exterior force also acts on the particles, he deduced a differential equation for the above- mentioned function of probability by a phenomenological method. Some time after Mr. H. C. Buraur *) deduced this differential equation following a method, which takes the essence of the function of probability more into consideration. Both deductions do not stand in direct connection with the mechanism of the Brownian motion; my object in this paper is to demonstrate, that starting from a relation which Mrs. pr Haas—Lorentz*) has used in her dissertation, to determine the average square of the distance accomplished, one is able to determine the function of probability of the Brownian motion. It is worth observing that the way in which different averages depend on the time may be calculated from the results obtained by Mrs. pr Haas— Lorentz by a slightly more careful transition of the limit than was necessary for the object she had put herself (viz. the determination of the stationary condition). First I want to determine these averages by a new method, which will offer the opportunity of demonstrating, that the opinion, from which Dr. A. SNETHLAGE *) starts in the theory of the Brownian motion that Einstnin’s theory is in conflict with statistical mechanics, is incorrect. Besides the function of probability for the distance I shall also deduce that for the velocity. The chain of thoughts which lead to 1) Compare e.g. M. v. Smorvenowskr. Drei Vorträge Uber Diffusion, Brown’sche Bewegung etc. Phys. Zeitschr. XVIL p. 557 1916. 2) H. C. Bureer, Over de theorie der BRown’sche beweging. Verslagen Kon. Ak. XXV p. 1482, 1917. 3) Mrs. Dr. G. L. pr Haas—Lorenrz. Over de theorie der Brown’sche be- weging, Diss. Leiden 1912. 4) Miss Dr. A. SNETHLAGR, Moleculair-kinetische verschijnselen in gassen etc. Diss. Amst. 1917. 97 the results given below shows great similarity to the deductions which Lord RayLuieH') gave utterance to already years ago. Kindred ways of regarding the stationary condition are also found in the work of Dr. Foxkrr’*) and M. Pianck’). $ 1. In the dissertation Mrs. pe Haas-—Lorenrz starts from the equation of motion for a emulsion particle, which she brings in the formula MW me... ae ee KE) Here u is the velocity of the particle, w=6 zgua the resistance which according to Srokes’ formula the spherical particle (radius a) would experience in a liquid with internal coefficient of friction pe. The force expended by the shoeks of the molecules is divided into two parts, of which one is that according to Srokrs, the second is quite irregular, so that #’=0. The determination of the average is to be understood in this way that it is to be taken at a given moment for particles which all have had the same velocity u, a time before. Now we are able to integrate the equation (1), if we introduce w — = 8, we have m t meet fer od. zere: ej | 0 where uw, is the velocity at the time ¢= 0. If then we determine the average of this equation in the way indicated, the result is Meerbeke uil oor ereen ml ERD or expressed in words: when we start from a great number of particles of given velocity, the average velocity decreases in the same way as witb large spheres; the damping coefficient also is deduced in the same way from radius and coefficient of friction of the fluid. Let us now calculate also the average of the square of the velocity. For this we find: 1) Lord RayreranH, Phil. Mag. XXXII, p. 424. 1894. Papers III. Dynamical problems in illustration of the theory of gases. 2) Dr. A. Fokker, Over de Brown’sche beweging in het stralingsveld, Diss. Leiden, pg. 523, 1913. 8) M. PranckK, Ueber einen Satz der Statistischen Dynamik u.s.w. Berl, Ber. p. 324. 1917. 7 Proceedings Royal Acad. Amsterdam. Vol. XXL 98 (4) t é ae 4,7 e2% + @—2Bt | fe F(t) dt 0 In order to determine the integral in the second member we proceed in the following way. We write for it fg [fF (8) F (1) BEN) dy dE. 0 0 Now F(8) F(y) is only differing from zero if and § differ very slightly, i.e. there is for short periods a correlation of the forces #. If we introduce 7)= § + y, it is allowed to replace 7 in the exponent by § and to split up the integral into a product of integrals according to & and wp where we may integrate from — oo to + oo. If then we assume + J FORE wart. rs which is a constant characteristic of the problem and if we perform the integration towards &, then (4) is transformed after substitution into a ae (leet) 9 uu, e—28t + —— Fane eee oe) a EN 28 When applying this equation for t= o, u’ = — and thus we get m kle = —2 m In the same way we are able to determine the average square of the distance accomplished. From (2) or by direct integration from (1) we get namely 2 (6) — B? s? == (u—u,)? + | fra 0 For the last integral we find in a quite analogous way ot If we calculate the first average with the help of (3) and (5) we obtain 99 grs — u (1 —2e—F*! 4+ e—2Al) 4 oe 3 — el + 4e—At) + It. (7) Consequently for very long periods we find Fe EE ae ee (8) B 3 ap which is the well-known formula for the average distance in the Brownian motion. If we determine the average of (7) with reference A . . ee chk en = en 1 to all possible initial velocities and if we consider that u,’ = aa? f we find for the average square of the distance accomplished as an arbitrary initial velocity : — f= eer a oS dbf) Ne eh eee As long as pt is large in relation to 1—e-# the formula of Einstein is thus the right one. For the cases, considered in experi- ments, the lowest limit for ¢ to be obtained in this way is of the order of 0.01 second. § 2. On the basis of statistical mechanics objections have been raised by Prof. J. D. v. p. Waars Jr. and Miss A. SNETHIAGE ') to the application of the division which has been applied to this case upon the example of Einstein and Hopr in their treatment of another problem. Starting from the supposition than in an ‘ensemble’. Ku=0 where A is the active force, they work out another fundamental formula viz. (with a slight variation in notation) BN 9 aT aoe Pe ee = ie 0) where w has to been taken zero. We can again integrate this equation and obtain then t i biede oh werpt binet fe Deine on ek KAAN Q 0. 0 If taking the average we get: == 7] u= U, cosg@t + — sin et Q The average velocity would in this way possess a definite period. If however we work out wu? we arrive at an incompatibility. 1) Cf. Versl. Kon. Ak. v. Wet. XXIV. 1916. p. 1272. 100 Because for wu? we get 7 fo (§) sin o(t—E) dS} . (11) 0 — u° 2 1 zn cos @ t + “sin et } “a Ee ° For the integral, if again we make a double integral of it and if we introduce the constant 6 +e | lo, = fe (S)w(E + w) dip . 2. ee we can write t ia ER ol a5 —— ae! + periodical terms ir C g 5 = 30 periodic rms. 0 Thus we find — Ot u = goe 1 Periodical terms . . | 3 ee This formula shows that u? increases indefinitely with the time, while it is evident according to statistical mechanics that uv? must el approach —. m Consequently if the equation (9) is treated as a differential equation we arrive at results which are not right’). §3. Miss Dr. SNETHLAGE and Prof. Dr. J. v. p. WAALs JR. have observed, that the theory of the Brownian motion must be in accordance with a general theorem of statistical mechanics. For the case that we consider a particle, that under the influence of the impacts of the molecules of the liquid executes a movement, the force which the molecules exercise does not depend upon the velocity, but only upon the coordinates. Consequently the product of force and velocity must on the average be zero, as well in a canonical as in a microcanonical as in a time ensemble. Now they are of opinion that Ersrxmn’s formula comes into conflict with this. I shall demonstrate that this is only the case to a certain extent. If we assume in a canonical (or micro-canonical ensemble) all systems selected in which the velocity of our particle at a point of time O is equivalent to 1, and if then we follow this group of particles, 1) An analogous question is treated by M. PranrcK (Ann. der Phys. 1912. Bd. 37 p. 462) where it is demonstrated that the energy of a resonator subjected to the irregular field of black radiation increases in proportion to the time; the f of PLANCK agrees here with v. p. Waats’ U. 101 we can work out an average of every arbitrary quantity for the group of systems which after a time ¢ has developed from the group considered at first. The value of the quantity considered varies for the different systems of our group (part ensemble), because the systems where at ¢=O the velocity of the particle is «,, may still show considerable differences, so that e.g. the impulses which the particle gets will be widely different. I shall call this average the case-average with a given initial velocity. Moreover the velocity u, may be varied and again the case-average may be worked out and then by making wu, run through all possible values all systems are taken into con- sideration in determining the average at the time ¢ If now the case-average of a quantity g(w) for wu, is g(u,) and if the number of systems of the group is N (u) then — if N represents the total number of systems in the ensemble — the quantity = N (u) g (u) N is the case-average for the entire ensemble. However, as the ensemble is stationary the case average for a quantity is equal to the average of the corresponding quantity in the ensemble. If in particular g(w) for every w is equal to zero, the average in the ensemble is also zero. I shall now demonstrate that if we start from EiNstriN's formula the case average of Ku for every initial velocity w, is zero, and from this it follows imme- diately that Einstein's formula does not come into conflict with the theory of the ensembles, more particularly that Aue=0, (—e means determining the average of an ensemble, which — is used every- where here for the case average). EINSTEINs equation comes into conflict with the theory of the ensembles if we select at t=O a group of particles with a given velocity u, from the ensemble. For if we determine the average of the equation it yields m AT K = — wu, whilst according to the ; t theory of ensembles K is independent of the velocity. If however we leave the selected group to itself and if we apply to its motion Einstrin’s equation which is not right in the first moment, it is evident that in the long run the group moves in such a way that in the long run Einstein's equation can be applied to it. Moreover from the group with particles with a given velocity u, those systems can be selected to which (1) applies. From what follows it becomes apparent that for this group in the long run the usual relations 102 with regard to the averages in a canonic ensemble become the right ones. ary du du Instead of Ku=0O we can also write m—u=0O or u— = 0. dt at If now we multiply the EinsrriN-Hoer equation by uw and work out the case average, we get , 2 =, — 8 u + u F dt Further we shall demonstrate that for sufficiently long periods the second member is identically zero. 9 9 For according to (5) we have got for the first term — = 8 or — = We shall now determine the second average. For this purpose we multiply (2) by F' and work out the average, and arrive at t uF =u, e—Pt F Hert r fen F (t) dt 0 In this formula the first average in the second member is zero. To determine the second average we must consider that /’ for the integral refers to a definite time ¢ And so only those parts of the integral where the argument differs only slightly from ¢ yield distributions to the average. In the exponent we can again take the argument equivalent to t‚ so that we can write for the second term t 0 [FOF Ena or fFOFE Da mrt) — 00 Now this integral is just half of the integral of (4a), as it covers half the region of integration, whilst the integrand is of course symmetrically with regard to &. And so in the end we find for the second member identically zero as the two averages first neutralize each other. If now we take (5) into consideration for shorter periods the second member becomes wore it — Dan t This result is also obtained by differentiating (4) with respect to the time. du ate The case average of U for finite times not large with reference re : to 5 is consequently not strictly zero. Now we can however determine 108 the average according to the initial velocity «, and so find the ensemble du average of u FTE Then we obtain C du ° — 0 À » bo Ie Ee ig A) dt 5 28 é . . . r . . possesses the equipartition value. Thus it is proved that also for short periods the average value of the case average does not come into conflict with statistical mechanics. aS UU," 4. I shall now deduce the law of frequention for distribution of velocity. If we integrate the equation (1) for a short time 7, we can write for it u—u,——Bfur+e or w=u,(l—fpr+ea. . (14) Ki where « =" F(t) dt and x? = Or. 0 Now there is for a law of frequention g(x), so that Sri +n +m ly (a) tara We on, fev (@) da = 0: and fr ple) dais Bt! 1615) If now a particle starts with a given velocity w,, the number of particles, for which in the time ¢ the velocity lies between w and u du, may be represented by f (u, wt) du or shorter f (u, t) du Let us now consider the distribution of velocity at the time ¢-+ rt and again fit our attention on the particles whose velocity lies between w and w+ du. These particles have had at ¢ a velocity u’ in such a way that u (1 — pr) =u —e or wa wl(l + Bela Fy a ee tas) eee (0) whilst an interval du’ = (1 + gr)du corresponds to the interval du). The number of particles that is at ¢ in dw and at ¢-+ 7 in du consequently amounts to f (ul, t) p (a) da du’ and thus we get 104 / fet + 1)du =(1 + Pr) u f RDE ee es ET) If now we work this out and retain the terms up to the first order in t and if we take (15) into consideration and if after division by t, we make t approach zero, we obtain df (u, t 0 ; odf it oer To the function of frequency for the velocity theextended equation of diffusion is thus applicable, where “ plays the part of coefficient of diffusion. The equation is quite of the same form as that for the Brownian motion under the influence of a quasi-elastic force (— u or —s) (cf. also § 4). If we apply (18) to determine the stationary condition we have (18) 7 0 9 077 =F a) = 3 Om from which follows er RN Pee PU ceo OE dl du. This last term becomes infinite for w — oo, consequently the inte- gration constant must be taken c, = 0. For the law of distribution we thus find the Maxwerr division of velocity quite independent of the initial condition. Moreover Ray.eicH has carefully investigated this question for his particular example. He has deduced a similar equation for a particle in a Ù highly rarefied gas, where only the constants 8 and 7 have another meaning (ef. loc. eit). It goes without saying that if one starts from the equation of v. D. Waars-SNETHLAGE, one arrives at the conclusion that the division after long periods is not that of MaxweLr, and that there does not even exist a stationary division of velocity. And on this point also these investigators thus come into conflict with the statistical mechanics of Gress, which is the starting-point of their reasonings. It may further be observed that for a particle beginning with a velocity zero, as long as w is still small with respect to the velocity of the particles, which collide against it, we get as Rarurran has demonstrated 3 of nay dt ADE 105 For the change in velocity we get then at each impact according to RayYLEIeHn ú tE an, where q is the relation of the masses of partieles and molecules, v the velocity of the molecules. Now the problem treated by Rayreicn in this way may be connected directly with the theory of the function of probability for the way in the Brownian motioh. If we take the velocity marked as vector, the terminal point is removed + qv after every shock. The terminal point of the vector consequently executes a Brownian motion at least according to the scheme which is often given of it (cf. e.g. Mrs. pr Haas—Lorerntz’ dissertation). It is certainly remarkable how Lord Rayueren had already so long ago deduced these results, which came to the foreground only by SmoLvcnowskI’s work, which opened so many new views. It may have its advantages now it has become apparent that Ernstein’s formula is the right one to say something further on the kinetic mechanism. Let us first direct our attention to a single shock of a particle of a great mass with a particle of-a small one. If the velocity for the first is before the shock w', after the impact u, the velocity of the small particle v and the relation of the masses g, where we have 181 then we get for every impact: uu (l—q) + q». If we assume then that again and again after a time ta collision takes place, then we have oom | Grane T T T for every impact. We can only make a differential equation of this equation of differences by taking t infinitely small, if q is of the same order infinitely small and then we get dE ni B de ve where # may be written for pv. Thus we see here by a (not very strict approach to the limit) Ernsrein’s equation arise as it were. If now we do not go to the limit, but avail ourselves of the following graphic representation, its meaning becomes even more clearly visible. On one axis we measure out the time (and to make things easier we take again equal intervals between the impacts), on the other the velocity. Between two collisions the velocity is then constant, at an impact the velocity suddenly jumps to another value and this jump consists in every case in two parts; one part pro- portional to the velocity of the particle before the shock with which 106 the velocity decreases and one part which may be either positive or negative (and in general may posses all sorts of values dependent upon the conditions of the impacts, which in the simple case investigated by Raytuien is + qv). The velocity-time curve is thus a discontinuous curve. If the velocity has become large it has the tendency to become smaller by shocks owing to the first part, whilst the second part exercises no systematic influence in a contrary sense. If now we imagine a combination of curves drawn starting from a given velocity, E:srntn’s equation will represent for each of these dis- continuous curves the differential equation. At the same time if we introduce the curve w= u,e—* into the scheme, this line will at all times be an average of the discontinuous velocity time-curves in the diagram. § 4. Finally I will deduce the function of probability for the Brownian motion under the influence of an external force. We take this force km, where & depends upon the place (s). The equation of motion for our particle is then the following du a ie ee ea: Ps) oil ae) GR dt If now a particle has in the time {—0 a velocity w’, if in a time t—r the velocity has become « and a way s' is accomplished, and if w and s represent these magnitudes in a time ¢, we get t t u—u' = — B(s—s') + {Pa + at 5 or We now consider the time so small that the way accomplished in that time is small enough to treat K in the last integral which depends upon s as a constant. We have thus t u—u' = — B(s—s’') + fra Cem nn ok, CU T= 15 Now we want As —=s—s’ and As’. In order to determine these we apply (3), this yields u—wu' =u, erft (Let) as we have to take the mean value of As for all possible values of u, the average of us being zero we get B Asterales oF til) 107 and in the same way | Aat zenne fens 22) In order to arrive then at the differential equation for the function of frequency we reason again in the same way as before. Let FJ (s,5t) represent the chance that a particle that at the time O has the coordinate s, possesses at the time t the coördinate s (with margin ds) where we determine the average according to the initial velocity. Now we follow the movement for a short time rt and build the function of frequention at the time {+ 1 from that on time s. If As again’ represents the mean deviation during the time 7, and p(As) the function of frequency, we know for this deviation that we have ac dsl, fe g (As) dAs = kr and (Ls y (Qs) dAs=drt (224) We then obtain JE sot Hv) ds =| gs alan eg) yt (ie) d An! eS (28) where s’ = 5s — As. If now we take (20) into consideration we find for the connection of ds’ and ds Developing according to (23) up to the first order with respect to t, we find òf 9077 —— 24 ae Tet ey oe If we introduce the value = 9 and 8 we obtain òf m m 09° 3 ehh Ee ied . (24 Òt 6x ua ds € D4 6 pa 20s , Ge) This equation agrees with that of Smorvcnowski, if we take D ryy (coefficient of diffusion of the Brownian motion ). The factor mua 5 is the w of SmoLucHowsk! i.e. the factor with which the force Jr LA mk must be multiplied in order to calculate the velocity which in a stationary condition was caused by this force. By DeByr and his pupil Dr. Tummers') a differential equation for 1) Degye, Zur Theorie der anomalen Dispersion. Verh. Deutsch Phys. Ges. X, p. 790. J. Tummers, Over electrische dubbelbreking, Diss. Utr. 1914. 108 the function of the frequention of the axes has been deduced for the case of molecules (particles) which turn in a liquid, which is acted upon by an external couple and by a couple resulting from the molecular impacts. The deduction of the results obtained there follows immediately from our formulae. We need only split up the couple exerted by the molecular movement into two parts, the one — ga (« be the angular velocity of the particle) o = 8 a ua? (a radius of the particle) and into a second part of which the average value (case average) is zero. For the motion of the axis we get then Pe=—oa Xx where P is the moment of inertia of the particle. : O xv If we take Gin, are = fF, we get for w equation (1), from O which appears that the function of frequention can be deduced from a differential equation of the form (24a). Finally it may be observed that it offers no difficulties to extend our considerations to the Brownian motion of coordinates in systems with an arbitrary number of degrees of freedom. "Wirech, Dee. 8 1917. Institute for Theoretical Physics. Physics. — “The Theory of the Brownian Motion and Statistical Mechanics”. By Prof. L. S. Ornstein and Dr. F. Zernike. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of January 26, 1918). Prof. J. D. v. p. Waars Jr. and Miss. Dr, A. SNeraracÊe have raised objections derived from statistical mechanics against the usual deductions from EiNsTEINs formula of the Brownian motion. These objections may be formulated as follows: Firstly: It is not right to introduce a resistance on an emulsion particle, which is proportional to the velocity of that particle, as according to a well-known result of statistical mechanics velocities and forces are independent of each other, as is appearent from WG =) MENEREN oen fl) Still more clearly this independence is visible, if one considers that the above equation is not only applicable to the average over a canonical ensemble, but even for any group of systems from that ensemble for which the particle considered has a definite velocity », so that for such a group K=0. | Secondly: It is not right to apply to this force of resistance the _ formula of Sroknrs, as it supposes that the liquid around the particle has a motion dependent upon the velocity of the particle. This comes into conflict with statistical mechanics, for these teach, that AAE 7 where e.g. for v the velocity of the particle, for v, that of a molecule (both eg. in the a-direction) in its neighbourhood may be taken. And so Miss SNrrnraarK bas assumed for the calculation of the persistence of a particle in_the Brownian motion, that the surrounding molecules have the usual Maxwellian distribution of velocity. The authors mentioned have tried to give a theory of the Brownian motion which escapes these objections, by starting from (1). In what follows we want to show, that the equations (1) and (2) are much less far-reaching than it seems so that the objections to the usual theory may be considered to have fallen away, and on the other hand the reasoning given is proved not to be the right one. In order to deduce tbe differential equation, which she wants to 110 put in the place of tne equation of LANGEVIN-EINSTEIN, Miss SNETHLAGE differentiates equation (1) according to ¢. This yields on Fe : U ap. == = Ah — We . . . . . ° 5 ( ) From whieh she rightly concludes that = is not independent of v. The nature of this dependence will be known, if for each value of » v dk one knows the average ae Fv). Then there may be written for C one system: Lae 4 pi == (v) +. Ww . . . . . . . . ( ) where w is an accidental quantity which on the average is zero v (w= 0). If we want to determine /’(v) it is apparently necessary to consider the group of systems with a detinite value of ». We shall further on indicate an average in such a “v-group”, the same as above by —v, i.e. the average over the systems where in one definite moment v has a prescribed value, whilst the symbol — will indicate the average for the whole ensemble. As is proved by the following calculation equation (3) in not applicable, as (4) and (2), to every v-group in particular; and this Miss SNernracr has left out of consideration. Let K represent the force acting in the «x-direction on the particle, v the velocity of this particle in that direction. Equation (1) is then found for the canonical ensemble, when A does not depend upon the velocities, and is exclusively a function of the coordinates HEART Let g, be the x coordinate of the particle, so that q, =v. Then we have dK OK OK. Ok zg eran and for the average at definite v dk’ OK” IR Br mn Rr gq, + etc. In order to reduce the last term we have made use of the well- known independence of the extension in velocity and configuration. - Dv . These terms fall out because g, = O. The same independence has as its result that 111 aK’ OK òg, ed 09, The last average is easily calculated from Gisps’ formulae. ye DK = e i) dq, ...4Qn dain Ga tiles | fe 0 dq... dgn Integrating by parts the denominater yields di as the integrated part falls out (e= o at the limits). Now de OK 1 == evan tierelore: —- == —- — kt 09, 09, 2) Considering that @ = Mv’, we obtain ae” i — <= -— ne RE (2 | dt M v? And so #'(v) has been found; equation (4) becomes dK K? (5a) — = — ——viw...... . oa dt M v? i.e. the very form given to this equation without further proof by v. D. Waars and Miss Snrruiace. (Miss SNETHLAGE equation 24, sees however the note of these Proc. 24, 1278 where a calculation remotely analogous to ours is found, without however our conclusions being drawn from it.) dy” dK” iy edad K° Phe tuetinde = <0 and — == — KS or —_ = —- —— = dt dt Mv’? dt? M* v has great inportance for the theory of canonical ensembles. If at a given moment one chooses a group of systems in which the suspended particle has a definite velocity-component v, then the formulae found are applicable to this group. Now one ought to consider, that, if one follows these systems in the time, the velocity of the particles does not remain the same for all of them, but that different velocities are going to arise. Moreover our formulae indicate that, if we take the average velocity a very short time t after the selection of the group, it has become smaller than ». By 112 substitution of the above results in a series of TarLor we find namely: a) Kk? e(t —— 2 M? vy? Now it is remarkable that, if we follow the systems back into the time, i.e. determine the average for a moment — rt for a group where at ¢—O the velocity of the suspended particle is v, exactly the same formula can be applied. So that we get a reversible process and questions analogous to the problem of the tops of H curves solved by Eurenrust arise. Our reasonings consequently also give in principle how the objections may be put aside, which ZerMELo has raised to the statistical mechanics of GiBBs (as well as to the molecular theories of BorrzMaNN concerning the H theorem '), The result obtained may shortly be formulated in this way: the properties of a group of systems, chosen so that in all of them the suspension-particle has a velocity v — a v-group —, are dependent on the time elapsed since the selection. aS We may also ask now after the change of vA with the time for the v-group selected at the moment ¢=0O. From the preceding calculation results that LE NT ) dt v — dt +- Vv 7 re it == RE e . e e ( ) from which it follows that the relation which is the right one for the moment in which the group was selected in the ensemble, is not right when this group is followed further. It is true that the average for the last member of (6) for the ensemble is equal to zero, as is necessary with regard to the stationary character of the whole ensemble, which was already used in the deduction of (3). Consequently we should be very careful in interchanging differen- tiation and determination of the average. So equation (5) will only be right for the first moment (just as (4)) and consequently also h One ought to bear well in mind that the series-development given here is only right for a short time after the selection of the v-group from the ensemble. - If one follows the group during a long time then the systems of which it consists will have spread themselves over the whole phase-extension with the density that belongs to a canonical ensemble. For the importance of the E1nsrein-Laneevin formula for this process compare the paper of one of us (ORNSTEIN), (preceding paper). 113 (Sa), which accordingly must not be looked upon as a differential equation, Or: if we do consider (5a) right for later moments, we do not get in connection with it w—0, as it is made use of by Miss SNETHLAGE. We are able to refute the second objection, viz. that statistical mechanics should not allow that the fluid moves with the particle in an analogous way. For this purpose we shall calculate the derivatives of vv =0 where v' is the z-velocity of an arbitrary molecule situated in the neighbourhood of the particle. We have Bet ae En Te Tj — as v’ as well as XK’ are zero. Further d? “dv dv dv' dv! Pwe KK' v dK' RE et Da a ht, En Bag tap a get Pe The average of the first term yields MM'' M' de" v yak ÒK, ee en KK SS Sa ee Ba et ae AS a eee M | òg ae Moy MO UM and of the third term mj RAK OEL | v? OK" v? KK! v? KK ia Utica! Zn OS FF SS SE SS M' | òg dg M' 0q M' © >? MM’ so that a eerie a: i v? 8 Ba aiid! mer . e . e . e (5) For the change of v' with the time we have according to the preceding d*y' KK' dit MM'v' Now K is the sum of the forces in the «z-direction, which all other particles exercise on the first, A’ the corresponding sum for the second particle. | If we develop the product of these sums, we shall obtain the average of the product of action and reaction that may be assumed 8 Proceedings Royal Acad. Amsterdam. Vol. XXI. tid to preponderate, so that we may expect that KK’ is negative. The second derivative of v’ consequently has the same sign as 7, i.e. the movement of the surrounding matter with the particle, which does not exist at the moment of selection, arises after a short time. According to (8) there is on the average no question of a movement of the surrounding particles with the Brownian particle in an ensemble, as may be expected. When we want to investigate statistically the qualities of a stationary system of molecules, we can make use of a statistically stationary ensemble and identify the qualities observed with the qualities of the most frequent system in this ensemble or with the corresponding averages. The system that we consider at an investigation of the Brownian motion — a liquid with a particle suspended in it — is, it is true, not stationary, but all the same it changes only slowly: the moving particle changes its velocity only slightly by a great number ot impacts. Consequently, in order to make use of ensembles for the study of the Brownian motion we must start from a “quasistationary”’ ensemble and deduce the qualities of the real Brownian motion from the properties of the most usual system in such an ensemble. The calculations given show, that the groups chosen with definite » from a canonical ensemble do not form such quasi-stationary ensembles. However it seems probable to us that such a group, when we follow it a short time, will get to fulfill the requirements, though it will be difficult to show this by direct calculation. The v-group, which has become quasi-stationary at a later moment, would then correspond to the ensemble selected from the canonical ensemble, by selecting those systems in which the particle has already got the velocity v during a short time. An indication with regard to the length of time required was found along another method by one of us in a former paper ’). The above quoted statistic-mechanical objections to the application of the law of Sroxes thus probably have no justification for a real system, but only for the first moment of a v-group, ie. at the very time when it cannot yet be made use of to represent the properties of a real system. Groningen. Utrecht, Institute for Theoretical Physics. 1) L. S. ORNSTEIN. l.c. — This time must namely be of the order of the time during which there is a correlation between the irregular impulses, i. e. F(©)F(s + 1) differs from zero. Physics. — “The Scattering of Light by Irregular Refraction in the Sun”. By Prof. L. S. Ornstein and Dr. F. Zernike. (Communicated by Prof. W. H. Junius). (Communicated in the meeting of April 27, 1917). By the investigation of Dr. J. Spikrrsorr in his dissertation “Scattering of Light, and Distribution of Intensity over the Disc of the Sun”, the supposition of Junius"), that besides the scattering by the action of the molecules also the scattering by irregular bending of rays owing to accidental gradients of the optical density must play a part in the origin of the distribution of light over the dise of the sun, has been corroborated. In this paper it is our object to show, that a mathematical treatment of the problem of the scattering of light by curving of rays is possible. For it is possible to put up an integral-equation for this phenomenon and to transform this with suitable and plausible suppositions into a differential equation with boundary conditions. The problem treated here is the counterpart of the problem of molecular diffusion in a flat layer of gas, which was solved by ScHWARZSCHILD. The deduction of the integral equation resembles his train of thoughts, but in our case we have the advantage, that the peculiar nature of the problem allows that a differential equation can be deduced from it, which makes mathematical discussion so much easier. It seems to us, that the irregular reflection and the irregular double refraction may be treated in an analogous way, which is important for the theory of the extinction of liquid crystals. In a medium, in which accidental gradients of the index of refraction arise, a ray of light will be curved in an irregular fashion. If a broad bundle of parallel rays runs through the medium, the different rays will be curved in different ways. And so, if the bundle is broad with regard to regions of a constant index of refraction, a bundle of initially parallel rays will be spread plume- shape. Now we fix our attention on the action of a volume-element on a ray, which goes in a given direction, and imagine that the nature of the irregularity of the index of refraction in the medium 1) W. H. Jutius. Verslag Kon. Akad. v. Wet. 18, 195 (1909) and 22, 64—75 (1913). 8% 116 is such, that after running through the element in question the diffusion of the rays has only taken place over a small cone. A characteristic difference with molecular scattering is, that with the scattering by refraction not a great part of the bundle goes on unimpeded and a small part spreads to a// sides, but that the chief bundle itself gets continually broader. And so, expressed in mathematical terms: let us follow light of a given direction over a length /, then, if this bundle has an intensity of one per unit of square, the intensity of the light which is found in a cone of the opening dw, of which the axis formed an angle « with the direction of incidence, will be possible to be represented after running through / by a function: x(al)dw or y(a)dw By making use of a particular image the form of the function x may be determined. This form will be analogous to the law of errors ; it is however unimportant for what follows. What is important is the supposition that the function y possesses a perceptible value only for very small values of a. : If we take the meaning of 7 into consideration, we see imme- diately that of course ide x (a) dw = 1 where the integral, just as everywhere else in what follows, must be taken over the whole unity sphere. Now we shall deduce the integral equations for the intensity of radiation, Let f(z, y, z, 9, p\ represent the intensity of radiation in a point (wz, y, 2), Whilst the direction is given by the angle 9 with the x-axis and p. If now we know the radiation in a point (, y, 2), we ask this quantity in a point that is situated / further in the direction of the ray 9, y. The coordinates of this point are: e+tleosd® , ytlsndcsp , 2 lsin d sin p And so the intensity of radiation may be represented by: fe + leos 9 . y + UsinHecosp , z+ lsndsing , 9, p). This intensity must now be equal to the intensity which by the bending of rays comes in the given direction. When 9 and y’ are the angles which determine a ray in 2, y,z, then, if « represents the angle of this ray with the ray 9, wp, the intensity in the second point will also be given by: | 4 (a) f (2, y, 2,9, pj) dw! 117 so that the required integral equation is: fe + leos ,...) =| x (a) f (a, y, 2, 9, pl) dw’. Now it is easy to transform this integral equation into a differential equation, if we bear in mind that y has only perceptible values for small values of « We express by « the integration-element dw’ and the angle yw, which the plane through (9', #!) and (9, p) makes with that through (Ap) and the v-axis. So that the value of du’ is: sin ada dy = a da dp For the difference of the angles / and /' and that of the angle p and g' we find going to the second order to « and after elementary reduction : . tod ans A9 == 9'—9=—a cos w + 3 sin 9 cos H Ap? = — acosw + a? ra 5 y in NP = ks : sin & Now we can develop in the integral /(x, y, z, 9’, #!) with respect to A9 and Ap and get in this way: Fey 29 ) J x (@) do’ + ki al Ad ay («) da dp + Si oer gel 9 +55 [fOr ae soy {feo Ap ay («) du a | Als zo { [eo ay («) da dp The first integral is equal to unity, the second yields aU = cot oe x(a) der, the third is zero as well as the fifth, whilst the fourth and the seventh yield: a feu da and safe x(a) da sin? Ù Now we can introduce the mean value of « according to we? = | a?y(a)do= 2a aby(a) da And thus we obtain at length, — if we also combine the first term of the second member with the first member and develop according to / — for the differential equation of the diffusion of light by irregular refraction : 118 Ow Oz 4l For the case important in practice that f only depends on « and + consequently : òf 94 Of 5 Of ee a? (cos 9 Of 2 id 1-40 = cos -— sind cos p + — sind sin p = — —= EE : oy vial i ht sin DOD A9? sin? BOG? dz Al 09 09? For this case the boundary conditions are, if we have at «=O a layer, which radiates according to the cosinus law and at d a plane layer through which no inward-radiation takes place: == for: eo) cos # > 0 {= for: Bead cos F< 0. sin oa cos } oF == at (co ml 2 sin 9 if ) 1) ar of df For 9==0 and all values of w: f and af continuous, ——- =0; for ; Oo 00° 0 ej 09 It is worth observing that the diffusion is determined by the cos 9 = 1 or— 1 oe ae quantity qe the average square of diffusion per length- unity. This magnitude is specific for the problem, does not depend upon the length, as a” is doubled in doubling the length /. The magnitude is related to the nature of the irregularities. It is a constant which still may be different for different layers of the sun. The study of the distribution of intensity over the dise of the sun will be able to supply us with the knowledge of the average value of the characteristic constant of the sun. Utrecht, April 1917. Institute for Theoretical Physics. 1) For the two dimensional problem: NE, 08 D=. de AL Oe? Meteorology. — “On the diffraction of the light in the formation of halos. II. A research of the colours observed in halo-phenomena”. By Dr. 5S. W. Vissrr. (Communicated by Dr. J. P. van per STOK). (Communicated in the meeting of June 29, 1918). In the first paper on the diffraction of the light in the formation of halos *) a survey of colours observed has been given on pag. 1175 taken from ‘Thunderstorms, optical phenomena ete. in Holland”. Prof. Dr. E. van Everpincen however informed me, that these records are altogether insufficient because only a small number of colour observations are dealt with in “Thunderstorms”. On his „suggestion I have studied a number of records sent in to the “Koninklijk Nederlandsch Meteorologisch Instituut’. In the first place I hope to have set right a neglect against the sincere voluntary observers of the Institute; in the second place this research gave valuable materials for the answer to the question how far and in what manner the diffraction works in the formation of halos. In this paper a survey of the research is given; the results will be discussed and it will appear, that indeed the diffraction has an important influence on the refraction of light in ice crystals. I started with all the colour records in the years 1913, 14 and ’15. Then the research was extended to the years 1911, ’12 and 713. In the first part I soon found, that great prudence was necessary. As an example I take the observations of “rainbow-colours’’. In the three years 19138—’15 I find “rainbow-colours” 12 times recorded by 9 observers. There are however 7 who have never sent in another record during all this time. They were evidently led by suggestion and fancy more than by observation power. The personal character also comes to the front. Therefore it was resolved to make a very careful selection and only to use records of those observers known to the Institute as wholly reliable. In this manner the notes are studied of eleven observers, who are mentioned in this paper with the numbers I tot XI. 1) These Proceedings Vol. XIX. 120 Omitted were all incomplete observations and those about the moon. 550 Observations were at my disposal, divided over the six years as follows: 1910: AI 1912 1913” 19t¢ 1915 “Doral colour-records 125 (6) 107(7) 114(7) 81(8) 66(8) 57(7) 550 total 473 480 399 2838 325 377 2337 Se 26 22 29 29 20 15 24 In parenthesis is mentioned the number of observers. The row “total” gives the total number of records taken from “Thunderstorms”. | The fourth part of all the notes gives reliable records. 31 Colours and colour-groups are mentioned, among which three very anomalous ones: blue IX; violet, red VIII ; red, violet, green V.. In the first communication are mentioned’) golden brown (red), yellow, green, violet (3 times by two observers); yellow, violet; golden brown, clear white, blue. All the colour observations (divided over the two circles, the tangential ares and the circumzenithic arc) are contained in the table I. (See table following page). In the table the personal character comes out strongly. 144 observations of red by 10 observers (among which 115 of VII and IX), and 1383 orange by 5 (among which 130 of V and X) where in most cases the same colour is meant, this clearly points to the personal estimation of this colour. The same is evident in the groups red-white and orange-white. The table II also gives an idea of this individual opinion. (See table II page 122). All the colour records of each observer, separately for the ordinary circle, the parhelion and the tangential ares have been collected without further observations in this table. From this it appears, that the records of V, VIL and X are limited to green (V has one observation of violet on a total number of 158). III] also mentions blue, but never violet; VIII notes neither green nor blue, but in 22 records 12 times violet. This phenomenon however interesting from a physiological point of view, greatly diminishes the value of the records, but without doubt green and blue colour shades often occur, as is further evident in the percentages of the separate colours in the following table (in which parhelion and tangential arc are taken together). 1) Le. p. 1830 seq. 121 TABLE I. circle 22 parhelion tang. aad circle 46 cae an | total | | | | | white SG | 7.0) 1 (1) Pes | oes OS) red (brown) ‚119(10) | 14 (3) Gia NEE Ie | 114 (lo) orange 123 (4) 72) Woah Be SA ee ih bo) yellow | 15(6) | 4(3) KUN |e | 20 (6) blue oo — — 1 (1) — 11) red, white (brown, white) | 20 (6) 8 (1) ray. 4 — | -- 29 (6) 2) orange, white be OF (1 BEER BEE Ile a ee 29 (2) yellow, white | 5 (2) —- | = | = | -=— | 5Q) red, orange (3-40) — En ity tee } 4(1) red, yellow (brown, yellow) 10 (4) ZU ON ee (1) — -- 13 (6) 3) red, green ne) Ee 2(2) | 38(5) red, blue 10 (3) RD! af. ON: | Gi — 15 (4) red, violet ras ACD) 2 ie Hee B) EE ntt AT orange, yellow ot, HED) en Gil Belge nie; 1 — 1 (1) orange, green 1 (1) — | = ana ae ol AEN) orange, violet i Wis eg 2 eN Ret, violet, red 1a} ee SEND OE EEND ae yh) red, orange, white mhr ig Tel) — | — | — | 10) red, yellow, white Fe 2k (1) Pye ee ee 3 (2) red, green, white de en BCR | BS ees eee 1 (1) red, orange, green 1 (1) — PE a= —— 1(1) red, orange, violet eee Oe ee he en 1 (1) red, yellow, green A Gye 2.2) 1 (1) a 4 (1) 11 (3) 5) red, yellow, blue Be = 1D | — Ee | 9 (4) red, yellow, violet KG) LD —- | = Nee 2 (2) red, green, blue ewe Be LL en | 6 (2) red, blue, violet BAER in Lye ate tS | 1 (1) red, violet, green Sed tot (1) uy, Sta a be eek OI red, yellow, green, blue Ls | zt abl 322) — — | 2 (2) red, green, blue, violet ee Mah, marily ahd) fli ee > Se rd., or., vl, gr., bl, vi. | 1D | 10) 1 (1) | — 20 | 59% | 436 | 65 26°. 108i din recl een 1) Among which brown 10 (2). 2) pr „ brown, white 11 (3). 3) . » brown, yellow 3 (1). 4) 1911, April 12: “on the outside reddish, inside violet”. VIII]. — April 23 VIII records: “on the outside common red”. 5) An observation of an arc of Lowitz by Hiss)nk 1910, Sept. 7 at Zutphen was neglected. 6) Colours mentioned 2 times. “All colours” once “rainbow-colours” twice. 122 TABLE Il. Circle {of 22° | Parhelion | Tangential arc Ahead ip Ie ile Ue yig|olv wire yt v bebe (eta a | Pk Ee ED HE | 28) | es rege |) 1 |— j= hee! bn 3|—|—|=| BE ed A ee Gg =| a A et MEA ej € phat sh dje A |e ae hennes | eapee Sleep) |e) 1 a6] 1) LA ese eet eet ALAN ed 2) VII} 23| 60 fl otal) OEE EN of ght alt set 4) genial vink 2) 13} oel ela sl 3 1 — kende 1 IX| 11] 50 7| 1]—|—|14} 23) 2) 4) 11 2] 2} 2) 6|—| 3}—| 1 Hesel ast ores al el 1} ils ee Kh Teile kde je: “lalate 105 213 158 46 33/22 10/18 45 | 12 141343 7| 4 | 30 5| 81311) 4 white red orange yellow green blue violet circle 22° 17.9 36.3 26.9 7.8 5.6 3.8 1 parhelion,tang.arc11.9 40.8 SEREEN EE 7.0 6.1 Surely the figures for violet are strongly. flattered (21 observations of violet among which 12 of VIII). Without the records of VIII the percentages for violet are respectively about 0.3 and 3.8. Evident is the great variety of colour of parhelion and tangential arc '). By adding red and orange, green, blue, and violet the personal influences may be destroyed to some extent. Then I find white red orange yellow green blue violet circle 22° 19 63.2 (se AE parh. tang. arc. 11.9 50.0 10.3 27.8 Against a decrease of white and red we see an increase of the other colours. In more than '/, of all cases colours are recorded approaching green and blue for the parhelion and the tangential ares. _ This also happens with one in nine ordinary circles, where colours are made mention of. 1) Without doubt in the first communication I have slightly misunderstood Pernter: the predomination of fixed crystal positions must at all events be very important. 123 Some colours and groups occur relatively often. Vell KEN OW, WHITE... i ERE ste ii es ol tes, 25 times; 6 observers fed, Yeuow., red, yellow; WILS n. .:5 oer 16 6 a red, green; red, blue; red, violet . . . . 64 „ 8 a red, yellow, green; red, yellow, blue; red, yellow) violet 4 + id Feu Sree DING REEKS i erase Sk la TO 2 m5 SPCC OULS EE AE EE ce kos olien ee DT 3 ie Green, blue and violet, to escape from personal influences, are again added. The yellow takes a peculiar place. Yellow circles seem to occur. It is clear, that the yellow is often missing between the red and the green, but on the other hand it is often met with. As regards the rainbow colours; 5 observations of 3 observers remain in six years. Separate mention deserve the estimations of breadth by Hemmes at Arnheim in the ordinary circle and the tangential are. 1911 Dec. 29 red 3° yellow 3° blue 1° 1912 Feb. tl March al AT EEE aah PE Nae May 10 March 8 > + „ ¥ especially at the top also blue. 1912 Jan. 6 1913 June 14 1911 - Dec. 3 red 4° yellow 4° green 4° blue 4. ‘red 4°, green +° blue 4° The fact that the breadth strongly varies also appears from the detailed tables on the circumzenithic are by Besson '): 17 times on 91 arcs BrssoN measured the colours. The distance from red to violet varied from 14° to 3° (14°: 3 times; 2°: 6 times; 24° twice; 3°: 3 times). Three times blue and violet are wanting; among these is one are, with which the breadth of the inside red to the green is 5°. Busson notes: “très large, tres brillant’. These variations of breadth are very important for the theory of diffraction. Summing up I find as the results of the research after eliminating the individual influences: 1. the pretty large wealth of colours, 2. the variation of colours, 3. the variation of breadth. 1) Sur la Théorie des Halo’s. Paris 1909. p. 62. 124 These results are not expected by the simple theory of refraction ; they demonstrate the action of diffraction. It is these very properties which, for the rainbow, made the ordinary refraction-theory insufficient. Evidently the conditions for the development of these phenomena of diffraction are present rather frequently. With great certainty this research has established the conclusions drawn in the first paper. The observations however difficult by the small power of the colours, which generally are to be taken as mixed colours, the records, however often delusive by personal influences sufficiently show, that in the formation of halos the diffraction plays an important part. Mathematics. — “Nouvelle démonstration du théor’me de Jordan sur les courbes planes’. By Prof. Arnaup Dunsoy. (Commu- nicated by Prof. L. E. J. Brouwer). (Communicated in the meeting of June 29, 1918). Le théoreme fondamental de Jorpan sur les courbes fermées peut g’énoncer ainsi: St les points d'un ensemble T et ceux d'un cercle se correspondent réciproquement et continument, chacun à chacun, l'ensemble TP divise le plan en deux régions. L’hypothese faite sur I caractérise une courbe de Jordan. Je me propose dans cette Note de donner une démonstration du théoréme ci-dessus énoncé. Je rappellerai d’abord certaines définitions et résul- tats connus. Nous caractérisons comme il suit les cétés positif et négatif en un point IL d'une ligne HIK formée de deux segments de droite H/, 7K, dont / est le seul point commun. Déerivons, dans le sens direct des rotations, un are circulaire inférieur a 2a, de centre /, ayant son origine sur ZK et son extrémité sur HJ. Cet are borne, avec HJ et JK, un secteur de cercle w. Soit 4 un ensemble continu, tel que, a l'intérieur d’un certain cercle c de centre / et de rayon inférieur a celui de w, L et HIK aient seulement / en commun. Nous dirons que, au voisinage de /, L est situé du côté positif de la ligne H/K (ou du côté négatif de la ligne K/H) si les points de ZL intérieurs a c et distinets de / sont tous dans w. | Il est aisé de voir que, si TA’ est du côté positif de HIK, LK est du côté négatif de HIK. Si J est un point non extrême d’une ligne brisée 2 simple (c'est- à-dire telle qu’ un point queleonque de la ligne n’appartient a deux côtés différents que si ce point est origine de l'un et extrémité de autre), pour définir les côÔtés positif et négatif de A en J, nous considérons un secteur de cercle analogue a w, limité au cÔté (ou aux deux cdtés) de 2 contenant /, et ne rencontrant ancun autre côté de 2. Soit P un polygone simple, défini avec son sens de parcours. On montre (voir Comptes Rendus de P Académie des Sciences de Paris, 1911) que P divise le plan en deux régions (nous les appelons respectivement positive et négative, et les désignons par P+ et PS), 126 telles que tout continu joignant un de leurs points J/ au polygone P, atteint celui-ci du côté positif pour P+, du côté négatif pour P-. A et B étant deux points de P, la ligne brisée décrite en parcourant P selon son sens, de A a Best l'arc direct A B de P. L'arc rétrograde AB est géométriquement identique a lVare direct BA, mais les sens de parcours des deux arcs sont opposes. Pour démontrer le théoréme de M. JorpaN, nous utiliserons le lemme suivant: Si, en parcourant une fois un polygone P dans un sens invariable, on rencontre successivement les quatre points A, B, C, D de ce polygone, et st (AC), (BD) sont deur continus joignant respectivement Aa CU, Ba D, et dont tous les points, sauf A, B, C, D, sont dans une méme région limitée par le polygone, ces deur continus ont au moins un point commun. Supposons d’abord que (AC) soit une ligne brisée simple. On peut toujours choisir le sens positif de parcours de P, de facon que la région de P contenant (AC) et (BD), sauf leurs extrémités, soit „Bi Considérons alors le polygone a formé de lare direct CA de P, et de la ligne (AC) parcourue de A vers C. (AC) atteignant P en A et C du cdté positif, Pare direct AC de P s’écarte de a du côté négatif en A et C. Done, D qui est sur cet are est dans 7—. Mais, Pet m ayant en commun are CA qui contient B, les côtés positifs de P et de a au voisinage de B coincident. Done le continu (2D) est, au voisinage de B, dans at. On en déduit que (BD) rencontre x en un point différent de B. Comme (LD) ne rencontre pas l'are CA, (BD) rencontre (AC). . Supposons que ni (AC) ni (BD) ne soient des lignes brisées simples. Si ces continus n’ont pas de points communs, leur distance minimum est un nombre positif «. On remplace le continu (AC) par une ligne brisée simple 2 d’extrémités A et C, située, sauf pour ces deux points, dans P+ comme l'est (AC), et ayant tous ses points a une distance de (AC) inférieure a «a. D’apres la premiere partie de la démonstration, 4 rencontre (BD). Nous aboutissons done a une contradiction. Done (AC) et (BD) se rencontrent dans tous les cas. Nous déduirons de ce lemme une proposition essentielle. Soit I” une courbe de Jorpan et O la circonférence de cercle correspondant ponetuellement a FF. Si un point déerit O dans le sens direct, nous dirons que le point homologue de I décrit Mdans le sens positif. On échange le sens positif de parcours de I en transformant le cercle O en lui-même par une symétrie par rapport a un de ses diametres. Cela pose, 127 Si A, B, C, D sont quatre points d'un polygone simple P, et A', B, C', D' quatre points d'une courbe de Jorvan Fne rencontrant pas P, si (AA'), (BB), (CC), (DD) sont quatre continus deur a deux distincts contenant respectivement les points mis en évidence dans leurs désignations et n'en ayant aucun autre de commun avec P ni avec VT, ordre des quatre points A', B', C', D' sur la courbe T, et celui de A, B, C, D sur P, Pune et autre parcourus dans le sens positif, sont identiques ow inverses. On voit sans peine qu’en échangeant entre elles, s’il en est besoin, les dénominations des couples associés A et A’, ete, et aussi en modifiant le sens positif de I, la proposition serait en défaut dans le cas unique où, A, B, C, D étant rencontrés sur P dans leur ordre d’énonciation, on rencontrerait sur I” successivement A’, C’, B’, D’. Mais alors le continu (AC) formé de (AA’), de (CC’) et de larc direct A’C’ de I, ne rencontrerait pas le continu (2D) formé de (BB’), de (DD’) et de l’are direct B’D’ de T. Or ces deux continus sont, a Vexception de leurs extrémités A, B, C, D Pun et l'autre dans la région de P contenant J. C.q. f. d. Rappelons maintenant que si Fon forme une subdivision du plan en carrés égaux (y) par deux familles de droites respectivement parallèles à deux directions rectangulaires, et, si lon considere les ensembles formés par les carrés ne contenant, ni intérieurement ni sur leur contour, nul point d'un continu £, ces ensembles forment des domaines (réunion d'un continuum et de sa frontière; un continuum est un ensemble connexe dont tous les points lui sont intérieurs) dont chacun est limité par un polygone simple appelé polygone d'approvimation de KE, relatif au quadrillage (y). Le sens positif d'un tel polygone a sera défini par la condition que / soit dans a. Tout point H de = est situé sur Pun (on sur deux) des cdtés d'un (ou de deux ou de trois) carré y dont l'intérieur appartient à m— et qui contient, intérieurement ou sur son contour, au moins un point de Z. L’un de ces points-ci 1’, est tel que la distance HH’ est minimum. Les points non extrémes du segment ////’ sont situés dans a et étrangers a &. D'ailleurs HH’ est au plus égal à la diagonale de y. Cela étant, soient M et N deux points, distinets on non, appar- tenant a une méme région limitée par PT, et P, QQ deux points de T tels que les segments J/ P, NV Q ‚aient en commun 1° avec 7, uniquement les points respectifs P et Q, 2° entre eux, éventuellement ‘et seulement certains de leurs points extremes (done si JM coincide avec N, P est distinct de Q et inversement). M et N peuvent être joints par une ligne simple 4 dont tous les points sont distincts de 128 T. Soit 4« un nombre inférieur à-la distance de 1 a IT, et à la distance rectiligne P Q. « étant moindre que a, considérons dans un quadrillage de côté e le polygone a d’approximation de FP, dont la région positive contient M et N. A partir de P et de Q, les segments PM, QN rencontrent a aux premiers points respectifs M, et N,. Soit 9 la plus grande des deux longueurs M, P et N, Q. & tend vers zéro avec «. Si 9 He respirations 45 sec. pause 40 sec. pause 13 respirations 11 respirations 40 sec. pause 70 sec. pause 20 respirations 11 respirations 62 sec. pause This observation has this advantage over the registration, because on account of the suspension the respiration of the frog varies temporally at least often. After this the respiratory curves were registered by suspension of the mouth- bottom, as LANGENDORFF did for the first time. The first 8 minutes after the suspension the respiration stopped entirely. Thereupon the groups reappeared again. Fig. 3 represents some of these. The bottom row was registered '/, hour, the top one 10 minutes after the suspension. When we compare the respiratory- curves of these groups with the curves 1) In order to control the experiment the two lenses were extirpated after the experiment and compared with normal extirpated lenses. The latter were clear and transparent, the former turbid and opalescent. 155 of the normal respiration, it appears that with each movement of the mouth-bottom one movement of the flanks takes place during these groups. Only the two first groups of the top-row set in with a separate expiration-movement, which is not followed by a movement of the mouth-bottom. For the rest all groups begin with an expiration, as appears from the fact that the lever descends in the beginning. During the normal respiration the frogs show one flank-movement with some movements of the mouth-bottom. During the groups of the Cheyne-Stokes’ respiration every movement of the mouth-bottom is almost always followed by one flank-movement. In this way the frog respires likewise when it is dyspnoeic. If the blood of frogs is replaced by normal isotonic solution of RinGer instead of hyperisotonic fluid of Ringer the mentioned phenomena do not oceur. The respiration remains normal, the lense does not become turpid. A more explicit discussion of the cataract and the Cheyne-Stokes’ respiration follows here. Cataract. In whatever way the fluids of the body of frogs may be made hyperisotonic, cataract oecurs always. The cataract disappears however again, when the osmotic pressure of the fluids of the body is made normal again. The cataract develops itself very slowly. When after a perfusion during 15 or 20 minutes the respiration has stopped already, the surface of the pupil begins only to become a little dim. If then one waits a short time without continuing to drain the ecirculation- apparatus, the dimness gradually increases. At last two vertical parallel white stripes are observed on the lense, between which there is a long dark stripe. It makes the impression as if one sees two white walls and between these a deep, dark moat. The direction is usually vertical, sometimes almost vertical. This vertical stripe will correspond to the frontal vertical suture of the lens, as it is described in Gaupp. (Anatomie des Frosches). One often sees white, thin lines proceeding in a radiary direction from this vertical line, corresponding to the so-called spokes of the human cataract. It is obvious that the origin of the cataract must be attributed to a congelation of the albuminous substances in consequence of an increase of the saltconcentration of the fluids of the body. As soon as the osmotic pressure of the fluids of the body decreases again, the process is likewise converted. In my opinion another explanation of the phenomenon is impossible. 156 It is obvious that also every other circumstance by which a con- gelation of albuminous substances is caused, can bring about lense- cataract. In my opinion however it is of importance that I have indicated, that a mere increase of the osmotic pressure of the fluids of the body can result in cataract. Cheyne-Stokes’ respiration. The periodical respiration was caused in my experiments by an increase of the osmotic pressure of the fluids of the body. The periodical respiration disappeared likewise again as soon as this pressure did not exist any longer. In order to study the origin of the Cheyne-Stokes’ respiration more accurately the desiccation of the frogs was continued in a series of experiments so far, that the jheyne-Stokes’ respiration had not yet set in. Thereupon the mouth- bottom was suspended and in a warm room the frog was exposed to further desiccation. Usually the Cheyne-Stokes’ respiration slowly set in there in the course of a few hours. During these experiments it appeared that besides a periodicity of the flank-respiration we must distinguish a periodicity of the pharyngeal respiration. In far advanced stages the two periodicities coincide, so that then during the groups movements of both the flanks and the mouth-bottom take place, whilst in the pauses the respiration stops entirely. As a transition to this complete Cheyne-Stokes’ respiration we find a stage in which the groups are equal to those of the complete Cheyne- Stoke’s respiration, but during this stage the movements of the mouth-bottom continue. It appears consequently that both ways of respiration are to a certain degree independent of each other, as appears indeed also from the normal respiratory curves. According to the examinations of LANGENDORFF the movement of the flanks comes off passively without a contraction of the pectoral muscles. If this is correct, then the movement of the flanks is after all made possible by an active opening of the glottis. With the Cheyne-Stokes’ respiration the periodicity of the movement of the flanks is determined by a periodicity of the glottis-muscles. An opening of the glottis is almost constantly followed by a move- ment of the mouth-bottom, this however is not necessary either. The first 2 groups of Fig. 3 set in with an expiration that is brought about by an opening of the glottis, which is however not followed by a movement of the mouth-bottom. One word more about the cause of the Cheyne-Stokes’ respiration in these experiments. As I explained already, this phenomenon 157 occurs at a hyperisotony of the fluids of the body. This hy perisotony leads in the end to a cessation of the respiration. Like the cataract this cessation of the respiration can be suppressed again by a decrease of the osmotic pressure. Consequently the setting in of the cessation of the respiration in hyperisotonie surroundings, just as the Cheyne-Stokes’ respiration that precedes it, and likewise the development of cataract originate in modifications that are reversible. Chemistry. — “On the Klectrochemical Behaviour of Metals’. By Prof. A. Smits. (Communicated by Prof. Zrrmay). (Communicated in the meeting of March 23, 1918). 1. Introduction. By application of the considerations on which the theory of allotropy is based to the internal state of the metals and to their chemical as well as to their electromotive behaviour, we are enabled to consider all the metals, also those which serve as so-called unat- tackable electrodes, from the same point of view. - These considerations rest on the more than probable assumption that every metal contains metal atoms, one or more kinds of metal ions, and electrons, which ean be in equilibrium under definite circumstances. When a metal is immersed in an electrolyte, then in agreement with Nernst’s views of the phenomenon of solution, the heterogeneous equilibrium between the metal and the boundary layer will be established with so great velocity, that it may be said that this equilibrium always exists. When we, therefore, restrict ourselves to the simple case that the metal consists of metal atoms, »-valent ions, and electrons, we may say, that when this metal is immersed in an electrolyte the following heterogeneous equilibria will at once set in. M, . My. v0, i i, Ae at Whether the homogeneous equilibrium will also exist in the two coexisting phases between the metal atoms, metal ions, and the electrons, depends on different circumstances. Whereas it seems that a metal in perfectly dry condition can assume internal equilibrium as a rule only at comparatively high temperature, this often takes place very quickly when in contact with an electrolyte, but it may also occur that the metal gets in equilibrium very slowly, or not at all, under these circumstances at the ordinary temperature. The velocity with which a metal assumes internal equilibrium under definite circumstances is undoubtedly one of the most charac- teristic properties of the metal. 159 2. The Potential Difference Metal-Klectrolyte when the Metal is Attacked. When a metal in contact with an electrolyte superficially assumes internal equilibrium with very great velocity, the infernal state in the metal surface remains unchanged, in whatever way the metal may be attacked. Let us suppose that we immerse zine in an aqueous solution of hvdroehlorie acid; then hydrogen generation takes place, because the electron concentration of the metal equilibrium 8 ns gk bs EJ in the solution is greater than the electron-concentration of the hydrogen equilibrium : Ede BO nei estote tld) Hence the electrons of the equilibrium (1) are removed, and through this the equilibrium is disturbed. It is now the question how the equilibrium can be restored. The concentration of the zine-atoms in the liquid is so small that even if the reaction constant of the conversion Zn, > Zn, + 207 was very large, yet only exceedingly few zine ions and electrons would be split off per second in this way. The only way in which the state of equilibrium can be restored is this that the metal sends electrons into solution, which is of course accompanied by zine ions going into solution, because zinc-ions and electrons, with a difference of only a very small amount, are always present in the same concentration. Through this process the internal equilibrium in the metal surface is disturbed, which can be restored again by the reaction: Zn, > Zn; + 20, . As the heterogeneous equilibrium in the boundary layer sets in with very great velocity, the question whether the metal zine during solution in an acid will be disturbed, comes to this, whether the internal equilibrium in the surface of the metal sets in with so great velocity that the concentration remains practically unchanged. This is actually the case for zine under certain circumstances. Mr. Hürrer S. J., who examined some metals at my request, found among others, that when the potential difference between zine and a solution of zine-chloride is measured during vigorous stirring, and then that between zine and a zine-chloride solution of the same 160 concentration acidified with hydrochloric acid, the potential difference retains the same value, notwithstanding a strong hydroyen-generation tukes place in the latter case. The metal zine is, therefore, not disturbed through solution in hydrochloric acid, and this result is in perfect agreement with what is found when zine is anodically brought to solution in a zinc- chloride solution. In this process, which likewise rests on the with- drawal of electrons from the metal, the potential difference, zinc- electrolyte, does not change appreciably, even for comparatively great densities of current, so that our investigations about the potential difference during the solution of zine in a hydrochloric acid solution, as well as the measurements of the potential difference of the same metal on anodic solution in a solution of zine-chloride lead to the result that the equilibrium in the metal zine in contact with the above-mentioned electrolyte sets in with a velocity which is very great compared with the velocity with which electrons and ions are withdrawn from the metal. 3. General consideration. When we .now consider the phenomenon in general, we can distinguish the following cases. : On immersion of a metal in an acid we have in the simplest case among others the two following equilibria in the electrolyte: Hi, = 2H; + 264. and My, 2 My, + Or The electron-concentrations of these two equilibria are in general different, and a consequence of this is that either the electrons of the metal equilibrium, in the liquid, combine with the hydrogen ions of the hydrogen equilibrium, which causes electrons + ions from the metal to go into solution, or the electrons of the hydrogen equilibrium with the metal ions of the metal equilibrium pass from the electrolyte into the metal. Let us first imagine the limiting case, viz. this that the internal equilibrium of the metal surface is established with great velocity, so that this velocity is very great with respect to the velocity with which electrons + ions are withdrawn from the metal or are added to the metal, then the metal surface will not change independent of whether one process takes place or the other, and the potential 161 difference metal-electrolyte will remain equal to the potential difference of the unary metal. In the second place the case may present itself that the internal equilibrium of the metal surface does not set in so rapidly as was supposed above, and then it will be possible to disturb the metal surface either in one direction or in the other, i.e. it may become either nobler or baser, hence the potential difference can differ from that of the unary metal in noble or base direction. “A third case, which like the first, represents a limiting case, is this that the metal is so inert that the velocity with which it assumes internal equilibrium is very small compared with the velocity with which the electrons and ions are withdrawn from the metal or added to it. In the first limiting case the potential difference is entirely governed by the state of internal equilibrium of the metal, and in the last case the potential difference is dominated by the electron concen- tration of the hydrogen equilibrium in the electrolyte. 4. Nickel as Example of an Inert Metal, the Inertia of which Increases under the Influence of the Dissolved Hydrogen. An example of the latter case with this particularity, however, that the just mentioned great inertia is only slowly reached, because the metal is converted to this state after some time through the negative catalytic influence of the dissolving hydrogen, is furnished by nickel. As was shown in a previous communication, the case presents itself that when this metal is immersed in such an acid solution that hydrogen generation would have to take place, this phenomenon does not take place to an appreciable degree, and the metal appears to be disturbed after some time so far in a noble direction that its potential difference has become equal to that of the hydrogen electrode. On that occasion we already gave an explanation of this pheno- menon, and pointed out that, nickel being so inert, the electron concentration of the nickel equilibrium in the electrolyte Ni, = Nit + 26; becomes equal to the electron concentration of the hydrogen equi- librium : He = A 4+ 26 so that finally (0), = Orn), _ This was demonstrated in the following way. We pointed out 11 Proceedings Royal Acad. Amsterdam. Vol. XXI. 162 namely, that on application of the electron equation for the potential difference, metal-electrolyte, for the derivation of the relation for the electromotive force of a circuit consisting of two metals immersed in the corresponding salt-solutions, we arrive at the following equation : Po bo, mn (O7) — ea me Te EEE (Or) so that, taking into consideration, that the first term of the second member denotes the Volta-effect of the two metals, which is a very small quantity, the electromotive force A,—A, will be zero in first approximation, when (47,) = (O7). In the case discussed here the metals 1 and 2 are nickel and hydrogen, and experiment has taught that Ani— An, was really practically zero, from which therefore followed (6v;)r = (Op) Through the inertia of the metal nickel, which inertia was still increased by the hydrogen dissolved in the metal, which is here a negative catalyst, as was already stated before, the metal could, therefore, be disturbed so far, that the electron concentration of the niekel equilibrium in the electrolyte had become equal to the elec- tron-concentration of the hydrogen. We may, therefore, also express ourselves in a different way, and say, that the nickel had been passivated by the acid. Finally the nickel phase and the hydrogen phase present the same potential difference, accordingly these phases, which are in contact with the same electrolyte, can coexist. As in the case discussed here the nickel will of course be covered by a layer of hydrogen, the found potential difference refers to the three-phase equilibrium Ni + H, + + electrolyte. 5. Unattackable Electrodes. As follows from the communication cited here, this disturbance is comparatively slowly reached for nickel. There are, however, metals for which this goes much quicker, and these are the metals of which tbe so-called unattackable electrodes consist, as the plati- num metals. These metals belong to the group of the most inert metals that we know. Even in contact with an electrolyte these metals do not get in internal equilibrium, but they are almost always in passive state, so that the potential difference of the unary metal is not even known to us. When such a metal is immersed in a solution of HCl or H,SO,, 163 and hydrogen is passed through, the electron concentration of the platinum equilibrium in the electrolyte Pt, 2 Pir + 401, has almost immediately become equal to the electron concentration of the hydrogen equilibrium, H, 2 2Hy, - 201 Wen corresponding to the pressure of the hydrogen that passes through, so that e.g. the platinum electrode has almost immediately become electromotively equal to the hydrogen electrode. For these metals, which behave ideally inert, the potential differ- ence is, therefore, governed by the existing electron concentration in the electrolyte. This is also the reason why these exceedingly inert metals may serve not only as gas-electrodes, but also for the determination of the so-called oxidation, resp. reduction potentials. When e.g. platinum is immersed in a solution in which the equilibrium : ye) ra Foal Or prevails, the electron concentration of the platinum equilibrium in the solution has almost immediately become equal to the electron concentration of the above ferro-ferri equilibrium, so that in the electron equation for the potential difference of the platinum RT Ko = — ln ——~ FF (6x) the electron concentration of the ferro-ferri equilibrium may be written instead of (Oy), in consequence of which we get: RT. Ky (Fe) In Is a) A Ee, as was already stated before. The peculiar feature of these platinum metals is therefore their extraordinary inertia, which causes them to behave ideally passive in most electrolytes. 6. Considerations in the Light of the Theory of Phases. It is clear that for the explanation of the phenomena discussed here, considerations as have been introduced by us of late, are indispensable. Phase-theoretical considerations are inadequate here, but all the same it may be of use to represent the obtained results graphically by means of A,a-diagrams. ELT 164 Let us first consider the case that the metal zine is immersed in a hydrochloric acid solution of ZnCl,; then it is the A-a-figure of the system Zn-H, that may serve for the graphical elucidation of the found result. Fig. 1. Zine. A A-v-figure holds for constant 7, P, and a constant total ion- concentration; for Z’ we choose here the ordinary température, and for P the pressure of 1 atm., the total ion-concentration being put here at 2 norm. The situation of the point C, which represents the electrolyte which coexists with the zine phase and the hydrogen phase of the pressure of 1 atm., is found from the equation: de K zn RL Kir Sh en 2k (Zn7) F (H7) from which follows: Kez ie (Zar) Ki Gi) 165 or also from the equations for the product of solubility of zine and hydrogen: Bin (Zn) (0)? and DH) (09° by putting (@)z, = (Or), in consequence of which: Lin (Znc) BENEL or Zn, 102x—35 es ( L) = 10°, (Hi) 102x—48 When we put for a moment (Zij) = 1, then: p= 40 We see therefore, that tbe point C lies so much on one side that practically it coincides with the zinc-axis. Hence the line d, c, e or the line for the three-phase equilibrium zinc-hydrogen-electrolyte lies practically on the same level as the point a, so that the measured potential difference of the zine, which contains a little dissolved hydrogen, and is besides covered with a layer of hydrogen, is certainly practically equal to the potential difference of the pure hydrogen-free zinc, the measurement of which is impossible here. Let us now suppose that we immerse zine in an electrolyte, the composition of which, as regards the zinc- and hydrogen ions, is 2, ; we then see, that zine cannot be in stable electromotive equilibrium with this liquid, but that hydrogen can. If, however, the hydrogen did not appear as a new phase, but only dissolved in the zinc, a metastable electromotive equilibrium would, indeed, be possible, viz. g f, but the potential difference would be more strongly negative than that of the three-phase equilibrium represented by the line d, c, e. This metastable electromotive equilibrium does not appear, however; on the contrary, we observe a generation of hydrogen, and we will point out here in a few words, how the experimental fact is to be explained that under these circumstances the potential difference zinc-electrolyte is equal to that which corresponds with a, ¢; é or what is practically the same thing, with a. The explanation is this: when zinc is immersed in the electrolyte of the concentration «,, the establishment of the three-phase equili- brium between the zine phase, the electrolytes, and the hydrogen phase takes immediately place in the boundary layer. 166 Accordingly the concentration c prevails in the boundary layer, whereas the total concentration of the electrolyte is w,. The hydrogen ions now diffuse in the boundary layer, where for the maintenance of the concentration c the reaction : 2H SH, takes place, in consequence of which, as we have already seen, electrons and zine ions from the metal phase go into solution. The zinc phase assumes internal equilibrium with great velocity, and consequently it remains unaltered during the hydrogen generation, and the measured potential difference is that of the three-phase equilibrium d,c, e, which practically agrees with that of pure zinc, a. When a platinum electrode is placed in the same electrolyte, it indicates the hydrogen-potential which corresponds with the line mn. The zine electrode and the hydrogen electrode present therefore entirely different potential differences in the same electrolyte. This is the graphical elucidation for our conclusion that the potential difference of zine with respect to a solution of ZnCl, acidified with hydrochloric acid is determined by the state of internal equili- brium of the zine. Nickel. Let us now proceed to the case that instead of zinc the metal nickel is taken; then it is worthy of note in the first place that under the same circumstances we then find for the composition of the electrolyte c Enea ss ONS Oe tee ‘08 KG ti) in oA) eee ae When we now put (Ni; )=1, weget(H )?=10 ‘or(H )=10~. Here, too, the electrolyte c has still a one-sided position. Let us assume that the adjoined figure 2 again holds for 18°, and a pressure of 1 atm. for a total-ion concentration of 2-norm.; then an entirely different phenomenon is observed on immersion of a nickel-electrode in the electrolyte of the concentration 2, than in the case with zinc discussed just now, because the internal equilibrium in the metal surface cannot maintain itself when electrons and nickel ions go into solution. The metal is more and more disturbed in noble direction, and the result is, as we demonstrated already, that the electron-concentration of the nickel equilibrium in the solution has become equal to the electron-concentration of the hydrogen-equilibrium in the electrolyte, 167 in which the potential difference of the nickel electrode has become equal to that of the hydrogen electrode. This may be graphically represented in the way as has been done in fig. 2. Ni Hef 2H Fig. 2. In consequence of the disturbance point d has got in point d/, and represents, therefore, the ennobled nickel phase which coexists with c’ and the hydrogen phase e’. It could be derived from our considerations how we have to proceed when we want to know the potential of the unary nickel, or in other words the equilibrium-potential. Then the nickel is to be brought into a solution with a hydrogen-ion-concentration, smaller than that in the electrolyte c. Then our A, X-diagram 3 shows that under these circumstances e.g. the electromotive equilibrium between the nickel phase d" and the electrolyte c” will be established, the potential difference of which practically coincides with that of the unary metal, which is perfectly free from hydrogen and indicated by a. It is necessary to point out that when a metal is in electromotive equilibrium with a coexisting electrolyte, the electron-concentration 168 of the metal equilibrium must always be equal to the electron- concentration which exists in the liquid in consequence of the other prevailing equilibria. In this case we may say, that the metal is really in equilibrium with the electrolyte. When the metal dissolves in an acid, or when a metal is deposited, the just mentioned equality of eleetron-concen- tration prevails only in the boundary layer between metal and electrolyte, and diffusion takes continually place in the boundary layer. The just mentioned equality of the indicated ele¢tron-concentrations must, therefore, also exist when in the case mentioned just now the metal nickel has got in equilibrium with the electrolyte. Of course there are always some transformations required for this, but these are soon over, and can, therefore, not give rise to a permanent disturbance, at least if the solution has been freed as much as possible from air and hydrogen by boiling in vacuum. We have acted upon this principle, and, as was communicated in the preceding paper by Mr. Lopry pr Bruin and myself, by this procedure the equilibrium-potential was found of nickel that contained 169 only a trace of hydrogen, so that the found potential difference will practically very certainly agree with that of the purely unary metal. It is supposed here that the potential difference between nickel and the nickelsalt solution with the exceedingly small hydrogen-ion- concentration, is measured after the electrolyte has been heated with the nickel electrode in vacuum, after the whole apparatus has been Niet T OH Fig. A. filled with the electrolyte and connected with the 1 N. Calomel electrode by means of a siphon and a liquid circuit, so that the pressure under which the electrolyte is, amounts to 1 atm. also in this case. An entirely different result is obtained when the foregoing measure- ment does not take place in vacuum, but in a hydrogen current. In this case the nickel electrode is disturbed, but the disturbance does not take place now in a noble direction, but in a base direction, and as we showed before the potential difference of the nickel has again become equal to that of the hydrogen-electrode. This result can again be brought to expression in an exceedingly simple way by means of a 4, X-fig. 4. 170 When we lead hydrogen through the electrolyte, of which the concentration «, lies on the lefthand of the point c, the potential of he hydrogen-electrode is indicated by the line c’ e’. In this mode of procedure the nickel electrode gets in contact with gaseous hydrogen, and in the boundary layer which is simultaneously in contact with nickel and hydrogen, the electrolyte c will be formed in consequence of the reaction: Hey > 2H, 4-20); while electrons and nickel ions (and a few hydrogen ions) are deposited on the metal. This renders the metal baser superficially and both the three-phase equilibrium dee and the point a rise. Nee Fig. 5: This disturbance in base direction continues till the concentration of the electrolyte has become equal to that in the boundary layer. This is the case when a three-phase equilibrium has formed of which the electrolyte possesses the concentration «,, hence at the place where the curve bc intersects the vertical which corresponds with this concentration. As fig. 5 shows, this takes place in point c’ 174 and the three-phase equilibrium, which therefore finally is established, is here indicated by the points d’c’e’. Accordingly also in this case the potential difference of the nickel electrode is equal to that of the hydrogen electrode. When we now consider the metals of which the unat- tackable electrodes consist, we need only remark that because as was just now demonstrated, these metals are ideally inert, the potential difference metal-electrolyte is in almost all cases exclusively determined by the electron-concentration in the electrolyte. Hence, when e.g. a platinum electrode is immersed in an electrolyte through which hydrogen is led, the platinum shows the hydrogen potential almost immediately, which was the case for nickel only after some time had passed. When we want to express this graphically in a A, X-fig., we get, of course exactly the same representation as for the case nickel-hydrogen. That in aqueous solutions we cannot determine the equilibrium potential of platinum, whereas this is still possible for nickel is owing to this that the electrolyte c has such an one-sided situation for platinum-hydrogen, that an aqueous solution of a platinum salt always possesses a concentration on the righthand side of the point c as regards the platinum and the hydrogen ions, so that a disturb- ance must always take place. In a subsequent communication I hope to enter into a fuller discussion of the phenomenon of the ‘‘super-tension”, which has already been repeatedly referred to in our considerations without having been named. Amsterdam, March 1918. = General Anorg. Chemical Laboratory of the University. Anatomy. — “On the Nervus Terminalis from man to Am- phiovus.” By Prof. J. W. van Wisuk. *) (Communicated in the meeting of April 26, 1918). Although hardly credible, it is a fact that a good three years ago — in 1914 — a new nerve, arising independently in the brain, was discovered in man. This is the Nervus Terminalis. Naturally it is not visible to the naked eye, but can be seen through the mag- nifying glass, especially through the dissecting microscope, with the aid of which its discoverer, the American Brookover found it. (Journ. of Comp. Neurology. Vol. 24.) It has its course through the pia mater, parallel and mesial to the olfactory bulb and tract, running over the middle of the gyrus rectus (vide fig. 1.) When a rectangular piece of the pia mater in this region is taken up and placed under the microscope, the fine fibres of this nerve can be seen. Here and there the fibres are retracted from each other to come together again later on. The nerve is independent of the olfactory tract and bulb, and in the opinion of Brookover enters the brain at the mesial root of the tract. A number of ganglionic cells, Brookover taxes their number at about 50, lie spread in the nerve in its course along the olfactory tract. The nerve can be followed not only along the tract but also some- what further distally along the olfactory bulb, but in this vicinity it is embedded in the dura mater, while it has here also partially pierced the former and lies on the lamina cribrosa. In the vicinity of the bulb the number of its ganglionic cells is considerably larger than is the case along the tract. It was estimated by Brookover at about 100 to 200 cells. Undoubtedly its branches pass through the mesial row of openings in the lamina cribrosa to the mucous membrane of the nasal septum, but the research did not extend as far as this. In adult man the course of the new nerve is as yet known in the brain-case only, not on its outside. *) 1) Lecture delivered before the meeting of the Neth. Zoological Society, Jan. 26, 1918. 3) Vide, however, the postscript at the end of this article. 173 As was to be expected, in the adults of the mammals the nerve was not first found in man. The dog and the cat (Me. Corrrr. 1913.) and the rabbit (Huser & Guinp. 1913.) were the first, but it is remarkable that in the embryonic stages of the mammals the human embryo was the first in which, although incompletely, the nerve was discovered. This was done by our countryman Ernst DE Vriks, who also observed it in the embryos of the guinea-pig. He described his research (published in the Proceedings of the Royal Academy of Sciences of April the 22"¢ 1905), which also drew much attention abroad, in an article of four pages, which proves that it is not ne- cessary or even desirable to be loquacious when one has found something of importance. Dr Vries found ganglionic cells spread in the course of the nerve which supplies the organon vomeronasale, (the organ of JACOBSON, or better the organ of Rvuyscn)') near the base of the nasal septum. He moreover found that the so-called olfactory ganglion, by him called the ganglion vomeronasale, does not belong to the fila olfac- toria, which are taken collectively as the true olfactory nerve. In his opinion it belongs to the N. Vomeronasalis, which supplies Rurscu’s organ, lined by a layer divided off from the nasal mucous membrane. As the vomeronasal nerve also enters the central nervous system at a different place — the area vomeronasalis — than do the fila olfactoria, pr Vries drew the conclusion that the N. Vo- meronasalis is not, as was the general opinion, a component part of the olfactory nerve, but an independent nerve, homologous to the N. Terminalis in the fish. é A serious difficulty to this explanation however is that, according to the illustrations of pr Vrirs, the N. Vomeronasalis issues from the olfactory bulb, while the N. Terminalis of the Dipnoi and the Se- lachii issues out of the true hemisphere and not out of the bulb. ’) This difficulty seems to have escaped pr Vries’s notice. On the first page of his publication he rightly distinguishes between the olfactory lobe and the hemisphere, which are separated from each other laterally by the fissura rhinica, and mesially by the fissura prima. On pages 3 and 4 he states that the area vomeronasalis, where the nerve of this name enters the brain, belongs to the he- misphere. According to his own communication and illustration, however, this area lies at the suleus cireularis bulbi, hence not on 1) Concerning Ruyscu’s organ see postscript at the end of this paper. 2) Entering and issuing out of a nerve are used in this address, indiscriminate of the direction in which the impulse moves. 174 the hemisphere, but on the olfactory lobe. In young embryos the tract is thicker than the bulb, later on this ‘relation is reversed. It was therefore very desirable that more light were thrown on the question whether the vomeronasal nerve should be considered as the homologue of the N. Terminalis of the fishes. This happened in 1913 in America, more especially through two publications viz. of JOHNSTON in the Journ. of Comp. Neur. Vol. 23. and of Huser & Gurp in the Anatomical Record Vol. 7. *) JOHNSTON examined embryos of the pig, the sheep, and of man. Besides mammals he also examined embryos of tortoises and a larva of Amblystoma. The elucidation which Jonnston brought, consists herein that (as he found) the ganglion and the ganglionic-cells do not belong to the N. Vomeronasalis, but to another nerve, which does not enter the brain in the olfactory bulb, but in the true hemisphere, near or in the lamina terminalis, as is the case in the Selachii. What pr Vries had considered as one nerve, was in reality two nerves which for the greater part cover each other; one is the N. Terminalis, the other is the true N. Vomeronasalis. The vomeronasal nerve has no ganglionic cells and arises out of the cells of a part of the nasal mucous membrane which had been split off (Organon Vomeronasale). In structure and development it is exactly similar to the bundles of the olfactory nerve. It also enters the brain in the olfactory bulb, just as the fila olfactoria, | which collectively form the olfactory nerve. It is true that it enters the bulb at a special place, on its mesial plane rising high up caudally, but then it is a specialised bundle of the olfactory nerve. The peripheral ganglionic cells and the true ganglion belong to the N. Terminalis. DE Vrins’ mistake is easily comprehensible; he used no special methods to make the nerves visible, could not expose his material of human embryos to this risk and was thus compelled to consider the proximal end of the N. Vomeronasalis (split into four bundles according to him) as a root of the Ganglion Terminale, by him incorrectly called the Ganglion Vomeronasale, which is as it were pasted up against it, while the true roots of this ganglion escape observation in cross section through their fineness. That it is possible to make mistakes even when using nerve-staining methods is proved by the work of DériKen (1909). He examined embryos of mice, rabbits, guinea-pigs and man. Following in the footsteps of pr Vries ') Further literature is found mentioned in these publications. 175 he also took the roots of the N. Vomeronasalis to be those of the N. Terminalis. Regarding the mouse he says “Die sog. mediale Riechwurzel von der bereits Casat, Kappers u. A. behauptet haben, sie sei nicht als eigentliche Riechwurzel zu bezeichnen, ist eine Wurzel des N. Ter- minalis’”. No wonder that he continues “Sie hat bedeutende Beziehun- gen zum Olfactorius”. The second important elucidation appeared, as has already been said, in a communication, also in 1913, of Huser and Gurp, who had come on this subject & propos of the work of JonnsronN, which had partly been done in HuBer’s laboratory. These writers examined rabbit embryos by the silver-pyridine method. They could fully confirm Jonnsron’s results that the N. Terminalis and the N. Vomeronasalis were two different nerves, and that the ganglion and the disseminated ganglionic cells belong, not to the N. Vomeronasalis, which is evidently a specialised bundle of the olfactory nerve, but indeed to the N. Terminalis. While Jonnston however was still of opinion that the peripheral ter- mination of the N. Terminalis was limited, principally in any case, to the region of the N. Vomeronasalis, these investigators discovered that this ending is to be found in the foremost part of the nasal septum, reaching caudally to the rear border of the Organon Vomeronasale. It is only a small part of the peripheral branches that reaches this organ and the true olfactory mucous membrane, the region of the fila olfactoria, was free from branches of the N. Terminalis. Through difference in tint the branches of the Terminalis could well be distinguished from those of the Trigeminal nerve (Nasociliary and Nasopalatine), which are also found in the mucous membrane of the nasal septum. As will presently become clear, it is of importance in following the nerve to Amphioxus, that the N. Terminalis does not branch in the olfactory mucous membrane. In 1912 and 1918 Mc. Correr published his investigations on the N. Vomeronasalis and the N. Terminalis. By means of the dissecting microscope, thus as it were at magnifying glass magnification, he found the latter in the adult dog and eat, but not in the rat, the rabbit, the sheep, the guinea-pig or the oppossum. That he did not find it is not to be wondered at considering his method. His opinion that the N. Terminalis ends peripherally at or near the vomeronasal organ is also comprehensible because the bundles here are thicker, the fibres of the N. Terminalis being strengthened by those of the vomeronasal nerve. 176 This much as regards the mammals, which I have considered somewhat more extensively as most, and to my mind the most accurate, investigations have been done on them. I can be brief about the birds, reptiles, and amphibians. There does not seem to be much known about the N. Terminalis in the birds. In the frog it was found in 1909 by C. Jupson Herrick, who also described its central termination more especially; its peripheral branches could not be traced accurately. This was also the case in the Urodela, where the nerve was observed by Mc. KrBBEN (1914), who could not however find any ganglionic cells in it. Some time later Jonnston succeeded in this. He says “In Amblystoma the nervus terminalis is ganglionated and supplies the vomeronasal organ, as in reptiles and mammals’. Concerning the reptiles he says that the peripheral termination takes place “in the turtle to a medial diverticulum of the nasal sac, which presumably corresponds to the vomeronasal organ or a part of it”. We now come to the fishes wherein, setting aside an isolated observation by G. Fritscu about one of the Selachii, the nerve was first found by Pixkus in Protopterus. His preliminary communication appeared in 1894 in the “Anatomischer Anzeiger’ and was followed in 1895 by his elaborate treatise “Die Hirnnerven des Protopterus annectens” in the ‘“Morphologische Arbeiten”. Pinkus found that his new nerve originates in the brain, places itself rostrally against the most mesial bundle of the olfactory nerve, takes its course over the nasal mucous membrane and is to be followed to the roof of the anterior nasal opening. The nerve consists of nonmedullated fibres and has in its course a cellular swelling, which is undoubtedly the Ganglion Terminale of later writers, although Pinkus could not convince himself of the ganglionic nature of the cells. Sewrrtzorr (1902) found the nerve in embryos of Ceratodus. He mentions the fact, of importance for the homologisation, that the nerve does not branch in the olfactory mucous membrane and that it terminates in the skin at the external nasal opening. Soon (1904— 1905) Bine and Burcknarpt described the nerve in the adult Cera- todus also. Concerning the Selachii the treatise of Locy, which appeared in the ‘“Anatomischer Anzeiger” after several smaller publications, is well known. In this treatise, which is accompanied by a large number of handsome illustrations, he described the structure and development of the nerve in Acanthias as seen in series of sections, as well as its course as this is to be seen, by means of the dissecting microscope, iy ’ in 20 genera of sharks and rays. At first he held the nerve to be a part of the olfactory nerve, but later on he recognised its homology to the new nerve of Pinkus, and called it the N. Terminalis. . In the Selachii the distance between the nasal sae and the olfactory bulb is small, hence the olfactory nerve is short. Immediately on its appearance out of the nasal sac it is separated into a lateral and a mesial bundle by a small groove into which the distal termination of the N. Terminalis enters. Seoliodon terrae novae alone has something peculiar. Here the two bundles are not only completely separated from each other, but the division also continues on to the bulb, and even to the distal (foremost) end of the tract, which usually is long in the Selachii. After the N. Terminalis of the Selachii has made its appearance out of the hemisphere, it takes its course along the mesial border of the tract, and when it has reached the bulb it forms a ganglion. In some species two ‘ganglia were observed in the course of the nerve. Locy assures that the nerve in its distal ramifications is principally limited to the olfactory mucous membrane, but to my mind he has not proved this. His method was not sufficient to do this, and considering the results of other investigators in other classes of animals this assertion needs corroboration by preparations treated with silver compounds. ; In the Ganoids the N. Terminalis was first found and clearly represented by Puetps Arris (1897, fig. 64) in Amia calva. He could follow it caudally up to the fore-brain. In the larvae he also found its ganglion. In 1910 Brookover described its development in these fishes. His investigation contains many new finds and interesting communications, but his conclusion that the nerve is a branch of the olfactory nerve cannot in my opinion be correct. In his work in 1914 on the nerve in Lepidosteus he also came to this conclusion. In the Zeleostei SHerpon and Brookover (1909) found the nerve in the ecarp and in Amiurus. According to them the roots of the ganglion enter the olfactory bulb to reach the hemisphere, contained in the tract. Here however the question arises whether they. have not made a mistake analogous to that of pr Vries in the embryos of man, as this is not the condition in the Dipnoi, in the Selachii, in the amphibians or in the mammals, nor either in man according to what Brookxover himself (1914) found in the last-named. Concerning the lungfishes I can here demonstrate to you two fine models of the fore-end of the brains, with the nerves arising there- from, of Ceratodus and Protopterus, both constructed by Dr. vAN DER 12 Proceedings Royal Acad. Amsterdam. Vol. XXI. 178 Horst in the Institute for Brain Research of Dr. Ariins Kapprrs, who was so kind as to lend them for this evening. One sees the N. Terminalis arising out of the hemisphere, and running rostrally quite independent of the olfactory lobe, as is also the case in man according to BROOKOVER (c.f. fig. 1). Finally I come to Amphtoeus, on whose cerebral nerves I published a communication in the meeting of the Royal Academy of October the 27th 1894. As is known the trigeminal nerve of the craniata forms a complex of two dorsal segmental nerves, the components being the N. Ophthalmicus profundus (N. Nasociliaris) and the rest of the N. Trigeminus. I found both these components in the two nerves, of which the one appears before and the other behind the first well developed myotome (which has morphologically to be considered as the second). Before the homologue of the profound ophthalmicus, however, there is in Amphioxus still another nerve which supplies the utmost point of the snout. On account of this and because it arises from the morphological fore-end of the cerebral ventricle I called it the N. Apicis. At first I thought that the N. Apicis would be aborted in the higher chordata, but shortly before the publication of my article the preliminary communication of Pinkus appeared (Anat. Anz. 1894), in which he reported the discovery of a new nerve in Protopterus, later named the N. Terminalis by Locy. This had to be considered the homologue of the N. Apicis consi- dering its course, ramification and origin, not from the infundibulum as I concluded out of the preliminary communication, but near the Lamina Terminalis as became clear when the more extensive treatise appeared the next year. I must acknowledge that I have later on sometimes doubted whether this homologisation were correct, when I read the investi- gations of Locy in the Selachii, of BROOKOvER and SHELDON in the Ganoids and Teleostei, and of Ernst DE Vries and DöLLKEN in the mammals, because all these writers assert that the peripheral ter- mination of the N. Terminalis is wholly or principally limited to the olfactory mucous membrane (or in mammals to the vomeronasal organ, which is covered by a split-off part of the olfactory mucous membrane). In Amphioxus on the other hand the N. Apicis stands in no relation whatever to the covering of the olfactory groove. After however reading the research of Huser and Guin (1913) this doubt was dispelled. Their illustration (c.f. tig. 2) shows the N. Apicis of Amphioxus in the rabbit —TI could almost say “in optima forma”, even to the J. W. VAN WIJHE: “On the Nervus terminalis from man to Amphioxus’’. Mensch Fis. 1. Shows the lower surface of the foremost part of the brain of man and the intracranial part of the N. terminalis. (According to a figure of BROOKOVER, slightly modified). a ee, Me vomezen wasalis chvonijn. sagittale snee door fiet mearssepteumt combinatie) nn Gen en is WL sekd J Fig. 2. Shows the mesial surface of the right olfactory lobe and of the contiguous part of the hemisphere with the nerves which radiate from this into the septum, after a combination of sagittal sections in the rabbit. (After a fig. of HUBER and GuILD, slightly modified). Proceedings Royal Academy. Vol. XXI. 179 disseminated ganglionic cells, which have already been long known in the N. Apicis. As the N. Apicis is an ordinary cutaneous nerve’), the relation in which the N. Terminalis stands to the olfactory epithelium in some of the higher animals must be of a secondary nature. It is even possible that the terminal ramification of the nerve has become principally limited to the olfactory mucous membrane, as appears to be the case in many fishes. Thus has the N. Terminalis completed its course through science in 20 years (1894—1914) beginning in the lung-fishes, [ may as well add in Amphioxus, and ending in man. It can no longer be doubted that we have here to do with an independent cerebral nerve and not with a bundle of the olfactory nerve. In most or all of the craniata however branches of both nerves run close alongside of each other, and on account of this it is difficult to distinguish their peripheral distribution. From Amphioxus to man the N. Terminalis is provided with disseminated ganglionic cells, which can partly be gathered together to one or more ganglia. On the other hand the olfactory nerve (including its specialised bundle, the N. Vomeronasalis of the Amniota) is distinguished by the complete absence of ganglionic cells. At the end of this summary I want here to express my thanks to Dr. Arténs Kapprrs, who was so kind as to send me for perusal a dozen treatises on the N. Terminalis, nearly all of American investigators, which have become the occasion of this address. POST? SCRIPT UM, Early in March Dr. Karpers sent me for perusal a copy of a new work by Brookover, which he had received a few days earlier: “The Peripheral Distribution of the Nervus Terminalis in an Infant” (Journal of Comp. Neurology Vol. 28 N°. 2). Brookover found the branching of the N. Terminalis in the nasal septum of the child analogous to that in the rabbit, according to Huser and Gurp, only much more strongly developed. In it he could count about 1500 ganglionic cells, not considering the Ganglion 1) It is a well known fact that ganglionic cells are found not only in the first but also in the second cutaneous nerve (N. Ophthalmicus prof.) of Amphioxus. De Quatreraces discovered them here in 1845 already, but held them for mucous cryptes, ‘‘cryptes mucipares’’. Incorrectly it is assumed that peripheral ganglionic cells are not present in the other nerves. | found multitudes of them in the nerves running under the atrial epithelium which covers the intestine and the liver. 12* 180 Terminale. This ganglion was a compound of 6 to 8 ganglia, com- bined by a net of nervous fibres. He mentions nothing about a N. Vomeronasalis, but found a stout nerve without ganglionic cells, which, with a branch of the N. Terminalis, passes through one of the hindmost openings of the Lamina Cribrosa to the nasal septum and anastomoses peripherally with the N. Nasopalatinus. Brookover considers the above-named stout nerve as a sympathetic anastomosis between the Ganglion Sphenopalatinum and the Ganglion Terminale. To my mind this nerve is the N. Vomeronasalis, which has then not been aborted after birth, in man, as was hitherto the general opinion. In case this conjecture is correct, it must arise behind in the olfactory bulb and supply the vomeronasal organ. This organ is generally present in the vertebrates higher than the fishes *), and seems to be a product of adaptation to terrestrial life. It first appears in the amphibians, and has been lost or is present only in the early stages of development in the higher forms which have secondarily become aquatic again (crocodiles, partly also the Chelonia, the Cetacea, and the Pinnipedia). Flying also seems to be unfavourable for the development of the organ (birds and some — not all — of the bats). The organ is usually named after JacoBsoN, who found it inde- pendently in a large number of mammals, and who also discovered the N. Vomeronasalis. His work became known through the report Cuvier made on it’). After the considerable praise which Cuvier bestows on the work, for a part done in his laboratory, one would expect at the end of his report to the “Institut” the advice to have the treatise of JACOBSON, “pensionnaire et chirurgien-major dans les armées de Roi de Danemark”, printed. The end of the report, however, reads as follows: ‘‘Nous croyons que le Mémoire de M. Jacosson mérite l’approbation de la classe [de l’institut] et que cet anatomiste doit être invité a continuer des recherches, qui-ont déjà fourni un résultat aussi curieux”’. This encouragement does not, however, seem to have had the desired result. At least it is not known that JAcoBsoN has published his treatise, enlarged or not. For the rest Cuvier makes a mistake in believing that nobody had observed the organ before Jacopson, and that it is not present 1) Cf. R. WrepersHeim, Vergleichende Anatomie der Wirbeltiere, Jena, 1909. 2) G. Cuvier, Rapport fait à l'Institut, sur un Mémoire de M. Jacopson intitulé: Description anatomique d'un Organe observé dans les Mammifères. Annales du Muséum d'Histoire naturelle, Tome 18, 1811. 181 in man. It has escaped his attention that Rurscr, who is cited by him a propos of the Meatus Nasopalatinus, (l.c. p. 414. He writes: Ruiscn) is the discoverer of the organ, and just in man in whom it is normally present as was corroborated later on. KOLLIKER*) and Herzreip*) found it regularly in children while it is seldom wanting in adults. When this was the case it had probably to be ascribed to former diseases of the nasal septum. The description of Rurscu*), who also gives a clear representation of the orifice of the organ, with a sound brought into it, on the nasal septum of a child, reads as follows: “‘In anteriore et inferiore parte septi juxta palatum in utroque latere foramen apparet, seu osculum cujusdam ductus de cujus usu et existentia nil apud authores legi; inservire muco excernendo existimo’’. Jacosson also, not knowing ReurscrH's work, is inclined to consider the organ as being secretory, although the powerful innervation pleads for a sensory function, but (le. p. 422): “quel agent exterieur pourroit aller se faire percevoir dans un réceptacle si cache, si profond, si peu accessible >” Cuvizr himself still thinks — under reserve — he has to accept a kind of olfactory perception and the later writers do this too. It is usually assumed that the organ serves to smell the food which has already been taken into the mouth; in mammals the odour would then rise up through the Meatus Nasopalatinus. This can however not be the case in the horse or the donkey (nor in the camel or giraffe), because here the Meatus is no longer opened to the buccal cavity, while their Organon Vomeronasale cannot be held to be rudimentary as is the case in man. The secretory function is evident on account of the numerous glands (Körraker, |. e. p. 11) which fill the organ with mucus, which streams out through ciliary motion, but the difficulties against accepting 1) A. Kouurker, Ueber die Jacobsonschen Organe des Menschen. Reprinted from the Festschrift für Rixecker, Leipzig 1877. 2) P. Herzretp, Ueber das Jacobsonsche Organ des Menschen und der Säuge- thiere, Zoologische Jahrbiicher, Abth. für Anat. und Ontogenie der Thiere Bd. 3, 1889. HERZFELD gives a summary of the mammals in which the organ had been found by him and others up to 1889, also in connection with the meatus nasopalatinus. He might have added that Jaconson had also already observed it in the marsupials (kangaroo). Later on it was also found in the Monotremata and Edentata. 3) F. Rurscr, Thesaurus anatomicus Ill, Amstelodami, 1724, p. 26, N°. LXI, 5. Illustration: Tab. IV, fig. 5. KOLLIKER (1877) cited the description, mentioned above, from an edition of 1703 p. 49; hence more than 100 years before CUvIER's report. 182 an olfactory function, already hinted at by JacoBsoN, and which K6LLIKER tries to evade in a peculiar manner, are not to be got out of the way. In this regard an observation of HrrzreLp in connection with the venous sinus, with a strong circular layer of nonstriated muscular fibres, which is found in the wall of the organ of the rat on the inner side of the bony capsule, — cartilaginous in the majority of the mammals — seems to me worthy of attention. He assumes that the air will be sucked into the organ through contraction of the sinus and the lessening of the volume of the wall, inside the rigid capsule, caused by this. If this appears to be the case in other animals also — the oppor- tunity for research will probably present itself in a veterinary college — then a sort of olfactory function would become comprehensible. It would then also become clearer why the Cetacea and Pinnipedia are nearly the only *) mammals in which the search for the organ *) of Ruyscn has been in vain. It is comprehensible that the Cetacea and Pinnipedia have lost the true olfactory organ, adapted to aquatic life in earlier fishlike ancestors, it became adapted to smelling in the air in later ancestors, which lived on land as mammals, When these, in a still later period, again went back to aquatic life, as Protocetacea and Proto- pinnipedia, the true olfactory organ could not undergo this change and became rudimentary or disappeared altogether. If the organ of Ruyscn in terrestrial mammals is always filled with liquid (mucus), and does not need to adapt itself to smelling in the air, then there is not the same reason for its disappearance in the Cetacea and Pinnipedia as there is for the degeneration of the true olfactory organ of the Cetacea. 1) One would expect the Sirenia here also. It is remarkable however that Manatus, according to Srannrus (Lehrbuch 1846, p. 399) possesses an exceptionally well developed Organon Vomeronasale. In some bats and catarrhine apes the organ has disappeared through some cause or other, as in the Cetacea and Pinnipedia. 2) The numerous morphological investigations on this organ have taught us very little about its function. On histological grounds a sort of olfactory function is not to be doubted, (c.f. amongst others M. von Lenuossex, Die Nervenursprünge und Endigungen in Jacobsonschen Organ des Kaninchens. Anat. Anzeiger. 1892). This is about the only result, concerning the function which we can, after about 200 years, add to the words of the discoverer: “Inservire muco excernendo existimo.”’ Microbiology. — “The significance of the tubercle bacteria of the Papilionaceae for the host plant’. By Prof. Briserinck. (Communicated in the meeting of April 26, 1918). As there is no reason to doubt of the accuracy of Hreuurimeer’s ') experiments, it appears certain that the bacteria of the nodules on the roots of the Leguminosae are indispensable for the fixation of atmospheric nitrogen by these plants.*) But I shall prove that the theory, at present generally adopted, according to which this process takes place only within the tubercles, cannot be correct. But previously some remarks on the occurrence of the tubercles and the cultivation of bacteria from them. For some plant species such as serradella (Ornithopus sativus) and the yellow lupine (Lupinus luteus), it cannot be doubted that only the tubercle-bearing specimens grow vigorously in nitrogen-poor soils and consequently, after the theory, fix the atmospheric nitrogen. It is therefore easy on poor heath fields to find languishing, stunted lupine plants, always devoid of nodules, amid the luxuriantly growing tubercle-bearing ones. Never did I find there well-developed lupine or serradella plants quite without them. But the number of tubercles is of no consequence, it evidently suffices if only few come to development. In garden experiments on open sandbeds, without supply of nitrogen, but where inevitably more nitrogen compounds occur than in heath soils, also in peas and beans (Vicia faba), plants with nodules grow better than those devoid of them. In fertile garden soil such as in the laboratory garden at Delft, yellow lupine and serradella do not fully develop, and especially their roots make the impression of sickliness; tubercles do not grow on them, not even when the soil has been abundantly provided 1) H. HervriegeL und H. Witrartn, Untersuchungen über die Stickstoffnahrung der Gramineén und Leguminosen, Zeitschrift für Rübenzuckerindustrie, Beilageheft November 1888. See further the excellent treatise of Hirryer, Bindung von freiem Stickstoff in höheren Pflanzen, in Handbuch der technischen Mykologie, Bd. 3, 1903—1905. 2) For the objective proof that here free atmospheric nitrogen is fixed see, besides HELLRIEGEL (I. ce. p. 191), Senvösine et Laurent, Fixation de l'azote libre par les plantes, Ann. de l'Institut Pasteur, Tome 6, pag. 65, 1892. 184 with the concerned bacteria. Whether the latter die in -the soil or are not attracted by the roots of the plant is not yet clear. Most other leguminous plants, such as clover, Vicia, peas and Vicia faba, bear also in fertile soil many nodules, and it is not easy to find specimens wholly devoid of them, unless the soil has been previously sterilised. On the roots of Genista anglica and Genista pilosa, growing on poor heath fields, I found after long seeking only very few tubercles, although they. and in particular the former, bore many pods with good seeds; the tubercles are, however, never quite absent. When sown in my garden at Delft or brought there as plants, they die after some few years. On the other hand, Genista tinctoria thrives as well at Delft as along the highway of Zutphen to Vorden and at both places bears a small number of nodules. For Robinia pseudo-acacia the favourable influence of B. radicicola only on the young plant, has been stated by Noppe.') As to full- grown specimens on poor heath soil at Gorssel I could after long digging find but few tubercles, while at a small distance, but on somewhat better soil more tubercles occurred, but still so little numerous, that nobody would attribute to them any direct signifi- cance for such a large tree, had not the fixation of nitrogen in the tubercles become an inveterate belief. Sarothamnus vulgaris and Ulex ewropaeus behave in the same way as Robinia. On Phase- olus vulgaris on sandy soils I found but few nodules, and then only on thin rootlets and nearly always enclosed by plant remains; in the pure sand the nodules are very rare. In garden soil at Delft Phaseolus produces no nodules, but it does in a there arranged sandbed; Lupinus luteus and Serradella behave likewise. When comparing the various mentioned plants, all noted in agriculture for their power of ameliorating the soil, as they contain in their dry substance nearly double the quantity of nitrogen found in other plants, for example the grasses, we come to the conclusion that only for lupine and serradella the number and weight of the tubercles is of some significance in regard to the whole weight of the plant. For other species they are of so little volume that even if within them free nitrogen were fixed with great intensity, only an extremely little quantity of fixed nitrogen could be expected, whilst in reality this amount is very considerable. Hence the theory, at present generally accepted, after which the fixation takes place in the 1) Hittner Le. Also Biisaen, Bau und Leben unserer Waldbäume, 2te Aufl., Pag 246, 1917. 185 nodules only, requires reconsideration. Also other experiences make this reconsideration necessary. But previously a few remarks on the isolation of the bacteria from the nodules and from other materials, and on the question of their specificity. A very conyenient medium for isolation was already described in 1888, ') namely pea leaves- or clover-extract-gelatin with 2°/, cane sugar. B. radicicola grows thereon in soft, white, non-liquefying colonies, while B. ornithopodis from Ornithopus perpusillus, O. sativus or Lupinus luteus, when isolated in the autumn or in March, liquefy somewhat, as does B. herbicola. *) As a solid medium, poor in nitrogen compounds, I recommend a plate of: Tapwater 100, agar 2, cane sugar 1, starch 1, bipotassium- phosphate 0,05, in which, because of the albuminous matter of the agar, enough fixed nitrogen is present to cause a distinct growth of B. radicicola, but the colonies remain small. Later a little saltpetre or ammoniumsulphate may be added locally, which makes the tubercle bacteria like the other saprophytes thrive well, showing that they do not assimilate the free atmospheric nitrogen. If on such a plate eventually germs of Azotobacter, which is able to assimilate free atmospheric nitrogen, are present, these will grow quite well if no nitrogen compounds are added. Such nitrogen-poor plates are also useful to recognise the spore-bearing soil bacteria, which almost constantly appear at the isolation of B. radicicola. I only call tubercle bacteria those species which develop mutually identie colonies by thousands or hundreds of thousands from the exter- nally well-sterilised and cautiously crushed nodules. These bacteria derive for the greater part from within the cells. I consider the deviating and less numerous colonies obtained at the culture experiments as the product of germs accidentally present in the intercellular cavities of the rind of the nodules. °) That the full-grown bacteroids cannot develop on the plates is well-known; hence bacteria may be expected from the tubercles only in the beginning of their development. It is an important and until now not yet sufficiently investigated circumstance that from the tubercles of the same plant not always the same bacteria are obtained. So I found for Ornithopus perpusillus 1) Botan. Zeitung. 1888 Pag. 764. 2) Occasionally a great number of colonies of B. herbicola are obtained from the tubercles; whoever is unacquainted with this species may make mistakes in the isolation of B. radicicola. But even with this knowledge the isolation of serradella- and lupine-bacteria is difficult. Good descriptions of these forms do not -exist. $) Besides B. radicicola B. herbicola can also occur within the living cells. 186 the bacteria I had isolated in March different from those grown in October, whilst the tubercles came from plants growing side by side and being in the same state of development. With the yellow lupine and serradella I had similar results. In most other cases, however, for example with Pisum, Lathyrus, Vicia, and Trifolium, the similarity of the various mutually independently isolated stocks is so complete and the image of B. radicicola can so distinctly be recognised, that the above observation requires nearer confirmation. But we cannot now enter upon this point. When trying to isolate B. radicicola from materials other than the nodules, for example from the soil and from the dying surface cell-lavers of the root, it proves very difficult to recognise this species amid the numerous other saprophytes, especially when the number of the germs of the different species is to be determined quantitatively. B. jluorescens liquefaciens causes much trouble by the liquefying of the gelatin plates, and yet it is necessary to use these plates as on them the colonies of all the species lie free froin one another, while on agar they are overgrown and rendered unrecognisable by 5. fluorescens, which extends strongly sideways. Concerning the question if only one or more species of tubercle bacteria exist the following. Already in 1892 experiments thereabout were made by the late HELLRIEGEL *) in the experimental station at Bernburg with pure cul- tures of the bacteria made by myself at Delft. Of his results HeLr- RIKGEL sent me two reports. In the first, dated 24 July 1892, he gives as “Augenblickliches Hauptresultat: ‘Es gelingt mit den Reinkulturen von B. radicicola var. Pisi oder von Vicia faba, die Erbsen und Bohnen, und mit denen des Bac. radic. var. Lupin. oder Ornithopodis Lupinen und Serradella erfolgreich zu infiziren und zum Wachstum resp. der Assimilation des freien Stickstoffs zu bringen, und das ist was unsere anfängliche Bebauptung bestätigt”. Already earlier HerLLRIEGEL had arrived at the conclusion that the bacteria of Lupinus and Ornithopus belong to a species different from that of Pisum and Vicia, which was also my own opinion. In later years many interesting experiments were made in this direction, especially by Hurner. Yet the evidence is unsatisfac- tory as it proved hitherto impossible in the sand cultures 7”) to bring Leguminosae to complete development by infection with 1) He died 24 September 1895 of a stomach disease and was already suffering when I visited him at Bernburg in 1892. 2) It is a well-known fact that the Papilionaceae, when cultivated in liquids, do not fix the atmospheric nitrogen indifferently whether they produce tubercles or not. 187 B. radicicola only and with exclusion of all other microbes. Such cultures are always at the end of the vegetation period rich in various other species, in particular in ZB. fluorescens liquefaciens and the nitrogen-tixing spore-forming Granulobacter (Clostridium) pasteurianem and Helobacter cellulosae. This observation holds good as well for the first experiments made by myself as for those of others, and this should never be lost sight of when reading the descriptions of the infection experiments with the so-called “pure cultures”. It had not escaped HEL LRIEGEL’s attention, and we see it in all the photographs of his above mentioned treatise at the film of the glass vessels, wherein he cultivated his plants (in bright daylight), which film consisted of Chlorophyceae and various other species of microbes, but he thought it of no consequence (l. ¢. p. 169). For myself I have observed in nitrogen-free sand, besides tle mentioned species, Chlorella and Cystococcus and sometimes also Palmella cruenta and many Cyanophyceae. Many of my later efforts to bring clover plants to complete growth on agar with nutrient salts and B. radicicola in large cotton-plugged ERLENMeYER-flasks, failed as the plants ceased to grow before they blossomed, although the nodules developed very well. The tubercle bacteria do not fix the atmospheric nitrogen when cultivated in nutrient media. I will now call attention to my chief subject namely the want of power of the tubercle bacteria to fix the free atmospheric nitrogen. They do this neither when cultivated out of the plant nor within the nodules. Regarding the first point the experiment is very simple. We have but to erush the nodules and bring the thus obtained material into culture soils used for the ordinary experiments to fix free nitrogen and then cultivate at 20° to 30° C.; or we use the pure cultures for infection of the same media. A convenient medium is: Tapwater 100, Glucose 2, Dikaliumphosphate 0,05, lime 2, fresh garden soil 2. This liquid, to which the garden soil is added as a catalyst, must previously be sterilised to kill the germs of Azotobacter, Gra- nulobacter and Helobacter; notwithstanding the sterilisation, the soil preserves its catalytic power very little impaired. The spores of the nitrogen-fixing Helobacter and Granulobacter often adhere to the nodules and, when present, fermentation phenomena show that the experiments cannot be relied upon, B. radicicola not causing fermentation. Commonly, however, these fermenting and nitrogen- 188 fixing microbes can be removed by thoroughly washing of the nodules with aleohol and water. In the course of many years [ have experi- mented in this way with numerous species of tubercle bacteria, and with many modifications in the nutrient media as well in the temperature as in the source of carbon. Moreover I have, as said, tried to grow pure cultures of the bacteria themselves in the liquid culture medium as also on solid culture soils of various compositions, and at first | thought I had observed a rather considerable increase of these organisms. This increase, however, proved to be really very slight, so slight that gain of atmospheric nitrogen is not proved, whilst the obvious augmentation of dry weight of the sown bac- teria derives from the formation of thick slime walls, that is of ni- trogen-free, cellulose-like substances. around the hardly augmented original protoplasmic material. *) Only when cultivating the microbes in plant extracts with cane sugar, wherein nitrogen compounds are evidently present, I could observe a very slight and by no means convincing increase of the total nitrogen rate of the liquid in consequence of the growth of B. radicicola. But when performing these experiments [ was not yet acquainted with the circumstance that laboratory air contains sufficient carbon and nitrogen compounds to be made percept- ible by the growth of microbes which can feed on them. This was afterwards demonstrated by Ir. A. vaN DerpeN and myself in our investigation on Bacillus (Actinobacillus) oligocarbophilus. *) There exists moreover an aérobic spore-producing bacterium®), hard to kill by sterilisation of the nutrient Jiquids, which fixes free nitrogen ; at that time it was still quite unknown to me and even now it is very imperfectly understood. It may have been present at my experiments likewise as at those of other investigators who think they have observed fixation of free nitrogen out of the plant in the pure cul- tures of B. radicicola. With sufficient precautions the results of such experiments are however always the same: The bacteria of the nodules do in no way fix the free atmospheric nitrogen. When the experiments are performed, not with 1) The slime formation is of importance for the explanation of the “slime threads” (erroneously called “infection threads”) within the nodules. See “Die Natur der Fiiden der Papilionaceénknéllchen.” Centralbl. für Bakteriologie. Bd. 15, pag. 928, 1894. 2) Ueber eine farblose Bakterie deren Kohlenstoffnahrung aus der atmosferischen Luft herriihrt. Centralbl. f. Bakteriologie 2te Abt. Bd. 10, pag. 33, 1903. 3) Bacillus danicus, T. Westermann and F. Léunis, Ceniralbl. f. Bakteriologie, 2te Abt. Bd. 22, pag. 250, 1909). 189 nutrient liquids, but with a solid medium, the results are quite the same : fixation of nitrogen does not take place then either. Stress must be laid on the latter fact as it seems impossible to fix free nitrogen by the Papilionaceae when cultivated in liquid media even under the best circumstances and whether tubercles are produced or not. So it seems probable that for this process a direct contact with the air is necessary, which cannot be realised in the liquid culture media, but very well in solid ones. Further it must be observed that the plate cultures of.some of the nodule organisms, ') for example the forms from Piswm, Vicia, and Trifolium, on glucose-agar-potassiumphosphate plates, in absence of purposely added nitrogen compounds, at superficial view make the impression of being quite able to develop, but here too, it is only the formation of much wall substance, as already described above, and not of nitrogen-rich protoplasm, which explains the voluminosity of the colonies.*) With other slime-producing bacteria, as B. radiobacter and Aerobacter viscosum, of which it is quite certain that they cannot live on the atmospheric nitrogen, extensive colonies may likewise be grown on the said nitrogen-poor medium with fit carbon food. By a better nitrogen nutrition such colonies may even be greatly reduced in volume, the wall substance then serving as food under a strong increase of the bacterial protoplasm, which gives rise to very interesting experiments. It is only when being acquainted with these facts by personal observation that we ean understand how in the literature so many statements can occur on the nitrogen fixation by the nodule bacteria, which does not take place. Within the nodules the atmospheric nitrogen is neither fixed. The preceding gives rise to the question, whether the protoplasm of the host plant might be the catalyst that enables the invading bacteria, in their bacteroidal state, to fix the free nitrogen. However improbable this hypothesis may appear, being in contradiction with the laws of heredity, still it deserves attention because the rate of 1) The wonderful “experiments” of Mazé (Annales de l'Institut Pasteur T. 11, pag. 44, 1897, T. 12, pag. 1 and pag, 128, 1898), who asserts that on broth gelatin plates at the same time ammoniumcarbonate is produced and fixation of free nitrogen by B. radicicola takes place, need not be considered, although they are taken up uneritisised in the handbooks of Plantphysiology. 2%) Likewise for the ordinary saprophytic bacteria the want of nitrogen compounds varies very much: the large-celled Bacillus megatherium requires very little, the small celled Bacterium fluorescens very much. 190 albuminous matter in the nodules is so very high. I myself found about 4°/, nitrogen, which is about 25°/, albumen in the dry matter of pease-nodules. Others found 5 to 6°/, nitrogen. It is noteworthy that the bacterial colonies on agar plates, grown out of the plant, contain but 1 to 2°/, nitrogen of the dry weight, which consists for the greater part of carbohydrates. So it is certain that the bacterial body is very much modified by its entrance into the plant cell as well morphologically as physiologically. Therefore it was tried gazometrically to state nitrogen absorption in the tubercles. If the hypothesis is founded it must be possible, with a great quantity of tubercles in a closed space and under favourable physiological conditions, easily to observe that absorption. For the number of tubercles, for example of the woody papilionaceae, being as said very small, while yet these plants are noted in agriculture for their considerable nitrogen-fixing power, the action of the tubercles must necessarily be very intense. To test the hypothesis we acted as follows. ') First small, later larger quantities of lupine and serradella tubercles were placed in wide glass tubes which could readily be connected with the gas burettes, then put in thermostats at about 25° C. The tubercles respiring vigorously we had to keep account with a rapid assimil- ation and supply of the oxygen. Further it was only necessary to determine the quantity of nitrogen still present after deduction of the carbonic acid and the oxygen. The only difficulty we met “with was that the nodules, which by their abundant content of albuminous matter are an excellent food for bacteria, when they touch each other and get moist, easily give rise to fermentations in particular by Bacterium aérogenes. Hereby hydrogen and much carbonie acid are produced, so that it is then necessary also to determine the hydrogen. But this fermentation may be prevented by introducing the material very loosely into the burette, so that there are but few points of contact between the nodules, and the air can freely pass between. Under such conditions there is no danger that free nitrogen will be formed; this only occurring through the action of the denitrifying bacteria on nitrates, which substance is in the nodules completely absent. Of the tubercles of yellow lupine we used in our experiments quantities of 100 grs. 500 grs, and later even of 1 kil. In some 1) In some of these experiments | was assisted by Ir. D. C. J. MINKMAN, formerly assistant to the Laboratory for Microbiology of the Technical High School at Delft. 191 experiments we had the root tubercles cut off, in others the roots with the tubercles were left united with large pieces of the stem, so that eventually formed nitrogen compounds might be able to flow into the stem. All our estimations, however, showed that not in a single case the slightest fixation of nitrogen by the tubercles was observable. As at first we doubted of the accuracy of our results obtained with relatively little material, we afterwards used the just mentioned larger quantities, but this did not make any difference either. Besides the two said species we still examined several times 10 to 20 ers. of the nodules of Vicia faba, and once about 15 grs. nodules of Robinia pseudo-acacia, but other results were not obtained. As our researches did not last longer than 12 to 20 days it might be objected that we have not sufficiently imitated the conditions of the plants in the field. Further, that in these experiments the growth of the tubercles, together with that of the whole plant, was excluded. Although these objections have not been refuted in the preceding, it is still highly improbable that nitrogen fixation would be associated with the growth of the tubercles and not with the augmentation of the bacteria out of the plant. Principal, however, is the fact that if within the nodules nitrogen fixation were to take place, which might have escaped our attention, the concerned quantity must certainly be extremely small. When we now consider how difficult it is to collect a few grams of tubercles for example of Robinia, it is clear „that if this material is to be of any significance for such a great tree, its nitrogen-fixing power must be enormous. The experiments, however, show that the tubercles are wholly inactive or nearly so, hence there can be no question of attaching to them any importance concerning the nitrogen nutrition, whilst yet nitrogen fixation by this tree is as certain as for lupine and serradella and even on a much larger scale. So the nitrogen nutrition of the Papilionaceae can only be indirectly connected with the bacteria of the nodules. In my opinion this relation can only exist in the herbaceous species and in the germ plants of the shrubs and trees of that plant order, but in full-grown specimens of the woody species such as Robinia pseudo-acacia the presence or the absence of the nodules is wholly indifferent. Likewise on tbe roots of shrubs, such as Sarothamnus vulgaris, Spartium scoparium, Genista anglica, and Gentsta pilosa in full-grown condition, the number of tubercles is so small, their volume so insignificant to that of the whole plant, that even if they were able to assimilate some free nitrogen their slight activity could not possibly explain the rich nitrogen store of the whole plant. Hence, the at present generally accepted explanation of the peculiar 192 behaviour of the Papilionaceae cannot be correct. New researches, especially with Phaseolus, are desirable. From the preceding follows: For various Papilionaceae, excelling by their abundance of nitrogen compounds, even when cultivated in media without such compounds, the number and volume of the tubercles is so small, that if only „within them the fixation of free nitrogen should take place, the intensity of the process in these tubercles must necessarily be very great. We have not, however, succeeded gazometrically in observing the process in the tubercles at all. Neither do the tubercle bacteria fix the atmospheric nitrogen when cultivated out of the plant in nutrient liquids or in plate cultures, nor enclosed in solid media. The contradictory statements in the hand books of Plantphysiology are erroneous. Physics. — “On the rotational oscillations of a cylinder in an infinite incompressible liquid”. By D. Costrr. (Communicated by Prof. J. P. KurNen). (Communicated in the meeting of May 25, 1918) The method to be followed in the discussion of the problem will be in the main the same as that used by Prof. VerSCHarFFELT in the analogous case of the sphere’). We consider the rotational swings about its axis of an infinitely long cylinder which executes a forced vibration. Our object will be to ascertain the motion in the liquid which will establish itself after an infinite time (in practice after a relatively short time’)) in order to compute the frictional moment of forces exerted on the cylinder by the liquid. For the sake of simplicity the calculations will be referred to a height of 1 em. The motion of the cylinder may be represented by « =a cos pt where a is the angle of rotation. An obvious assumption to be made is that the liquid will be set in motion in coaxial cylindrical shells each of which will execute its oscillations as a whole. On this assumption it is not difficult to establish the differential equation for the motion of the liquid. Let o be the density of the liquid. u the viscosity of the liquid. w the angular velocity of a cylindrical shell. r the radius of the shell. The frictional force per unit surface of one of the shells will dw then be #= ru an and the frictional couple on a cylindrical surface jk w of radius r: 27 1° yu ae = Taking a shell of thickness dr its equation of motion will be dw 22 r* dr i So dw : — _ aw rn —} dr, dt Or f 0 r which reduces to 00m dw rt 3 dw uot, Or? r Or 1) Comp. Proceedings 18 p. 840. Sept. 1915. Comm. Leiden 1485. *) Comp. Comm. 1485. pag. 22 footnote. 13 Proceedings Royal Acad. Amsterdam. Vol. X XI. 194 It is important to note that equation (1) may also be deduced from the general equation of hydrodynamics without its being necessary to neglect the second power of the velocities, as is the case in many problems of that kind. For an infinitely long time of vibration i. e. for uniform rotation (1) simplifies to Zw 3 dw dr? r dr (2) Cc . . . The solution of (2) is o=—-+c,, c, and ec, being integration- Ts 8 constants. If the solid cylinder (radius /) rotates with uniform speed Ke) “= @ in an infinite liquid, the result will be w = —_, giving for the pe frictional couple as is well known the expression =A RO ron (eee. aten In order to arrive at a possible solution of (1) we have to make our assumption regarding the motion of the liquid a little more definite by assuming that the angular displacement of each shell is represented by BHP) c08 (Pl — @ (r):)- 7 aan 3, fs, SN We may also consider (3) as the real part of the complex function uel’, where w is a function of 7 the module of which gives the amplitude of the oscillation and the argument the phase-shift (7). Ou Remembering that w = = equation (1) may be reduced to t d'u 3 du io pu AU en Den dr u “) Equation (4) is closely related to the differential equation of the cylindrical functions. Indeed by the substitution y= zv Bussn1’s d*y 1 dy l ; equation of the 1s* order = REE +(1——]y=0, changes to LZ zZaz 2 dv 3 dv tente 0=> 0. = dz? z dz It follows that the general solution of equation (4) is 1 u = {AF (of) PBN 2. ks SG if ep where c= Ea Ey Aand B being complex integration-constants. u 195 J, is the cylindrical function of the 1st kind and 1s order, N, that of the 2°¢ kind and 15 order '). As regards c an agreement must be come to. We shall choose the in root with the negative imaginary partie.c=ke +, where k= |c| = eee u As a first boundary-condition we have Limra,=0®). As this r= 00 relation must hold for all values of ¢, it follows that lm ru = 0. ro The cylindrical functions with complex argument all become infinite at infinity with the exception of the so-called functions of the 3rd kind or Hanket’s functions H,D and H,@. Of these H,) disappears at infinity in the positive imaginary half-plane and on the contrary becomes infinite in the negative half, whereas the opposite is true for H,. By our choice of c in the negative imaginary half we are led to the function H,®. For the integration-constants in equation (5) this gives the relation 4 — —7A%), so that (5) becomes A u == — H,") (er) ps PONS) eee ee, a (6) r For the determination of A we have to use the 2rd boundary- condition ap == acospt, R being the radius of the cylinder. We therefore assume that there is no slipping along the. wall. ak Hende. A St HCR) so that ak A?) (er) H2(cR) r The symbol R is intended to indicate, that the real part has to be taken of the function which stands after it. If we had chosen for c the root with the positive imaginary part, we should have had to utilize the function MD. It is quite easy to verify that this would not have made any essential change in tbe solution (7). . RE ES IL nd 1) Comp. Jaunke u. Empe. Funktionentafeln pp. 90 and 93. Nriersen. Cylinderfunktionen. Instead of NV Nriersen uses the symbol Y. 4) Prof. Verscuarrett puts Lim xr = 0, which in my opinion is not quite correct, r= @ as the linear velocity has to disappear at an infinite distance, Comm. 1480 p. 22. 5) Between J, N, and H a linear relation holds. Comp. J. u E. p. 95. | 13% 196 For large values of «x (real, positive) H,>(a Wi) approaches asymptotically to Ean tf & x e V2 ( aay ) EE e V2 : u ine therefore for (£ F) sufficiently large: kr ak e V2 ’ ka a F Ee . —- ——_——— cos | pt — —— - — : 4 VAE KT tam ©) where y = argument /7,')(c R). From (8) it appears that damped waves are propagated from the cylinder to infinity, the velocity of propagation being Asp: 2: 2 2 jl wag A ts ADN Pp k op The frictional moment on the wall of the vibrating cylinder is Ou Ow x ; Oar 2a uk*| —| where w ——. First we determine | — from (7) 0 R dt Or R â dar — Sint | ac fie Ee eipt REE Or IR ien le IT, ') (cR) For the reduction of the 2"¢ part on the right hand side of (9) we make use of the well-known recursion-formula of the cylindrical functions: dH ,\*) (z) » ane ae = Herz = ia Vedi (z) By its application (9) obtains the form 0a, Eire 2a ee _ Af? (cR) int 10 Fras Pas Rp KCT | | NTR giving for the frictional couple Ka 2 agu R de = — 4auR?w+R 2au KR ac Hie Nee ept} (11) Or |R ORE Hf, '?) (ck) For an infinite time of swing, i.e. » =O, but with a rotational velocity differing from 0, |¢,= pe becomes (). In that case the u | 197 second term on the a of (11) disappears on two grounds: 7, (2) (cR (1) becausec = 0 (2) Lim —— of PE =O; only ‘the first term then oardiele ic (cl remains, which agrees an oR Moreover HGR) |.» — — 2. „1m Fr Fn iks A (2) (ch) It gts is Bia the accompanying graphs ’*) of the module and (ek) aoe argument bre that this limiting value i is practically reached at | 2x V2 esb Ef. 8')| \cR| =k. R=10 (12) The condition |ch|210 means, that the radius of the cylinder must be about equal to or larger than the wave-length. If A is small compared with À the second part of the frictional couple is negligible. For |cR|210 the 2°¢ term on the right-hand side of (10) becomes i (n+) Ze — act ePt=—ake ke 455 since ¢ = & e ‘) Hence equation (11) now becomes: d 9 Km Aaron takt (aon (+7) ) . (58) where s d t) == — (a cos pt). w aA cos p The frictional couple thus divides into two parts, one which does not contain the density of the liquid and another, in which it occurs and which therefore refers to the emission of waves. In the transition to the limit of uniform rotation the first part only remains. In the discussion of the 2°¢ part of the frictional moment the Gx, ; quantity £= ae is an important factor. If we take a time of u oscillation of 2. seconds, so that p= 1, we have k= ga u This gives the following values for 4. 1) Comp. J. u. E. |. c. 2) Tables for Hy(1) and Ho? will be found J. u. E. p. 139, 140. 199 Water 16° 0.011 9.5 Atm. air 0° 0.0013 0,000171 2.8 Air 0.01 atm. ') 0.28 Air 0.001 atm. I) | 0.09 Hydrogen 1 atm. 0° | 0,0000898 | 0.000085 1 From this table it appears that, except for dilute gases, A has to be relatively small in order that the 2nd part may be neglected with respect to the first. For instance for atmospheric air with Rk = 0.5 c.m. AR = 1.4 and ROIS) of the frictional couple is still 56 °/, of that of the first (see equation (11)), every thing calculated for a time of oscillation of 2 a seconds. There is a further special limiting case of equation (13), which is of some interest. Let A become infinite, and let a at the same time disappear, in such a manner that Ra converges to a finite limit 6. We thus approach the one-dimensional problem of the oscillation of an unlimited flat plate in its own plane in an infinitely extended liquid. The frictional force per unit of surface is found from (13) to be pe ee ‘+ F)) 14 ae se ae ere ee NE) a formula which is well-known from hydrodynamics’). A term analogous to —4 au R* w does not occur in the one-dimensional problem, the reason evidently being that with a uniform translation of the plate a condition of equilibrium does not arise, until the whole liquid away to infinity proceeds with the velocity of the plate. Finally it is of importance to ascertain for what frequency the amplitude of the forced vibration becomes a maximnm, in other words to what frequency the system cylinder-liquid resounds, if the cylinder is urged back to the position of equilibrium by a quasi- elastic force. = 0.80, so that the amplitude of the 2rd term 1) At these pressures jy has not become much smaller. Comp. Kunpr u. War. BURG. Pogg. Ann. 1875 Band CLV. 3) Comp. Lams. Hydrodynamics, 34 edition 1905, p. 559. 200 The differential equation for the forced oscillation in complex notation is as follows: 0 +L + Mam Ee. Sine: Leases eee Here in our case L is a complex quantity 1 = L'+ 7L", where L'=(4ruR? + Y2auk R*) ES Vera ee: If we only concern ourselves with the particular solution of (15) which gives the forced oscillation, we can also write (15) in the form : (4 + >) as 4- mee + Mas Berts ods «sy i(l6) p) de dt We see therefore that in consequence of the motion of the liquid an apparent increase of the moment of inertia arises. Putting 6+—=—80 Ve the particular solution of (16) becomes: E = OPE) V (M—6'p?)? + L" p? in which the phase-angle p is determined by the constants of the differential equation. Resonance occurs for J — 6' p? =0 or a Op Lip MSO iy ek SS er = Now L" is proportional to 4 and k= PAS so that we may {4 conveniently write £" — Np}, N being a constant. (17) is now replaced by Oper Net Me) see This equation which is bi-quadratic in Vp determines the frequencies to which the system resounds. On closer examination there appears to be but one resonance-frequency. Naturally we are only concerned with the real roots p of equation (18). There are found to be two of such, one for which Wp is positive, and another for which Wp is negative. Now it follows from our calculation that we have assumed Vp, which occurs in & to be essentially positive. For if we substitute a negative value for V/p in our equations, we obtain a system of waves which moves from infinity towards the cylinder. 201 But the amplitude of this system is infinite at infinity, so that our first boundary-condition would not be satisfied. We may also choose our boundary-conditions differently. We may for instance imagine the liquid limited on the outside by a second cylinder co-axial with the first and at rest. It is then advisable to write the general solution of equation (4) in the following form 1 w= —{C Aer) + DA @(er)} . . . . . (19) Hie At a sufficient distance from the axis of the cylinders two systems of waves then arise, one of which is propagated outwards and the other inwards. At the surface of the exterior cylinder we obtain reflection with reversal of phase, so that the liquid there is at rest. For the determination of the integration-constants (and D we obtain comparatively complicated relations which may be omitted here as they do not yield anything of further interest. The problem of the free oscillation does not now give any further special difficulties. We must now seek a solution of equation (1) of the form Cm Gy On LCOS (ek — p (8), which for r= A becomes ap —= ae ktcosk"t. Again we may write a=uet, where n = — kl + th", The same method of solution may now be followed. Instead or (7) we obtain: he fi Eaten) ie (20) HO(ER) +r | Tr 2 = R . . - where c = Las a if for c' the root with the negative imaginary oi { part is chosen. Hence da, 2a FT, '?) (c'R) = — — e+ ad ERE. As OP NEN) Beute ok HO (CL) H,?) (c'R) 10 den lc’ R| =o A) (c k) da, 2a ne — ——_gnt_q EN F7) dr |R R u if for Uae we take the root with the positive real term. u Theretore: The frictional moment now becomes: 202 da no | - 2au ale teer el IES The differential equation for me free vibration is: Padi sir: giving for the natural ene of the system the equation Kn +in+mM=0. ape = + Ma = 0 The quantity L here contains Vn. If we put L= P+ QVn, where P= 4 pR' and Q = 22 uRR va 8. ul (24) assumes the form: Kvit+(P4+ Qynn+M=0 (25) Equation (25) is bi-quadratie in z—Vn. On further examination it is found to have 2 complex roots z in the right hand portion of the complex plane and 2 in the left portion, only the former of which we can use (comp. equation (22)); hence the system has but one natural frequency. Further 2? = 7 is found to contain a negative real term, as indeed could not be expected otherwise. Chemistry. — “Investigations on Pasrrur’s Principle concerning the Relation between Molecular and Crystallonomical Dissym- metry: V. Optically active complea-salts of Iridium- Triovalic Acid’. By Prof. F. M. Jareur. (Communicated in the meeting of June 29, 1918). § 1. A short time ago I published') some data about the properties of racemic Potassiwm-lridium-Ouwalate: SK, Ir (Cy Od} + + 43 H,O0*), and on that occasion I announced experiments under- taken with the aim of splitting this compound into its optical anti- podes. It was our purpose to gain in this way the necessary infor- mation to enable us to indicate the correct configuration in space of these optically active complex ions, in comparing it with that attributed to the corresponding rhodium-derivatives, in consequence of the arguments brought forward on that occasion. At the same time I hoped to investigate in this way, what influence the substitution of the central rhvdiwm-atom in these complex ions by the homologous zridiwm-atom appeared to have upon the magnitude and the specifie character of the optical rotation and its remarkable dispersion. It may be considered of importance, of course, to know the relation existing between the two functions just mentioned, especially in connection with our former studies on the analogously constituted complex salts of ‘cobalt and rhodium combined with three molecules of ethylenediamine, where the problem arose as to the true configurative relations between the salts of these homologous metals of the eighth group of the periodic system, when rotating the plane of polarisation in the same direction’). In the following the fission-experiments mentioned and the results obtained by them are recorded in details. Thus for the first time the possibility of a “partial” asymmetry‘) has been proved, in the case of iridium as the central atom ; the series of the metals showing 1) F. M. Jarecer, Proceed. R. Acad. Amsterdam. 20. 263. (1917). 2) C. GrALDINr, Rend. Acad. d. Linc., Roma, (5a), 16. Il. 551. (1907); Proceed. Acad. Amsterdam, loco cit. p. 278. 5) F. M. JArcer, Proceed. R. Acad. Amsterdam, 17. 49. (1915); 20. 244. (1917); conf. Zeits. f. Kryst. u. Miner. 55. 209. (1915). 4) F. M. Janaer, Lectures on the Principle of Symmetry, Amsierdam, (191%), p. 285. 204 this phenomenon being herewith extended to chromium, tron, cobalt, platinum, rhodium, and iridium. With respect to. the dispersion of some of these salts, we hope yet to furnish some new data in the near future. § 2. The required racemic Potassium-Lridium- Oxalate: K,{Ir(C,O,} + +4! H,O was obtained in the following way. A 3°/, solution of pure, hydrated ridiumchlortde of commerce (HerArus) is treated by a solution of potasstumhydroaide in excess. A dirty brownish precipitate is formed, which dissolves in the excess of KOH to form potasstum-iridiate. The alkaline solution is heated, and then some perhydrol (30°/, H,O,) added: the colour changes to dark blue, and the principal part of the iridium precipitates as r(OH),, Another part of it remains in the solution as a colloidal suspension of great stability, not being precipitated or coagulated from it, even after addition of electrolytes. These solutions are therefore better evaporated, and the residue transformed into ammonium-chloro-iridiate to be used afterwards in other experiments. The blue precipitate is washed by decantation with water slightly acidified by means of oaalic acid; the filtrates and washings are also later converted into the mentioned compound. When the removed liquid gets colourless, the decantation may be considered complete, and the precipitate is brought into a round bottom flask, the super- fluous liquid removed after some hours, standing, and a hot, concen- trated solution of ovalic acid added. The contents of the vessel are boiled under a reflux-condenser during 30 or 40 hours; a part of the ovalic acid is thereby decomposed, and the tetravalent dudiwn reduced to trivalent according to the equation: 217r(OH), + C,O,H, = 21r(OH), + 2H,0 + 2C0,, while ¢ridiwm-triowalic acid is then formed from the derivative of the trivalent zridiwm, conforming to the equation: 21r(OH), SCO FTC 0) 6H, 0. The gold-yellow solution finally obtained is filtered, and almost perfectly neutralised by means of KHCO,; it is then concentrated on the waterbath, and the successive fractions of the crystals formed are separately collected. Almost pure potassium-ovalate is first deposited, and afterwards, besides this, the orange crystals of the salt required, which is very soluble. These crystals have to be separated mecha- nically, and they are afterwards recrystallised for purification *). The racemic compound erystallises in pale orange-coloured erystals, 1) Dr. J. Kann has aided most effectively in the preparation of a part of this racemic compound, and in the troublesome working up of the tridiwm-residues. 205 which have already been investigated and described in a former paper’). They are triclinic-pinacoidal, and completely isomorphous with the corresponding racemic rhodtum-salt, so that the direct isomorphous substitution of the metals Rho and Zr, also in their complex salts, has been hereby definitely proved. As we shall see, this proof has now also been given in the case of the optically- active components of such complex salts. §. 3. Fission of the racemic Potassium-lridium- Oxalate (+-4'/,H, 0) into its optically-active components. 27,5 Grams of pure strychnine-nitrate are dissolved in 1300 cem. boiling water; then a solution of 15,5 grams of the racemic salt in 250 ces water of 60° C. is rapidly poured into the boiling solution under perpetual stirring. The flask with the golden yellow liquid is allowed to cool slowly for twelve hours to room-tempera- ture. The deposited, highly yellowish coloured, felty-like crystals are sharply sucked off at the water-pump, washed with some strong aleohol, afterwards with some ligroine, and dried at room- temperature in an air-current. The mother-liquid is evaporated on the waterbath to about ’/, of its original volume; on cooling highly yellow, needle-shaped crystals are again deposited, which are treated in the same way. While the first fraction, however, represents the strychnine-salt of the deatrogyratory component, — the (aevo- gyrate antipode was immediately obtained from the second precipi- tate. On further evaporation of the filtrate, some paler coloured fractions are consecutively obtained, all of which give the laevo- gyratory potassium-salt. The sixth and the seventh fraction finally con- sisted of pure strychnine, accompanied by some of its nztrate, while in the last fractions crystals of the free, racemic potassium-salt together with some of the laevo-salt, and also some potassium- nitrate, appeared. The rotation of this last fraction as a whole, after removing the potassium-nitrate, was negative, amounting only to about '/, of the rotation of the pure laevogyrate salt, so that a considerable amount of racemic salt is evidently admixed. Probably a partial hydrolysis during the repeated evaporations has taken place, so that the free potassiumsalt accumulates in the last fractions. The strychnine-salt of the dertrogyratory component has the formula: {17(C,0,),}3(C,,4,,N,0,), + 34H,0; it appears as pale yellow, very fine needles. For a series of wave-lengths the rotation of this strongly active salt was determined; the solution used contained 0,4763 grams of the hydrated salt in 100 ces. of the liquid. ') F. M. JAEGER, Proceed, R. Acad. Amsterdam, 20, 278, (1917). 206 The following data were found : ROTATION-DISPERSION OF STRYCHNINE-d-IRIDIUM-OXALATE (+ 31/2 H20). Bis ie : Observed Rotation : Molecular Rotation: 5105 | + 1.02 | 4. 20997° JE | 0.86 | 17703 id | 0.65 | 13379 5610 | fag ar | 5800 | 0.27 ed | 6020 | 0.21 ee | ment | 0.17 3499 | | 6530 | 0.155 | Jone na In the same way tation in fig. 1: tbe composition of the corresponding strychnine- salt of the /aevogyrate component appeared to be: {/7(C,0,),§Ci, Aa, N,O,); + 3H,0. This salt too erystallises in needles, somewhat thicker than those of the first. The substance is strongly daevogyra- tory, and ifs dispersion is smaller than in the case of the other salt, as may be seen from the following data, and the graphic represen- ROTATION-DISPERSION OF STRYCHNINE-/-IRIDIUM-OXALATE (+ 3 H,0). te ee Observed Rotation: | Molecular Rotation: 5105 BENE | — 172360 5260 0.28 | 14624 ae 0.24 12535 5610 0.19 9994 5800 0.15 Se | 6020 0.125 wr: | 6260 0.09 4701 6530 0.08 A178 | 207 This solution contained 0,1886 grams of substance in 100 grams of the liquid. It is remarkable that the laevogyrate strychnine-salt has a some- what greater molecular rotation than the corresponding potasstum- salt itself; the dextrogyrate strychnine-salt rotates more strongly than the free pottssium-salt. 00 a ns lang W 5100 5260 5430 5610 5800 6020 6260 6530 Wgslronv. lacks. Fig) MatecularhotulionVispersion of Syychinine-a-Jridiun- F Oxalale C6 KO nf ghana Cha GHD. $ 4. The different fractions were mixed with about six times their weight of finely pulverized potasstum-iodide, and then ground together in a mortar, some cold water being added to the mixture. When all potassium-iodide has just been dissolved, the yellow liquid is sharply sucked off from the white precipitate, this last washed with a very small quantity of cold water, and the yellow filtrate precipitated by the addition of an excess of 95°/, alcohol. A yellow deposit is formed, which is sucked off on the Bucuner-filter, washed with alcohol, and recrystallised from water. On heating on the waterbath the solution does not autoracemise notably. By slow crystallisation at roomtemperature, beautiful trigonally-shaped, orange crystals are formed, which are dried between filterpaper. They can grow in their mother-liquid to considerable size. The optically-active components are extremely soluble, more than the very soluble racemic compound ; on this account the crystallisation of the active components was executed in smaller and somewhat deeper crystallisation-dishes. 208 § 5. The rotation-dispersion of these optically-active zridiuwm-salts was measured in the same way as formerly by means of a great polarimeter of Scumiprt and Haxrnscu, with threefold field, and equipped with a monochromator. The tube was always 20 cm. long, and determinations were made for a whole series of wave-lengths. Even in thin layers of the liquid the spectral region of the transmitted light appeared to be appreciably limited by absorption, also in the case of not very concentrated solutions. Waves larger than 6850 A.U. were never transmitted to a sufficient degree, while even in a solution of 1°/, no exact determinations could be made for wave-lengths smaller than 5300 A.U. To investigate the shape of the dispersion-curve also for shorter wavelengths, it was therefore necessary to use very dilnte solutions, of 0,2°/, and 0,1°/, or less. We have used four such solutions for this purpose, containing respectively one grammolecule of the hydrated salt in 14,57 Liters (a) of the solution, in 57,73 Liters (6), in 228,86 Liters (c), and in 418,7 Liters (d). In the following table the mean values are taken into account, and the molecular rotations calculated from them; in fig. 2 these results are moreover graphically plotted, in their relation to the light used: Wioteculerpulatow Lv Degrees f 4790 4920 5020 5100 51805260 5340 5420 5500 561057005800 5910 6020 £140 6260 6380 6520 6660 6800 Ungstror = Wils. Fig. Molecular Iotedin Dis wsion fte hilly active’ Jolasstumn/= Jon Tables : | The values obtained with the different solutions agreed very well mee 209 together. In the table are therefore quoted those values which approached nearest to the curve of the mean values. The dispersion- curve shows a tendency to get more and more horizontal for wave- lengths beneath 5100 A.U. | ROTATION-DISPERSION OF DEXTROGYRATE POTASSIUM-IRIDIUM-OXALATE | (+ 1 H,0). Wave-length Observed Rotation | Molecular Rotation Wi ae Ui he in Degrees: in Degrees: | 4790 | 4 0-79 (d) | + 163400 | | 4920 0.785 (d) | 16237 | 5020 0.78 (d) EA ie HOL 5100 0.78 (d) | 16134 5180 0.77 (d) | 15927 5260 0.75 (d) | 15514 5340 L228: (€) 14647 5420 4.36 (5) | 12586 | 5510 14.42 (a) | 10508 | 5610 11.94 id. | 8699 | 5700 9.88 id. | 7198 5800 8.17. id. | 5952 | 5910 | 6.86 id. | 4998 6020 | 5.49 id. | 4000 6140 | ATS td. | 3446 6260 | 3.86 id. | 2813 6380 | 3.32 id. ee 2446 | 6520 2.61 id. | 1901 6660 2.33 id. | 1698 | 6800 B: TO 14, 1530 | Although the dispersion of these orange-coloured solutions is extraordinarily strong, the slope of the curve is quite different from that found in the case of the corresponding rhodium-salt. This fact proves the preponderant influence of the special nature of the 14 Proceedings Royal Acad. Amsterdam. Vol. XXL 210 central metal-atom on the specifie light-absorption (colour) of these salts and on the whole character of the rotation-dispersion. The yellow crystals obtained on evaporating the original mother- liquid of the strychnine-salts to */, or '/, of its volume, gave after treatment with potassium-iodide a fraction which appeared to be the pure laevogyrate salt. The following measurements, made with a solution containing one grammolecule of the hydrated salt in 42,97 Liters of the liquid, may make this clear: ROTATION-DISPERSION OF THE LAEVOGYRATE POTASSIUM-IRIDIUM-OXALATE (+ 1H,0). Wave-length Observed Rotation Molecular Rotation in A. U.: | in Degrees: in Degrees: 5340 | — 6.66 — 142870 5430 ale 12289 5520 | 4.89 10506 5610 | 3.86 8293 5700 | 3.26 1004 5800 2210 | 5801 5910 2.28 4898 6020 1.95 4189 6140 1.63 | 3502 6260 | 185 2900 6340 1.07 | 2299 6520 0.88 | 1891 6660 0.74 | 1590 | 6800 0.70 1504 | | | A comparison of the rotations for the same wave-lengths in the Salt: case of the corresponding rhodium-salt with* the here described Molecular Rotation: Atomic Volume of the metal: Ks {Rho (C2 O4)3} +H20. | Magzo = 14200°; Msgo0 = 790°; Msoro = 0°; Meseo = — 1215°. | Kg {Ir (Cy O4)3} + HzO. | Miogo = 16230°; Msgoo == 5952°; Msg70 = 4500°; Mgceo =-+ 1698°. 8 50 8.61 | 211 wridium-salts, teaches us, that the rotation of the zridiwm-salt is always appreciably greater than that of the rhodiuni-salt, although the atomic volume of zridtum differs only slightly from that of rhodium, and even exceeds it by a small amount. If we were able to demonstrate later, that for {/r(Aetne),}/, the rotations are smaller than those of the corresponding rhodium-salt, then we should have proved that the influence of the atomic volume on the magnitude of the rotation, may be in this or in the opposite direction, according to there being either basic or acid substituents attached to the central-atom. $ 6. DEXTROGYRATORY PorasslUM-IRIDIUM-OXALATE : GAO AA Lil, O. Beautiful, rather large, orange-coloured, and very lustrous crystals, which are commonly regularly developed in the shape of flattened, triangular bipyramids. They are well built and geometrically easily determinable, allowing very exact measurements. The deviations of the angular values from those found with the corresponding rhodium- Dextrogyratory Potasium-Iridium-Oxalate. Fig. 3. salt are more appreciable than ordinarily stated in the case of rigorously isomorphous crystals. The symmetry of the two series of crystals is however exactly the same, and their form-analogy is sufficiently great, to consider the optically-active salts of the two series as quite isomorphous, also with respect to the doubtless isomorphy between the racemic salts of the rhodium-, and the zridium-series. The more deviating values of the angles and axial parameters are probably connected with the rather great difference of atomic weight of the metal-atoms. Analysis teaches us, that also these optically-active salts crystallise with only one molecule of water. Trigonal-trapezohedral. a:c=1:0,9520.- (Bravais); a = 100°20’. (Mrrer). Forms observed: P = {1122} [521], as positive trigonal bipyramid, 14* 212 predominant and yielding good reflexes; o = (1011! [100], positive rhombohedron of the first kind, well reflecting, but in most cases only with rather small facets ; a= {1101} [212], a negative rhombo- hedron of the first kind, commonly much broader than 0, somewhat less lustrous; y=={2114} [421], a negative trigonal bipyramid, about as large as 2, but much smaller than P, commonly yielding good mirror-images ; 7 = {1102}, as a narrow obtusion of the edges of the rhombobedron 0, ordinarily absent, but in the other case very lustrous. Sometimes a very narrow and rudimentarily developed prism m = {1120} was observed, truncating the basal edges of P. The crystals are mostly very regularly developed as flat trigonal bipyramids ; but occasionally more or less deformed, table-shaped individuals are met with, showing the same combination of forms. The faces of / are often striated parallel to the edges P:o. (Fig. 3). Also erystals are found, where o is about twice as large as P; in this case the striation on P was observed in all cases. Finally we met with individuals showing only P and o in about equal size, P having its characteristic striation; besides them also y was found occasionally, but very small and subordinate, especially in the case of the dextrogyrate component. Angular Values : Observed : Calculated : P: P"=(1122):(4212) =* 4320: Le Py P= 411222 (1122) = 92 49 92° 49' P:o =(1122):(011)= 21 40 Fi) AS a:o —=(1101):(011)= 43 28 43 26 w:y =(1101):(2111)= 28 .20 2834 joe Odd de OLDE 73 OA 28 Por = de DE dE 60 48 60 “ 503 0:0 =(1011):0111)= 79 30 79 40 o:r =(1011):(1102)= 39 43 39 gO) m:P =(1120):(4122)= 46 30 46 25 No distinct cleavage was observed. Optically uniaxial, without noticeable circular polarisation. The character of the birefringence is negative. The specifie gravity of the crystals at 20° C. was: 2,734; the molecular volume is therefore: 217,77, and the topical parameters are: y:w — 7,0618 : 6.7230, if calculated with respect to hexagonal axes, and y =p = w = 6,1321, with respect to rhombohedral axes. 213 § 7. The crystals of the laevogyrate antipode ordinarily showed only the bipyramid P, which must be considered as a /efthanded bipyramid here, because all phenomena are in agreement in this case with Pasreur’s law, as is proved beyond doubt by the hemihedral symmetry of the crystals. Therefore to this bipyramid must be attributed the symbol {2112} [512]; besides the forms o, w, and y, y having the symbol {1121} [412], appear subordinately here. Because o and we were in most crystals about equally large, and could uot be discerned in any other way, the external habit of these lefthanded crystals was not different from that of dextrogyrate salt, the latter being brought into the same position as the lefthanded by a rotation through 60° round the trigonal axis, with the only difference, that the forms w and o are thereby interchanged. However, if & and o are of unequal size, the occurrence of mirror-images could be seen immediately in the crystals. The zonal relations may be made clear by the subsequent stereographical projections (Fig. 4). Just as in the case of oppositely rotating rhodiwm-salts, a non- superposable hemihedrism accompanies here the contrary power of rotation. Fig. 4. Stereographical Projection of the Crystalforms of d- and l- Potasstum-TIridium-Oxalate (only the top-ends of the crystals). $ 8. The specifie gravities of the formerly investigated rhodium-.- salts, were determined at dy — 2171 for the racemic compound (44 H‚,O), corresponding to a molecular volume of: 260,34; and at die = 2,255 for each of the optically-active salts (+ 1H,Q), which corresponds to a molecular volume of 222,70. From this the topical axes of these salts are calculated at: 214 x: Ww: w = 6,8980: 6,4274: 6,6306, for the racemic rhodium-salt, and y: w = 7,2660: 6,4944, for the optically-active salts, with respect to hexagonal axes, and x= w= w = 6 1856, with respect to rhom- bohedral axes. Comparison with the corresponding parameters of the here studied tridium-salts *): Topical Parameters: racemic K3 { Rho (C, O,4)3}, 4/2 H20. racemic K3 { Ir (C O4)3}, 44% H20. | optically-activeK3} Rho (Cz O4)3}, H20. optically-activeK,\ Ir (Cy O4)3 }, H20. YX? Ws W = 6,8980 : 6,4274 : 6,6306. Y: WW: W = 6,7454 : 6,2626 : 6,5162. %: w = 7,2660 : 6,4944; y' — 6,1856. {: W = 7,0618 : 6,7230; x’ = 6,1321. the substitution of the central /ho-atom in the complex oxalate by the isomorphous /r-atom, produces a diminution of the topical parameter w in the case of the racemic salts, but a slight increase, in the case of the optically active antipodes, although the values for y and x’ are in this case smaller with the correspond- ing Fho-salt. In the same way as in the case of the previously described rhodium-oxalate, all phenomena observed in the fission of potassiwm- tridium-ovalate are really in full agreement with the sense of Pastrur’s law. teaches us, that Laboratory for Inorganic and Physical Chemistry of the University. Groningen, June 1918. ') The specific gravity of racemic Kf Ir (C,04)3 } + 41/2 H,O was at 182 determined at: dy = 2,688; the molecular volume is thus: 243,82. Chemistry. — “J/nvestigatiuns on Pasteur’s Principle concerning the Relation between Molecular and Crystallonomical Dissymmetry : VI. On the Fission of Potassium-Rhodium-Malonate into Its Optically-active Components.’ By Prof. F. M. JarGER and Wurm Tuomas. B. Se. (Communicated in the meeting of June 29, 1818). 1. Some time ago one of us’) described the crystalform of racemic Potassium: Rhodium-Malonate: K,{Rho(C,H,O,),{ + 3H, 0, and hinted at the possibility of separating this salt into its optically-active com- ponents. In the following we are now able to describe the results of the respective experiments, which have led to a positive result, and to give a review of the highly remarkable rotation-dispersion of these new salts. The racemic salt nesessary for these experiments was prepared in the following way. A 3°/,-solution of pure Na, RhoCl, + 9H,O was heated to 40°C, and then precipitated by means of a 10 °/, solution of caustic potash, so much of the base being added, that the liquid showed a feeble alkaline reaction. The precipitate is separated from the excess of potash as well as possible by repeated decan- tation in high cylindrical vessels; it settles down extremely slowly, so that this operation takes much time. Then the precipitate is brought into a round bottom distilling-flask and heated under a reflux-cooler some forty hours with a solution of the calculated amount of potassiwm-bimalonate: KHC,H,O,, and some free malonic acid, until the precipitate no longer diminishes in quantity. The red coloured liquid is then filtered, and concentrated on the water- bath: on slow evaporation at room-temperature there soon appear red flat crystals of the complex malonate, which are once more recrystallised from water for purification. The residue in the flask is again changed into the complex sodtum-rhodium-chloride: Na,RhoCl, +9H,O in the usual way, and afterwards precipitated as described in the above. § 2. After a series of attempts we succeeded in separating this salt, which erystallises in beautiful monoelinie crystals’), into its optically-active components by the aid of its cinchonine-salt. 1) F. M. Jaraer, Proceed. Kon. Acad. Amsterdam, 20. 276. (1917). 8) loco citato, p. 277. 216 For this purpose the potassiumsalt is first converted into the bariumsalt, by adding a strong solution of 3 molecules of barium- chloride to a concentrated solution of 2 molecules of the potassium- salt: a yellow precipitate is formed, which dissolves rather easily in hot water, but which can be almost completely precipitated from its aqueous solution by the addition of 97 °/, aleohol. This barium- salt was now dissolved in water at 50° C, and then a solution of one equivalent cinchonine-sulphate ), also heated to 50° U, was added to it. The solutions need not be too concentrated, because the cinchonine-rhodium-malonate will otherwise partly precipitate, as it is only sparingly soluble. The bariumsulphate formed is carefully sucked off, and by washing with water of 45° C. all the included yellow cinchonine-salt eliminated. On standing for 24 hours in a large crystallising-dish, the liquid begins to deposit beautiful, pale yellow and often in rosettes united needles of the cinchonine- salt of the dwevogyrate component, as will soon be shown. On repeated partial evaporation of the mother-liquid on the waterbath, the successive fractions were separately collected and investigated. The first three fractions appeared to contain the lefthanded com- ponent; the fourth fraction gave the almost pure deztrogyrate antipode, the fifth and sixth fractions the pure dextrogyrate com- ponent immediately. It is a remarkable fact that the cinchonine- l-matonate and the cinchonine-d-malonate ave both deztro-gyratory, notwithstanding the very large rotation of opposite sign of the complex ions present therein. This peculiar behaviour was checked by us by a special control, namely by preparing the free potassium- salts again from the conchonine-salts used in the polarimetric measu- rements. We could easily prove in this way, that the salts thus obtained, really represented the right and left antipodes. From the pure daevogyrate potassiumsalt we once more prepared the corre- sponding cinchonine-salt by means of the bariwm-salt; the rotations determined with this especially prepared salt proved to be positive, and they agreed very well with those formerly found. We have also investigated the influence of the addition of three molecules of cinchonine to a solution of the optically-active potassium-salts, and the rotations found with these solutions were compared with those 1) Originally we tried to reach our purpose by means of the strychnine-salt, as in the case of the rhodium-oxalate. However, these experiments had no result, the potassiumsalt prepared from the carefully fractionated strychnine-salt by potassium-iodide being always optically-inactive. It is difficult to say whether racemisation or partial racemism is the cause of this; but only after several failures we passed to the use of cinchonine. 217 of cinchonine itself: the observed rotations appeared to be practi- cally identical with those of the cinchonine-l-rhodium-malonate, so that evidently the influence of the three molecules of cinchonine far outweighs that of the laevogyrate rhodiwm-malonate-ion itself. The last fractions of the erystallisation-series of the cinchonine-salt finally gave pure cinchonine, a small amount of the dextrogyrate salt and a certain quantity of the racemic salt remaining in the last mother-liquids. Evidently also here the repeated evaporation on the waterbath, just as in the case of the corresponding owalate, seems to cause a partial hydrolysis. Analysis taught us that the cinchonine- d-rhodium-malonate crystallises with 3 H,O; the corresponding cinchonine-l-rhodium-malonate with '/, H,O. This last mentioned salt could not be heated above 100° C, being less stable than the right- handed salt, it is rapidly decomposed with formation of a dirty brown powder. For the rotation-dispersion of the cinchonine-salts we found tbe following values: ] | I. CINCHONINE-d@-RHODIUM-MALONATE (+ 3 H,0). Observed Rotation ‘Melecular. Rotation et in Degrees: in Degrees: | 5105 | EE Gn | + 300109 | 5260 | 1,23 26943 | 5420 | 1,14 | 24972 | 5610 | 1,07 | 23406 | 5800 | 0,99 21686 6020 | 0,94 | 20591 | 6260 | 0,88 | 19277 6520 0,84 | 18400 | | | The solution investigated contained 0,3070 grams of the hydrated salt in 100 grams of the liquid. The results obtained are plotted in the figure 1. it shows us, that both curves are situated above that of pure cinchonine, notwithstanding the fact, that one of them contains the strongly negatively rotating complex rhodium-malonate-ion; of a simple superposition of the ‘optical activities there is therefore no question. The transformation of the cinchonine-salts into the corresponding potassium-salts was carried out in the following way. The pure 218 rN IL. CINCHONINE-/-RHODIUM-MALONATE (+ !/. HO). Wave-length in A.-U.: | Observed Rotation | Molecular Rotation in Degrees: | in Degrees: | lis + 0,65 | + 166470 | 5260 0,61 15622 5420 0,57 14598 | en 0,52 13384 | 9600 0,45 11525 | 6020 0,40 10244 | 6260 0,36 9220 | 6520 0,32 8195 The solution used had 0,2538 grams of the hydrated salt in 100 grams of | the liguid, | Nave -length in 7000 : 5000 5105 5260 5420 5510 5800 602 6260 6520 CE Fig Molec etalon Lisson vrien ipegndne ay 219 cinchonine-salt is ground with about ten times its weight of potassium- iodide, and a small amount of water added to the finely pulverized mass. The mixture is allowed to stand for 24 hours at room- temperature; the yellow liquid is then sucked off as sharply as possible from the precipitate. The reddish yellow filtrate is precipitated by 97° -aleohol, and the pale yellow precipitate of potasstum-rhodium- malonate thus formed recrystallised from a little water. During the evaporations on the waterbath a noticeable racemisation does not occur. It is advisable to add as little water as possible during the transformation of the cinchonine-salt by means of potassium-iodide, as otherwise the precipitation with alcohol is very incomplete. 3. The optically-active components are, like the racemic salt, but in yet higher degree, very soluble; at ordinary temperatures the racemic form is therefore doubtless the stabler phase in comparison with the optically-active components, so that there is no chance to execute a fission by spontaneous crystallisation *). The solutions possess a beautiful orange or bloodred colour. For a series of wave-lengths the rotations were determined in the case of both antipodes; the values obtained agreed completely in both cases with exception of the algebraic sign. The concentrations of the solutions used in these experiments must be varied over wide limits, if measurements are to be made over a greater spectral range, because the absorption of the light in layers of 20 em. is very intensive. In the visible part of the spectrum no distinet absorption-bands occur; but at both ends it is abruptly cut off: a 1,5 °/, solution allows the transmission of waves from 5190 to 6800 A. U.; a 0,75°/, solution the transmission of the whole red, yellow, green, and blue part of the spectrum to 4870 A. U.; a 0,37 °/, solution in the same way to 4420 A. U.; etc. With a 1,48°/, solution these limits were found at: 5020 and 6900 A. U. For the polarimetric determinations we used solutions which contained respectively 1,503 grams (A), 0,511 grams (B), and 0.305 grams (C) of the /aevogyrate anhydrous salt in 100 grams of liquid; in the case of the dertrogyratory antipode we used a solution containing 0,804 °/, of the anhydrous salt. The results of these measurements are reviewed in the following table, and in fig. 2 they are plotted in a diagram.The data have been calculated with respect to the anhydrous salt. 1) Conf.: F. M. Jarcer, The Principle of Symmetry and Its Applications to All Natural Sciences, Amsterdam, (1917), p. 209, 210. ‘NF UL U}8uaj-aAe MA 4130 4870 ‚5020 5105 5180 5260 5340 5420 5515 5610 5700 5800 5910 6020 ‚6140 6260 6380 ‚6520 6660 | 6800 220 ROTATION-DISPERSION OF LAEVO- AND DEXTROGYRATORY POTASSIUM- RHODIUM-MALONATE. | Observed (4) and Molecular (M) rotations of the left salt: Observed (2) and Molecular (M) Rotations of the right salt: a: M: — 1,50 | —2621° 1,53 | 2673 1,59 | 2778 1,61 | 2812 1,63 | 2847 1,55} 2708 1,47 | 2568 1,44 | 2516 141 | 2463 1,38 2410 1,35 | 2358 B. a: | M: — 0,49 | — 25240 0,49 | 2524 0,50 | 2575 0,50 | 2575 0,51 | 2627 0,52 | 2678 0,54 | 2781 0,55 | 2833 0,56 | 2884 0,53 | 2730 0,50 |_ 2575 0,49 | 2524 0,48 | 2472 0,47 | 2420 0,46 | 2369 2317 0,45 | ee + 0,86 0,87 0,88 0,90 0,92 0,93 0,89 0,85 0,82 0,81 0,74 0,78 + 26142 2645 2675 2136 2797 | 2827 | 2106 2584 2493 2462 2402 | 2371 | From fig. 2 the very remarkable shape of the dispersion-curves may be seen, which at a wave-length of about 5800 A. U. show a maximum. For wave-lengths smaller than 5800 A. U. the rotation of the plane of polarisation increases with increasing wave-length, while for those greater than 5800 A. U. it diminishes with increasing 221 wave-lengths, as in ordinary cases. In the neighbourhood of 5800 A. U, the absorption-spectrum, however, does not manifest a single line or band. However the occurrence of such an anomalous rotation- Tae 00° ; 4590 4730 4870 5020 S105 5180 5260 5340 5420 5515 5610 5700 58005810 6020 6140 6260 6360 BSZ0 6660 6800 Ugs wom - ats Fig2 2 Molecular Soi Zh Leggersion fdg and laonapyralory dispersion under these circumstances seems to be theoretically expli- cable, if the assumption may be made that at least two kinds of active ions are present *). Besides this anomalous rotation-dispersion, the whole character of which is in sharp contrast to that of the regular one, the absolute activity of these salts appears in general to be appreciably smaller than that of the analogously constituted owalates, unregarded the passing through the zero-point in the case of the oxalate at 5970 A.U., formerly mentioned. The substitution of the oxalic acid-ions: COO’ COO’ | by the three ions of the malonic acid: CH, around the COO’ | COO’ 1) Drupr, Lehrbuch der Optik, (1900), p. 382. 222 central rhodium-atom, is evidently followed by a very radical change of the character of the optical rotation of the molecule, which affects not only the magnitude, but also the algebraic sign of the molecular rotation for a number of corresponding wave-lengths. The special chemical nature of the substituents placed dissy mmetric- ally round the central atom therefore appears to have as much influence on the magnitude of the rotation, as the chemical nature of the central metal-atom itself. § 4. After many attempts we were able to obtain the crystals of the optically-active salts in a measurable form. The laevogyratory component set free from the cinchonine-salt of the first fractions, appeared, as already mentioned, to be extremely soluble ; the solutions manifested a strong tendency to supersaturation, By this circumstance the formation of well measurable crystals is severely hindered; and, as gene- rally occurs in such cases, the crystals finally obtained appeared to be badly formed. Because of Fig. 8. Laevogyratory the vicinal facets present, most Potassium-Rhodium-Malonate. erystal-faces yield multiple mirror- images, causing the angular values to oscillate often more than 30’ round their mean-values. Hence it was at first thought, that triclinic crystals were present here. But the repeated determinations, in con- nection with the optical investigation proved to us finally, that the salt erystallises monoclinically, and more especially in forms differing from their mirror-images. The analogy of the parameters of the optically-active salt and those of the racemic compound is most remarkable, as becomes clear, if the directions of the a- and c-axes in our former determi- nations are interchanged *). Monoclinic-sphenoidical ab: e= 10637 Ae: 1667; RSD AT: Forms observed: ce = {001}, predominant, and mostly very lustrous ; 6’ = {010}, broad and lustrous; 6 = {010}, very narrow, often absent and always yielding good reflexes; 0, = {111}, broad and lustrous; w, = {111}, narrower than o,, yielding multiple reflexes; 0, = 11} DE. M. Jaraer, Proceed. R. Acad. Amsterdam, 20. 277. (1917). There the ratio a’:b: c’ was equal to: 1,0783: 1: 1,2309 ; with g = 86° 36’. 223 and w, = {1 1 1}, about equally broad, giving sharp reflexes; s = 101}, broad and well reflecting, but as all faces of the orthodiagonal-zone, often showing oscillating angles; = {101}, extremely narrow and dull; a= {100}, hardly observable, in most cases totally absent; g = {021}, very narrow and dull. The external habit is that of hemimorphic thin plates parallel to {001}, with a slight elongation in the direction of the b-axis. No distinct cleavage could be found. Angular Values: Observed: Calculated : b': w, = (010): (1141) = *50° 46' ze c:w,=(001):(411)= *60 14 ik c: 0, = (001): A11) = *55 45 _ ene = (OO wt 0 Op= 85 41 os a da ce: s —=(001): 401) = 50 4 50 8 0: w!, = (411): (111) = 64 5 64 1 s:w,—= (101): (114) = a9 Jae. Bode gee =(012 (OOI cast 0 Beld b': o', = (010): A11)= 52 49° Fag 5S On {001} the directions of extinction are orientated parallel to and normal to the orthodiagonal. The crystals are not appreciably dichroitic; their birefringence is feeble. The optical axial plane is {010:, with a feeble, inclined dispersion; one axis emerges on {001} at the border of the field. The crystal-form of the corresponding dextrogyratory antipode is reproduced in fig. 4. =< The specific gravity of the crystals was at 18° C. found to be: dias 7; the molecular volume is therefore: 238,76, and the topical parameters Fig. 4. Dextrogyratory become: v::w—6,1471: 5,7790: Potassium-Rhodium-Malonaie. .6 7423, Analysis proved that the salt contains 14 H,0; on heating at 120° C. it is decomposed, assuming a brown colour. If the specific gravity of the racemic compound be also taken into accourt, (Tee — 9,251; V — 257,80), it appears, in comparing it with the corresponding potassium-rhodium-ovalate*), that the sub- 1) The topical parameters of the racemic malonate, after interchange of the d- and c-axis, becomes: /: yw: w = 6,2484 : 5.7947: 7,1329. 224 stitution of the malonic acid for the oxalic acid, causes a diminution of the topical parameters in two, but an enlargement in the third direction as well in the case of the racemic as in that of the optically-active compounds. | At all events this investigation has brought full evidence of the fact, that the salts of the complex rhodium-trimalonic acid may also be split into optically-active components, and that the phenomena observed in their study are in agreement with Pasrrur’s law in its fullest scope. Laboratory for Inorganic and Physical Chemistry of the University. Groningen, June 1918. Chemistry. — “Investigations on Pasrrur’s Principle concerning the Relation between Molecular and Crystallonomical Dissymmetry : VII. On optically active Salts of the Tri-ethylenediamine- Chromi-series.” By Prof. F. M. Jararr and Witttam Tuomas. (Communicated in the meeting of June 29, 1918). $ 1. Some time ago it was already found') by one of us, that racemic tri-ethylenediamine-chromichloride: {Cr( Hine),}Cl, + 3H, O, was completely isomorphous with the corresponding cobalti- and rhodium- compounds. We prepared this salt according to a method indicated by Preirrer ®), from the tripyridyl-chromi-chloride : {Cr(Pyr.),}Cl, by heating this product with ethylenediamine, and subsequent purification. Then it was separated into its optical antipodes by means of sodiwin- a-camphor-nitronate*), and these were obtained in this way as the pure 2odides. In this fission 6 grams of the racemic salt were dissolved in 20 eem water, and a solution of 6 grams of pure sodiwm-a-camphor- nitronate in 15 eem water subsequently added. A pale yellowish precipitate of d-triethylenediamine-chromi-d-camphornitronate is form- ed; it is sucked off and to the mother liquid 2 more grams of sodium-a-camphornitronate are then added, and the solution allowed to stand for a few hours, when some more of the precipitate is separated. After filtration the mother liquid was used for preparing the corresponding laevogyratory component. The precipitate, thoroughly washed with alcohol and ether, was ground in a mortar with an excess of finely pulverised sodiwm-iodide, some water added, and the dark yellow liquid sucked off from the precipitate, which was well washed with alcohol and ether, dissolved in a small quantity of water, and again precipitated by an excess of sodiwm-iodide. The mother liquid formerly mentioned, containing the camphor- nitronate of the /aevogyrate salt, was precipitated by addition of 5 grams sodium-iodide. The precipitate formed appeared, after being thoroughly washed, to be the racemic iodide. The remaining 1) F. M. Jaecer, Proceed. R. Acad., Amsterdam. 20. 247. (1917). 4) P. Preirrer, Zeits. f. anorg. Chemie, 24. 282, 286. (1900). 3) A. Werner, Ber. d. deutsch Chem. Ges. 45. 865. (1912). Proceedings Royal Acad. Amsterdam Vol. XXI. 226 mother liquid, however, was now treated in an analogous way with 8 grams of sodium-iodide ; the precipitate appeared to be this time the /aevogyrate triethylenediamine-chromi-iodide. It is difficult to obtain these iodides in well measurable crystals, and they are moreover ordinarily very small. § 2. The rotation-dispersion of these salts was determined in the usual way, already frequently indicated. As the orange coloured liquids already manifested a very appreciable absorption of the trans- mitted light in layers of 20 ¢.m. thickness, the measurements for the limiting wave-lengths had to be made with very dilute solutions, These measurements agreed very well with those made in the case of more concentrated solutions, so that for all solutions we have given the mean values of the molecular rotations obtained. In the case of the dextrogyratory component solutions were used, containing 1,0133 grams (A), 0,5070 grams (B), 0,2535 grams (C), and 0,0325 grams (D) of the anhydrons salt respectively in 100 grams NMideculir Rotation Me Legrees ; 4 4120 3020 _ 5180 I sooo irae 440 sd ard 5100 5268 ab ko 3800 6028 6250 sar Fig. dar (euderr-Yrtadiorn Disp 227 ROTATION-DISPERSION OF THE OPTICALLY-ACTIVE TRI-ETHYLENEDIAMINE- CHROMI-IODIDES. Wave-length | | Molecular Rotation: | | Observed Rotation : | in A: U.: (positive and negative), 4260 0-30 (D) | 282639 4320 0 29 id. 27321 4420 0.27; 0.35 (D, K) | 25385 | 4480 0.26; 0.34 id. 24552 | 4570 0.25; 0.33 id. | 23619 | 4640 0.23; 0.31 id. | 22053 4720 0.22; 0.29 id. | 20858 4790 0.20; 0.28; 0.27 (DER) 18652 4860 0.18; 0.26; 0.24 id. 17610 4920 0.16; 0.23; 0.21 id. 15128 5020 0.14; 0.40; 0.21 (D, H, 1) 13267 5100 0.97; 0.36; 0.18 (CE Tj 11714 5180 0.88; 0.32; 0.16 id. 10579 5260 1.60; 0.79; 1.07 (B CG) | 9647 5340 1.43; 0.71; 0.95 id. 8578 5430 1.27; 0.64; 0.84 id | 1634 5520 1.12; 0.57; 1.46; 0 72 (B, C, F, G) | 6692 5610 0.96; 0.48; 1.22; 0.62 id. 5741 5700 1.63; 0.81; 0.41; 2.18 (A,B, C,E; FG) 4891 5800 1.33; 0.67; 0.33; 1.85 id. | 4093 5910 1.15; 0.56; 0 28; 1.55 id. fi. 3422 6020 0.96; 0.47; 0.24; 1.30 id. \ Se | 2912 6140 0.86; 0.43; 0.22; 1.16 id. | FO 5-2 | 2621 6260 0.77; 0.38; 0.19; 1.04 id. Eat | 2328 6380 0.70; 0.35; 0.18; 0.94 id. ( PEEL | 2133 6520 0.65; 0.32; 0.16; 0.88 id. ee 1951 6660 0.61; 0.30; 0.15; 0.82 id. | MG Sar | 1820 Lo 228 of the liquid; in the case of the laevogyrate antipode the six different solutions employed contained 1,3512 grams (£), half or a quarter of this (7, G) in 100 grams of the liquid, and 0,0927 grams (H), 0,0464 grams (/), and 0,0232 grams (K) respectively of the anhydrous salt in 100 grams of the liquid. The dispersion-curve for the molecular rotation, shewn by mea- surements is plotted in the diagram (fig. 1). It has much analogy with that of the corresponding cobalti-salts, but only a slight analogy with that of the ¢triethylenediamine-rhodium-compounds. Probably the magnitude of the rotation for corresponding wave- lengths in the case of these analogously built complex ions greatly depends on the magnitude of the atomic volume of the central metallic atom, in such a way that the rotation appears higher, if the atomic volume of the metal is smaller. As for instance: | | ATOMIC VOLUME | COMPLEX SALT: MOLECULAR ROTATION OBSERVED: | OF THE | | METAL : | Tj i ae : 5 | oe | {Co (Eine); } Is + HO. | M5goo = 7230°; Ms100 = 21580°; Megoo = 2114? 6.76 | iCr (Eine), | Is + H,0. | Mszg00 = 4093°; M5100 = 11714°; Meeo0 == 1880° | eae | {Rho (Eine); I; + H20. | Msgoo = 3125°; M5100 = 3965 ; Meeoo = 2243° 8.50 | The values for 2— 6600 A. U., are mentioned at the same time for the purpose of demonstrating that this antiparallelism of rotations and atomic volume is surely nod true for all wave-lengths: for rays of great wave-length, as e.g. in the visible red part of the spectrum, — the rotation of the /ho-salt surpasses even that of both the other salts; only in the domain of appreciable dispersion, is the said regularity met with. As regards the absorption, we were able to state the following. In a layer of the solution of 20 ¢.m., a liquid containing. 1,1212°/, of the salt, allows the passage of all red and yellow rays up to those of 5380 A.U. OHGOG0 Fn man = | oe Reet. he S220A O 28080 nt on 5 5 i NEN peer ” if n » y» 2030A.U. 0, 1402 %9 » ” „ ” ”» ” ” ” ” „ ” ” ” ” » 4850 A.U. 0,0701 Do „ ” ” „ ” „ LJ » ” „ ”„ ” ” » ” 3940 A.U. § 3. Numerous attempts were made to win these chrom-salts in well measurable crystals, and to investigate the validity of PasTnur’s law also in this case. But a heavy impediment in reaching this aim was created not only by the facility with which those salts decompose in solution, especially under the influence of the light, — but also 229 by the great solubility of these salts, inducing us always to work with only small volumes of concentrated solutions, from which good crystals are ordinarily deposited with difficulty. For the same reason the transformation of the iodide into the chloride or bromide could not be of any use, so that these for our purpose so very important salts, could not be made use of in this case. Racemic TRIETHYLENEDIAMINE-CHROMI-IODIDE. {Cr (Hine),} I, + 1H, 0. On slow crystallisation this compound presents itself in the form of very small, orange, apparently octahedral crystals. Crystallisation must occur in the dark, because this salt, in the same way as all the triaethylenediamine-chromi-salts, becomes violet under the influence of the light. Also increase of temperature must be avoided, because the solutions change from an orange colour to a dark reddish violet by the transformation into salts of the violet aquo-type. The crystals measured were not larger than a pinhead, and often they were disfigured and distorted in rather a strange way. Some of them showed under the microscope the appearance of fig. 2a, without Fig. 2. Racemic T'riethylenediamine-Chromi-lodide. (+ H0). it being possible however to determine the Mirrrrian indices of their facets with complete certainty; the crystals pictured in fig. 26 and 2c manifested however some measurable forms. Rhombic-bipyramidal. a:b:c=0,8632 :1:0,8652. The crystals are pseudo-tetragonal, and perfectly isomorphous with the corresponding crystals of the cobalti- *), and of the rhodiwm-’) compound, just as we were able to prove this before in the case of the trigonal chloride of this series*). The colour of the crystals was orange or red; by partial loss of water of crystallisation, they sometimes get locally yellow and opaque. 1) F. M. Jarcer, Proceed. R. Acad. Amsterdam, 18. 62. (1915). 3) F. M. JAEGER, Proceed. R. Acad. Amsterdam, 20, 250. (1917). 3) F. M. Jarcer, Proceed. R. Acad. Amsterdam, ibid. 247, (1917). 230 The forms observed are: 0 ={111}, great and very lustrous; c = {001}, small, but well developed and yielding good reflections; m = $110}, broad, but commonly with curved and rudimentary facets, and thus practically not well measurable. Probably also a form q = {021} occurs, and in the case of the erystals of fig. 2a doubtless : a = }100}, as a broad pinacoidal face, and r= {ho k}. Angular values: Observed : Calculated : c:0= (001):(411) = 52° 56’ ay 0:0 = (441): A11) = 62 49 = 0:0 = (111):(111) = VA cd) 14 8! od hee Ge oh) — za 27 tae) No distinet cleavability was found. There cannot be the. least doubt about the complete isomorphism with the corresponding Co- and Rho-salt: Cr-salt 4. :. . a:03¢ 90de: 08052, Co-salt..... a: 6:6 =) 8700. & 0.8699. Rho-salt....a:6:c = 0,8541 :1 : 0,8632. Up till now we have had no opportunity to prove this iso- morphism also in the case of the optical antipodes, because no suitable crystals could be obtained. There can be however no doubt, that the said relation also exists in this case. Laboratory for Physical and Inorganic Chemistry of the University. (rroningen, June 1918. Anatomy. — “The Involution of the Placenta in the Mouse after the Death of the Embryo”. By Dr. A. B. DROOGLEEVER Fortuyn. (Communicated by Prof. J. Boeke). (Communicated in the meeting of June 29, 1918). In various species of mammals which are pregnant with several embryos at the same time it accidentally occurs that one or more embryos die before birth. The subsequent fate of the placenta has been controlled in only a few cases and it appears to be intimately connected with the structure of the placenta. Now this structure in the mouse considerably deviates from that in. many other mammals. So it seemed to be worth while to investigate in this animal too, as has not yet been done, the involution of the placenta after interruption of the pregnancy. For this purpose the uteri of 8 mice were at my disposal containing together besides many normal egg- chambers 20 egg-chambers without an embryo. Judging from the degree of development of the normal egg-chambers one of the 8 mice had been killed on the 13 day of the pregnancy, one on the 15%, four on the 16%, one on the 17 and one on the 18 day. The 20 empty egg-chambers are more fully described in a paper that l offered to the “Tijdschrift der Nederlandsche Dierkundige Vereeni- ging’. Here I shall only communicate the results in a general way. Never was any other trace of the embryo left than some free cells which could not be duly recognised. Many portions of the foetal membranes survived the embryo, but they did not all do so during the same time. So among the empty egg-chambers some groups could be recognised with more or less remainders of the foetal membranes. In the first group the giant-cells (in the mouse trophoblastic cells which are greatly enlarged and have become independent) and the membrane of Reicuerr were left and moreover parts of the ecto- placental cone and of the proximal or distal entoderm of the yolk-sac or of both. The proximal entoderm of the yolk-sac could be well recognised by the appearance of the cells, but it had always been broken into pieces. The distal entoderm of the yolk-sac sometimes lined large pieces of the membrane of Ruicuert internally; besides free cells of it occurred. The ectoplacental cone 232 never inclosed embryonic blood-vessels, but sometimes some con- nective tissue of the allantois entering the ectoplacental cone together with the large blood-vessels. Always the cells or the syncytium of the ectoplacental cone could be recognised. In some cases they changed into young giant-cells, which in normal circumstances too can originate from cells of the ectoplacental cone. Often spaces filled with maternal blood lay between the cells of the ectoplacental cone, as is the case in normal egg-chambers. RricHErt’s membrane could easily be recognised as the homogeneous membrane that develops beneath the trophoblastic epithelium when it changes into free giant-cells. After the disappearance of the embryo the contraction of the uterine wall had pressed the greater part of the ectoplacental cone into the space previously occupied by the embryo. Moreover this contraction bad folded Reicuert’s membrane. Sometimes this membrane had much diminished in size, but it always showed the aperture through which the cells of the ectoplacental cone previously cohered with the allantois. Generally the giant-cells very clearly showed their power to ingest erythrocytes and other portions of the maternal decidual tissue, but they had hardly changed in this group. This was not so in the second group of empty egg-chambers where, as to the foetal elements, only the distal entoderm of the yolk-sac, the membrane of RrtcHEert and the giant-cells were left. There several of the latter cells had grown out till they reached dimensions that were extraordinary even for giant-cells. In normal egg-chambers it is the task of the giant-cells to attack the decidual tissue and the maternal blood and to leave part of the ingested food to the embryo. As soon as they have been loosened from the trophoblastic epithelium or the ecto- placental cone they lead an independent life. After the death of the embryo the only change is the fact that of course the giant- cells can provide no longer any food to the embryo. They keep all to themselves and consequently thrive extraordinarily. In all directions they acquire the same dimensions, as the pressure of the embryo which in normal egg-chambers flattens them much, has been suspended. Therefore the space occupied by the giant-cells is much larger than in normal egg-chambers. Their number only seems to me to be larger, because they are not dying away so soon asin normal egg-chambers, not because more of them would have devel- oped. Yet here too the fate of the giant-cells is to die away. This is more conspicuous in another group of empty egg-chambers where giant-cells are the only foetal element that is left. Especialiy here one sees the body of the giant-cell losing its affinity for the dyes 233 and dissolving, leaving the naked nucleus behind. Afterwards the nucleus submits to the same fate. The giant-cells have not been able to consume the whole layer of decidual tissue before they disappear. Yet this layer must be removed if the normal situation of the uterine-wall is to return. Therefore the giant-cells are supported by another type of cells, apparently amoeboid wandering cells with phagocytal qualities. These cells are of a hitherto unknown kind and in normal egg-chambers they do not occur, not even post partum. Their shapes and sizes are very variable. They have a nucleus which generally lies ex- centrically and sometimes two or more nuclei. Their cytoplasm stains remarkably intenseley with eosin dissoluble in water, whereas eosin dissoluble in alcohol stains them, it is true, but not extraordinarily. I propose to call these cells eosinophilous phagocytes. About their origin nothing is known to me, but I think that they are maternal cells. The eosinophilous phagocytes were lacking only in one of the twenty empty egg-chambers, and this one obviously had been preserved within a day after the death of the embryo. In the first place they appear in small groups between the group of giant-cells and the layer of unattacked decidual tissue. These groups enlarge into constantly thicker layers, which are always situated either between the decidual tissue and the giant-cells or between the former and the uterine cavity. The eosinophilous phagocytes attack only the maternal decidual tissue and not the giant-cells and they continue to do so after the disappearance of the giant-cells. So a fourth group of empty egg-chambers exists where one sees no foetal rests at all, but only eosinophilous phagocytes which remove the layer of decidual tissue, which has in the mean time greatly diminished in size. 1 could not observe the disappearance of the eosinophilous phagocytes. As is known, in egg-chambers of the mouse the uterine cavity disappears at the mesometrical side of the embryo to extend at the antimesometrical side of the embryo starting therewith from the portions of the uterine cavity that are lying between the egg-chambers. Before the parts of the new uterine cavity reach one another in the middle of the egg-chamber, which occurs on the 17 day of the pregnancy, a more or less thick partition of decidual tissue in the egg-chamber separates the parts of the new uterine cavity, which approach one another. Now this partition can be found in many empty egg-chambers, but in some it has been ruptured, in others it is attacked by eosinophilous phagocytes, and in still others it has been removed prematurely by eosinophilous phagocytes. 234 I cannot even guess the cause of the death of the embryo, but I observed that the embryo may perish at different ages. At least I think I have met with a case where this occurred on the 8 day and with another where this occurred on the 16'h day. He who: disposes of younger specimina (my youngest embryos were of the end of the 13 day) probably will also find eggs that have perished before the 8" day. Moreover it appears that in one and the same: uterus embryos may die away at very different ages. I discovered in the same uterus one of the empty egg-chambers with the smallest: and one with the greatest quantity of foetal rests, and I conclude that one embryo had been dead a much longer time than the other. Leiden. Anatomisch Kabinet. Physics. — “The Limit of Sensitiveness in the String galvanometer.” By Prof. I. K. A. Wertnem SALOMONSON. (Communicated in the meeting ‘of June 29, 1918). In EINTHOvEN’s stringgalvanometer the deflectional constant is subjected to the same law as holds good in the movable needle- and movable coil galvanometer: it is proportional to the square of the periodic time of the movable part. In the string instruments the duration of the oscillations is modified by altering the tension of the string. The sensitiveness finally depends on this tension as well as on the dimension and material of the string, and lastly on the strength of the magnetic field. The tension of the string can only be altered within certain limits. The upper limit is given by a tensile stress exceeding the elastic strength. The lower limit is the total absence of tension. But even when no pull is exerted, the string can still vibrate transversally. The frequency of the vibrations it then makes, is a function of the dimensions of the wire and two properties of the material i.e. density and the elasticity-modulus, and may be represented by ; m? d E N ZE Ze 73 —=e . - . e . a (1) 8a l 9 in which N denotes the frequency, / the length, d the diameter of the string, g being the density and £ Youne’s modulus, whereas m is the smallest root of the transcendental equation cos m cosh m= 1. The value m = 4.730... (Rayteied On sound I. Art. 174). As we may discard the influence of temperature on the elasticity, this formula gives the lowest frequency for transverse vibrations obtainable in strings, which in a definite material and with given dimensions cannot be lessened. We may therefore say that the periodic time of the string in the EiNrHoveN galvanometer, and consequently the sensitiveness of the latter is limited by the impos- sibility to lessen the frequency; and as the elasticity of the material is responsible for transverse vibrations which might occur in a perfectly relaxed wire, the true limit of the sensitiveness is to be found in the elasticity, and as we shall see also in the density and specific resistance of the material of the strings. With the formula (1) we can always calculate the minimum of 236 the frequency of the transverse vibrations of the string, if we know its dimensions and the material of which it is made. In table I a few numerical data are given regarding various materials which may be used for making strings. In this table the length is taken as 10 cm and the diameter as one micron (10-4 centimeter). TABLE I. | Ero g | Nise. T Ge SUE, OE 11000 | 8.9 0.3100 sh ADE re NN 1500 | 10.5 „2356 4.25 Ald Moh ti. et GE 7500 Enid „1724 5.80 Ala eileen ee 6750 | 2:7 .4408 2.27 Rhy sa EE 16500 | | 4.4 „2448 4.08 | | | It is very difficult to measure directly the vibrations per second in wires of 1 micron diameter. The air resistance in this case is so considerable as to cause the movement to come to a dead stop. We should have to examine such strings in a perfect vacuum. Furthermore we are not able to make wires of 1 micron except in platinum, and perhaps in aluminium and gold. Silver wires of 1 micron are as yet not atlainable. But the dimensions used in this table and the figures in the last columns permit us to calculate in a simple way, e.g. with a slide rule, any periodic time when other dimen- sions are given: the vibration time being proportional to the square of the length and inversely proportional to the diameter. A wire clamped at the end without any tensile stress will sag under the influence of a load P, uniformly distributed over the total length. The maximum deflection / will be Pe? hs Se (2) ET 384 where / represents the axial moment of inertia of a section of the 4 : 5 8 uw wire. As for a round wire / = ac Pe get al en 62 Ed‘ In the string galvanometer the transverse load P is equal to Hil Dynes, if H be the strength of the magnetic field in Gausses, 2 (3) 237 the current strength in Webers (== 10 ampere) and / the length of the wire in centimeters. If we put this value for P in 3): De es 0 (4) 62 Ed giving the deflection of the middle of a string clamped at the ends without tension, and of a length / and a diameter d, placed in a magnetic field H, as soon as a current of 7 Weber passes through the string. We need hardly insist on the fact that this formula gives the absolute limit for the sensibility for small currents. The next table II shows this limit for strings made of different materials. For the dimensions of the string we again take 10 cm and 1 micron, for the current-strength 10—1% Weber (10~—!? ampere), for the field intensity 10000 Gauss. The last column gives the deflection in millimeters if the absolute deflection of the middle be magnified 1000 times. We can use the figures of this column, if we wish to calculate the possible deflection with strings of other dimensions in a field of a different strength, and when observed magnified to another scale. The deflections are proportional to the fourth power of the length and the inverse of the diameter of the string. TABLE II. Ean ae En (pee gn | 67E | (f= 10cm, iS EBER J Cu | 11000 __\___4916.10-20 | 49.16 mm ze 7500 | 7210.10-2 | 72.10 , in oJ 7500 | —-7210.10—20 | Teds: AI 6750 8012.10-20 80.12, pe | 16500 | 327.102 | hl | The figures in the last column may also serve to calculate figures for existing strings. With an aluminium string of 2 microns and 56 millimeters’ length in a field of about 16000 Gauss, | found a deflection of 0.40 millimeters for 10—!? ampere, which took place in about 20 seconds. From the figure in the table we calculate that the deflection ought to be 1.20 millimetres. In EiNrnoven’s publica- tions on silvered quartz fibres we equally find figures about the possible sensibility, the order of which does not disagree greatly with the theoretically possible deflection if we take for the sensitive- ness with silvered quartz fibres the same value as in the case of silver 238 wires. The same may be said about my own observations with quartz fibres. Generally the deflection actually observed is some 3—5 times smaller than we ought to expect from the theory. The explanation is found when we consider the behaviour of silver strings of 16.5 microns and of copper strings of 15 microns diameter. These wires still give vibrations when the tension is reduced as far as possible, but in every case the frequency is about 1.5 to 2.1 times greater than that calculated from formula (1). With a silver string of 16.5 u and 53 mm length I could not reduce the frequency under 20 per second instead of 14 as calculated. When magnitied 47 times 1 microampere caused a deflection of 1.91 millimeter; the string being placed in a field of 14000 Gauss. The theoretical value is 3.7 millimetres with 14 vibrations, which would come to about 1.8 millimeters with 20 vibrations. As there may be a slight differ- ence between the figure taken for the diameter and the actual diameter, the agreement may be considered not unsatisfactory, the more so as the value of / must also be considered as merely an approximate one. Finally we must state that the string was not an entirely straight one, and that in being mounted it had probably retained a slight torsional stress. In a few other observations of the same kind with wires of different material I found a deflection of 8.1 mm where 9.1 mm was expected; also one of 36 mm, where 40 had been calculated. Generally speaking, the agreement was by far the best with the thicker wires. Yet in all cases the agreement was close enough to allow an extension of the theory to the sensitiveness for small potential differences. From the formula (4) we find an expression for the sensibility for small potential differences by dividing both parts by w, the resistance of the string: h Ht uw 60 Ed*w ME: This formula gives the deflection in centimeters of the middle part of the string when a potential difference of 10 Volts is applied to the terminals. But with this formula we have not taken count of the damping. The movement of the string in the EINTHOVEN galvanometer is damped partly by air friction, partly by electro- magnetically generated counter-electromotive force. In the following cases we shall consider only the electromagnetic damping, which with thick wires greatly exceeds the air resistance. As the electro- magnetic damping is caused by the number of the lines of force 239 cut by the string during its movement, the form taken by the string in its deflected condition is of the greatest significance as well as the strength of the magnetic field. A string under tension deflected in a homogeneous magnetic field takes the form of a parabola. In that case the damping factor is HU? w. 10?” (6) Whenever the string would take another form whilst being deflected, the factor 3 would take another value. This factor represents the mean deflection taken over the whole string as compared with the maximum deflection. A perfectly relaxed string, clamped at the ends and uniformly loaded takes the form given by the formula en Pyle ot a* ; Pome ee py a in which y gives the deflection for a point at distance z from the end. If we put «—3/ we get the deflection for the middle of the string, which has already been given in formula (2). In order to tind the mean deflection we integrate (7) over the whole length: | lik Ea Fendt ' 0 Comparing (2) and (8) we find l def 5 &/Ymax Tr y daymar = 3 0 so that we may state 8 op io 15 w. 10* . . « . ° . . . (9) Taking M as the mass of the string, we can always represent . ° . . ol ( ) if A be a lateral force and if we suppose the damping to be very slight. If the string should make critically damped vibrations, the damping would be DS AME BaN is Shere der args (11) Eliminating K from (10) and (11) we get, in connection with (9) aa al Anr MN = — mat dbte ahe at LE) 15 w.10° 240 In (12) we substitute for the mass M = -zd*/g and for the resistance Bp DS ae giving: EE Tbr DOK GONE Gee ge ED which shows a simple relation between the allowable and necessary strength of the magnetic field, the frequency and the density and resistivity of the material. If we were at liberty to choose any figure for N, the length and the thickness of the string would seem to be of no consequence. But we started from the premise that the frequency should be as small as possible with a string of predetermined length and thickness, and elasticity. Hence we must put the value for N taken from (1) in (13) giving ag NT es ae Now we can substitute this value for H in (5) and by likewise Hee el Jen substituting Dn Ne, MER at a formula for the sensibility for A potential differences: h ia - g : — = 6040 —_.. ES eee tw dy E* 9? This expression for h/w gives the extreme limit for the sensibility of a completely relaxed string in a magnetic field of a strength exactly calculated to render the movements of the short-circuited string critically damped. The volt-sensibility increases by /* and decreases by dj/d. It also depends on the density, resistivity, and elasticity of the material. In table III we find the constants for different materials and in the fourth column the comparative “material-factor” for each material. These have been multiplied by 10° so as to indicate deflections per microvolt with strings of 1 u TABLE III. | ‘non aly aoe | (age E.58100000| 2:1 coo 10 H N/sec ——— Ca ie 8.9 | 11000 1.62 | T14 325 | 0.3100 Ag | 10.5 | 7500 1.75 | 1034 320 | 0.2356 Au | 19.2 7500 2.20 | 1073 | 417 0.1724 NE A GaSe 2.87 622 284 0.4408 Pt | 21.4 | 16500 9.40 295 1078 | 0.2448 241 diameter and 10 e.m. length with a microscopic magnification of 1000 times. The next column shows tbe strength of the field H in Gauss, and the last column contains once more the frequencies. Gold fibres, if critically damped, will give a larger deflection than strings of any other material, but of the same dimensions. The time of vibration, and consequently the time of deflection is larger than with other strings. Hence we cannot easily compare the results This table is only useful if we wish to calculate the possible deflection with strings of other dimensions. In order to get compar- able figures we shall have to consider strings of the same diameter, which have the same vibration time. A formula for this case can be given by calculating h JP 1 eT IPA AN) tse Me Ao tgs Ww Vd FP Eo’g This formula represents the deflection caused by 10 Volts through a string of a diameter d, completely relaxed, and vibrating once a second, whilst placed in a magnetic field of a strength, sufficient to cause the movement of the string to be entirely damped. In this case the length is predetermined for-any material by the condition that the frequency is one per second. In table IV we give a few figures which can be calculated by this last formula. (16) TABLE IV. 5400 | Deflection per u V. with d = lp. a H EE Se baken g | | and V = 1000 X | | Cie 2410 |. 500! |) 5.56... 1150 241 mm Ag | 2448 | 658 | 4.85 | 1080 | zit, rian eis ore | dete” 1168 | 188 „ Al 2151 | 426 | 6.63 | 2430 | Zs 5, 5 | | | Pt 726 | 2199 | 4.95 | 5040 ee From this table we see that aluminium is the best material for strings in an KHINTHOVEN galvanometer if used for the measuring of small potential differences. A string of 1 u, completely relaxed and 6.65 em in length, gives vibrations of one second. With it we can get a deflection of 276 mm for 1 microvolt, if the field be adjusted at 997 Gausses; and the microscopic magnification amounts 16 Proceedings Royal Acad. Amsterdam. Vol. XXI. 242 to 1000 times. If the string were placed in a perfect vacuum the movement would be critically damped. Silver follows with 245 mm deflection if the length be 4.85 em. Copper requires a longer string viz. of 5.56 em and gives about the same deflection. Practically we shall have to make our choice between aluminium, silver, or copper, whenever we want a high sensitiveness for small potential differences with a critically damped movement. From the formula we conclude that with a given material the thinner the string the higher will be the sensibility for small potential differences. Finally we shall have to consider one other possibility for ren- dering the voltsensibility as high as possible. We take again the case of a perfectly relaxed string, clamped at the ends. If the weight P be uniformly distributed over the entire length /, we must use the formula (1). But if the string is loaded in the middle only with the weight P, the deflection will be exactly twice as large: Pte A= — —_. EI 192 If we put the string, of a length / in a stronger magnetic field H’ but of a very short length 2 so as to make Hl— H'2, and if we suppose 2 to be very small as compared with /, we shall come very near the conditions represented by the last formula (17). Especially if we use strings of not too small a diameter there will be scarcely any difficulty of making the magnetic field 10— 20 times stronger and the string !0—20 times longer than the field. In this case we practically double the deflection, but at the same time the damping will bave become too great. Tbe damping factor will have become nearly 1.0 instead of 8/15. Hence the magnetic field must be made V2 times weaker. Finally the sensibility for small potential differences will become only W2 times greater. (17) Anatomy. — “The egg-cleavage of Volvow globator and its relation to the movement of the adult form and to the cleavage types of Metazoa.” By Dr. H. C. Drisman. (Communicated by Prof. J. Bork). (Communicated in the meeting of June 29, 1918). For the zoologist still more than for the botanist Volvox is an interesting object. Already in this organism, where it is still dubious whether we have to consider it as a plant or as an animal, we see indicated the main lines along which the phylogenetic development of the Metazoa has taken its course. Bitscant’) rightly observes that Volvox is no longer to be considered as a colony of Protozoa, but as a pluricellular organism of simple structure. Not only do the cells communicate with each other by plasmodesms, forming thus one single mass of protoplasm, but also there is a difference between mortal somatic and potentially immortal propagation cells as is characteristic of Metaphyta and Metazoa. Between these two Volvox holds an intermediate position, reminding one more of the former by the possession of chlorophyll but pointing more in the direetion of the animal kingdom by the rest of its organisation. Long ago the first stage of development in Metazoa, the blastula, has been compared to Volvor and was termed by Huxrer®”) e.g. “the animal Volvox”. The resemblance afterwards appeared to be still greater than HuxLey could have suspected, for Volvox is by no means a homaxone sphere rotating indiscriminately in all directions, but shows a distinct opposition between an animal and a vegetative pole. The line joining them can be described as the main axis of the organism, which is not strictly globular, but a little elongated in the direction of the main axis. With the animal pole directed forwards it swims with a rotary movement round the main axis just as is the case with the pelagic larvae of lower Metazoa and also still of Amphioxus. At the animal pole the cells are smaller and further from each other and contain also less chlorophyll than those at the vegetative pole which are darker green, by reason of the higher 1) O Bürscuu, 1883—1887, Protozoa II, p. 775, in Bronn’s Klassen und Ord- nungen des Thierreichs. *) T. H. Huxtey, 1877. The Anatomy of Invertebrated Animals, p. 123, 678. 244 proportion of chlorophyll, and communicate by more numerous and broader plasmodesms. The cells at the animal pole each contain a red stigma as characteristic of flagellates sensible to light (to which also the Volvocinea belong), whilst those at the vegetative pole are lacking them. The two kinds of cells pass quite gradually into each other. The propagation cells are restricted to the vegetative half. Any one having an opportunity to study Volvo can easily verify all this. The propagation occurs either by means of egg-cells and sperma- tozoa, or parthenogenetically by so-called parthenogonidia. The latter mode occurs, just as in Rotatoria and Infusoria, during a number of generations, the former mode at the close of such a period, the encysted egg being the result. The cleavage stages of the egg and of the parthenogonidia in which development proceeds in a similar manner, exhibit again a striking resemblance to those of Metazoan eggs. The figures given of these stages for Volvor, Pleodorina, Eudorina, Pandorina, and Gonium, remind one especially of stages of the spiral cleavage type, which probably we may designate as the original cleavage type of the Zygoneura or Protostomia, and which is still found with Polyclads, Nemertines, Polychaetous Anne- lids and most Molluscs. It therefore seemed to me very interesting to find out how far the cleavage of Vo/vow corresponds to the spiral type. The statements made by former investigators appear to be insufficient and too contradictory to answer this question in a satis- factory way *). Fig. 1. Volvox globator, parthenogonidium, four-celled stage, seen from the vegetative side. When, therefore, the opportunity presented itself to study more closely the cleavage of the parthenogonidia in Volvo, which appeared 1) Statements on the cleavage of Volvox are found in: J. GOROSHANKIN, 1875, Genesis im Typus der palmellenartigen Algen. Versuch einer vergleichenden Morphologie der Volvocineae. Mitt. Kaiserl. Ges. naturf. Freunde in Moskau, Bd. 16 (Russian, an extract is found in Botan. Jahresber. f. T3135, ps 27). E. Overton, 1889, Beitrag zur Kenntniss der Gattung Volvox. Botan. Centralbl., Bd. 40; p: 171. L. Krein, 1890, Vergleichende Untersuchungen über Morphologie und Biologie der Fortpflanzung bei der Gattung Volvox. Ber. naturf. Ges. Freiburg, Bd. 5, p. 15. 245 to occur in considerable number in the Victoria regia-basin of the Leyden botanical garden, I readily seized it. The study was made on living material. During development the parthenogonidium, which continues to communicate by plasmodesms with the surrounding cells, considerably increases in size’), so that the older stages are often easier to study than younger ones, for which the use of oil- immersion as a rule is to be preferred. By two meridional cleavages the parthenogonidium is first divided into four equal cells, which each will give rise to a quadrant. The eight-celled stage has already been figured repeatedly for Volvon and other Volvocinea, but not the transition of the four- into the Fig. 2. Beginning of Fig. 3. Transition 4—8, the third cleavage, animal vegetative side side eight-celled stage. Figs. 1, 2, and 3 teach us that during this cleavage a torsion amounting to 45° occurs between what we may call for the sake of shortness the four vegetative cells and the four animal cells. In the terminology of the spiral cleavage type we should call this torsion a dexiotropic one since, if we look at the egg from the side of the animal pole, the four animal cells appear to lie to the right of the four lower cells. It seemed to me interesting to make out if this third cleavage always takes place in the same way or if, as could equally be imagined, it is sometimes dexiotropic and sometimes laeotropic. In the cleavage of Balanus, which shows a similar torsion, I found eg. both possibilities occurring indiscriminately *). In the spiralcleavage type the third cleavage is always dexiotropic with the exception of inversely wound Gasteropoda where the whole cleavage proceeds in an inverse manner. So not only the adult form but equally the earliest cleavage stages present the reflected image of what we find in dextral Gasteropods. 1) Ail the figures in this article have been drawn the same size. 2) H. C. Detsman, 1917. Die Embryonalentwicklung von Balanus balanoides Linn Tijdschr. Nederl. Dierk. Ver. (2), Dl. 15. 246 1 found that in Volvor the third cleavage always proceeds in a dexiotropic manner, and the suggestion lies at hand that here too some peculiarity of the adult form might stand in a certain relation to this phenomenon. What, for example, is the direction in which Volvox rotates round the main axis, is this always the same or at one time dexiotropie and at another laeotropic? As has been already observed by earlier investigators and as I ean confirm here once more, the rotation always oceurs in this way that, seeing it from the animal pole, we may designate it as clockwise, i.e. in the direction of the hands of a clock or dexiotropic. It lies at hand to suggest a relation between these phenomena, as has been stated equally in Gasteropods. That in the latter there can be no question of a direct causal relation between the torsion of the adult animal and that of the cleavage cells will be evident at once if we bear in mind that the spiral cleavage type occurs equally well in forms that are not wound at all,.as Lamellibranchiata, Chitons, Polychaetous Annelids ete. We will revert to the question whether possibly in Volvor we might think of a more direct relation between the torsion during cleavage and the direction of the rotation during movement. In the eight-celled stage (fig. 4, 5) which has been figured already Fig 4. Stage 8, Fig. 5. Stage 8, animal side. vegetative side. more than once, the four vegetative cells alternate with the four animal cells. They constitute together a little cell-plate representing phylogenetically the Gonzwm-stage, but which at the border already begins to curve in. This curving in accentuates itself during the transition into the 16-celled stage and in Volvo evidently manifests itself somewhat earlier than in Pleodorina, Eudorina, and Pandorina, where also in the stage 16 the cells are still lying in a concave little plate, while in Volvor it has then already passed into a hollow globule with an opening, the “phialoporus”. The eggs always have the vegetative side, with the phialopore, directed to the surface of the maternal organism. 247 The passage of the eight- into the sixteen-celled stage, which in the spiral type is always performed by a laeotropic cleavage, is characterised in Volvox by a progressive torsion of the cells of the vegetative side with regard to those of the animal side and this in Fig. 6. Beginning of the fourth cleavage, vegetative Fig. 7. ‘Transition 8—16, side. animal side the same direction in which it has already manifested itself in the foregoing cleavage, which is what we may call dexiotropic. This expresses itself in the shape of the cells immediately when the fourth cleavage sets in, as fig. 6 teaches us. The torsion here has already become a little greater than 45° as becomes evident if we compare the situation of the inferior parts of the vegetative cells (a,—d,) . with regard to the cross of the cleavage furrows at the animal pole. The fourth cleavage (fig. 7, 8), therefore, under the influence of Fig. 8. Transition 8—16, Fig. 9. Stage 16, vegetative side. animal side. the above torsion, must be described as dexiotropic. For this reason, and in regard to the further cleavage, 1 think it inadvisable to apply here the nomenclature proposed by Conkuin for the spiral type, but will modify this a little. I call the cells of the four quadrants resp. a, b, c, and d, and to their descendants I give each time the’ exponent 1 to the cell that lies to the animal side, and the exponent 2 to that lying to the vegetative side. Thus all the cells with the letter a are descendants of the cell a of the four-celled stage, forming 248 together one quadrant which, moreover, I have surrounded with a thick line in the figures. Now fig. 9 answers wholly to the image presented by a 16-celled stage of the spiral cleavage type, however, one would expect the cells a'’, a'?, 67! and 67* to represent together one quadrant. This hasbeen shown not to be the case, and if one takes a view of figs. 8 and 9, the dexiotropic torsion that has occurred during the cleavage, at once strikes the eye. Between a‘ and a’? this torsion now amounts to between 45° and 90°. The phialopore is surrounded by the cells 4°*—d*? and a'?—d’’, the former consti- tuting the four longer, the latter the four shorter sides of the octangular phialoporie border. Sometimes one of the cells a?—d*' also reaches the border, the latter then being formed by nine cells. The fifth cleavage, leading to the 32-celled stage, is again laeotropic, as is shown as well by a view from the animal (fig. 10) as from the vegetative side (fig. 11). Thus the dexiotropie torsion is again Fig. 10. Transition 16—32, Fig. 11. Transition 16—32, animal side. vegetative side. continued here, and as fig. 11 and especially fig. 13 shows, this torsion of the vegetative extremity of each quadrant with regard to the animal extremity (the cell a’'') now amounts to nearly 90°. While tbe cell a*'? forces itself between a’? and a?*, as was already the case in the 16-celled stage, so that a°* is pushed aside a little (figs. 11, 13), a‘? while dividing is pushed to the phialoporic border by a, which also divides. As a consequence a‘??—d'*? now form the longer, a??*—-d?** the shorter sides of the octangular phialopore. Often also all eight sides are of equal length. The arrangement of the cells in the 32-celled stage is so regular (figs. 12 and 13) that no doubt one would not suspect from it the torsion stated here by watching the cleavages. The last cleavage studied by me is the one that leads from the 32-celled to the 64-celled stage (figs. 14 and 15). The direction of 249 the divisions gradually becomes subject to more variation, yet the equatorial direction just as in former cleavages — though with a At art Fig. 12. Stage 32, animal Fig. 13. Stage 32 (the same side. egg), vegetative side. deviation caused by the torsion — continues to predominate. That the torsion still proceeds is evident from fig. 15, which shows that it is already more than 90°. ; At the beginning of this investigation I almost expected to find that Volvox divides according to the spiral cleavage type. The figures given by some investigators seemed to me to point in this direction. No doubt this result would have been interesting with regard to the derivation of the different cleavage types of Metazoa and their mutual relation. A more direct relation between Volvor and the lowest forms with a spiral cleavage type would not then appear improbable, since, as shown above, there are other points of agreement. That Volvox possesses chlorophyll would be no insuperable obstacle Fig. 14. Transition 32—64, Fig. 15. Transition 832—64, animal side. vegetative side. since it can hardly be donbted that animals must descend from organisms with chlorophyll. No production of organic from inorganic substance would have been possible otherwise. 250 We have seen, however, that the cleavage of Volvox may not be counted as belonging to the spiral type in the form in which‘it occurs in Metazoa, though the arrangement of the cells is more in a spiral than it is with the latter. Though there are certain points in common 1 yet refrain from further speculations in this direction. In another respect, however, the results reached seem to me to be interesting. We have been able to state during the cleavage a progressive torsion of the vegetative cells with regard to the animal cells which becomes especially manifest from a comparison of figs. 2, 4, 7, WAO SIM and dE andor ngs 3, 0, 6, 8; 11, 13;and te: So we have every reason to assume that in the adult form also a similar arrangement of the cells prevails. In the spiral cleavage type the succeeding dexiotropie and laeotropic divisions nearly annul the effect of each other, so that in the blastula the cells belonging to one quadrant nearly occupy an area situated between two meridians distant 90° from each other, as is represented fig. 16a. Fig. 166 shows the situation of the cells belonging to one quadrant in the case where the torsion does not surpass 90°. How great the latter has become in the adult Vo/vor cannot be made out. Now in the same dexiotropic direction also the rotation occurs, as we have seen, and it seems to me probable that in this case we may look for a more direct relation between the two phenomena than with the torsion of Gasteropods. Let us assume to this end that not only the colony but also each of the cells of Volvoe has a certain polarity and thus a main axis in the direction from the animal to the vegetative pole of the colony. This polarity of the cells e.g. manifests itself in the corresponding direction in which all the flagella beat causing a water current from in front backwards, which makes the organism move Fig. 16a. Fig. 160. in the direction of the animal pole. If a Volvov be pressed between a cover-slip and an object-slide so that it cannot move anymore, this 251 water current can easily be demonstrated by watching the little particles suspended in the water. If further we assume that by the dexiotropic torsion during the cleavage the direction of the main axis of the cells undergoes a dexiotropic deviation and the flagella thus beat in the direction of the arrows in fig. 165, then the dexiotropic rotation of the colony follows directly from this assumption. That indeed the flagella beat in this way needs no further proof, but follows from the rotation itself. It would be interesting no doubt if a variety of Volvox globator rotating to the left, were discovered. It can hardly be expected otherwise than that the cleavage bere will equally belong to the inverse type. Have we accounted now for the rotating movement of Volvo by the torsion presenting itself during the cleavage? In a causal sense we have, if our suggestion is right. But how is the torsion of the cleavage cells to be accounted for? Phylogenetically now I should feel inclined to consider the torsion during the cleavage rather as a consequence of the rotation of the adult animal than as * its cause. The study of ontogeny ever anew teaches us that we must not consider the structure of the adult animal phylogenetically as a product of the developmental processes, but we rather must account for the latter by the structure of the adult animal. Thus | would see also in the torsion during the egg cleavage of Volvow nothing but a very precociously appearing character of the adult form related to the movement of the latter. This character, which cannot be demonstrated in the adult form, could be revealed only by the study of its development. nna, 2 ie ; i 7h > iis, Ged eee ' bek wy yey nakie (Po tr dga TAA Af „. vil AKL J : a al fd) Uae colt sit ee 3 96 he E A Neer te iat 6 MEN AN, thei Mi me Bain sid | se ee RDR = ait iA gf fae Fy GRRL hil rs pon a ' Enid 8 a ae cae me de dt Mk AU, Hain IMA 1 pag “ALLA te : Enk aiff auverbe: en item ; aU ple Sse oh Beh il Tt A COBRA? 0 ed avi Es: ple lk: Senet fal DOr reir Qe en srt -2 Zalig Te, AO B AL ; ' 7 . : ti) Tent i ae gi: seit j oboe Ki ATIE : Duet, 7 ‘ : ‘ wer 1 'e ar we fas ive say Ae it be! | AP |. hl ui ai Poe emma, Biro) eT lord ad, nn 2 ite i 1 4 ye | * : ifs ri ze rah "ds i krt b it then ut ‘ iat krt ; a ‘ J y : _ = 4 ~ iy kj ’ » * Li \ : zie ir si r 4 baz ~ LS an s KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS VOLUME XXI N°. 3. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling," Vol. XXVI and XXVII). CONTENTS, W. J. H. MOLL and L. S. ORNSTEIN: “Contributions to the study of liquid crystals. III. Melting- and congelation-phenomena with para-azoxy-anisol”. (Communicated by Prof. W. H. JULIUS). p. 254. W. J. H. MOLL and L. S. ORNSTEIN: “Ibid. IV. A thermic Effect of the Magnetic Field”. (Communi- cated by Prof. W. H. JULIUS), p. 259. J. G. VAN DER CORPUT: “The primitive Divisor of xk_1”, (Communicated by Prof. J.C. KLUYVER), . 262. H: é BURGER: “On the Evaporation from a Circular Surface of a Liquid”. (Communicated by Prof. W. H. JULIUS), p. 271. L. BOLK: Oe topographical relations of the Orbits in infantile and adult skulls in man and apes”, p. i JAN DE VRIES: “Null-Systems in the Plane”, p. 286. JAN DE VRIES: “Cubic involutions of the first class”, p. 291. JAN DE VRIES: “Linear Null-Systems in the Plane”, p. 302. JAN DE VRIES: “Null-Systems determined by two linear congruences of rays”, p. 309. E. D. WIERSMA: “The Psychology of Conditions of Confusion”, p. 312. J. A. SCHOUTEN: “On the direct analyses of the linear quantities belonging to the rotational group in three and four fundamental variables”. (Communicated by Prof. J. CARDINAAL), p. 327. S. WEBER: “The thermal conductivity of neon”. (Communicated by Prof. H. KAMERLINGH ONNES), p. 342. J. E. VERSCHAFFELT: “On the shape of small drops and gas-bubbles”. (Communicated by Prof. H. KAMERLINGH ONNES), p, 357. ; J. E. VERSCHAFFELT: “On the measurement of surface tensions by means of small drops or bubbles”. (Communicated by Prof. H. KAMERLINGH ONNES), p. 366. A. SMITS: “The Phenomenon Electrical Supertension”. (Communicated by Prof. P. ZEEMAN), p. 375. A. SMITS and C. A. LOBRY DE BRUYN: “On the Periodic Passivity of Iron, II”. (Communicated by Prof. P. ZEEMAN’, p. 382). (With one plate)... “oyi' ' A. SMITS and J. M. BIJVOET: “On the System Iron-Oxygen”; (Communicated by Prof. S. HOOGE- WERFF), p. 386. “a A. SMITS and V. S. F. BERCKMANS: “On the System Ether-Chloroform”. (Communicated by Prof. S. HOOGEWERFF), p. 401. A. J. BijL and N. H. KOLKMEIJER: “Investigation by means of X-rays of the crystal-structure of white and grey tin I’. (Communicated by Prof. H. KAMERLINGH ONNES), p. 405. A. WICHMANN: “On Tin-ore in the Island of Flores”, p. 409. H. ZWAARDEMAKER and H. ZEEHUISEN: “On the Sign of the Electrical Phenomenon and the Influence of Lyotrope series observed in this phenomenon”, p 417. H. B. A. BOCKWINKEL: “Observations on the development of a function in a series of factorials”. (Communicated by Prof. H. A. LORENTZ), p 428. TH. DE DONDER: “Le tenseur gravifique”. (Communicated by Prof. H. A. LORENTZ), p. 437. J.C. KLUYVER: “On the evaluation of Z(2n +1)", p. 446. 2mi = th N. G. W. H. BEEGER: “Ueber die Teilkörper des Kreiskorpers K Ik ), (Erster Teil). (Communi- cated by Prof. W. KAPTEYN), p. 454. 17 Proceedings Royal Acad. Amsterdam. Vol. XXI. Physics. — “Contributions to the study of liquid crystals. III. Melting and congelation-phenomena with para-azoay-anisol.” By Dr. W. J. H. Morr and Prof. L. S. Ornstein. (Communicated by Prof. W. H. Jutius). (Communicated in the meeting of September 29, 1917). In our second paper on the extinction of liquid crystals we observed, that among others with para-azoxy-anisol there is, with regard to the extinction, a difference between the liquid crystalline condition which arises by melting the solid crystals (‘‘exsolid”) and that which arises in cooling the isotropic liquid (“exliquid”). Where with para-azoxy-anisol we had stated the existence of two solid phases, the question lay at hand whether in exsolid and exliquid we had perhaps got two different liquid crystalline phases. To make this out a research after the exact position of the points of transition was desirable. Method of Research. There we made use of a kind of radiation calorimeter, which is schematically represented in figure 1. The substance to be examined (about + em.) half filled a gold beaker M with thin walls, against which on the exterior on the one side a brass wire, on the other a constantane wire hasbeen soldered. The two wires form a thermoelement, by the help of which the temperature of the beaker and its contents is measured *) and serve at the same time to hang it within a brass cylinder. This < Se A 1) Before we came to this way of fixing up the “thermobeaker”’, we had followed the usual method for our determination of the temperature, i.e we had placed a thermometer and later on a thermoelement within the substance to be melted. Then a number of “Schmutzeffecte’” were produced, which on close investigation had to be ascribed to convection currents. Our method is of course quite free from this disturbance. 255 cylinder is closed by a cork at the top and one at the bottom and, that it may be heated electrically, it is provided with a layer of isolated wire. The whole stands on a little table under a glass, which only allows of passage to the wires of the heating-current W and those of the thermocurrent 7’. The research consisted in the determination of the temperature- time curve with a constant heating current. If this heating current is closed, the temperature first rises quickly, then more slowly and asymptotically approaches a limit value. If now (also before the limit-temperature is reached) the heating current is weakened and then kept constant, the temperature first falls rapidly, then more slowly till a second lower limiting temperature is reached. When within the temperature-region in observation the substance in the beaker melts, congelates, or in general undergoes some change of phase, this will be observable on the 7-¢ curve. So during the supply of heat to the beaker the melting will appear as a sharp twist in the ascending branch. The place of the twist indicates the melting-temperature and that with a much higher degree of accuracy, than would even be possible with a measuring of the temperature within the substance. The second contact place of the thermo-element was in melting ice during the time of observation. The thermo-current was measured by a quick-indicating galvanometer of Morr. To keep the deviations within bounds the thermoforce was first for the greater part com- pensated with the help of a constant current-source and a shunt, and besides the sensitiveness of the galvanometer was strongly reduced. The deviations of the galvanometer were registered and the 7-t¢ curves thus photographed. The figures 2—6 are reduced repro- ductions of our original photos. The figures put underneath give in an arbitrary measure the temporal value of the heating-current. An abscisslength of 8 em. corresponds to a quarter of an hour. The dotted line indicates the same temperature of about 118° in the different figures. Discussion of the Results. Fig. 2. The two ascending branches fully agree and at A the first point of transition shows itself sharply, i.e. the melting point of the solid substance. This temperature, which amounts to about 118°, we have always chosen as point of departure. The strong oscillations the 256 of temperature at B are without importance for our investigation and may be left apart’). 62-130) G- a- ne enone nee - oF §- - ~~ ~~~ - =o Ge 1 30--2 9 Fig. 2. The second point of transition is difficult to observe in the ascending branch on account of its steepness, in the descending branch it appears more clearly at 5. Further we observe in the descending branch a strong undercooling to far below 100° and then a sudden development of heat and congelation. The highest temperature reached in this process (13°,5 below the zero-line) is the point where a very unstable solid phase congeals, which we shall call phase II (phase I has the melting point at about 118°), and which after a short time spontaneously and under the development of heat passes into another 1) They have their cause in the fact that the volume of para azoxy-anisol changes considerably in melting. The internal sidewall of the beaker gets for this reason detached from its contents and can temporarily rise to a higher temperature, so that a drop, dripping from the solid centre, again occasions a sudden falling of the temperature. 257 phase. Probably directly into phase I, for at a supply of heat the same melting point of about 118° shows itself. Fig. 3. After the same initial development as in the case represented by fig. 2 the same deep undercooling is again followed by the appearance of the solid phase Il. But now we have to take care that immediately after this phase arises, heat is supplied by streng- thening the heating-current, and that to such value that phase IL is melted but the arising liquid crystalline phase remains undercooled. Whilst at # its temperature has become constant, we have, in order to hasten the process of congalation, reduced the heating-current, and in result of this the temperature has scarcely fallen a few degrees before under the development of heat the liquid crystalline substance congeals and now at a temperature of 2° below the zero-line. We call this new condition solid phase III. Fig. 4 also gives the origin of phase III from the undercooled 20° mer PIO rare Cane sen eeeere omaree enon acne ns-- ~~~ liquid erystalline phase. The heating-current during the preceding cooling was chosen in such a way that the formation of phase II was excluded. Fig. 5 gives once again the origin of plate III with the exclusion of phase II. But where the undercooled liquid crystalline phase in fig. 4 CD5=-) €--0 arrow cin ovne vanen pnneranns seme anne Ai neess soos canteens asso es shr ps dm ene eid a) eee | Fig. 5. was ex-liquid, we have taken it exsolid in fig. 5. Besides this figure also gives the melting of phase III and there the remarkable pheno- 258 menon presents itself, that phase III melts at a temperature which we know as the melting point of phase I. Thus it is shown that in this way phase IIT, during its formation, gradually passed into phase I. Fig. 6 is reproduced as it allows of studying the second point of transition. At J” ex-solid, at G ex-liquid pass into the isotropic 30° EE) IL] CIO nO P Cene 1Ode- 2 owen ereevenss IJOeeverme ed G+ wee eeee- lo) eer) Fig. 6. phase under the absorption of heat, AH gives the phenomenon of transition while heat is developed and the three transitions FG and H happen really at the same temperature. CONCLUSION. Whilst thus, as far as the situation of the points of transition is concerned, we have found no indication of a difference between exsolid and exliquid with para-azoxy-anisol, we have discovered several phases in the solid condition of aggregation in our investigation. Beside the three phases which we have distinguished as I, I] and III, there certainly still exists a fourth *), with a melting-point of about 108°. It seems that this phase, which during this investigation never once showed itself, can only exist in capillary layers (between glass). This short, more or less schematic: summing up of the most striking facts, which present themselves in an investigation of the melting- and congelation phenomena of para-azoxy-anisol, must suffice, however interesting a closer examination of this substance and perhaps of other substances according to the method indicated may be from a standpoint of phase-theory, for our interest is greater for problems of a different nature. Physical Laboratory, Institute for Theoretical Physics. Utrecht, September 1917. 1) Cf. our second contribution. Verslag Kon. Acad. v. Wetensch. XXV, p. 1114. Physics. — “Contributions to the study of liquid crystals. IV. A thermic Effect of the Magnetic Field”. By Dr. W.J. H. Morr and Prof. L. S. ORNSTEIN. (Communicated by Prof. W. H. JULIUS). (Communicated in the meeting of March 23, 1918). The fact that the particles of liquid crystalline substances are directed by a magnetic field, justifies the question whether perhaps the action of the field, may entirely or partially manifest itself as heat. We shall in this paper develop the results of an investigation into this matter. We made use of an arrangement for this investigation, which in the main points is the same as that described in our previous paper *) on the subject. The p-azoxy-anisol was again heated in a little gold beaker, being within a little oven which was heated by elec- tricity. But our little oven had to serve in this case as thermostat, and above all it was necessary, that the temperature inside of the oven was as far as possible equal everywhere, so that a temperature gradient within the substance to be investigated would be excluded. Instead of our original, very primitive little oven we fitted up as such a brass tube 10 cm. long and 2 cm. wide, provided along pretty well the whole length with a single widening of their manganin wire, closed off at the bottom by a brass plate, at the top by a brass screw-stopper, in which only a narrow opening to leave pas- sage to the thermo-element. To present a current of air along the heating-wire, the latter was wrapped up in chenille, and closely around this there was a double brass mantle, through which water circulated of the temperature of the room. This arrangement was put (with the axis of the little oven in a vertical direction) between the poles of a Dubois-magnet, and its (horizontal) field may be looked upon as homogeneous at the place of the beaker. The magnet could be turned round a vertical axis. The electromotive force of the thermo-element was almost fully compensated, the changes arising were registered by photography. In our investigation as to whether the connection of the magnetic field has a thermic effect, we arrived in the beginning at results, 1) Compare the foregoing paper. 260 apparently in contradiction to each other; until at last with the help of a thermo-element of a peculiar structure we have come to understand the phenomena which dominate the effect. In fig. 1 this thermo-element is represented, as it is hung within the gold beaker 5. (The wall of the little beaker must be thought transparent). 6 is attached by two pins S to the screw-stopper of the little oven; ZD are two wires twisted together, brass and constantane, which are soldered by their ends to a thin plate of silver-plate Z(5 X 7 mm., 0,02 mm. thick), which is immersed for somewhat more than half in the p-azoxy- Fiel, …antsol With the help of this thermo-element we have indeed been able to state the thermic effect, but learned to distinguish between the effect of a transversal field (with the lines of force perpendicular to the silver-plate) and of a longitudinal field (with the force-lines // the silver-plate). TRANS V, TRANSV. LONGIT. <-- 75 GRAAD CELSIUS" "* Fig. 2 Fig. 2 may make this clear; a curve, taken by photography, is represented. At A a transversal field is connected with the result that the temperature of the silver-plate rises slowly; 15 sec. later at B the field is broken up and a still slower cooling of the plate is the result; 15 sec. later, at C (before the temperature has as yet regained the original value) the transversal field is put on for the second time; 15 sec. later at D it is again broken up; and at last 15 sec. later a longitudinal field is put on at # and this causes a quicker reaching of the thermostat temperature. A new connection of the longitudinal field remains without effect’). The principal conclusion from our experiments is this, that the effect observed must have its origin in the immediate vicinity of 1) It need hardly be mentioned, that in the isotropic phase the connection and disconnection of the magnetic field offers a thermic effect. 261 the metal wall, and that within the liquid crystalline matter a magnetic field offers no thermic effect or only a very slight one. For then the effect of a transversal and a longitudinal field would have to be equal. Where the influence of a magnetic field means in the end the direction of the more or less stretched particles of the liquid crystalline substance we come to the conclusion that against this directing there is only important resistance to be overcome near the wall. Now we have accepted an action of the wall for the explanation of some extinction phenomena’), and that in such a way that the wall directs the particles parallel to itself. The resistance which is to be overcome for a transversal field is quite in accordance with this way of seeing the question. Then as regards the nature of this resistance, we might imagine it to be of elastic origin; the particles would then get another form at the cross-action of field against wall (heating), and return at the disappearance of the field again elastically to their original form (cooling), in which proces the longitudinal field hastens this return. Conclusion. With para-azoxy-anisol the thermoeffect of a magnetic field is investigated with the help of a thermo-element of peculiar structure. The investigation shows that an effect is only apparent at the wall of the thermo-element, which effect probably has its origin in the elastic change of form of liquid crystalline particles. Utrecht, Febr. 1918. . Physical Laboratory, Institute for Theoretical Physves. ') See our first and second papers. Mathematics. — “The primitive Divisor of xk—1.” By J. G. VAN DER CorputT. (Communicated by Prof. J. ©. Kiuyver). (Communicated in the meeting of September 29, i917). This paper is an extension of the article of Prof. J. C. Krurver: “The primitive Divisor of a"—1.” (These Proceedings, Vol. XIX, page 785. mie Definition 1. If & be a positive integer, the product H(a—ek), c extended over all the values o of a reduced rest-system, modulo &, is called the primitive divisor /(«) of «*—1. Definition 2. If # be a positive integer, y = (4) represents the number of positive integers =, which are prime to 4. Proposition 1. If 4 be a positive integer, then the primitive divisor of «*—1 is a polynome of the degree gp. Definition 3. The numbers 4, (754350) are defined by the relation Bae) Sky a, i=0 k being an arbitrary positive integer. Definition 4. In the functions x, (n‚k) of the variable integer n (k being a positive integer), which are called the arithmetical characters of n, modulo &, » represents an arbitrary integer, prime to hf. The functions ,(7,4) and ¥,(7,4) are identical or different, according as a and rv are mutually congruent or incongruent, modulo &. Hence it follows, that there are different arithmetical characters x(n,k), modulo & and these functions possess the following properties: lL. wnd OE) bm I. y¥,(n,4) = y(n,k) if mn (mod. &). III. The modulus of y,(n,k) is equal to O or 1, according as n and & are commensurable or incommensurable. IV. y,(n,£4) is equal to O or 1, according as n and & are com- mensurable or incommensurable. ; 1 V. y-1(,h) is equal to the symbol (+) of LEGENDRE. WA Sky T. 263 , . MIE = yx (n, k) = 0, if vat=1 (mod. k). il VIII. 1 (nb) ge eS (tr B if & be equal to the product of the two incommensurable integers k, and &,. Definition 5. y(n,k) and (nk) are {wo conjugate functions; they are, therefore, identical, if y,(x,k) is real and they are conjugate imaginary, if y,(n,4) is an imaginary function. Proposition 2. ¥,(n,/:) is an arithmetical character of », modulo Z. Proof. Each function of , satisfying the conditions 4.1, IT and III, is an arithmetical character of n, modulo 4. Definition 6. The functions a,(n,k) of the variable integer x, wv and & ‘being incommensurable numbers) are defined by the relation 2 imn 2 dm 5S Yo (m, he Ok a, (n, k) 5S 4, (m, ke wk mil M= Proposition 3. If % be the product of the two incommensurable numbers 4, and &,, each of which is prime to the integer », then we shall have dy (ns kj) ay (a, He) =a, (n, Z). Proof. In the expression by Qramyn ke Qrimy A Zil Koehn Ey (nk, Je HS Vy y,(m, ‚kje ky = 5 3 ly (mk) Ly (mk )e k ml mol Pee Mol We Mave M= 12, a and Mm, => Lid, Se ae ky We mar make mik, + m,k, congruent to m (mod. k) and k2 m21. Then we Wave nm =d ok and % (m, k) = Yo (mm, k,) po Om, zn (according to 4, VIII) = Yo (m,k,, ky) x (mok, ke) (according to 4, II) = 1) (kk) 4) (45%) Ly (mk) Yo (m,,%,) (according to 4, U). Consequently Qrimyn 2rüngn k 2rimn re) (ae 5S Yo (mk, je ua oy Ho (Mm, sky) e Ie ee = Ve) (m, k) e k ml mal m=1 and (make 7» = 1) a 7 ra tape EE in lg Ve (hay Rid hr (BE) 5 1 (m,,k,)e % = hv (m,,k,)e ke Fos 4, (m, he k 1— Mog mz The first two factors occurring in these formulae, are according to 4,III not equal to zero, because 4, and k, are incommen- 264 surable. If the last two formulae be divided one by the other, we shall obtain, according to definition 6 a, (k,, n) a, (ko, 2) = a, (A, n) - Proposition 4. If the following conditions are satisfied : k, and k, are two squareless incommensurable numbers and their product is equal to 4, n is an arbitrary integer and v an arbitrary number, prime to &, Dis the’ GO. Dof pr Ask andes D, is the G. C. D. of v—i, n and &,, Dis the GG) Dior web nand 2 then we have k k u(D,) p D) x» (» i) a (D,) p(D.) yo (« i) == (DP (DE (» 5). Proof. D, and D, are incommensurable, because 4, and 4, are incommensurable. The numbers D, and D,, therefore also D, D,, are divisors of Dand Disa divisor of D,D,; consequently D,D, = D. Hence it follows u(D,) u(D.) = u (D), g(D,) PD.) = oP) ( k, ) ( k, ) >) ts | hy WON Eel BRON es dD: Dp: D according to 4, VIII. Proposition 5. If » be an integer, prime to the positive integer k and the integers n and n’ satisfy the relation nn’ =1 (mod. &), and then we shall have y, (7,4) =, (n’,h). Proof. From the relation nn’ =1 (mod. k) it follows that n and n’ are prime to & and according to definition 5 and 4, II] 4 (n,k) and y,(n,k) are two conjugate functions with modulus 1. Consequently yo (n, b) Yo (rn, B) = 1. Moreover we have, according to 4. I, II, and VI, (sb) Yo (n, b) = Yo (nr', | =H (1, A) = 1, hence urls) =H (n'. k). Proposition 6. If / be a squareless number, prime to v and D represents the G. C. D. of v—i, n and &, then we have k a, (n,k) = u (D) p (D) wo (». 5) 3 E 265 Proof I. & is a prime number. a). k is a divisor of v—1 and n; consequently D = &. k uD 9 Dn (m5) == DH OD = —(k—1), according to 4. IV, 2Qrimn Bi Boe nr (Why RYE. Perm br hed m==1 m= and Meh 2rim a anim Diy, (my ke # sek ==, m= ml consequently, according to definition 6, 2. (4 = — EN = UD) 0 (Dn (m5). D B). If k be a divisor of n and not of y—1, we shall have | D=1, k ; Yv (». 5) = x, (n, k) = 0, (according to 4, IIb), Qrimn i > plm he k = = ym, k) ml m1 hf (according to 4, VID, consequently ay. (2; 4) =O, so that now both members of the sought relation are equal to zero. y). Let & be no divisor of 7, so that we may make nn! = 1 (mod. k), mn=m' (mod. k), k2m' 2 |. According to proposition 5 Yo (n, k) = % (n', k), consequently %, (n, k) Ys (m', B) = y(n’; b) yo et) Ki). = fo (mn, hi) = 4 (m, fy), because m'n’ = mnn' =m (mod. k). Hence it follows alt 2Qnimn ates 2m’ > y(mbhe k =y rhb = y(m, ke *, m—) | consequently k a, (n, k) = fv (n, k) =u (D) p (D) vo (». 3) ; because D = 1. Il. & is the product of two different prime factors. Take k = kk, 266 (k, and k, are two different prime numbers). According to I we have k a, (n, k,) = u (D,) p D,)% (x i) 1 k a, (n, k,) aN (D,) Pp (D,) hy (>. zE) » D, being the G. C. D. of »—1, n and 4; D, being the G. C. D. of »—1, n and &,. If these two formulae are multiplied, we shall obtain a,(n, k) = wD) gD) 7, (x. 5) according to the propositions 3 and 4. HI. & is the product of three or four different prime factors. Take & equal to the product of the two incommensurable numbers k, and k,, each of which is equal to a prime number or to the product of two different prime numbers. The proof is given in the same way as in II. etc. Proposition 7. If & be a squareless number, prime to the integer nedd MES represents the GGD. of wt, n-- ) and 4, then the in definition 3 defined coefficients A) satisfy the relation B sek afk = Au Das) YL (Dr) % | n+A, —— | =—0, ji Da whatever be the value of the integers 2 and rv. Proof. Qximn 2QriAm gam kerb 2 Aye BeSS05 10 for, if m and & have a common factor, x, (m, £) = 0 according to 2 im 4, IIl and if m is prime to &, e © isa primitive root of the equation tk—1 = 0, i.e. a root of the equation P Fy. (a) = 2 Ava = 0: A=0 so that then the last factor is equal to zero. Hence i. 2rimn Aim O=E yn he k FS Aje * m=1 A0 ° jk 271m (n4-)) =E AS yplmhbe kk , ==() m=1 consequently ? k en ay Oe A WD) Gr) ae ee = yx (mke*® |, M1) n |. 4 \ ml L according to definition 6 and proposition 6. 267 The last factor is not equal to zero, for then the sum ee 2Qrimn rh Ito Zim (ms ke = =p(D,) PD @ =] = x (m ke k nh n/mn=sl would be equal to zero if n—=1,2,3,..., k, that is the equation fe =. ¥ (Gn, hi) at = 0 mzzl 2 min would possess the root zero and & different roots e * , and this is impossible, since the coefficients yx, (m, hi) are not identically equal to zero, according to 4, IIi. Consequently we conclude ? k = A) D;,, » fp DE >) Yv Ay See PSN cS u (Ds) Pp (Dats) % (14 a Proposition 8. If & be squareless, then we have 5 A) u (D'n+-a) f (D'r) = 0, 10 D', +x being the G. C. D. of n + A:and ke. In this formula n may be any integer number whatever. (This formula is to be found in the above mentioned article of Prof. J. C. Kruyver). Proof. Make in proposition 7 » ‚ then we shall have Da =i Da and | stapt) Jo | n 5 == Uit Are in h Da ; Dia according to 4, IV, since n-+ 2 and aie have no common factor, nA because & is squareless. Hence it follows if can #6) u(D'n43) f (Drs) =, EET (25) Le 41=0 k n-+-4 (=) being the symbol of Legendre. In this formula n may be Proposition 9. If £ be an odd squareless number, then we have any integer whatever. (This formula is also to be found in the article of Prot... C: Kivyver). 268 Proof. Make in proposition 7 » = — 1: then we have D,4, = 1, since » — 1 —— 2 and & is odd. Hence (Dan) == A p En = p(l) = 1, 2 Xv (Hi zn 1 (n HA, k) = (+ ) according to +, V. Consequently Definition 7. If & be an integer > 2, the coefficients B, and C; G SS 0) are defined by the relation 2 nia =a ne gk jie DS B; a a he) and Qnit f nl) ion cs ==) in which the products are extended over all the values o and rt of 9 a reduced rest-system, modulo &, for which (5) + and (ent Proposition 10. If & be an odd squareless number, we have = Il M ve le oe rel = 0 and 2 Ci Cn) = 0, I | B bea" Vk k A n=u(s wij (7): als (1)? Vk, D', being the G. C. D. of 2 and &. In these formulae n may be any integer whatever. | ~ if we make and Proof. D',, is greater than 1, if (5) == 0, according to 4, IV; consequently 269 = u(d) zm > uld) = 0 if Gi 0. dk UID, k d/m g m E lia If be) =1, the equation © B, 2’ = 0 has a root e * , hence ; pl) £ Qrims - d BPR 0 it (7) ; dsl) k Consequently a 2m) m 1+ (=) 2 Byé © - Sa (=, kN i=0 Els for the third, second or first factor is equal to zero, according as (2) is equal to 0,1 or —1. e k Hence k 2rimn a ri 2nimd Sea. grenst gk 1+ (EE Be Sao m=1 k ]\j=0 d\k lon he : a 2rimn 5 2nima Se FS Be = mld) mI 1=0 d\k d|m 9 k Zin) 2 d Ks = 26 elder U (make m = od) 20 d\k p=! À k d! 2ric(n—-d) k = By (5) meee ( make == 1) == d'\k d fl d Me] 2rie(n4) Because the sum > e € — is equal tod’ or O, according as (n +2) o=l is divisible or not by d’, we have is 2 k GERE u(5)-4 41=0 d'\k -d d'| (nà) n : k A Since k is squareless, — — and we have no common factor, so nd C k k Dita : ai D'n tl ae ; : s reek i en je MOre- iat u (3) u (aes 7 ) is equal to u (rea al 7 ) More over we have 18 Proceedings Royal Acad. Amsterdam. Vol. XXI. Da ! 1 =| tu d= y (D+), consequently g 2 je Die mes SB) > o( a A=0 D n+) / q! | D'n d k _ BS B, u Sn PP D' +). (5, ) ( 2 ) ì=z=0 m ; If m and & have a common factor, (7) =O and in the other case we shall have = ud) =u(l)=1, dik dim hence me m € dik ) k dl m and k 2Qrimn a £ 2Qnimd Ss == = 2 k @rezx k = u (d) m=1 1=0 d|k d|m k a 2rim (n-+-2) ; ae RN msi k il) k according to the theorema of Gauss. Hence we conclude di . Dr % i 0 nn S, st S, —— 2B B) loen) @p(D'n+2) + (5) za (k—1)? val == Da k ww | vo |-6 Ul M ro |-s > M | M > A0 By changing the symbols of Legendre every where into their opposite values, we find in the same way the relation ah 2 = C; A = 0: Nee Physics. “On the Evaporation from a Circular Surface of a Liquid’. _ By Dr. H. C. Burerr (Communicated by Prof. W.H. Junius). (Communicated in the meeting of December 29, 1917.) In a publication recently published Miss N. THomas and Dr. A. Frrauson') communicate observations concerning the evaporation from circular water surfaces. These observations are made under different circumstances viz. in a dark, very quiet room, in a lighted room and in the open air. It appeared, that in every case the quantity of water evaporated in unit time, might be represented by: E= Km, in which 7 is the radius of the water surface and K and n are constants that, except on the external circumstances, also depend upon the distance of the surface of the liquid and the rim of the basin in which this is contained. Now while, as the writers remark, usually in the literature the opinion is found, that the evaporation is proportional to the area of the surface, i.e. that m= 2, it was shown by their experiments that this exponent was always between 1 and 2. Now Srrran*) has treated the evaporation from a circular surface of a liquid, supposing that the vapour diffuses in the space above the plane in which the level of the-liquid lies, while at the liquid the concentration of the vapour is a constant. The result of the computation is, that the speed of evaporation is proportional to the radius of the surface. So it is apparent that in the experiments of THomas and Frreuson the conditions that STEFAN supposes in treating the problem, are not fulfilled. As | have already been engaged for some time upon the theoretical and experimental treatment of the diffusion in a flowing liquid *), it was of importance to inquire whether my results agreed with the above mentioned investigations. For this purpose we must extra- polate the values of the exponent 2 for the case that the surface of the liquid is on a level with the rim of the basin. When this is 1) Phil. Mag. XXXIV p. 308, 1917. 2) Wied. Ann. XVII p. 550, 1882. 3) My principal purpose in this is to investigate whether the solution at the surface of the crystal is saturated or if perhaps, when the solving takes place sufficiently rapidly, an undersaturation arises. 10 272 not the case it hardly seems possible to apply the mathematical analysis to the problem. For the three cases the extrapolated exponent is resp. 1.4, 1.5 a 1.6 and 1.65. In the last case, in which we are most certain that the air above the liquid is in continuous movement, 72 proves to agree quite sufficiently with the theoretical value °/, = 1.67, which will be deduced hereafter, so that therefore in this case we may be sure, that the air-currents effect the evaporation. In experiments in more quiet air, the values of m approach the value n=1 more closely, which value is found by STEFAN. In the following sections we will give a theoretical treatment ot the diffusion in a flowing gas. As the evaporation from an arbi- trarily formed surface is easily deduced from that of a rectangular one, we firstly choose this last shape, We imagine the space above the plane z=0 filled with a flowing gas, while the plane z=0O itself is formed by a fixed wall, of which a part consists of a surface of the liquid. Let this part have the shape of a rectangle with its sides parallel to the axes of w and y, situated at positive y and bounded by the axis of «. Further we will choose the velocity of the gas to be parallel to the axis of y and to be proportional to z, SO v, = az. As namely the gas at the plane z= 0 through external friction must have a velocity equal to zero, we may put: Vy 2 Oke by ee ess and we may neglect the second and following terms of this series when as will generally be the case, the vapour is concentrated in a thin layer above the plane z= 0. When we put c for the concentration of the vapour and D for the coéfficient of diffusion then, as is easily seen, c fulfills the altered equation of diffusion : 0 „=De div (ed) Se we eee Tee 1 7 Further we suppose that c at the surface of the liquid fulfills the boundary condition : 1) The last term in the second member may be explained in this way: In the element of volume dx dy dz flows through the element of surface dy dz an amount dy + y Oe OE these amounts also for the axes of y and z, we get for the total amount that flows outward div (cv) dx dy dz, when v is the velocity, considered as a vector. of vapour : vx dy dz inward and (Cvx) | dy dz outward. By computing 273 cms C.*) in which C is the concentration of the saturated vapour. dc Now we will suppose that the state is stationary, i. e. that SS 0. Then c satisfies the equation: D Oc Ò% | cal We ee Oa? | dy? dz? Biak Ont ROD while v, = vz= 0 andv, = az. In this equation we will further take dc le Of course this is only approximately true for the values U of w that concern points within the rectangle. For points beside the rectangle c will be very small only when pel is small with res- a pect to the dimensions of the rectangle, which we will always suppose. So we will treat the problem as a twodimensional one, i.e. as if the rectangle has an infinite breadth in the direction of X*). So we 3 will neglect —. dz? 3 a C Finally we remark that D being large, consequently D aga may Y 2 be neglected with respect to az = One might object to this when y c is zero or very small, but then is c= C or at least then c is approximately a constant, so all terms of the differential equation 2 : 5 : c are zero or very small, and so it will be allowed to omit aca That y? 0? D sr may not be neglected, notwithstanding the small factor D, z is caused by the fact that the evaporated substance will be concen- , ; dc trated in a thin layer, so that c varies rapidly with z; a, ond & therefore are large. After these simplifications the differential equation for c becomes: dc a dc dz? D oy As, in consequence of a sufficiently rapid stream, a diffusion II) 1) When by the rapid evaporation an undersaturation arises, this will probably be proportional to the speed of evaporation. 2) Experiments with crystals that solve in a flowing liquid, have confirmed this supposition. 274 against the stream is impossible, we suppose with regard to the fact that the arriving gas is free from vapour that ¢ = 0 for y = 0. For- the same reason we may further assume that the surface of the liquid extends from y=0O to y=o, while for arbitrary y the concentration will not be influenced by the presence of liquid at ‘the boundary z=0O at greater values of y. As was already said we take for z=0O the boundary-condition c= C, while for z=o c of course must be zero. Problems of this kind may be solved in a general way by making the range in which z may vary finite, further by constructing a solution with the aid of a series of proper functions, and finally by going to the limit, whereby the range is made infinite. I hope to explain this method at length in my dissertation; here however it may suffice to give a much simpler treatment, because the purpose is only to find how the quantity of liquid evaporated in unit time depends upon the length of the rectangle, i.e. upon y. When we introduce in (III) as a new variable: a 5 SSS gr == 9 D this equation assumes the form: 07¢ : 0c UH ) —— oF. a OG Go Oy The boundary conditions of c are here: CU for y= 0 Cas Se c=) 5 S= The solution of the transformed equation will not contain a or D, because these quantities occur neither in the differential equation nor in the boundary conditions. Therefore is: 1 =v Gn =o (- B ‚). The quantity of the liquid that evaporates in unit time from the part of the surface between y—O and y is found by computing the quantity of substance that flows through a plane perpendicular to the axis of y. As the velocity of the gas is az, the quantity of vapour that flows in unit time through a unit surface perpendicular to the axis of y, is azc; so the total mass of vapour that flows away per unit breadth in the direction of 2, amounts to: 275 B= fas cdz =a f: y (: bn, yt 0 0 a When now we introduce again 5 = 2 P/ D this quantity becomes: zg DN | IE (e =) fs p(&y)d =a'k Dh p(y) . . (LVa) a 0 We may transform (III) also by putting: D i Ye a Then we get the equation: Oey ude pe dz? SSS Oy . ° . Cs . . ° ( ) To this belong the boundary conditions : CaN Tor tags ce ke a= == 0 ná 2 => aor Here again the solution will be independent of a and D viz.: D =fen=f(arz): From this we find: = hr at 8 D == azede=afes(z¥ de == ak v=) hl Eel7D) a a 0 0 When now we compare the found values of H, (IVb) proves to agree with (IVa) only when: F (p) = A. ph, where A is a constant. So 4 becomes: vhs Dl; iB) = ia; A ES A Dale ee oe al a EV) a's Of this result the fact that in the first place interests us is that E proves to be proportional to ys. To deduce from the acquired result what 2 becomes for a surface of an arbitrary shape we imagine that this surface is divided into narrow strips with the long sides parallel to the axis of y i.e. to the current. As the breadth of these strips may not be taken too small when we wish to apply the acquired results, but on the other hand may not be too broad when we want to consider them as 276 rectangles, it proves that the circumference of the surface of the liquid may not be too irregular and also that the linear dimensions D of this surface may not be too small with respect to V—. a Then however for each of these rectangles /# is proportional to the breadth and to the */,”/ power of the length. The total value of fF is found by integrating over the whole surface, and it is easily seen that this quantity for conform figures is proportional to the ‘/,’¢ power of the linear dimensions, of which this exponent °/, as it were refers to the length and °/, to the breadth. As ali circles are conform it is proved by this that the evapora- tion from a circular surface of a liquid is proportional to the */,rd power of the radius as is also found by Miss Tuomas and Dr. Frerauson, when the circumstances were in agreement with those that are used at the theoretical treatment given above. The theory that is given here I have found confirmed by expe- riments of the solving of erystals in a flowing liquid, which will be treated in my dissertation. The quantity of the solved substance proved to be proportional to the */,"¢ power of the velocity of the liquid, with the breadth and with the */,"¢ power of the length. Institute for theoretical Physics. Utrecht, December 1917. Anatomy. — “On the topographical relations of the Orbits in infantile and adult skulls in man and apes’. By Prof. L. Bork. (Communicated in the meeting of March 23, 1918). In the Proceedings of this Academy of 1909 two papers by the present author were published, dealing with the position, shifting and the inclination of the Foramen magnum in the Primates. In these papers it was shown that the topographical relations of this Foramen in the infantile skulls of the Primates and more parti- cularly with the Anthropomorphous apes present only small deviations from those in the human skull. It is only in their subsequent growth that a difference between the development in man and the Primates becomes apparent. This difference comes in the main to this that in man the original topograhical relations, such as are found in the infantile skull, are permanent, the skull retaining infantile characteristics; in the remaining Primates, on the other hand, and especially in the Anthropoid apes, these juvenile conditions are replaced by others. The chief phenomenon, which may be briefly stated afresh here, is that in infantile skulls of man and anthropoid apes the foramen magnum lies in the middle of the cranial base, and during growth is shifted backwards over a longer or shorter distance in the direction of the occipital pole of the cranium, while in man it remains situated in the anterior half of the cranial base. It is difficult to reconcile this result of my investi- gations with the conception, often met with in literature, that the more occipital! position, as found in these apes, would be the original one, so that it would be in man that a forward shifting would take place. Now of such a forward displacement, presumed on theoretical grounds, nothing appears during individual development in man. On the contrary. From about the eighth year, i.e. in conjunction with the commencement of the loss of the milk-teeth, also in man a slight backward shifting is stated, which is not of much significance, however. So the characteristic difference between the human and anthropoid skulls is that in the former infantile, not to say foetal, characteristics are retained. While the infantile skulls of man and anthropoid apes thus show a great similarity in this respect, the adult skulls grow dissimilar, and it is not the 278 human but the antropoid skull which deviates more and more from its original shape. The object of the following communication is to draw the attention to an analogous phenomenon in an entirely different part of the skull, namely in the orbital region, and regarding more particularly the following question: what are the topographical relations of the orbits in infantile and adult skulls of Primates? The answer to this question gives an insight into the phenomena of growth in this border-region between the cerebral and faciaf skull. These are well fitted to give a definite shape to our conception about the morpho- genetic relation between the human and anthropoid skull. In this communication the main points only will be stated, the more extensive paper will be published elsewhere. For the present purpose the best starting-point is a form in which the differ- ences in topography between the infantile and adult skull are as large as possible, their character thus being clearly revealed. The Gorilla skull serves this purpose well. We shall mainly deal with the topographical relation of the orbits in regard to the cranial cavity. The easiest way of surveying this Fig. 1. 279 is by means of horizontal sections, passing through the middle of the orbits. In fig. 1 such a section is sketched through the skull of a young Gorilla child, in fig. 2 through that of an adult male individual. In the lateral wall of the orbit in the infantile skull two parts may be distinguished, an anterior one which borders the orbit outwardly and forms the free outer wall of the orbit, and a posterior one forming a partition between the orbit and the fossa media of the cranial cavity. Between these two parts the lateral wall of the cerebral cranium is connected with the lateral wall of the orbit. This arrangement implies that the cranial cavity partly extends laterally of the orbit, in other words that this cavity partly enters into the Cavum cranii, so that there exists a common partition-wall between the Cavum orbitae and the Cavum cranii. Upwards in the direction of the roof of the orbit this partition-wall between the two cavities becomes larger, as the cranial wall frontally more and more joins the supra-orbital ridge. The free exterior wall thus becomes smaller and is entirely lacking near the roof of the orbit in the youthful Gorilla skull, as the cranial wall is attached to the orbital roof along the supra-orbital arch. Thus the whole orbital roof bas become the partition between this cavity and the Cavum cranii. This means that in the infantile Gorilla the orbits lie entirely under the cranial cavity. How is this in the adult skull? It appears from fig. 2 that now on the lateral wall of the orbit the just-deseribed two parts can no longer be distinguished; the posterior intracranial part has disappeared, since the lateral wall of the skull is attached as far backward as possible to the lateral wall of the orbit. The whole lateral wall has become an outer wall. From a topographical viewpoint this means that the orbit no longer enters into the cranial cavity, but has come to lie before it. This conclusion is confirmed by a closer examination of the orbital roof. In the infantile skull the frontal wall of the cranial cavity is attached to the orbital roof along the circumference of the orbit, which means that the whole roof of the orbit forms a partition between the cranial and orbital cavities and does not form a free exterior wall. In the adult individual, on the other hand, the cranial roof is attached to the orbital roof very much towards the back, as is seen from fig. 38, representing a sagittal section through the orbit of an adult Gorilla. The roof of the orbit has here for the greater part become a free exterior wall. From this short comparison it already appears that the topogra- phical relations of the orbit with regard to the cranial cavity are 280 very different in the young and the adult Gorilla. This difference may be briefly summarised as follows: in the young individual the orbit for the saute part enters into the Cavum cranii, in the adult individual it lies before the cranial cavity. So there is a forward dis- placement during growth, caused by lengthening of the orbit in a forward direction only. By the aid of figs. 1 and 2 this can easily be proved if the Septum orbitale is particularly kept in view. In both figures the see- tion passes exactly above the Lamina cribrosa, i.e. through the anter- ior extreme part of the cranial Pig: a, cavity. In the septum orbitale of the infantile skull three parts may be distinguished, a middle one, formed by the Lamina cribosa, an anterior and a porterior part. Also in the adult skull these three parts are visible in spite of the pneumatising. Comparison now shows that the lengthening of the septum is almost entirely brought about by the increase in length of that part of it which lies before the lamina cribrosa. One has only to compare the dotted lines in the two figures, indicating the plane through the anterior edge of the Lamina cribrosa. These lines are also serviceable for gaining an insight into the forward shifting, resulting from this mode of growth. In the small young skull almost the whole of the orbital cavity lies behind this line, in the adult skull only the posterior part. Thus the growth of the skull of Gorilla has an evident influence on the position of the orbits with regard to the cranial cavity. That this is accompanied by a considerable change in the shape of the orbital cavity, is also perceived by comparing figure 1 and 2. In the adult skull the posterior part of the orbit has been drawn out in the shape of a funnel or canal. The change of position of the orbit caused by growth can be illustrated in a simple manner by projecting the outlines of this cavity on the median plane, which is easily done by means of the well-known Martin pantograph. Fig. 4 shows such a projection taken from the skull of a Gorilla child in which the tooth-change had commenced (the medial incisors have been changed; fig. 5 a similar projection of the skull of an adult man’). The cranial base is partly, the 1) Fig. 5 is on a smaller scale than fig. 4. oe A. SM PLATE I. AWA ent in fig. 1. Potential measured 1.5 cm. from the bottom. 8-43 milli-ampère/cm?. 1 period 2.7 seconds. N \ i Aha KAD AAn A j A f h A | N Kn din iI | ale laa fun A HENEAANAN PON VEV En En Fig. 5a. Electrode, solution and les tential measured arrangement siphon for potential nt density 30—34 measurement as in fig. 5. Current iod 2.2 seconds. density 28—32 milli-ampére/cm’. V Hf NA oA AR | nie VU liv dali AV UV VA Fig. 3 ution as in fig. 5. Potential measured 1 mm. under the krent density 47—53.5 milli-ampére/cm?. 1 period AC Fig. id long of a diameter of 3 mm. electrode spirally wound \tential measured at the second winding from the top. ‘rent regularly periodical at + 0.5 ampere. A. SMITS and C. A. LOBRY DE BRUYN: “On the Periodic Passivity of Iron. II”. PLATE I. EERENS ANNAN AANA AY en RI Wy UV Ui UV UE U VUT IUI WE VREHEN Ld Bane Fig. 1. Sealed-in iron electrode 1.5 cm. long, of a diameter of 3 mm. Solution contains per Liter 0.72 gr. mol. FeSO, + 0.014 gr. mol. FeCl,. Potential measured halfway the height. Current density 28 —33 milli-ampère/cm?. l period 6.15 seconds. Difference of the extreme values + 1.7 Volt. DA AMIN OLINE ENNE DS PMN NINES, fA MUU UU IV HUD UD UL Fig. 2. Electrode, solution and arrangement siphon for potential measure- ment as in experiment in fig. 1. Current density 25—30 milli- ampérecm?. | period 7.8 seconds. DANA AANA nyo ANN KV il) vn ne Fig. 3. Electrode 1.4 cm. long, of a diameter of 3 mm., not sealed in, solution as in the experiment in fig. 1. Potential measured at the bottom of the electrode. Current density 32—36 milli-ampére cm? 1 period 5.25 sec. Fig. 4. Electrode 5.1 cm. long of a diameter of 3 mm., not sealed in. Solution as in the experiment in fig. 1. Potential measured 1 mm. from the bottom. Current density 24—39 milli-ampére cm?. 1 period + 5.1 seconds. PLATE II. eae ed Pelee eh Hy | POCO HE COLTER ERECT RCIA ESRI Fig. 4a. Iron electrode 5.3 em. long of a diameter of 3 mm. not sealed in. Solution as in the experiment in fig. 1. Potential measured 1.5 cm. from the bottom, Current density 38—43 milli- ampérecm*. | period 2.7 seconds. Wired HN NN Vane tie ey } Wi ide| 2 URIS AS YS OER Pa tA vv Fig. 5. Iron electrode 5.1 em. long of a diam. Fig. Sa. Electrode, solution and of 3 mm. not sealed in. Potential measured arrangement siphon for potential halfway the height. Current density 30—34 measurement as in fig. 5. Current milli-ampère/cm?. 1 period 2.2 seconds. density 28—32 milli-ampére/cm?. \ EAA Ni) id AVA Wilf NITEM Wall dv Fig. 6. Electrode and solution as in fig. 5. Potential measured 1 mm. under the liquid level. Current density 47—53.5 milli-ampére/em?. 1 period 3.65 seconds. Men Fig. 7. Electrode + 60 cm. long of a diameter of 3 mm. electrode spirally wound in 5 windings. Potential measured at the second winding from the top. Strength of the current regularly periodical at + 0.5 ampère. 281 outline of the cranial cavity entirely indicated. Position and direction of the lamina cribosa are also shown. To the transformation of the cranial cavity during growth, chiefly consisting in a flattening, han ES ; : attention may be passingly drawn. These figures require little explanation, the change in the topographical relation of the orbits with regard to the cranial cavity is seen at a glance. It should only be pointed out that the shifting of the orbits quite before the cranial cavity must be regarded as the direct cause of the origin of the very strong bony ridge characterising the anterior part of the cerebral skull of Gorilla. This bone-ridge is, as also appears from fig. 3, nothing but the necessary upward enclosure of the orbital cavity, the newly-grown roof of this cavity. Without this bone-ridge the orbit would lack an upper bony enclosure. Before proceeding to a description of the conditions in man, we shall briefly sketch those in the two other anthropoids by means of a few projection figures. Figures 6 and 7 refer to a young Orang still in possession of its complete milk-dentition, and to an adult individual of this genus. More strongly still than was the case with Gorilla the topographical change of the orbits with regard to the 282 cranial cavity appears in these two individuals. This is mainly the result of the circumstance that the little skull of the Orang child was so much younger than that of the Gorilla child. With this Fig. 7. very young Orang the orbit is still entirely enclosed by the cranial cavity, the whole roof of the orbit is here still the floor of the anterior cranial cavity. In the adult Orang the orbit has come much more forward. So here also a considerable forward shifting has taken place. In orang this was not accompanied by the formation of a ridge as in Gorilla, firstly because the orbits and in particular their roof did not advance so far before the cranial cavity, and secondly because the anterior cranial wall in Orang had thickened evenly. The changes in the topographical relations with Chimpanzee appears when we compare figures 8 and 9. With this genus the forward shifting is smaller again than with Orang, although still considerable. The projection in fig. 8 has been taken from a little skull with complete milk-dentition, that of fig. 9 from an adult skull. 283 From this short summary it appears that the three anthropoids agree in this that as the result of certain phenomena of growth the topographical relation of the orbits with regard to the cranial cavity - AE 2 stef MRP ER sdk Dg Fig. 9. is altered. The chief change is that in the infantile antropoid ape the orbits lie under the cranial cavity, in the adult individual more in front of it. This is most strongly seen in Gorilla, where almost the whole orbit lies before the cranial cavity. The sagittal sections Fig. 10. Fig. 11. 284 through the orbit in fig. 10 (Chimpanzee) and fig. 11 (Orang) when compared with those of fig. 3 (Gorilla) show this difference in shifting with the three Anthropoids very distinetly. What is now observed in man? We refer in the first place to figs. 12 and 18. In 12 a horizontal section is given through the orbits of a new-born infant, in 13 through the orbits of an adult individual. In both figures a dotted line indicates as before the frontal plane passing through the anterior edge of the lamina cribrosa, i.e, Fig. 13. through the anterior border of the cranial cavity. When therefore we wish to answer the question whether the orbits are also in man shifted during growth, and, if the answer is affirmative, to what extent this happens, we have only to compare the position of the orbits in both figures with regard to this line. It then appears that there is no evidence of such a shifting. For in the infantile as well 285 as in the adult skull nearly the whole orbit lies behind this line. As to the topography of the orbits with regard to the cranial cavity, in man no change is observed during growth, such as was found with the Anthropoids. We come to the same conclusion when com- paring the anatomy -of the lateral wall of the orbits in the two figures. When dealing with the Gorilla skulls it was pointed out that in the infantile skull two parts could be distinguished in this wall, an intracranial part, partitioning the orbital and cranial cavities, and an anterior part, bordering the orbit outwardly. Between these two parts the cranial wall joins the orbital wall. In the adult Gorilla the intracranial part has disappeared, the cranial wall is attached to the posterior part of the orbital wall: In man nothing appears of these altered anatomical relations. As well in the young as in the adult skull the intracranial part is found, which means that in the adult as well as in the infantile skull the posterior part of the lateral wall of the orbit has remained a parti- tion between this and the cranial cavity. In man the orbital cavity always enters into the cranial cavity, which is moreover proved by the fact that the frontal wall of the cranial cavity is attached along the anterior border of the roof of the orbital cavity, as well in infantile as in adult skulls. Thus in regard to the phenomena of growth in the orbital region of the skull there is a very noticeable difference between man on one side and the Anthropoids on the other. This difference is that in man infantile topographical relations remain permanent. In their juvenile stage these relations are the same in man as in the antro- poid apes. While in these latter they are replaced by other relations, however, so that the adult skull becomes very unlike the infantile one, the human skull retains its infantile cranial characteristics. As has been stated in the beginning of this paper, the same holds good for the Foramen magnum. From this ensues that when we compare the human and anthropoid skull those of the anthropoid apes may not be considered as primitive forms from which the human skull should be derived. 19 Proceedings Royal Acad. Amsterdam. Vol. XXI. Mathematics. — ‘““Null-Systems in the Phine’. By Prof. Jan DE VRIES. (Communicated in the meeting of January 26, 1918). 1. In a null-system N(a, 8) a group of « straight lines » passing through a point N is associated to that point; to a straight line » belongs a group of 8 points .V lying on m. A point is called singular, when it is null-point of oo null-rays; a straight line is called singular if it has oo null-points. The null-systems, for which « or 3 is equal to 1 (linear null-systems) are characterized by the fact that they always bave singular null- points if «=—=1, always show singular null-rays if 3 = 1. Considerations concerning the case «== l are to be found in my papers “On plane Linear Null-Systems” (These Proceedings vol. XV, page 1165) and “Lineare ebene Nullverwandischaften” (Bull. de Acad. des Slaves du Sud de Zagreb, July 1917, Auszug aus der im Rad. Bd. 215, S. 122 veröffentlichten Abhandlung). That a non-linear null-system does not necessarily possess singular elements, appears among others from the consideration of the null-system N(3, 8n—6) formed by the points of inflection and their tangents appearing in a general net of curves of order ” *). Only for n= 8 we have in general a group of 21 singular null-rays, viz. the straight parts of the binodal figures. 2. Let us suppose that a Ne, B) possesses 5 singular points $, which are singular null-points on each ray drawn through them, and o, singular points S,, which replace two null-points on each ray ®). We further suppose that there are o singular rays s and 6, Singular rays s,; the latter are characterized by the fact that they represent two coinciding null-rays for each of their points. If the straight line m is’ caused to revolve round the point P, the 8 null-points MN describe a curve (/) of order (a + 8), which has an «-fold point in P. Analogously the null-rays , which have a null-point NV on the 1) See my paper “Two null-systems determined by a net of cubics” (These Proceedings vol. XIX, page 1124) 4) In the linear null-system formed by the tangents and their points of contact of a pencil (cr) the base-points are singular points S,, the nodes singular points S. 287 straight line p, envelop a curve (p) of class (a a 8), of which p is a B-fold tangent. Through a point S pass (a+ 9) tangents of (p); from this it is evident that the null-points on the rays of the pencil S form a curve (S)*+% Now, S is always one of the null-points, so that an arbitrary ray of the pencil bears only (8—1) points NV outside S. Consequently ($+? has an (a + 1)-fold point in S. Analogously we find that (spp has the straight line s as (3+-1)- fold tangent, while a straight line s, is a (8+-2)-fold tangent of the curve (sy)z+,. 3. The curve (P)7+# is of class (a+ 9) (a + 8 —1) — a(a—). Through P pass therefore (Za + 8) (8—1) more tangents, which touch it elsewhere. To them belong evidently the straight lines PS,, as S, represents two coinciding null-points. Consequently the null- rays bearing a double null-point envelop a curve of class (2a + B) (B — 1) — 6, The Aertel enveloping ligure contains moreover the Ox class- points Sy. | It is of course possible that the enveloped curve breaks up. This e.g. happens with the null-system that arises if each tangent of a pencil (c”) is associated to the (n—2) points, in which it moreover intersects the c” (satellite points of the point of contact). We have to distinguish then between the envelope of the stationary tangents, which each bear one double null-point, and the envelope of the bitangents, which each contain two double null- points. The curve (P) is now the so-called satellite-curve '). | In a similar way we find: The locus of the points N, for which two of the null-rays n have coincided is a curve of order (a + 28) (a — 1) — ao 4. The curves (pars and (q).4g have the « null-rays of the point pg in common. To the remaining common tangents the singular rays sand s, evidently belong *). There are therefore («-+-)?—«—o—o, rays n, a null-point NV of which lies on p, another null-point NV’ on q. This number has another meaning yet. If _N describes the straight 1) Cf. my paper “On linear systems of algebraic plane curves” (These Pro- ceedings vol. VII, page 712) or “Fuisceaux de courbes planes” (Archives Teyler, série II, t. XI, p. 101). 2) If 8=1, (p) and (q) have, besides the « null-rays of pg, only singular rays in common; consequently we have g-+ ox =a?+a+1. The tangents and points of contact of a tangential pencil provide an example of this. 19* 288 line ‘p, the remaining null-points V’ of the null-rays n borne by N will describe a curve (V’),. Its order is evidently equal to the number of rays », which have a null-point on p and another on g. Let us now consider the points that (V’), has in common with p. Each of the 3 null-points of p is associated to each of the remaining (?—1) null-points, and therefore is a (’—1)-fold point of the curve (N’). The remaining points NV’ lying on p are evidently double null-points on one of the null-rays determined by them. Hence: The locus of the double null-points is a curve (N,) ef order aap git 8 oes The consideration of the curves (P) and (Q) produces analogously : The double null-rays envelop a curve (n,) of the class B* + Zug + + a—B—o— oy. 5. By means of an arbitrary conic gp? another null-system may be derived from a given null-system. Let MN be one of the null- points of the ray n, N* the intersection of nm with the polar line of N with regard to g*. A new null-system arises now if on each straight line m the null-points NV are replaced by the corresponding points N*'). The number @ remains intact. In order to find what a passes into, we observe that the null-rays n of the new null- point V* must have one of their old null-points MN on the polar line p of N*. The null-rays 2 of the points of p envelop the curve (p)et+s- On each of the (« +) tangents which it sends through NV * is N* one of the new null-points. By the harmonical transformation \ («,6) is therefore transformed into a N* (a + B, B). If N lies on g? while one of its null rays touches at p°, N* becomes an arbitrary point of m, and 7 a singular straight line of N*. In order to determine the number of these singular rays, we associate to each tangent 7 of g* the 3 tangents p, which meet n in its 3 null-points NV. The envelop (pps determined by p has evidently 2 (@ + ) tangents in common with g°. Besides the straight line p, which, as B-fold tangent of the envelope (p), replaces 8 common tangents, (2a -+ 8) rays n are associated to p. The correspondence between p and nm has 2(« + 8) coincidences; on g° lie therefore 2(a + 8) points N, of which one of: the rays » touches at p°. In other words NE (a + 8,8) has Ue + 8) singular rays more than N(a,B). ') The “harmonical’ transformation dually corresponding to this I applied formerly to a N(1,3) (vide “Plane Linear Null-Systems’’). 289 By the dual’ transformation %(«,8) passes into a N*(a,a + B), which has 2 (« + 8) singular points more than %. 6. The harmonical transformation may be replaced by a more general transformation in the following way. The polar curve 2 of a point N with regard to a given curve m+! intersects the null-ray m in m points N*, which we shall consider as new null-points of 7. In the new null-system N* each straight line has then me null-points N*. As N* lies on the polar curve a” of N, NV belongs to the polar line p of N* with regard to "+1. Now (a+) tangents of the curve (p) pass through N*; they are the null-rays of N* for N* — Le. Na, B) is transformed into a R*(a+p8,mp) by the new trans formation. In opposition to the harmonical transformation this transformation produces no new singular straight lines. 7. If we write «=1, B=1, m=2, we find from a bilinear null-system a NA, 2) for which the three singular straight lines of M (1,1) are also singular. We may indicate the bilinear null-system by V'S 8, = Ya" Sa Ss LES, and the curve ®* by Bet Ende Loe we, 0. The polar curve of (4) is then expressed by yy («. + 2,3) + 4, (v, + @,%,) + Ya (e, +, w) = 0. For the null-system % (2,2) we have therefore S, es asc vs ©) 5 Si Ss es vy &;) =e 51 5. (+ vy ©) =v (1) 5% + Ss, By S; ts = 0 In order to find the equation of the curve (P)* we have to com- bine these two equations with Pi 51 + Ps =, + Ps 5: — 0, Elimination of E, then produces for (P)' Piets — Pat) (Pa, — Pot) (7 + 24%) + (pv, — pts) (Pat — Po) (a,” Fis a, 5) 8 (P37, Ten P25) (Pits — p,#,) (a,” =i PT) — 0. The equations (1) determine the two null-points of the straight line (§) as intersections of (&) with a conic. As a condition for the coincidence of the two null-points we find after some reduction the equation Ut ke el “A ne os Weke en it dad Sy eae yee os Eros thy bees; Ss ==0. 290 It shows that the rays that bear two coinciding null-points, envelop a curve of the 6*" class. From this it ensues that the curve (/’)' has no other singularities outside the node P. Combination of (1) with the equation Tat, a ey ey 0 produces for the curve (p), by elimination of x, the equation (ar, &, — 7, El rr a REEN [(ar, Ee San Us sig zr (x, Ss ES Ss Ty 5.) | 5 53 =r Ta) (To 5, — 7, Sal == — 3s 5) Ts En ER Sa) | 5, Es This is always satisfied by 57 0, &;= 0. This was to be expected as the straight lines O,O0,, O,0;,, 0,0, must be singular rays. ir Mathematics. — “Cubic involutions of the first class”. By Prof. JAN DE VRIES. (Communicated in the meeting of February 23, 1918). 1. By the “c/ass” of an involution in the plane we understand the number of pairs of points on an arbitrary line. In a paper printed in volume XVI’), I have proved that the cubic involutions of the first class may be reduced to six principal species provided that it be supposed that there are no collinear triplets. I will prove now that these involutions, with a few exceptions, may be determined by nets of cubics. Let a net [c*] be given with sv base-points Cz. All c* that yet pass through a point X, form a pencil (c*), have therefore still two points X' and X" in common, which form with A a group of an involution /,. On an arbitrary straight line [c*] determines a cubic involution /?, of the second rank; the neutral pair consists of two basepoints X', NX", consequently is /, an involution of the vst class *). To [c°] belongs the y°, which has a nodal point in Cy. If X is chosen on this nodal y°, one of the points X', XY" comes in C;; so Ck is a singular point that forms groups of the /, with the pairs of an J,, lying on the singular curve y*,. Each of the two points of 7°; lying in Cz, belongs to a pair of the /,; from this it ensues that the pairs of this /, are lying on the tangents of a conic (curve of involution of the /,). To [c*] belongs also the figure formed by the conic y,?, which y contains the points C,, C,, C,, C,, C;, and a certain straight line c,, on which C, lies. As [c*| determines on ¢, the pairs X, X’ of an I,, e, is a singular straight line. The involution I, has therefore sia singular points and siv singular straight lines. The points X’’, which complete the pairs of the /, lying on c, into triplets of the /,, lie evidently on y?,. Let Y’’ be the projection of X’’ on c,, out of a fixed point of 7°,; there exists a relation (2,1) between )’’ and X, so that Y’’ coincides three times 1) “Cubic involutions in the plane”. These Proceedings XVI, 974—987. *) If therays XX’, XX” are associated to each point X, a null-system % (2,2) arises. 292 with Y. From this it ensues that the rays XX’ envelop a curve of the third class, which evidently has C, as bitangent. The straight lines «= NX’ X’’, which are indicated by the triplets of the Z,, form the triplets of an involution of rays 7,. For this involution too, cz is singular, as it belongs to oo’ groups; the straight lines x’, «’’ form an d,, for which yz’ is the curve of involution. 2. When a point Y describes the straight line p, the rays 2’, 2", which connect X with X",X', envelop a curve (p), of the fourth class, which has p as bitangent. The curves (p), and (q), have 16 tangents in common; to them belong the rays 2’, 7", which emanate from X = pg, and the six singular straight lines cz. There are conse- quently 8 straight lines 2", for which X lies on p, and X' on q. In other words, if X describes the straight line p, X' and X" describe a curve p°. The latter intersects p in the first place in the pair of the 7, lying on p, and further in six points X, which have each coincided with a point X', consequently are coincidences of the /,. The coincidences of the 1, form therefore a curve of the sixth order, Y°. If two base-points of a pencil meet in a point B, there is a curve that has a nodal point in B. So y° is at the same time curve of JacoBt for the net [c°], has consequently nodal points in the six base-points Cz. In each of these points it has the tangents in common with the nodal curve y*s. Outside the points C the lines y° and y*, have only two more points in common; they are the coincidences of the involution (Y, X') lying on y'r. The curve (p) is of order 10, is therefore cut by p in 6 points. For each of these intersections Y, w" coincides with 2’, consequently X’ with X’’. The locus of the “branch points’, the “complementary curve’ is consequently also a curve of the sixth order, 2°. It has nodal points in the singular points Cx, because 77° bears two coin- cidences. The curves y° and x° have besides the 6 points C more- over 12 points in common, they are united in pairs into triple points of the /,. So there are in /, six groups, in which the three points are united in one point. The above mentioned curve p° has a triple point in Cp, because yi has three points X in common with p, for which X’’ lies every time in’ Cy. The pairs of the /,, which are collinear with an arbitrary point P, lie on a curve (P;*,‚ which passes twice through / and contains the singular points C'). So p° and (P)* have in Cx 18 points in common; 1) For Cx this curve consists of yx%° and the straight line Cz. 293 the intersections X of p with (P)* supply further 4 common points X’. The remaining 10 points which they have moreover in common form 5 pairs X’, X’’, of which the line of connection « passes through Z. In other words, if AX describes a straight line, w envelops a rational curve of the fifth class. 3. Let us now consider the case that three base-points B,, B,, B, of a [c°] lie on a straight line a, while the remaining three, C,, C,, Cs, have been chosen arbitrarily. To the net belongs a pencil, each curve.of which consists of the straight line « and a conic that passes through C\, C,, C, and a certain point A. These conics determine an /, on a, the pairs of which are completed by A into groups of the /,. So A is a singular point, a a singular straight line. To the singular points C,, C,, C, the nodal curves y;? are again associated as before; to the singular points b,, B,, B, now belong curves 8’, which pass through the points C and A. Bach 3,” forms, as is known, with a the net-curve that has a node in By. On the pair of lines ACC, C,, [c'] determines a system of groups of the J,, a point of which lies every time on C, Cj, so that AC, contains an /, of pairs X, X'. The three straight lines cr, = AC, are therefore singular, they form with the singular straight line a the curve (P)* of the point A (see $ 2). For Cy. (P)* consists of yz° and cz, for A, of B’, a and a singular straight line bj. There are consequently seven singular points (A, Bj, Cr) and seven singular straight lines (a, bj, x). The straight line a is component part of the Jacobian, the curve of coincidences is now a y° that passes through the three points 5 and bas nodes in the three points C. The curves (p), and (q), have now only 7 tangents xv’, in common, which connect a point X of p with a point XN’ of g. In connection with this p* is now replaced by a p',‚ which passes three times through C;, twice through Bz. Between the points X of p and the points X*, which are every time produced by the intersection of & on p, a correspondence exists, each coincidence of which is at the same time a coincidence of the I,; hence a envelops a curve of the fourth class, when X describes a straight line. 4. Let us now suppose that one of the six base-points of [c*] is collinear with the base-points 5, 5,*, and with the base-points 294 B, B,*; let this base-point be indicated by A,, while the sixth base- point will be indicated by C. Now [c?]| contains a threeside formed by a, = A,B,B,* a,=A,B,B,* and a straight line a,, which contains C and forms with y? the curve (P)* of C. The singular straight line a, bears an /,, of which the pairs are completed into groups of the /, by A, To a, belongs again (as in $ 3) a pencil of conics, the curves of which are completed by a, into figures c°. This (c*) has as base- points B, b,*,C and a point A,, which is singular, because it forms groups of /, with the pairs of the /,, which (c?) produces. by the intersection with «,. Analogously there is a singular point A, to which an /, belongs placed on a, To the pencil (c’), which is associated to a,, belongs the figure formed by a,= B,B,*, and the straight line CA,; the latter is therefore identical with the third straight line a, of the threeside mentioned above. Analogously a, and a, form one of the conics that are associated to a,. From this we conclude that the singular points C,A, and A, are collinear, and lie on the singular straight line a To the pencil associated to a, belongs also the pair of lines C'B,, A,b,*; on the second of these lines the net determines an /* or pairs (X, X’), which are each completed into triplets by a point of „B 0 the lines A,5,*, 4,8, A,B,* and A,B, ate singular; we may indicate them by 6,*, 6,, d*,, 6,. Finally there is moreover a singular straight line c, which passes through C and forms with the tbreeside a,a,a, the curve (P)* of C. It contains an 4, of pairs X, X’, which are every time base- points of pencils out of [c’|. If we now take two arbitrary fixed points M and MM’, and if we associate the two c’, which each of the pencils in question sends through J/ and J/’, two (c*) are on account of this made projective. As any two homologous c’ intersect each other in three points of c, and the two pencils have a curve c*, in common, the figure produced by them consists of c,*, the 0 line c, and a conic 7°; the latter therefore is the locus of the point X". Summarizing we find that this /, bas eight singular points and eight singular straight lines. Its coincidences lie on a y*, which passes through the points B and twice through C. In an analogous way, as in § 3, it appears that X envelops a curve of the third class, when X describes a straight line. 5. Let us now suppose that the base-points B,, D,, B, are respec- 295 tively lying on the sides A, A,, A, A,, AA, of the triangle which has the base-points A; as vertices. In the same way as with the preceding /,, there belongs to the straight line «w,*—A,A,B, a (c’), the base of which consists of A,, B, B, and a certain point: A,*, which is again singular and belongs to an /, lying on a,*. Analogously there are moreover two other singular points, A,* oad A,*, which are related to involutions LOW ende It appears now from the consideration of the threeside formed by a,*,a,* and by the straight line 7,, which must pass through B,, that a, contains the points A,* and A*, (see § 4). Evidently the singular straight lines a,,a,,a, belong respectively to the singular points A,, A,, A, The nine singular points are now placed in Sch a way, that each point Bj is the intersection of the lines aj, and a/*; the triangles A,A,A, and A,*A,*A,* are consequently circumscribed to the points BBB, Besides the six singular straight lines aj, a;* there are moreover three singular straight lines by, = ArAr*. For, on the pair of lines A,A,*, B,B, [c'] determines groups of the /,, of which every time one point lies on B,B,, while the other two form a pair on 5, The curve of coincidence is now a y*, which passes through the points B. To a straight line p a p® is associated, while the straight line X envelops a curve of the second class, when X describes the straight line p. For B, the curve (P)' consists of a 9,° (A,B,B,6,A,*) and the lines a,,a,*; for each of the remaining singular points it consists of four lines easily to be indicated. For further particulars I refer to my paper mentioned above. 6. We now consider a net end that has the vertices Az, of a fourside, with sides aj, as base-points. To the straight line a, a (c®) is associated, which has as base-points A,,, A,,, 4,, and a certain point’ A,; each of these c? forms with a, a figure of the net. To these figures belongs the threeside that is composed of a,, a, and a third straight line a,,, which must pass through A,,, but cannot but contain the singular point A,. But this threeside may at the same time be considered as compound of the straight line a, with a pair of lines of the (c*), which has as base-points A,,, A,,,A,, and a certain point A,; consequently the third straight line a,, passes through A,, and A,. The singular straight line a,, contains therefore the three singular points A,, A,, A, Besides the siz singular points Aj, which have each a straight 296 line a, as corresponding singular line, the /, has as appears from the above, moreover four other singular points A, which are in pairs collinear to the points Aj, and that in such a way that A, and A, are connected with A, by the singular straight line ap. In other words, there are ten singular points and ten singular straight lines, which form a fourside and a complete quadrangle, in which the former is inscribed in such a way to the latter that a configu- ration 10, of DrsarGurs has arisen. *) The curve of coincidences is now a conic, as the four straight lines a, form part of the Jacobian. This may moreover also be confirmed by paying attention to the common tangents of the curves (p), and (q),; they have besides the two straight lines w indicated by the point pg and the 10 singular straight lines, moreover 4 straight lines w in common, which each connect a point X of p with a point X’ of g. To a line p as locus of A corresponds therefore a curve p* as locus of the pairs X’,X" and the latter intersects p in two coincidences. It is easy to find now that the straight line X == X’X" describes a plane pencil. The /, here described has been known longest; it may properly be called the involution of Reve. 7. With this fwe of the involutions /, found in the above mentioned paper have been deduced from nets of cubies. The sixth /, is obtained if each c? passing through the points ZF, F,, F, is intersected by each c? passing through the points /, G,, G,, G,. This /, was amply discussed in my paper “A quadruple involution in the plane”. (These Proceedings XIII, 82—91). When the base-points B, B, B, of a |c*] lie on a straight line Dina and the base-points B, B,, B, on a straight line 6,,,, this net contains a pencil, each figure of which is composed of the two straight lines mentioned and a ray s ofa plane pencil whose centrum be indicated by A. On each rav s [c*|] determines an /,; here we have therefore a cubic involution in the plane, which contains collinear triplets only, and consequently was excluded from the investigation mentioned above. Neither is it of the first class, for on an arbitrary straight line does not lie a single pair. The Jacobian of this net consists of the lines 6,,,,6,,;, and a ') In a more symmetrical way the points and lines of the 103 are indicated by the symbols kl and kim; the points kl, km, lm, lie on the straight line klm (k, 1, m to be replaced by 1, 2, 3, 4,5). 297 curve y‘, which contains the coincidences of the cubic involutions lying on the rays. Analogous results are arrived at by considering the net of which the six base-points lie on a conic. 8. Let a net of nodal cubics be given, which all pass through the base-points B, B, and have their node in D. To ),= B,D belongs a pencil of conics passing through D, B, and two other points A, and A,*. Analogously to 6,= B,D a (c°) with base D, B,, A,, A,*. The two pencils [c?] indicated by this have the threeside in common, which consists of 6,,6, and a third line d. From this it ensues that d must contain the points A,, A,*, A, and A,*. On the singular straight line d, [c'] determines an /,; here too we have consequently a triple involution, which was excluded in the investigation mentioned above, because it has collinear triplets. On the pair of lines DA,, B, A,*[c*] determines groups of the /,, which have each a point on DA, and a pair of points on B, A,*. The last mentioned line is therefore singular, and the same holds good for the lines B, A,, B, A, and B, A,*. Taking into consideration that the curve of coincidences is a y* with triple point D, we can now deduce from the combination of two curves (p), that besides the jive singular straight lines mentioned there can be no others. For, (p), has d as bitungent, so that d represents four common tangents of (p), and (g),. And, as to p, on account of y*, a curve p° is associated, as locus of X’, (p), and (q), can only be touched yet by four singular straight lines. | As none of the singular lines passes through D, the curve (P)‘ for P= D will have a triple point. On this 6*, which passes through B,, B, and the points A, lies an /,, of points X, X’, for which X" is lying in D; the straight line X XN’ envelops a curve of the 3" class. For B, the curve (P)* consists of a conic 8,* (which contains an /,) and the straight lines B, A,, B, A”. The singular points A,, A,* form triplets with each of the points of 6,; to them no singular straight line is therefore associated. For A, the curve (P)* consists of the straight lines A, B, and b, together with the twice to be counted line d. The curve (p), is evidently of order 8 (two bitangents) ; it is conse- quently intersected by p in 4 points. Consequently the complementary - curve is of the fourth order. As it has nodes in D, B,, B,, it can have besides these points but 16—2 x 3—2 «2 or 6 points in common with 7°. In this /, only three groups occur of which the three points have coincided. 298 It possesses seven singular points and five singular straight lines. 9. In § 7 there was a reference to a triple involution that has only collinear groups. Another /, with only collinear triplets is determined by the projective nets ka + 1b," + me,' = 0, kA, + UB, + mr: = 0. Each triplet consists of base-points of a pencil (c*) belonging to the net [c*] indicated by | « B 7 | Fy | a A oe | Ae By I which has thirteen fixed base-points Sj. For the curves a,* B,= 6,2A, and a, C= c,* Ay have in common the three points indicated by a,’ = 0, A; = 0, and they do not lie on the net-curve 6,’ C,=c,* B,. The curves of [c'] pass therefore through 13 fixed points. Any straight line contains three base-points of a pencil (c*). If it is represented by 4A, + (6, + mC, == 0, which is always possible, the pencil in question is found by writing ka + (8 + my = 0 in : | ka +8 + my B 7 | | kas + loi + mc," be € [=O Ar + IB. ml Be Co | Then we find the pencil | B =| | De kA. Ge | = kA, B. se} | 3 and it has as base-points the intersections of > fas = 06 witha eo, = 0E) The thirteen points Sr are singular, for each point S forms a triplet with each of the pairs that is produced by the intersection of the pencil with centre S; on the nodal curve o,', which has Sr as node and belongs to [c*]. The groups of the /, that are collinear with the point P lie on a curve (P)*, which passes through the points S, consequently also belongs to [c*]. 1) An arbitrary net [ct] has 12 base-points at most and intersects a straight line in the groups of an involution 1? (of the second rank), which has three neutral pairs. Here the three pairs are replaced by a neutral triplet. 299 Any net-curve contains a point P, for which it serves as curve (P)*. For the Jacobian y°, at the same time curve of coincidences of the /,, has nodes in S, and intersects a c* of the net consequently moreover in 10 points /?, which must be coincidences of the /,. Let A, be one of those points; the tangent in R, at c* has two more points in common with that curve; one of them forms with R, a triplet of /. Let P be the second of those points. The (P) belonging to P has now in common with c* the 13 points S, the point P and the triplet of the /, determined by A; but the two curves are identical then and the tangents at c* meet in the 10 points Ze in P. From this it ensues at the same time that the dines t containing the coincidences of /, envelop a curve tr of the tenth class. 10. In Sj, six tangents of oz meet; each of the tangents in Sz replaces two straight lines ¢, so that + has a node in Sj. If (P)* has a node D, PD replaces two straight lines ¢ and P. is a point of t. If (P)* has two nodes D, and D,, Pis node of t and PD,, PD, are the tangents in P. Analogously t has a cusp in P, if (P)* is a euspidal c*. Consequently r has besides the 13 nodes S;, moreover 225 nodes and 72 cusps. *) Hence we find further that r isa curve of order 27 and of genus 15. It must correspond in genus to the curve of coincidence y’; in fact the latter is also of genus 15, because it has 13 nodes. As of, contains six coincidences besides S;, the complementary curve « has a sextuple point in S¢. On each (P;* lie 10 points of x, viz. on the straight lines ¢, which meet in P. So (P)' and w have 10 + 136 points in common, z is consequently a curve of order 22. The curves y’ and z°* can only touch outside the points S; and in each of those points of contact the curves of a pencil (c*) have an osculation. From 9 x 22 —-13 x 2% 6—42 it appears therefore that /, has 21 groups of which the three points have coincided. 11. Let us now consider the case that the curves indicated in § 9 by a, =0, 0°; =0, cz = 0 have a node in S,. The net [c*] may now be represented by ') A net [c:]) without multiple base points has 3/, (~—1) (n—2) (8n®—3n—1]1) binodal and 12 (nm—1)(m—2) euspidal curves. (Cf. e.g. my paper-in volume VII, p. 631 ~ 633). 300 a Az? #, + az’ : sie B Do, Hb a —0 | | a a2 2 ’ 3 ’ re Py Cf tex a, in which a,’ ete. are functions of we, and «,. All c* have a node in S,. The groups of the /, on the rays passing through S, consist of the point S, twice to be counted and a point of the curve of, indicated by which has a triple point in S,. As (P)* has a node in the singular point S,, P bears eight straight lines ¢ so that r is now of class 8. The curve of coincidence y° intersects (P)* in the points of contact of the 8 straight lines ¢ and twice in each of the 9 singular points S; (single base-points of [ots from this it ensues that y° passes five times through S,. We now consider two arbitrary pencils of the net [c“| and associate to each c* of a pencil the curves of the other, which curves intersect it on y°. The product of the pencils that consequently are in an (8,8) , eight times the c*, which consists of the twice counted curve y° the pencils have in common, and the complementary curve w. From 64 — 2 x 9 — 8 Xx 4=— 14 it now appears that « is a curve of order 14. The curve 6% belonging to S; has nodes in S; and S,; consequently is Sj quadruple point of wv. A combination of (P)* with «'* now leads to the conclusion that z'* possesses a sevtuple point in S,. We now find by the combination of y° and «'* that /, contains 12 groups in which the three points have coincided. The characteristic numbers of t are easy to find, as this curve corresponds in genus to y’, and has the 12 points of contact of y and 2 as points of inflexion. It appears to be of order 20. 12.1 lian a Oa U, B b,* Le ll) / Cx vs a,’ ete. again represent functions of 2, and a,, all the curves of [e*] in O, = S, have a triple point. The groups of the /, are now determined by kar? + lb? + me =0 and ke, + le, + me, =0. The first of these equations shows that the rays have been arranged by S, into the triplets of an involution of the second rank. 301 If two rays of a group coincide, we have’). ka, + lb, + me, = 0 ka, + lb, + me, = 0. We find, therefore, for the curve of coincidences | Gs kB | dr 7G: an ie. a y® with quadruple point S,. This result was to be foreseen; for the net [c*] has moreover 4 single base-points Sp; the JACOBIAN has consequently 4 nodes S and an octuple point S,, breaks up, therefore, into four rays SS and a y°. If the three rays of a group of the involution /,? coincide, we have Un 6, Ch | je Ön Gre |= 0. Ay ae Ce There are consequently three groups of the /, in which the three points coincide; their lines ¢ are stationary tangents of the curve tT. As (P)* has now a triple point in S,; P bears only four straight lines ft. The curve rt is consequently of class 4; as it must be of the genus null and has 3 stationary tangents, it is a curve of order three. The /,? has a neutral pair; these two straight lines form a c* with the conic that passes through the five singular points. A 13. The net determined by | a ax” A | | B Ob Be | 3 Cy" GE has 12 base points, consequently produces an /,. lf, however, the 6 conics corresponding to the 6 quadratic functions, all pass through a point S,, the curves of {c*| have a node in S, and pass further through 9 fixed points besides. The variable base-points of the pencils (c*) form now an /,. This triple involution of the third class 1 have fullly investigated in a paper, printed in volume XVII, p. 134 of these “Proceedings”. In a paper published in volume XVII, p. 105, a triple involution of the second class is to be found; its groups are arrived at by intersecting any conic of a pencil with any curve of a pencil (c*); the two pencils viz. have three base-points in common. == i() 0a B 1) By akis meant ——, by ak! the form ——__. |. Ad ae ; 20 Proceedings Royal Acad. Amsterdam. Vol. XXI. Mathematics. — “Linear Null-Systems in the Plane”. By Professor JAN DE VRIES. (Communicated in the meeting of April 26, 1918). 1. A linear null-system ® (lm) may be determined by two equations of the form 6,4) 558,74 £4, —0 EA, HEA, + &,4,=0, where Aj, indicates a function of order m, in ap. When the straight line 2 revolves round the point P(yp), its m null-points iV, viz. the intersections of &-=0 with the curve EEA, = 0, describe a curve of order (m+ 1). As &=0, this nul/- curve (P)"+1! has as equation, a, Ver ee | t, Tee, eet | | Ay. Ags | The curves (P)"+1! form a net that is represented on the point- field by the points P; for each netcurve belongs to a definite point P. The net has (m? 4m +1) base-points. For, if for the sake of brevity its equation is written in the form „B tu B+ y, B=0, it appears that the curves B, =O and B,=0 have in the first place the points indicated by a, —=0O, A,—=O in common, which, however, do not lie on the curve B, =O. For the (m? + m +1) points S;, which they have moreover in common, we have the relation Andr Ay sas ear These points lie consequently at the same time on B, = 0. Each of the base-points Sp, bears op! null-rays 2, is therefore a singular point of the null-system. Two null-eurves (P)"+!' and (Q)"+! have in the first place the m null-points of the straight line PQ in common; the remaining intersections must be singular as they bear each two null-rays; they are therefore identical with the (m* + m +1) singular points S. If the point O, is laid in one of the singular points we have to write Ay —= at mrt +..., where a indicates a linear function Ole And 303 We find then for null-curve of O (vz, a@) — aw, a@) a ml ...=0, from which it is evident that the null-curve op"+t of Sj has a node in Sp. This result was to be expected, but of course holds good only in the case of S being single null-point for an arbitrary ray passing through S. 2. If a point N describes the straight line p, its null-ray n envelops a curve of class (m + 1), which will be indicated by the symbol (p)n41- For the null-curve of an arbitrary point Q intersects p in (m1) points N, of which the null-rays pass through Q. Evidently p is an m-fold tangent of (Plm The null-curves (Pm and (Q)n41 have a common tangent in the null-ray of the point pg. Each of the remaining common tangents is a straight line ”, of which one of the null-points NV lies on p, another null-point V’ on q. If N describes the straight line p, the remaining null-points MN’ describe consequently a curve (NV) of order (m? + 21). Each of the null-points of p is to be considered (m—1) times as point MN’, so that (V’) in those null-points has m({m—t) points in common with p. In each of the remaining 38m intersections of p with (V’) a point N’ coincides with a point N into a double null-point N°) of the corresponding straight line n. In a double null-point the curves (?) of a pencil have a common tangent; one of the pencil-curves has a node there. The locus of the double null-points (curve of coincidence) coincides with the Jacobiana of the net of the curves (/’). As the latter is in general a curve of order 3m, the conclusion may be drawn from the above made statement that the null-system possesses in general no singular straight lines. For, if a straight line has each of its points as null- point, it is common tangent of null-eurves (p)n41 and (q)m44- The curve of coincidence y?" has, as Jacobiana, (m* + m + 1) nodes Sz. This may be confirmed as follows. Through P pass (m? + m—2) tangents of (P)"+!: their points of contact are double null-points, consequently points of y3”. The remaining Smun + 1) —(m? + m—2) intersections of (/?) with y must lie in the singular points, but then y must have a node in each point S. 3. Let us now consider the locus « of the groups of (m — 2) null-points, lying on the null-rays ¢, which possess a double null-point. 20* 30+ Through each point S pass (m? + m—-6) tangents of the null- curve omt of JS; as they bear a double null-point each, S is an (m? Hm — 6)-fold point of the complementary curve x. Besides the points S, x has moreover the groups of (m—2) null-points in common with (P)mt!; these points lie on the (m? + m— 2) straight lines t, which meet in P. The two curves have consequently in common (m? + m + 1) (m? + m — 6) + (mm? + m—- 2) (m— 2) points. For the order of x we find from this (m*—+ 3 m* — 5m? — 9m — 2): (Mm 41), i.e. m*> + 2m? —7m — 2, or (m — 2) (m+ 4m- 1). 4. The straight lines ¢ envelop a curve t of the class (m + 2) (m — 1). If a curve emt! of the net has a node D, DP replaces two of the rays ¢ meeting in P; P is then a point of t and PD the tangent in P at that curve. If P lies on a binodal ee, with nodes D and D', PD and PD’ replace each two straight lines ¢ and are tangents in a node of r. If a ct! has a cusp in K, PK replaces three straight lines t, and P is a cusp of r. Now the net [e”+'] contains according to a well-known proposition 2 m(m—1) (B m? + 3m — 11) binodal and 12 m(m— 1) cuspidal curves. If we moreover take into consideration that the base-points S are nodes of r, it appears that + possesses 3 (9 m* — 40 m? +35 m + 2) nodes and 12 m (mm — 1) cusps. We can now determine the remaining characteristic numbers of r. From the formula » = 2 (n—1) — 2d—-3r it ensues at once that the order of r is 3m’. From 3n—r= 3v—o we deduce for the number of points o7 inflevion 3(m—2)(2m-+-1). The genus of vr is equal to that of y%", viz. equal to $m(7m—11). And we now finally arrive from g = 4 (v—1) (2) — (6 + 9) at the number 4(m— 2) (m—8) (m?-+7m-+4) of betangents. It appears from the results arrived at that X(1,m) has 3 (m—2) (Qm+1) rays with triple null-point NS and 4 (m—2) (m—38) (m?@+7m+4) rays that have two double null-pomts each. By means of these two numbers it would be possible to determine again the order of the complementary curve. For the curves y and x will touch in the triple null-points and must intersect in the coupled double null-points; they have further in each singular point 305 2(m?-+-m—6) points in common. Taking this into account we tind indeed for the order of x the number arrived at above. 5. Till now we have supposed that the singular points are all single and different, but moreover that each point S is single null- point on a ray arbitrarily drawn through S. An example of a Nm), of which the singular points are partly double null-points, is furnished by a pencil of curves c’, when each straight line is associated to its points of contact with curves of the pencil. A ray passing through a base-point of (c’) is touched outside that point by 2(7—2) curves, while an arbitrary straight line has 2(7—1) null-points; so each base-point is to be considered as double null-point. The remaining singular points of this null-system %(1,27—2) lie in the nodes of the nodal curves c”; they are evidently single null-points on the straight lines drawn through them. We shall now suppose that (1,7) has s, singular points S@), which are double null-points of their rays. As a ray passing through S2 outside that point bears (m—2) null-points the nz//-curve 6) has a triple point in S?). The complementary curve now consists of the s, null-curves 6°?) and a curve x* of order (m—2) (m?+4m+1) — —(m-+1)s,, while the curve t has been replaced by a curve rt“ of class (m-+-2) (m—1) — s, and the s, class-points WS 2). If it is taken into consideration that @{2) contains all singular points S) and S, it is found that »* passes through each point S with (m?+m—6—s,) branches and with (m?--m—8—-s,) branches through each point S®), 6. In order to arrive at a determination of the number of triple null-points NW), we associate to each point MN) of a ray ¢ the (m—2) null-points NV’ of ¢, and consider the correspondence which arises in consequence of this in a plane pencil with centre 7. As the points N@) lie on the curve yöm, the points N’ on the curve *’, the characteristic numbers of this correspondence are evidently 3m(m—2) and (m—2) (m?+4m+1) — (m+1)s,, while any ray ¢ passing through 7’ produces an (m—2)-fold coincidence. The number of the remaining coincidences amounts to 3m (m—2) + (m—2) (n° + 4m + 1) — (m + 1)s,—(m + m—2—s,) (m—2) i.e. (m—2) (6m + 3)—3s,. There are consequently 3(m —2) (2m + 1)—3s, null-rays with a triple null-point. In order to find the number of coupled double null-points NM) we associate to each point N' of a ray ¢ each of the remaining 306 null-points " of ¢. The involutory relation which arises in conse- quence of this in the plane pencil 7’ has as characteristic number [(m—2)(m? + 4m + 1)— (m + DS, Jon —3); any ray ¢ passing through T represents now (m—2) (m—3) coincidences. The remaining coin- cidences to the number of 2(m— 3) [(m—2)(m? + 4m + 1) —(m + 1)s,| — (Mm? + m—2—s,) (m—2) (m—38) form pairs of double null-points. There are consequently 4(m—2) (m-—3)(m? + Tm + 4) — Alm) (m+ djs, rays which each bear two double null-points. A null-system N(A,m) with (m? + m + 1) simple singular points has therefore 3(m— 2) (2m + 1) null-rays with a triple null-point and 3(m—-2) (m—38) (m? + 7m + 4) null-rays with two double null- points. With this the results of § 4 are confirmed. For the null-system Jt(1, 27—2) mentioned above s,=77; the number of triple null-points amounts therefore to 3(77?—22r + 12). Forr==3 we find from this 27. For each pencil (c°) each base- point is point of inflexion on three curves c°; the number 27 conse- quently arises from the fact that the 9 base-points serve each on three null-rays as triple null-point. As this observation holds good for each pencil (c’) the number of points N® outside the base- points will be equal to 3(67?—22r + 12). In such a point a cr has four coinciding points in common with its tangent. In general a pencil (er) has therefore 6(r—3) (87—2) curves that have a point of undulation *). 7. If the curves: Arp=0 (§ 1) have an r-fold: point in Oi S,= 0, is an r-fold null-point on each of its rays. Outside the singular null-point S, there are then moreover (m? + m +1) — 1? simple singular null-points S. The null-curve of S, has as equation A, 2,—A,«,—0; hence it has in S, an (r + 1)-fold point. The null-curve (P)"+! has in S, an r-fold point, consequently sends through P (m? + m—2)— (r?—r) tangents ¢, of which the points of contact lie on the curve of coincidence y. The latter has nodes in the points S; so of its intersections with (PH! there lie in S, 3m(m + 1) — (m? +m — 2 — 7? Jr) — Qn? + m + 1 =r) = = (3r—-1)r_ points. From this it ensues that y has in S, a (87—1)-fold point. In order to determine the order of the complementary curve, we consider two pencils of null-curves (c,”+1) and (c,”+*), and associate 1) Another deduction of this number | gave in “‘Faisceaux de courbes planes’. (Archives Teyler, sér. Il, t. XI, p. 99). 307 to each c,"+1 the (mm? + m — 2 — 7? +7) curves c,”+!, which it intersects on y?", outside the points S. The figure produced by the pencils coupled in this way consists of twice the curve y, of (m? + mm —2— —r? + 7) times the curve e”+!, which belongs to both pencils and of the complementary curve z,. We now find as its order (mm? + m —2—r +r) (m+ 1) — 6mi. e. (m — 2) (m? + 4m + 1) — (m+ 1)r (r — 1). With regard to §3 we conclude from this that the null-curve of S, is to be considered 7 (# —1) times as component part of x. Applying the method of §6 again, we now find the number of triple null-points from 3m (m—2) + (m2) (m? + 4m H 1)—(m + 1) rr (1) m2) (m? + m—2—7? Hr) Le. (m—2) (6m + 3) — 3 r(r—1). Analogously we find for the number of null-rays with two double null-points bm 2) (m— 3) (m? 4+ 7m + 4) — 5 (m —3) (m4) r (r—1). 8. A very particular linear null-system is obtained by supposing -that the functions Az ($ 1) only contain w, and w,. In that case EVA, + £, Ay + E;Ap=0 represents an involution of rays of the second rank, of which the oo? groups, each of mm rays, correspond projectively to the straight lines of the plane. The null-curves have now in S, = 0, an m-fold point, are conse- quently rational; the null-eurve of S has degenerated into (m + 1) rays, which each contain one of the simple singular null-points S. If the derivatives of A; with regard to w, and a, are indicated by (Ap), and (Ap),, we find for the locus of the double null-points the equation U, vs v, | (4), (A,), (As); =='0 Ag, tay | This curve of order (2m—1) has in S, a (2m — 2)-fold point. By the (m+ 1) rays S,S; it is completed into the Jacobiana of the net of the null-curves. The rays ¢ with the double null-points envelop a curve rt of class (2m — 1); for (P)"+! is now of class (m + 1)m —in(m— 1) = 2m. The triple rays of the above mentioned involution are indicated by 308 (A). (A4 (A): (A) (Ad) Aids (=O. (Ads: (Ads (Aas | Their number amounts therefore to 3 (m— 2). There are consequently 3 (i — 2) null-rays with triple null-point; they are evidently stationary tangents of the curve t enveloped by the null-rays 4. 3 Analogously the bitangents of that curve are intersected in their points of contact by the pairs of double rays that occur in the groups of the involution. Their number, as is known, amounts to 2 (m — 2) (m — 3). For the order of t we find now m; it has no cusps, but + (im — 1) (m — 2) nodes. It is, just as y?”~—!, rational. The involution has $(m— 1)(m— 2) neutral pairs. Each pair belongs to oo’ groups and corresponds projectively to a. plane pencil of null-rays. In connection with this the null-curve of the centre of that pencil consists in the corresponding neutral pair of rays and a curve of order (m — 1), which has an (m— 2)-fold point in S,. The null-curve of a singular point Sz: consists of the ray S,pS, and a curve of order m with (m— 1)-fold point 5, Mathematics. — ‘““Null-Systems determined by two linear con- gruences of rays”. By Professor JAN pr Vrins. (Communicated in the meeting of April 26, 1918). 1. A twisted curve. «’ intersected by a straight line a in (p—41) points, determines a linear congruence (1,p), of which each ray wu rests on a and on «”. Analogously a curve #9 intersected by the straight line 6 in (q—l) points determines a congruence (1,q), of which the rays v rest on 6 and s7. Through the point N pass in general one ray w and one ray v. If the plane »==wv is associated as null-plane to N a null-system arises in which a plane » has in general pg null-points, viz. the intersections of the p rays of w with the g rays of v. If N describes a straight line /, the rays w and v deseribe two ruled surfaces, which are successively of order (p + 1) and order (q +1), and intersect along a curve (/) of order (pg—-+p—+q). An arbitrary plane » passing through / has with (/) the pq null-points of » in common, and moreover (p + q) points lying on /, which belong each as null-point to a definite plane rv. In other words, the straight line / is (p + q) times null-ray. In R. Srurm’s notation the null-system has therefore the characteristic numbers a =1,8= pg, y¥=p+gq, may consequently be indicated by N(l,pq, p+). . 2. If v coincides with uw, any point of that straight line has any plane passing through that straight line as null-plane. Now, the congruences (1,p) and (1,qg) have in general (pq +1) rays in common. There are consequently (pq 1) singular straight lines s. The curves av and 7 are also loci of singular points. Through a point A* of a passes a ray v* and a plane pencil of rays u. In any plane passing through v* lies one ray v; so A* is null-point to any plane of a pencil that has v as axis. The straight lines v* form a ruled surface of order p(q +1); for a plane passing through 6 contains p rays v* and a point of 6 bears pq rays v*. Finally the points of « and 6 too are singular null-points. A point Ay of a bears one ray v, and oo * rays u, which form a cone of order p with (p—1)-fold generatrix. Any plane passing through v, contains p rays u, so that A, is to be considered as p-fold null-point. The rays v, form a ruled surface of order (q +1). A straight line wu 310 (or v) is null-ray to any of its points; in connection with this the curve (/) degenerates for -=u or Ev. 3. If a plane r continues to pass through fhe point P, its null- points describe a surface (P) of order (p+q-+1). For a straight line / passing through P bears (p+ q) points N, which send their null-plane through P. The straight lines « and v, which intersect in P, lie on (P); for each of their points sends its null-plane through Z. On (P) lie further the (pq +1) singular rays s and the singular curves a, 87, while the singular straight line a is evidently a p-fold line, the singular straight line 6 a q-fold line. The surfaces (P) and (Q) have, in connection with this, the singular lines s, a, b, a and 8 in common and intersect further along the curve (/), which belongs to = PQ. 4. As the straight line / intersects the ruled surface (v*) in p (q + 1) points, the eurve (/) contains evidently p(q +1) singular null-points A* and thus g(7-+1) singular null-points 5*. There are further (g-+1) planes passing through /, which bear a p-fold null-point A, each, and consequently (p +1) planes each with a qg-fold null-point By. Let R be a point outside the straight line /. To the intersections of the surface (2) with the curve (/) belong in the first place the pq null-points of the plane /R. Further the p(q +1) points A* and the g(p+1) points B*. The remaining common points to the number of (p +q +1) (ptq+tpg)— pgr 1)--q(p+1) i.e. r(¢+i)+teg(p +1) must be lying in the (7+ 1) points A, and the (p-+1) points By. As a ou (B) is a p-fold line each of the (q +1) points A, must be a p-fold point of the curve (J). Analogously has (/) in each of the (p +1) points A, a q-fold point. The curve a? is rational, sends consequently 2(p— 1) tangent planes through /. In each of these tangent planes two rays w coincide, so there are q double null-points, so that the plane is q-fold tangent plane of (1). Analogously 87 sends through / 2(q—1) tangent planes which are p-fold tangent planes of the curve (/). As / is intersected by (I) in (p+q) points, the rank of / is equal to 2(p —Aq+2(q—) p+2(p+q), ie. 4pq. 5. Let us inquire in how far the results arrived at are altered when the congruence of rays (1,g) is replaced by the congruence (1,3) of the bisecants v of a twisted cubic 3’. 311 Let B* be a point of 3’, u* the ray which the congruence (1,p) sends through that point. Any plane passing througn w* contains two straight lines v, which intersect in B*; B* is consequently a double null-point. The surface (P)? ++ has consequently 3? as nodal curve; it further cointains the curve of, the (8+ 1) singular straight lines s and passes p times through the singular straight line «. The ruled surface (v*) is of order 4p, the ruled surface (u*) of order (8p-+ 3), while the straight lines vy, as bisecants of 9°, form a ruled surface of the fourth order. If the congruence (1,p) is also replaced by the congruence (1,3) of the bisecants of a curve «’*, a null-system à (1, 9,6) arises. The surface (P)’ has « and $*° as nodal curves and contains 10 singular straight lines s; (P)' and (Q)' have moreover a curve (/)!® in com- mon. The ruled surfaces (w*) and (v*) are of order 12. 6. For p=1, g=1 we have a bilinear null-system ® (1, 1, 2), in which the rays wu rest on two straight lines a,a’, the rays v on two straight lines 6,6’. The singular figure consists then of the straight lines a, a’, 6,6’ and their two transversals s,s’. For each singular point the null- planes form a pencil; the axes of those pencils form four quadratic systems of generatrices. The surface (/)* has a triple tangent plane *) in the null-plane of P. ) Cf. my paper “On bilinear null-systems” (These Proceedings, vol. XV, p. 1160). Experimental Psychology. — “The Psychology of Conditions of Confusion”. By Prof. E. D. Wirrsma. | (Communicated in the meeting of April 26, 1918). The contents of our consciousness distinguish themselves by their intensity. When attentiveness is directed on them they have a high grade of consciousness. When our attention is scattered over many psychical contents or there is a weakening or depression of the attentiveness, we speak of a generally low grade of consciousness, by which we have to understand a condition in which external impressions or also our own thoughts can not, or with difficulty, cross the threshold of consciousness; in which associations do not; or incompletely, come to pass; in which the formation of syntheses is hampered, in other words, a condition in which the precision, the clearness and the velocity of conception of the contents of consciousness is diminished. Such depressions of consciousness occur in many forms, normally as well as pathologically. The momentary weakenings of consciousness cause normally the phenomena of depersonalisation and of ‘‘fausse reconnaissance”, as was proved by the investigations of Heymans, and pathologically the epileptic fits as the psychology of epilepsy teaches us. More prolonged depressions occur normally in dullness, exhaustion, sleepiness, and sleep, and pathologically in the conditions of acute confusion as we meet them in or after acute infectious diseases, in some intoxications, and sometimes in meningitis. These processes can make their appearance in many different forms. At one time the stupor is more pronounced, then again the confusion and desorientation, strong disturbances of memory, hallu- cinations, delusions, and motor restlessness. In whatever form the disease presents itself the characteristics of a lowered grade of consciousness are always clearly present. The constant presence of this one symptom with the great change in all other phenomena, makes it probable that the former is primary to those other symptoms. This opinion is strengthened by the fact that all the symptoms of confusion disappear for a moment if we are able to obviate or lessen the intensity of the depression of attentiveness. In a raving fever patient, in a patient with delirium tremens, with uraemia or 313 with meningitis one can often let all symptoms disappear for a short time by heightening the psychical.level or by concentrating the attention on -something. The patient is then no longer confused, gives the right answers, knows his bearings, and has no more hallucinations. Moreover there is so strong a correspondance between normal depressions of consciousness such as sleep and the dream, and the acute pathological conditions of confusion, that of old a comparison was readily made between these conditions. If now the low grade of consciousness or the depression of the attentiveness is the cause of the various symptoms of acute conditions of confusion, then it is to be expected that such symptoms will also make their appearance, albeit in rudimentary form, in normal and pathological conditions in which the grade of consciousness has sunk. To determine this I have made a series of investigations on persons of whom it could with certainty be assumed that the groups in which they were classified would show large differences in attentiveness. Among these were patients with obvious intellectual disturbances, sufferers from melancholia with strong obstruction and depression, hysterics with a narrowed consciousness, and normal persons. After the grade of consciousness had been determined by an examination of the attentiveness, several other psychical functions, which are more or less disturbed in acute conditions of confusion, were further investigated. In a number of other subjects I limited this examination to a few psychical functions only, viz. to the annihilation of weak impressi- ons by later stronger ones. The results of this later examination I shall mention immediately after describing the arrangement of the test, while the results of the first experiments, in which various methods of examination have been used, will follow after a descrip- tion of the methods has been given, so that in this way a better survey is obtained for comparing the results. It is well known that there are many good methods for measuring the voluntary attentiveness, which gives us an idea of the grade of consciousness. | have made use of two of these viz. the Esthesio- meter, which was first used by GRriesBacu *) to determine fatigue, and the marking method’) as this has been used in the determi- nation of the psychical after-effects of school children. 1) GriesBAcH Ueber Beziehungen zwischen geistiger Ermüdung und Empfindings- vermögen der Haut. Arch. f. Hygiene Bd. 24. 1895. 2) Wiersma. Psychische Nachwirkungen. Zeitschr. für de ges. Neur. u. Psych. Bd. XXXV H 3. 314 Opinions vary strongly as regards the value of the esthesiometric method, but I do not want to dwell upon that now. I have used the method as described by Biner') and for the sake of brevity I refer to the original description. The subjects of the experiment were touched on the back of the hand with the ends of two blunt needles of a certain thickness, which were fixed at various distances from each other in pieces of cardboard. The distances between the needles were 0 (one needle only); 1; 1.5; 2; 2.5 and 3 em. The needles must be placed on the skin simultaneously and always with the same amount of pressure. With such a set of needles the subject is touched in irregular order, but with the same distance equally often. These experiments were repeated on five consecutive days at the same time of day. Then the percentage of double and single touches was determined for each distance. For the purpose of judging the attentiveness | used as a criterion the fact that a touch with one needle had always to be felt as one point, and with two needles 2.5 or 3 cm. apart always as two points. This has been proved by a prolonged investigation of various normal people. If a mistake is made here it has usually to be considered as a disturbance of the attentiveness. By computing the average number of mistakes it was possible to get an opinion of the attentiveness. It was evidently necessary not to reckon with the border values only, because some patients suf- fering. from’ dementia and often also those suffering from Melan- cholia, always answer over all distances with 2 or with 1. In these patients one would come to very misleading results. On the other hand it has been proved, also by investigations of Binet, that it must not always be ascribed to inattentiveness when a touch with one needle is felt doubly. A high degree of attentiveness could sometimes be the cause of this. There is therefore no doubt about it that this method does not always yield trustworthy results, but it is serviceable for measuring larger differences of attentiveness, as are found in pathological cases. The second method of investigation consists of the marking tests. It is accurately explained to the person experimented upon what he is expected to do. A large piece of paper, on which there are printed 50 lines of groups of dots, is placed before him. These groups, of which there are 25 on each line, consists of three, four, or five dots. The order of the various groups, which are more or less equal in number, is extremely irregular. 1) Biner. An. ps. XI. 1905. 315 The person to be examined was now instructed to mark the groups of four dots with a vertical and those of three dots with a horizontal line, in pencil, in as short a time as possible. The end of every minute was notified by the investigator and had to be recorded by a line. At the end of three minutes there was an interval of two min. and then the work was started again for three minutes, but then with this difference that now the marking was reversed, the groups of four being indicated by a horizontal, and those of three by a vertical line. This was repeated on five consecutive days, however so that on the even days the reversed marking had to be done in the third minute before the interval. The standard of attentiveness could be determined from the results of this work in different ways: 1. By the number of dot groups that had been examined by the subject. 2. By the number of mistakes. After the grade of attentiveness had been determined in this way I have investigated whether the phenomena of confusion were to be found in the persons when the grade of consciousness sank. The memory was examined by the following method: The so-called “Treffer” Method of Müuurr and Pinzecker, This consisted herein that during five days eight pairs of words, which were typed on a piece of paper and between whom an associative connection had been avoided as much as possible, were laid before the subject on each day. One of the lists follows here ; Poplar Air Clock John Grey Willow Jacob Sleep Match Chestnut tree Violet Charles Letter Garret Earth Brown These words are slowly read aloud in pairs and are then with- drawn from the subjects observation. The first word of each pair was then mentioned by the examiner whereupon the subject had to name the corresponding word. By computing the number of correct, incorrect, and missing answers one could form an idea of the memory. 2. The above named pairs of words were chosen in such a way 316 that three ideas, which could be combined in the same general conception, appeared three times on each list. In the list above there are three trees, three christian names, and three colours. In other lists there are limbs, coins, birds, names of cities ete. A quarter of an hour after the examination with the “Treffer” method, during which other tests had been made, the subject was asked which trees, christian names, colours etc. were on the list he had seen. Here again the percentage of correct, incorrect, and missing answers was computed. The further investigations were for the purpose of determining the faculty of inculcation, and the annihilation of freshly received impressions by later and stronger ones. Inculeation and reproduction of numbers of two figures. A row of five numbers, which has been carefully selected so that in each test numbers of the same tens, the combination of the game figures, and round tens, were avoided, was placed before the person to be examined. After he had read them aloud slowly twice he had to repeat them after an interval of one minute. Then the same test was repeated, but now with this difference that additions of two figures had to be done as quickly as possible during the interval. This test was repeated during five days and the results with the subsequent impediment, and without it, compared. Recognition of numbers of two figures. The test described above was afterwards made in a modified form. The numbers which had been observed and read aloud had now not to be mentioned, but were to be selected from a list three times as large. The percentage of good bad and nil-achievements in both tests was computed. A comparison between the reproduction through association and through recognition, and between the annihilations of subsequent work in these psychical functions was hereby possible. Ineuleation, Reproduction, and Annihilation of Observations of simple little figures. The person to be examined is placed in a dark room before a box in which an opening of 10 cm? has been made on the foreside. In the box there is an electrie lamp, which throws its light on the opening. Small glass plates to which small drawings on white paper, 317 that has been blackened on the back-side, have been attached, are pushed into this opening. In front of the box there is another lamp, which is lighted during one second automatically. As this light falls on the drawings they are exposed to the subject during one second. The subject has been told to remember the order of the drawings. The number of exposures necessary before he can do this is a measure for the faculty of inculcation. Then the same test is again put, but with the drawings in a different order and with this modification that the lamp in the box is turned on immediately after the exposure of the drawings, so that the full light out of the box falls on the eyes of the subject during several seconds after observation of the drawings. By determining how often under these circumstances the observation had to take place for the subject to be able to name the correct order, one could now determine the retrograde annihilating influence of the strong light. Persons of various ages were examined by this method. The number of observations, the average of two tests, necessary to determine the order of the drawings, without and with the subsequent strong light, is expressed in the following table: Age _ Number Without obstacle | gen 10—15 years | 19 eo | 4.2 HGR AOS AE nak 22 Bal 2.4 4.3 above 40, | 12 | 2.6 | 5.2 The children have thus to see the drawings 2.3 times. In each observation they then remember 3$ or 48.5 °/, and with a subsequent strong impulse 4% or 23.8 °/,. When expressed in percentages we get the following table: Age Number | Without obstacle | dee 10-15 years) 19 43.5 egeees: 16-40 , 22 41.7 | 23.3 above 40 ,, 12 40 19.2 Proceedings Royal Acad Amsterdam. Vol. X XI. 318 We now get a measure of the obstructions by expressing the differences in percentages of the amounts of inculeation. Age | Obstruction | 9—15 years 45 9, 16—40 , | 44.1 % over 40 „ | 52 9, From these tables it is apparent that this investigation proves what is to be expected, that the inculcation is smallest above 40 years of age, and that the destruction of impulses received is then also strongest. These lamp-tests were modified in such a way that this investigation became serviceable for clinical purposes. Just as with the former tests we experimented on five different days, now however not in a dark room, but in broad daylight. Four round coloured dises were pasted on a piece of grey cardboard. These colours were shown to the subject during 2 seconds. Then, after an interval of 15 sec. he had to name the colours in the right order. If the answer was not correct, the colours were exposed till the correct answer was given twice in succession. The tests were then again repeated, but with this difference that, after the observation of the colours, the light of an electric lamp, in a little box, of which the cardboard with the colours formed the foreside, was exposed by the removal of the cardboard and the light allowed to shine in the eyes of the subject during 15 seconds. The influence of the subsequent strong light could be determined by investigating how often the test had to be repeated to get the correct answer twice. This test was subsequently repeated in precisely the same way with four figures e.g. XOAQO which are drawn in pairs next to each other on the cardboard, and afterwards also with three colours and three figures, which were drawn in such a way that there was a colour next to each figure. The result of these tests was such that in normal people a very slight destruction was caused by the subsequent impulse, but in persons suffering from dementia this was the case to an important degree. At my instigation these experiments have been repeated in a slightly modified form and the results obtained will be published in a thesis. 319 Sound tests. In a quiet room the ticking of an electric bell was deadened, by distance and by wrapping the bell in a box with cotton- wool, to such a degree that the intensity of the sound was only just above the border value. On the table in front of the subject, who regularly heard the ticking, there was a hubbub-maker of Barany, which could be set going automatically through an electric contact, immediately after each tick. By having observations made after the tick, with and without subsequent noise in irregular order, it can precisely be determined how often the weak impulse is lost by the retrograde power of the stronger noise. In thirteen normal persons and in three suffering from dementia (2 dem. paralytica and 1 dem. arteriosclerotica) twenty tests were made daily on each person during five days. Average number of observations. Without subsequent With subsequent. Number | strong sound. | strong sound. | Normal | 13 | 97.1 | 88.9 | | Dement ä | 50.0 0. The great disturbances in the observation and the enormous destruction in the sufferers from dementia are immediately apparent. Touch tests. It can be easily verified that the observation of a slight rough- ness, which one feels by stroking the fingers over a flat surface, disappears when the observation is followed by a strong touch impuise. It is not necessary that the strong subsequent impulse should act on the same locality, but the preceding weak observation also disappears if the subsequent strong impulse acts on another part of the finger, or even when it acts on one of the fingers of the other hand. Tests were put in the following way, on one half of a smooth dise a layer of paper 3 mm. thick was pasted. When the dise revolves swiftly, the fingers, resting on the disc, clearly feel the unevenness. If now a larger elevation is placed at some distance from this unevenness so that the finger will collide with this eleva- tion during the revolution, after it has passed the smaller uneven- . 21*. 320 ness, then the first and weaker impulse will not be felt if the subsequent stronger impulse is not too far removed. from it. By regulating the distance between the two impulses one can determine the retrograde destructive influence of the stronger impulse. In experiments made with people of various ages it became clear that there are strong individual differences. These tests are excellently suitable to demonstrate the retrograde influence of strong impulses, but there are so many sources of errors that I shall give up the description of the individual differences. A much better method consists herein that the observation of a weak electric impulse on one hand, which is noticed regularly, disappears when it is followed by a strong electric impulse on the other hand. After the description of the tests and the communication of the results I shall communicate the results of the examination of the subjects on whom the various methods of investigation had been applied. Fifty-three persons were examined viz. 14 normals, 9 neurotics, 13 melancholies, and 17 with intellectual defects. This preliminary communication would become too extensive if 1 were to give a more detailed description of the subjects. I want only to state that in the group of the neuroses there were 4 sufferers from hystery and 5 from psychasthenia, while the latter also exhibited hysterical stigmata. The melancholics were obstructed and depressed and some of them had micromanias which were not present in others. Among the sufferers from dementia there were patients with senile dementia, dem. paralytiea, dem. praecox and dem. epileptica. The intensity of the dementia was strongly varying, but in no case was it so great that it caused any difficulty in this fairly long investigation. In accordance with the aim of the research, to acquire more knowledge concerning the influence of a depression of consciousness on the various psychical functions, it is sufficient for the present to communicate the differences which appear in the various groups of subjects in which there was a very large difference in the grades of consciousness. Esthesiometer. The attentiveness of the melancholics and especially of the patients with dementia is considerably worse than of normals and of the neurotics. The good achievements of the neurotics prove that the narrowed Percentage of Number | correct answers. Normals 14 96.9 Neurotics 9 98.1 Melancholics 13 87.9 Dements 17 16.9 consciousness of the hysterics and the psyehical disturbances of the psychasthenies caused them no difficulty in concentrating their attentiveness for a short time on work which interests them. Underlining tests. ‚Normal under- | Normal under- | Reversed under- Reversed under- | lining average lining N°. of lining average lining N°. of ‚work per minute mistakes °/o9. | work per minute mistakes "/o9 Normals 86.0 4.1 rde | 17.4 Neurotics 14.6 5.5 59.9 23.8 Melancholics 66.8 : Tek 52.9 91.0 Dements 54.8 26.5 42.6 11047 The great differences in attentiveness is apparent from the number of normal and reversed underlinings as well as from the number of mistakes. The automatic after-action, the perseverance, is increased strongly simultaneously to the diminution of the attentiveness. This appears out of the stronger influence of the normal underlining on the quantity as well as on the quality of the reversed during a depression of the attentiveness. Perseverance, continuing to cling to observations, conceptions or actions is a phenomenon that frequently occurs in dreams and in acute confusion. It is also clear that the neuroties now, as opposed to the esthesio- metric test, achieve considerably less than the normals, probably on account of the circumstance that their attentiveness had now to be settled on a work for a longer time. Test with the method. Parallel to the descent of the grade of consciousness the number 322 Percentage of answers. Correct | Incorrect No answer mma Normals 43.2 | 24.2 | 32.6 Neurotics | 35.3 31.2 33.6 Melancholics | 24.5 | 30.5 45.1 Dements | 23.0 35.7 41.2 of correct answers decreases and the number of incorrect ones increases. The lessening of the fixation of associations must be considered as an indication of the very defective conception in acute confusion, in which it is often a striking symptom so that, no matter what trouble the patient takes, it is not possible to digest, to assi- milate the external impressions. The inereasing number of incorrect answers indicates a loosening of the associations which must be considered as a rudimentary form of a lack of the connection between the conceptions in the same way as this makes its appearance in acute confusion. This falling out of the associative connection is so essential in this disease that this has been named after it. In our subjects the difficulty of fixation of impressions causes disturbances of memory, as are found in acute confusion of a very high degree, and which correspond to the depression of consciousness. Reducing specialised conceptions to general ones. Percentage of answers. | | Correct Incorrect | No answer Normals 67.3 4.4 28.3 Neurotics 63.9 5.2 30.9 Melancholics IS 8.3 37.8 Dements 42.2 | 12.8 44.7 While the associations by contiguity and by simultaneousness were examined more especially in the preceding test, the association by agreement plays the greatest role in this the last research, i.e. the reduction of a special to a general conception. 323 Here again it is apparent that the impulses are more strongly fixed in the presence of a better attentiveness. The increasing number of incorrect answers and the strong lower- ing of the grade of consciousness indicate that the paramnesias, which occur in acute confusion in a much higher degree, so that they then often occasion confabulation, are dependent on these. Inculeation and reproduction of numbers, with and without subsequent work. | Without With | Correct | Incorrect None Correct | Incorrect | None | | | Veale) | Normals | 68.3 | 280 | 2.9 54.3 | 33.1 12.6 Neurotics 66.2 | 26.7 71 52.9 | 27.6 | 19.6 Melancholics | 60.9 Sled 1.4 38.8 41.2 | 20.0 Dements | 44.0 38.1 17.9 26.4 «| 48.0 | 25:6 | | It is in the first place apparent from these tests that the number of correct reproductions, with as well as without obstruction, here decreases sharply with the stronger lowering of the grade of cons- ciousness, and in such a way that the minute lessening of the attentiveness in the neurotics is accompanied by a slight disturbance of memory, while the much stronger depression of attentiveness in the sufferers from dementia is accompanied by a much stronger one. If we compare the correct answers with and without obstruction it is clear that the destruction of remembrances is caused by the subsequent work. When we consider the decrease of the achieve- ments in the percentages of the correct answers, with and without subsequent work, we come to the following table. Destruction by subsequent work. Normals 20.5 Neurotics | 20.1 Melancholics | 36.3 Dements | 40 The retrograde destruction by subsequent work thus increases in accordance to the lowering of the degree of consciousness. This 324 phenomenon makes its appearance in much stronger measure in acute confusion, in which often nothing is remembered. Freshly received impressions were immediately destroyed by subsequent psychical contents. There is another phenomenon worth mentioning. The number of incorrect answers increases in accordance to the lowering of the grade of consciousness. This has also been proved by the preceding tests. The neuroties however form an exception to this rule. Of all the people examined they give the smallest number of incorrect answers. This phenomenon is explained by the characteristics of the psychasthenies, who are withheld from giving an answer by all sorts of scruples unless they are absolutely certain. The normals on the other hand will guess at a number if they have remembered one figure only. In tracing all the answers separately this becomes clear. The stronger inclination of the psychasthenics to keep silence rather than give an incorrect answer also becomes clearly apparent if we compare the percentage of incorrect and nil-answers of the total number of answers which were not correct. : Answers that were not correct. Percentage | Percentage incorrect | unanswered Normals 90.9 - 9.1 Neurotics | 19.0 21.0 Melancholics 81.1 18.9 Dements | 68.0 32.0 The number of incorrect answers is smaller in the dements than in the melancholics. Recognition of numbers, with and without subsequent work. | Without With Correct | Incorrect | Unans Correct Incorrect | Unanis- wered wered | Normals 80.6 19.4 0 18.0 DE 0.3 Neurotics 82.2 13.3 | teed 72.9 20; 0: dee Melancholics 69.5 2513 > 1? eee 62.8 30.5 | 6.8 Dements i Pel wees Se Se 6.6 ee ae a a 9.4 325 An important influence of the grade of consciousness is here perceptible. There is hardly any difference between the neurotics and the normals. The power of raising conceptions of remembrance through observations is thus lessened in accordance to the loosening of the attentiveness. If this phenomenon makes its appearance in such a degree that the observations remain independent and that no conceptions of remembrance can be brought into connection with them, then desorientation takes place, a phenomenon that is usually present in acute confusion. The retrograde influence of subsequent work is here not so strong by a great deal as was the case in the preceding test. This is especially apparent from the following table. Destruction by subsequent work Normals 32 Neurotics PS) Melancholics 9.6 Dements 15,2 In the normals the retrograde influence is nearly absent, in the dements on the other hand it is very clear. The uncertainty of the psychasthenies is apparent in these tests in the same way as in the preceding and again becomes clear if we compare the percentages incorrect answers mutually and the nil-answers mutually, of all the answers that were not correct. Answers that were not correct | Without subsequent With subsequent obstruction obstruction | percentage | percentage | percentage | percentage incorrect unanswered incorrect | unanswered Normals 100 | 0 98.6 1.4 Neurotics | 1421 | 25-3 13.8 26.2 Melancholics | 82.6 | 17.4 So ge 18.0 Dements 84.6 15.4 81.7 18.3 The number of incorrect answers is smallest in the neurotics, while here the number of nil-answers is largest. Inculcation, reproduction and destruction of the observation of simple figures. These tests were not made on the 53 subjects mentioned above, 326 but on 41. They were repeated during five days. The method of experimenting and of computing the results was the same as that which has been described more in detail above. Number quent light quent light Destruction Normals 12 81.1 38.4 Le Pe Neurotics 3 | 66.7 31.2 53.2 Melancholics 6 58.6 29.5 56.7 Dements 10 | 29.5 1150 | 60.4 It is remarkable that the inculcation decreases regularly as the depression of consciousness becomes larger, and that the dements especially achieve much less. The destruction of the newly received impulses increases as the grade of consciousness becomes lower. By this research it is apparent that the phenomena of acute confusion are present in the bud in the normal and pathological subjects examined, and that they increase as the grade of con- sciousness becomes lower. Mathematics. — “On the direct analyses of the linear quantities belonging to the rotational group in three and four fundamental variables’. By Prof. J. A. Scnouren. (Communicated by Prof. CARDINAAL). (Communicated in the meeting of September 29, 1917). Quantities and direct analyses. By a (geometric or algebraic) quantity existing with a definite transformation-group we mean, according to F. Kier, any complex of numbers (characteristic numbers of the quantity), that is transformed into itself?) by the transformations of that group. Quantities only have any signification and only exist with definite transformation-groups and may be “disturbed”? as such with other groups, whose trans- formations do not transform the characteristic numbers into themselves. They are completely determined by their mode of orientation, i.e. the mode of transformation of their characteristic numbers, The variables of the group are called fundamental variables and are the characteristic numbers of a fundamental element. MF the group is the linear homogeneous one in » variables, the simplest quantities are those, whose characteristic numbers are transformed as the determinants in a matrix of p fundamental elements independent of each other, p — 1,...,. With a homogeneous interpretation of the fundamental variables they correspond to the linear /,,,-complexes in Ra, provided with a number-factor. All the quantities, whose characteristie numbers are transformed in that way under the trans- formations of the rotational group, we call linear quantities. By a direct analysis we mean a system of an addition and some multiplications by means of which we can express the relations among quantities of a definite kind left invariant under the trans- formations of a definite group. Every quantity is in the analysis a higher complex number. Till recently suchlike analyses were brought about by choosing for multiplications some characteristically distri- butive combinations conspicuous in geometry or mechanies, and uniting them into a system as well as might be. Owing to the great number of existing combinations of this kind arbitrariness could not fail to arise, and this led to the formulation of many systems, the adherents of which have been involved in a violent polemic for these twenty five years. y e. g F. Krein, Elementarmathematik vom höheren Standpunkte aus. Leipzig (09) II p. 59. 328 Application of Kuxin’s Principle of Classificatton. The author of this paper observed in 1914+) that it follows from the application of Kruis principle of classification to analyses belonging to definite quantities, that to a given group of transfor- mations and given quantities belongs a completely determined system, which may simply be computed. This was practically done for n= 3, the rotational group, and quantities up to the second order inclusive. In a more exhaustive investigation contemplating four different sub-groups of the linear homogeneous group the same was executed for arbitrary values of 2 and for quantities of an arbitrary degree’). We shall briefly state some results of this investigation bearing on linear quantities, in particular for n=3 and n= 4, founded on the: rotational group (a,7-+...-+ a,” invariant, det. = + 1) and availing ourselves of the: orthogonal group (a? +... + a,’ invariant, det. = + 1) specialafyin. group (in. hom. with det. + 1) equivoluminar group (lin. hom. with det. + 1) linear homogeneous group for further classification of the quantities existing with the rotational group. General symmetrical and alternating multiplication. Three multiplications of fundamental elements exist with all the sub-groups of the linear homogeneous group and for all the values of n, viz. the general, the symmetrical and the alternating one. The general product of p fundamental elements has 7” characte- ristie numbers, being the products of the characteristic numbers of the factors. Their mode of transformation is entirely determined by this definition. We express the product in this manner: | le) AS ARO nne i A Bn ie Fe A ‘ . 2 5 . (1) o By isomers of a,....a, we mean all the general products that . : o can be formed by permutation of the factors from a,....a,. An even respectively odd isomer is concomitant with an even resp. odd permutation. The symmetrical product of a,....a, is the sum total of all the isomers divided by their number p/: en ‘ne 5 dt: dd enk EE n= Ais or ine ee The alternating product is the sum of all the even isomers dimi- 5 Grundlagen der Vektor- und Affinoranalysis, Leipzig (14). 2) Ueber die Zahlensysteme der rotationalen Gruppe. Nieuw Archief voor Wiskunde 1919. 329 nished by the sum of all odd ones divided by p/ and may be expressed as Cayleyan determinant: Ai iss’ Koos - (3) ai aa iene trated eN a, =a. Reh hy) SS (to. be developed according to rows). Ars OD The alternating product of p fundamental alas is a linear quantity for p Sax. For p >>n it is zero. A symmetrical product is never a linear quantity. The Associative Systems R,. Classifying up to the lin. homog. group inclusive, the system belonging to the linear quantities is R', which is an associative system, entirely determined by the rules: ej + es = es +e = eij ') e’; 4 e’; As 7 Bn e’; = Cry ei a ej =k er1e; =k ° bj] » 2 J J BBN SEL = eije AR Seay C12..1= 1 e122 =P , 7 ; bd ’ ’ 7 Cro) Gar... ey Le “era Ere Wer e= ep vie se, F WSE AE Lat kT oe as aa ee: n(n—1) (are €,,...., Ee, are the covariant fundamental units, i.e. units of a fundamental element, and e’,,...., e, are the contravariant funda- mental units belonging to characteristic numbers, transforming themselves contragrediently relative to the fundamental variables. When classifying up to the equiv. group incl, the system A, is constituted, being obtained from the preceding one by the identification Fl and being entirely determined by the rules: ei 4 ej = —Ej 4 & = Gij | e‚ +ei=k Men eo cam 4 CLOG mear WEL Ri jes ol Ane ti | B=las I=! Quantities, whose units, apart from an eventual factor I, do not contain two equal fundamental units as factors, exist unlike the 1) In a more exhaustive investigation “Die direkte Analysis zur neueren Relativi- tatstheorie”, Verhand. der Kon. Akad. v. Wet. Sectie | Deel XII N’. 6 we consider : at : eje;—eie; a. also not linear quantilies and we write ej ej = eij ande: + ej = — ae = eij etc. For more convenience we write here ej ej = ei j. 330 others with the lin. homog. group too, and are called projective quantities. Then they are of the sub-degree (Dutch : ondertrap, German : Unterstufe) p, when the number of the factors of the units is p, p= Ii + on Om, and we write them „a. The others are called orthogonal quantities. All linear quantities may be composed of projective ones and powers of K. ; When classifying up to the special affin. group inclusive, for n odd the system A, is obtained from the preceding one by the identification : ew CT oe Erik The sub-degree p, pen coincides with the “ae es bn and forms the degree (trap, Stufe) p. For „even no system is feasible here, because kien edet . (6) hence identification of | with an ordinary number is Kakes When classifying up to the orth. group inclusive, R, arises out of Re by the identification KE eg eet ee wee EN The „system makes no difference between projective and non- projective quantities. The sub-degree p, pSn coincides with the sub-degree (2n—p) and forms the by-degree (neventrap, Nebenstufe) p. When classifying up to the rotational group inclusive, for n odd, ; R! arises out of Ry by the identification VEREN oe oe a ee ee Neither does this system make any difference between projective and non-projective quantities. The sub- degrees p, (n—p), (n -- p) and (2n—p) coincide and constitute the principal a Hauptstufe) p ; n—l n PENN == ‚for n odd and n’ = for n even. In all these ad systems the associative product of dissimilar fundamental units is equal to the alternating one. The systems A, are the products of original systems and principal rows *) according to the general formulae: Rn = Oz 2 n—1L Ri, — H‚ Oz 2 EEE Ne, n—1 ( ) RE =H; Oo 2 i) Cf. Grundl. pages 11-—18. 331 \ for n odd and ; a fre == Og 2 | (10) Rv =H, O22 for n even, where Q; denotes an original system of the order 2 and H; a principal row of the order ¢. But for some divergence in + and — signs the systems R? are identical with Crirrorp’s n-way algebras *). If none of the units is privileged the choice of the numbers occurring in the identifications is altogether determined by the dualities existing in the different groups. There are four altogether, and we shall call them: a- n—18 a-p a- nia Oy a- 2n—1a a-Jd a-a’ (8 From the mode of transformation we conclude for the existence of these dualities as subjoined: Duality: | Gr — bj | Cry ad a-é& Group: | n even n odd | n even | n odd n even | n odd 5 DR eel ee A EN My ee As RD ed linear | | homog. Fo zi 4 ae 7 | re ay Pat ae Sie Tee Oa Pee — — l — equivo- ed ee ie En sieke A for 2=2)| IOF Al za-d lumin. identity | identity | | ial | | | | | i te special. | sarees Ot eae Ln ee Cre En t ap | a a affin | En sb ‚identity — identity | a=| { B | | Are. R N | orthogon. | —a-y i— a3 identity identity | identity | | rotation | identity identity | identity | identity … identity | identity | identity | | eS € | + = existing, — — not existing. 2) 1) Crmrrorp's systems have been worked out by J. Jory, Proc. Roy. Ir. Acad. 5 (98) 73—123, A manual of quaternions (05) 303—309. He gives geometrical applications after the manner of the quaternion-theory without decomposition of the product. A. M'Avray has elaborated this -matter as well, Proc. Roy. Soc. Edinb. 98 (07) 503—585. These papers do not aim at a foundation on the theory of invariants or a closer investigation of the fundamental groups. 2) The squares of the dualities not founded on contragredience have been indicated by blacker demarcation. These dualities only exist when » is even. 332 The associative Systems R3 and Rg. If we call the unities of the sub-degrees (n—1), (n+-1), and (2n—1) corresponding to e;:e;, e; and e; and the contragrediént unities e;’, the rules of calculation for n= 3 are: a = — ees e3 = — €'1l e123 = I e123 = di l R; —el = e 23 a-y | —eg3l’ = e': ud i re PI ah: See en =k ei ae ei 4e'1 = e14e sf Cycli, er = | — €23 l= —e I= —e'r3 I es = ei = es = —e; I | e123. = | — e931 = — €123 I= e193 =| | a-B:— (11) Rs oe) e033 = e1 = €’23 bf Ge a-d: + — e23 I= |—e1 I= — €23 I= e’1 EE ei3l= | —E'123 = — e123 = e193 I =[?=— k®=-+1 cycl. 1,2,3. en =k ei =—k?| en. =k esa) = kt | | Ei Pay al ' —kegs=|e1 =~ | €23= eer Be ded ei = | k?e’23 a-Bit R | N id 3 | eer es iden- | e13= | E123= =I=ki= 1,01: tity (12) | Buiken == Kel a-d=a—6. | i | ; | Ml a—-&=a—/? ; AG Cycl 1:2, 2: | e123 I= | — e123 = ke =d |\— e3 I= |e1 = =e I= — €23 I een | | | Yi 18 Ril —ei I= | e23 = eg 5650 raid el / . vn 1 | | É | | a-d: iden- ei3= \—egsl= =] tity. ft i a—e: identity. EEE | rn | cycl1,2,-8. 1) «Oycl 1,2,3,...;n” means that the numbers 1,...,n may be substituted by any even permutation of these numbers. 333 a-B identity fee a-y ” Sy aie ear ern IE) a-d ” rs d-e ” eyel. 123. With a non-homogeneous rectangular interpretation of the funda- mental variables e@, is a polar vector, e', an axial bivector, e, an i axial vector, e’, a polar bivector'), I a projective, and k an ortho- gonal ‘pseudoscalar’, ke, a polar, and k’e,, an axial versor (qua- ternion with tensor 1) without scalar part. 3 includes and discri- minates all these quantities, R3 identifies polar quantities with axial ones and I with an ordinary number, /3 identifies all the polar quantities and all the axial ones as well, and k with a common number, whereas in &3 only the difference between vectors and ordinary numbers exists. The rules of calculation for n= 4 are: a = e234 I | E12 = ae see | ea = —ex I 6234 = ex I €1234 = 71234 ae ae Gila e'234 dee Dsl e’34 | Pairing — €34 Ì e'12 | OS EEE | per a Pie ket | x= + 1 hon ri PE evel. 123,4 “1 = | e111 =k’ | Ore = e14e1 =+ | 1) In space these quantities have the symmetry-properties of a line-part with direction, a plane-part with rotative direction, a line-part with rotative direction and a plane-part with + and — side, all conceived as parallel removable with respect to themselves. For odd it holds good that polar quantities change their sign, when the + direction of all axes is inverted, and that axial ones do not change their signs. 22 Proceedings Royal Acad. Amsterdam. Vol. XXI. fos ays rear | e1 = |—Ile’234 = —ia I= e 234 I | | or ae | el her hee enn ei =| e’s4 I | en = ete Ive eee ENA . | ed rs ap 4 e234 = * | — e's = | -—Jees4 [= ex: 1 ag: + (compli- €1234 = €'1234 = C1234 = e'1934 =] a—-y:-+-) cated) at = ad: Ri \ ig tte De —lee34 I= —-1e4 = e'034 tg (15) | et : a-&=a-d. TI = melt —— C190) AS € 24 cycl. 1, 2 an -e34 I= E12 = |— ess I= ez easa I= |—ie'sx4 = | —ieos4 = ey | e12341= e'1234l= | ~— er 2341= e034] =[?=kt=+1 eit, =Ki ear =K*) em =k en =k | | iM | 1934 1 = e1234 1 = | itt | e234 1 = | e1 = — i€234 = —ie, | | | : „gg: 4. (compli- —e34 1 =| e12 = ein = — €31 I Eede cated) ; | a-y =a— 16 R$ |—ew l=jea = e4 = — e121] Y 7 (18) | ee ey a—0: identity — 9% — — = — l €9: . : € | €234 ed peese™ | © (a =e = identity | e134 = C1234 = |=I cycl. 1, 2, 3, 4. eu = =+1 en =+1) The dualities «a — 3 are complicated ones in this case, i. e. dualising leads say for e-p from e; to e;, from e; to —e;, from — e; to — e'‚ and from — e; again to e;. This complicated duality always exists for n even’), as long as one of the units is not privileged. If one of the units is privileged, or, to put it otherwise, if we derive the system belonging to the group, leaving invariant the quadratic form a a,” li a,” = bee &,—1" we find, when classifying up to the orthogonal groups inclusive, the system: 1) The complicated duality exists also in GRASSMANN's Ausdehnungslehre for n even. eo = €123 e1 =— €023 E1 =— e023 €01 =— eo1 E12 = + E12 ea ate eoo =+1 eoo =+1 a (17) é | a-y=a—B ce en =—1 iden- WE a-J: tit e0i23 = Ì e0123 = — I ae A am iden- eo I= eo —eo I= eo OK tity. Bare ead mt eycl. 1.2, 3: eor- [= E23 = €23 —eo I= E23 = €23 ei2 L=— eo3 = e03, I2?=—1, —e1e I= — E03 = €03 | with non-complicated duality. This system may also be obtained from the preceding system Ry (page 334) by the transition e, > e,, — 1e, > @,, ete., e, See —> e.. etc. “I it noteworthy that, for n= 4 the theory of relativity (for the space-element) exactly corre- sponds to this more simple system. For non-homogeneous rectangular interpretation of the fundamental variables e,, and @,,, are a vector, resp. a trivector of the first kind and Ie, and Ie, are the corresponding quantities of the second kind’). I is a projective and k an orthogonal pseudoscalar. Ri contains and distinguishes all these quantities. Ri identifies a vector resp. a trivector of the first kind with a trivector vector of the second kind and k with an ordinary number. resp. a Decomposition of the Associative Product. The associative product of two projective quantities of the sub- degrees p' and q and the principal degrees pand q, p,q E12 — EZ ei X E23 = — €3 | dual ER ei . €234 =l ei I= lei? = — esa . ei X e123 = e23 e123 X e234 = — e14 (compra ey ei l=—Ie1 =e234 = — le e123 . e123 = — cated) e12 Xe34 = 1 e234 | = — I e234= 1 Ear and for ©,, @;, ©;, €;: (See for formula (23) page 339). The quantities of an even by-degree form a sub-system with 8 units and the rules: i1 * ig =— ie * iy =i ji * je =—je* fr =—1s ij jo =—je * ii =js hs ig =—ito « ji =|3 i; ° ii =— 1 ji 3 ji =+1 cycl. (24) lik =i T=— fa In =j = +i: la baie Ah =! cee 4 ji = €01 But these are the same rules as those for the units e€,, €,, 5, /e,, ie,,7e, of 3 with ordinary complex coefficients, so that the free rules for A also hold good for quantities of an even by-degree of Rs if, instead of X and. we introduce the symbols * and ea « ob = quantity of the second by-degree | oa X eb =scalar in I and 1 aa X (ab # 2C) =2a « 2b X 2 | za x (ab x 2e) — (2a X 2b) 2c — (2a X 2C) 2b 2a (eb x 2¢ X 2d) = (2a X 2b) (2c * 2d) +.... (ca x 2b) « (ac * 2d) =(2b X 2c) (2a * od) +.... ab) X (2e « 2d) = (eb X 2c) (2a X 2d) +... (sa x ab) (2c * 2d X ze) = (ab X 2c) (2a X 2d) ze + (aax 2b X c)(2dx ze X 2f)=(2¢ X deb X 2e)(2a X of)+.... Hence these rules may be written down from memory, as well as the others. (za « |- (25) | The System Ry and the theory of relativity (in an element of four dimensional space). Fragments of A have been used by various authors‘) on the theory of relativity. With them five products occur and two of these 1) H. MiNKowsKi, M. ABRAHAM, A. SOMMERFELD, M. LAvE, PH. FRANK. 339 — [-~=10° la [+= 99° 09 619 = 619 = deed 69 — = 69> [9 109 — = [109 = 09x Ig === lox 09 1—= (I —) P= "al £09 = £09 — = 319 J — = | 219 — ‘E39 = 29 = 109] — =] 109 — ‘t0a= £09 —= Slay = | STO ‘659 — £29 = 109] =| 109 | —= 8&9 ° sto ‘; += 100 ° 109 | — = 8le° ela ‘| += 109 - 109 Alet == GO) Erde) 619 = G19 = TE9 + €59 209 —= 209 — — 319 x 109 ‘cl9 — == EIJ —= 209 x 109 209 — 209 —= S19 x 109 ‘219 —-=— 619 —= 09 * 109 ] — = &o X 109 | = £9 x 108 Ig —= 8609 = TA, =] Lo — Ig — = £209 = Ta, =| 09 = EI = 09 | =] 09 — 09 = &c19 = 09] —=] 00 £090 —= 809 = 69 X 19 £09 = p= 0D SEE Ta 85 — = £29 — = l9 XX 09 €29 — = £29 —=10X 09 [== to * Ta —= 09: 09 [= 3° To T=" 09 1g = 109 * 09 ‘09 —= 019 © To ‘49 — = cT9 * TO 16 = 109° 09 © 09 —= 019 => 1a) ror == 419) ata S= 8109 = El 09 ‘09 = 5619 = E69 & Ia Eg — = 2109 = 219 & 09 "09 == 8819 == HESS: 59 | -=13° 19 ‘T+= 09 ° 09 éla = 619 = las 68 —= CX Ta 109. — = 109 = 09 B= Ta 09 :£9 ‘2g ‘Ta ‘09 IO} PUB 340 said products are doubled by introducing the “dual” bivector (dualer sechervektor)*). EB. Witson and G. Lewis have further elaborated the system and obtain all the products, but three*). All these conclusions are founded on analogies with the common vector-analysis and the multiplications form no parts of the associative multiplication. Therefore the free calculation-rules cannot immediately be put down from memory according to the transvection-rule, but in so far as they exist they only allow a use by means of a table. The names scalar and vectorial too, have been divided over the existing multiplications by analogy and not in agreement to the duality e—y. WILSON-LEWIS | SOMMERFELD, LAUE, etc. +aX b | aX b= ec [ab], vectorial product te EE [ab], scalar je + aX ob a X eb = 3C Ic= [a 2b*], vect. pr.w. dual bivect. +a. eb | a. eb=c — [a zb), vect. pr. Ebita=-ak al=—Ia=3b*)| a eee | : = + aX 3b | a. 3b=40*) | NT aXsb=ec |l + 9a X 2b oa X ab = 4C*) Al I se = (ga 2b*), scal. pr.w. dual biv. oa « ab = ec s [ga 2b], vector pr. (G. Mie) —ea . eb ja ge =| — (za 2b), scal. pr. tkea=teak zal =l2a—zbt) | — ab = + 2a* kk=-1 =+1 *) +k sa =+ 2a ke | ‘za 1 = — 1 3a—b*) aa 3C — 3a.eb | 3a X ab = C +3a. b or Neer | dE 3a 2< b =2c 1) This is not a proper duality, because in the only duality existing with the orthogonal group, a-y, a bivector e.g. @), is not dualistic to the “dual” bivector Ie‚s, but to ej» itself. Nh The connection with an associative CLIFFORD algebra and the absence of three products has already been briefly pointed out by J. B. SHaw, ‘The WILSON and Lewis Algebra for Four-Dimensional Space” Bull. of the int. ass. for quat, (13) 24—27. 341 Therefore this duality does not attain expression, not even in the system of WirsoN and Lewis, though they use units of the kind @,, ,,@,, e,. The foregoing table (subjoined p. 340) presents a summary of the products used by various authors. The table has been arranged dualistically. Each product has been indicated by an example. For the multiplications we used in the columns 1 and 38 the anthor’s own notation, but for the quantities we used all through the notations adopted in this paper. The dual bivector only has been written with the customary asterisk, while the commutative scalar of Wirson and Lewis has been indicated by &. The products marked with *) do not correspond exactly to the other systems, because these systems do not contain the non- commutative scalar I. The system AR? contains the existing fragments and all the existing multiplications and rules, and owing to the free rules of calculation (21 and 25) it is eminently suited for practical purposes. The system R° and the elliptic and hyperbolic geometry in three dimensions. With a homogeneous interpretation of the fundamental variables R° corresponds to a projective geometry in three dimensions, a non degenerated quadratic surface being invariant. If the units are selected according to (16) the equation of the absolute surface in point- resp. plane-coordinates is: rage ae eee ie aie aire u + u Hu, + u = 0 and the geometry is elliptic. If, on the other hand the units are selected according to (17) the geometry is hyperbolic. The free rules of the system are the same for both cases. To a fundamental element a point with a number-value corresponds, to a quantity of the second degree a sum of linear elements (Dyname) and to a quantity of the third degree a planar element. The sub-system of the quantities of the second by-degree is a form of biquaternions, which was first mentioned by Crirrorp') as a system of linear elements in a non-euclidic three-dimensional space. Hence the system A? completes these biquaternions to a system which also contains points and planar elements. 1) Preliminary sketch of biquaternions. Proc. Lond. Math. Soc. 4(78) 381— 395; Further notes on biquaternions. Coll. Math. Papers (76) 385, 395. Physics. — “The thermal conductivity of neon” By S. WeBenr. Suppl. N°. 426 to the Communications from the Physical Laboratory at Leiden. (Communicated by Prof. H. KAMERLINGH ONNEs). (Communicated in the meeting of Febr. 23, 1918). § 1. Jntroduction. In a communication by Prof. H. KAMERLINGH Onnes and myself’) attention was drawn to the considerable devia- tion from the law of corresponding states which shows itself in the comparison of the viscosity of argon and helium. This circumstance brought out the importance of an investigation of the viscosity ot neon down to the lowest temperatures to be reached with this substance. In this connection we also planned an investigation of the heat conductivity of these gases at the lowest temperatures to which measurements can be extended. Indeed according to the kinetic theory a very close connection exists between internal friction and conduction of heat. The two are only distinguished by a factor (specific heat — numerical factor) and for monatomic gases, where the theory as regards viscosity is confirmed in many respects, this factor is independent of the temperature. According to the theory of heat conduction and viscosity the same law of dependence on the temperature will therefore be found in the two cases and it is im- material which of the two quantities is submitted to investigation. If both are measured, the results afford a means of mutual control. Personal circumstances allowed me, before the research above sketched out could be carried out at Leiden, to undertake the in- vestigation of ihe heat conduction of gases in the physical labor- atory of the Puirips Zncandescent Lamp Factories. It was there, that the research contained in this communication was carried out. Only that part which refers to the lowest temperatures will still have to be performed at Leiden. The neon required for this investigation was put at my disposal by Prof. KaAMERLINGH Onnes, who had prepared it from a large supply of gas-residue rich in neon presented to him by G. CLAUDE *); I am glad to offer him my sincere thanks. According to a commu- 1) Leiden Comm. N°. 134c April 1913. ®) Leiden Comm. N°. 147c p. 38. 343 nication from Prof. KAMERLINGH Onnes the gas probably still con- tained a trace of nitrogen. For this reason I purified it once more by GexiHorr’s method *). Whichever of the known experimental methods’) one may choose for the investigation of the heat conductivity of gases, there are always two sources of error which will have to be specially con- sidered: the difference of. temperature between the surface and the gas in contact with it and the convection. With diminution ot pressure of the gas the influence of the convection becomes smaller, that of the temperature-drop greater. For the latter, similarly to the analogous quantity in the internal friction, the slipping along the wall ®), depends on the ratio of the mean free path to the dimensions of the apparatus. Whereas it has been found impossible to calculate the influence of convection‘) on the heat conduction, M. Knupsen *) and M. von Smo.ucHowskt *) have been able to bring the theoretical investigation of the temperature-drop to a successful issue. _In accordance with Kunpr and Warsure’) the temperature-drop 4@ at the solid wall is defined as follows: AO=—y. id dn where n represents the direction of the normal and © the tempe- rature. Kunpr and Warsure by their experiments established the fact, that y is proportional to the mean free path 2. Von Smonucnowsk1 based his first investigation on the kinetic theory as developed by Crausius and was led to the following approximate formula, in which I have introduced the accommodation-coefficient as defined by KNUDsEN *). 4(1-—a) a Wi, == 000 4 (7) Later on SmoLucHowski made a new calculation of y, in this 1) GentHorr, Verh. d. D. Physik. Ges. 13. (1911) p. 271. 2) Comp. A. WINkKELMANN, Handbuch der Physik III, 1906 p. 525. 3) H. KAMERLINGH Onnes, C. DoORSMAN and S. WeBer. These Proc. XV (2) p. 1386 4) A. OBERBEEK. Wied. Ann. VII, 1879, p. 291 and L. Lorenz, Wied. Ann. XIII. 1881, p. 582. 5) M. Knupsen. Ann. d. Ph. (4) 3 4, (1911), p. 655. 6) M. von SmorvenowskKi, R. v. Smotan. Wien. Sitz. Ber. [2u] 107, (1898), p. 304 ; 108, (1899), p. 5. 1) A. Kunpr and E. Warpure. Pogg. Ann. 156, (1875), p. 177. 8) M. Kyupsen. loc. cit. p. 608. 344 case starting from Maxwen1’s hypothesis *), that the molecules may be looked upon as centres of force which repel each other with a force proportional to 7—>. In this way he found 15 2—a U =; ; Qn 2a UL) In these formulae 2, represents the mean free path as determined in Crausius’s theory, therefore : iV 22 1 a= = — 4 p-@ If the mean free path, as found by O. E. Mever’s®) method of calculation, Le Ws 1 1 8 0,30967 Vp.e is introduced into formula II, we find: 2—a 2a § 2. In a paper’) which has appeared in the Annalen der Physik and to which the reader may here be referred, the absolute value of the heat-conductivity at O° C., K,, has been investigated for a number of pure gases. In the experimental determinations SCHT.WIERMACHER’S *) method was used modified in such a manner that it was possible to eliminate the influence of convection on the heat conduetion. Simultaneously the value of the temperature-drop at 0° C. was determined for the same gases and an excellent agreement was found between the experimental value and the one calculated from formula II, if for a the results obtained by KNuDsEN*) were used. Amongst the gases experimented on was the same distilled neon, with which the present experiments were made. The result of the measurements for neon was K, = 0.00010890 gr. cal. grad. sec. em. and 7/, = 2.391. For pure neon and bright platinum Knupsen found a = 0.653, hence: DQ (IIL) ty Seer iy = 2.82 . 2a 1) J. Gu. Maxwett. Scientific papers Vol. Il, p. 23. *) O. E Meyer. Kinet. Theorie der Gase p. 111. It makes no difference whether Meyer’s calculation or a different one is followed here, seeing that the factor which has a different value in the various results drops out from the final result by the introduction of the pressure. 8) SopHus Weser. Ann. d. Ph. (4), 54, (1917), p. 342. 4) A. SCHLEIERMACHER. Wied. Ann. 34, (1888), p. 623. 5) M. Knupsen. Ann. d. Ph., (4) 46, (1915), p. 641. 345 The value of K, is certainly accurate to 2°/,, and agrees well with the measurement made by BANNAW1Tz *) on neon, which Professor KAMERLINGH Onnes had drawn for him from the same vessel from which he had supplied me with the gas I used. § 3. For the determination of the temperature-coefficient of the heat conductivity for neon the same apparatus could not be used as for the absolute measurement, and on this account I resolved to apply GoLpscuMipt’s?) method. This method introduces another important improvement into SCHLEIERMACHER’s method. The loss of heat at the ends of the wire is eliminated in a simple manner by making measure- ments first with a long wire and then with a short one of the same diameter, and heating the wire in both cases with the same electric current. The difference between the amount of energy developed in the short and in the long wire gives the energy which is lost radially by a wire of the same section and of a length equal to the difference of the two experimental wires. The first apparatus which was used in testing Go.p- SCHMIDT’s method is represented in the figure, the con- stants and the dimensions being collected in table I °). The figure shows that the thin platinum wires are stretched along the axis of the glass tubes by means of platinum springs. In the measurements the two wires and a normal resistance of 1 2 are connected up in series. When the condition has become stationary, the potential-differences between the terminals of the long and the short wires KE, and Ex are measured, as also the difference at the terminals of the normal resistance I. The potential differences were measured with a compensation-apparatus by Woxrr which is free of thermo-effects, possible thermo- forces outside the apparatus being eliminated by com- mutation. From the resistance the mean temperatures of the platinum wires t; and tp are calculated. Using these results the following expressions 1, E. Bannawitz. Ann. d. Ph. (4), 48, (1915), p. 577. 3) R. GoLpscHMipr. Physik. ZS. 12 (1911), p. 418. 3) The value given here for 27) was found by weighing, since it is only used in the correction for the temperature-drop and not in the calibration itself, as this was carried out with atmospheric air (see further on). 346 for L and S may be computed : L = 0.2388 . Lj 1 Wiardi 02888. L bbl ke de If the loss of heat at the ends could be neglected, L and S would represent the radial loss of heat per degree and per cm for the long and for the short wire respectively (in a surrounding of 0° C.). In that case ZL and S as well as the quantity would all be equal. Attending now~to the difference in length of the two platinum wires we may according to GorpscHMipr assume, that the heat given off by this portion of the wire is not influenced by the heat con- duction of the terminals. If D represents the loss of heat per unit length of a wire of the same section in an infinite cylinder of the same shape with a temperature-difference of one degree with the outside at 0° C., and if the loss of heat may be taken proportional to the temperature difference, ¢, being the temperature-difference of the uniformly heated wire with the surroundings, we have W wf Ww. >) a(W ,—,) (ae ta From the value of D the mean conductivity A on the way which the heat follows between the wire and the wall may be cal- culated according to the relation *). nk 1 l os) 1) M. von SmotucHowsk1. W. A. 64, (1898) p. 101. D defined below ty = and D = 0.2888 (E,— Ei). 1. D= av) ln — TABLE 1. | App. a App. b a | Diameter of the platinum wire | 2ro = 0.005246. cm | 2r9 = 0.005246 cm Length ,, ,, i x | a —=11843 „Ik =3.198 am Electric resistance at 0°C of the platinum wire Wo = 5.4187 12 “Wo = 1.4481 2 | | Temperature-coefficient of Wo, zo —100 X%o-100 = 0.003888 | %o-100 = 0.003888 | Conductivity of the platinum wire x = 0.1649 hg = 0.1649 Diameter of the glass tube 2R = 1.449 cm 2 R =1449 cm | Je where 4 = section of the platinum wire ee = 1.2039.10—6 | ye = 4.5437.10—8 Zn = ud ae | ee aes = Al ¥ eee 347 where y is the coefficient of the temperature-drop at the wall. Using this apparatus | have made a few experiments to test GoLDsCHMIDT’s theory. In a set of measurements with the apparatus filled with dry pure carbon dioxide the temperature of the bath being 0° C. the following values amongst others were found for ¢), tz, L,S, ta and D. p=21.61.cm ‘ti =S 409 ie 4 630 ts =—5.693 Leativea4n3 O12 107 8 ARo SLO? Bb dOr be ee LOT Gi Sars Se OO p=6.28 cm ty —= 400 th =4.669 fist Ag A OO ee TOS ml ol end U) ren dl te koke Mi el U lined Loor =388.7. 107 Soy. =888.2.10 7 _ These measurements show very clearly that entirely erroneous values may be arrived at for A, if the loss of heat along the ends of the wire is not taken into account. [t can now be shown by means of a simple calculation that the ‘value found for L or S after having been corrected for the heat carried away along the ends agrees with the value of D. The quantities of heat Q, and Q, which in the stationary condition are conducted away through the surface and the ends of an electrically heated wire respectively (apparatus a) are given by *) Q, : 4 Q, eT ga Q x? ———— = €(# . = ae —— ang 0 > — are c(a?-+ m?) ie. ( i n Pen ef m? + is 1 —— Tox 1-—T9x & A.x 0.2388 tr where c = and m° = ————_._ « W,./*, a being an auxiliary c quantity which is determined by the third equation. In. these equations Q is the entire quantity of lieat developed in the wire (appa, Q=L./.t), t the mean temperature of the wire, t, the temperature of the glass wall, / the current and W,, A,x and / the resistance at ¢,, the section, the conductivity and the length of the heated wire respectively ; Z7'gw stands for the hyperbolic tangent of w. When the values found for Zl and Sk are corrected in this manner, the figures given under Lo, and So are obtained; they are seen to agree very well with D. In this way it appears, that the application of GoLpscumipt’s method is allowable, if the dimen- sions of the apparatus are chosen correctly. 1) S. Weser. Ann. d. Ph. (4) 54, (1917), p. 169. 348 § 4. In the determinations with neon an apparatus was used the dimensions of which are contained in table 2. In order to be able to use the apparatus at the temperatures of liquid air it is necessary to compare its resistance with the resistance of a platinum thermometer, whose resistance is known. If this thermometer is calibrated, so that from its resistance the absolute temperature on the Kelvin-scale can be deduced, it becomes possible from the resistance of the conduction-apparatus to determine the corresponding absolute temperature. For this purpose [ have chosen the platinum thermometer ZP, the standard thermometer of the cryogenic laboratory at Leiden. For this thermometer there is a table) which gives the relation between JW, or properly speaking W —, and the absolute temperature. ia TABLE. 2 Aer 7 8 App. 1. | App. 2. 2ro = 0.005240 cm 2r, . ~= 0.005240 cm | l = 9,992 + | k = 3.373 5 | 2R = 1.526 Ee | AR == ‘i Wo AGE Wo = 15416 42 %—100 = 0.003891 | %—100 = 0.003891 | have carried out the comparison in the following manner: in a closed cryostat, provided with a stirring arrangement and filled with pure liquid oxygen, the double conduction-apparatus and an oxygen-thermometer according to Stock’) are mounted side by side. When the condition had become stationary, the resistance of the apparatus w’ — W —w was measured and simultaneously the vapour- pressure of the oxygen-thermometer was read. The following corresponding values were obtained in this way. Vapour-pressure of oxygen p= 742.35 mm and w’ = 0.75828 2. From p according to KAMERLINGH Onnes and Braak *) the absolute temperature 7 of the oxygen-bath is calculated by means of the following relation which holds from 83 and 91° K: 1) G. Horsr. Leiden Comm. N°. 1484. 8) A. Stock and C. Nissen. Ber. d. D. Chem. Ges. 39 (11), 1906, p. 2066. 3) H. KAMERLINGH Onnes and C. BRAAK. Leiden Comm. N®. 107a, comp. Horsr loc. cit. 349 369.83 TT 6.98460 —log p This gives 7’ = 89.896° K. From the table for Pt; the following mutually corresponding values are found: w' W T w‚l Gelk ty 0 89.896 0.24988 0.25079 The two platinum-thermometers can now be compared at each temperature with an accuracy sufficient for our purpose ') by means of Nrrnst’s formula W w' i W Wr LG MT Pe, Introducing the above value in this formula we find «= 0.001221. Using this value for @ it is now possible to calculate the value Ld Al f (=) corresponding to each value of as measured and hence W, Eer W 4 by means of the table for Pf7 to determine the temperature on the Kelvin-scale. The apparatus is then placed in a bath of finely ground ice and distilled water and by means of dry air free of carbon dioxide and R bees | of pure neon the denominator in eq. IV, dn tilt) is i, re 12 : Ri, ni oS determined, which gives —, whereas Ris found by calibrating the 0 tube with mercury. When these measurements are completed, the apparatus is put in a bath of solid carbon dioxide and benzene, and new measurements with neon are made; this time, however, the measurements are conducted in the following order: first the resistance W—w is measured without the wire being heated, whereby the temperature of the bath is determined; then the conductivity measurements are made, first at higher pressure, then at lower pressure and again at the higher pressure, as shown in the tables, and finally the resistance of the wire is determined once more without heating. The temperature of the bath was found not to bave changed during the measurements. The correction for radiation is calculated from the dimensions of ') See G. Horsr Leiden Comm. N°. 148 and P.G. Carr, H. KAMERLINGH ONNES and J. M. Burcers Leiden Comm. N°. 152c. 23 Proceedings Royal Acad. Amsterdam. Vol. XXI. 350 the apparatus and the formula for the complete radiation of platinum *). The corrections as used were as follows: Ryo = 1.70. 10-§, Ry — 0.48.10-§, R_7g = 0.13.10-* and R_1s3 = 0.005.10-°, A being the radiation per degree. The measurements are collected i the following tables. In these tables column I gives the pressure pan in ems Hg; reduced to 0° C. and 45° latitude; column II At, the temperature- difference between the central portion of the long wire and the bath ; where Q is the difference of the quantities column II] D= TT ay! of heat given off by the two wires expressed in gr.cal./sec.; column Atm. air at 0° C. | | a eee Mk = | Pom | cay | D D’54=17.50 | D corr. | | | | | 28.355 | 17.428 0.0004358 0.0004353 | 0.0004367 | 13.069 | 17.495 | 4342 | 4337 4367 { | | 3.010 | 17.860 | 4258 | 4251 4378 Pee ves be fie sen | Neon at 0°C. | Pom | | D Pa = 9.20 | Dorr. 33.791 9.029 0.0008165 0.0008162 | 0.0008222 | 20.182 | 9.077 8122 8118 8219 | 10.181 | 9.192 | 8023 8018 | 8215 | 5.729 | 9.367 | 1878 7812 | 8215 | 1.894 \ 10.252 | 1220 7206 8160 | 0.709 12.726 5868 | 5840 7900 0.217 | 23.109 | 3349 | 3292 1078 | | 1) In the paper quoted above (Ann. d. Phys. (4) 54, 1917, p. 330), where the complete radiation was investigated for platinum and tungsten, it was pointed out, _that the correction for radiation cannot be determined by a separate experiment in vacuo. This is due to the fact that the distribution of temperature along the heated wire is quite different in vacuo than in a gas. Neon at 99°.81 C ge Pom at | D D' ,4=12,00 D’ corr. | | 31.454 ‚ 9.945 0.0009965 | 0.0009971 | 0.00010077 20.225 « 10.044 | 9867 9872 10067 7.413 | 10.389 9547 9547 10057 | 4.290 | 13.795 | 9241 9205 10064 Ct Tis inion die 8123 | 8072 | 9952 0.673 | 21.362 6081 6000 | 9570 | 9.329 | 13.136 9690 9661 10075 | 21.262 | 12.830 9914 9888 10074 | 38.973 | 12.721 | 9998 9973 10072 | | | es me — aie 2 = J = | KLA De ee f | Neon at 194°.72 K. (Solid carbon dioxide in benzene). Pom iy B D'st=8.02 | Peorr. | ] ] —T 39.915 | 7.934 0.0006584 | 0.0006583 | 0.0006608 | 21.649 | 7.966 6557 6556 6603 13.453 | 8.000 6530 6529 6606 6.844 | 8.102 6451 6450 6597 5.046 | 8.169 6398 6397 6597 | | 40.167 J 7.944 6576 6575 6600 Neon at 99°.0 K. (Liquid oxygen). Ë cm | Af | D | D af=3.54 D ‘corr. | | 32.342 | 3.481 | 0.0003740 | 0.0003740 00003746 20.543 | 3.480 3741 3741 3751 10.309 3.510 3110 | 3710 3718 | 5.127 | 3.514 3706 | 3706 3146 2.642 | 3.558 3663 — 3663 3739 0.826 | 3.728 3506 | 3505 | 3739 | 32.165 | 3.490 3730 | 3730 | 3737 4 SS 352 IV D’=D—R, where A is the radiation (as Af is not quite constant, — R has been reduced to the same temperature-difference) and column V D'r. arising by the correction of D’ for the temperature-drop at the wall. The latter correction is made by the formula (comp. formula IV on p. 346): 1 te l 1 1 R 7 Dr Dil ere y= Den. Pem R In — ‘ r 0 With the differences of temperature which are used we may assume with sufficient accuracy, that A corresponds to the temper- At ature ¢, + Pe. hence we have: Atm er od eet el BD = D' corr. = 0.0004371 neon: ,, = 273.1 + 4.60 5 0. 0008218 ee agg 4 Bee BBN re ULNA Ry » 22 194.72 + 4.01 « == 0:0006602 ei mn B add 0000 LN Hence taking the temperature-coefficient of the conductivity for air as 0.0038, the following results are obtained: Atm airs fe 273 Dorr. = 0 0004248 neon =o » oz -0,00038135 For the conductivity at O° C.') of dry air free of carbon dioxide I have found ‘A, = 0.00005680; using this value A’, for neon is found as follows: 8135 "4248 in good agreement with my previous determination A,’ = 1089.10—7. This result shows that the two calibrations of the apparatus are in good mutual agreement; the following results are now obtained for neon: di . 0,00005680 = 0.0001087 arcel./se grad.em. T | K |X ‘cai. S) Kau? Se | | 273.09-+ 105.81 | 0.0001344 | 0.0001344 | 0.0001364 | | 273.09 | 1087 1087 1087 | | 273.09 — 74.37 | 0879 0869 0869 273.09 — 181.43 | 0499 0468 0505 1) S. Weper. Ann. d. Phys. (4) 54, (1917), p. 352. 353 Columm I contains the absolute temperature 7’; column II the conductivity found A’; column IIT the values computed by means of SuTHERLAND’s formula. The value of C in this formula, 57.5 for neon, was derived from the first two measurements *). It appears, therefore, that SUTHRRLAND's formula cannot represent the dependence on the temperature of the conductivity for neon at the lower temperatures. That SurarrLanND’s formula is not satisfactory at low temperatures, was proved before by investigations on the 1) In connection with the value of C and the high viscosity of neon 7, = 2981 . 10-7, it is of interest to calculate the diameter c of the neon-molecule. Using Cuapman’s formula (London Phil. Trans. A. Vol. 216, 1916, p. 279) c.o n= 0.491 (1 + &,). eS Suk RA : vane n(145) where the small correction ea is determined by C, we find, with = 2.77 . 1099, o = 2.32 .10—8. An approximate value of gc may also be obtained by means of the critical constants. From VAN DER WAALS’s formulae in the notation of H. KAMERLINGH Onnes and W. H. Keesom: Die Zustandsgleichung, Comm. Supplem. N°. 23 Fussn. 284 it follows, using Pea nekt ali and Ruf - Vp NM = buf buf tl b XK, R Tr jat : wf == =>. Awf- — Ot be K. According to VAN DER WAALS (see Fussn. 459 Le) is approximately equal to the theoretical value '/,, hence: Using N — 62.10% and by means of the critical constants pj, = 26.86 (intern. atm.) and 7), = 449.75 K. (H. KAMERLINGH ONNEs, C. A. GROMMELIN and P.G. GATE: Comm. NO. 1515) we find « = 2.36. 108. If we use the isothermals for neon at 0° and 20° U. (H. KAMERLINGH ONNES and C. A. GRoMMELIN Comm. NO. 147d) and assume, that @ in VAN DER WAALS’s equation of state is independent of the temperature, one finds (comp. H. KAMERLINGH ONNEs, Comm. NO. 102a p. 5) B,,—B, by === 0°001398. 20 R hen the value of the virial-coefficient C4 at 0° C. one obtains patel SE TA — 0:00136 Using the former of the two values hs a is ¢ = 2.90. 10-8. 354 viscosity of helium and hydrogen (comp. H KaAMERLINGH ONNEs and Sopnus WeBeER: Comm. N°. 134). | have therefore tried, whether an improvement is not brought about — as appeared to be the case in z 1 Comm. N°. 134 — by using a formula of the form zr) Rag £, i. i i As shown by column IV ¢=5 gives a very good agreement. According to Maxwerr’s theory in a more general form *), where the forces between the molecules are taken proportional to r—", we should have to take for neon 28 + 1=n=—11. The measurements give for the temperature-coefficient between 0° and 100°C., B)—100, 0.00226; this agrees very closely with the temperature-coefficient of the viscosity, for which Rankine’) found0.00225. $ 5.. From the experimental values of /) and the corresponding pressures p the values of D'..,. and y, can be determined according to the relation Dorr pa SE th p results for y, were obtained: ) In this manner the following if | 71 | | | | | | “Veale. | Neale. | 384.90 | 0.400 0.363 | 0.408 282.29 0.250 0.250 0.250 202 Tel 0157 0.168 0.154 a4) sl 405055 0.0676 0.055 The values found for D’,,,, are given in the 5 column in the tables on p. 350 and p. 351. In the tables for 0° and 100° C. D'.,.. will be found to become too small below about p=4 em.; this is quite intelligible seeing that the theory about the temperature-drop is derived under the assumption that the mean free path 2 is small compared to the dimensions of the apparatus. For neon at 0° C. and p=4 em. 4=0,000375 em, 2r, the diameter of the experimental wire being 0,0005240 cm., hence a —1,4. It appears therefore that the theory for the temperature- drop given by Kuypr, WARBURG and Smorucnowski is applicable 1) S. CHAPMAN. London Phil. Trans. A. 211, (1912), p. 433 and 216, (1916), p. 279. 3) A. O. RANKINE. Physik. Z. S. 11 (1910) p. 497 and 745. 855 over a wider range than might have been expected according to the kinetic theory. in) = r a It follows from y.p ————.y, that y.p=——; at 9°.2 C. ws from y. p ; y, that y.p 68.44 a NRA 0 when the pressure p is measured in dyne/em* and 2 in cms. we find according to O. E. Meyer (p. 344) from the viscosity T \12 pi=1s.93.(=): This gives at the temperature of the wire Pe 99210, 7/, = 2.46 Hence with a = 0.653 2-4 ¢/) ==-2.38 . ; The agreement is not so close as it was with the value found previously, but the deviation is not larger than can be explained by accidental errors. It appears from the table that y, changes with the temperature ; this was to be expected as 2 depends on the temperature according T\L2 to the relation 2 ==, . (=) . Calculating the values of y, which have 0 to be expected at the various temperatures, the results y, calc, given in column 3 are obtained. On coinparing these with the experimental results the latter are seen to ehange more rapidly with the tem- perature. This can be explained by the assumption that the accom- modation-coefficient a is not independent of the temperature. The same assumption is also rendered probable by the results for hydrogen; Knupsen') found that in this case a had a negative temperature- coefficient —0.001. Assuming the value —0.00076 for the temperature- coefficient of a for neon we find for y, the results given in column 4 under y,calc.- $ 6. By the aid of the principle of “similar motions” as given by H. KAMERLINGH OnNEsS?) a comparison may be made between the heat conductivity of different substances for which the conduction through the molecules themselves may be disregarded. It is found that at equal reduced temperatures we must have: 1) M. Knupsen. Ann. d. P1. (4) 34, (1911) p. 632. 3) H. KAMERLINGH Onnes. Verh. Kon. Akad., 21, p. 22, 1881. 356 ve EE 1 2 ut Pi, 1E ut Pe Pis «ata a Barban hy where PP and 77 are the critical constants. When a comparison is made in this manner by means of the experimental results between the conductivity of helium and argon with that of neon ©, it seems in the mean time as if the reduced heat conductivity of neon changes in a different manner with the reduced temperature from that of argon and helium; in order to obtain more evidence on this point it beeomes even more important than before, also in view of J. J. THomson’s theory (that neon would consist of two isotopic elements with molecular weights of 20 and 22 respectively), to determine the conductivity of neon at reduced oxygen- and neon-temperatures and that of helium at reduced hydrogen-temperatures; as was mentioned in the beginning of this paper, it is the intention to carry out this research in the cryogenic laboratory at Leiden. In conclusion IT am happy to express my sincere thanks to Dr. Ir. G. L. F. Pairs for his kindness through which I was enabled to carry out this research. I also wish to thank Mr. H J. Micntensen for the excellent manner in which he assisted me in the measurements and the calculations. Physical Laboratory of the Philips Incandescent Lamp Factories. 1) S. Weper. Ann. d. Ph. (4) 54, (1917), p. 460. Physics. — “On the shape of small drops and gas-bubbles”. By J. B. Verscuarreir. Supplement N°. 42c to the Communications from the Physical Laboratory at Leiden. (Communicated by Prof. H. KAMERLINGH ONNgs). (Communicated in the meeting of June 29, 1918). § 1. It is well known that the meridian-section of a liquid drop or gas-bubble (which we shall suppose to be bodies of revolution) cannot be represented by a finite equation by means of known functions. The differential equation to the section 1 ; Een ie xy’ uh Wa R, | R, «de Via —k(h+ y)*)Q) 2 has as a first integral the equation msn p=tkhe' -+—u, . (2) 2 where p represents (he angle which the tangent forms with the z-axis Fig. 1. (fig. 1; OY is the axis of revolution) x and w= 2 [rydi *), but the computation of wv and consequently o (u, —u)g oO 1) In this equation k stands for the expression , « being the surface tension, “,;— the difference of the densities below and above the surface in its top, g the acceleration of gravity; k is therefore positive or negative according as the liquid is below the top of the surface, as with a drop resting on a plane, or above it, as with a hanging drop; y is the height of a point of the surface above . 2 the tangent plane at the top. h is determined by kh = R? Ry being the radius ° of curvature at the top; Ry will be reckoned as positive when the surface is hollow upwards, negative in the opposite case. 2) uw is evidently the volume of the body which is originated by rotation of the surface OAA’O (fig. 1) about the y-axis. Equation (2) may be written in the form 2220 sin pp = (u,—U,)g (T27*h+u), . . . . . (8) 358 the further integration of the differential equation can only be carried out by successive approximations or a development in series. In the case that the drop or bubble deviates little from the spher- ical shape, y is small compared to h'). In first approximation y may thus be neglected by the side of /, i.e. we may puty = 0; asa second approximation a circular meridian section is then obtained ; if the expression for y corresponding to this as a function of « is substituted in wu, a first deviation from the sphere is found as a third approximation, ete. *). ° which is also found directly, when, for instance by applying the so called *weight- method”, the rise in a capillary tube is calculated. The contradiction found by A. Fercuson (Phil. Mag., (6), 28, (1914) p. 128) between the result of the integration of the differential equation and that of the application of the weight-method is merely due to an error of computation in the approximation of equation (2), owing to which Fereuson’s formula (7) is incorrect. Equation (2) can also be written as follows k x sing =tke' (ht y)— Feely nel Gh At (2') art where v =z x*y—u represents the volume arising by the rotation of the surface OAA”O. (2) gives: 2x sin p = 7 (u,—,) gu? (h + y) — (u.—#,) 9, « + (3) which expresses for instance, that the resultant of the forces acting along the edge of a section of a hanging drop makes equilibrium with the hydrostatic pressure on the section and the weight of the portion below it, in other words the surface tension does not balance the weight of a hanging drop alone, a fact which may also be derived from a simple consideration of the equilibrium (cf. on this point Tu. Lonnstein, Ann. d. Phys., (4) 20 (1906) p. 235). 9 1) Hence Ry is also small compared to h or to ER , that is KR,? is a small 0 number. ) Cf. for instance A. WinkeLMANN, Handb. der Physik, 2e Aufl. 1 (2), 1148 — 1144, 1908. Putting y = Ryp—Y Ry?—x*?+<2, where z is considered infinitely small as compared y' Vijg may be developed in a series, which gives, if 2, represents the first approximation of z: to y, and supposing that 2’ is also infinitely small compared to 4’, sin ip = RP Nr ip VR bs ——d 4 od — j 6 0 oy) 0 g IR as is also found by FerGuson (loc. cit.) although in a somewhat circuitous manner. This expression, however, does not hold near «= Ry, as 2’; is there no longer infinitely small with respect to y’, but of the same order of magnitude (viz. of the order (KR,2)—+; this fact has been overlooked by leRGUSON (loc. cit.). 0 359 $ 2. The introduction of polar coördinates, choosing as origin the centre of curvature at the top M (fig. J), gives the advantage that . . . T . there is no discontinuity at 7 = oM that case e=osn3 and y= Rk, —eecosd Mf con de ere) and the equation (1) becomes 0 sin 9 —o! cos D + Spb sin 9 (9? +0'°)" i (0° Koen sty: If we now put oil) and ter, terde ee (|) where 1,, rt, etc. represent the successive approximations to the infinitely small quantity +, we can, as long as t and r’ are infinitely small, separate equation (5) into a series of other ones, the first of which being x" sin BHT, cos 9H2r, sin PAR," (1 — cos 9) sin 9; . . (5) 1 + cos > = pth | eo 2) +2 cos 3 log 5) el an expression which remains valid from 7 =0 to / =a throughout. + k(R,—g cos 9). . (5) hence *) § 3. The result of the third approximation is as follows u—trR,*(1 — cos 0)? (14-2 cos D) + Fark (1 — cos 9)? cos? JH = akR.§ sin? 9 (1 —2 cos 3+ 2 cos? Y) lo (5) 7 5 athe Pin ee t g 9 a” Vast At’) and v—$nrR,* (1 — cos 9)? (24 cos 9) — 4 ak,‘ (1 — cos 9)" (2 + cos 9) — 1 Ù — gak, sin* 9 log a \ (7') § 4. Between the angles 2 and p the following relation holds: @ sin J Pa cos 1} —=snt JT cost +....3 snif = 5 In order to integrate these equations we sg to bear in mind, thal cos 9 (vt! sin F HT cos & + 2rsin J) = — = [sin F(t sin + T' cos F)| and did \ cos 1 The integration does not offer any special difficulties, but the calculations are long, that of r, being already very laborious; for that reason we have confined ourselves to 7. It is easily seen, that Rory = 2% cos S. d rsnd Jr cos 1) = cos? F — a) 360 putting therefore pee CL Won be CR a OT we find in first approximation (for 9 < zr) | =S e WV, 1 o Dy, yt, = kK,’ sin Feat — 2 log (= )| Pa i= | COs ¢ Hence, as long as p is not too near 2, equation (2) in connection with (7) gives (1 — cos p)(1 + 2 cos p) x= R, sin p — £ kh sin p 1 + cos sp LKR. si 1 — cos p ETE C sn. op ———_———— i 8 . (1 + cos p)? — HAR sin p log ( (1—3 cos p + 6 cos? p + 8 cos* p) — 1 + cos p\') mey) ° § 5. In fig. 2 OAB represents the meridian seetion of the capillary surface for k>0, OA’B’ gives the section for k< 0; both have been drawn for a positive 2, (for R, nega- tive the diagram must be turned upside down about the z-axis); the dotted curve between the two is the circle with radius /, (corresponding to k=O). In both cases x goes through a maximum (in A and A’ respectively), but, whereas in the first case the curvature keeps the same sign all the way, so that y passes a maximum (in B), re a minimum ete. ($ 7), the curve in the second case has a point of inflection (in B’), beyond which Fig. 2. xt becomes minimum etc. HK The maximum value of 2 is obtained by putting y= = in. (9); the result being: valea — Bend BRE dB (6 log 2e Oe 1) This degree of approximation (the 4th) is one higher than what is obtained by simply using the relation (4). and correspondingly 3 wa HER (2log24+1)) . . (109 and yA(ya') = RR, Pa Rh, FAR (2log24+ 1). . (1 0") § 6. From the equations (6) and (8’) it follows that in the neigh- bourhood of #= 2, putting 9 =a — se, 8° 1 =~ PAR, (lS - 1] and bee iia ge ies AKE) in order therefore that these equations may still be valid in that region, seeing that t’, has to be small, it is necessary, that e must remain large with respect to £#,?. This is still the case in B, where y has its maximum, for (comp. 4, 6 and 11) VRB ( lo 7 1), dated heave tk) dy so that it follows from oF = 0 that 5 ‚€ eg = V 3kR,', yp = 2K, [1 + HER log (HER ") — AIR] . (12) and, also to a third approximation, She Bien UIR oe ve ve (12) These coordinates are only real, when & is positive. If & is negative, p has a maximum in B’ (fig. 1) corresponding Ee: f dp dw to a value of e which is determined by 0 = a =1+ Dn (see eq. ' DP, 8); this gives: ER = WV ikR *); hence ys = 2R, [1 + SAR log (—4&R,?)] ~ (13) op =R VHR and pp =27—2V—2kR. . (13) § 7. It is possible to go a step further in the analysis of the meridian section of the capillary surface. Close to 9 == 17 the curve has a sharp bend (fig. 3): BCD with a double point Z for k>0, C 1 d. 1) Obviously this expression is also found by putting 5 == i 362 B'C’D’ with two points of inflexion B’ and D’ for k < 0; the dotted line (two circular ares) represents the transition between the two cases for kl. In that region the equation to the surface may be written in the form 5 : ke (h a Ri weed (4c + 4) „ar (4 where Av =h-+ yo, yo being the ordinate of C (or C’), and 14 = y—yc. In the region under consideration, however, 1% is small compared to #,, so that in second approxi- mation % may be neglected with respect to h. and therefore with the same degree of accuracy to which hitherto the deviation from the circular shape was calculated we may write: 1 l =d zkh Constant Art. Tern I 2 In third approximation BCD is thus a part of the curve which was called nodoid by Prarrau, b’C’D’ a part of an onduloïd. The equations of these curves are known *); but in our case they Fig. 3. 2 may be materially simplified. Putting ho = —”*) the first integral of pi (14) in the case of the nodoid (sin p = 0 for Ben will be PE APD. ei ea If wv, and w, (=a) are the maximum- and minimum-values of « corresponding to sin p= l and sin p= —1 we have approximately since wy is very small with respect to r, (see eq. 12’) EB 2 ne ie *) vy VN En ig joke? he & (16) 0 Further it follows from (15), as long as x is small with respect to 7, *) : FER 1 ical al ene ee ery il Aaa ie ©, 1) See for instance WINKELMANN, loc. cit., p 1150. 4) In first approximation 7) = Ry; in second approximation kh = k (h + 2Ry) = 9 = = SE = (1+ KR), so that ro = Ry (1—KR,?). B R, 8) Here x, belongs to the nodoid and has thus not the same meaning as ZA in § 5. 4) Since in that case dy eB a eR — x" Ree. a? de Vote epe) Vreren Vata? 363 This gives ar 45 1B == — FER log (RR) —24kR,. 2... (18) whence YC = YB — 1B = LR, + FER log (GAL) — EER ), . (16) and similarly, if rp, yp and the eoördinates of D, ap =ep=R VER 8 ypo=yco— B= 2R, + KR log 2 kR,*) (19) § 8. Im the case of the onduloid, where sin p goes through a minimum in 4’, we have PAE ID pas, BRE a ae aay oh hen nl 7 ot. (20) The maximum- and minimum-values of «(sing =1) are now approximately À ee D= i, = 8 SS ne en ee | R, Moreover in that case iz a 1 mt oe Hae log ED ie — +. CY a — hee … (22) ©, 2R, whence NB = — HRe log(— ERR) HAER? . 5 . . (23) yoo = YUB NB =2R, + ZAR log (— HAR) ARR . (21) ay = fk, V—3 kh,’ ID 2k, at kR,* log i 6 kk,*) Ee 5 kR,* (24) §.9. lt follows from (7’) that the volume of a drop from the top to the horizontal plane passing through B or 5’ (9 = a — ®), in second approximation is given by GEER Cee lal Wee ae aaa (21) With the same degree of approximation this is also the volume of a hanging drop up to the level of the neck; indeed the volume 1) If x is large with respect to x, we have 22 oe + n=, log — — —, . ‘ 45 OR so that the equation to the branch CBE (fig. 3) is (177) 2 2 v wv Y=Ye + ny = 2k, — kh,’ + EkR,? log ER SE DR, in agreement with (11’) | since € —= Beth B. From this the abscissa of the node E(yg= yC) is found to be BE == 2 kis log (ke. 364 between the planes passing through point of inflection and neck is found to contribute a negligible amount to the total. In connection with this it follows from eq. (2’) in fourth approximation : ag=V23kR,? (1—ER,?), On wt, = EER (1—$ER2) . , (199 the upper sign corresponding to the upper index. § 10. Starting from the points D and D’ (fig. 3) the analysis may be further continued in a manner similar to the one used above. Indeed the meridian curve of the complete capillary surface consists approximately of a series of nearly semi-circular arcs connected each time by parts of an onduloid or nodoid *). The centres of these ares are situated at the heights Zi, 3 A, 5 A, etc. successively ; with each (ntb) are we therefore place the origin in the corresponding (nth) centre and as in § 2 write: e=osn) , y==(Qn—1)R,—ocst? , o= B, (l—r) . (26) t is determined by: | t" sin J + T cos 9 + Ar sin J) =kR,? (2n—1—cos 9) sin 9, (26) whence it follows, introducing the condition that the arcs and interme- diate pieces form a continuous curve: 4 (n—1) e= [$+ 4@—l-— + Hlog 2 HJ Frn(n—-l)— Z “go log (n—1) — —2n(n—1) log (+ HAR cos 9 + cos ‘} log (1 + cos “*) — A, : ‘ a emma toe dee Terme. For the connecting curves equations (17), (22) and 2, = ac¢-=—. are each time satisfied. The successive arcs and their connecting curves can only be realised in separate parts, for instance between two horizontal plates or between two vertical coaxial cylinders. Not every surface, however, obtained in that way is a part of the surface whose meridian-section was analysed above by approximation. As an instance, if the surface is formed between two cylinders which are moistened by the liquid, (En &A)n cylinders and this fraction cannot in the analysis of § 10 assume any arbitrary (small) value, as long as n represents a whole number. Still, putting 2(m—1)—=ea and admitting an arbitrary (positive or the fraction represents the ratio between the radii of the Db Cf. WINKELMANN, loc. cit, p. 1141, fig. 404. 365 negative) value for «, the equations (26) and (26’) remain valid and t=hkh,* [a + Had boos + £(& — a) cos log (1 + cos 7) — —+4(4 —a) cos JP log(1 — cos?)],. . . . . (28) where a and 5 are integration-constants. /, is still undetermined, as also A, which remains connected with A, through the relation 2 ‘ 5 nd as regards the value of a, this may be chosen at will 5. @ With small values of ‘ the curve shows a minimum for y or a point of inflexion?) according as (&—a)k>>0; for a value of 9 which differs but little from a the curve has a maximum for y, if (E—a)k>0 or a point of inflexion, if (& — a) k <0. ®) $ 12. Here again the meridian-section consists of a series of curves which, however, now extends indefinitely upwards as well as down- wards. For £>0O the higher curves in the series show maxima and minima for 4, the lower ones points of inflexion, as represented diagram- matically in fig. 4. For £>0 on the other hand the upper curves have points of inflexion | ae | | —— | | | | | . . . | and the lower ones maxima and minima of y, | . Ld te | which case is obtained by turning fig. 4 upside | and maxima of w satisfy the relations é | ! | | | | down. Putting 4 — a = 8 the successive minima 2n a Eni = (3 ay 5) kR,° | | / 1 2n+1 2 Omar. == R, ee DI a—B + ay kh, e (29) | 2n ay Ro At the point where 8 + = changes its sign Big (smallest value of #4) is the transition between 2m the two kinds of curves. If accidentally 6 = 4 e ‚m being a whole number, the smallest value of x,,;, becomes zero and the case reduces to that of the meridian-sections discussed in § 10. 1) Supposing for instance the meniscus to be formed between two co-axial cylinders which are moistened by the liquid, the radii of the cylinders being R and 7, where 7 has to be small with respect to R. a and Ry are determined by the conditions xc=r and #4= FR; « and 5 may still be chosen at will; one might for instance take x = 0, while determining b by putting yp = 0. 2) In general therefore in this case the presence of a minimum or maximum for y is not, as in the section 6 sqq, bound to k>0 or the existence of a point of inflexion to k <0. 24 Proceedings Royal Acad. Amsterdam. Vol. XXI. Physics. — “On the measurement of surface tensions by means of small drops or bubbles.” By J. EB. Verscuarreit. Supplement N°. 42d to the Communications from the Physical Laboratory at Leiden. (Communicated by Prof. H. KAMERLINGH ONNES.) (Communicated in the meeting of June 29, 1918). § 1. The usual methods for the determination of surface tensions by means of small drops or gas-bubbles, to which properly speaking the method of the capillary rise also belongs, are based on the measurement of the difference of hydrostatic pressure between the two media inside and outside the drop or bubble; indeed the sur- face-tension is given by the formula Gl) Rand = * 3 e oe where »,— u, is the difference of the densities of the two contiguous media, g the acceleration of gravity, A, the radius of curvature at the top of the meniscus and / the pressure-difference on the two sides of the surface, measured as a column of the liquid in the surrounding medium. If the drop (or the bubble) is so small, that it may be considered as spherical, we may take for A, half of the diameter of the drop (or bubble), or the radius of the capillary tube, in which the liquid ascends, at least if there is no angle of contact; in order, however, that the approximation obtained in that way may be sufficiently close‘), the radius must be taken so small, that as a rule the relative accuracy of the measurement of the radius remains far behind that which can be reached in the measurement of h, whereas naturally it is desirable to know R, and 4 with the same relative accuracy. In order that this may be possible, it is necessary to make the measurements on drops or bubbles which are not too small, in which case at the same time the necessity arises of a correction on account of the deviation from sphericity. *) Ws Tu (ef. eq (2)). 1) The relative error is of the order kR?, where k = 4) Ry can also be measured directly by an optical method (cf. H. SIEDENTOPF, Diss. Géttingen, 1897); it may also be determined by measurements on photographs (cf for instance A. Ferauson, Phil. Mag., (6), 23, (1912) p. 417. A high accuracy is, however, not obtained in that way. 367 § 2. Putting 24 =~7 in eq. (10) of the previous communication (Suppl. N°. 42e) it reduces to By Bli 2 BY a, Bh from which R, may be calculated, when r is given and # is known approximately. This value substituted in (1) gives 9 hg br} hr (2 log 2—I). PRE ee or f =H) etsen]. entel formulae which are already known *) and by means of which the surface tension can be calculated to a third approximation from the capillary rise h in a tube of radius 7, which is completely moistened by the liquid: *) These equations, when proper account is taken of the signs of the various quantities, are applicable in every case, where the width of a drop or bubble can be measured as also the pressure necessary to form it, As an instance, when the liquid does not moist the wall (mercury) the liquid may be forced up by an excess of pressure from a very wide into a narrow communicating tube, until it pro- trudes from the narrow tube in the form of a drop; / then is the height of the liquid surface in the wide tube above the top of the meniscus on the top of the capillary. *) Similarly when the capillary is moistened by the liquid, the meniscus may be forced down by the pressure of a gas, until a bubble is formed at the bottom of the capillary. *) 1) See for instance A. Winketmann, Handbuch der ‚Physik, 2e Aufl, I, (2), 1144 and 1159, 1908. 4) For the case, when there is an angle of contact 7, the following relation is found by putting «=r and p= 5 —i in eq. (9) of the previous communication (Suppl. N°. 42c) 6 = Hu, —u,) ghr secr [ta Jd > sec! 2 (1— sin 2)’ (1 + 2 sint) + i r? 1 + sini + $=, sec’ 1 (1—sin 7)? (1 + sind + 2 sin? i) + re sec” ilog 5 aa (5) ’ 5) In this case eq. (3) and (4) remain valid without any modification, as both h and PR, therefore also r, change sign (see previous comm.). This simple method, which is independent of the angle of contact and which allows the capillary surface being refreshed by removing the drop, does not appear to have been ever applied to mercury. 4) Cf. A. Winketmann, l.c., p. 1162. See also further down in § 9. In this manner, however, it is not the surface tension of the pure liquid in contact with its vapour which is determined, but that of the binary system liquid gas. 24* 368 $ 3. In dealing with a hanging drop, u,—w, and h change sign (see Suppl. N°. 42c) and eq. (4) becomes . , r 7 p=) gleed He Blog 2}... (A) The practical application of this equation is not so simple, however, as that of eq. (3), as a hanging drop formed at the end of a capillary which communicates with a wide tube is not in stable equilibrium '). But the equilibrium may be made stable by also taking a narrow tube for the one with which the capillary communicates, say by making the drop hang from a single capillary, as in Sentis's method’); in that case, however, account must be taken of the curvature of the meniscus in the narrow tube. If h, is the distance between the tops of the two menisci, and A, the ascension of the liquid in the narrow tube (which can be obtained from a separate measurement), it is evident, that in eq. (4’) the substitution : h=h,—A,. has to be made. § 4. Let us return to the case of a drop, say a mercury drop, forming on the top of a capillary under the influence of an excess of pressure from the liquid in a very wide communicating tube (section 2). When the mercury by raising the liquid in the wide tube has reached the edge of the capillary, the meniscus protruding above it begins to curve more and more as the liquid gets higher, so that the difference in level 4 between the two tubes, which had remained constant so far, now increases. Soon the meniseus attains the maximum-curvature and at the same time the difference in level 4 reaches a maximum. This maximum occurs at the moment that 2, has its smallest value *) which happens when /, is all but equal to r, the radius of the capillary (for simplicity A, is here taken as positive). Putting 1) An imperceptible fall of the level in the wide tube is sufficient to give an appreciable increase to the radius of the hanging drop, through which h becomes larger, while the capillary counter-pressure diminishes; consequently the liquid flows continuous. With a lying drop on the other hand h diminishes and within definite limits the equilibrium is stable (see also §§ 4 and 7). *) Sentis, J. d. phys. 6 (1887) p 571. 2 $) Seeing that between f and PR, the relation kh mre holds (see previous comm. 0 in these proceedings). 369 - MI Dj Sm where # has the meaning given in $ 2 of the previous communication and w represents an infinitely small angle, equation (9) of the same communication, in view of (see eq. 8 and 8’ l.c.) peat was 4+ LER, (2 log 241) + wo, takes the form: r= R.—4 R, [w+4kR,? (2 log 24 IAR + ty HR (6 log 2—1) . (6) It follows, that the minimum of A, is reached for Ope OR Qi Dye ae wettest enon (GO) Tv therefore for ae that is: exactly when w4=7. The surface tension is thus given by the relation (4), when 7 now represents the radius of the capillary and A the greatest difference in level of the mercury in the wide tube above the capillary; conversely for given 7 and & the greatest difference in level is given by equation (3). § 5. The mercury can still be raised toa higher level in both tubes. The radius of curvature &, at the top of the drop then again increases, so that / becomes smaller. All the same the mercury continues to rise in the wide tube, that is: the height H=A-+ y of the liquid in the wide tube above the top of the capillary (/ represents the height of the drop and is therefore here taken with the positive sign) still increases. But this height also soon attains a maximum. Jt . Putting again 7 sca a flea we have (eq. 4 of previous comm.) € 2 nn mr 280 RTE AP A ( from which, joined to the condition «const =r, it follows that H is a maximum when w, = FER (1 — log 2), (Ro), =r + Fhr? — ag hr’ (12 log 2—17), (9) so that 2 nf” + Brit dake? a a a te 0h) r whence r r? 6 = Hu, —u) 9Hnr Lt : tr ener LAA § 6. To test the use of the method sketched out in sections 4 and 5 a few trials were made with mereury in contact with air. The wide tube was so wide (+ 2 cm in radius), that the meniscus could be 370 considered as flat; the radius of the narrow tube at the top which was sensibly flat was 1.090 mm. Through a rubber tube the wide tube was connected with an adjustable funnel filled with mercury; by slowly raising the funnel the moments are easily marked at which first 4 and then MH attain a maximum’). The maximum-values of A and MH were found to be to a bigh degree dependent on the slowness with which the drop was being formed. From these experiments therefore a definite value for the surface tension mercury-air did not follow. As an instance for a drop which had been exposed to the air for a very long time hn = 0.490 (at ¢=17°.3) was found, which by means of eq. (4) U, — u, = 13.55 g = 981) leads to 6 == 0,355 (1 + 0,074 0,006) = 379, whereas immediately after the formation of a new drop the observ- ation gave 4 = 0.592, whence o = 454. Similarly an experiment where the mercury ran over about 1 min- ute after the surface being renewed gave H,, = 0.708 (at 18°.2 C.) whence (eq. 11) 6 = 513 (1 —0,102 —0,004) = 459, whereas for a drop which did not flow over till after half an hour Hn = 0,659, c.e. 6 = 423; after some hours these values had even gone down to H,, — 0,619, « = 393. *) | h The maximum of 2 can be very easily observed by using a micrometer with moveable cross-wire; the fixed horizontal wire [ is set on the meniscus in the wide tube, the moveable wire II, also horizontal, on the meniseus in the narrow, tube. The funnel is first moved up until the mercury protrudes above the narrow tubg as an almost hemi-spherical meniscus: by now raising it very slowly or by adding mercury a drop at the time, so that J and II rise slowly, the distance I—II is seen to increase slowly and attain a greatest value. After that I and II continue to rise, but the distance I-—II now diminishes. At the same time the drop above the narrow tube is seen to bulge out more and more, to exceed distinctly the half-sphere and finally fairly suddenly to swell and tlow over the edge of the tube; at that moment the level | falls very rapidly, so that H has gone through. a maximum. 2) This diminution of the surface-tension of mercury which is exposed to the air was first observed by Quincxe (Pogg. Ann. 1 (1855) p. 105). Similarly GRÜNMACH (Ann. d. Phys, (4) 28 (1909) p. 247; method of capillary waves) found a much higher value of > for fresh surfaces (¢ =491,2 at about 18°) than for surfaces which had been exposed to the air for half an hour (¢ = 405,0). See also WINKELMANN, loc.eik:, p.. 1168: Volatile vapours in the air also appear to lower the surface tension of mercury very considerably; it was found in the experiments of § 6 that it was sufficient to bring a piece of blotting paper soaked in benzene or alcohol near a practically hemispherical drop in order to make it flow over at once. 371 > The usefulness of the method is sufficiently demonstrated by these experiments. Moreover they can easily be so arranged, that the surface tension is determined in vacuo, in which case probably a gradual change of 6 with the time would not show itself). § 7. Instead of forming the mercury drop on the top of a capillary it is possible to make it form at tbe bottom. Tbis can be done by closing a wide tube at the bottom by a plate with a small circular hole; mercury being poured in, a small hanging drop is formed at the orifice, which gives way at a definite maximum height of the mercury in the tube, after which the mercury runs out. From the observed maximum height and the radius of the opening the surface tension of the mercury may be derived. If AH represents the height of the liquid in the wide tube above the opening, H = h’--y, h’ being the height of the mercury in the wide tube above the bottom of the hanging drop and y again representing the height of the drop. In this case & is negative (see Suppl. N°. 42c), hence h=-—=— A’ and H=—(h-+ y). The condition for the maximum of H at constant ws —=r then leads to the same eqnations as in $ 5, except that 4 and A have to receive the negative sign. lt follows in the first place that w, (eq. 9) is negative; that JT is: the drop begins to fall before p has reached the value > (see $ 3), so that A in this case cannot reach its maximum value (§ 4). In the second place according to eq. (11), A, being reckoned as positive, r 7 ==. (1,9 Hn r(1 + Lr — tas) Er vn Cen $ 8. By this method also a determination was made of o for mercury. For this purpose a tube of 1.5 em radius was closed at the bottom by a plate, through a hole of which a short piece of glass capillary (r==0.522 mm) had been stuck. When mercury was put into the tube a drop at a time, a drop was formed at the lower end of the capillary which gave way before the hemisphere had been reached. Here again the value of H,, was very strongly dependent on the time elapsing while the drop was being formed; the greatest value 1) In a vacuum Fürtu (Wien. Sitz. Ber. |2a), 126 (1917) p. 529) found + = 440 to 445 at 18°C; in this no trace of a change of c with the time was observed. 372 observed was H,, = 1.230 which gives: 6 = 427 (1 + 0,028 — 0,003) = 438 When the tube was filled to a smaller height and then left to itself, the drop could be seen to bulge out more and more and finally give way in consequence of the diminution of o. §9. When in a capillary tube, in which a liquid ascends, pressure is exerted by means of a compressed gas, so that the meniscus is forced down, until a gas bubble is formed at the bottom of the capillary, the bubble is found to escape ata definite maximum value of the difference between the gas-pressure and the hydrostatic pressure at the bottom of the capillary. From this maximum of the pressure- difference the surface-tension of the liquid (in contact with the gas) may be derived *). The phenomenon is of entirely the same nature as the one deseribed in § 5 and the theory may be developed in the same manner’). If H represents the said difference of pressure, whereas 4 = na again 6 represents the capillary pressure at the lowest point of the meniscus and y the height of the bubble, then, as in $ 5, H=hA-+y and, as k and R are also positive, the same equations are obtained in this case as in section 5%. In this case / also obtains a maximum-value, which might also be used as the basis for a determination of the surface-tension; in that case eq. (4) would again apply. But the measurement of H is simpler than that of A and therefore preferable from a practical point of view. § 10. Several observers have derived surface tensions from measure- 1) The first to use this method was Simon (Ann. d. ch. et d. phys. (3), 32, 5, 1851), who assumed without sufficient proof, that the maximum pressure-difference is determined by the capillary rise, which is only correct for very narrow tubes. Simon's method was used by several other. experimenters later on (see WINKELMANN, Le, p. 1162). 2) See also: M. Cantor. Ann. d. Phys. (3), 47 (1892) p. 413; R. Feusrer. Ann. d. Phys. (4), 16 (1905) p. 61; A. Frerauson, Phil. Mag., 28, 1914 p. 135, and E. ScHrÖpiNGeR, Ann. d. Phys, (4), 46 (1915) p. 413. 8) In accordance with what was found by Scaröpineer, (le.). It is not astonishing that Cantor, Frusren and Frreuson find an incorrect expression for the second correction-term in these equations, seeing that — apart from errors of calculation by Cantor and Fereuson — the authors in their reductions assume a spherical shape for the drop, although the second correction-term is actually determined by the deviation from the spherical shape. ments on drops without pressure-measurements.') With small drops the surface tension is then derived from the deviation from the spherical shape; in that case principally equations (10) and (10") of the previous communication (or (2) of the present paper) are to be applied, which lead to the relations: : ir 5 = (U) | 1 — Bed (B log2—2) |, . . (12) — 1 nf r R,— 3 et Pe ed a La ee eR ee ee) r_ being the largest radius of the drop (its half breadth) and y the distance from the top to the plane of the section with radius 7. Seeing that here the determination of o depends on the exact knowledge of the numerical value of terms which only served as correction-terms in the method of the pressure-measurement, this method cannot but give much less accurate results than the previous one. But its use seems indicated for liquids which can only be obtained in very small quantities. § 11. A third manner of determining surtace-tensions by mean of small drops consists in measuring the weight of small falling drops. It follows from the equations (25) and (19’) of the previous com- munication, that the volume of a small constricted hanging drop is. Jaro r' nlt) eld | see Att Ae sAE (14. U) K, r’ being the radius of the cireular neck. When the drop is made to fall from a very thin rod — this would be the method, if the liquid moistens the wall — or from a very narrow tube — in the opposite case — °) 7’ is not equal to the radius r of the rod or tube, but the difference is very small. Indeed the drop does not fall at the moment, when +7’; before it falls away the drop 1) See WINKELMANN, loc. cit., p. 1160. See also J. E. Verscuarrert and Cu. Nrcarse, Bull. Acad. de Belg., 1912. p. 192. 2) Properly speaking equation (14) only holds for a drop hanging in equilibrium and not for a drop forming from a tube while flowing (cf. WINKELMANN loc. cit. p. 1162). It appears from § 7. that a constricted drop cannot hang in equilibrium from the opening of a tube, if the drop is in free connection with liquid in a wide tube. A strongly constricted drop is only possible, if the connection with the free surface in the wide tube is broken, for instance by the interposition ofa tap; by opening the tap very little the drop may be made to form very slowly, until it falls: at any moment it can then be looked upon as in equilibrium and its further deformation may be prevented by closing the tap. Similarly a strongly constricted drop may form at the end of a long capillary, through which the liquid flows very slowly (cf. also § 3). 374 contracts a little more, the volume thereby increasing slightly, until it reaches a maximum. For according to eq. (17) of the previous communication, when 27 differs very little from t=", the volume contained between the circles of radii r and r’ is equal to ar? Vp! (r—r’), so that the volume of the drop up to the plane of suspension is equal to Vern, (te, — ue) 9 and this is a maximum, when ne oe The maximum-volume will still be represented with sufficient accuracy by eq. (14), if 7’ is replaced by r. When the maximum is reached, the smallest further supply of liquid must necessarily make the drop break off. If G is the weight of the drop >), it follows from eq. (14) with r instead of r’ that G = (1 of re Ee rj == —(1 bie be yao Quer R,/ Zar 5 2ar | v This is therefore the formula which in the case of a very small drop has to replace the simpler expression used by QUINCKE. ’). 1) G=(ug—“) gv; G is therefore the apparent weight, not reduced to a vacuum. 2) It is perhaps not superfluous to point out, that the expression (16) may be deduced in the following simple manner. The molecular forces (surface tension) along the circular neck of the drop make equilibrium with its weight and the hydrostatic pressure on the plane of the neck; hence according to (8) of the previous communication 26 Jaro = G + —. ar, 0 where the term — zr? (u,—,) gy has been neglected. This equation agrees with the relation which is found in calculating the capillary rise by the so-called weight- method (see previous comm.); in a certain sense, however, it must be considered as its opposite: in both cases the surface tension balances a hydrostatic pressure and a weight, but whereas in the case of the capillary rise the weight is introduced as a correction, here on the other hand the same is true for the hydrostatic pressure. Seeing that for mereury == 30 about and for water k= 18, r must not be greater than 0,07 to 0,11 mm. in order that the correction-term 5 be 0,1: Im order that this term may be still further reduced, as is necessary for the accuracy of the method, in view of the further unknown terms which have been neglected, much narrower capillaries would have to be used and this would diminish the accuracy of the measurement of 7. This shows that the method of the weighing of falling drops is not a very suitable one for the determination of surface tensions. 3) Pogg. Ann., 134, (1868) p. 365. See also WINKELMANN, loc. cit, p. 1147 and 1161, and Ta. Lounsremn, Ann. d. Phys., 20, (1906) p. 238. Chemistry. “The Phenomenon Electrical Supertension”. By Prof. A. Smits. (Communicated by Prof. P. Zeman). (Communicated in the meeting of June 29, 1918). It has already been pointed out in a previous communication *) that the metals which furnish the so-called unattackable electrodes, differ from the other metals in this that they are ideally inert, so that the potential difference of such a metal electrode with respect to an electrolyte is governed by the prevailing electron-concentration in this electrolyte. Let us now suppose that a smooth platinum-electrode immersed in an aqueous solution of hydrochloric acid, is made cathode, it is then easy to see what will happen. The two equilibria that are to be considered here, are: Bie Diggs Ge Ao and „EL 2 2E + 20 the former of which is entirely determined by the electron concen- tration of the hydrogen equilibrium. When we immerse a platinum electrode in a solution of hydrochloric acid, the platinum ion-concentration in the electrolyte will be imperceptibly small. Yet we can speak of a platinum equilibrium in the electrolyte, which, as was already remarked, is entirely determined by the electron-concentraticn of the hydrogen equilibrium. In virtue of this it may be said that platinum is a hydrogen- electrode from ihe very first, but so long as the hydrogen has not yet appeared as second phase, the platinum will be a hydrogen- electrode, corresponding with a hydrogen pressure smaller than the pressure under which the electrolyte is. When we make the platinum cathode, there are electrons added to it, and a consequence of this will be that hydrogen-ions from the electrolyte are deposited on the metal surface, and are dissolved in it, from which it appears that the hydrogen is of course not immediately present as a new phase. When the internal equilibrium : H, 22H’ + 20 sets in very rapidly on the metal surface, this internal equilibrium would already have been established in the metal surface in spite 1) These Proc. 376 of the supply of electrons. But above a certain current density, which evidently lies very low, this is no longer the case, and the metal-surface will contain more hydrogen ions and electrons than corresponds with the internal equilibrium. In consequence of this the potential difference, as appears from the formula RT : Kir (Hs) A= — —11 = F (Hr) will be more negative than when internal equilibrium had been established. Let us now suppose that the current density is continually increased, the potential difference becoming continually more negative, then at a given moment super-saturation of hydrogen will set in in the metal surface, and at a certain degree of super-saturation hydrogen will be generated as second phase. When the current density is kept constant, the potential difference can now diminish a little, but on increase of the current density the potential difference will now also increase further, because, even when hydrogen generation takes place, this process can yet be accompanied with an increase of the concentration of the hydrogen ions in the surface of the electrode, and besides because the formation of the gas bubbles through the diminution of the surface of contact metal- electrolyte, causes the current density to increase very greatly. As at the moment that the hydrogen begins to separate as second phase, the metal surface contains more hydrogen ions and electrons than corresponds with the internal equilibrium, the potential difference at his moment will be more strongly negative than corresponds with the state of internal equilibrium, which is in accordance with the above mentioned formula. This internal equilibrium sets in when without passage of a current, hydrogen of a pressure of 1 atmosphere is conducted round the platinized platinum electrode. The difference between this equilibrium potential of the hydrogen and the potential difference, at which during the passage of the current, the hydrogen begins to separate as second phase on the unattackable electrode for the first time, is called ‘‘swpertension”. It is clear that in the light of the newer views this phenomenon is not distinguished from the phenomenon of the cathodic polarization in any respect. The supertension of hydrogen is, accordingly, nothing but a consequence of the retardation in the establishment of the internal equilibrium during its electrolytic separation, and the supertension in case of all the other gas-generations can be explained in exactly the same way. It has been found that the amount of the supertension for the same current density is still dependent on the nature of the metal 377 electrode; nor is this strange in the light of these considerations, for the different metal electrodes will exert a different catalytic action on the establishment of the internal hydrogen equilibrium. But -not only the nature of the electrode, but also the condition in which a certain electrode is, will be of influence on the supertension. A polished platinum electrode or a platininized platinum electrode do not give the same result; in the latter case the supertension is practically zero, which can be explained by the fact that the much larger surface of the catalyst causes a rapid establishment of the internal equilibrium, to which is added that the actual current density is much smaller than is supposed, exactly in consequence of this larger surface. Finally also the electrolyte can exert influence on the setting in of the internal equilibrium, and thus we see that the polarization phenomena at gas-generations can be surveyed and accounted for with all other electrolytic polarization phenomena from the same point of view. Considerations in the light of the theory of phases. A so-called unattackable metal as hydrogen electrode. In my preceding communication “On the Electromotive Behaviour of Metals” *) I have already treated the unattackable electrodes and their efficiency as gas-electrodes. In this | have demonstrated that the result of these considerations can be given in a A, w-fig. in a lucid way. That a platinum electrode, immersed in an acid solution, and surrounded by hydrogen of one atmosphere indicates the hydrogen potential in correspondence with,this pressure, is elucidated by the adjoined fig. 1, which holds e.g. for atmospheric pressure and con- stant total-ion-concentration. _ Though the equilibrium-normal-potential of platinum is not known to us in consequence of its great inertia, yet it may be said with certainty that this potential of the equilibrium, if it could be mea- sured, would be very strongly positive with respect to the hydrogen. The concentration of the electrolyte ¢ would therefore practically quite coincide with the axis for the hydrogen. With a view to lucidity I have however purposely not made the point c coincide with the H,-axis in this schematic drawing. Let us now imagine that a platinum electrode is immersed in vi ag Pa 378 an electrolyte of the concentration «,, and that the electrode is sur- rounded with hydrogen of a pressure of 1 atm., then our conclusion from the preceding communication that namely the platinum equi- librium in the liquid is governed by the electron concentration of the hydrogen equilibrium in the electrolyte, or in other words that Fig. 1. the platinum electrode becomes hydrogen electrode, has the follow- ing meaning: It appears from the A, w-fig. 1 that the potential difference of the hydrogen with respect to the electrolyte w, is indicated by point c!, lying on the metastable prolongation of ac. Now it follows, however, from the considerations given here that platinum will present the same potential difference as hydrogen in the experiment mentioned here, and that the electrolyte will, there- fore, not only be electromotively in equilibrium with hydrogen, but also with platinum. This means therefore that c’ does not only lie on the prolongation of ac, but at the same time on a line that has taken the place of dc. The line he referred to the electrolytes which coexist electromotively with platinum in infernal equilibrium, whereas we now have to do with a curve that indicates the electrolytes that can coexist with a state of platinum disturbed in a base direction; hence this curve lies above be, and is here indicated by bc. 379 The potential difference, which we therefore measure at the pla- tinum electrode in the case supposed here, is the potential difference for the three-phase equilibrium dc'e, in which «' represents the hydrogen phase, c the electrolyte, and e’ the hydrogen-containing platinum phase. It is clear that in this binary figure it is in fact impossible to indicate the composition of the platinum electrode as the electrode contains atoms and ions of platinum and hydrogen as well as electrons. The composition of the electrode is in consequence of this indicated in platinum and hydrogen in total. As was already said c lies practically on the hydrogen axis, and as in the case that an attackable electrode is used as hydrogen electrode, the unattackable electrode is immersed in an electrolyte which is practically free from the ions of the electrode material, the concentration «, lies likewise entirely on the hydrogen side, so that like c also the point c’ will practically coincide with a, i.e. the different unattackable electrodes, applied as liydrogen electrode, will practically present the same potential difference under the same circumstances. The Supertension Elucidated by Means of the A,a-Fig. When we immerse a smooth platinum electrode in a large quantity of an electrolyte of the concentration 2,, and when we then make it cathode, fig. 2 gives the successive states. Before the platinum electrode is made cathode, we have electromotive equilibrium between the electrolyte m and the disturbed hydrogen-containing platinum phase n. As soon as the platinum becomes cathode, platinum- and hydrogen ions are deposited on the metal surface, and as the establishment of the equilibrium in the metal surface cannot keep pace with the ion-separation, we get a platinum surface that is still more greatly disturbed, in which there are more platinum and more hydrogen ions and also more electrons present than corresponds with the state of equilibrium. Hence a moment after the passage of the current the point m’ indicates the potential difference and the com- position of the disturbed, hydrdgen-containing platinum electrode, so that now m’ and n’ represent the coexisting phases. With increasing density of the current the electromotive two- phase equilibrium moves continually upwards in our A,r-figure, and it might be thought that the hydrogen can be separated for the first time as phase at the very moment that the line indicating the elec- trolytes that can coexist with a platinum electrode of definite 380 disturbance, passes through the point ec”, or in other words at the moment that the disturbance of the platinum electrode has increased to such an extent that the potential difference is indicated by a horizontal line passing through ec". Fig. 2. This would, however, be the case when the hydrogen could have assumed internal equilibrium in the metal surface, and when there was, therefore, no supertension. As was already said the supertension is just to be explained by this that also the establishment of the internal equilibrium of the hydrogen cannot keep pace with the ion-separation. Hence the hydrogen appears as second phase for the first time not when the potential difference of the metal phase has risen to c°‚ but to a higher point, e.g. c. In correspondence with the concentration of the electrolyte, the curves a'c' and 6"c' intersect in this point, which curves refer to the electrolytes which can coexist with a disturbed hydrogen phase, resp. platinum phase. The hydrogen phase d', which therefore is generated, is a disturbed hydrogen phase, as it contains more hydrogen-ions and electrons than corresponds with the state of internal equilibrium. The supertension can now directly be read from the figure; it is equal to the distance c'c". ; Now it should be borne in mind that the point c practically coincides with the hydrogen axis, and that when a platinum elec- 381 trode is immersed in an aqueous solution of an acid, the concen- tration x, practically coincides with the hydrogen point, and conse- quently the point of intersection c’ will likewise practically lie on the hydrogen axis. It is clear that the considerations given here are general, and will, therefore, also apply to the supertension of other gases. As was demonstrated there is no essential difference between the phenomenon of supertension and that of polarization. The former is only a little more complicated in so far that here also an unattac- kable electrode has been inserted into the system. When, however, we consider the phenomenon of supertension at non-unattackable electrodes, every difference with the ordinary phenomenon of polarization bas disappeared. Amsterdam, General and Anorganic-Chemical Laboratory June 18, 1918. of the University. 25 Proceedings Royal Acad. Amsterdam. Vol XXL. dene sal EE Chemistry. — ‘On the Periodic Passivity of Iron, IT’. By Prof. A. Smits and C. A. Losry pr BRUYN. (Communicated by Prof. P. ZEEMAN). in Ee ee (Communicated in the meeting of June 29, 1918). Periodic passivity in experiments with sealea-in iron electrodes. In a previous communication’) on this subject we have shown how we have succeeded in calling forth the phenomenon of periodic passivity on anodic polarisation of iron in a solution of 0,473 gr. mol FeSO, -+ 0,023 gr. mol FeCl, per litre. In these experiments we made use of an iron electrode 0.8 em. long with an area of +0.3cm?, which was sealed into the short leg of a U-shaped tube by means of shellac. The considerations that led us to these investigations were the following. During the anodic solution of iron in a solution of FeSO, the internal equilibrium in the metal surface above a certain density of current, can be disturbed so greatly that passivity appears. When into the solution Cl, Br., or l-ions are introduced in a sufficient concentration, which need, however, be only exceedingly small, activation of the iron suddenly makes its appearance. It follows from this that for a definite density of current, given by the velocity of solution of the iron, it must be possible to find a halogen-ionconcentration, for which at a definite moment the chance that the iron remains passive, is equally great as the chance that it becomes active. | When at this moment the density of eurrent is slightly diminished, the transition passive-active is sure to take place. The iron anode in the passive state will dissolve only exceedingly little, the iron, which has now become active, will, however, go very greatly into solution. In consequence of this the contact of the halogen-ions with the iron will diminish, and as the iron is now almost entirely with- drawn from the catalytic influence of the halogen-ions, it can again pass into the passive state. Since, however, as has been said, the passive iron dissolves very little, and the processes which now take place at the anode consist 1) These Proc. 383 of the discharge of the SO" -ions with the subsequent O,-generation, and further of a concentration increase of the halogen-ions, activation will again make its appearance through this latter process at a given moment ete. This surmise was perfectly confirmed, and using Dr. Mou1’s excellent galvanometer, we photographed some exceedingly regular periodic curves, the maxima and minima of which differed 1.74 Volt in situation. These graphs were, however, still incomplete in so far that the lines of time still failed. In our further researches we made use of a photographic registra- tion arrangement with time-signal-apparatus manufactured at this laboratory, so that also the time-lines are visible on the new photos, and accordingly a better idea of the regularity of the phenomenon can be formed. We intend to answer several other questions by means of this arrangement, but before proceeding to do so we will first give a photographie representation of the phenomenon of the periodic passivity, under about the same circuinstances as before, bat now with registration of the time. This photograph is given in Fig. 1. The potential difference again ranges here from about —0,3 Volt with respect to the 1 norm. calomel electrode in the active state, to about + 1,4 Volt, in the passive state, the current density retrogressing from 33 m.Amp. to 28 m.Amp. per em’. Since the time-lines, which are at a distance of 3,3 seconds from each other, are now also drawn, the regularity of the phenomenon can be much better observed than before. The maxima lie 6,15 seconds apart. Fig 1 shows further that the iron was only a short time active, and comparatively long passive. The electrode was sealed in as before, and 1,5 cm. long, and the siphon of the auxiliary electrode was halfway of the height of the electrode. The solution contained 0,72 gr. mol. FeSO, and + 0,014 gr. mol. FeCl, per litre solution. The content of FeCl, was, therefore, much smaller than before, hence the periodicity appeared here already at a smaller current density. Fig. 2 refers to an experiment with the same electrode, but taken with a slightly smaller carrent density, viz. 30--25 m.Amp. As is very apparent from this photograph, this has caused the periods to become longer, and the time during which the iron was in active state to become about equally long as the time in which the iron was passive. It is remarkable that when we endeavour to proceed in the same direction, and try to make the active state last still longer 25% 384 by diminishing the current density still more, this can only be main- tained fora short time, and a state soon sets in again as reproduced here. Periodic passivity in experiments with iron electrodes that were not sealed in. The following experiment was made with an electrode that was not sealed in, but in which an iron electrode was simply immer- sed 1.5 em. deep into the electrolyte, the siphon of the auxiliary electrode being placed quite at the bottom against the iron electrode. In this case there was always an activating influence, starting from the iron at the height of the liquid level, but in preliminary experi- ments we had already found that this activating influence did not, however, prevent the iron at the bottom of the electrode, which was 1,5 em. long, from exhibiting pretty regular periodic passivity. Fig. 3 shows the result obtained in this experiment. The pheno- menon is, indeed, not quite so regular as with the sealed-in elec- trodes, but the difference is not great. Periodic passivity at different heights under the liquid level. We will now examine what is the behaviour of a non-sealed electrode at different heights under the liquid level. For this purpose experiments were made with an electrode which was immersed much deeper, viz. more than 5 em. under the liquid surface. When the auxiliary electrode was again placed quite at the bottom, a pretty regular periodicity was observed, just as in case of less deep immersion; this is shown by Fig. 4. When the auxiliary electrode was placed 1.5 em. above the lower end against the iron electrode, the activating influence exerted from above, was already very clearly noticeable. Thus Fig. 4a shows that though the pheno- menon is still regular, the character of the curve has been greatly modified. The periods are much shorter and the passive state lasts very short, and what is very remarkable, now a longer duration of the activity than of the passivity can be maintained. In the following experiment we have placed the auxiliary elec- trode halfway up the immersed part, hence + 2.5 cm. from the bottom, and under these circumstances still greater modifications were found, consisting in this that the iron did not always become equally strongly active, and that regularly two less active states were followed by a more active one, or that alternately a more ‘active and a less active state followed, as is clearly shown by Fig. 385 5 and 5a. The next figure 6 refers to the phenomenon that occurs when the auxiliary electrode is placed only 1 mm. under the liquid level against the iron electrode, and from this we see how greatly the activating influence issuing from the iron at the level of the liquid surface, disturbs the periodicity ; the regularity now consists only in this that the most active state recurs at pretty regular times. As might be expected the strength of the polarising current was perfectly regularly periodic. Influence of the area of the surface on the periodic passwity. In conclusion we have examined what is the influence of an enlargement of the immersed surface. For this purpose we have made an experiment with a spiral, of which 5 windings, with a joint length of 60 cm. were immersed into the electrolyte. The cathode was placed inside the windings, the auxiliary electrode being placed against the second winding from above. While the strength of the current was again regularly periodic, the .potential difference exhibited very irregular oscillations, as Fig. 7 clearly shows. The irregularity was such that even the most active state did not recur regularly, and the whole curve, therefore, shows the periodicity under the influence of great disturbances. Hence it could be clearly perceived when observing the iron electrode, that this was never passive resp active throughout the whole area at the same moment, but that different parts were activated at different times. This curve is a very fine demonstration of the fact observed by us already before that a piece of iron can be passivated with the more difficulty as the surface is greater. . Amsterdam, General and Anorg. Chemical June 27, 1918. . Laboratory of the University. Chemistry. — “On the System lron-Oxygen”. By Prof. A. Smits and J. M. Bisvorr. (Communicated by Prof. S. HOOGEWERFF). (Communicated in the meeting of June 29, 1918). The equilibria to which the reactions between iron-oxides and reducing gases as carbon oxide and hydrogen give rise, have already repeatedly been a subject ef a scientific research. Thus of the gas phase of the three-phase equilibria FeO + Fe + G (CO) (H,) . am Teen: > was studied ’). (CO) “*? (HO Three-phase systems of three components were studied, i.e. systems that were monovariant at constant pressure. In this there was, however, no need to keep the pressure constant, because the above-mentioned and Fe,O, + FeO + G the ratio relations are independent of this. As result the researches with CO as reducing gas yielded two equilibrium curves, which may be called three-phase curves for the homogeneous equilibrium in the gas phase which coexists with two solid phases, namely one for FeO + Fe + G, and another for Fe,O, + FeO + G, of which SCHEFFER *) showed that they had to intersect in virtue of the heat- effect of the conversions. Researches with H, as substance of reduction did not only give the situation of the three-phase line for Fe + FeO + G, but also that for Fe + Fe,O, + G. The latter was made probable by REINDERS, who also computed the situation of the three-phase line for Fe,O, + FeO + G in this system from the corresponding line for the reduction with CO by the aid of the water-gas equilibrium. When we trace the three-phase curves for Fe,O, + FeO + G and for FeO + Fe + G for the case G = CO + CO,, we get the following figure when log K is drawn as function of T, in which figure a third three- phase line for Fe + Fe‚O, + G must start from the point of inter- section, which is here a quadruple point as ScHErrer has noticed. ') A survey of the literature of these researches has been given in REINDERS’ paper on: the equilibria of iron-oxide with hydrogen and water-vapour. Chem- Weekblad 15, 180: (1918). %) These Proc. Vol. XIX, p. 686. 387 Loe K pn Fig 1. On the mixture of the solid phases Fe,O, and Fe,O,. Now the question presents itself whether there exists also a three- phase line for Fe,O, + Fe,O, + G’). SosMAN and Hosrerrer*) think that they have to derive from their determinations about the tension of dissociation and the diffraction of light of mixtures Fe,O, + Fe,O, that the oxides Fe,O, and Fe,O, in the solid state are miscible if not in all proportions, yet very near the concentration F,O,. If there really existed a continuous mixed crystal series bere, there would not appear a three-phase curve for Fe,O, + Fe,O, + G, and the figure discussed here would be complete. It is, however, the question whether on the ground of Sosman and Hosrerrer’s researches we may conclude to a continuous mixed erystal series. When we draw up a p,t-section of the system oxygen-tron corresponding to the temperature 1100°, on the assumption that Fe,O, and Fe,O, are only miscible to a limited degree in the solid state, we arrive at the schematic representation drawn in fig.. 2. In this p‚v-seetion, in which it is assumed that the oxides present a certain mixture in the solid state, the line df represents the mixed crystals that are rich in Fe,O,, and which coexist with the vapours be, the line gh referring to mixed erystals rich in Fe,Q,, which can coexist with the vapours eh. | A point on the line df, here p, corresponds with the concen- tration Fe,O,, and thus a point of the line gh, viz. q, corresponds with concentration Fe,QO,. It follows immediately from this what curve we must get, when 1) These Proc. 19, 175 (1916) RetnpeRs has supposed the existense of such an equilibrium but the results of the experiments of SosMAN and HostTeTTER, were unacquanted at that time. 2) Journal Amer. chem. Soc. 38, 837 (1916). 388 we start from Fe,O,, and every time take away a quantity of the Fig. 2. vapour phase at the constant temperature of e.g. 1100°. The total concentration will then change in the direction from Fe,O, to Fe,Q,, and in this the pressure will also be subjected to a change. First the pressure will gradually descend from p to /. During this decrease of pressure two phases coexist, viz. mixed crystals rich in Fe,Q, and vapours consisting almost exclusively of oxygen. When the pressure has fallen to that of the three-phase equilibrium e fg, a mixed erystal phase g rich in F,O, will be deposited by the side of the mixed erystal phase / rich in Fe,O,, and a three-phase system arises of which the phase rule demands that the pressure remains constant in case of equilibrium. On continued withdrawal of a part of the gas phase, during which the total concentration continually moves to the right, the pressure therefore remains constant until the last trace of the mixed crystal phase rich in Fe,O, has entirely disappeared. At this moment only the vapour and the mixed crystal phase yg rich in Fe,O, coexist, and on further withdrawal of the gas phase the pressure will again descend regularly, in which the solid phase moves downward along gh. When we now draw the vapour tension as function of the total concentration, theory predicts that , on partial mixing of the two oxides Fe,O, and Fe‚O, in the solid state, P £ a broken line as is schematically represented in fig. 3, will be found, the middle part of which runs 1 horizontally. FE,0, A ¥E,0, This is the theoretical curve, Fig. 8. and now it is directly to be seen, 389 in what the experimentally determined curve will differ from it. In the first place it is self-evident that through all kinds of disturbing influences of these small pressures, as e.g. the presence of traces of adsorbed gases or contaminations, and the slow pro- gress of the dissociation, there is a great chance that the middle part will not be found to be horizontal, but more or less sloping; f é : P : g 1 FE,0; x Fe,04 FE,0, F304 Fig. 4. Fig. 5. and in the second place the transition of the two sloping parts to the horizontal part will not be found to be discontinuous, but always continuous, especially when many observations are made in the immediate neighbourhood of f and g. Instead of the above given broken line the continuous curve of fig. 4 will, therefore, be found in the most favourable case. When with these curves we compare the lines found by Sosman and Hosterrer, which have been reproduced in fig. 5, we see that the found curves closely resemble those which theory led us to expect for only partial mixture of Fe,O, and Fe,O, in the solid state. Everything depends on this whether the non-horizontal course of the middle portion is essential or not, for if this zs essential and the observed pressures correspond with the states of equilibrium, this course of the isotherm would really plead in favour of the existence of a continuous mixed crystal series. SOSMAN and HosTETTER see a confirmation of the view that the mixing of Fe,O, and Fe,0O, is continuous in the fact that the indices of refraction of the mix- tures change far from proportionally with the quantity Fe,O, between hematite (e= 2,78) and magnetite (n = 2,42). They give namely the following results. 390 Concentration of the mixture | & for 700 gu Hematite 2.74 0.58 9/9 FeO 2.74 5.60 ” ” 2.73 12090 eer 2.72 1G UE ys Me 2.71 LEASES LY ve | 2.71 Magnetite (31.03%) FeO) | n=2.42 It seems to us that they overlook in this that in case of unmixing it is by no means impossible that the phase rich in Fe,O, of the equilibrium of unmixing would show a much stronger refraction in consequence of its content of Fe,O, than the pure magnetite. This possibility is by no means improbable, because it already follows from the above determinations that independently of the fact whether or no unmixing is assumed, the refraction must diminish much more rapidly somewhere in the optically not investigated region than on the Fe,O, side. It should besides be considered that, as also SosmMan’) remarked, if we assume a continuous series of mixing between the hexagonal hematite and the regular magnetite, this would be an instance of a continuous mixture between non-isomorphous substances, which has not yet been experimentally observed in a single case. Now if we assume that from the p,w-figure at 1100° and 1200° we mug actually conclude to a continuous mixed crystal-series between Fe,O, and Fe,O,, the said difficulty can still be obviated by the assumption that at this temperature the two oxides are iso- morphous; as SosMAN and Hosterrer found that the bomogeneous ~ mixed crystal phases are bi-refringent, magnetite would have to possess a point of transition below 1100°, above which point the regular form is metastable. This not very probable change of crystal class has, however, not been observed, and can besides not render the continuous mixing plausible for temperatures below that of the point of transition. However this may be, the existence of a continuous series of mixing of Fe,O,-+Fe,O, does not seem proved to us, and we 1) Journ. of the Washington Ac. of Science 7, 10 (1917). 391 deem it, therefore, desirable to consider the possibility that in fig. 1 there should be added another three-phase curve, viz. that for Fe,O, + Fe, 0, + G lying under that for FeO + Fe‚O, + G. This situation gives rise to the question, whether this new three- phase line can intersect another. If it intersected the three-phase line for Fe + Fe, 0, + G, the mutual relation would be as given in fig. 6. Loek ETI —— The conclusion that the three-phase lines for Fe + FeO + G and Fe, O, + FeO + G intersect, and that this point of intersection indicates, therefore, the lowest temperature at which FeO can occur stable by the side of the gas phase G, is entirely in accordance with the sign of the conversion which must take place in this point on withdrawal of heat, viz.: 4 FeO — Fe + Fe, O, + cal. *) When also the three-phase lines for Fe + Fe, O, + Gand Fe, O, + + F,O, + G intersected in the way indicated here, then the conversion : 3 Fe, 0, — Fe + 4 Fe, O, would have to take place in this point of intersection on withdrawal of heat, but this is in contradiction with the heat-effect of this reaction. It follows namely from the measurements that: 3fe,0, — Fe + 4Fe,O0,—b cal. °) The supposition expressed in fig. 6 should, therefore, be rejected. Now there remain two possibilities, namely these that the two three-phase lines for Fe, O, + FeO + G and Fe, O, + Fe, O, + G intersect at higher temperature, but that melting sets in, before this intersection takes place. In this case we get a situation as has been schematically given by fig. 7. ‘) Comptes Rendus 120, 623 (1895). %) loc. cit. 392 | Lock Pig... 7. Another possibility is this that the just mentioned intersection does take place in the stable region, and then the situation of the lines is represented in fig. 8. *) Fig. 8 It will be pretty easy to decide experimentally which of these two figures represents what really takes place. We will draw the attention on the fact that the transition point of iron is intensionally not considered here. The Blast-Furnace Process. What precedes gives time a survey of the three-phase lines and ) It is clear that when HFe,O; and Fez0, become miscible above a definite temperature in all proportions, the line for FegO, + Fe,0;-+ G ends abruptly. 393 the two-phase regions in the system Fe—CO—CO, (resp. Fe—H,— H,O), and this has rendered it possible to elucidate the reduction processes, which e.g. take place in blast-furnaces, from beginning to end by means of a graphical representation *). For this purpose we choose one of our last two figures, e.g. the Fig. 9. more probable one, fig. 7, and draw in this the line pQ for the equilibrium CO, + C 2 2CO as this is situated in the blast-furnace. Thus arises fig. 9, and the processes that take place in the blast- furnace are read from the graphical representation bearing in mind that then the course of this line of equilibrium should be followed through the different regions given here. We then start from point P and end in point Q. In this way we see that theoretically the reduction from Fe,O, to Fe,O, takes place for the first time in the point a, then in 6 the reduction from Fe,O, to FeO, in c that from FeO to Fe, and finally in d melting of the iron. Hence we shall remain in each of these 1) The Figure can easely be completed considering the formation ot cementit but this is omitted here intensionally. Reinpers (Proceedings 19, 175 (1916) has already indicated partly the equilibriums with cementit. . 394 points of intersection which represent four-phase equilibria on addition of heat, until one of the phases has been completely converted. It is known that especially the equilibria with carbon are com- paratively slowly established below 800°, which is the reason that in experiments with flowing gas, the just mentioned stage-wise reactions are found at temperatures above those corresponding with the points a, b, and c. Derivation of the P,T-Figure of the System O—Fe from the Equilibria of the Tron-Oxides with Reducing resp. Oxidising Gases. When it is borne in mind that an equilibrium as the following: Fe‚O, + H, Z3Fe HO: . . 2h. ONE may be conceived as consisting of the two equilibria: 2FeA yee G¥ oO JO pad re ee and 2H OPS ZH, 0e ee and likewise Fe,0, + COV 3FeO + CO, .\. .-. (3) as consisting of the equilibria: 2 e,0; = 6GleO' RON 2 a a eee and 2C0, = 200 =. 0, . oe At ee it is clear that from the equilibria (1) and (Lb) resp. (2) and (26) the situation of (la) can be derived, and the same thing may be said in reference to the other equilibria that are considered here. It also follows from this that where the three-phase line for FeO + Fe,O, + (CO + CO,) was studied, also FeO + Fe,O, + O0, were in equilibrium with each other. Hence the shape of part of the P,T-projection of the system O-+ Fe can be devived from the situation of the determined lines of equilibrium. Thus it is immediately seen that, the three-phase lines for Fe + FeO + (CO + CO) and for FeO + Fe,O, + (CO + CO) intersecting this must also be the case with the three-phase lines of Fe + FeO + G and of FeO + Fe‚O, + G in the system O—Fe. When we express this in a diagram, and when we also assume the existence of the three-phase line Fe,O, + Fe,O, + G, we arrive at the following P,T-projection (Fig. 10) on the assumption, as was also supposed in fig. 7, that the three-phase lines FeO + Fe,O, + G and Fe,O,—Fe,O, + G do not intersect in stable points. Accordingly this projection presents the peculiarity, which up to now has never yet been observed in a case like this, that 395 namely two three-phase lines for two solid phases and vapour, intersect, without inverse melting taking place in the system. In case of inverse melting such an intersection must take place, as was before demonstrated by one of us’); the case of such an Fig. 10. intersection without inverse melting, had however not yet been considered, so that the system O—Fe teaches us something new here. When we now consider the possibility of a stable intersection of the two three-phase lines for FeO + Fe,O, + G and Fe,O, + Fe,O, + G Fig. 11. according to the supposition in fig. 8, we get a shape for the P, T-projection of the system O — Fe as indicated in fig. 11. In this case the three-phase line for Fe, O0, + L + G would, therefore, be metastable. 1) A. Smits. Zeitschr. f. Elektr. Chem. 18, 1081 (1912). 396 Calculation of the Oxygen- Pressure of the Dissociation- Equilibria. When we know the constant of equilibrium of an equilibrium like M 4 CO, 2M OO CEN) resp. M + HSO MOS DE EE) and likewise the constant of equilibrium of the equilibrium : 2CO; = 200 4 Orr en NE resp. 98.0 HE 0 scion. ook at the same temperature, the oxygen pressure follows immediately from these data. From (3) follows namely : Pco Po: and from (4) P'coPo, Kro, Eta: so that Kroo, == "Po, or Kp CO Po. = O = € hence log Po, = log Kroo, — Adog Wert hijes > EEN In this way the oxygen pressures for the equilibria of dissociation : 2FeO 2 2Fe + O,, 2Fe,0, = 6FeO + O, and Fe,0,@ 3Fe-+ 20,, have been calculated by us between 400° C. and 1000° C. We had to use for this an equation of log Kp for the CO, dissociation equilibrium. Wishing to apply this equation for tempe- ratures between + 500° and + 1000 °, we have substituted the heat-effect corresponding to the temperature of about 800°, viz. — 133000 cal. for E in the equation: din Ke E on RTE and put Zre, =O, so on integration we got: 133000 4 log Ke = — zont © Mt Rana os he ARN or: 133000 log Kp = — Feng eelt ein Sag mn oN) 397 Then C’ is chosen so that agreement is obtained with the experimental data’). Thus was found e.g. when p is indicated in atmospheres : log Kop = 18000) = — 13,45. When we substitute this value in our equation for log Ky, we find: Dik and then equation (7) becomes: 133000 se oe ee SM a aaah or 29100 | log Kp = — pe + foe FT -o,8-. 2 or (9) When we now calculate log Ky, by means of this equation for temperatures between 400° C. and 1000° C., we find what follows: TABLE I. ; log sj (p in Atmospheres) 400 — 34.6 450 — 31.5 500 | — 28.9 550 — 26.7 600 — 24.6 650 — 22.8 100 — 21.1 150 — 19.6 800 — 18.3 850 — 17.1 900 — 15.9 When at the same temperatures and pressure we now also know the values of log K for the equilibrium in the gas phase of the different three-phase equilibria in the system Fe — CO — CO,, then follow from equation (5) the values for the oxygen tensions at the different temperatures. The values of log K for the equilibria Fe + CO, 2 Fe O + CO, 3 FeO + CO, = Fe,0, + CO and 3Fe + 4 CO, 2 Fe,O, + 4CO, have 1) Apeae, Handbuch III, 2. 183. 26 Proceedings Royal Acad. Amsterdam. Vol. XXI. 398 TABLE II. Fe + CO: = FeO + CO 3 FeO + CO, rad Fes0, —- CO log Kg log 2 ~2logK;= log Po, 2 log Pre FeO +0, (in atmospheres) 3 Fe + 4 CO, = Fe3O4 + 4 CO | log Kp — 2log K, = log Po, CO, — 26.5 — 24.3 — 22.3 — 20.5 — 18:9 — 17;5 — 16.2 — 14.9 log Preo + Fe,0,-+ 0, (in atmospheres) —2logK's=logPo C02 3 8498 | = 4188 — 28.8 ee log Pre + FesO, + Op | (in atmospheres) 399 been borrowed from ReiNDERs ’s paper on: “The Equilibria of Tron and Iron Oxides with Watervapour and Hydrogen” *). The results of these calculations of the oxygen tensions have been expressed in the following table 2. (See table 2 pag. 398). In this table we find, therefore, the oxygen-dissociation tensions of the equilibria: 2FeO @ 2Fe + O, - 2Fe,0, € 6FeO + O, and Fe,0, 2 3Fe + 20, and we-see from this that these expressions are very small, as was to be expected. No importance is, of course, to be attached to the absolute values of these pressures, from which we should have to conelude to the presence of one gas-molecule in many litres, when we continue to consider the ordinary gas-laws as valid, because the formulae which we used in our calculation rest on the supposition that we have to do with a great number of molecules. Yet at the lower T the real oxygen pressures corresponding to these calculated numerical quantities will be so exceedingly small that the question suggests itself whether the oxidation of the reducing gas, which in this case proceeds with pretty great velocity, can still be considered as a homogeneous gas reaction. *) P,T-Projection of the System O—Fe. By the aid of these data we are now able to indicate part of the P,T-projection of the system O—Fe, when we put the found oxygen-pressure equal to the total pressure. When in this projection we also indicate the points p and g which would follow from Sosman and Hosterrer’s observations for the vapour pressure of the equilibrium Fe,O, + Fe,0, + G, corre- sponding to the temperatures 1100° and 1200°, starting from the supposition that the almost horizontal part of the isotherms of disso- 1) lie. 2) Entirely analogous questions suggest themselves in the study of the mechanism of the reactions between e.g. solutions and salts, or between metals with very small solubility-product and electrolytes. Especially in the latter case the numerical values, which denote the electron concentration in solution, can be exceedingly small, as one of us showed already. *) *) Zeitschr. fiir physik. Chemie 92, 1 (1916). 26* 400 ciation found by them actually refers to this three-phase equilibrium, we get the P, T figure (12). 400 600 809 1066 2200 rans Fig. 12. Of course these two last mentioned isotherms are perfectly inadequate to determine the direction of the three-phase line for Fe,O, + Fe,O, + G with any certainty; we will, therefore, only point out that the situation of this three-phase line, as it would follow from these two data, with respect to other three-phase lines, would be in good agreement with the expectation expressed schematically in fig. 10 or fig. 11. General and Anorganic Chemical Laboratory of the University. Amsterdam, June 28, 1918. Chemistry. — “On the System Ether-Chloroform”. By Prof. A. Suirs and V. S. F. BERCKMANS. (Communicated by Prof. 5. HOOGEWERFF). (Communicated in the meeting of June 29, 1918). 1. Our investigation on the system ether-chloroform was prompted by a remark of a physician addressed to one of us. He namely drew our attention to the comparatively great generation of heat which occurs when these two substances are mixed, a phenomenon that was first observed by Gurnrin'), and had given occasion to the assumption of the existence of a so-called molecular compound of the composition (C,H,),0.CHCI,, because the heat-effect just reaches a maximum value for a mixture of this concentration, as follows from the adjoined figure 1. Gurure thought he found a further support for this assumption in the results of his researches on the volume contraction and the vapour tension of ether- chloroform-mixtures, and it seems Us io Lees eo that he has also tried to test his _ OL. ec, 5 Ede assumption by means of deter- Fig 1 minations of the point of solidific- ation. He namely says: “The liquid of the said concentration solidifies below 0° to a white crystalline mass at a constant tem- perature, which I shall state when I shall have determined it accurately”. GUTHRIE has, however, not come back to these determi- nations of the point of solidification. Afterwards Do.ezaLEK and Scuurze’s’) researches on the generation of heat and volume contraction led them to the result that these phenomena are maximum for an equimolecular mixture of ether and cbloroform. In the conviction that these phenomena were to be ascribed to the formation of a compound they have tried by fractionated crystallisation to separate this compound and found in 1) Phil. Mag. 5 18, 508 (1884). ‘ 3) Zeitschr. f phys. Chem. 83, 45 (1913). 402 this way about —80° for the point of soliditication, when using a pentane thermometer. Like Gururie before them, DorrzaLEK and Scuurze arrive in this way in virtue of their observations at the conclusion that in the system ether-chloroform an equimolecular compound makes its appearance, which is more or less dissociated in the liquid phase. 2. In order to be able to answer with perfect certainty the question whether or no a compound is formed, it is necessary to determine the melting-point figure of the system, and this is the reason why this inquiry has been taken in hand. The results recorded in the adjoined table were obtained by means of Dr Leeuw and ZerNIKe’s quick and sensitive resistance thermometer, a manufacture of the Amsterdam anorganic-chemical laboratory. When these results are represented in a 7'X-diagram, we get the following figure. (ETHER. xXx CHLOROFORM. Fig. 2. This figure (2) shows in the first place that the supposition of Gutariz, DOLEZALEK, and Scuurze is correct, and that in this system 403 ‘Final point of 2 Weight in Molecule Ist point of Baal grammes percentages solidification | solidification SE | Repetition E CHCl; |(C,H;),0, CHCl; | (C‚H5)20 | with old Zz | | | mixture i BAN iro ge U OT = — 66.5 2 | 21.545 | 1.479 | 90.04 Oday ee 12.6 | = 12.3 3° Pot .500° | HB | 16-25 | — 16,7 4 | 8.358 | 2.223 | 70.00 | 30.00 | — 93.5 — 95.3 5 | 15.807 | 4.226 | 69.89 | 30.11. | — 94.2 | — 93.8 zel 6. |; 8.419; | ATI 68.00 |. 92:00, | — 93.6. | bal Seidl -2-311 |) 68.10.41 32-007 || — 93.5 047 \ 1 | 7.952 | 2.468 | 66.66 | 33.33 | — 93.5 Ta | 7.952 2.468 66.66 33.33 — 93.3 Sl. 1.761 2.504 =| 65.00.) 35.00 | — 93.6 | — 93.6 Sa | 7.761 | 2.594 | 65.00 | 35.00 | — 93.9 9 | 7.283 | 2.890 | 61.00 | 39.00 | — 96.0 | — 98.1 10 7.164 | 2.964 | 60.00 40.00 | — 96.4 be ana 11 | 14.328 | 6.022 | 59.62 | 40.38 | — 96.8 | — 96.4 12 | 6.567 | 3.335 | 55.00 | 45.00 | — 95.1 5 abh 13- | 11.947 | 7.414 | 50.00 | 50.00 | — 94.4 | — 94.4 i3a | 5.910 | 3.705 | 50.00 | 50.00 | — 94.4 | {dee} £1956 Shc 9.02% 0 45-132)" Balazs | — 95.8 15 | 4.716 | 4.446 | 40.00 | 60.00 | — 99.7 | 15e | 11.41. | 10760901 40,03 {59.97 | = 90.9 | — 99.81] 16 | 4.179 | 4.817 | 35.00 | 65.00 | —104.8 | —105.1 | {6a 4.179 | 4.811 | 35.00 | 65.00 | —104.9 | 17 | 3.916. | 4.935 | 33.33 | 66.66. | —108.1 | 18 | 7.798 | 11.266 | 30.05 | 69.95 | —111.4 | —110.5 | 18a | 3.582 | 5.196 | 29.97 | 70.03 | —111.4 | | 186 A bep 206° | 20:84 TAG |l. 4 | Rt seas We a0oe 1: 87.00) 99 00-77 114,3 | 20 | 2.985 | 5.558 | 25.00 75.10 | —114.7 20a 2.985 | 5.558 | 25.00 | 75.10 | —114.3 21 | 4.798 | 11.888 | 20.03 | 79.97 | —117.2 | —117.5 2la | 2.388 | 5.928 | 20.00 | 80.00 | —117.6 | 23143 |) 11-717 18.65 | 81.35 | —118.4 |. —118.3 —121.7 ) 23 | 1.791 | 6.302 | 14.99 | 85.01 | —121.5 —121.7 24 | 1.901 | 11.113 | 10.01 | 89.99 | —119.9 | —120.1 —121.6 J 24a | 1.993 | 11.133 9.99 | 90.01 | —120.2 | 25 | 0.616 \ 7.063 NK: Pe ey EE DEN =a md 100.00 | —116.4 | —116.4 404 an equimolecular compound with a point of solidification lying at — 944°, is actually present; but —, and this particularly gives evidence how necessary the study of the melting point figure is to obtain full certainty about the appearance of compounds, — we see at the same time, that besides the mentioned two other compounds appear, viz. of the concentration (C.H), 0.) CHC and (Can): O.2CHCI,; the latter of these has a point of solidification at —93,3°, and the former does not present a stable melting-point, but before this temperature has been reached, viz. at — 113,8°, it is subjected to the following conversion : 2(C,H,), 0.CHCI, — (C,H,), 0.CHCI, + L. It is now interesting to consider in connection with this the p,x-figures of the same system, as they have been found first by KonnstamMM and van DALFSEN, and then by DorwzALEK at 33°,25; 66°, and 100°. (fig. 3). ees: —> molbt Chace, Fig. 3. It is now very probable that the peculiar shape of these p‚z- figures, especially that for the lowest temperature, is influenced not only by physical forces, but by the side of them also by chemical forces. General and Anorganic-Chemical Laboratory of the Unwersity. Amsterdam, June 26, 1918. Physics. — “Investigation by means of X-rays of the erystal- structure of white and grey tin. V’. Communication N°. 1 from the Laboratory of Physics and Physical Chemistry of the Veterinary College at Utrecht. By A. J. Bir and N. H. KorkMrIJER. (Communicated _in behalf of Prof. W. H. Kessom, Director of the Laboratory, by Prof. H. KAMERLINGH ONNES). (Communicated in the meeting of June 29, 1918). Till now three methods of investigation by means of X-rays of the inner structure of crystals have been used. That of Frivpricn, Knippinc and Laver’) and that of the Brace’s’) can only be used, when rather large, well-formed crystals, of which the erystal-system is known, can be procured. On the contrary the use of DeBijE and ScHeRRER’s *) method requires a great number of differently orientated minute crystals‘). For that reason this method is the most adapted for the investigation of the crystal-structure of those materials that are not or only with difficulty obtainable in larger erystals, e.g. of those metals — eventually of those modifications of those metals -— that are only known in a micro-crystalline state. Another example of the probable applicability of the method of DrBije and SCHERRER is found in the solidified gases, which till now are not to be obtained in sufficiently large crystals of exactly defined orientation. The investigation of different modifications of the same material, e.g. allotropical states of an element, is specially interesting. This investigation has already been initiated by Onm and Bir, who published lately in these Proceedings their interference-photos of grapbite and diamond.*) In the meantime DeBijr and SCBERRER °) investigated the same materials and also so called amorphous carbon 1) W. FriepricH, P. Knippine and M. Lauw, Sitz-Ber. d. K. Bayer. Akad. d. Wiss. Math.-Phys. Kl. 1912, p. 303. 2) W.H. and W. L. Braga, X-Rays and Crystal-Structure. London, 1916. 3) P. DeBije and P. ScHERRER, Nachr. d. K. Ges. d. Wiss. zu Göttingen. Math. Phys. Kl. 1916, p. 1. 4) A. W. Hutt, Phys. Rev. (2) 10 (1917), p. 661, evidently arrived independently of DeBiJE and SCHERRER, at rather the same method. 6) J. Orie and A. J. Bur, Proceedings Roy. Ac. Amst. XIX (1917),p. 920. 6) P. DEBE and P. ScHerRER, Physik. ZS. 18 (1917), p. 291. 406 and succeeded in determining the structure of these modifications of carbon. ; In connection with the investigations, which have already been made of silicium, lead, and the modifieations of carbon, we have in this investigation set ourselves the task, to determine the structure of still another element of this group of the periodical svstem, the tin namely in two of its modifications, the grey and the white ones. The X-rays were for the greater part of our photos furnished by a Röntgen-tube, constucted after a drawing, which Prof. DeBije and Dr. Scuerrer kindly procured us, and which must be considered as a modification of the tube, constructed by Rauscn von TRAUBENBERG °). The anticathode was a copper one, the parallel-spark-length (point- disc) measured 6 cm. The X-rays left the tube by an aluminium window of 0,02 mm. thickness, afterwards passed a lead screen, thickness 34 mm. with an opening of 2 mm. diameter and entered then a cylindrical camera of 27,3 mm. radius. In the axis of the camera the material, to be investigated, was placed in the form of a bar of 2 mm. diameter. The white tin bar was filed from a thicker bar, which for the diminution of the erystals and in order to obtain irregular orientation of these, was beaten for some time with the hammer. The grey tin was pressed into a small bar. We obtained this material through the kindness of Prof. CoHeN, to whom we therefore render our thanks. A photographical film (thickness 0,2 mm.) was stretched along the wall of the camera and pressed against it by springs. The exposition-time was 4 hours. During the preparative experiments we observed that in excess of the interference-lines of the tin, still others were obtained, of two different kinds. The first resembled hyperbolas, the centre of which was in the point, where the beam of characteristic rays, after having passed the investigated preparation on both sides, fell on the film. They must be ascribed to the interference of rays, which are diffracted by the silver-bromide of the film. The second kind had the same direction of curvature as the lines, which originated from the preparation in the axis of the camera, they intersected these lastmentioned however at some points, which indicated, that here too the origin was not to be sought for in the preparation. We got the evidence, that the origin of them was to be found in the back-opening of the lead sereen. which opening was in the wall of the camera. We got rid of the first kind of undesired lines by making a circular hole in the film, at the place, where the beam 4) H. Rauscu von TRAUBENBERG, Physik. ZS. 18 (1917), p. 241. 407 \ fell upon the film, as indeed Degijr and Scuerrer already did too. The coming up of the second kind of undesired lines was avoided by extracting the lead screen a little from the surrounding brass tube, so that the rays coming from the lead, were screened from the camera by the remaining wall of the camera, or by widening the end of the sereen, which was directed to the film so much, that the rays, emerging from the back edge of the not-widened part of the screen, were arrested by the widened part of it. *) On the photos, obtained by us of the white, as well as of the grey tin there appear points on the lines, as were also observed by Degijn and Scuerrer on their photo of silicium. They must be ascribed to larger crystal-parcels. *) In order to facilitate the comparing of the results for white and grey tin, we have indicated in the drawing underneath, one under _the other for both these materials, by vertical lines the places, where the interference-cones intersect the film in a plane, perpendicular to the axis of the preparation. The numbers at the bottom give the distances of the lines to the intersection of the axis of the penetrating beam with the film in mm. By greater or smaller thickness of the lines at the top and the bottom respectively the intensity of the lines is given, as it was estimated by us in five degrees, namely : 1) Perhaps it is worth while, to state, that the criticism, exerted by TAYLOR, Physik. ZS. 17 (1916) p. 316 on articles of Laus, Physik. ZS. 15 (1914) p. 732, 844, seems not to be founded, now that we too found, that characteristic X-rays when going through a leaden screen, can give interference-lines, originating from the edges of the screen. 2) The length-direction of these oblong points is the same as the direction of the interference lines. The explanation of this is as follows. On a single point of a plane out of the atom-net of such a crystal-parcel there falls, on account of the imperfect parallelism of the beam, which has passed the screen, a convergent conical beam of rays (first cone). If the net-piane, mentioned, has such a position that for a direction of rays in the axis ofthis last beam the condition for favourable interference is fulfilled, then there are in this beam still other directions of rays, for which this condition is just as well fulfilled The collection of these directions is found by describing a cone, with the normal on the plane mentioned, as axis and the axis of the first cone as one of the describing lines (second cone). All the describing lines of the second cone that are within the first cone, fulfill the condition for favourable interference. After their reflection all these rays form together a diametrically opposite part of the second cone. Just there however the second cone is tangent to the third cone, formed by the reflected rays, which are obtained by letting the reflecting plane take all possible posilions favourable for interference. Now, where the film intersects with both the last cones, the intersection-lines will be tangent to each other. One of these intersection-lines is an interference line, the other such an oblong point. 408 very feeble, feeble, moderate, strong, and very strong. On the axis of the drawing the greatest error, that possibly is made at the readings is indicated by the distance of two little lines (to the left). From the photo of the grey tin it follows in the first place that this material is crystalline, which fact, so far as we could find, at present was unknown.') Further there appears to be a difference between the erystal-structure of grey and of white tin. We intend soon to communicate the composition of both the atom-nets. In concluding we thank Prof. Kersom very much for the kindness with which he placed all the requisite apparatus for the experiments at our disposition and for his interest and cooperation. 1) [Note, added during the translation]. After this communication had been printed in the Dutch edition, we found, that von Founton (Jalirb. d. Kais.-Kön. Geol. Reichsanst. Wien. 34 (1884), p. 867) as well as Frirscue (Ber. d. d. Chem. Ges. 2 (1869), p. 112 and 540) assert, that grey tin is erystalline. It does not appear in what manner these scientists have verified the crystalline condition. Geology. — “On Tin-ore in the Island of Flores”. By Prof. A. WICHMANN. (Communicated in the meeting of June 29, 1918). Some years ago | maintained on the ground of geological studies that tin-ore does not occur in the island of Flores, at all events not to an amount worth mentioning’). As appeared from a memoir brought forward last year by Prof. 5. J. Vermaus, the author is otherwise-minded ®). I think it worth while to study the author’s arguments, which he pretends to be based on the doctrine of the deposition of ore, on ethnography and on metallurgy. Geological researches have not been made in Flores of late years, so that in this respect there was no need for revising my paper. Nevertheless Prof. VermaAns supposes he has been fortunate enough to make a discovery, which throws a new light upon the matter *). This finding appears to be nothing else but a piece of tin-ore, exposed in the Colonial Museum at Harlem, “weighing 131 grms and composed of chlorite and tin-ore with fissures, in which occurs some kaoline. Besides the two mentioned minerals a single grain of chalcopyrite is also noticeable’. Prof. Vermars altogether fails to see that the finding place “Gunung Rokka”, indicated on the label must be fictitious, since the mountain — the Inije Rije of the natives — is a voleano and even now is in the condition of solfataric activity *). Indeed, not one of the researchers has ever found the least indication of the occurrence of ores on that mountain and even C. J. VAN SCHELLE, who never shrank from a bold hypothesis, has wisely refrained from making inquiries after tin-ore in the volcanoes of Flores. He looked 1) “On the tin of the Island of Flores”. These Proceedings Amsterdam. Vol. 17, 1914, p. 474—490. 4) Tinerts op Flores. De Ingenieur. 32. ’s-Gravenhage 1917, p. 584—590. 3) l. c. p. 584. 4) J. J. PANNEKOEK VAN RHEDEN. Overzicht van de geographische en geologische gegevens, verkregen.... van het eiland Flores in 1910 en 1911. Jaarboek van het Mijnwezen in Ned.-O.-Ind. 40. 1911. Batavia 1913, blz. 219—220, — Eenige geologische gegevens omtrent het eiland flores. Jaarboek van het Mijnwezen. 39. 1910. Batavia 1912. Verhandelingen, p. 135— 136. 410 for it, at haphazard in a district supposed by him to contain tin- ore, which was partly overlaid by voleanic formations *). As to his arguments based on metallurgy, it strikes us that Prof. Vurmars has deemed it unnecessary to inquire further into the matter as regards tin, or he could not have written as follows: “He (WicHMann) also cites what he wrote before, namely that tin-ore could not be reduced by burning grass. I have often seen alang- alang burning, still I would not make bold to say on metallurgical grounds what Wichmann presumes ”)”’. A geologist will not confine himself to merely ‘see’ an alang-alang field “burning”, but will also try to watch the effect such a fire has on the components of the soil under it. | myself experienced that the voleanic sand and the lapilli of augite-andesite at the foot of the Batu angus bara in the Minahassa did not change a bit. Nor could anything else be expected, for the grass (/mperata cylindrica Beauv.) furnishes such an insignificant quantity of fuel that it is burnt away in a trice. This short burning process does not even extend as far as the roots, so that when the West Monsoon sets in, the grass begins to sprout again. Prof. Vermars continues: “We read, however, in VAN SCHELLE’S report: “““When the forests are on fire, part of the ore seems to be reduced... .”’, after which Prof. Vermars concludes: “If, therefore, a mass of tin-ore is imbedded near the surface in the root-leaves of a large tree and, after felling the tree, a pile of combustible materials is kindled at the stump, there ts no doubt but that the tin-ore is reduced to metal)". First and foremost I wish to quote a passage from P. van Dixst’s well-known work on Banca: “The remainders of a charcoal-furnace *) are identified by the natives as the spot near which it is supposed that tin was first discovered in Banca, that is after the burning of part of the forest near the spot’). The belief in those stories is negatived even more, when we reflect that the heat produced by a burning pile of tree-trunks is not adequate to reduce tin-ore with- out a certain amount of coal being mixed with them, especially 1) Verslag van het onderzoek naar het voorkomen van tinertshoudende gronden op Flores. Extra-Bijvoegsel der Javasche Courant. Batavia 1890. N°. 10. (Uittreksel: Tijdschr. voor Nederl.-Indié. Zaltbommel 1890. 2, p. 79). 2) The italics are mine. 8) The italics are mine. 4) On the Sambong giri hill near the Lindjoe mine. 5) Tradition says this happened in 1710. (F. Epp, Schilderungen aus Archipel. Indiens Heidelberg 1841, p. 134; J. H. Croockewir. Banka, Malakka and Billiton. The Hague 1852, p. 134). 411 not in the case of the coarse-granular ore at the foot of this hill >)”. This assertion is apparently founded on the fact that tin-ore is one of the minerals that are difficult of fusion, but that it is easily reduced to tin by the addition of charcoal. This assertion does not satisfy us any more than VERMAKS’s pronouncement that “there is no doubt but that the tin-ore is reduced to metal’, by means of a burning pile of wood. An experiment does not seem to have ever been made. The melting point of tin-ore as established by R. S. Cusack at 1127° C.*), seems to me to be tov low, as bigher points are found °) for much more fusible minerals, e. g. for augite l100°—1200° (according to C. Doeurer®)) and for plagioclases (labradorite to oligoclase) 1130°—1300°*). Anyhow Cusack’s melting point (1127°) being even higher than the heat produced by the burning of living wood — the only wood we have to deal with — it is obvious that a burning wood cannot reduce tin-ore to tin. Moreover, whereas alang-alang fires may occur repeatedly every year towards the close of the West-monsoon, forest-fires are decidedly the exception, so that even on this account the required amount of tin could not have been produced in this way. Furthermore, it still remains to be seen whether the expected result can be obtained even at high temperatures. In modern mine- ralogical textbooks and manuals — with a few exceptions — the hypothesis is advanced that tin-ore does not undergo any change, when the blowpipe is applied. This squares entirely with the results most inquirers are capable of achieving in connection with the difficulty of mastering some facility in handling the blowpipe. Years ago Burzeuius wrote: “Das Oxyd verändert sich und schmilzt nicht, aber von einem starken und anhaltenden Reduktionsfeuer kann reines Zinnoxyd ganz und gar ohne Zusatz zu Zinn reducirt werden. Dies erfordert indessen eine Gewohmheit das Löthrohr zu gebrau- chen”’®) This is in character with C. FK. PrartrNeRr’s opinion, who, 1) Bangka, beschreven in reistochten. Amsterdam 1865, p. 68. *) On the melting points of minerals. Proceed. R. Irish Acad. of Se. (3) 4. Dublin 1896—98, p. 413. 8) Beziehungen zwischen Schmelzpunkt und chemischer Zusammensetzung der Mineralien. Tschermaks Miner. petrogr. Mittlg. 22. Wien 1903, p. 399—311. CG. Doetter, Handbuch der Mineralchemie. 1. 1912, p. 663. 5) The above mentioned melting-points are somewhat too low, as the author himself has acknowledged afterwards (Handbuch der Mineralchemie 2. 1. Dresden — Leipzig 1914, p. 579). 6) Von der Anwendung der Léthrohrs in der Chemie und Mineralogie. Uebersetzt von H. Rose. Niirnberg 1821, p. 113—114. At present it is extremely difficult to ascertain whether any writer before Berzetius has obtained the same result. 412 however, adds that in this process a white layer of tin-oxide is formed.') W. A. Ross, on the contrary, maintains that the ore does not melt “aber eine weisse Ausbliihung kommt hervor” *). Groreio Spezia again believes that the ore changes, but ‘non par fusione ma par consumo” and he tries to account for the behaviour of the tin-ore by stating that in consequence of the intense heat a reduction takes place indeed, but that it is incontinently followed by an oxidation evolving the white layer.*) Should this interpretation be correct, there cannot possibly be any question about reducing tin- oxide by heat alone. Now which of us is, to quote from Prof. Vermars, the “metallurgist of Flores” who has indulged in fancies *)? And when the same writer continues: “WicHMANN ought to have considered that such utterances cannot but be fatal to the upgrowth of a mining concern, of which many experts anticipate great success’, I feel urged to say that it is rather disappointing to find that still in the year 1918 one is obliged to appeal to the timeworn maxim that the man of science does not ask whether or no anything is fatal in its effect on a mining concern but that he considers bis sole task to be to find the Truth. Apart from this, the effect of science can never be fatal, at all events not for those who know how to study it; on the other hand it is always inspiriting, even when an inquiry yields a negative result. If tbe mining industry had paid more regard to science, they would have been spared many dis- appointments in the island of Celebes and they could have saved many people’s capital. Presumably they will not have become wiser by this time, in spite of all this. Furthermore, if we reflect that tin foundries in Flores cannot be imagined without charcoal furnaces and agglomerations of tin-slags of which no trace was ever found, we are safe to say that Prof. VerMaAks’s endeavours to prove the occurrence of tin-ore on metal- lurgical grounds have utterly failed. We shall have to dwell more at large on his arguments derived from ethnography. In estimating his material Prof. Vermaxs has entirely neglected to ascertain whether the premiss from which he started was correct, which is a common mistake among ethnologists. 1) Probirkunst vor dem Löthrohre. 5. Aufl. bearbeitet von Tu. Ricurer. Leipzig 1878, p. 136. 2) Das Löthrohr in der Chemie und Mineralogie, übertragen von B. Cosmann, Leipzig 1889, p. 161. 8) Sulla fusibiltà dei minerali. Atti R. Accad. delle sc. 22. Torino 1886—87, p. 422. 4) Le. p. 588. 413 An early illustration of this error was afforded by J. H. Croockrwit, who considered the absence of tin-objects in Billiton — those that were found there had been imported from Banea — to lend support to his hypothesis that no tin-ore was to be found in that island. ?) Jonversely U. J. v. SCHELLE'sS whole argumentation rested on/y on the fact that natives of Flores were found in possession of tin objects ete, on which fact also Prof. Vermars set so high a value. This is the logie of a Papuan, who, judging from the knives and axes he gets from the merchants in exchange for his birds of para- dise, believes that Holland is rich in iron-ore. The track to which the natives attach great value does not only serve as an ornament, but also as a form of investment, as some- times occurs iu Europe also. There is even among uncivilised nations a liking for capitalization, especially among the more intelligent part. This tendency increases with the degree of personal safety. That is why the government of a Western Power has always encouraged “capitalization”. The natives’ choice of articles of investment is very limited compared with that of Europeans, who prize stocks and other paper value so highly. Such articles must be proof against the influences of the climate and moreover be gaudy and showy. ‘In districts where Europeans have settled or in not too insignificant commercial centra coined money, “rijksdaalders’,*) and especially gold coins are greatly in favour. If the soil does not produce the desired objects, as is the case in nearly all the islands of the Timor Archipelago *), the native is obliged to look about for foreign objects. Next to weapons and other iron tools all the native tribes set great value upon the “muti tanah”, dirty-coloured orange-red glass beads. They are skilfully wrought, but not beautiful and owe their value rather to being ‘‘antique” and to the fact, that they were not imported after the pre-historic period, i.e. after the arrival of the Europeans. 1) Extract ee: the report of a journey across the island of Billiton (Natuune. Tijdschr, Ned. Ind. 3 Batavia 1852, p. 401. *) Dutch coin worth 4/2. 5) It is true, gold and copper occur in Timor, but by far not sufficiently to meet the demand for those metals. Moreover the occurrence of ores does not prove at all that the natives are skilled in metallurgy. The inhabitants of Billiton eg. were entirely unacquainted with the art of reducing tin from tin ore, whereas from time immemorial they are quite familiar with the more complicated process of working iron, and it was only recently that F. Sarasin declared: “Die Kunst Metalle zu bearbeiten, haben die Caledonier trotz des enormen Reichtums des Landes an solchen, speziell von Eisen, nie gekannt und auch heute noch nicht gelernt” (Neu-Caledonien. Basel 1917, p. 83). 27 Proceedings Royal Acad. Amsterdam, Vol. XXI. 414 It is remarkable that in Flores the same legend about their origin prevails as regarding tin, viz. that they have been formed in the soil itself in consequence of the burning of alang-alang; they are accordingly called ““muti tanah”, that is: earth-beads. About their origin we are still as much in the dark as about that of the glass objects found in the South-Sea islands. Beside a marked concordance in their taste for beads the islanders of the Kast-Indian Archipelago evince none the less a vast difference respecting the other favourite objects for capitalization. While “moko-moko”, peculiar kettledrums made of brass, are in vogue in the Alor-islands, elephants’ tusks are generally in favour in the Solor-islands coming next to them in a western direction. In West-Flores there prevails a fancy for tin- ornaments, while the inhabitants of Rotti prefer chains made of gold-wire. The reasonings of C. J. van SCHELLE and of Prof. Vermags would lead us to conclude that there are coppermines in the Alor-islands, herds of elephants in the Solor-islands and gold-diggings in Rotti. It would be throwing words away to say more about it, but we wish to say a few words more about the tin objects of Flores. __ Prof. Vermars might adduce the argument that there is not a single record extant to support the assertion that tin or tin objects were imported into Flores. But the same argument could apply to the elephants’ tusks of the Solor-islands. In the second half of the previous century cast moko-mokos were introduced into Java from Grissee, but the natives soon found out that they were imitations. In Rotti, where not a single grain of gold has ever been found, an old branch of industry has revived in consequence of the sale of horses to Australia, which brought a large number of sovereigns to the island, which were wrought into gold chains *). To support his argumentation Prof. Vermars has added to his memoir not only a number of fine illustrations, but also numerous analyses of the metallic objects found in the island. | hope they will prove most interesting for the ethnography of Flores, but they are not relevant to the origin of the metals. I can imagine the possibility of establishing through analysis that a table spoon has been procured by such and such a firm, but hundreds of analyses cannot enable us to establish the source of the silver used to make the spoon. 1) The Rottinese are very superior in civilization to the people in Alor and Flores, who possess but little skill in working metals. J P. Freyss says about them: “the art of forging is very little advanced among the inhabitants of West-Flores.” (Reizen naar Mangarai.... Tijdschr. Ind. T. L. en Vk. 9. Batavia 1860, p. 511). 415 It strikes us that in former times neither export nor import of tin objects was ever thought of’). In the long run such a trading possibility could not have escaped European enterprise, no more than the trade in “Billiton-axes” and “Tambuku swords”. It attracts our attention, however, that the tin objects in this poor island are found in the southwestern part in the possession of the natives. This might be due to the presence of tin-ore in that part; however the conclusion might also be drawn that the population disposes of more truck and consequently obtains possession of such objects as their countrymen in other parts must do without. According to GopinHo DE Erepia ‘‘cinnamon’’?) was exported from these parts already in the time of the Portuguese settlement *) and the fort in Nusa Endeh was certainly not built only for the purpose of protecting the Dominicans. That also this product attracted the notice of the East-Indian Company is borne out by the report of P. A. Leupr on the discussions at Batavia in 1757, which says: “They had still to contrive a means to get possession of the cinna- mon-wood Rokko in Endeh *)’, and according to J. C. M. RADEMACHER it was in the year 1756 that “the Company permitted the natives of Makassar to trade on Endeh and the Mangary provided no wild cinnamon was exported, on the penalty of confiscation of ships and cargo” *). Nearly sixty years ago J. P. Freyss still wrote: The gathering of wax and cinnamon constitutes the chief commercial resource °). The natural result of this trade was a comparatively higher degree of prosperity than was enjoyed by their countrymen, whose income, derived from woodproducts, was smaller and who therefore had 1) As 1 mentioned before only in 1871 J. A. van per Criss made mention of the export of tin arm- and leg-rings from the Rokka district. (Tijdschr. v. Nijverheid en Landbouw in Nederl.-Indié 16. Batavia 1871, pp. 158—159). 3) No doubt Cassia was meant. (J. G. Fk. Rreper The island of Flores or Pulau Bunga. Revue colon. internat. 1. Amsterdam 1886, p 66). 8) Antonio Lourengo CAMINHA, Ordonacôes da India do Senhor Rei D. Manoel de eterna memoria. Informagao verdadeira da Aurea Chersoneso feita pelo... ManoeL GopinHo pe Erepra. Lisboa 1807, p. 143 (written in 1599). 4) Besognes der Hooge Regeering te Batavia gehouden over de commissie van Paravacini naar Timor in 1756. Bijdr. t. de T., L. en Vk. (4) 1. ’s Gravenhage 1877, p. 479. 5) Korte beschrijving van het eiland Celebes en de eilanden Floris, Sumbawa, Lombok en Baly. Verhandel. van het Batav. Genootsch. van K. en W. 4. Batavia 1786, p. 252. 6) Reizen naar Mangarai en Lombok in 1854—1856. Tijdschr. Ind T., L. en Vk. 9. Batavia 1860, p. 512, ai" 416 to be content with bartering their articles for necessaries of life, whereas the Rokkanese could also acquire artieles of luxury. What, however, were the events that stopped the import of muti tanah and the objects made of tin, will long remain a puzzle, perhaps for ever. Prof. Vermars bas prefixed to his memoir the following quotation from CROOCKEWIT, as a motto: “1 feel justified in concluding from these inquiries, made in three different ways, that the ore found in Billiton does not contain tin oxide’ '). The tendency of this motto was to stigmatize my being mistaken with regard to Flores as Croockrewit had been with regard to Billiton. As appears from the foregoing Prof. Vurmans has not sueceeded in demonstrating that tin-ore occurs in Flores; the comparison therefore halts, and was at the very least premature. He has mistaken the persons also in another respect. [t was not I, but van ScHELLE who, just as Croockewir, started from faulty premisses; it was not 1 but vaN SCHeLLE whose inquiries, just as Croocknwit’s, led to wrong conclusions. No wonder that both failed. 1) Extract from the report of a journey through the island of Billiton. (Natk. Tijdschr. Ned.-Ind. 3. Batavia 1852, p. 401. Physiology. — “On the Sign of the Electrical Phenomenon and the Influence of Lyotrope series observed in this phenomenon’’. By Prof. H. ZWAARDEMAKER and Dr. H. Zeunuisen. (Communicated in the meeting of June 29, 1918). I. In a previous publication ') we have established that the nebulae of salicylic acid salts generated by spraying, owe their electrifying power to the contained anion. Fig. 1. Influence of 0.1 n. NaCl and 6 °/, canesugar on the charge of salicylas natricus. 0,000! vv. 0,00/ vv. 0,08 vv. 0, ! Ww, pv. eu EE dace as EE cai a on ag i 5 t wo ' x $ om © 1. Curve of the Charge of salicylas natricus alone. 2. Curve of the Charge of salicylas natricus + salt. 3. Curve of the Charge of salicylas natricus + sugar. Along the top-abscissa the normal concentrations, along the bottom abscissa the logarithms of the normal concentrations of salicylas Na. are given; along the ordinate the deflections of the electroscope in scale-divisions. 1) These Proceedings Vol. XX, p. 1272, 1918. +18 As set forth loco citato, the attending cation lessens the negative charge of the nebula of this acid. Consequently salicylic acid and salicylates, in weak concentrations, determine the electrifying power of the nebula in quite the same way and with a negative sign, only with quantitative differences in such a sense that the salt produces a lower charge than the acid, and can be sprayed in higher concentrations on account of greater solubility in water. This also holds good for other salicylates that are soluble in water. The negative charge of salicylas natricus lessens with the increase of the concentration, so that it gradually approaches the zero-line, and ultimately crosses it. The charge, then, is positive. For a long time there is no charge at all, as is ‘well seen in Fig. 1, which Fig. 2. Negative and Positive phase of the Charge of salicylic acid and the influence of 0.0025 n. NaCl and of 6°/, sugar on the charge. ' opoolvy 0,00). m/ ieee 1. Curve of the Charge of Salicylic acid alone. 2. Id. of salicylic acid + salt. Duala: sor ge „ + sugar. Division of abscissa and ordinate as in Fig. 1. 419 also shows the positive part of the charge of salicylas natrieus, and gives the logarithms of the strengths of the solutions along the abscissa. Addition ot sodium-chloride to these strong solutions of salicylas natricus had no more influence upon these positive charges than an addition of sugar had. On the other hand, under the influence of the same substances, changes were brought about in the negative charge of weaker solutions, as the figure illustrates. The charge of the salicylic acid itself also first increases with the rise of the concentration, afterwards it decreases (Fig. 2), to rise above the zero-line a little before the point of saturation. The effect of salt and sugar upon the charge is very slight at this moment, just as it is with salicylas natricus; after crossing the zero-line, however, it comes forth again, but in a positive sense. The facts just described or illustrated seem a chaos at first sight. Some order is discovered, when we reflect that in dilute solutions, which are the only solutions we have to deal with in spraying, cations and anions act separately. The effects of cations and anions are superposed. In spraying salicylic acid and caproic acid (fig. 2 and 3) only the anion is of importance. In spraying salicylas natricus there is an action of salicylic acid and sodium; in the combined spraying with salts and sugar there is a combination of effects of all cations and anions present. It is not possible as yet to account for these phenomena; they belong, indeed, to the field of physics proper. Provisionally we are able to state only that strong solutions of salicylas natricus yield, on spraying, dense nebulae, forming large droplets; the latter are so large that sodium-chloride and sugar particles do not apparently affect them. When the salicylas natricus solution is diluted the charge is affected first by the sodium-chloride and only much later by the sugar. It is not clear why with sodium-chloride this influence reveals itself first in a negative sense, and, with a weaker salicylas natricus solution in a positive sense, while with sugar that influence man- ifests itself later and only in a negative sense. This negative sodium- chloride phase is altogether lacking with salicylic acid, while the charge-curves of this acid with and without sugar run together for some time in the neighbourhood of the zero-line (as was the case with salicylas natricus) and after this yield a positive sugar-effect. Considering these results, it was interesting to ascertain whether there are also other substances presenting charges of opposite signs in different concentrations. We fixed upon the acids of the’ fatty 420 acid series, that are soluble in water. The alcohols of fatty acids behave in a remarkably uniform way‘). With all of them the charge of the solutions, if there is any, is positive. With ethyl- and propyl- alcohol it begins in stronger concentrations than with butyl- and amyl-alcohol *), as was established before in the physiological labo- ratory when studying the charges of homologous series; all these charges are raised by sodium-chloride, the more so as the concen- tration of NaCl was taken higher. The fatty acids present quite a different electrical phenomenon. Leaving formic acid, which gives only a very low charge, out of consideration, we find that acetic acid always gives a positive charge in all the concentrations in which it gives any charge at all. Pro- pionie acid, butyric acid, valerianie acid, caproic acid on the con- trary have a negative sign in those weaker concentrations, in which they produce a charge, as will be seen in fig. 3. The lowest negative charge occurs with caproic acid ; with valerianic acid the curve is less deep, with butyric acid still less, while with propionic acid it is extremely level. The behaviour of the latter acid is very strange indeed; its negative bend keeps very close to the zero-line and proceeds over a considerable distance, the positive line rises suddenly. up to a maximum. So the curve of this acid also runs between those of acetic acid and butyric acid; its positive zone is much steeper than that of butyric acid *), whereas its nega- live zone is much less deep than that of butyric acid and accounts for the absence of a negative zOne with acetic acid. All this may be seen in Fig. 3, without further description. With none of these acids, except with acetic acid, could we find the descending portion of the positive line, because their solubility was less than that of acetic. acid. | Taking them all together these various acids present the whole negative and positive phase, with this restriction that practically part of it comes to nothing; of the positive phase with the higher terms of the group on account of too little solubility ; of the negative phase with the lower terms for reasons of which we are entirely ignorant. Il. In the second place*) we watched the influence on the charge 1) We must perhaps except methylaleohol on account of a probable complication. *) With ethyl- and propylaleohol in 0.01 n.; with butyl-alcohol in 0 002 n.; with amyl-alkohol in 0.0005 n. 3) This is not noticeable on the (logarithmic) curve. 4) These Proc. Vol. XX, p. 1272. 421 Fig. 3. Phases of negative and positive charges of terms II—VI of the fatty acid series. -40 z 4 a =i 22 =-% e +3 1. Acetic acid, 2. Propionic acid, 3. Butyric acid, 4. Valerianic acid, 5. Caproic acid. Division of abscissa and ordinate as in fig. 1. of salicylic acid of alkali-cations added in the form of neutral salts. Here it was proved that the inhibitory influence of these cations on the charge of salicylic acid was effected according to the lyotrope series | Li 5) s 5 5 yi a = am fF aoe Te eee ae AN ED x Ben = 5 + + Oo v 5 de} SE a. © S08 = ae | 5) u KS) 8 5 SS a a ë © 5 SS © AE Kn | - a oe Ss vt cus het ke’ oem ae a = ON © aos = be RO Reken Tee oS «a Ww eS a ge |e ee EES es. B 2 3 - 3 eN ana + 19 © = z © jaa) >) . e 2 5 rey 5 = = — = © 2 a © = —- + TN Oe a Ge OE Ste ae ce st Oe Sane Ql x) 99 | 0 le} vo = Lae =! . = 8 ™ =O N= = eae — Re Seep ag ee pe = = a = 9. > So fol a + a Ga is v a uN ~ = = bs o Ww a) dT ss — MD CY) se) [est - — So Ol _ we x D zet: s a Pia Ee = 5) 2 2 5 Ned is] = ro - 5 = = es ro) Ee = B ieee a =e Se A as = OC Ne 5) a Ss MSN) =) rn ae 5 oo WO 5 Sne a S Soo lors nl Pliiger’s Archiv 168. 351—71, 1917, p. 360, 361). True, this was the case with most of the solutions examined by Backmann, with some the charge was decreased under the influence of salts, nay even arrested. This could also be established a new by us for some substances. 425 A distinct lyotropia Na < K << Am was established; *) to a certain extent also Mg < Ca < Ba? The increase of charge through alkali- earths was, in lower concentrations, lower than that of alkalis. It was somewhat difficult to determine the charge because the solutions of the acetates of the alkali-earths of themselves impart a weak charge to a sereen placed in the nebula. Magnesium again oecupied the same peculiar place that it took up in the case of salicylic acid, TABLE III. Strengthening influence of Cations on the charge of acetic acid (0.1 normal). Factor increasing the deflection of the electroscope, when instead of 0.1 n; acetic acid being sprayed alone, 0.1 n. acetic acid is sprayed with addition of the acetate of the subjoined alkalis and alkali-earths. (NaCl as a control). Concentration De Acetas | Acetas | Acetas | Acetas . | Acetas | Acetas Cation | | Na | K Am. | Mg !) Ca ') | Ba | | | | | | | 0.01n | 4.4 B.S |) #855. (4.1 4,0: | 24.6 0.005 n. Poel Sal BE TTN TEN 0.0025 n. azo ads 2.3. 3.8 AB 2.36 | 2.35 0.002n. | 18 2.2 Si ter Pp ber 51.95 0.001 n. 1.55 | 1.4 Ck 2.204. bao at BAS | 21.46 0.0005 n. | 1.35°|: 1.3 135 | 18 4.08 | 1.29 | 1.20 0.00025 n.| 1.24 | 1.1 de: 0.0002 n. ‘Te 1505 |, 1.05 | 0.0001 n. | 1.05 | 1) Charge of these acetates: 0.01—0.02 n. charges 0.4—0.7; 0.002 n. charges 0—0.2 scaledivision. Charge of NaCl and of the alkali acetate = 0. The increase of the charge of the alkali-earths is not greater than that of the alkalis; in weak concentrations (0.002—0.005 n.) even smaller. it was almost equal to sodium, its action was decidedly weaker than that of potassium and ammonium, and it also here constituted the transition from alkalis to alkali-earths. The lyotropia of the anions appeared to be more intense and more general than that of l) In another set of experiments with alkali-chlorids the series Li < Na, K < Rb, < Ús < Am was obtained. 426 501 | | | | | u_1000°0 SII sol 82° I ge” 1 oot | ARE | | cel ‘u 2000°0 cl el be'l Gpr | Srl 881 | nares | cel i eet SRS se .OPSI ce" u §2000'0 Et 81 ce" 26° 1 eg’ 1 [Set | 61 | KT | gren Al 7 Hel | bo) a 091 | ‘u g000°0 ec Sa. ic SS I O8 LE IL? | Eee Ga | Bes aL | 08E | 50 081 | ‘U 100°0 O's Ce en 00 } oot se €6°C | LG'2 19’ | LEE | OEZ | go"? | Oss ors | u 200°0 e's cies |= O8 oee | 89E | 9s | Loe | Be LL*S | OL’? | 00'€ | 00'S OL'Z | ‘4 54000 cL’Y bP | are GK pk =ST-S | G Sa.) 1. <5 A 6€ °F See €9°€ | c6°¢ | soe | GSE | u ¢00°0 KO Sak | Les ev 'g 68'G | age 2) | ‘u 10°0 EN EN ajeydins, azen | ydsoyd ayesysey = | | uolue ‘ache’ pine | IBN | ae Ee | 8 Ken ‘OH IN| JAM | IM | DI | ONM | SNIA ese nuastion ‘Sips WAIPOS °q ‘SJDS WNISSDJOJ 'D ‘pakeads st spioe paulofqns ay} Jo s}jes (EN pue) 4 a4} Jo SUONIPPE AUF UHM pioe alzooe ‘u ['Q ‘auoye pokeids Suloq proe dija0e ‘u ['Q JO peagsur vaym ‘adorsorjoajs a4} JO UOTPaYep AU} SUISBAIOUL JOEF “(u [°0) plow a0" fo aSuvys ay} uo suOIUE fo souanpyful Suluayisuags “AL ATAVL 427 the cations (Table 4). It is evident that chloride takes up the lowest place in the series of anions, citrate stands at the top: Cle CNS, NO, < Br,J.< C,H,0, Tarter. Phosph. << Sulph. < Citr. This series of anions differs only very little from that of salicylic acid. With acetic acid as well as with salicylic acid the differences between the extreme terms of the lyotrope series appear to be larger with the anions than with the cations. Sb) Neha RY. 1. Salicylic acid, if not in a completely saturated solution, gives a negative charge, that is weakened by cations and anions in a lyotrope-series, in the direction from Li to Cs and from chloride to sulphate. 2. Acetic acid, when dissolved in water in a sufficient concen- tration, gives « positive charge increased by cations and anions, for the greater part in a lyotrope-series in the direction from Li to Cs and from Chloride to Sulphate. 3. Acetic acid weakens, resp. arrests, the negative charge of salicylic acid. In higher concentrations with a positive electrical phase if prevails, as an acid, over salicylic acid sprayed in weak solutions, which gives a negative charge. As appeared already sub. 1, acetates also have a weakening effect on the charge of salicylic acid, whereas, of themselves — at all events the alkali-acetates (the acetates of the other metals give a very slight charge) — they produce no charge. Mixtures of salicylic acid showed the electrical property of acetic acid, so long as the latter acid predominates. If the concentration of acetic acid in the salicylic-acid acetic acid mixture decreases to such an extent that such an acetic-acid solution gives of itself hardly any charge at all, the charge of the mixture will at one moment be —0; in still lower concentrations of acetic-acid the charge appears again, this time of a negative sign. There is a moment when the favourable action of the cations on the acetic acid is arrested by the inhibitory effect of these cations on the salicylic acid. Mathematics. — “Observations on the development of a function in a series of factorials”’. 1. By Dr. H. B. A. BockwinkEL. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of June 29, 1918). 1. In his book “Theorie der Gammafunktion” N. NierseN gives the necessary and sufficient conditions for the development of a function in a series of factorials. According to him the following proposition holds: The necessary and sufficient condition that a function 2e) may be developed in a series of factorials, is that (rv) may be represented as a definite integral of the form 1 M= f yy At dt bal Bedels ve uren es TEAN 0 where ¢(t) is a function with the following properties: 1. g(t) is regular within a circle, centre the origin and with a radius no smaller than unity, so that it may be expanded in a power-series : GG) =a at os, aa ee oe ot Oe 2. If yt) is the first of the derivatives of p(t) which is infinite for t= 1, then there is a real number 4 such that lim giP\(t) — (1—t)—P-4P)) 2 ww we 8) t==1 1) By this notation we mean that lim oe vp) (t) = 0 or oo, according tl as ò>0 or 3 <0. We may express this by saying that -W)(t) is, for ¢=1, equivalent to (1—t) CTP), Since (pl) is supposed infinite, we have A+p>0. Further it follows from a well-known proposition (Dini, Grundlagen für eine Theorie der Funktionen einer reellen Grösse, p. 104) that, if -(4) is itself finite for ¢=1, Le. if pS 1, we must have A+p <1. For, if we had A+p>1, then ok) would, according to that proposition, also be infinite, being equivalent to (lt) “T?—), which is contrary to the hypothesis that +\/(¢) should be the first of the derivatives infinite for t—=1. We have therefore OZAnkls. for Ste on Bee eee cee from which it follows that A is never positive in this case. If, on the other hand, g(t) is itself infinite for t—= 1, it is, according to (3), equivalent to (l—t) 4, and then A is not negative. 429 3. There exists a real number 2’ with the property that, corre- sponding to any assigned number e, however small, an integer MN may be chosen, such that we have, wniformly in an interval 0 << J, which, for the rest, we may think as small as we please PO (I — yet ———S f n VAD eet meee reat) | > =| 7 e es according as R(x) >>’ or R(x) < 4’. (Re) means the real part of 2). The series of factorials corresponding to an integral of the form (1) is a, a, L 2/ a, re ‘ di ni An : x vlet) a(w+1) (w+4-2) ae TPL (aon) ee) where a,,@,,... are the coefficients of the power-series (2). This series converges, according to NierseN, for the values of w satisfying the two conditions VE REEN CAV a oii cise eames Mod ea dD and if at least one of the two characteristics 2 and 2’ is not negative, the series (6) will represent the integral (1) for the values of x mentioned *). From the first of the inequalities (5), applied for ¢= 0, and from the consideration that p'"(0) an == n! OUA Tt te ene, AO it may at once be derived that the series (6) converges absolutely for Re) <1’ H1. In connection with (7) NierseN therefore infers that the number 2 is at most equal to 2’ + J. The thing to be remarked in the statement of NiersuN is however 1) If A and A’ are both negative, so that v(t) is finite for ¢= 1, the integral (1) has in general only a meaning for R(x)> 0, and the development in question is valuable for these values of x only. But then we may consider the integral 1 > g (pt) (1 — Eet dt sof FO ) ea ee u(e+1)...(e¢+p-1) which has a meaning for R(x) >A. To this integral a certain remainder of the series (6), viz. Miah tis ay! von ARNE (8) let). (ap) 2(e+1)...@@tp tl) °° corresponds and the integral would then be equal to this remainder for the values of x determined by (7). 28 Proceedings Royal Acad. Amsterdam: Vol. XXI. 430 that, if A <2’ +1, there is a certain domain on the left of R(x) = 4’ +1, where the series (6) converges conditionally. If A Lae this convergence takes place in a strip of the plane determined by A< R(e) 4’ + 1 and A(z) >a-+1. The example given by NierseN of the develop- ment of an integral such as (1) in a conditionally converging series is exact, and it is not difficult, as it will be seen, to add others to it. But in reflecting on the subject we have finished to doubt of the general validity of the special theorem of NiELSEN. In any case, for the strong proof of it investigations of a farther reaching extent would, in our opinion, be necessary. 2. The mode in which NrerseN arrives at the development (6) consists in integrating by parts the integral (1): this gives immediately 1 {vo (ltr dt = a 4- El Je. + Em a + R‚(10) E au(v+1) au(v¢+1)..(e¢+n—1) 0 -where pr 0) (1 git) (ltr ie „fr To From the condition 2° mean derives that the first of the ine- qualities of (5) will be valid for an interval d 4. Here the quantity Jd may be conceived arbitrarily small; it must of course be understood that the choice of the integral number N is influenced by it, and that N will increase indefinitely with 1/0. ') Rendic. d. R. Ace. d. Lincei (1908, 2e Sem.). 431 For this reason NirrseN has to add the new condition under 3° expressing that the inequality (5) is also valid for an interval -O 2’ +1. We shall even restrict the latter condition a little: not assume that the inequality occurring in it is satisfied for a certain small interval (0,d) of ¢, but only for the endpoint t= 0. Since the values of the different derivatives of g(t) in =O depend on the coefficients of the power-series of that funetion by means of the formula aes Gy == af (13) this supposition amounts to the following as to those coefficients themselves : 28* 432 3a°. The coefficients a, of the power-series for p(t) satisfy the condition Tis, gc Oe n= 0 oo for d< 0 which we shall denote shortly by writing haaien (aoe a NRE (14’) n= 09 and by saying that the upper limit of a, for 2 =o is equivalent to n”. We write intentionally the sign of upper limit, because this will do for our purpose; it is not necessary to suppose that the coefficients a, have a “croissance régulière”. From the only supposition 8a° it may already be derived that the part of the earlier condition 3° corresponding to the first ot the inequalities (5) is wntformly satisfied in the whole interval (0,1) of the variable ¢ provided we add a factor (l—t). To prove this we compare, for large values of n, the nt derivative of (4) F(n +2 | i 1 pet) = in Flags a Grt Fra ek te nae Di L .. (15) é 8! with the nt derivative of the function F (24-641) LO (LP HAL that is ' (A! 4 dn 1) (n)(t) — = rs (‘) (1 LLL FA -Ed jl halves | TUe ip eel) 0 8: Here d is a certain positive number. However small we may choose this, there is always, on account of our condition 3a° corre- sponding to any arbitrarily small number ¢ an integer MN such that for all values of s5 0 | T'(A'+d+n+s +1) Qnts| SE P'(n+s+1) Thus we have uniformly in the interval O< t<1 (1 as t)’+e+n+1 gpr(t) (A 4-dA-n 41) here p(t) means the natural majorant of p(t) determined by g(t) = la,| + |a,lt + + lanl m+... The same inequality a fortiori holds for p(t) itself, and since J can be taken arbitrarily small, it follows that the integral (11), Zetor en Met sons See 433 denoting the rest of the series of factorials in question, has zero as a limit for n =o, if R(w)>4’+1, and thus that for these values of « development of the integral (1) in such a series is possible. That this series for these values of w converges absolutely, may, as we had already occasion to remark, as well immediately be derived from 3a’. 4. We make the following remarks: 1st. The proposition proved just now may be compared with a result due to Crsarò *), according to which it follows from the condition 3a° that the inequality (17), for lin t= 1, is already valid from and after the value n = 0. 2nd, In our result, and in that of Crsarò, is included that the number 2, introduced by NierseN, is for the natural majorant of git) exactly equal to 247+ 17), whereas for the function itself it is in any case not greater; thus we have Whe te Pe andeeeh er ow ats a was. stacey AON if 4 is the number in question for p(t). This result,.derived by NIELSEN from his theorem, can therefore, if our opinion with regard to the inexactness of NieLsEeN’s proof is right, no more be regarded as deduced by him. 3rd, We may also easily prove that inversally every series of factorials such as (6), if it converges at all, is equal to an integral of the form (1), where p (¢) satisfies the conditions under 1° and 3a’. In fact, if such a series converges for a certain value a + 78 of xe, the limit of the terms must be zero for that value. Now we have ne 1 lim en =de). (eden) ne from which it follows that the coefficients a, of the series of factorials satisfy the condition lim an — 7, n= 00 where 2’ is a certain real number which is at most equal to «, but may be equal to — oo. If now we form with the coefficients a, a function g(t) as in (2), this function has the properties expressed in the conditions 1° and 8a’, and, as we proved in the foregoing paragraph, the given series of factorials is equal to the integral (1) for Riv) > 2’ +1; at least, if 2’ + 1 > 0, for if not, we should say 1) Acc. d. Scienze fisiche e matematiche di Napoli, 1893. See also: Boren» Legons sur les séries à termes positifs. 3) At least when dn has a “croissance régulière”; otherwise it may be smaller than A/—+ 1, but no smaller than A/. 434 that a certain remainder of the given series is equal to an integral of the form (8) in the footnote of p. 429. 4. If the coefficients a, in the power-series for q(t) are real, we can assign one detinite case in which the series converges conditionally for Ra) > 4’, viz. if all derivatives of an order higher than some definite number have the property of conserving the same sign throughout the interval O<¢< 1, and this in such a way that always two immediately succeeding derivatives are of vpposite signs. This result can be deduced from the equality n! an ele +1)... (en) where A, has the meaning given by (11). First let « be equal to a real number «. Then F, and — Pe, have the same sign throughout the interval of ¢, so that either of them is smaller, in absolute value, than the series-term in the right-hand member of (19). If therefore the latter has zero for its limit, then also #,, and this is RR = (19) the case for a > 4%’, since lim a, — n”. If « is complex —a- 28 and a >2’, we have 1 =| OAN AN als | is 5 ( ) | at fae a | a(a+1)...(a+n—1) | g@™(t) (1—t)tr—-1 | @+ip)...(a+ip+n—l) | a(a+1)...(a+n—1) — 0 The latter integral is equal to A, for «=a and therefore, as we showed, zero for n= os; the factor by which this integral has to be multiplied evidently has a modulus smaller than unity; thus FR, approaches to zero as 7 increases indefinitely. As an illustration we take the example of NIELSEN nl ey Oe es sags a(x) = En dt, 0 1 ! POE NE NR This function p(t) satisfies the special condition mentioned in the present remark, and on account of this circumstance we may deduce the possibility of developing the function a(z) into a series of factorials for Ru) > 2’, this series converging but conditionally if R(x) < 2’ +1. Other examples to illustrate his theorem, especially such that do not where 435 satisfy the special condition mentioned here, are not given by Niersen, but may easily be imagined. 5. Finally we want to make a remark on another manner to derive the development (6) from the series (2); a way which has been followed both by Pincnerue and by Nistsen. We may write p(t) (L—t)*¥-1 = a, (1—?t)*—-! + a, (1 —t)?—1 +... + ant (1 HE 4+... , and thus 1 1 [ro (1 — ¢)* -' dt = [tac —tr-t4..,4a4,t(1 —t)r—l +... ] dt. 0 0 If this series is integrated term by term between the limits 0 and 1, we obtain the series of factorials required. In case the latter con- verges, NieLsEN (p. 239, Handbuch) derives from it that this series is equal to the integral in the left-hand member of the preceding equation; his reasoning is based upon a certain proposition of Dini (Grundlagen, p. 523). If this reasoning were right, then, besides the special case treated of above, we should have a more general one, in which the integral (1) may be developed into a conditionally con- verging series of factorials: viz. always when the series of factorials to be derived from that integral, whether by means of integration by parts or in the manner described just now, is a converging one. _ But, in our opinion, the proposition of Dint in question has been ‘applied in a wrong manner by Niersen. The fact is that, in applying that proposition, we should first integrate the funetion in question over the interval O 4’ +1, this is realized, because the series (20) then converges at the same time wnzformly in the closed interval O0 indique une som- ab mation étendue aux 10 combinaisons avec répétition des nombres 1, 2, 3, 4 pris 2 a 2. En dérivant / par rapport aux Jab» et en permutant les indices, on obtient le résultat suivant: ge (gre gi — glt gh) dl Eab hl H 4 = k(—g)'b (1 Ad me Ss =|" | {gba (gr gl = gl” gia) ; Adab,u 2 hel a i | en, g” (get gi an gi gi) En dérivant de même / par rapport aux gas,,i et en réduisant les termes semblables, on obtient: dl ey -==k(—g)'!2 (1 mn a¢ = 5) (2giv gro—agar-gt —gbgt), dgab.ni Substituons ces expressions dans (1), et utilisons la formule: d(—g)''s da; =} (gh EE g%Pgaai = — kg) FS J Guag**. a f ar fg Après quelques réductions obtenues en permutant les indices, on a enfin: bi el + CMS TTL > [Lgthhgud-gadshghe +. gyi, ngm@geg" |gab,r— | Clee th er (3) aa hd (—g)} ae [g#! g26— g2g® | gars | a 1 439 Cette expression de 4, se simplifie encore en vertu de |’équation complémentaire /=— 0. Il. Valeur explicite du tenseur by + Tip. Rappelons la valeur du tenseur électromagnetique *): TE (erp dae ee dy Mais en vertu de Pidentité): (lt) J 1 Mg)" BB gl (ih, 12) — Mp) Cori expression (4) peut s’écrire: Tip = — Fpl + hl—g)h EEE grighl (ih,lay. . . . (5) ND Rappelons aussi que la parenthese a 4 indices de CHRISTOFFEL a la valeur suivante: h Ut AAL (th, 1A) = Hgn nt —Jhit— Githa + Ghia) + EEG? = . a 8 a B a ej Retournons maintenant a la valeur de f, et utilisons la formule: göh—=— EEgeng gt... Af ae Pe AEB) g T Rapprochons la valeur de ¢,, ainsi obtenue (3,6), et la valeur de T„ (5): tous les termes de ti, renfermant des dérivées secondes des potentiels gravifiques se retrouvent, changés de signe, dans le tenseur électromagneétique TT. Apres quelques réductions provenant de permutations d’indices, on trouve enfin: | Jona” gh! — gig] Nee S294: Ohe 9 x (gih q3l —gqlk gi inte ye eae ge) en hl i 42g (at gx — gihg2t) | Se 29°} (gel gf — g% 98!) Hl. La force géneralisce F, est nulle. La force généralisée MH, (A= 41, 2, 3, 4) satisfait aux relations suivantes °): d(ti + Tryp Ee en de, Ao 3i 4. 1) Tu. De Donper. Archives du Musée Teyler. Sér. 2. T. Il. 1917 (voir spéc. pages 94 et 99). *) TH. De Donner. Voir ma note citée ci-dessus; équations (10). Voir aussi mon mémoire, Archives Teyler, Haarlem 1917; équations (347). 440 Substituons dans (8) les valeurs trouvées (7) pour le tenseur t. + T;,. Apres dérivation et permutations (indices, on voit que tous les termes se détruisent deux a deux; on aura donc: Py =O) sa. tee A (0) Ce résultat est indépendant de expression choisie pour le tenseur gravifigue t,. C'est ce que nous allons démontrer au moyen de Videntité de HuBert *), qui peut s’écrire avec nos notations: ay d UO bED(l Hem) gt) L=— BE 5,7 (A Hs) genes Ue ENGEN , py my avy En vertu du principe généralisé de Haminron, les équations diffe- rentielles gravifiques sont *): OG (LO ou ee Or, on a’): wy LL Ev ) Lbs <= (3 — 1) 2 gk Bye VON Sees Wy Ey, NS Ol yg Te . V me (: 9 ) na fae . . . . (13) et inversement : To Dl eg Tse ee IJ. En vertu de (11), le premier membre de l'identité de Hu.Berr (10) peut s’écrire: dL dee )M D'autre part, en vertu de (14), le second membre de cette iden- tité (10) peut s'écrire: On a donc: bh D. HrrBerr. Nachrichten Königl. Gesellsch. d. Wiss. Göttingen. Math. phys. Klasse, Heft 3. 1915. (Berlin 1916). 2?) TH. De Donner. Archives Teyler, Haarlem 1917. (Voir équation 339). 5) Voir équation (353) de mon mémoire, Archives Teyler. Or, en vertu de (346) '): (Fet Ko En d'où Gah, ROTE A Remarquons que tout ce qui précede peut Être généralisé immé- diatement en remplacant / par une fonction covariante plus générale, par exemple: /—10(—g)'1, où y est une fonction de w,, 2, 2, #,; on obtiendrait ainsi nos équations généralisées °) du champ gravifique renfermant des masses. IV. Autres tenseurs gravifigues. Les seize fonctions 4, dont l'ensemble constitue le tenseur gravi- fique ne devant, jusqu’a présent, satisfaire qu’aux quatre equations aux dérivées partielles °): dt) « = se K), „don hal, 25-8; 4, il en résulte qu’il existe une infinité de tenseurs gravifiques diffé- rents. Le développement ultérieur de la théorie de la gravitation montrera probablement que le tenseur gravifique doit être déterminé d'une manière wrivoqgue par des conditions aux limites et des con- ditions initiales. En se reportant aux relations (841 a 345) de mon mémoire *) (Archives TrYLer), on verra aisément que les seize fonctions suivantes : [ (Hed) = | Ed BELT On =eul— Kn ix ien ae Me vagen 2 4 de; (15) ly ; bki a5 2 2 (Treat) dgb 9 ; déterminent un tenseur gravifique. 1) Voir équation (346) de mon mémoire, Archives Teyler. 2) Voir la dernière page de mon mémoire, Archives Teyler. 3) Voir équation (344) de mon mémoire, Archives Teyler. 4) Voir aussi notations (348 à 352) de ce mémoire. 442 Grâce A la théorie des invariants diffêrentiels, ou par un calcul direct, on trouvera que: git (92 pls 2g°%t + 97%) ; ee a — 9g”! (92 — "sä ( ) On remarquera que ces deux tenseurs gravifiques renferment les q en mémes dérivées secondes des potentiels gravifiques. On aura en outre: d(?\—t, „ Oty) B da y 1 1 t—t,, + Hg) hk DEE gee) | Zot == |B En vertu de nos équations (8) et (9), on pourra introduire le tensew gravifique —T7,,; M. Lorentz") a rencontré ce tenseur gravifique au cours de ses recherches. Quand on adopte le tenseur gravifique de M. Lorentz, le tenseur ty, + Th, est identiquement nul. Plus récemment, M. Ernstein*) a trouvé un tenseur gravifique qui ne renferme aucune dérivée seconde des potentiels gravifiques. Nous allons indiquer une méthode nouvelle pour obtenir ce tenseur gravifique (corrigé). L'invariant de courbure totale de Riemann peut s’écrire: ESS Zed, or) gf o* NT ao} d d oO == hI > > aN af = = eee ay dx. Sa CMG. Cet | Brela 0 a ia oO 5 ll en résulte que L=kC(—g)* vi sate — VS d — Ns eo af i g*P [* [=4k UEEE gh) | 9% 9) | Pel a> OTE a pat dag | 0 Vite où nous avons posé: *) d = Iho lt Yale [Gpeg] deg — ode gf jer (ep st ( TOR ee pve : . ab » On vérifiera, par un caleul direct, que le lagrangien d d'une ao 0) (18) : OT lag28 1) H. A. Lorentz. Voir la dernière page du mémoire cité. (Verslag Amsterdam 1916). 2) A. EiNsreiN. Sitzungsberichte Akad. der Wissenschaften Berlin (Séance du 26 octobre 1916). 8) Les termes qui figurent dans la première ligne du second membre de (18) ont été omis par M. EINSTEIN. 443 dérivée partielle par rapport à une des variables x,, x, %,, %,, d'une fonction quelconque de ces variables et des potentiels gravifiques est identiquement nul. -Par conséquent (17, 18): b BN: tn ER En heet e450.) C419) Posons maintenant *) : À [* 2) uth ab,) 20 ke 3) me dn ine We - e 5 ; 5 ( ) On aura’): dt B eg Me ee as) 7 da, ah (0 )s ( ) ou, en vertu de (19): dt, An dit eN EN ab). 99 B = 2() 1) ie: a a hate ea ab ou, a cause de (339) ainsi que de (343. 344) (voir mon mémoire, Archives TEYLER): dt i" x Kine ate eee ee p Ady On aura encore (voir fin du paragraphe III): Typ+t," PUES de eee me ali eer A pe de, On pourrait construire aussi le tenseur gravifique: al* Oh, =p *— 2 OREN Tabee Megha: aay ah W9ab p et un calcul simple montrerait que ty, = ¢,’”. IV. Champ gravifique d EINSTEIN —-SCHWARZSCHILD. On sait que les potentiels gravifiques du champ d'EINSTEIN-— SCHWARZSCHILD *) peuvent s’écrire: In FT ATS (Ra)! de == R (las Is = — R? (1—<,’) ° . . . (26) 9, = R-! (R—a) Dp = 0 ou Ayu — ie 2, 3, 4, et = u 1) Comme a |’équation (341) de mon mémoire, Archives Teyler. 2) Comme a |’équation (342) de mon mémoire, Archives Teyler. 3) K. ScHARZSCHILD. Sitzungsberichte Akademie d. Wissenschaften Berlin (Séance du 3 février 1916). Voir spécialement pages 191 et 194. 444 On a posé: R= (Ge. en On Rappelons enfin que « représente une constante. En substituant les valeurs (26) et (27) dans (3), on obtient, apres de nombreuses réductions, le résultat suivant: tous les t,, (A,u—=1, 2,3, 4) sont nuls, sauf t,, qui vaut — Rh. Les calculs se trouvent grandement simplifiés si l’on remarque que g se réduit & — 1 dans le champ considéré. En dérivant ce déterminant par rapport a 2; et 2), on obtient la relation : Egt gd Eg gaa» 2 + » » (28) a b a b Grâce A (28), le tenseur gravifique (3) pourra s’écrire: tye = b Bilgi ged + 397 gei gar, + 9 gl gary - (29) ih lie Pour s’assurer si le tenseur 45, d’Einstein est différent du tenseur ti,, il suffira de calculer /’,,, par exemple, relatif au champ d’ EinsTein- SCHWARZSCHILD: tous calculs faits, on trouve (25) que t’,, = *#—kR-®. Or ¢,, est nul; done, ces deux tenseurs sont différents. VI. Valeur explicite de = by: 4 En vertu de (3) et (6), on obtient, en permutant les indices’): k . k . cor (—9)'/2 3 gqx,i Gha,k(— 29%? 9% gk? + g*Bgh gat + gekgai geh). Cette expression se simplifie considérablement si l'on remarque que invariant de courbure C peut s’écrire: C= k= grrgi (g* 9% — gl 91") + Agf ga gkh pe 3978 gkt gah + } X Jqx,i Jha k — 2gzk gt gah at 2g% gêagih — 2qt8 gi” gi: A (30) Si l'on se rappelle la signification de / (voir $ I), on trouvera immédiatement, grâce à (30), que: ede ren + 2gke ght gs ete 272% ght g8k k Bt) = 3l + = (go) ls Egg Ghat 1) Dans les formules qui suivent, le signe X représente des sommes séparées portant respectivement sur les valeurs 1, 2, 3, 4 de tous les indices qui suivent. 445 En vertu de l’équation complémentaire /=0, on voit (30) que Et est une forme quadratique des dérivées premieres seules. d VII. Covariance du tenseur gravisique by. Effectuons un changement que/conque des variables w,, x,, x,, w,, et représentons par 2’; (t= 1, 2, 3, 4) les nouvelles variables. Le tenseur gravifique prendra une nouvelle valeur #5, (4, u = 1, 2, 3, 4) fournie par la relation (1) où tontes les lettres auront été, au préalable, affectées d'un accent. Rappelons que '): OG. v= a.) O(a’, si Grâce a cette relation (32), il sera aisé de comparer t’;, a tu; de cette comparaison, il résulte que pour tout changement linéaire des variables x,, z,, #,, £,, On aura: zi Un cis ia) Or, Òz', Un ZE — En Ò(z', ee A 5 Ys 02’) Our bit aha, 1008) Lars , autrement dit, pour tout changement linéaire des variables «,‚z,‚r,‚x,, le tenseur gravifique ¢,,, est cogrédient au tenseur électromagnétique *) 7; Il n'en est plus de même pour un changement quelconque de vari- ables*). Un fait analogue se présente pour les forces généralisées *) F, et Ks: la force généralisée gravifique K, n'est cogrédiente a la force généralisée électromagnétique /', que pour les changements linéaires des variables 2,, z,, 23, ©. Le 30 avril 1918. 1) Voir équation (364) de mon mémoire, Archives Teyrer. 2) Voir l’equation (319) de mon mémoire, Archives Tevyrer. 3) M. Lorentz, avait déjà fait remarquer que tm n'est pas cogrédient à Tiu dans le cas d'un changement quelconque de variables (Verslag Amsterdam, 24 Juni 1916). 4) Voir les équations (321) et (323) de mon mémoire, Archives Tevyrer. 29 Proceedings Royal Acad. Amsterdam. Vol. XXI. Mathematics. — “On the evaluation of §(2n-+1)’. By Prof. J. C. Krurver. (Communicated in the meeting of September 28, 1918). By means of a characteristic and very general method Markorr ') oo oo transformed the very slowly convergent series Zn? and 2n-? into : 1 1 other series, that converge more rapidly, and J. G. van DER Corput’) described a special method of transformation applicable to the series o Xn @htD for larger values of A. 1 propose to deal anew with the 1 transformation of these series and to add a few results to those previously obtained. In order to appreciate the increase of convergence resulting from the transformation, | will consider D’ALEMBERT’s ratio for the transformed expansion, which I will call its index. For the series given by Markorr the index is yy, and I will show that a lower index can be attained. In the first place | base my deductions on the properties of the function n= 0 2 ous) = Yo —, pmm n where & denotes a positive integer. In order to uniformise pz(z), it is convenient to regard the right line (+1, + ©) as a cut in the complex z-plane, and with this convention we may enunciate the following properties of p‚(z) in substance deduced by ABEL: Pp, (2) + Y, (l—2) = — log z log (1—z) + $ (2), 1 p‚ (2) + gp, (=)=- 4 log? z + nilogz 4 26(2) | z k ei! ER y, (2) — gp, (=) = 4 log’ z — log z log (1—z) + § (2) | z Ps (z) 4 fs (2) — 4 Ps (z°) | Obviously in these formulae we have to take real values for log z and for log (1—z), when z itself is real and between O and 1. 1) Comptes rendus, t. 109, p. 934. 4) These Proc. XIX, p. 489. 447 As in general we have d dlogu zz Pri) = Pe ekeren. it is possible to extend partially the above relations to the function ¢,(z), and indeed it follows that 1 ~, (2) — Ps (=) = — flog’ z + } wmilog’ z + 26 (2) log z, Ls (; tnt + p‚ (2) = Hlog' 2 ad } log? z log (l—e) a), (2) + $(2) loge + (3), fs (z) tE Ls (—z) Te 2e Ps (2*). dl For £>3 a linear relation between ;(z), #,(1—z) and pk (=) z no longer exists, there only remains, besides the relation ] PE) Pe mnd = og Pe (z°) an equation of the form 1 log z pr (2) + IE gx (=) = — (2 mi) zel 5: ), 2 2ai where gz(w) denotes the differential coefficient of BrerNourrr’s polynomial /;(2). Another expansion valid for all positive and integer values of 4 is the ne ge (4) = — yk! by 1 Goal ttt Au + gy — loge) + ores BEN Real 4D n=0 Here the right line (0,—o) must be considered as a barrier in the complex y-plane and log y is real for positive values of y. The accent in ZX’ denotes that the term with the index n= k—1 is to be excluded. As for the numerical values that the ¢-function takes for the value zero and for negative integer values of the argument, we have S(O) — 4 , $(—2A)=0 , 5(1—2n) = (1) Therefore after a certain stage the coefficients in the expansion at the right hand side of (3) are expressible by BerNOULLI'S numbers and the radius of convergence of the expansion is evidently 27, 29* 448 as we might expect since equation (3) may be established by integrating repeatedly both sides of the equation n= 00 n= GO = aay 1) B = ae ee log: ba mas 0 2 pie ogy + dy + Zi ‚(0 cae? Dsl a 5 and substituting y= and also y—a in (8), in order to obtain 2 expansions of p‚(a), p‚(4°) and @,(a’), we infer that 5) = El apr v B & eet liso Ye a)? tle hy EBD Ts ea Ss (1B, ate 2) =3 nea 5 (2) je { Dae On! rary a mass (—1)"-! Be a 2n+2 (3) =$ out ae —. =). Pome! 2n 2n4-2 The index of the first series representing &(2) the other two series it is less than +. is about z+, for Again it is possible to convert the series into definite integrals. From the power-series in «a we find 450 i dE = i say =z log(f + VIE), be} 0 “4 dé ote A (3) = 10 f° Flog (e+ VIFF). 9 If we wish to expand 54) for k>3, we must proceed in a different manner. We shall use the general identity | + Perl) = De ps(er) . (4) 152) + ers (20) + 175 (20°) +. Ani where p is an integer and 0 =e? Denoting the series n= o MNT x COS S 3 —= ———— n=1 ns by Un, we get by substituting in (4) ni DB en parti 1 2U, + ie Te gee 1 Ot RE ijn and hence n= 0 nt 1 1 NX COS Tl ns From equation (3) we deduce by taking k=—= 2h+1,y=w Ss cosnv_(lfek/T 1. 1 Ue eae pa Ee Ves Wale ee gS i EEN se yt ole 8 1 a! ane Lm) gn) = tat ’ we may conclude to (lito? it 1 | 1 \ ( as ee ae Sille hence writing Aan +10 (2h + I= on 5 =h—1 = 8 (— 1) i— v2 mg Bn yen+2h de s Ee ~§(2h + 1- 2n) + (—1)*— pay ae SEL Dr n=! JT where v now stands for 5 By means of this equation ¢(24 +1) is expressed in terms of ay a en ht and taking successively 4=1,2,3..., we get 3, §(5), §(7),... expressed by a linear BERLIN of expansions ite ae i which is ge '). A slight transformation of (6) is possible. By using and by effecting some other reductions, it may be shown that (6) can take the form Aan —1 ì hea gees =(1— h(Q2h— + Jen = tact n=h—1 rx (A —n) (2h + 2n — 1) (— Int on a 2e h (2h — 1) RT G(2h + 1 2n) + (— Ito (Ahl <= ‘Bn Qn + 4h — 1)v?” ah (h—1) ) 21 =: pare a ee + )v ‘ (=4) If we put h=1 and A=2, it will be seen that 3 n= B IND ee as 2 3)y2n+2 $ (3) Sn Pi yv 137 oa se Wi ieee Sey a ee a I will now proceed to show that for each of the quantities S(2h + 1) there exists a linear combination of expansions with an index less than 3. For this purpose [ use se the identity (4) and writing 7 x COS ww — — 15 = 1) Similar results were deduced by Mr. vAN DER CorPur in the paper quoted. However, in the fundamental expansion of the quantity J(n,a) on p. 1464 by a slight inadvertence the factor 22k has been omitted in the general term, hence in all the subsequent expansions the general term should be multiplied by 22% and the index of the series on p. 1470 is #- and not 47 452 I make in (4) the substitutions ps 2t, ee" end 2 p=3 , zzetiw, p= St These substitutions give the four equations 1 0, + Ugs Us a pe gi der U, 5 i U, Kid Oe? U, ate 2U, Ea BU ee man Un. and eliminating U,, U,,, U,, we get eee eee te tik ln Bark geiko ee si) gee ae 1 1 Now, ‘taking s—= 2h +1, we may expand U,, U,, U,, U, by applying equation (5) and we will get U7, = ¢(24-+ 1) expressed in terms of the quantities C(3), &(5),..,6(2h—1) and of four power- series in w, the indices of which are respectively to, sts, Tho zis: Since the formulae become somewhat complicated, I will consider only the simplest case h 1. Then we have 336 4(3) = — 900 U, + 225 U, + 800 U, + 900 U, and hence n= 689 5(3) = 45010(36-24 | Biord-sior se Ln DS w a Oow - 0 ¥ 0 ge gu g g3) 2x asset an n=l! ) AE eres E(6—1) #) wo D die Diseriminante von K ist, d die von & und d, die relativ-Discriminante. Das Element CO = (2-2), ge ane, nee Am zie Weil D eine Potenz von / ist, *) kann die Relativ-Discriminante dr nur teilbar sein durch { und die Relativ-Differente D, nur durch £; also auch @@) nur durch £¢. Aus der Form des Elementes. €” folgt, 1) H: Satz 39. 2) H. Satz 38. 5) H. pag. 205. 4) H. Satz 121. 461 dasz alle ihre Zahlen teilbar sind durch £ und dasz sie nicht alle teilbar sind durch &?, also OS ae DD, = eh) andes el n(d,-) — [)b-1 oe Pe the 2) — dele —1 Hieraus folgt der Satz. $3. Die Klassenanzahl des Teilkörpers k. Satz 6. Wenn die Zahl 6 gerade ist, und der Teilkörper also reell ist, se hat man für die Klassenanzahl des Teilkörpers den Ausdruck : 4 hs R wo | log €, WOE ee. a, log &—1 | (Ie Des) | loge, LOG Ee en log & = ( | | log&—1 loge... log &2-—3| e= ali! gte 7 (b—1)c : a7 On Vial tis EE | i ai EE ZEE Gan Se SL R ist der Regulator. Wenn der relative Grad 6 ungerade ist, wenn also der Teilkörper imaginär ist, ist Qrutt re e © (ret ripe +... J ride) u t=1 ag Dise [Joke Diva wo das Product über alle ungerade Werte u < c laüft. | log &, loge, log Ey. | loge, loge, -..….loge'he A = (—1)'sla 2X04) | | | log €y,c—1 log &,¢ ...-. log &~3 : =|/— rae EME : (i — _ 79 ey 1—_Z Ee) Sa (le iv 30 Proceedings Royal Acad. Amsterdam. Vol. XXI. 462 log n, log n, lognyse—1 1 1 1 a ln lo, = log njh -2) log nh 2) log nj” Mi 1, ---- ist ein System fundamentaler Einheiten und ® sind ihre Conjugierten Beweis: Um diesen Satz zu beweisen, benützen wir den für alle Körper giltigen Ausdruck bne Dal! ye 1) (1) ESET ODA Ni SSS . . ° e id! Nt n(})s und wir beweisen zuerst den Hülfssatz: 1 a Pp bu er u(i— =e ta pret @ Das Product linker Hand ist zu erstrecken über alle verschiedenen Primideale, welche in der Primzahl p enthalten sind; das Product. rechter Hand iiber alle Zahlen der Reihe Of ise eae nh. Beweis: Das Symbol | *) wird definirt durch die Gleichung Qmp't Pli l= ( 1) wo p’ der kleinste Exponent ist, für welchen 7?’ = p(mod/*) ausfallt ; r ist eine Primitivzahl nach /'. Es möge d der gröszte gemeinsame Teiler von p’ und /'~1(/—1) sein, dann folgt aus der letzten Con- gruenz, dasz i Aq) tly r dd ==p oder prey l=p d En U(l- 1) Hieraus ergibt sich, dasz TER der kleinste Exponent ist, für welchen die Potenz von p der Einheit congruent ist. Daher ist tan ee 1) H. Satz 56 en 8 27. a) He § 116. _ 463 und '' — Das Symbol ist also eine f-te und nicht eine niedere Einheits- men | : ; ‘ Pp : : : ; us : wurzel. Folglich ist E ein Lip und nicht eine niedere Einheits- t g wurzel, wenn g der gröszte gemeinsame Teiler ist von f und 6; g ist also der gröszte gemeinsame Teiler von uA (ll) vee oak gps Es sei q eine Primzahl, die bis zur m-ten Potenz in [—i(l—1) aufgeht, in e bis zur n-ten Potenz und in al! bis zur n’-ten Potenz. Sie geht dann in f und 0 bis zur (m—n)-ten, resp. zur (m—n’)-ten Potenz auf. Hieraus folgt, dasz q in g aufgeht in einer Potenz deren Exponent der kleinsten der beiden Zahlen m—n und m—n’ gleich ist. Weil dies für alle Primzahlen q gilt, ist (LI) 1 Heinste gemeinsame Vielfache von e und al! tk—1(11) | Bres. also g = ——. Wir haben daher gefunden dasz A | ens v fv v : : e= —-te Einheitswurzel ist und keine niedere. Ull) se Fassen wir nun das Product rechter Hand aus (2). ins Auge. Weil 2] : (+5) mj Eik [h ia a v ealk—! v so. folgt hieraus, dasz jede —te Einheitswurzel a/*-1!; — = ome é Mal in dem Producte auftritt, und da fat os p bu —s, ek =|5| gies wird das Product rechter Hand von (2) dem Ausdrucke ce v \v ——s fos , he gleich sein. Hieraus ergibt sich der Hülfssatz, wenn man sich-des Satzes 3 § 2 bedient. Aus (1) folgt nun: 464 wo die Producte sich erstrecken über alle früher genannten Werte von u und über alle Primzahlen p, mit Ausnahme von J. Auf Grund des Satzes 4 ist n(!)—/ und es ist also 1 1 h =— lim (s—1) IT - le 1 sl p Tp 5 peu P bu ä [A] >= Das erste Product erstreckt sich über alle Primzahlen p, wie auch das zweite; denn wiewohl der Wert p=/ zuerst ausgeschlossen war, darf man den Factor l 1 — Fe fir Ht hinzufügen, weil er den Wert 1 besitzt. Das dritte Product erstreckt sich über die angegebenen Werte van wu, wobei aber der Accent angibt, dasz u = 0 jetzt ausgeschlossen st. Wir wissen, dasz 1 lim (s—1) IJ —-— = 1 gs > leapt und wir finden schlieszlich felt f 1 — ed x s=1 p u bu in. 1—|7 po Es werden nun die zwei Productzeichen verwechselt: und wir entwickelen jeden Factor in eine DrricaLet'sche Reihe oo | n |t 1 HW She ae “uw. nl Uh ns In dieser Reihe setzen wir das Integral ein: 1 1 ee J ae tee ys—le-—nx SET ae x 0 en bu] bu DS == heer > NI, Sal s=rof Ela Nun ist | == | wenn x= (mod Ih) also 465 jh_y bu / DD, ke e7nr AO poe hy -gi"@—1 eet = erf ER ee een n=1 l n=1 n= Wijhe Wir haben daher zu betrachten Weil Esser FQ) = =. n= h d [Pay bu Ww | tf". Das zu betrachtende Integral wird jetz 1 Ft [29a at } lt!) Die Integration lässt sich durchführen nach Verteiling in rationale — Briiche : Qhre 1 py U1 i dt ae fas Denk el me os, und nach der Integration ergibt sich: eN kri kri kr ri a TET 1)? = pha gun ——| J Fre" J log ————_ + lh ll t dea h Qhri h Qkri Mt! ian mit medek. + F\el j— = = kF\e! _ . Fa pe = iat P i ae icra . De ‘ty Helge, Ato: em, raw ewig! toiling he aides a € Ae P Crt Ne py dl ut 7 ie ï fi Kr dane tdi k . C3 tpi re < | ebi 4 4 § es 7 we ha : t tent it LA fT % 5 . Ld Ta “x \ Sat brei lrigoht? sbiucid arto uh | iy ieee } ‘ Me 4 ’ à x ze nov: ticuuiiolie’/ ei werdt gip A eee inva Pegs 1 4 k A , Oe 4 ' ; i wed? md. 6 a Arn MR : mete 4 i i, Zeb et EC oe Wal } -. ‘ enk k imal : wk veh RE Re en ER Er eee é ‘ a « L ' i 4 ' 7 — « te = ~~ VEL aA Pe 2 B: 3 Et KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS VOLUME XXI N°. 4. President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: “Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,’ Vol. XXVI and XXVII). CONTENT Sn: « C. SCHOUTE, F. A. VAN HEYST and N. E. GROENEVELD MEIJER: “An instrument to be used by the pilot of an aeroplane to measure the vertical velocity of the machine”. (Communicated by Dr. VAN DER STOK), p. 468. G. GRIJNS: “Is there any Relation between the Capacity of Odorous Substances of Absorbing Radiant Heat and their Smell-Intensity ?” p. 476. . A. F. C. WENT: “On the course of the formation of diastase by Aspergillus niger’, p. 479. . J. BYL and N. H. KOLKMEIJER: “Investigation by means of X-rays of the crystal-structure of „white and grey tin II]. The structure of white tin, p. 494. Ibid II]. The structure of grey tin.” (Communicated by Prof. H. KAMERLINGH ONNES), p 501. A. D. FOKKER: “On the equivalent of parallel translation in non-Euclidean space and on RIEMANN’s measure of curvature”. (Communicated by Prof. H. A. LORENTZ), p. 505. W. J. A. SCHOUTEN: “The Distribution of the absolute Magnitudes among the Stars in and about the Milky Way”. (First Communication). (Communicated by Prof. J. C. KAPTEYN), p. 518. J. A. SCHOUTEN: “On the arising of a precession-motion owing to the non-euclidian linear element of the space in the vicinity of the sun”. (With Addendum by Prof. W. DE SITTER). (Communi- cated by Prof. LORENTZ), p. 533. Miss. M. A. VAN HERWERDEN: “Effects of the Rays of Radium on the Oögenesis of Daphnia pulex”. (Communicated by Prof. C. A. PEKELHARING), p. 540. H. J. HAMBURGER and R. BRINKMAN: “The conduct of the kidneys towards some isomeric sugars”. (Glucose, Fructose, Galactose. Mannose and Saccharose, Maltose, Lactose), p. 548. A. SMITS and J. M. BIJVOET: “On the Significance of the Volta-Effect in Measurements of Electro- motive Equilibria”. (Communicated by Prof. P. ZEEMAN), p. 562. J. F. VAN BEMMELEN: “Androgenic origin of Horns and Antlers”, p. 570. A. W. K. DE JONG: “On the estimation of the geraniol content of citronella oil”. (Communicated by Prof. VAN ROMBURGH), p. 576. H. B. A. BOCKWINKEL: “Observations on the expansion of a function in a series of factorials”. Il. (Communicated by Prof. H. A. LORENTZ), p. 582. W. H. KEESOM and Mrs. C. NORDSTROM-VAN LEEUWEN: “Deduction of the third virial coefficient for material points (eventually for rigid spheres), which exert central forces on each other”. (Communicated by Prof. H. KAMERLINGH ONNES), p. 593. W. H KEESOM and Mrs. C. NORDSTROM-VAN LEEUWEN: “Development of the third virial coefficient for material points (eventually rigid spheres), which exert centrai attracting forces on each other proportional with rt or r ~~.” (Communicated by Prof. H. KAMERLINGH ONNES), p. 602. J. A. SCHOUTEN: “On the number of degrees of freedom of the geodetically moving system and the enclosing euclidian space with the least possible number of dimensions”. (Communicated by Prof. J. CARDINAAL), p. 607. FELIX KLEIN: “Bemerkungen iiber die Beziehungen des DE SITTER’schen Koordinatensystems B zu der allgemeinen Welt konstanter positiver Kriimmung”, p. 614. O. POSTMA: “On friction in connexion with Brownian movement”. ‘(Communicated by Prof. H. A. LORENTZ), p 616. > 7 Proceedings Royal Acad. Amsterdam. Vol. XXL. Meteorology. — “An instrument to be used by the pilot of an aeroplane to measure the vertical velocity of the machine”. By Dr. C. ScHourr, F. A. van Heyst, and N. E. GROENEVELD Meyer. (Communicated by Dr. van per Stok). (Communicated in the meeting of June 29, 1918). In many cases it is important that the pilot of an aeroplane should get direct information about the velocity of climbing or falling of his machine, and in the greater part of these cases it is of a bigher value that he should know the vertical speed with ~ respect to the medium, the air, than in relation to the earth. The latter especially when he wishes to know the velocity of ascent or descent to be able to judge the effect of his altitude-steerage, as well in relation to the safety of flight as where either an extreme quickness or an extreme slowness of the climbing or falling is required. As a rule the vertical speed is determined by measuring the quickness with which the atmospheric pressure varies. Therefore barometric readings can be executed after certain intervals. In that case however we do not get instantaneous values. The quickness of the changes in the pressure of the air can also be measured by means of the so called variometer, consisting of a vessel that com- municates with the onter air by a narrow opening or a capillary tube and is provided with another tube in which a liguid-column can move to and fro when notwithstanding the “leakage” an under- pressure Or an over-pressure is generated in the vessel. All these determinations based on pressure-changes give the vertical speed in relation to the earth and not to the medium, and moreover the results are not independent of the height at which they are obtained; so that generally a correction, and often a considerable correction, has to be applied. This complication does not arise when the determination is made by vertical cup-anemometers. The idea suggests itself, to get indications of the vertical speed electrically by means of a dynamo-armature mounted on the axis of a vertical anemometer. The more so because indications at a distance are wanted, as the anemometer requires to be mounted in undisturbed air, which as a rule is not to he expected immediately in front of the pilot-seat. In such measurings however, the result would be spoiled by the friction, which is un- avoidable in the anemometer, and which cannot be slight in this 469 case, because of the strong lateral pressure on the axis during the rapid motion. The rapport between the vertical speed V,, and the number of revolutions V can be expressed by the relation V3 = A+ BN in which A and B are constant. In the case V, is smaller than A the anemometer stands still. Further it would already be a difficulty to obtain a permanent vertical position of the axis. | In the instrument the description of which follows and which we have called scansimeter the foresaid disadvantages of the vertical anemometer have been avoided without the advantages having been abandoned. Therefore electric currents are measured which are generated in a dynamo armature, mounted on an axis which has the direction of the line of displacement of the machine instead of being kept vertical. This directing as well as the rotation of the axis are obtained by an airscrew fixed on the hind part of the axis, which is made to rotate by reaction during the rapid motion through the air, and thus gives the correct position to the well-poised axis. Therefore it was practically sufficient to fix the axis pivoting around a line perpendicular to the body of the machine, which line must be horizontal when the aeroplane lies flat in a lateral sense. This simplification was applied for the advantage in the construction only. In what follows this line is always dealt with as a horizontal one, which does not introduce an appreciable error. The total speed of the machine V; can, in perfect anology to what has been said about the vertical velocity be represented by Vi = A+ BN In this case the influence of the constant of friction A is much smaller because V, is always considerable, so that when the axis is made to run lightly between ballbearings the number of revo- lutions N is approximately proportional to V,. In order to generate a current which offers a measure, not of the total speed with respect to the air, but of the vertical component of that quantity, V,= V, sin a (fig. 1), the magnetic field in which al the dynamo-coil rotates is dressed in wae pee ne special way, viz. in such a way * that the direction of the field, which a we suppose to be homogeneous for Bee the sake of simplicity, is always kept horizontal and besides that parallel Fig to the vertical plane through tbe axis of rotation. i fe 470 In order to realise this in the first model slits have been made in the permanent magnet, which was to procure the magnetic force. This magnet was suspended between horizontal pivots, in such a way that the lines of force were always kept horizontal. By inserting the axis of rotation througb the slits, this axis can be made to be parallel to the lines of force when the direction of the flight is horizontal, while when the machine climbs or falls the angle between the lines of force and the axis corresponds to « in fig. 1. The later models had two magnets instead of one, placed parallel to each other on both sides of the axis. In both cases the horizontal axis of suspension of the magnet was made to coincide with the line around which the armature axis pivoted. If the axis AA (fig. 2) be parallel to the lines of force, no difference of potential will arise between the ends of the rotating Fige winding W. The electromotive force in an element ¢ of the coil is exactly counteracted by an equivalent E. M. F. in the opposite sense, generated in the element ¢& lying in the position of a reflected image with respect to the axis. If the axis AA (fig. 3) makes an angle « with the lines of force, an electromotive force will be generated in W proportional not only Figs to the speed of rotation and the strength of the field, but caeteris paribus also to sin «. The magnetic field can be considered as being 471 the effect of a cooperation of two fields, one in the direction of the axis of rotation and another parallel to the vertical plane through the axis and perpendicular to the latter, which two fields bear a proportion as for the strength as cosa: sina. Of those fields only the latter can generate a current in W. When the changes in the electromotive force in W in consequence of changes in « shall be proportional to sin «, the condition must be made, that the magnetic field be unvariable when the relative position of axis and lines of force changes. Therefore 1 the material for the axis of rotation must be a non-magnetic one, and 2 if the homogeneity is not to be relied upon, and if a weak iron core shall be used, the form of the core must be globular. By combining the use of a dynamo axis driven by reaction by an air screw with the simusdynamo described here in principle, a system is obtained by which currents are generated, proportional as well to the total speed VV, of fig. 1, as to the sinus of the angle « under which the machine climbs or falls, (fig. 3) so that the strength of those currents offers a direct measure of the vertical velocity with respect to the medium, which measure is independent of the height of observation and of all quantities connected with the machine, so that no correction whatever has to be applied. In fig. 4. the mechanism of such a sinusdynamo is illustrated schematically. The armature has been executed in the most simple way: the coil W has been imbedded in two parallel circular grooves on both sides of the axis aa. In the magnet M the slits G allow the axis to take different positions with respect to the magnet. The axis pivots around Q, the point that at the same time determines 472 the horizontal axis on whieh the magnet is suspended. As by con- tinuous currents only the direction of the motion is given, so that it is possible to discriminate between rising and falling, a two parted collector C has been made use of, with insulated (4) brushes B, fixed on the frame R. Fig. 5 represents the scheme of the scansimeter. Besides the parts noted in fig. 4 by analogous symbols this figure shows the screw &, the ball-bearings A and the circular cover Omh. sideways slightly convex. Ga is what is shown of the support in which the common axis of the magnet and of the frame A are adjusted. The currents generated in the dynamo are measured with an instrument, fixed in front of the pilot. It is preferable to choose am instrument of a special form, so that a rising of the hand corre- sponds to a climbing of the machine and a lowering of the former to a descent of the latter. Meters of the usual Desprez d'Arsonval type are very practical for these determinations. To avoid the proportionality of the excursion to the speed of rotation from being disturbed by reaction from the armature, it is advi- sable to keep the current as weak as possible, consequently to give the measurings the character more of determinations of potential 473 differences than of the. strength of currents, and therefore, to take a small resistance for the dynamo-coil in comparison to that of the current meter. On the other hand it is desirable to take a resistance for the dynamo coil of at least some times ten Ohm, in order to exclude a too great influence of transit-resistances. The following can be recorded about the preparatory experiments carried out with this instrument. One of the specimens of the scansimeter showed the following unexpected effect, which the others showed less or did not show at all. After the axis of rotation had been carefully placed horizon- tally in the laboratory, and the position of equilibrium of the magnet had been regulated in such a way that a rotation of the axis did not generate any current, a current could be produced arbitrarily in both directions by changes in the position of the brushes only. This phenomenon, rather startling at first sight, can be explained as follows *). Evidently the magnetic field cannot be considered as resulting from two fields, one after the axis and another perpendicular to the axis, but here a third component has to be accepted, which we can assume to be horizontal and perpendicular to the axis. If the brushes are fixed in such a position, that the component in the vertical plane comes to ful development, i.e. that the commutating of the current takes place at the moments the difference of potential is zero (fig. 6), the effect of the horizontal component vertical to the axis is likewise zero, because for this component the commutation comes at the moments the E. M. F. has its maximal value (fig. 7). When the windings are distorted with respect to the collector or the brushes with regard to the magnetic field, then this horizontal component is no longer inactive, so that a current can be produced, at will in either direction (fig. 8 and 9). The wryness in the field, which this specimen showed, can easily occur, when the centering of the armature between the poles of the magnet resp. the poleshoes, as well sideways as in the direction of the axis, is deficient, which can very well be the case in un air gap of. not more than some tenths of a millimeter. This explanation requires, as was confirmed by the experiment, that, when the relative position of collector and coil is the right one, such 1) We may neglect the armature reaction because it was very small, given the weakness of the currents. Besides, by taking into account this reaction hardly anything changes in the reasoning. 474 an unsymmetrical distribution of the magnetic force does not cause any error in the determinations. Some difficulty arose in finding the right form and the right material, that allows the current to be taken off from the collector if / Mf, Yj V, Yj / Lh, uninterruptedly during a long time. In one respect the conditions are not very exigent: fluxions in the strength of the current by changes of the resistance at the collector up to one percent can be allowed in practice without any reserve. On the other hand the reliability of the working must be guaranteed at a minimum of care and looking after. By making the diameter of the collector not to exceed some millimeters and by taking silver both for the collector and for the collector springs a method of transmission of the current was arrived at which gave excellent results during a long time at a streteh, without any greasing or oiling. A freely suspended magnet cannot be prevented from executing slight oscillations in consequence of incidental accelerations and 475 retardations in the air, nor the frame which carries the axis of rotation from being set swinging around its pivots. To avoid an inconvenient agility of the hand under influence of such oscillations, care was taken to secure a slow indication of the currentmeter, which was obtained by making the instrument over-deadbeat by dumping, the oscillations of the magnet being made as short as possible, besides which a damping by mechanical friction was applied for so far as coexistent with the requirements of an accurate working. If the instrument was adjusted sideways on the aeroplane out of the “propeller-wind”’ about halfway between the two decks of a biplane an ascension was indicated when the machine flew at level keel, while an ascending was strongly exaggerated and a descent was not at all or hardly indicated. This shift in the direction of an ascension in the indications, which was observed in a higher or lower degree in a number of places on the aeroplane is caused by air currents generated by reaction from the aeroplane in its quick motion through the air. When fixed at the end of a wing at some distance sideways, the instrument gave undisturbed indications. In this position however the troubles of the oscillations mentioned above were felt more strongly. lt would be a valuable result if by means of this instrument a systematical investigation could be made of the vertical velocities of those reaction currents produced in the neighbourhood of an aeroplane in flight. It is highly to be regretted that there is no possibility in our country of calibrating the scansimeterscale in an artificial air current. When the electrical resistances, the strength of the magnetic field and the sensitiveness of the currentmeter are known, this calibration consists in the research concerning the number of revolutions the screw makes at various velocities relatively to the air. Since the air-tunnel set up on a small scale by the “Kon. Nederl. Vereeniging voor Luchtvaart” could no longer be used, such a testing cannot be executed in our country. Physiology. — ‘‘/s there any Relation between the Capacity of Odorous Substances of Absorbing Radiant Heat and their Smell-Intensity?” By Dr. G. Grins. (Communicated in the meeting of September 29, 1918). For a thorough knowledge of the apperception of our senses we must first of all find out the special character of the stimulus to the receiving of which the organ is adapted. The study of light and sound teaches us that the quality is determined by wavelength, and its intensity by amplitude. With regard to smell and taste we do not know as yet what property of the odorous or the gustable substance determines the sensations they arouse. Many attempts have been made to establish a relation between smell and certain qualities. Up to the present without success. The chemical properties which, indeed, were thougbt of first and foremost, appeared to yield no satisfactory interpretation of all the peculiarities. ZwaAarDEMAKER'), therefore, reaches the conclusion in his ‘Physiologie des Geruchs”, that no direct relation exists between the chemical constitution of a substance and its smell. LiÉerors ®) pointed out that a large number of odorous substances, when put on the surface of clean water, present the same phenom- enon as Prévosr*) described for camphor and even ealls it the “odoroscopic phenomenon’. However, it turned out that a number of inodorous substances also present this phenomenon. So there is no argument for correlating smell with surface-tension phenomena, to which vaN MENSBRUGGE attributes the behaviour of camphor on water. ZWAARDEMAKER’S finding *) that a large number of odorous substances evolve an electrically charged vapour, when sprayed in aqueous solution, gave rise to the supposition, that electrical charges come 1) H. ZWAARDEMAKER, Die Physiologie des Geruchs. Leipzig 1895. 3) Ligceois: Sur les mouvements de certains corps organiques a la surface de l'eau. Arch. de Physiol. 1868 T. I p. 35. 3) Prevost: Annales de Chimie et de physique XXI p. 254, XXIVp 31, XLp 1. *) H. ZWAARDEMAKER, These Proc. XIX p. 44, 334 and 551. 477 into play here. Zennvuizen'), however, has detected that also saponins and antipyretica, which do not smell, produce an electrical charge. This, therefore, is not a property peculiar to odorous substances alone. TYNDALL’) previously observed that several odorous substances possess a great capacity of absorbing obscure rays. We might, therefore, expect a correlation between these two properties, as it is quite possible that odorous substances derive energy from their surroundings to transmit it to the olfactory organ. If so, we must expect smell-intensity and absorbing capacity in different odorous substances to evince, if not proportionality, at all events parallelism. | On Prof. ZWAARDEMAKER’s suggestion | undertook an investigation in this direction. | Since I purposed merely a preliminary orientation with regard to this problem, there was no need of being very accurate, so that many difficulties could be avoided. l determined the absorptive capacity of dry air that had passed through a slightly curved tube filled with the fluid under examination and compared it with the capacity of pure dry air by a differential method. On either side of a Nernstlamp a glass tube was placed closed at both ends with a plate of rocksalt. Behind these tubes were arranged Morr’s thermo-piles connected oppositely with a Morr. galvanometer. Air or gas could be sucked through one of the tubes by means of a spirometer. In front of the other tube an irisdiaphragm was mounted, the area of which could be read from a large pro- tractor. The width of the opening, corresponding to every scale- division, was determined by measurement. The deviation of the galvanometer was measured by the movement of the reflected image of an illumined copperwire on a millimeter- scale. For every determination dry air was first passed through the one tube, and then the diaphragm adjusted so as to make the galvanometer point to zero. This could readily be done, as the apertures in double screens, interposed between the lamp and the measuring tubes, could be covered or uncovered at the same moment. Subsequently air was sucked for about ten minutes through the fluid under examination, and through the measuring tube. After this the diaphraghm was so adjusted that again removal of the 1) These Proc. XX p. 1272. 3) J. Tynpatt: Heat as a mode of motion p. 366. 478 screens did not cause a deflection of the galvanometer. Next the tube was rinsed for a quarter of an hour, or longer if need be, with dry air and the zero-point determined again. If the two zero- points are approximately equal and the determination with the odour-containing air deviates, the ratio of the diaphragm apertures is that of the quantities of light transmitted, from which the absorption can be calculated. In the same way dry air, sucked through the fluid under exami- nation was led through a glass vessel provided with an opening, which could be closed by a glass plate, or to which ZWAARDEMAKER’S diaphragm odorimeter could be adjusted. By means of this apparatus, a description of which will soon be published, odour-containing air can be diluted to a known proportion. Into a cylindrical smell-chamber, from which the content is inhaled through a glass tube, the odorous air enters through an irisdiapbragm at the bottom, while free air can enter through lateral adjustable slits. We estimate the dilution in which the scent almost fails to be recognized. So the absorptive capacity and the smell-intensity of a given mixture of odorous substance and air were known. We now calculated fur every examined mixture how many times it had to be diluted to absorb just 1°/, of the incident light, and from this how many olfacts such a dilution would contain. The subjoined table shows the results thus obtained : Substance Number of Substance Number of examined | olfacts | examined olfacts Methylic alcohol 12 | Bromoform 16° Ethylic alcohol T:2 | Benzol 72 Ether eae We ee) Toluol 16° Amylic acetate 20°° | Xylol 94 Glacial acet. ac. 36° Toluidin 51 Propionic acid 22° | _Eucalyptol > 3400.7) Chloroform 14° | Eugenol | > 1490 2) If smell-intensity were related to capacity of absorbing radiant heat the above values might be expected to be approximately equal. Since they are so widely different such a relation must be denied. 1) The dilution of ether-containing air was beyond the capacity of my measuring instrument. 2) In both these cases the absorption was less than !/, perc., consequently so minute as to elude measurement with my equipment. Botany. — “On the course of the formation of diastase by Asper- gillus niger’, by Prof. F. A. F. C. Want. (Communicated in the meeting of September 29, 1918). It is already a long time that I have intended to carry out an investi- gation on the influence which external conditions exercise on the formation of enzymes, more especially in Aspergillus niger. In a certain sense this would be a continuation of an earlier investigation, which however related to another fungus, Monilia sitophila*). That my attention has now been turned to Aspergillus niger has its explanation in the fact that Karz has expressed opinions about the formation of enzymes in this fungus, which seem to me hardly probable, but. with regard to which a sure judgment can of course only be formed after a renewed investigation’). Other work has prevented me from carrying out my intention, and seems likely to do so in the near future; moreover in the meantime another investigator, Harar.p Kvyuin has repeated Katz’ investigation by a better method’). It might thus appear that there is no reason for publishing a paper, if my preliminary experiments had not given some results, which seem to me of sufficient importance to warrant publication even though the investigation is incomplete. This preliminary inquiry dealt with the question how the for- mation of diastase by Aspergillus niger depends on its age. It seemed to me that in all comparative investigations insufficient atten- tion has been given to this important point, namely the age of the cultures. Generally cultures of differing nutrition were all after the same length of development compared with one another, it being however evident, that the comparison was here made between unlike things, because the nutritive values of the solutions differed widely. It seemed to me that it would be necessary to trace the whole course of the formation of enzymes in suecessive intervals of time in each nutrient medium. I have not extended the investigation to this length, but have 1) F. A. F. C. Wenz, Ueber den Einflusz der Nahrung auf die Enzymbildung durch Monilia sitophila (Mont.) Sace. Jabrb. f. wiss. Botanik. 36 1901. 2) J. Karz, Die regulatorische Bildung von Diastase durch Pilze. Jahrb. f. wiss. Botanik. 31 1898. 3) HARALD Kyuin, Ueber Enzymbildung und Enzymreguiation bei einigen Schimmel- pilzen. Jahrb. f. wiss. Botanik- 53 1914. 480 only endeavoured to solve the problem in one definite case, how the formation of diastase takes place. For the culture-fluid a solution was always used containing 5°/, glucose, 0.5 °/, NH: NOL O40 Sn K,HPO, and 0.05°/, MgSO,. This solution was sterilised by heating it on three successive days for half-an-hour to 100° C. The inoculation was carried out by means of a platinum loop with fluid containing the conidia of Aspergillus. It is known that the latter remain floating on the sarface of the fluid and that one cannot keep them submerged. One can take care to get a fairly uniform distribution on the surface of the fluid and then always inoculate the same amount with a loop. Thus one does not indeed obtain a complete uniformity of the number of conidia in the different culture flasks, but the differences are so small as to exercise no influence on the final result, at most a slight difference in development is observable in the first two days. It should be possible to ensure a greater uniformity of inoculation-material, but this would be fraught with so many diffienlties that it is not worth while in view of the very small advantage it would yield. It needs no demonstration that the sowing of a single conidium is here wholly impermissible, because then one would have to reckon with great individual differences. These differences can only be compen- sated for by inoenlation with a large number af conidia. The fungi were grown in glass (Erlenmeyer) flasks and for com- parative experiments the same quantity of culture-fluid was always placed in similar flasks. The latter were kept in a room at a constant temperature of 24° C. with variations of 0.5°. The cultures were in darkness; artificial light only was used for inoculation and observation. At first daily and later after two, three and more days determi- nations were made of the amount of enzyme present in the culture- fluid and in the fungus-mass of one of the flasks, but generally this determination was made for 2 or 3 flasks. Then at the same time the fungus-mass formed in one of the flasks was collected on a tared filter and weighed after it had been dried, so that an idea was obtained of the quantity of dry material which had been formed. The other fungus-mass was ground fine in a mortar with the help of a little kieselguhr, and afterwards extracted for one hour with the culture-fluid and subsequently this fluid was filtered off and examined with regard to enzyme: now if we know the quantity of enzyme present in the culture-flaid, which had therefore diffused outwards, then we have only to subtract this from the quantity found in order to ascertain how much enzyme was present in the mycelium. 481 The quantity of diastase was determined by finding how long it was before the starch had completely disappeared out of astarch-solution of definite strength, which had been mixed with the enzyme-solution, whence the quantity of enzyme must be inversely proportional to the time. The presence or absence of starch was investigated by the aid of an iodine-solution of known strength. Theoretically this is not the best method; it would be better to determine the quantity of sugar which is formed in a definite short time from the starch and then to consider the quantities of diastase proportional to the figures thus found. But this is impossible in the case of the diastase of Aspergillus niger, because there is here often such a very small quantity of enzyme, that the sugar could not be determined. For this reason it was thought preferable to divide the enzyme-solution to be examined, into a number of equal parts, and to mix each of these with the same small quantities of very dilute starch-solution and successively to test the mixtures after a definite interval for the presence or absence of starch with the aid of the same very dilute iodine-solution. Preliminary experiments showed that when a solution is used which contains in 100 gr. water 62.5 mg. iodine and 62.5 me. potassium iodide, 1 eem. of this is enough to colour a starch-solution distinctly blue, when this contains 1 mg. soluble starch in 10 cc. of water, whilst when this amount is only 0.2 mg., a definite red- violet colour still results. There are formed from starch, before it is completely hydrolysed by the action of the diastatie enzyme of Aspergillus niger, erythro- dextrin-like intermediate products, which with iodine are coloured red; so long as such bodies were present it was assumed that the reaction was not ended; it was only when after mixing with the iodine solution, the colour remained yellow, that the starch was considered to have disappeared completely. Naturally all the fluids were under the same condition. For this reason 5 c.c. of the fluid to be investigated was always mixed in test-tubes with 5 c.c. of a solution which contained 0.08 pe. of soluble starch and with some drops of toluene; the latter was for the purpose of preventing any develop- ment of fungi or bacteria in the experiments of longer duration. The tubes were then shaken for a short time; and then placed in the dark at a constant temperature of 24° C. After a certain time the fluid was filtered off from one of the test-tubes and 1 ce. of the above mentioned solution of iodine was then added. If the colour was blue or red, this process was repeated some time later, until the colour was decidedly yellow. 482 The question was whether in this way, by repeated trials, trustworthy results could be obtained. In order to ascertain this, some preliminary experiments were carried out. From a culture of Asper- gillus niger which was 5 days old, the fluid was filtered off and divided into two equal portions. One half remained unchanged, the other was again divided into two, and one of the halves was boiled for a short time to ensure destruction of the enzyme, and after cooling it was mixed with the unboiled portion, so that there were two fluids of which one must contain double the quantity of the enzyme in the other. 5 cc. of each of the fluids was mixed in a test-tube with 5 ¢.c. iodine-solution and after certain intervals each tube was tested for the presence of starch. It was found that the undiluted solution after 95 minutes was still coloured some- what orange yellow by iodine, whilst after 100 minutes the colour was pure yellow, so that the colour-change had taken place after 97.5 minutes. In the twice-diluted solution the change had taken place in the interval between 180 and 190 minutes, therefore in about 185 minutes. The proportion of 97.5:185 is 1: 1.9, whilst the quantities of enzyme were to one another as 2:1, so that the error in determination was not more than 5 °/,. In another case the starch was lydrolised by the undiluted. solu- tion in 495 minutes, by the twice-diluted in 907 minutes and by the 10 times diluted in 4365 minutes. These figures are as 1:1.8: 8.8 instead of as 1.:2: 10. Lastly there was a third case in which the enzyme-solution had been diluted 2, 4, 8, and 16 times. The result is shown in the following table. Still coloured red | Pisappear- Propor- i ance of Average 9 with I. Sen tion Undiluted solution After! h. 45 min. After 2hours) 1 h.52 min. 1.— Twice diluted 8 EB er D0 hp A OLON jen 4 times : OE SO. te al ae Rea aot Sry, ' ; Ties Bhs je CG aga! veld A 1.9 16: >, k E „A , 00 fn Br fd When it is remembered how tentative these estimations were then the agreement between the determination and reality must be considered very satisfactory; the difference here amounts to much less than 10°/, in most cases. 483 One can therefore conclude that in this way an idea can be obtained of the quantity of enzyme which is present in certain fluids. This only holds good on condition that these fluids have for the rest quite the same composition. Now it has unfortunately to be stated that the culture-fluid of each fungus changes in the course of the development, partly because certain bodies from the fluid are taken up by the developing mycelium, partly also in consequence of secretions by the fungus. In this respect therefore the cultures of Aspergillus at different stages are not quite comparable. The concentration of the H-ions could be made equal, by the addition of acid or alkali, but that would not completely meet the case, because other bodies may certainly be present which hasten or retard the reaction and which, at least at the present time, cannot be determined. There is indeed another method conceivable which would consist in mixing culture-fluids of different stages with one another, after part of them had been boiled to destroy the enzyme. By this means one would then be able to trace whether in a given solution sub- stances were present which hasten or retard the enzyme-action. From some preparatory experiments it appears that something may perhaps be obtained by such a method. So, for example, the fungus from a culture 7 days old, was finely ground up, and then extracted with its own culture-fluid, and the solution was then diluted with an equal volume of a. its own culture-fluid, after this had been boiled to destroy its enzyme, 6. a culture-fluid similarly boiled from a culture which was 17 days old. and c. a like solution of a culture which was 3 days old. On investigation it was found that when 5 c.c. of the above mentioned solutions were mixed with 5 ec. of a solution of soluble starch of 0.08°/,, 1320 minutes elapsed in the cases of « and 6 before all the starch had disappeared, whilst in the case of c, this period amounted to 900 minutes. Hence there was in c either present an accelerator, or the solutions a and 6 contained retarding substances which were wanting in c. For the rest this method was not worked out further on account of the condition mentioned at the beginning of this paper, and the figures obtained must therefore be received with a certain reserve. It will, however, be seen that they nevertheless give some idea of the course of the formation of diastase in Aspergillus niger. Before I further mention the results of the investigation I will first indicate by means of an example, exactly how the figures were obtained. Three flasks, each containing 75 e.e. of culture-fluid were investigated, after a culture of Aspergillus niger had been in them 32 Proceedings Royal Acad. Amsterdam, Vol. XXI. 484 tor 3 successive days. From flask 1 the culture-fluid was filtered off, the fungus-mass ground fine in a mortar with kieselguhr, and afterwards extracted with the culture-fluid. Subsequently it was filtered and the filtrate distributed in test-tubes, so that each tube got 5 ce. of fluid; then to each was added 5 c.c. of a 0.08°/, solution of soluble starch. After 60 minutes the fluid of one of the tubes was examined in the manner already described with iodine ; the colour was blue. After 180 minutes the second tube gave a pale claret-red, after 225 minutes the colour of a third tube was light-yellow. It may therefore be assumed that the hydrolysis of the starch had taken place in about 200 minutes (1). A second flask was treated in the same way; the reactions with the iodine solution can be seen in the following table: After 60 minutes. ..... blue Be be! Ree EN: reddish-violet … 240 te gion Cicada te 4 Mi ate SS are he pale claret-red „ 295 ar ai) enen reddish-yellow „ ot0 a ON as yellow so that the time taken for hydrolysis was about 315 minutes (2). The fungus-mass in a third flask was colleeted on a tared filter, then washed and dried; the dry material weighed 69.5 mg. The culture-fluid alone was examined for enzymes by the method de- scribed for the other flasks. The following table shows what reactions were obtained with the iodine-solution. SATII „Lommel: blue al NN blue Loge mee at ee bluish-violet „485 tbl weet reddish-violet 1055 ER sere reddish-yellow alow ed, ita eee pale reddish-yellow 400 ee ME eee ee yellow It may therefore be concluded, that the hydrolysis of te starch required about 1160 minutes (3). If the quantity of enzyme which is necessary to hydrolyse the soluble starch in 10 cc. of a 0.04°/, solution in 100 minutes is put equal to 10 then the Ee ve the test-tubes in case (1) is 100 POOT EN, 500 > LO sing i2) 315% 10, in (9) ae 10, therefore the whole 75 quantity in the 75 em. is 5 er 15 times greater. 485 Therefore the calculation in the case of (1) gives a quantity of diastase of 75,0, in (2) 47.6 and in (3) 12.9. If it is assumed, that in flasks 1 and 2 the same quantity of enzyme was present in the culture-fluid as in flask 3, then there remains for the enzyme in the mycelium for (1) 62, and for (2) 34.7. These figures appear to diverge widely, but nevertheless we shall soon show, that there is a great regularity in the quantity of enzyme found on daily investigation. The experiment which is here described in detail was the be- ginning of a whole series of determinations. Altogether 42 similar flasks were inoculated with Aspergillus, and after a given number of days three of them were each time examined in the manner described. The result is given in the appended table. In column I the number of days is given, during which the culture lasted, in column II the quantity of enzyme in the mycelium and culture- fluid together, in column III the same for the other flask, in column IV the mean of II and III, in column V tbe amount of enzyme in the culture-fluid alone of the third flask, in column VI the difference between .1V and V, and finally in column VII the dry weight of the crop of fungus in flask 3. Wherever the sign < is found in the table, this indicates that the last test-tube still gave = sae - | | | — Pe Il ' Ill f IV | V | VI | VII | | | | | & 3 | 15.0 Amon dte GS | 12.9 | 48.4 | 69.5 mgr. 5 44.1 44.1 44.1 4.2 39.9 319.0)" 5. 7 66.6 41.8 54.2 3.7 50.5 398.0, 8 21.7 30.3 26.0 3.0 23.0 538.5 „ 101 -< 6.0 < 6.0 < 6.0 2.3 <31 549.0 12 4.1 1.5 5.8 1.8 4.0 | 699.0 , (ar 38 < 2.8 <08 trace 1 | £2:8 939.0 , 17 25 3.3 2.8 trace | 2.8 936.5 , 19 | <1.6 a < 1.6 sek el 943.5, 23 1.0 2.6 1.8 2.0 0.0 899.0 , 26 6.2 10.0 8.1 2.5 5.6 100.01 5 30 16.7 4.5 10.6 3.3 13 182.00, 33 11.1 29.3 20.2 16.7 3.5 742.5 , 37 16.6 50.0 33.3 11.1 2220p (134,5 , 486 a reaction with the iodine-solution, so that the limit of hydrolysis had not yet been reached. Wherever “trace” is written, the amount is less than 0.5, although a change in reaction was still clearly observable. The figures of columns II and III appear at first sight to diverge widely, but on nearer inspection a certain regular course is however noticeable, by which column IV is justified. The amount of enzyme in the culture-fluid is generally very small, so that the course of column VI is searcely different from that of column IV. There are a few exceptions to this, but I think it better to give the complete observations first and then only to begin a discussion of the figures found. A second series of figures had reference to a number of cultures which had been made at the same time as those already described, and which only differed from them in that instead of 75, 150 ec. of culture-fluid was used and in larger flasks, in consequence of which the surface of the culture-fluid amounted to 47 square em. instead of 24 as in the foregoing series. There was indeed the possibility that on this account the development of Aspergillus might continue longer, Enzyme in fungus and Enzyme in culture-fluid culture-fluid only after 3 days 65.2 | 6.7 ia BS, SS 75.0 | 5.4 ie ee: 15.0 3.6 we. 48 15.0 | 23/0 ge ad 30.0 | AP Pe EN 30.0 | < 1.9 Boe tame 13.6 1.5 Ee Wad 11.0 ez y AQF aS : ‘Gs | 0.0 Stee HA 4.0 0.0 2008 4.2 | 0.0 „Der | {1.3 | 0.0 „3m | 44.1 | 10:0 Bes ea | 4.5 ik 0.0 487 and with it also the formation of enzyme. Of these flasks half were directly examined as to enzyme, the others after they, had stood for an hour with the finely ground fungus-mass and had then been filtered. The results obtained are shown in the fore-going table. In order also to trace during a longer period the course of diastase formation, a new experiment was begun with 99 flasks each con- taining 300 ¢.c. of nutrient solution and treated exactly in the same “way as the first series with this one difference, that this time account was taken of the quantity of liquid which remained when cultivation was over. In consequence of consumption by the fungus and of evapo- ration this quantity diminishes, a fact that in the above experiment of 37 days could not have so strong an influence on the results as in those lasting no less than 149 days. In consequence of this the calculation was more complicated, but still sufficiently simple, and after what has already been said, it will not be necessary to give the slightly modified calculation here; we may only remark that the figure of column VI is therefore not exactly the difference between IV and V. With this qualification the different columns in the following table, have the same significance as in the first table already given. The figures of column IV are represented graphically in fig. 1. In this figure the age of the culture (in days) is measured on the axis of 25 9 i 1619 2326 3033 ITO GAT SI 58 65 T2 9 Bb 23 100 107 Ik 2 128 136 et Ies Fig. 1. the abscissae, whilst the amount of diastase is respresented by the ordinates. In figure 2 the figures of column VI are represented in the same way. Valid objections may perhaps be raised against the latter figures, because in these cases the amount of enzyme was deducted, which occurred in the culture-fluid of a culture distinct 100 107 114 121 128 135 142 149 488 2 2 6.57 4. 489 from that in which the amount of enzyme in the mycelium was 60 £0 30 20 ° 2 579 \2 1618 2326 3039 ST 40 4h WT 61 68 65 T2 20 66 93 WO WT Ue 2 128 135 IA 149. Fig. 2. determined. To this must be ascribed the fact that such cases occur as. that after 79 days, where in the fungus plus fluid an average of 3.45 enzyme occurs, but in the fluid alone 4.56, i. e. more. This is of course absurd, and if its own culture-fluid could have been used, such figures would not have resulted. Nevertheless I have included these figures and represented them in figure 2, in order to show the general course of the presence of diastase in the fungus- mass itself. It is evident that this curve is, generally speaking, exactly identical with that of figure 1. This is easily understood when it is seen from the figures of column V that in general the cul- ture-fluid contains no very appreciable quantity of enzyme; it is only when much enzyme is present in the mycelium, that a quantity can be found in the surrounding fluid, which is not inconsiderable. How it is that the enzyme occurs there, must be left an open question. It might be thought to have arisen from cells already dead, but on the other hand a somewhat greater quantity is only found in very young mycelia. The enzyme’ found in the culture-fluid can therefore be left ania further consideration in order that attention may be concentrated on the diastase which is found within the mycelium of the fungus. From all the tables and also from figures 1 and 2 it is quite evident that after germination a considerable increase of enzyme is observable, which very quickly reaches a maximum and afterwards shows an almost equally rapid diminution. There will naturally be an inclination to consider this increase in relation to the development of the 490 mycelium, but the figures for the fungus-crop in the different tables show that this cannot be so. Moreover in figure 3 a representation 180 900 A6 O12 16IP URG JOIN BT AQ HH 47 51 58 os ta 1e 86 9% 100 107 ib 12 128 138 142 169 Fig. 3. is given of the development of the mycelium in the last-mentioned experiment. There the duration of the development, given in days, is measured on the axis of the abscissae, whilst the ordinates give the dry-weight of the fungus-mass. In the first days a strong increase of the crop is evident, which is followed by a decrease, probably caused by the preponderance of the processes of dissimilation over the assimilation. On working up the mycelium for the enzyme a large portion of the older cultures was found to be dead, a pheno- menon quite easy to establish, because the hyphae felt soft and flaccid and no longer elastic, as in the young cultures; but this could only with certainty be observed, when the fungi were some months old. It is however quite clear that no proportion exists between the amount of dry matter of the fungus and the quantity of enzyme 491 formed. Moreover it is seen that there is indeed in the beginning of the development an increase in the dry matter as well as in diastase, but that the increase in enzyme comes to a standstill, whilst the fungus-mass continues to increase for many days. Do the experiments described indeed afford a proof that the amount of diastase in the fungus increases during the first days and then again undergoes a diminution? May it not be that the culture-fluid, which is used for the extraction of the enzyme, under- goes a slow change of such a nature as to accumulate a substance which destroys the enzyme or at least opposes its action? Or conversely may not there be some accelerator which is present at the beginning and first increases, and later diminishes? A decided answer to such questions can only be given when the method has been further worked out which was briefly described at the beginning of this paper. But nevertheless there are facts which make tbe explanations suggested here very improbable. | therefore point out that in all series of experiments the decrease in the amount of diastase does not take place regularly, but that later again a sudden increase is observed. It is not readily conceivable that the compo- sition of the culture-fluid should suddenly undergo such a change that the sudden change in the figures could be thus explained. One can only think of an increase of the amount of diastase within the cells of the fungus. And when in this case nogreat influence on the figures for the diastase can be assigned to the culture-fluid, then this cannot be assumed in the other cases either. It may therefore be expected with fairly great certainty that the general course of the curve of figure 1 or 2 gives a picture of the actual quantities of diastase whieh oceur in the mycelium of Asperyillus niger. If we may accept the above conclusion as correct, then there follows more from it. Firstly, if we disregard for a moment the later irregularities of the curve a very quick initial rise is observable, followed by an almost equally strong fall. The former would be explained by the constant formation of more fresh enzyme, but the fall? There hardly remains anything but the conclusion that des- truction of enzyme is always going on within the cell; this des- truction then in the first days is compensated for and exceeded by the new formation, which latter very quickly comes entirely to a standstill or becomes so slight that it is far from being sufficient to keep the quantity of enzyme at a level. I now return to the irregularities which are to be seen in the falling part of the curve. It is seen from the first two tables that after about one month the amount of diastase in the fungus suddenly 492 shows a considerable increase. These experiments did not last long enongh to show what the further course would be, and for this very reason the last series of flasks was started, in order to trace for a longer time the behaviour of Aspergillus niger in this respect. It was found that here also after about one month the rise of the enzyme-content occurred; thereupon there followed again a decrease and then again a rise, whilst after about three months the decrease became permanent, so that finally practically no diastase remained. The oscillations of this curve are explicable by considering that the great rise in the first 3—4 days coincides with a vigorous develop- ment of the young mycelium. The same takes place again later when the fungus has formed new conidia and these, having been shed into the culture-fluid, germinate there; each time that this happens a sudden increase in the quantity of the diastase will be observable. It is self-evident that it cannot always be predicted at what time in the course of the culture this will occur, but we may safely argue that the culture-fluid gradually will become less and less suited for the germination of the conidia, and that therefore this phenomenon will gradually stop altogether. If this explanation is correct it would therefore be possible to make the formation of diastase in Aspergillus niger go on for a much greater length of time by renewing the nutrient solution. The phenomena which have just been described are thus of a secondary character and all study of the formation of enzyme in Aspergillus niger must be limited to the first stages of development. One cannot indeed say whether the course of the formation of enzyme would be the same with different nutrition, but this may however be expected. It is therefore not only impermissible to draw conclusions from the quantities of enzyme which occur in cultures which are e.g. some weeks old, but one must try to follow accurately the course of the enzyme-formation during the first days of development of the fungus in the case of each kind of nutrition; then only can conclusions be drawn respecting the influence of a given nutrition on the enzyme-formation of Aspergillus niger. It is evident that these conclusions only concern the formation of “diastase in Aspergillus niger; but they will nevertheless oblige one to be careful about conclusions as to other enzymes and other fungi; in further investigations the possibility must be borne in mind of similar results occurring. Summing up, it is found that in Aspergillus niger during the first days after germination a great quantity of diastase is formed in the 493 mycelium (on feeding with glucose as source of carbon and NH,NO, as source of nitrogen), and that in addition, destruction of this enzyme takes place, which at first becomes negligible in compa- rison with the formation, but which soon makes itself so evident that the total quantity quickly decreases, after it has reached a maximum about 5 days from the commencement of germination. Into the nutrient fluid there passes never more than a very small part of the total quantity of enzyme, occurring in the mycelium ; this is perhaps partly derived from dead cells. Utrecht, August 1918. Physics. — ‘‘Jnvestigation by means of X-rays of the crystal-structure of white and grey tin. Il. The structure of white tin.” Com- munication N°. 2« from the Laboratory of Physics and Physical Chemistry of the Veterinary College at Utrecht, by A. J. Bir and N. H. Korkmeijer. (Communicated on behalf of Prof. W. H. KErsom, Director of the Laboratory, by Prof. H. KaAMERLINGH ONNEs). (Communicated in the meeting of September 28, 1918). In Communication N°. 1 (June 1918) we communicated, that we had taken Röntgenograms of white and grey tin by the method of Desise and Scurrrer and we gave a description of some particulars about the arrangement of these measurements. Moreover we showed in a drawing for both states of tin mentioned the piaces of the inter- ference-maxima in a plane, perpendicular to the axis of the bars, indicating at the same time the intensities of the interference-lines. The photo of the grey tin indicated, that this material is crystalline, and on comparing the photos for the two states, it was evident, that it possesses a crystal-structure, which differs from that of white tin. We have now determined from those photos the crystal-structure as well of white as of grey tin, and shall communicate in this paper our results for white tin, reserving those for grey tin for commu- nication N°. 26 (these Proceedings). In table I the intensities of the interference-lines are inserted in the first column: vf means very feeble, f feeble, m moderate, s strong and vs very strong. In the second column are inserted, expressed in tenths of mm, the distances of the intersections of the interference-lines with a plane through the axis of the incident Röntgen-beam and perpendi- cular to the axis of the preparation, to the point where that first axis, prolonged, would meet the film, measured on the film when developed on a plane. In the third column the values of sim’? '/, 0 are ranged, as computed from the above-meutioned distances and the data of the apparatus, which were given in Comm. N°. 1. Now the values, given in the third column must undergo a 495 TABLE IS 0 pn un ” >| ES sin?!/,6 | corr. | 5 Zi EN ER 8 ee sin2'/,g correc- | sin? a | 3 5 zZ € | z < ai boom #| 22/29 24 = ; | 9 0.0924 0.0790 vf 208 | 0.1393 0.1243 Ee toe 1.0.1. vs 2285 0.16635 0.1502 | 8.7 9 ese Eotyt f 280 | 0.2424 0.2253 | 13.0 13 | 3.2.0 s | 311 | 0.2928 | 0.2762 | 15.9 | 16 | 4.0.0. | 3.1.1. lf | 322 | 0.3114 | 0.2945 | 17.0 | 17 | 4.1.0 vf| 3495| 0.3591 0.3433 | 19.8 20 | 4.2.0 m| 362°] 0.3813 0.3653 | 21.1 | 21 | lvs| 390 | 0.4319 | 0.4173 | 24.1 | 24 3.3.1. | 0.0.2. vi|. 416 | 0.4794 0.4666 | 26.9 | 27 2.0.2 m| 4385] 0.5208 0.8079 | 29.3 | 29.5 fs | 464 | 0.5675 | 0.5565 | 32.1 | 32 | 4.4.0. | 5.1.1. 2.2.2. 's | 5005] 0.6332 0.6243 | 36.0 36 6.0.0. vs! 5385| 0.6990 0.6922 | 39.9 40 | 6.2.0. 5.3.1. | 4.0.2. vf} 554 | 0.7248 | 0.7182 | 41.4 | 41.5 ‚s | 5865] 0.7687 | 0.76405] 44.1 44 “zi vf| 597 | 0.7923 | 0.7878 | 45.4 45.5 | \vs| 627 | 0.8352 | 0.8325 | 48.0 | 48 f | 640 | 0.8525 | 0.8499 | 49.0 | 49 7.0.0. RE ml 600 | 0.9114 | 0.9104 | 52.5 | 52.5| 6.4.0. £ | 7215| 0.9415 | 0.9406 | 54.2 | 54 | m\ 7575| 0.9686 | 0.9686 | 55.9 | 56 BEET 4.4.2. vf} 7905] 0.9863 | 0.9863 | 56.9 | 57 | 496 correction?) in connection with the thickness of the preparation. This correction is not needed, when the absorption of the charac- teristic rays in the preparation can be neglected, which was the case for many of the materials, which were till now examined by the method of Desue and Scnurrer. This does not hold however in our case (Cu-radiation), in which the radiation penetrated only to a very small depth, practically not at all in the preparation. The corrected values of sin?'/,@ are found in the fourth column of the table. During the computations for the drawing up of a formula for sin? '/,@ it appeared that the majority of the values in the fourth column, in dividing them by 0,01734, give a quotient, which differs only little from a whole number. These quotients are found in the fifth column. The values of these quotients, smoothed to half a unity, are inserted in the sixth column. Minter?) has measured the erystal-form on crystals obtained by electrolysis. It appeared to belong to the tetragonal system with a proportion of axes a:a:e = 1:1:0,8857. So it lay at hand, to try to find agreement of the fourth column with a formula of the form: sn? JO —= Ah? +h’) + Bh? = ACR? + A,’ + gh,’); A3 in which A= Zee and A Pt By « and ¢ is meant now in this a C the edges of the elementary parallelepiped or of the elementary cell. It lies also very much at hand to choose for A the above- given value 0,01734, and so to seek further agreement of the numbers in the sixth column with an expression h,* + h,* + gh,’, in which evidently q was then to differ little from a whole number. By choosing q = 6, a proper agreement appears to be obtained. In order to see this, it is advisable to compose a table of the values 1) Godt also P. Degie, Phys. Z.S. 18 (1917) p. 5. For the value of this correction we found (to -diminish the value of sin? 1/, 6 eee pin? | + + cos @ + VAS zi © cos 8 see Here r means the radius of the circular cross section of the bar, & that of the camera, d the distance of the centre of the camera to the opening of the screen, by which the rays enter. Second and higher powers of r/R are neglected. (Added during translation: The deduction of this formula has in the mean time been inserted in A. J. Bir. Thesis for the Doctorate, Utrecht 1918 p. 22). 3) W. H Mruver, Ann. d. Phys. u. Ch. 58 (1843), p. 660. See also H. von FouLLon. Jahrb. d. Kais.-Kön. Geol. Reichsanst. Wien 54 (1884), p. 367. > 497 ‚of hi? +h, for the different combinations of h, and h, ranged in the succession of increasing values of 4,7 + /,?. On comparing (his series of values of h,* + h,* with the numbers of the sixth column, the index-triples with h, = 0, given in the seventh column (not printed in italics), are found to satisfy. By adding 6 to all the values of A? +h,’ and comparing again with the numbers of the sixth column, in the same manner the index-triples with A,—1 of the eighth column are found. In a similar manner the ninth and tenth columns are composed. Apart from the first four lines, in connection with the deviations which exist for the small valnes of @ (comp. p. 498), of the remaining 19 lines, there are only 2, which do not belong to index-triples (not printed in italics), of which not all the indices are either even or odd. On considering whether all the lines to be expected, which fulfill this condition, are present, it appears that 5 of them still ought to appear. The index-triples meant here, are collected and printed in italics (columns 7—10) and assigned to those lines, of which the place is nearest to the place expected for these lines. The difference between the places, where these lines are found and those, where they would be expected, is in all those cases minute, and can for two cases be ascribed to the fact, that the line meant, cannot be seen separated from a neighbouring line and in all cases does not exceed (in connection with the small value of A) the uncertainty of observation. There is evidently indication enough for accepting a structure that is in accordance with the exclusive occurrence of interference- lines, which must be ascribed to reflections on net-planes with only even or only odd index-triples. ~The absence of index-triples of which not all indices are either even or odd, points to centered sides. From the atomic weight (119,0), the specific weight at 18° C. (7.285), AvoGapro’s number (6,06.107*) and the wave-length of Cug, (1,541,10~%) follows with A=0,01734 and ¢=6,06 ') (which value later on appeared still better to satisfy than 6), for the number of atoms per cell the value 3,02. Now, we come to the number 3 by placing atoms in the corners of the cells and in the centra of the prism-faces. 1) [Note added during translation]. By a somewhat different method of calcu- lation A. J. Br, lc. p. 33 obtained the value 5.988. This fact points to a greater probability of g=6 accurately. In this case the projection of all the net-points on a diagonal-plane through the smallest edge of the elementary.cell would be a net of equilateral triangles with side 2,37 . 10-8 em. (Comp. A. J. Brit, le. p. 34). 498 This is, so far as we know, the first example, in which atoms are found in the centres of the prism-faces, not however in those of the bases. In excess of the above-mentioned triples, still other triples must in the case of this structure give occasion to interference-lines. In this case namely the structure-factor becomes S= | + et lat he) 4 + gris th), The value of this is 3, when all indices are either even or odd and in the other cases 1 or — 1. So it is evident that planes the indices of which do not fulfill the condition mentioned, will be present as interference-lines, though, under equal circum- stances for the rest, with */, only of the intensity of the other lines. So one cannot wonder that a few of this kind of planes can give sufficient intensity to the interference-lines and that their index- triples are present in colums 7—10 of table I. In order to be able to judge about the agreement between the place and intensity of the interference-lines, which are to be expected on the basis of the given structure and the observed ones, a table has been drawn up, in which the values of sim? 40, as well for Cux.-, as for Cuxs-radiations, are collected, as computed for all the index-triples that come into consideration, with their relative inten- sities. Of this table I] forms an extract in which are omitted in the first place all the triples for which the structure-factor not equals 3, and in which for the rest out of each group of index-triples which can be esteemed to produce lines which lie so near to each other that these together will give on our Röntgenogram only a single line, only that triple is noted that is the origin of the most intense line of that group. If such a group contains a- as well as B-lines, then the strongest only of both kinds is given. *) Deviations between computed and observed values of sin? */, 6, which are larger than corresponds to the degree of accuracy of the measurements, only occur in table [l at the first three lines. About this it must be remarked that these lines are very hazy, which, in connection with the veiling of the film as a consequence of the action of white radiation in the Röntgenbeam makes the reading 1) During Comm. Nr. 1 being under press we learned, that Dr. ScHERRER and Prof. JoHNSEN had taken a Röntgenogram of white tin and Prof. JOHNSEN proposed to us, to exchange the observed values mutually. The results, then communicated to us cordially by Prof. JOHNSEN, give, especially at lower values, all of them some- what smaller values of sin? !/,4 than ours. Perhaps this might be ascribed to the correction applied by us for thickness of bar being somewhat too small. In the notes added to table II a few of the differences between the results of SCHERRER and JOHNSEN and ours are discussed. | . | 2% goaded Sees |e” ,| ge% © le“ |8e8| « | B on jd zal 4 = 28 B sjat 5 ae eats << E 8 x= Bo 2 5 u <= Sao | 2 5 v = Sia oe ae Meg ellie oe [EES | 3) One 3) BN) a4 a4 om | vf 124 113 | VS 150 140 9.0 Pit | | f 225 | 225 | 9.0 8 iyi s 276 278 9.0 Si 280 3.6 420 f 2945 feeble z-line vf 343 347 3.6 420 Slk \3.0 331 m | 3652) | feeble Z-lines | vs © 4173) | 417 3.0 331 | vf | 467 road aac ER om 508 4) 490 | 2.6 202 504 1.0 600 s | 5565 556 | 4.5 511 561 3.6 531 ie 624 6e TD 600 G20 REE Ee 422 | vs | 692 695 3.6 531 | vf | 718 joepie eats 640 | } s | 764 fol Sr ceo a A | | | vf | 768 | | 15 | 2.6 Ta. | vs 8325 feeble @-lines Sl f 8505) AA, NT 12 602 | ‚_m 9106) 992) Joindgei (640 897 2.25 TE | | 900 2.25 622. | 904 2.25 313 | f | 94 | | 953 kel B om 969 - Maker ee a 711 ad | vf | 986 | feeble @-lines 1) In SCHERRER and JoHNSEN’s results this line is preceeded by f 57, which may be considered as 56 9 200 2. The values which S. and J. give for the first three lines, vs 72, m 116 and vs 143 namely, give better agreement with the places of the strongest lines, calculated, than ours. 2) Lines 343 and 365 are almost not to be separated on our film. They are given by S. and J. as one line, s 348. 3) S. and J. give 4045, presumably by stronger forthcoming of 202 2 (at 396) intensity 2,6. 4) s 491 of S. and J. 5) At the lines 8325 and 850 the film shows almost uninterruptedly points from 819 to 866. S. and J. denote the intensities with s to m and f. 6) S. and J. have s 906. 33 Proceedings Royal Acad. Amsterdam. Vol. XXI. 500 somewhat uncertain, and that further the correction for thickness of preparation brings with it a relatively large uncertainty for these lines. Mig: 1. From the given values of 4 and q, there follows for the edges of the elementary-cell: a — 5,84.10-% and c = 0,406 a= 2,37.10 8 em. In fig. 1 a representation of an elementary-cell is given. In the space-netting, built of these cells, we see alternating equidistant layers with distance 1,19.10-# em. The first layer has a netting of squares with side 5,84.10°% em, the next a netting of squares with side 4,138.10 8 em., which last squares are just above the squares symmetrically inscribed in that of the first layer. The dense crowding of these planes indicates a strong force- exertion in a direction, perpendicular. to the layers; perhaps the occurrence of needle shaped combinations is connected with this fact. Röntgenometrical investigations teach us to cast a look in the structure of the erystal and may for that reason lead us to a more rational choice for the system of erystallographic axes than the former erystallographie methods. The white tin procures an example of this. So e.g. the bipyramid, accepted by Miner as (111) (proportion of axes a:a:c=1:1:0,3857) would, conform to the system of axes formed from the edges of the elementary cell proposed by us, be indicated as (403); just so the planes (110), (100), (101) of Miner by (100), (110), (223) resp. according to us. Physics. — “Investigation by means of X-rays of the erystal- structure of white and grey tin. IM. The structure of grey tin”. Communication N°. 25 from the Laboratory of Physics and Physical Chemistry of the Veterinary College at Utrecht, by A. J. Bui and N. H. Korkmever. (Communicated on behalf of Prof. W. H. Kersom, Director of the Laboratory, by Prof. H. KAMERLINGH ONNFS). (Communicated in the meeting of September 28, 1918). In this communication we shall deduce the crystal-structure of grey tin from the Röntgenogram, taken by us and represented by a drawing in Communication N°. 1 (June 1918 p. 408) at the same time as that of white tin. With this aim in the first column of table I are given the distances of the intersections of the interference- lines with a plane, through the axis of the incident beam ot X-rays and perpendicular to the lengthdirection of the preparation, to the point where this axis, prolonged, would meet the. film, measured on the film when developed on a plane and corrected for thickness of preparation (Comp. Comm. N°. 2a, p. 496). The inten- sities are denoted by vf very feeble, f feeble, m moderate, s strong, and vs very strong. The second column contains the values of sin*+9 deduced therefrom, (radius of camera 27,3 m.m., to be diminished by half the thickness of the film, that is by 0,1 m.m.). By multiplying these values by 0,808, the square of the proportion of the wave-lengths of Cu,, and Cux the values which might originate from g-radiation, can be separated, and so a list can be composed which contains «-values exclusively. By using a general method of computation, in which we proceeded from an arbitrary system of axes and on which we shall perhaps publish another paper the grey tin appeared to be regular with great probability. In that case our values must fulfill the equation : oie’: 3 FS sin® 5 = Al EM ER) = i (h,?4 h,*+h,’), in which A is a constant, h‚‚h,, and 4, indices of Miter, 4 wave- length of the Röntgen-radiation, « edge of elementary cube. So we have only to deduce the constant A and from that the number of atoms in the elementary cube a’. A provisional calen- 33* 502 TABLE I. a Cu K. -radiation | Cu x -radiation E : a = =) - ! (an) Mer. Sea = 4 en SZ ” En OR Bn == 2 ze Mi 8 |Een sl Steele ien = 5 ple Bees Pe al ue a s Es = ST a ee a7) H 9 re aL ee a A ES f 112 41 a kad f 1445 69 43 RDE kt vf 168 | 92 92 \220 m 191 … 118 114 15 |220| 8 126 |311 m 221 | 156 156 EERDER ee vf 239 | 181 | 184 |400 f 212 230 231 0.4 400 16 m 300 2745 270 0:6 (3311719 2155 422 s 342 346 341 1.0 |422| 24 | m370 | 3955 | 384 | 06 SET) 27 || 402 [5 at | £406 461 455 0.4 1|440| 32-1) een m 426 498 497 0.1 1531 35 | s 464 567 568 0.6 |620| 40 | £488 611 611 0.3 1533| 43 f 509 648 | 643 |642 £530 684 682 0.2 |4 44 | 48 en 9 3 2 Arlen) | | m 554 ‘ 7125 725 055) Pl vs 5906 ~—s 7921 796 020°) Bee ling (55 3 vs 626 834 838 Doude 1450 555 vf 655 8715 862 7 5 1 £678 898 909 0.11800 | 64 £694 915 918 840 "m 7305 949 952 0.2 6733 61 hy? + hy? +-h3? wo 11 16 24 35 40 59 15 80 503 lation pointed to about eight atoms per cube, which suggests the structure of diamond. So we tried to get agreement of our interference- lines with the structure of diamond. Using the atomic-weight (119,0) and the specifie weight of grey tin at 18° C. (5,751), AvoGapro’s number (6,06.107*) and the wave- length of the A-series of copper-radiation (« 1,541.10 8, 3 1,885.10-5) A, = 0,0141 and Ag = 0,0114 are found. Nearly in agreement with this we find as a mean from the observed lines A, = 0,01422 and Ag—0,01149. The third column gives the values of sin? '/,/, computed with the last-mentioned values of A,, so far as they give occasion to interference-lines. From DesiJe’s structure-factor it can be deduced namely, that for the structure of diamond interference-lines can only be got, when /,,A, and A, are all even or all odd, and when in case they are even, their sum can be divided by 4. The expression h‚* + h,? + h,? then becomes 3, 8, ll and so on, see the sixth column; in the fifth column the corresponding values of 4,,, and 4, are given. The intensity, given in the fourth column is found by dividing 64 (A? + ht + A,°) in the product of the number of planes and the square of the absolute value of the structure-factor. To this intensity only so much value must be attaebed that only the inten- sities of three successive lines must be compared. The seventh, eighth, and ninth columns relate to g-lines. On comparing the observed and computed values of sin? '/,0 a satisfactory agreement appears to exist, also in connection with the intensity. The two smallest squares of sines are not satisfactory '). About this Comm. N°. 2a p. 498 gives information. When it is examined whether all lines, which must be expected on the ground of the structure mentioned, are present, it appears, that of the expected a-lines not a single is lacking; some of the B-lines however appearingly have too small an intensity, and so are not observed on the Röntgenogram. From the values of A, and Ag, deduced from the observations, we found for a, the edge of the elementary cube 6,46.10-8 cm. at 18° C. For the distance of two nearest atoms 2,80.10-S em. is thus deduced. In the grey tin with its diamond-structure, which silicium pos- sesses too, the tetravalency clearly makes its appearance, whilst in the tetragonal modification, in which each atom is surrounded by 1) Comp. P. Degije, Physik. ZS. 18 (1917), p. 488. Note. 504 two nearest ones with smaller distances than those of the rest of the surrounding atoms, two valencies especially come to the foreground. In this connection it might be interesting, to investigate, whether the deviating conduct, observed by Grinetsen') in the course of the conductivity at low temperatures for cadmium, tin, and mercury, and which he ascribed to the metals not being regular, is not found with the grey tin, or whether the diamond-structure still causes a deviation in the conductivity at low temperatures from the conduet of the metals crystallising in cubes with centered faces. We heartily thank Prof. Kersom for the great interest, which he always showed us. 1) E. Griineisen. Verh. D. physik. Ges. 20 (1918), p. 36. Physics. — “On the equivalent of parallel translation in non- Euclidean space and on RiwMann’s measure of curvature.” By Dr. A. D. Fokker. (Communicated by Prof. H. A. Lorentz.) (Communicated in the meeting of April 26, 1918). 1. Introduction. In the following pages I shall try to give a mental picture of some ideas recently developed by prof. J. A. SCHOUTEN before the Mathematical Society at Amsterdam which will help to illustrate the meaning of a “system of axes moving geode- sically”, and the “geodesic differential”, together with a few applica- tions.') The great point will be to realise in a new way wat kind of displacement in non-Euclidean space must be considered to corre- spond to a parallel translation, this being an operation indispensable in vector-analysis to compare vectors in different points. One of the characteristic properties of pure translations is this, that all points of a rigid body are thereby transferred over an equally long distance. This property might be used to define a parallel translation, provided the rigid consists of a number of points exceeding a certain minimum. If, for example, in three-dimensional space, we give a prescribed displacement to one of the points of a rigid system consisting of two or three points, it is not enough to demand an equal displacement for the other point or points to define a translated position without ambiguity. But in a Euelidean space of 2 dimensions other motions than pure translations are excluded, if for a rigid body of no less than (2n—2) points we want all points to run through equal distances. This will be our starting-point. We know, however, that in general no body of finite dimensions can move in curved space without changing the mutual distances of its points. In order to retain the idea of a rigid body we shall have to confine ourselves to bodies with dimensions of the order of an infinitesimal «. Another and more serious difficulty arises from the faet, that we cannot get all points to shift over exactly the same infinitesimal distance A. We cannot but leave a margin of the order of Ae? for the separate distances. Here the question arises whether in a certain 1) Cf. a treatise offered by Prof. ScHOUTEN to be published in the transactions of the Kon. Akademie: * Die directe Analysis zur neueren Relativitdistheorie’’. 506 direction only one displacement can be effected in which this approxi- ‘mation to the exact equality is realised? This, however, cannot be expect- ed, since in the special case of Euclidean space not only pure transla- tions but serew-displacements too are allowed by leaving this margin. Therefore a second property of pure translations is required, fit to exclude these screw-displacements. This property is found in the fact that the shifts are not only equal, but also parallel to one another. This amounts to a certain reciprocity between translations in different directions. Consider two translations, by which a point P is transferred to neighbouring points Q and A respectively. The first translation will carry point & to the same place where the second translation will carry point Q. This property indeed excludes screw-displacements. In the following pages we shall first give a summary of the results arrived at in this paper, and afterwards (§ 6) give the ana- lytical formulae. For examples we will mainly take those of three- dimensional space. The results, however, will hold good, independent of whatever number (nx) of dimensions we choose to ascribe to our space. 2. (reodesic displacement. Let us define an intinitesimal rigid as an aggregate of particles,which keep their mutual distances unchanged during their motions. One of these points we may chvose as a central, and imagine the other points defined by the ends of infinitesimal vectors from this central point, these vectors having constant lengths (of the order «) and including constant angles. The number of points must be no less than (22—2), hence the number of vectors (2n—3), no n of them being situated together in a space of (n—1) dimensions. We imagine this rigid to execute motions so as to shift the central particle from a starting point P to neighbouring points over distances of the order A. It appears possible (§ 7) to indicate a certain variety of motions in which, firstly the shifts of all the other points of the rigid, up toa margin of the order Ae’, equal the shift of the central point, and, secondly, there exists a certain reciprocity which becomes apparent when we observe two arbitrarily chosen motions belonging to the variety, which shift the central particle, let us say, from P to Q and from P to R, and when we notice the displacements of the particles having their starting points in R and Q respectively. The particle from R in the motion (PQ) will reach the same point attained by the particle from Q in the other motion (PR). The two conditions specified determine without ambiguity a variety 507 of motions which we may call ‘geodesic displacements’ of the infinitesimal rigid. They are the substitutes for parallel displacements’) in Euclidean extensions. We may assign the name ‘“compass-rigid” ‘to a small rigid body that cannot move but in the geodesic manner defined. It must be understood that a compass-rigid which, after a displacement, returns to its starting-point by the same way, will on arrival be in its starting-position too. If, however, it returns by a circuit, it generally will not be in its starting-position again on arrival. 3. Geodesic differential. If we want to compare two vectors in neighbouring points P and Q, we can proceed as follows. We put “& compass-rigid with its centre in ZP and by marking one of its points we delineate the vector in it. Now displacing the compass- rigid to Q it is reasonable to say that the marked point defines the vector displaced geodesically from P to Q. By comparing this vector with the one present in Q we immediately see the meaning of the geodesic differential of a vector. If this is known, it is clear what CHrIsSTOFFEL’s covariant differentiation means. In the same way we can displace our vector-units from P to Q. In general these will differ from the vector-units in Q. A set of geodesically displaced vector-units is what Prof. Scnourrn defined as a system of axes moving geodesically. 4. (Geodesic line. We can easily imagine what we have to do in order to prolong a given line-element geodesically. We put the centre of the compass-rigid in the starting-point and mark the end of the line-element by an arrow in the compass-rigid. After the centre has been displaced along the line-element, the arrow will point in another definite direction. This is the geodesic prolongation of the element. Continuing to move the compass-rigid in the direction of the arrow, the centre will gradually describe a geodesic line. In this case, during displacements along a geodesic line, vectors moving -geodesically will continue to include constant angles with the geodesie (cf. Luvi-Civita’s article), these angles being fixed angles in the compass-rigid. 5. RigMann’s measure of curvature. Let us now suppose that we 1) Taking another starting-point, T. Levi-CrvirA arrives at a definition of parallelism which comes to the same thing: “Nozione di parallelismo in una varieta qualunque, e conseguente specificazione geometrica della curvatura Riemanniana’. Rend. Cire. Mat. Palermo, XLII p. 1, 1917. His geometrical explanation of the measure of curvature, however, is totally different from the one we shall give in section 5. 508 make a compass-rigid describe a small circuit, e.g. along a vanishing (quasi-)parallelogram. We already pointed out that in general it will not return to its starting-position. The difference between the two positions is such as might have been produced by an infinitesimal rotation around the starting-point. The amount of this rotation is proportional to the area of the circuit described, the orientation of the “axis” of rotation (which in higher extensions is of (n—2) dimensions) being determined by the orientation of the plane of the circuit. The rotation is intimately connected with the curvature of space. When this rotation of curvature, as it may be called, vanishes in all points for every arbitrary circuit, then the » space is Huclidean. *) « The components of the operator by which from the data of the area included by the circuit the rotation of curvature for the compass-rigid can be derived, are Rimmann’s four-index-symbols, of the second kind. Further — to confine ourselves to three-dimensional space — if we take the length of the axis of rotation equal to the amount of the angle of rotation, and construct a parallelepiped with this axis and the parallelogram of the circuit, we can consider the ratio of its volume to the square of the parellelogram as a measure for the curvature of space. Indeed, in the limit, for a vanishing circuit, this ratio is just the number indicated by Ritmann as the measure of curvature of the space with respect to the plane of the circuit considered. 6. Now we shall proceed to give the required formulae. We take the length of a line-element.as defined by ds?== = (ab) garde? det , det de? representing increments of the coordinates along the line- element dx, gas (= gia) being regular functions of the coordinates of the starting-point, each index in the sum going through all the values from 1 to n, where mn is the number of dimensions of space. The algebraical complements of g,. will be denoted by g°, so that ) ‘Ihe fundamental idea of a recent article by H. Wey (Gravitation und Elektrizität, Berl. Sitz. Ber. May. 1918) may be considered the hypothesis that a small rigid (= “Vektorraum” after turning about an infinitesimal circuit of “trans- lations’ (of a somewhat more general kind) not only will have got in a changed position, but in general will have changed its dimensions as well. In four-dimensional space-time the linear dilatation of the (4-dimensional) rigid would be half the scalar product of the alternating electromagnetic tensor and the area included by the circuit. (Note added during the revisal of the proofs). Kd 509 Zie tot na, TO for n =a For the sake of brevity we shall write g for the determinant formed of the gy. Further we shall avail ourselves of CHRIsTorrEL’s well-known symbols: lm OGan … Odtard es Ots Im | 3 pa == ne SS sya! b | i Ee + dom = dx | | b \ (9 a The definition of the line-element entails the definition of the > (b) gas gr” length of a vector v, with components v*: v? = (ab) gas v4 vb, and the definition of the angle between two vectors v and w vw cos (vw) == (ab) gap vt wl. 7. Let the points of a small rigid be given by their coordinates relative to the centre: ue, vt, wt... (a = 1, 2, 3, n), these being the components of vectors u, v, w... which are of the order of a vanishing quantity «. If the centre shifts from P to a neigh- bouring point Q, determined by the infinitesimal increments in the coordinates det (of the order A), then we require the new coor- dinates of the points of the rigid relative to Q in order to satisfy the definition and first condition of section 2: the points are to be points of a rigid, and must each cover an equal distance. Denoting the new relative coordinates by wt + du", vt + dv" … etc. it is easy to formulate the latter half of the condition. For the increments of the coordinates of the point designed by u will be dat + dut, and the starting point of the line-element through which it runs lies beside /, at a distance defined by u. So, if this line- element is to be equal to that from 7 to Q, ie i= = (ab) Jap dx" dx, “we necessarily must have Ogab 0 = ZS (ab m) aa = gm If the aggregrate of points is to form a rigid, both the lengths of the relative vectors u, v, w.... and the included angles must be constant, and expressions such as ‘um da? dot + 2 (ab) 2q.yde¢ dub . . (U) u® = (ab) gum ut U”, -, uveos (uv) = & (am) gam ut v™, must have the same value in P and Q. This implies le am 0=—2(am Dail delut um + S(am) 2gam ut dum, . . . (2) 510 and NN (a 75 ee Jam dalut ym 4 (am) Jam Sut dom 4. ym dua br (3) These are the ah ae which must be applicable to du”, dv... etc. in the translations mentioned. It is not diffieult to find espressions satisfying the equations. We can add to (1) the identity 0 aim a) m 0 = = (alm) & ha en ze) dat dielur, Al and a similar identity to (2): Ogai:\, Gln 0= 2 (alm jut dalu», e ) (5 Oud Replace at the same time the index b in the first term, right-hand side of (1) by 4 and m in the second term, right-hand side of (2), by 6, and we get gam d a 0 m ; ate | > vm (ee ee Me Ee) deel un + 2(6) agu de | der, Oem One and 0 am 00a d m es (ay llen aa OER | det wm + (6) Bony dur? | wt Ow” dx = Dividing by 2 we can reduce the equations to the form Je Pp dn ‚Lm d 0 = 2 (ab) gay dat | dub 4 ie) 5 dakum sb, ed nes tal) l | 0 = & (ab) aan ut | du> + (lm) a dav! u Ati: (2') Similarly, we can put for ‘the third equation hoes Glee fw [deb 4 =} daten Tor [dut 4 >) ae wn 1| (3') So we can satisfy the imposed condition by taking du’ — — zm || daly. OW ak m2) ls and similar expressions for dv’, dw”. The equation (4) is covariant. It will retain its form wathever be the choice of coordinates. | It defines the position of the points of the small rigid when, by a first approximation, they have all covered the same infinitesimal distance. It is seen, from (1), that in developing gas into a series we have neglected terms with produets wer u". The squares of the distances covered therefore can differ from PQ by an amount of the order e? 4*, so that the distances may only be taken as equal up to an amount of the order «7 4, which we shall neglect. | 511 8. The “corrections” given by eq. (4) are of the order Ae. In order to see if the solution defined by them is the only one, we may ask if we can satisfy the equations by “corrections”. differing from dut, such as du* + du“, where dut is of the same order as du. If these are to satisfy eq. (1), (2), (3), we musi evidently have. 0 = Zab) gas dx du? , 0 = Sab) gas ut dub, 0 = (ab) gas ut dv? + gay v* du , etc. If the rigid, besides the centre, consists of p points, we shall have 2p + 4 p(p—1) equations for pn variables dut, dv... ete. For p = 2n—3 we have as many homogeneous linear equations as there are variables. If no set of n vectors u,v,w... are situated together in an (n—1)-dimensional space, then these equations only permit a solution of the form (e.g. for 2 = 3): Bis Pee ie ee agar | eee (5) V9 | Io Ges | V9 uy u where w is an arbitrary number, and by 6,c,a, we mean a set of three indices which form an even permutation of 1, 2, 3. We denote by dx, and u, with lowered index the covariant combinations: da, = Say det, wy, == 2) Jij Uj. It can easily be ascertained that the expression given for du", together with similar expressions for dv, dw“... satisfy the equations. They must define the displacements in the case of an infinitesimal rotation about dx as an axis'). For all vectors u,v,w... keep their lengths unchanged and both the angles included with dx and the mutual angles remain unaltered. Since the conditien imposed thus far appears not to be sufficient to define a displacement without ambiguity, we must recur to the condition of reciprocity of section 2. Shifting the centre of the conipass-rigid from P to Q the particle designed by u might come from a position A into the position defined by the coordinates a, + det + ut + due + due, or, w | dz, de, | V9} uu If we now shift the centre from P to R, and we then wish to find what will be the new position for the particle from Q according Um af + dat + ut — (lm) | dtm + a 1) Through an angle »/|da'. 512 to the same displacement-law, we only have to interchange the vectors dx and u. Now we see, the determinant changing its sign, that this position will never be the same as that reached by the former particle from R, unless w = 0. So the application of the condition of reciprocity excludes screw- movements *). 9. Now in the following way we can see that the condition of the body’s rigidity and the equality of the covered distances together with the condition of reciprocity are sufficient to define the variety of geodesic displacements without ambiguity. From eq. (1) and (2) we learn that the required ‘“corrections”’ du“ must be proportional both to the components of the displacement dx and of the vector u. Therefore let us put dut = Shy det uly > | eee Ch ee ee (4’) Now, according to the condition of reciprocity we must apparently have =— his . Substitute (4) in (3), and we get Oy = Yam mt mn a 0 = X (alm) Ral dx! ut vm + J (amst) Jam ths ut das ot 4 hey ve" das uj}, v Taking other indices and putting es b hat = 2(6) Jab him (hain = hal F we get 0 r=. (alm) dal ut om ir == hain Sn hina U In this equation we may regard the forms in brackets as unknown vari- ables. Because of the svmmetry in the indices a and m there are 4 n° (n—1) of them. As the equation is to hold for an arbitrary combination ') Dr. DRosTE remarked to me that a serew-motion might be excluded in a different manner. Let PQ be part of a geodesic. In P and in Q take two infini- tesimal planes perpendicular to the geodesic. Draw the geodesics perpendicular to the first plane and intersecting it in a line-element PR. These together form a “geodesic strip”, which will intersect the second plane in an element QR’. PR and QR’ can be called “parallel” and in the same way each pair of elements in the same geodesic strip including equal angles with the geodesic PQ. In our chain of thought, however, geodesic lines are defined by making use of the idea of geodesic displacements (see section 10), and so we cannot avail ourselves of the above suggestion. 515 of three vectors dx, u, v, we conclude that the variables must vanish. Thus Odam = —— + ha lm + hina. Oa! i Similarly 0g ¢ == = 4 hima ab hail Òg 1m . = re + An ai + ham . By adding the first two, and subtracting the last, and considering that Aram == Alano ete. we find “lm h Hi = sae Alls Now we know l l hee a geb ham > ge | A En —| a a b and so from (4) we see that our values for duê: mx im du) = — (lm) 5 EO eg re (cS constitute the only solution consistent with all conditions. 10. To explain the applications of sections 3 and 4 we proceed as follows. Suppose a vector V, in point 7, be marked in the compass-rigid. After a geodesical displacement to Q the marked vector will have got the eomponents lm Va— & dal Vm, a Now if we have in Q a vector with compunents Ve + d Ve, where dl“ now represents some increment of the component Ve, then obviously the components of the geodesic differential are lm dVe + E(lm) (a Vm, a This geodesic differential wiil be a vector itself, being the difference of two vectors, while dV“ are no vector-components. If the line-element PQ itself is drawn in the compass-rigid as a vector with components dv“, and displaced geodesically, then in Q the arrow will have got components det — 3 (lm) | „ ydeldem. This arrow we have called the geodesic prolongation of the 514 element. It is easily seen that this entails for the geodesic the equation l 0 = d'art + 5 (lm) 4 da! da a This (covariant) equation coincides with what we get from the familiar definition of a geodesic as the shortest line between two points. 11. We shall now displace geodesically a particle P’, which, with relation to P, is defined by a vector u, to a point S’ near 7, by shifting the compass rigid in two steps from P to 7, along PQ and Q@7. Then, a second time, we displace the particle to S" near T, taking the steps along PK and KT, the quadrilateral PQ7'A being a (quasi-) parallelogram with sides dx (PQ and K7’) and dx (PK and QQ 7 After the displacement along PQ the coordinates of the particle considered relative to Q have become (lm ut — =e dx! um , a At the second step we must be careful to take the values of CrISTOFFEI’s symbols at point Q, so that after the displacements along PQ and Q7' the coordinates relative to 7’ are lm lm Pq 0 yt 5 datur — EX Ja! wn — > aot, ere a a m \ Ou If the displacements had. been performed along PK and KT, the coordinates relative to 7’ would have been. 0 \Pq tern | — ae m “RP lm . dardalu” . a lm ud ioe devdaelur. a lm ut 5 a dln — > een da! & <= =) a Taking the difference we find l l Se = Li(lmp) E a — 2 (n) | a i | (da! der —derdel) un. or Be = 4 Emp) ò (lm) B pra, pnjylm) > | ln pm ke dup | al elle Ar |» Last m X (da! dev dards!) uP On mona ian utter ae) Se The first factor is seen to be a RigMann’s four-index-symbol, of the second kind. Availing ourselves of his notation we can put (da! dap —darda'jum. . . . (6) Gt == } Y(lmp) {ma, lp The "(a= 1,2,...n) are the components of the displacement- 515 vectors which would become manifest after a geodesical displacement of the compass-rigid about the circuit PK PQ7. The displacement cannot be anything else but a rotation, the lengths and angles remaining the same. We see how Rigmann’s symbols determine the rotation in terms of the components of the area of the cireuit. This rotation is characteristic of the curvature of space. 12. It now remains to prove the statement of section 5 as to the interpretation of RigmMann’s measure of curvature. The measure of curvature with respect to the plane of dx and dx is defined to be’) Za lp mq) gag ma, Ip} (da! dar — dada! (dem daa —daxidu™ ) (lp mq)(GimIpq-- Gq Ipm)(del der — dar? dae! \(dam dur — dv ada”) The denominator is four times the sqnare of the area of the parallelogram formed by dx and dx. For by changing indices with- out changing the sum we get four times ) | sim 9lg | Ipm Ipa Writing d for the length of dx, and d for the length of dx, we find for the denominator E (lp mq da! dam derdat, d? dd cos (dd) | dd cos (dd) cS this being four times the square of the area of the parallelogram. We shall discuss the numerator for the case of three-dimensional space and show that it represents four times the volume of the parallelepiped formed by the axis of rotation and the parallelogram. Proceeding with some caution, the analogon in more-dimensional cases is readily found in the same way. We will put for the numerator ’ 22 (am q) Jag Rm (der dra—daaden) … … «ee (0) denoting by A, the coefficients of the rotation of curvature (6): be — E(m) Ru”. or, ceo — Bj) Bj wi. How are the numbers A; related to the components of the axis of rotation? If we suppose the components of the latter equal to HS then the rotation is represented, as will be presently shown, by 1 Jar Ybi e= Dj Vg Jaj Ibi 1) Cf. for example Brancut, Lect. on Diff. Geometry, section 319. REE OH opt (8) 34 Proceedings Royal Acad. Amsterdam, Vol. XXI. 516 Here we mean by c that index, which with a and 5 forms an even permutation of 123. By pgr we shall denote a similar set: abe'(=) par (=) 12s. We already saw that displacements of this kind constitute a rota- tion. To inquire whether the angular amount of the rotation is equal to the length of 1, we must observe the displacement of the end of a vector u which is perpendicular to 1, so that = (ad) 9748 ll. =D. ee This displacement ought to be u multiplied by |I. Let us cal- culate (ele LC? = Ter) gere &, 1 * | da shad Ici gaan = — (rijv w) | aj Jj oj | ud. | Á | Yar Ybr Jer Ip Iqu leu wo The summation with respect to c has been effected by writing in full the first determinant. If we want to sum up with respect to v, we notice that the determinant vanishes but for one special value of 7, which is different both from 7 and j. If 1=p, J=g, then the determinant becomes + g, if t=g, J = p, then we get —g. In both cases we may write Ipv Iq» C= B(pq vw) | lel’ uf uw, ~ IpwIqu and, by (9): : Ca a So the correctness of formula (8) has been shown. But then we are justified in putting =d zl 9 H |B, ; V9 Jaj Gbj and we can subsequently show that (7) represents four times the parallelepiped mentioned. We write (7) with a slight alteration of indices and we get: 25 (¢) k) gek R; (dai dxk—da*dx J) = 9 | Jat Ibi Aci | =— (Uk) gas 91 Gej U (dei dak_—dakdei ). Jak Ibk Iek Now if j and 4% assume all values, a set 7,4 furnishes just as much as a set #,), the determinant taking the value +g or —g according to the combination #7, % being an even or odd permutation 517 of 123, and vanishing for other combinations. So we get for the numerator ony Ag) dei dei dxk|, . , ue Be ALO} | dx! ded dek this being four times the volume of the parallelepiped formed by l, dx and dx. This sufficiently explains section 5. We may remark that formula (8) for the displacements of rotation implies a convention as to the direction in which the axis of rotation has to be drawn. The axis of rotation must be orientated in a manner to ensure that the direction of ¢ is correlated to the directions of 1 and u, i.e. a paral- lelepiped constructed from 1, u and &, in the order thus specified, must have a positive volume: ag |e el Vg | ut ub ut | = positive, Peer DEE iy This amounts to the same relation which exists between the directions of the positive axes of Y, Y, Z. One sees from (10) that the measure of curvature will be positive if the direction of the axis of the rotation of curvature bears the above-mentioned correlation to the directions of dx and dx. Similarly, in four dimensions, if the axis of a rotation in a special case’ be a parallelogram on the vectors 1 and m, then the rotation is given by | Ja: 9bi dei | LE RV nt | Jak Jk Iek | where abcd (=) 1234, and. the direction of $ is correlated to the directions of 1, m and u, ie. Perg ieee mag Ca m= mo me md | Vg | BED edel en pecans Ca Sb Se Sd | My thanks are due to prof. J. A. ScHoureN for his kindness in allowing me to read his treatise on Direct Analysis, which is to be published soon in the Transactions of the Kon. Akademie. 34 Astronomy. — “The Distribution of the absolute Magnitudes among the Stars in and about the Milky Way.” (First Communication), By Dr. W. J. A. ScrHoureN. (Communicated by Prof. J. C. K APTEYN). (Communicated in the meeting of October 26, 1918). 1. Introduction. One of the most important problems of statistical astronomy is to examine, how in each part of space the stars with bright and faint absolute magnitudes are mixed and of which percentage of stars the luminosity lies between definite limits. Luminosity means the apparent brightness, which a star would have at the unit of distance, and for this unit we will take, in imitation of Professor Kaprnyn, the distance corresponding to an annual parallax a = 0".1. The first determination of the luminosity law was performed by Pro- fessor Kapreyn and published in Publ. Groningen N°. 11°). After- wards several astronomers, employing different methods, have repeated this investigation. Besides the studies of Comstock and Watkey, who availed themselves of measured parallaxes, and whose results therefore are not much to be relied upon, unless great cautiousness is used, the researches of SkELIGER and SCHWARZSCHILD are well-known. In our dissertation?) we have discussed the three methods mentioned, and we have compared the results that were found. It appeared that serious objections may be raised against SERLIGER’s investigation, so that we cannot attach much value to the frequency-function of abso- lute magnitudes found by him. We think we have also demonstrated in our work just cited that the method of Kaprgyn is for various reasons greatly to be preferred to that of SCHWARZSCHILD. After we had finished this inquiry an earnest wish arose in us to establish the luminosity law according to the method which we think the best. It was known to us, that such a determination had been 1) Also in these Proceedings, Vol. III, p. 658. *) On the Determination of the Principal Laws of Statistical Astronomy. Am- sterdam, Kirchner 1918. 519 in course of preparation at the Astronomical Laboratory at Gronin- gen for a long time already. In this preparation are included the countings of stars of deter- mined magnitude which are published in Publ. Groningen Nos. 18 and 27, and the mean parallaxes of stars of determined magnitude and galactic latitude, which will soon be published and have been kindly lent us for our use by Prof. Kapteyn and Dr. Van Rain. It is a matter of course that the preliminary results obtained by us, should be replaced by those of the definitive solution as soon as these are available. Yet we thought we might publish our results as they are partly based on other data, because they give a notion of the exactness that is to be obtained now already, and our preli- minary results might be of some service perhaps until the time that the definitive shall bave appeared. In this communication we mention the results that we found, when we tried to determine the luminosity law according to Kaprryn’s method for the whole sky and for five zones of different galactic latitude. We intend to publish in a second communication the results that are found, when ScHwarzscHILD's method is applied to the same data. At the same time we hope to compare our results with those of other investigators. In this article we also gratefully tender our sincere thanks to Prof. Kapteyn for his kind help, which favoured our investigation. 2. The Data of Observation. In applying Kaprryn’s method we have to take from the obser- vations the following data: 1. the numbers No ie. the numbers of stars of determined apparent magnitude and proper motion, 2. the mean parallaxes zr, of stars of determined magnitude and proper motion, 3. the value of o, the probable error of the error curve log. */x, in which a is the real and z, the probable parallax. In our investigation we divided the sky into 5 zones. The galactic latitude we shall indicate by 4. Zone | = part of the sky between b = —10° and b = + 10°. Zone Il = , » » y with 6b from —10° to —30° and + 10° to + 30°. PAE RE ns iy Ta , 0 % —BO 3 n0P--,) = 30°", .-+- 50°: Be IN os oo wien git One ver Ok on. Jer GU? , +- 70°. Pre og ok Oe Ne 7D 90°. 520 We have derived the numbers No, from the numbers of stars of determined magnitude and galactic latitude by performing countings in catalogues of stars with known proper motions. We have computed these numbers NV, for the different zones from the tables in Publ. Groningen N°. 18, as these have been corrected in Publ. Groningen N°. 27. The results that Van Ruin published in the work last-mentioned, are very reliable, as they were confirmed by several independent studies. The results of Norr and Srares, indeed, agree fairly well with those of Van Run and the numbers of CHAPMAN and MELoTTE too correspond to the Groningen countings as soon as they shall have been corrected for a mistake in the method of reduction which has been pointed out by Van Ruy. In our countings the stars were divided according to their magni- tudes into groups of 1—2.9, 3.0—3.9,4.0—4.9,.... 12.0—12.9. According to their proper motions they were counted between the limits O—2".9, 3".0—4".9, 5".0—7".9, 8".0—9".9, 10". 0—14".9, 15".0 — 19”.9, 20".0—29".9, 30".0—49".9 and >49".9. The following catalogues have been used: 1. LL. Boss. Preliminary Catalogue of 6188 Stars, 1910. 2. A. Auwers. Catalog der Astronomischen Gesellschaft, Zone + 15° bis + 20°, 1896. 3. W. G. Trackeray. Greenwich 1910 Catalogue of Stars, Zone + 24°0’ to + 32°0’. Analysis of Number and Percentages of Proper Motions. Monthly Notices 77, 204—212, 1917. F. W. Dyson and W. G. Tuackrray. The Systematic Motions of the Stars between Dec. + 24° and Dec. + 32°. Monthly Notices 77, 581 —596, 1917. - 4. F. W. Dyson and W. G. Trackeray. New Reduction of GROOMBRIDGE's Catalogue of Circumpolar Stars, 1905. 9. J. C. Kapreyn and W. pr Sirter. The Proper Motions of 3300 Stars of different Galactic Latitudes, derived from photographie plates, prepared by Prof. ANDERs Donner. Publ. Groningen N°. 19, 1908. 6. J. C. Kapreyy. The Proper Motions of 3714 Stars derived from plates taken at the observatories of Helsingfors and the Cape of Good Hope. With the co-operation of Dr. H. A. Weersma. Publ. Groningen N°. 25, 1914. 7. Dr. A. A. RamBaur. A photographic Determination of the Proper Motion of 250 Stars in the Neighbourhood of = 443. Monthly Notices 78, 616—630, 1913. 8. A. van MaaneEN. Remarks on the Motion of the Stars in and near the double Cluster in Perseus. Report of the Nineteenth Meeting 521 of the American Astronomical Society. Popular Astronomy 25, 108-—110, 1917. ; 9 W. G. Tuackrray. Notes on some Proper Motions derived from a Comparison of CaRrrINGTON’s Catalogue with the Greenwich Places for 1900. Monthly Notices 67, 146—148, 1906. Boss’ catalogue was only used for stars brighter than 95”.8. Auwers’ catalogue was only used to determine Nix» tore the whole sky. The proper motions that have been derived from the Cape plates in Publ. Groningen N°. 25, are not very accurate. The plates had not been originally destined for deriving proper motions from them, and they had been measured absolutely. We have, therefore, in determining the mean values attached but very little weight to them. No corrections have been applied in order to reduce our results for the different catalogues to the same scale of magnitudes or to one fundamental system or in order to correct the proper motions for errors of observation. These corrections may in our opinion be neglected, considering the comparative inaccuracy of the numbers. Moreover, we have always expressed the numbers Nn, for every magnitude in percentages of the numbers V,, and these percentages appeared to vary only little with the magnitudes. Our countings in determining Ns for the whole sky include 38818 proper motions, while moreover in the five zones resp. 8273, 10857, 6981, 3144 and 1488 stars were counted. Now we are able to determine the numbers of stars of determined magnitude and proper motion. The results we found for the whole sky, are mentioned in table 1. In our further research we did not use the numbers of stars with u > 50". It is very difficult to determine these with sufficient accuracy from the data of known proper motions, so scanty as yet. It will appear that in consequence of this limitation we could not extend the luminosity curve found by us to the faintest stars. The numbers AN, now being known for the different zones, we may also examine how the stars with determined proper motions are distributed over the sky with reference to the Milky Way. In order to do so we have calculated the numbers of stars with P.M. resp. > 10", 5", 3" and 0" for every magnitude per 100 square degrees. It appeared that the numbers of stars with u > 5” do not evince any galactic condensation, except perhaps for stars, fainter than Gm. The numbers with u > 3" very clearly show a condensation for all magnitudes in the direction of the Milky Way, although to a less high degree than the numbers of stars with u > 0". 522 00909S1 Op9prg ILSOPC 9626 o0zIE €S201 TIE L06 £92 66h < | 6°6h—0°0€ ‘AYs aJOYUM ay} JO} UOIJOW sadosd pue apnzyiusew peulw 6°62 —0' 02 189P 6LSC Sole 086 OES 676 OFI G:ot0 SIG F001 L69EP 606PL 11L382 8969EP | veesoL | 6-10 1 E9967 60062 | 610961 EOLZLI | SPLOLZ | 6°0I—0'0I 216 ozs | SLHE cei. | LeZ80l | 66-06 GILS | LTRS) <= | > COUT LISLZ | 9%66vv _ 6°8-0'8 91LI geor | op 8028 POOpL | @ L—O'L SEL | 79g gze1 gi9z | 6107 6:9—0'9 67E betrad | Lg c99 0601 | 6'S—-0'S Eu os |_gel LOI 606 (6'b—-0'F 6E 61 6E Lp 8h 6'E—0'€ vl | 8 el 9% Ie 64) ARE DiE a 0S | BP OE (60 AS ee eae eae ee Jaap Jo siejs yo SJaquinu ayy ‘| ATGVL 523 This result agrees with that found by KarrryN in 1893 and with the conclusions, which Dyson and TuHackuray found lately from a discussion of the Greenwich 1910 catalogue. The mean parallaxes zr, for the whole sky and for the different galactic zones have not been determined by us directly. In our opinion they may be reduced for our investigation, preliminary as it is, with sufficient accuracy from former researches. Kaptgeyn found in Publ. Groningen N°. 8 for these mean parallaxes ne ere in which a = 0".0038, 6=0.71 and e = 0.905. Use was made of the value 16.7 Kilometers a second for the velocity of the sun. If we accept the modern value, 19.5 Kilometers a second, the mean parallaxes a, used by Kaprryn in deducing his formula, are lessened by 14°/,. If we now suppose — as VAN Ran also does *) — that this correction for the mean parallaxes zr, changes only the value of a in ar, we find a = 0".0032°. Kaptryn’s parallaxes corrected in this way agree with the values, deduced by Dr. Van Ruin in his dissertation in an investigation, based on other and more modern data than those used by Kapreyy, in which he partly also applied another method. Hence it follows that the parallaxes zr ‚as given in Publ. Groningen N°. 8, are very reliable. In order to derive from the parallaxes which may be applied to the whole sky, the z,,, for the different zones, we have assumed, that the zr, of the various galactic zones are related in the same way as the z,, of these zones. This assumption is rather arbitrary. Most probably it cannot be proved to be correct in all respects. There is reason to believe, however, that the error, made in this way, is not very great. Therefore we thought we might use this hypothesis in our preliminary research. We have already mentioned that we were able to use, thanks to the great kindness of Professor Kapreyn and Dr. Van Ravn, the mean parallaxes of stars of determined magnitude and galactic latitude, which will soon be published in Publ. Groningen N°. 29. These parallaxes may be respresented by the following formulae: Zone | log. mn — 8.883 — 0.142 m II = 8:904 — 0.142 m Ill = 8.957 — 0.142: me IV = 9.024 — 0.142 m V = 9.066 :—- 0.142 m Whole sky 8.943 — 0.142 m 1) Diss. Groningen 1915, p. 35. 524 if we call the parallaxes, used formerly calculated Tv We have if for convenience’ sake we indicate and ’ by K APTEYN, ™ KAPTEYN “ hoe J DY MT those meant above by AN RHIN nay =} fax a id mM 2 (=) | | = 020° P10" 010° 800° ¢900° 1900" | 9800: ‚_6100: ON 8 eo: | 910° | = \210" 600° LLOO* | 900° | $600" | %200° 9°01 = L080. 610° p10: | 110° 0600° — 1L00° 0800: 200° 9°6 ee) | * | See | 620° zo: 910° Z10° €010° 1800° LS00° 6200" 9°8 = | Peo: C20" 610° P10" LII0° €600: | 9900" €£00° | OL & L10' GEIO' L010' | 9200" | Leoo’ LR) | GPO" EO: C20" 020° | Sto" €Z10° L800° £600" | 9°g 160: LEO" 620° €20” 8L10° ZPI0" 0010" 6700: MG 650" Tro: €€0" 920° 9020° €910: C110" LEO | 98 | | $90°,0 640° ,,0 Ee0270 || S060 ED L€z0".0 L810°,,0 GEl0.70 ¢900° 0 9'z ees * SE GLI cal 0°6 Gron alte OF ie eG | i eae A ml ces ‘Kys aJOUM ay} JO} uONOW Jedosd pue apnyusewW paululiajap Jo ses Jo saxeyjesed ueow UL ‘o ATAVL for the 5 zones by multiplying the numbers of table G of Publ. ler) 8 ls 6) 5 al N o 525 Fm Groningen N°. 8 by a factor — VAN REIN, The parallaxes, which may Tm KAPTEYN be applied to the whole sky, are given in table 2. We ourselves have not tried to determine once more the probable deviation of.the error curve log. */x,. Moreover we have supposed that @ does not vary with the galactic latitude. We made use of the value 0.19, found by Kaprryn both in our solution for the whole sky and in that for the 5 galactic zones. It has, however, been proved in Publ. Groningen N°. 11 that the value of @ has only little influence on the result. 3. Applying Kartuyy’s Method. We have applied Kaprnyn’s method without any modification. For a description in detail of this method we refer to his treatise in Publ. Groningen N°. 11, which we cited already above. We shall limit ourselves to a short discussion of the hypotheses that have been made and an explanation of the tables mentioned. in this essay. The hypotheses, made in Kapreyy’s investigations, are three in number : 1stly the density is only a function of 7; Qndly the luminosity curve is independent of the distance from the sun and there is no absorption of light in space; 3¢y the quantities z—=log.*/x, are distributed according to the law of errors. The first hypothesis is necessary if we want to derive frequency- functions that may be applied to the whole sky. We seek for mean values for the unknown quantities and so we cannot take into account the variations in the values with the galactic latitude and longitude. Kaprryy’s method may, however, be used just as well, if we want to reckon with the influence of the galactic latitude by making separate solutions for the different galactic zones. The second hypothesis can hardly be dispensed with. If we have certainty that the frequency-function of absolute magnitudes is everywhere the same in space, KaAprryn’s method offers the means to examine, whether there is a perceptible extinction of light and, on the other hand, when we know that there is no absorption we can examine if the luminosity curve varies with the distance from the sun. As neither one thing nor the other, however, is certain, we are obliged for the time being to make the supposition in question. If we establish the frequency-function for the different galactic 526 zones separately, this difficulty is diminished considerably. We can then compare the luminosity curves, found for the various zones. If now, it appears that these curves correspond, it is highly impro- bable that the distribution of luminosities varies with the distance to the sun. If the luminosity law depends on 7 and not on 5, we should find a fan-shaped composition of the sidereal system, which is not inconceivable, but highly improbable all the same. If we may assume that the distribution of the luminosity is independent of the distance to the sun, Kaprryn’s method enables us to determine the absorption, or at least to examine if it has any influence on the distribution of stars in space which we found. The third hypothesis was, indeed, when Kaprryn made it for the first time a rather arbitrary assumption, and it must be conceded that other results would have been found, if another hypothesis had been made. The hypothesis was made, because at that time there were still too few measured parallaxes, to enable us to deduce the form of the frequency curve log. */;, directly from the data. Perhaps this will be possible when applying the method once more. It is, however, of little importance to discuss in the present stage the question whether this hypothesis could be justified or not, as SCHWARZSCHILD has proved that the above-mentioned relation exists for a special form of the density law and the luminosity law, which form obtained a great amount of probability owing also to his investigations. We have deduced the luminosity law from the data of observation mentioned in § 2. Space was divided into a number of shells the radii of which had been selected for convenience’ sake in such a way that log.r increases with 0.2. Afterwards the mean parallax was determined for each of the numbers No, that had been found by means of the formula for zr Then with the aid of the value found for the probable deviation of the error curve log. */,,, it was calculated which part of tbe numbers found occur in every shell. The numbers N,,, which we found for the whole sky are given in table 1, and the corresponding parallaxes z,,, in table 2. In table 3 we have mentioned, how these numbers of stars are divided over the various shells. Tables in accordance with this we have calculated for the five zones. Now we have derived from these tables others, indicating the numbers of stars of every absolute magnitude per unit of volume in the different shells. For the whole sky we communicated our results in table 4. Between the fat-faced lines the numbers 527 ee 6E06GG1 | S66EFI | ogsobZ | SLLZ6 GLOIE | ILIOL | Ooze | 988 | MZ | zl os Bu eetl ry pre — del SRE NE | L 0 | 0 | 0 0 ee | I oa 0 OS a kh ger < 79 or | 0 i: I nk ab See ane as of Orel “faite 851" — 0001’ Is | Lv cp | 88 | zo |b | eenden ELO 0001” _— 1690: eoig | zoe | So | Le foe Ves SEPT RT) OG “92 Te —egFO" ef DRE ores | tree | 9082 | zei | zel eu ge lier leo | oz | st | 960: | 86e0' — 1920: zez6e | OLLI | 8011 | ZOLS site | ocor | 908 | ose |1er | oP | Iz | L8IO 1820: — 8610" . 6oeco1 | oosch | esgre | osiar | 9689 cete | oep |-cos |op | Ly | ve | gio: | gsto: — 0010: ocosee- | ooveer | zoo | vies | za | tar | oer | zes | ost | se | on | ero 0010: _— 18900: 196Lpv | IEiveg | GUGEET | OBE | «GEIST _0OL9 LS6I | OLS LP | 98 | Pl | 69v00' 1¢900° _—86600° Lb6STe | 6PTE OSI | LOZ | EPOOz eero | ett «| Sev | 66 | vi | L | 96200: | 86EOO' —I9200" O6LOzs | zEIIFE | F89ZIT | bhp Voer | eo | ot | ue les | 9 |e | wer | 1gz00" —8c100: zeste | zoeerz | 0060s | obese | 9096 | cece | 69 | vo gee an _81100:,0 | gct00" —00100: 66v68E | OlZzIz | goccc | SHOLI | Seb €8 sez ise |9 I 0 | 00100" ,0—00000".,0 oe mor | Ot | 901 9°6 9°8 o°L oo | oe | or | oe | oz | Pe x | "Sew en a 5 oe Sauna vonde AEN ifs WE NE a 26 ATAVL 528 |9CP'O 6L6°6 | €08'0 | FPSO €0S°0 | SPE'0 SPE'O 6L6°6 — |6L6'6f6L6'6 coro | #08:0 | | | gLg"0 | 99S°0] 229°0 69°0 SLS"O ILb'O | FES"O EEIO 911:0 (ep1”1 8601 P89°0 | ZPE'O | 8201 | LeL*o} 9¢¢"0 ZL9'0. 6PP'0 | 0220 0 Ol 0°6 | | | | | b02" | | | Sil £06'6 | 189°6 | | | | ero” 160°0 | OSL"6 6970" | 9866 | 91S°6 | S116] 6L8°S | | ‚960 600°0 €69°6 981°6 | 99L"8 frs | | | | | isto" | Z00°0 | 8S9°6 202°6 Loo's SLI'S] €88"L_ | | 8110: 910°0 | LLG"6 E16 6L9°8 |6LO'8 €8h'L €hL00° 696°6 29S°6 6216 G6S°8 990°S OLP'L 69700" 6LL'6 0zr'6 810°6 | c6v°8 FS6°L. IPs 'L 96200" 9€9°6 spofseL:s 66¢°8 99L"L | ILL L8100: vase | zopefveo'L sort 120"L €09°€ | 81100°.0 0e | oz | OF | 0°0 |0°I—|0'z—|0°e—| 0°— | 0"s— | 0'9— isl oe ‘“OWIN[OA jo Jun ese jo feat init So] UL Rd + ATAVL 529 of stars of the apparent magnitudes 3.0 to 10.0 are found. These are most reliable. The densities in the different shells were determined by comparing each time in two successive shells the numbers of stars of a deter- mined absolute magnitude. The relative density of two successive shells was found as the average of three determinations, based respectively on the stars of the apparent magnitudes from 3.0 to 8.0, 3.0 to 9.0 and 3.0 to 10.0. The density in the fifth shell, vzz. the one for which the mean parallax is 0”.0296, was supposed to be one, after which the density at every distance from the sun, expressed in this unity, is known. The average density for the whole sky varies about in the same way with the distance to the sun as Kaprryn found. In zone I the density at great distances is considerably more than the average, in zones III, IV and V on the other hand it is much less. By means of the densities found we now calculated from table 4 and the similar ones for the galactic zones the number of stars per unit of volume, after the density had been reduced to the one for a = 0".0296. In order to do so each number in the last-mentioned tables was diminished by the logarithm of the density of the shell pertaining to it. In this way for instance table 5 has been calculated from table 4. From the tables that were found last of all, the luminosity curves for the whole sky and the 5 zones may be deduced at once. The numbers, standing between the fat lines in each column, correspond pretty well. In table 5 we took the averages of these numbers (not of the logarithms), and noted down the logarithm of these averages in the last line of each table. In taking the mean equal weight was ascribed to all numbers, except to those of the first four shells. These numbers are not very reliable, because they are small, but especially because we were obliged to exclude from our investigation stars with a proper motion > 50" per century. Consequently the luminosity curves that have been found are only of value up to mm It is of interest to point out that our result for the whole sky corresponds beautifully to Kaprryy’s found in Publ. Groningen N°. 11. Furthermore it is remarkable that the curves found for the various zones differ only a little. We already observed that from this we may conclude with a certain probability that the luminosity curve does not change with the distance to the sun either. In table 5 and the corresponding tables for the 5 galactic zones 530 ‘L¥9"0 860°I * OL8"0 L¥6'0 L28'0 009°0 662°0 Loro 6296 zere 619°8 opte €L8°9| OPI'9 | LOES | Lpb'p IPTH UEOW i oor P= eS a he a are 6811 ZIL"O ZIL'0) — “ZIL'OFZIL'O | | | 402" 686°0 | sp6-0 G6L'0 | €6L°0 0990 | 6PT'Of GHT 0 | | | iors) 8L8°0 9F6'0 | 928°0 | ZZL"0 8S°0 FEO OPS 6f ZE6"6 | | | SPL" 680] 89L°0 $¢9°0| 1L6°0 | $6¢"0 962°0 | 9L6°6 | $L9°6] 692°6. | | 6970 | 880" 1 | TST" 06L 0 | PES"0| LEZ"0 986°6 9IS°6 SII°G] 6LS"S | | | | 9620: | | vpe” 1 | P11 ]098"0 | 8ES°0 | LIT'0 010:0\#69'6 L81'6 LOL'8| 9tb'S | | | L810" | por 1 {Leo 1fe0L“0 | 198°0 120°0 | LLO"6 ¥22"6 989°8 ¥61'8|206°L | | | sito | pro” 1 | epgro]zop-ozo0:0 | €29°6 6ST"6 | SZL'S gole | 62S" Lest "L _ €bL00° | | | oeL“o | Leeof LPO"0 | FE9°6 | LOT'6 | ELE" PPI'8| SPSL |OLL'OfLZE'O | 69400" | | pol 1106 | $89°8 | LP'S, HEG'L | 968°9 zporof mee 96200: | | |2€0°0 | 1bS°6] PSI"G | 69°8 | ZOI"S | LOS'L L68°9 | 6ET'9 ‘Liz's | 916" 18100: | | = 4 zege | 001°6]Z69°8 | ZEI"S G99°L Z16°9/ ITO |ZhH'S | Ch HT IPI 81100°.0 Ozi | O'1N | O'ol | 0°6 | oe) O'L | O'9 | O'S | OF [ o8 | Oz | 0! Ose ete a Dee ae OP OF =| 09 | One | Weare ‘96Z0',,0 = & 0} paonpas ‘auinjoa Jo pun Jad saejs Jo Jaquinu ay} Jo “Soy UL °C JIGVL OLEN LOO et EHEER Ne EEEN NE AIC ANS NANA INA AANKON ACE EF ENSONSNNE NEE) BENNINENNNNSE EEENSNNAAINNG ARRAS: RAS: CNRS Sea SN | ~ r ‘ t Proceedings Royal Acad. Amsterdam. Vol XXI. 8 532 the numbers in each column, as has been already observed, agree with each other. If our conclusion that the frequency curve of the absolute magnitudes does not vary with r, is right, we may derive from the agreement of the luminosity curves in the different shells that there is no perceptible absorption of light in space. In figure 1 the frequency curves that were found have been drawn. The six lines in the lowest part of the figure refer to the, determination of the luminosity curve, discussed in this essay. The six lines in the upper part refer to an application of ScHWARZSCHILD’S method to the same data, which will be explained in a following communication. The line representing our determination of the luminosity curve for the whole sky, indicates the logarithm of the number of stars of every M per unit of volume in the neighbourhood of the sun. For the other curves we added, in order to make comparison pos- sible, a constant amount to each number. Amsterdam, June 1918. Mathematics. — “On the arising of a precession-motion owing to the non-euclidian linear element of the space in the vicinity of the sun”. By Prof. J. A. ScHourenN. (Communicated by Prof. Lorentz). (Communicated in the meeting of June 29, 1918). If & be an curve in an nm dimensional space X,, of arbitrary form, there will be in the euclidian space ieee = dimensions, into which X, can always be placed without changing its linear element, a euclidian space Y,, i.e. a space Y, developable on a plane space tangent to X, along 4. If in the euclidian space Y, a system of n mutual 4 directions be moved with its origin along & parallel to itself, we find that these directions in X, define a ‘“geodesically moving system” *). If two arbitrary spaces are tangent to each other in a curve k, it follows from this definition, that a system geodesi- cally moving along 4 for one space, will geodesically move for the other space too. A volume-element covered with mass can move in X, as a solid body, but for some infinitesimals of a higher order. If a suchlike element always remains at rest with regard to a geodesically co-moving system of directions we will call it compass- body. Hence the compassbody mechanically realizes the geodesically moving system. If k be a closed curve, the initial position will as a rule not coincide with the final position, if X„ is non-euclidian. Thus the position of the compassbody is changed with every rotation. Now according to the investigations of K. SCHWARZSCHILD *) the space in the vicinity of the sun is not euclidian, but very slightly curved. The linear element is of the form de UR ERO Bint Od nr (PP MD 1) Cf. for a more detailed exposition of the geodesically moving system: “Die direkte Analysis der neueren Relativitiitstheorie’. Verh. of the Kon. Akad. v. Wet. Vol. 12. No. 6 and “On the number of degrees of freedom of the geodesically moving system and the enclosing euclidian space with the least possible number of dimensions”. Proc. of the Kon. Akad. v. Wet. May 25, 1918. 2) Ueber das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theorie, Berl. Sitzungsber. 1916, p. 189—196. 35* 534 L. Framm?) has indicated how a space with that linear element can be realized. The parabola situated in the wz plane: 2) == 4ole—a@)* ne eee has the linear element: hh == => under be a je v If this parabola is rotated about the z axis, in the xyz space, there arises a rotation-surface with the linear element : dh? ds? = ——_ + RR? dy? ATR vei kee. ae (4) a REE R g being the rotation-angle, measured from the yz plane (fig. 1). fs Fig. 1. A fan of directions in the centre of the sun determines a system of oo geodesic lines, together forming a diametral surface. Such a diametral surface can as regards the parts beyond the surface of the sun be developed on the rotation-surface (4) without changing its linear element. A may approximately be equalized with the 1) Beiträge zur Einstein'schen Gravitationstheorie 17 (16) 448—454. 535 | naturally measured distance relative to the centre of the sun, while a = 2.945.10'§ cm. The circle described by a definite point P: ee yi FR ces aaa consequently represents a circle in the diametral surface, having the same centre as the sun. If the sun be looked upon as a globe, filled with an incompressible liquid, a diametral surface within the sun will have the same linear element as the globe-surface, which touches the described rotation-surface in a parallel-citcle with a radius R,. This radius A, too may approximately be equalized with the astronomically measured radius of the sun. If the described rotation- surface (4) is rotated in the four-dimensional «yzw space around the yz plane, there arises a curved three-dimensional space with the linear element (L) when @ is the angle of rotation, measured from. the yzu space. We shall now investigate the motion of a compassbody moving ‚in the circle (5) around the sun. For this purpose it is sufficient to find a space, tangent to (1) in (5), and in which the geodesic motion can conveniently be indicated. We now make the tangent line PQ rotate with the parabola. That tangent line describes a cone witli the linear element: (5) dk? EAN ed pe ae ae cos* ¥ in which equation x only depends on the definitely selected point P, and therefore is a constant. With the second rotation there arises from this cone a space with a linear element : de == dR? ’ Ode U oat ea, ia ORO ee arne ve (A) cos* y in whieh y is once more a constant. The linear element of a eucli- dian space may be expressed (in polar-coordinates) R', g', @': ds? = dR" + R? dO" + R? sin? O'dp? … . . . (8) and by the substitution: R= Kf cosy eed hen” HOE Oe tae A ee MT KE) a | a= cosy (7) passes into: ! 6 ds? = dR* + R' d6" + R sin? — dg? . . . (10) cos ¥ 536 On the curve (5) we have cos@=0O. With the -exception of quantities of the order 7‘ the tangent space (7) behaves along the curve (5) as the euclidian space (8). Hence we need only trace the movement of the geodesically moving system in (8) along the curve corresponding to (5). As the coordinates y and 6 have obtained a factor cosy according to (9), (8) can be realized by the part of the euclidian space wyu that remains, when a rotation-cone having for axis the y-axis and a top-angle of 2x (l—cosy) is taken away. (fig. 2). Fig. 2. The part of the circle: tte | ma) EPEN 8 8 ae? + oh == “uO situated within this space viz. the part extending in fig. 2 from A via a point in the negative part of the y-axis to B, will then corre- spond with the entire curve (5). This result may also be obtained by replacing the cone (6) by its unfolded mantle, laid down in the ye plane symmetrically to the y axis, being a sector with an angle 27 cosy. The curve (5) will then coincide with (11). During the revolution about the yz 537 plane the cirele-sector describes the space-section described. By this method it does not at once become obvious that now the obtained euclidian space may indeed replace the originally existent non-euclidian tangent-space along (5). The motion of the compass- body can now be easily traced. In the euclidian space wyw the compassbody always moves parallel to itself. If a constant direction in A in this body has the direction of the radius, then that direction in B forms with the radius an angle 27 (1 — cos x) situated in a plane // at the zy-plane. Now = is very small, hence: il 12 cos X = “tgp? vies Keri kobe . ( ) so that the total deviation d in one revolution amounts to: nee (13) ZE == . e . . . . . . . R A compassbody moving around the sun, as its central point, in a circle with a radius equal to the average distance from the earth to the sun will show according to this formula after one revolution a deviation of 0.013". If the radius is equal to the average distance from Mercury to the sun, the deviation amounts to 0.0328". If the radius is equal to the radius of the sun, the deviation amounts to 2.73". If, from another cause, the compass- body already has a revolution around an axis, which is oblique relativaly to the plane of the orbit, there will set in, merely on account of the deviation described, a precession-motion, which in the first of the above-mentioned cases would result in a complete revolution of the equinoxes after + 100.000.000 years. It is note- worthy that the effect described is of the same order as the deviation of a ray, passing the sun at a distance A from the central point. According to Einstein this deviation indeed amounts approximately 2a to —. R Whether the deviation computed of the precession motion for the earth will indeed set in, depends on the question to what extent „and to what approximation a mass of the quantity and the com- position of the earth has the proporties of a compassbody. In order to answer this question it is necessary to make definite suppositions as regards the physical qualities of the earth, in particular the mutual attraction of her parts, and starting from these suppositions, to integrate the four-dimensional dynamical equations of motion. SS 538 ADDENDUM. Prof. pe Sirrer, to whom [ have communicated the above, writes: A precession of 0".013 per annum of course comes well within the reach of observations, since the observed value of the precessional constant is trustworthy to about O”.0010. The point is therefore with what accuracy the theoretical value can be computed. Now the lunisolar precession (the planetary precession can be taken as completely known) is given by a formula of the form p, (P+ QE fae ee eee where P and Q are known numbers and u is the mass of the C—A moon (expressed in that of earth + moon as unit) and Hi 5 depends on the moments of inertia of the earth. The uncertainty of uw causes in p, an uncertainty of about */,,.. of its amount, thus, if H was exactly known, p, would be uncertain to the extent of + 0.025 or twice the new precession. A better determination of « may be expected from the opposition of Eros in 1931 *). However the uncertainty of H is of much greater importance. In 19157) [ have with the aid of the hypothesis of isostasy, derived the ellipticity « of the earth from H, this latter being determined from p, by (4). To invert this order it would be necessary, in order to get a p.e. of + 0".005 in p,, to know « to about */,..., Of its amount. The direct determinations of € at present do not go beyond about */,59. To increase this accuracy seventyfold would in my opinion be beyond the forces of geodetical science, at least in the near future. We can determine H with greater accuracy from the constant of nutation, which is given by N= Bme Se Be ae eee where R is again a known number. From (1) and (2) we derive eh p= A EN Tr ren where S and 7’ are also practically exactly known. The uncertainty of the multiplier owing to the uncertainty of fe is now about 7/459, or ‘/;o0., and it will probably be reduced to */199.. by the new determination of u in 1981. 1) The figure of the carth and some related astronomical constants. The Obser- vatory Aug. 1915, page 322, 2) On Isostasy, the moments of inertia, and the compression of the earth. These Proceedings April 1915, Vol XVII, p. 1291. 539 The result of Newcoms’s discussion in 1891 of all available deter- minations of the constant of nutation has determined its value to about */,15. Of its amount. To get a p.e. of + 0”.005 in p this accuracy must be increased nine- or tenfold. This certainly is no easy task, but it would be preposterous to say that it exceeded the forces of astronomy. It will of course require very refined and prolonged observations and discussions. I may be allowed to remark that it still remains to be investigated whether the new precession is the only effect of the new gravitational theory, and the equations (1) and (2) are not affected, i.e. whether EiNsTEIN's theory gives exactly the same equations for the motion of the axis of rotation of the earth with reference to the geodetically transported system of coordinates, as are found in Newton’s theory relatively to a “fixed” system. This cannot be asserted without a special investi- gation, which so far as I know has not been undertaken, and it might even happen that the precession of the geodetically transported system of coordinates was exactly cancelled by a small change in the precession of the earth relatively to that system. Physiology. — “Hifects of the Rays of Radium on the Oögenesis of Daphnia pulex’. By Miss M. A. van Herwerpen. M. D. (Communicated by Prof. C. A. PEKELHARING.) (Communicated in the meeting of Sept. 29, 1918). For a considerable time I have been prosecuting the effects of radium radiation on a race of Daphnia pulex bred in the laboratory, with whose method of reproduction I had become thoroughly ac- quainted during a study of rather more than 8 years. My original purpose was to evolve, if possible, parthenogenesis in the sexual period, or conversely, to impart to the parthenogenetic females the faculty of producing a sexual offspring. The importance of this experimentation was sufficiently borne in upon me in connection with the view adopted by many researchers that by radiating the organism with radium, enzymic actions are accelerated or diverted. A short or a prolonged radiation with 6,7 mgrs or 3,1 mgrs of radiumbromide, at my disposal, never resulted in any effect upon the sexual or the parthenogenetic stage, whereas after a radiation with a stronger preparation the animals succumb. ') Nevertheless | considered it a point of importance to continue the experiments, since they throw a peculiar light upon the resistance of the proto- plasm in the several phases of the oögenesis and of the embryonic development under the influence of radium rays. Daphnia pulex affords extremely fit material for such experiments. Besides being fairly transparent and easy to watch under the micros- cope, it also enables us not only to follow in the living animal the development of the parthenogenetic eggs located in the broodpouch, but also to determine the degree of maturation of the eggs in the ovary. Sometimes the amount of yolk in the maturing eggs enables us to foretell correctly to a few hours, when they will leave the ovary. There is always plenty of material for control, as several young ones can be expected at every parturition. Over and above, the rapid succession of generations grants a comprehensive survey not only of the animals under examination, but also of a large progeny. 1) Verslag “Koninklijke Akad. v. Wetenschappen” Deel XX p. 20. 541 In the experiment I placed Daphnia pulex, embedded in a drop of ditehwater, immediately on the micaplate of the radium-prepara- tion. In the following descriptions I shall designate the preparations used, containing 0,7 and 3,1 mgrs. of radium-bromide respectively as capsule A and capsule B. Animals belonging to different age-periods were radiated separately. It immediately appeared that the adult Daphnia is much less respon- sive to the radium rays than the new-born animal. An exposure of 18 hours on capsule A did not affect the animals so as to kill them. If the animal kept alive after this prolonged radiation, it was sometimes seen to succumb later on at the periodical ekdysis of the chitin shell, which was often attended by an abnormal chitin-formation. This then pointed to some injury to the ectoderm. Still, even when the animal remained perfectly healthy, it had become sterile for the rest of its life. In the case of only a few hours radiation without any yolk-rich mature eggs being noticeable in the ovary the repro- ductive faculty is not interfered with: the eggs leaving the ovary later on develop normally and the young generating from these eggs reach sexual maturity in the regular way and produce a healthy offspring. However, when radiation takes place, while the ovary bears large yolk-rich eggs, or when the eggs have only just entered the broodpouch, a radiation of 25 minutes on capsule A, or of some minutes on capsule B, will suffice to cause the eggs to develop ab- normally, so that they are destroyed already in the blastula-stage and are resorbed in the mother. Susceptibility varies individually, without any apparent connection with the age of the mature females. Whereas the eggs that have only just entered the broodpouch are as susceptible as, or more susceptible than before leaving the ovary (abortus sometimes occurs already after 1'/, minutes’ sojourn on capsule B), the resisting power of the embryos is seen to increase during the development. Viable young were developed even after a three hours’ radiation on capsule A in the gastrula-stage. Yet in these cases the brood did not seem to be always in good condition, since one of the young emerging from such a radiated gastrula — the only one of this lot that reached sexual maturity — produced a brood of an anomalous morphological structure. It appears then, that the future germ-cells of the gastrula in this case must have been injured already during the radiation. A greater resisting power is displayed by the almost viable young, as they can stand a 20 hours’ radiation on capsule A, without any prejudice to their future fecundity. If, however, these young ones 542 leave the brood-pouch during the radiation, they invariably perish, every one of them; a sojourn on capsule A of a couple of hours only will kill them. After an hour’s radiation the cardiac action is weakened and irregular and they die soon after. In the brood-pouch they are presumably walled off by the chitin shell of the mother and by the fluid in which they are swimming, which protection, however, is not sufficient for the young embryos, which are so much more susceptible to the radium rays. A short radiation of a female with maturing egg-cells in the ovary, leads indeed to destruction of the eggs; but it leaves the mother unhurt. Afterwards it even causes sometimes a more numerous Offspring, which phenomenon is analogous to that seen in the action of various poisons on Daphnia pulex *), on which a small dosage of the poison acts as a stimulus. The resistance of eggs from one and the same lot sometimes differs very much, as among the eggs that are being destroyed occasionally a single normal young may be seen that does not appear to have suffered at all from the noxious influences | that threatened it before the embryonic stage, and later on may possibly possess a normal faculty of reproduction. An anomaly in the structure of such a young occurs only rarely. It cannot be said to be typical for radium-radiation. Generally these monstra (with abnormal profile or defective intestine) are few and far between. As a rule, therefore, radiation yields a normal embryonic develop- ment or none at all. This is the reason why I never succeeded yet in breeding mutations of Daphnia pulex by radiation with radium, as Morean achieved on a large scale with the Drosophila fly. The few abnormal specimens never reached maturity, one excepted, which recovered completely and produced a normal offspring that was still healthy after four months. Indeed my eight years’ expe- rience with Daphnia pulex have convinced me that this race though highly modifiable, shows only slight mutability. Other researches in experimental embryology also show that eggs from the same lot vary as to their resisting power to noxious effects. [ here call attention to the researches of PEARL ®) on the difference in degree of resistance evinced by embryos of the domestic fowl to intoxication with alcohol, in which case also the eggs, liable to reach full development bring forth normal chickens. It might be objected that in the radiation experiments the eggs are differently exposed. This may occur with a numerous brood, in which the inner eggs are shielded from the noxious rays by the 1) ess, 9) Proceedings of the National Acad. of sciences. U. S. A., Vol. II, p. 380. 543 external eggs. But the exposure will presumably be the same for the Daphnia moving about freely in the waterdrop on the radium capsule, when, as in our case, there are only 4 or 5 eggs located in the brood-pouch. What is the cause of this unequal resistance of the Daphnia eggs ? A reduction-division of the chromosomes, which might be responsible for the unequal distribution of paternal and maternal hereditary units does not oceur in this parthenogenetic development; the egg retains the number of chromosomes of the mother. A similar difference in the reaction of the eggs to the noxious influences | previously detected, when treating the Daphniae with phenyl- urethan *). Whereas with a short radiation of a female with mature egg- cells only the first brood suecumbs, and the succeeding broods are normal (even very abundant), with a longer radiation also the eggs in a younger stage of development are seen to be damaged, until ultimately the Daphnia becomes completely sterile. After one single radiation a Daphnia may, after many abortus, produce quite unex- pectedly normal young again, if namely the younger oögonia are not damaged. Such an after-etfect 1 observed up to the 6" of January 1917 of a radiation for some hours on capsule A on the 234 of November 1916. After the first date again young were born that were completely normal. | My experience that eggs of Daphnia pulex in the last stage of maturation are most susceptible to radium rays, and that only after a prolonged radiation also the younger egg-cells and at last the oögonia are injured, accounts for PAcKARD'’s *) experience that after radiation of the Drosophila larvae, the young flies become sterile for some weeks, and afterwards become fertile again. If we bear in mind that results with mammals also favour the theory that especially the mature egg-cells are very susceptible, we are justified in presuming that this holds for the whole animal kingdom. How to account for the fact that the maturing eggs are more susceptible to the radium rays than the immature and the remaining cells of embryonic and maternal organism? Again, what molecular trans- formations occur in the protoplasm under the influence of radium radiation? The view adopted by several researchers that enzymic actions are accelerated or diverted, prompted me to compare the embryonic development of radiated and non-radiated sisters, which as to temperature and diet had been bred under the same conditions. 1) Le. page 1. 2) Journal of exp. Zoology. XIX, p. 332. 544 Up to the moment the eggs left the ovary these sisters were kept in the same culture glass, animals being selected for this experiment whose eggs entered the brood-pouch in the same hour. As soon as the eggs began to develop one Daphnia was radiated for fifteen minutes: on capsule A. After this the conditions were made equal for either animal. In ease a normal brood was developed after this radiation, embryogeny was neither retarded, nor accelerated. At the same hour the heart’s pulsation became visible in the brood of the two animals; at the same time the development of the limbs commenced and the first eye- pigment developed itself; in the same hour the young left the parental organism. A similar observation was made at a second radiation. If only the radium action does not pass the physiological boundary there is neither acceleration nor retardation of development. Is it the alpha-, the beta-, or the gamma-rays to which the egg- cells of the Dapbnia are particularly responsive. In radiating on the capsule with radium-bromide the alpha-rays are screened out by the mica-plate of the capsule, which they cannot penetrate and conse- quently they do not reach the animal. When separating the Dapb- nia from the radium-preparation by a leaden platelet of 3 m.m. thickness, the beta-rays do not reach the Daphnia, while the secon- dary beta-rays are allowed to resorb through a mica-platelet of 50 u thickness on which the Daphnia is placed. In this way Daphniae with maturing eggs in the ovary could stand a radiation with the gamma-rays from 0,7 mgrs. of radium-bromide for 24 hours, without abortus, which proves the harmlessness of the gamma-rays. When applying the radiumpreparation of 3,1 mgr., which, as has been seen, will destroy the eggs within a few minutes, a radiation of 24 hours with the exclusion of the beta-rays, could be borne without delete- rious influences. In a few of the latter experiments, however, the first brood is aborted. It is, therefore, possible that to a stronger concentration of the gamma-rays (not obtainable with these prepa- rations) the Daphnia egg-cells prove to be sensitive, a sensitivity, however, that is not to be compared with that to the beta-rays. The antagonistic action between uranium and radium, demon- strated by ZwaarpeMakerR') for the frog’s heart, induced me to radiate Daphnia pulex in a drop of uranylnitrate and to determine whether resorption of the eggs stayed away in this process. A concentration of 600 mers. of uranylnitrate pro L. is tolerated for some hours without inbibiting the development of the brood; with 1) These Proc. XIX p. 1048. > 545 < a higher concentration the Daphnia itself succumbs in less time. When a Daphnia with maturing eggs in the ovary is placed in a solution of 500 mgrms of uranylnitrate pro L. and after half an hour again in a drop of the same fluid on the radium capsule of 3,1 mgrs, the radiation may be continued for } to 4 hours in a series of experiments without causing the brood to abort, while, under the same circumstances, the eggs, when placed in water, are fatally injured already after some minutes. Sometimes, however, the protecting influence of uranylnitrate was not at all discernible. Up to the present I have not been in a position to account for these various results. This also applies to the lower concentration of urany nitrate. More than fifty Daphniae were examined in microscopic sections and compared with non-radiated specimens. Morma/ maturation of eggs in Daphniae has already been described by Künr *). Broadly speaking my findings for the normal eggs are in agreement with his. A prolonged radiation did not enable me to detect in the maturing eggs any change either in the chromosomes, or in the nuclear body, or in the egg-plasma. Only in one polar spindle (for the formation of the first polar body) was the number of chromatin rods larger than could be anticipated with twice the number of chromosomes. Any possible alteration in the shape of the chromosomes is diffienlt to detect owing to the small dimensions. Not before the blastula-stage, that is about the time when also in the living animal under the low-power microscope the embryos are seen to succumb, well-marked alterations take place in the nuclei, characterised by a collapse of the chromatin into coarse granules. The injury to the eggs, however, has been done long before the aided eye can detect it. ‘ Though microscopic examination did not put us in a position to ascertain whether the noxious action of radium-rays has initially affected the nucleus, the cell-plasma or both, the high degree of susceptibility of the egg in a period when also considerable evolu- tions take place in the nucleus (formation of the polar spindle and decomposition of the large nuclear body) is indicative of a noxious effect of the beta-rays, especially on the nucleus. The fact that the first cleavage proceeds regularly and only at the close of it degene- ration manifests itself, may be explained, when we call to mind Boveri's *) investigations, which demonstrated that with the Sea- 1) Arch. f. Zellforschung, Vol. I, p. 538. %) Jenaische Zeitschr. Vol. 43, 1907. 546 urchin it is only at the close of the blastula-stage that the various properties of the chromosomes manifest themselves. The period of maturation, in which the egg is so extremely susceptible to the radium rays, also proves to be the critical period for a poison as phenylurethan in a certain concentration, as discussed by me in an earlier paper '). Just as some eggs of the brood sometimes escape death after radiation with radium, and develop into perfectly normal young, a Daphnia was il occasionally developed after treatment with 79000 n phenylurethan. When transmitted to water it produced a normal offspring. This again proves that the resisting power of the eggs to the danger, threatening them from the outer world, was occasionally very dif- ferent, even with these parthenogenetic animals. But if they succumb in the struggle, the method of reaction in the two series of experiments is widely different. Whereas with a treatment with radium radiation this reaction leads irrevocably to degeneration at the close of the blastula-period, a treatment with phenylurethan evolves fully devel- oped monstra, in consequence of a deleterious influence, exerted in the same period of susceptibility. These monstra, however, are not viable after birth; they are not resorbed, but are expelled from the parental organism. Summary. The egg-cells of Daphnia-pulex are most susceptible to radium radiation in the last stage of maturation. The resisting power in- creases in the embryonic stage. In one and the same brood individual differences of susceptibility to the rays of radium is frequently noted. The egg that resists the deleterious influence often develops into a perfectly normal animal, which itself becomes fertile. The rare samples with morphological abnomalies seldom become adults. Only once did we succeed in breeding from such an abnormal young a stock without morpholo- gical anomalies. A long-continued radiation from 0,7 mgrs of radium-bromide does not endanger the life of the sexually mature Daphnia, but only its fertility. It depends on the duration of radiation whether only the maturing eggs, the oöcytes, or also the oögonia are injured. Large progenies being easy of observation afford an opportunity to study this in every special case. Minuaes ps. 547 Prior to maturity Daphniae resist radium radiation for a long time. Only after a sojourn of many hours on the capsule with 0,7 mers of radium bromide the future ripening of the odgonia is also endangered. A microscopic examination of the ovary and the embryos reveals that the deleterious effect of radium manifests itself only towards the close of the blastula-stage by an abnormal behaviour of the chromatin, also when the egg-cells were affected when lying still in the ovary. If the beta-rays are eliminated through filtration the deleterious effect of radium is arrested or highly diminished, which proves the beta-rays to be mainly responsible for the destruction of the eggs. 36 Proceedings Royal Acad. Amsterdam. Vol. XX1. Physiology. — “The conduct of the kidneys towards some isomeric sugars (Glucose, Fructose, Galactose, Mannose and Saccharose, Maltose, Lactose)” By Prof. H. J. Hampurcer and Dr. R. BRINKMAN. (Communicated in the meeting of September 28, 1918). It has been proved by former researches '), that the glomerulus epithelium of the kidney of the frog is able to hold back glucose if the solution which is passed through the vascular system has a suitable composition. [f one passes through the arteria renalis of the frog the following Ringer’s solution: NaCl 0,7 °/,, KC1 0,01 °/,, CaCl, 0,0075 °/,, NaHCO, 0,02 °/,, in which 0,1 °/, glucose has been dis- solved, then an artificial urine is excreted containing 0,07 °/, glucose ; 0,03 °/, glucose has thus been retained by the glomerulus epithelium. If however the Ringer’s solution contains 0,285 °/,, i.e. a quantity that corresponds to the titrational alkalicity of the serum of the frog, then much more sugar than 0,03°,, is held back and not seldom the urine is free from sugar. This phenomenon proves that the glomerus membrane, which is permeable to salts, is under physiological conditions impermeable to the also crystalline glucose. In order to come to an explanation of this remarkable and useful contrast it seemed interesting to investigate how the glomerulus membrane would behave towards laevulose, galactose and mannose, all isomeric to glucose, and also towards the mutually isomeric saccharose, lactose and mannose. Let us begin with the four first-named. As is well known, the structural formula of the monosaccharides (C,H,,0,) can be represented in the following way: !) HAMBURGER and Brinkman: Proceedings of the Royal Acad. of Sciences Section of Jan. 27, and Sept. 29, 1917. Also: Biochem. Zeitschr. 88, 97, 1918. 549 CH,OH CH,OH CH,OH CH,OH | | | fC — OH =Car H-—C—OH H—C—OH | | | | fe. OH |g ES LE OH—C— et | | | | OE CH OH— CH OH OHHH | | RT | H—C—OH C=O ho — OF Ys ee eS | | | | OH CH,OH On eee Oa d-Glueose Fructose d-Galactose d-Mannose (laevulose) The experiments were made in exactly the same way as was described in the articles cited above. The perfusion liquid was of the following composition : NaCl 0,5 °/,, NaHCO, 0,285 °/,, KCI 0,01 °/,, CaCl, 0,2 °/,. This solution was prepared by mixing 50 eem. NaCl 10 °/,, 50 cem. NaHCO, 5,7 °/,, 10 eem. KCl 1°/, and 40 eem. CaCl, 5 °/, and adding boiled distilled water up to 1 Liter. Certain quantities of the sugars') were dissolved in this solution, but still the reduc- tive capability was estimated before each experiment. This was done in view of the possible errors in weighing or unknown differences in the amount of water contained in the sugars. Bane’s method (1916) was used for estimating the reductive capability of the perfusion liquid as well as that of the urine excreted. The reductive power of the various sugars was expressed in the percentage of glucose contained. As is well know the final titration is an estimation of lodine with the aid of amylum; the amount of glucose contained is then computed from the quantity of Iodine necessary, by means of the formula (a—0,12) : 4, in which “a” is the number of c.c. solution of Iodine used. A. Laevulose (Fructose). Experiment 1 (July 11, 1918) The perfusion liquid contains 0,1 °/) laevulose. The reduction, expressed in glucose, amounts to 0,21 9/,. 0,1 eem. urine from the right kidney needs 0,87 ecm. lodine solution, which 0,87—0,12 corresponds to mi SIE 0,0875°/, glucose. 1) We are indebted to Jhr. W. ALBERDA van EKENSTEIN, Director of the Jaboratory of the ministry of Finan e and by Prof. H. J. Backer, for several of the sugars. 36* 550 0,1 eem. urine from the left kidney needs 0,88 ccm. Iodine solution which 0,88—0,12 corresponds to ea tea J = 0,190/, glucose. Retained by the right kidney 0,21—0,1875 = 0,0225°/,,. Retained by the left kidney 0,21—0,19 = 0,02°/). These quantities are so small that one may say that practically all the laevulose is allowed to pass by the glomerulus epithelium. The following experiments affirm this result. Experiment 2 (July 12). The same solution as used in experiment 1. Reduction (0,1 ¢.c.) urine of right kidney 0,21250/,. Reduction (0,1 c.c.) urine of left kidney 0,215°/). Retained by right kidney 0,21—0,2125 = 0. Retained by left kidney 0,21 —0,215 = 0. Result: Mo laevulose retained by the glomerulus epithelium. ’ 8 p Experiment 3 (July 13). Experiments 1 and 2 were repeated with fresh solution. Reduction 0.1 c.e. of solution passed 0,18"/ . Reduction 0,1 c.c. urine of right kidney 0,18°/,. Reduction 0,1 ec. urine of left kidney 0,18°/,, Result: No laevulose retained. Experiment 4 (July 14). The same solution passed as in experiment 3. Reduction 0,1 c.c. urine of right kidney 0,1825°/,. Reduction 0,1 c.c. urine of left kidney 0,1825° 9. Result: No laevulose retained. It has probably not escaped attention that the laevulose solution used above causes about twice as strong a reduction as a glucose solution of 0.1°/, viz. 0.18°/, on an average. Where it could be possible that a solution with a so much larger reductive capability could, just in connection with that faet, be allowed to pass by the glome- rulus epithelium, we experimented with a solution in whieh there was 0.05°/, laevulose instead of 0.1°/,. Experiment 5 (July 15). RinGer’s solution in which 0,05°/) laevulose has been dissolved. Reduction 0,1 c.c. of perfusion liquid 0,095°/). 0,1 ec. urine of right kidney: 0,5 ¢.c. Iodine solution; reduction 0,0950/,. ,1 c.c. urine of left kidney: 0,52 e.c. lodine solution; reduction 0,1°/,. 0,1 ec. urine of right kidney: 0,52 c.c. Iodine solution; reduction 0,19/o. Retained by right kidney 0,095—0,095 = 0. Retained by left kidney 0,095—0,1 = 0. Retained by right kidney 0,095—0,1 = 0. 551 Result: The diluted laevulose solution also passes completely through the glomerulus epithelium. It seemed interesting to investigate whether laevulose could influence the retention of glucose. In order to ascertain whether the circumstances, as regards the manner in which we had formerly worked with glucose, had indeed remained unaltered, several experiments were made with glucose aloue. B. Glucose and a mixture of Glucose and Laevulose. ‚Experiment 6. (July 18). Perfusion liquid in which there was 0,1°/, glucose. 0,1 e.e. of solution, reduction 0,0975°/,. 0.1 c.c. urine of right kidney: 0.3 c.c. Iodine solution; reduction 0,045°/). 0,1 c.c. urine of left kidney: 0,31 c.e. Iodine solution; reduction 0.0475°/,. Retained by right kidney: 0,0975—0,045 = 0,0525"/,. Retained by left kidney: 0,0975—0,0475 = 0,05°/). Experiment 7 (July 18). The same perfusion liquid as in experiment 6. 0,1 c.c. urine of right kidney: 0,26 c.c. lodine solution; reduction 0,035°/). 0,1 c.c. urine of left kidney: 0,29 ec.c. lodine solution; reduction 0,0425°/). Retained by righ! kidney: 0,0975—0,035 = 0,0625°/,. Retained by left kidney: 0,0975—0,0425 = 0,0550°/,. Experiment 8 (July 20). Perfusion liquid contains 0,07°/, glucose. Reduction by 0,1 e.c. of solution passed 0,065°/, 0,1 c.c. urine of right kidney: 0,22 e.c. lodine solution; reductidn 0,025°/,. 0,1 c.c. urine of left kidney: 0,22 c.c. lodine solution; reduction 0,025°/5. Retained by each kidney: 0,065—0,025 = 0,04°/,. Experiment 9 (Sept. 20). At the same time an experiment was made in which the perfusion liquid contained 0,20/, glucose. Reduction 0,1 c.c. of this solution 0,22°/. Reduction by 0,1 cc. urine of right kidney 0,095°/5. Reduction by 0,1 c.c. urine of left kidney 0,11250/,. Retained by right kidney: 0,22—0,095 = 0,1250°/). Retained by left kidney: 0,22—0,1125 = 0,10%5°/,. From these experiments it follows that now, as formerly, a quantity of glucose is retained, which is physiologically present in the blood of the frog. A remarkable contrast thus exists between the permeability of the kidney to glucose and to laevulose. With this marked difference it appeared to be of importance to ascertain whether laevulose was perhaps capable of altering the permeability to glucose. 552 Both these substances were therefore dissolved in the perfusion liquid. Experiment 10 (Sept. 20). The perfusion liquid contains 0,1°/, glucose and 0,05°/, laevulose. Reduction 0,1 e.e. of solution passed 0,205° 9. 0,1 c.c. urine of right kidney: 0,58 c.c. lodine solution ; reduction 0,11750/,. 0,1 c.c. urine of left kidney: 0,60 cc. lodine solution: reduction 0,12%/ Retained by right kidney : 0,205—0,1175 = 0,0875°/). Retained by left kidney: 0,205—0,12 = 0,085°/,. Experiment Il (Sept. 20). The same perfusion liquid as in experiment 9. Reduction 0,1 c.c. urine of right kidney 0,155°/). Reduction 0.1 c.c. urine of left kidney 0,15750/,. Retained by right kidney: 0,205 —0,155 = 0,050/,. Retained by left kidney: 0,205—0,1575 = 0,0475°/,. Experiment 12 (Sept. 20). The perfusion liquid contains a mixture of glucose and laevulose. Reduction of solution passed 0,205°/,. Reduction 0,1 c.c. urine of right kidvey 0,115°/). Reduction 0,1 c.c. urine of left kidney 0,120/,. Retained by right kidney : 0,205—0,115 = 0,099/,. Retained by left kidney : 0,205 —0,12 = 0,085°',,. Experiment 13 (Sept. 21). The perfusion liquid contains 0,07°/) glucose and 0,05°/, laevulose. Reduction 0,1 e.e. of this solution 0,175°/ 9 (average of 3 homonymous experiments), 0,1 ce. urine of right kidney: 0,65 ce. Iodine solution; reduction 0,12750/,. 0,1 ec. urine of right kidney : 0,62 ¢.c. [odine solution; reduction 0,125°'). 0,1 ee. urine of left kidney: 0,65 cc. lodine solution; reduction 0,13259/,. Retained by right kidney : 0,175—0,1275 = 0,0475°/ . Retained by right kidney :0,175 —0,125 = 0,05°/o. Retained by left kidney : 0,175—0,1325 = 0,0425°/. Experiment 14 (Sept. 21). The same perfusion liquid as in experiment 13. 0,1 c.c. urine of right kidney: 0,64 c.c. lodine solution; reduction 0,139/,. 0,1 c.c. urine of left kidney: 0,64 ee. Iodine solution; reduction 0,1275°/). Retained by right kidney : 0,175—0,13 = 0,045°/,. Retained by left kidney: 0,175—0,1275 = 0,0475°/,- It is apparent from these experiments that the glomerulus epithelium which, as we have seen, is completely permeable to laevulose, has held back a quantity of glucose, which was also retained when the solution perfused contained glucose alone. The laevulose while passing itself does not or hardly influence the retention of glucose. With a little exaggeration one might this say that the kidney separates the glucose from the laevulose by means of filtration. 553 C. Galactose. Experiment 15 (July 18). The perfusion liquid contains 0,09°/, galactose. Reduction (0,1 c.c.) of solution passed 0,079/,. 0,1 c.c. urine of right kidney: 0,35 c¢.c. Iodine solution; reduction 0,0559/,. 0,1 c.c. urine of left kidney: 0,33 ¢.c. Iodine solution; reduction 0,0529/,. Retained by right kidney: 0,07 — 0,055 = 0,015°/,. Retained by left kidney: 0,07— 0,052 = 0,018°/,. Experiment 16 (Aug. 22). The perfusion liquid contains 0,1°/, galactose. Reduction 0.1 ¢.c. of solution passed 0,07°/). 0,1 ee. urine of right kidney: 0,3 ¢.c. Iodine solution: reduction 0,045°/,. 0,1 ec. urine of left kidney: 0,28 e.c. lodine solution; reduction 0,040/,. Retained by right kidney: 0,07 —0,045 = 0,025°/). Retained by left kidney: 0,07 — 0,04 = 0,03°/,. Experiment 17 (Aug. 23). Reduction 0,1 e.c. of perfusion liquid 0,055°/,. 0,1 ee. urine of right kidney: 0.25 ec. lodine solution; reduction 0,03250/,. 0,1 c.c. urine of left kidney: 0,15 e.c. lodine solution; reduction 0,0825°/,. Retained by each kidney 0,055—0,0325 = 0,0225°/,. Experiment 18 (Aug. 23). The same perfusion liquid as in experiment 17. 0,1 c.c. urine of right kidney: 0,25 c.c. lodine solution; reduction 0,03259/,. 0,1 c.c. urine of left kidney: 0,25 c.c. Iodine solution; reduction 0,0325°/). Retained by each kidney: 0,055—0,0325 = 0,0225°/,. All these experiments show that the kidney retains a slight quantity of galactose. We shall now record a few experiments with a perfusion liquid the reduction of which approximately agrees with that of 0.1°/, glucose. Experiment 19 (Aug. 23). The perfusion liquid contains 0,15%/, galactose. Reduction 0,1 c.c. of perfusion liquid 0,0975°/,. 0,1 c.c. urine of right kidney: 0.4 ¢.c. lodine solution; reduction 0,07°/,. 0,1 cc. urine of left kidney: 0,4 c.c. [odine solution: reduction 0,079/,. Retained by each kidney; 0,0975—0,07 = 0,0275”/,. Experiment 20 (Aug. 23). The same perfusion liquid as in experiment 19. 0,1 c.c. urine of right kidney: 0,33 c.c. Iodine solution; reduction 0,0525°/). 0,1 c.c. urine of left kidney: 0,32 c.c. lodine solution; reduction 0,05°/,. Retained by right kidney : 0,0975—0,0525 = 0,045°/,, Retained by left kidney: 0,0975—0,05 = 0,0475°/,. Here again it becomes clear that some galactose is retained. In the last experiment (20) the quantity is even comparatively large. 554 That galactose is retained is efficient in the same way as is the case with glucose. While glucose is a source of energy for muscular contraction; galactose helps in the formation of the cerebrosides. As has been remarked earlier, frogs often present not inconside- rable differences in their capacity for retaining glucose. The time of year also has some influence. Therefore experiments were again made with frogs that had lived under the same circumstances as those of the experiments described above. Experiment 21 (Aug. 24). The perfusion liquid contains 0,1°/) glucose. Reduction 0,1 c.c. of perfusion liquid 0.10%)... 0,1 cc. urine of right kidney: 0,24 cc. lodine solution; reduction 0,039/,. 0,1 c.c. urine of left kidney: 0,22 c.c. [odine solution; reduction 0,025°/,. Retained by right kidney: 0.10—0,03 = 0,079/,. Retained by left kidney : 0,10 —0 025 = 0,075"/, Experiment 22 (Aug. 24). The same perfusion liquid as in experiment 21. 0,1 cc. urine of right kidney: 0,25 c.c. Iodine solution; reduction 0,0325°/, 0,1 cc. urine of left kidney: 0,22 e.c. lodine solution; reduction 0,025°/,. Retained by right kidney : 0,10—0,0325 = 0,0675 /). Retained by left kidney : 0,10 —0,025 = 0,075 /,. It is clear that these frogs, which were placed in the same cir- cumstances as those of experiments 16—20, retained a much larger quantity of glucose than of galactose. It now seemed desirable to investigate whether the retention of galactose although this occurred in a much smaller degree than was the case with glucose, was governed by the same conditions as regards the composition of the Ringer-solution, as had formerly been found to apply to glucose. For this reason the following Ringer’s solution was used: NaCl 0,7 °/,, NaHCO, 0,02 °/,, KCl 0.01 °/,, CaCl, 0.0075 °/,. With the application of this solution glucose was at the time retained to a maximum of 0.03 °/,. Only when the quantity of NaHCO, was increased above 0.09 °/, on account of which the urine was no longer acid, there could be retained much more sugar; the urine could then even be free from sugar. What would now be the result with galactose if the Ringer’s solu- tion also in this case contained 0,02 °/, NaHCO, only? Experiment 23 (Sept. 14). The perfusion liquid with only 0,02°/, NaHCO, contains 0,1°/, galactose. Reduction of 0,1 c.c. of perfusion liquid 0,08°/,. 0,1 cc. urine of right kidney: 0,42 c.c. lodine solution; reduction 0,0759/,. 0,1 c.c. urine of left kidney: 0,43 c.c. Iodine solution; reduction 0,0775°/,. Retained by right kidney: 0,08—0,075 = 0,005"/,. Retained by left kidney : 0,08—0,0775 = 0,0025°/,. 555 Experiment 24 (Sept. 14). The same perfusion liquid as in experiment 23. 0,1 e.c. urine of right kidney: 0,43 cc. Iodine solution; reduction 0,0775°/). 0,1 ee. urine of left kidney: 0,44 c.c. lodine solution ; reduction 0,080/,. Retained by right kidney: 0 08—0,0775 = 0,0025°/, Retained by left kidney : 0,08—0,03 = 0. We see thus that with galactose as well as with glucose, the quantity of NaHCO, in the perfusion liquid is of great importance for the permeability of the glomerulus epithelium. A solution con- taining a small quantity of NaHCO, causes galactose to pass in toto; if the perfusion liquid contains a physiological quantity of NaHCO, (0,285 °/,) then on an average 0.025 °/, galactose is retained. We say on an average, as frogs present individual differences. Here follows a table which gives a summary of a series of other experiments with galactose in which however the quantity of NaHCO, was, accidentally, 0.2 °/, instead of 0.285 °/,. As had also formerly been found with glucose such a modification was of very little importance. Retention capability of kidney for galactose. | ‘ | eduction 5 | 2 A __.. ‚ Reduction urine Retained Reduction urine © Retained perfusion liquid, | | | | containi ae af | by | ot Ke n | 8 Mee right kidney right kidney) _ left kidney _ left kidney | 0.1 °/, galactose | | u } | | | | | 0.0825, | A 2-95 to o49 | 0.0525) | | 0.04755 0.033 % | A 9. 9475\0-050 0.0325 °/, 0.0775 | | __ 0.0825 | LB 6:08 40-078 008 Bo:ogas0-0625 | 0 ~~ 0.0675 | 0.0675) | | | C 6:065 40-0663 0.063, C 97067549-0675 | 0.05 , | | | 0.0725 ,, ‚_D 0.05 0.0225 „ | D 0.0525 |: 002, | | | | |E 0.045 | 0.025 „ | E 0.048 0.045 , ‚_F 0.047 0.0250 „ | F 0.045 0.0275 , When we make a study of this table it becomes clear ; 1. that there is a difference in the power for retaining galactose in the various frogs A, B,C, D, E, and F. 2. that the individual differences range between 0 and 0,033°/,. 3. that the power of retention is more or less the same for the right and the left kidney of the same frog. Here again as in the former experiments ut is clear that galactose 556 does not, like laevulose, pass through the kidney altogether, but is generally retained to a slight degree. This indicates in any case that galactose is used in the body. We know indeed that lactose is built up from dextrose and galactose and an article of Ersr HirscuBerG *) from WInTERSTEI’s laboratory has just appeared in which a certain affinity, that is wanting in laevulose, becomes clear between glucose and galactose in connection with the spinal cord. We repeat, that the amount of galactose retained is in all the experiments expressed in the percentage of glucose. This is also the case with other sugars. D. Mannose. Experiment 25 (Aug. 20). In the suitable perfusion liquid consisting of NaCl 0,5°/), NaHCO, 0,285°/,, KCl 0,01°/, and CaCl, 0,02/,, 0,19/) mannose is dissolved. 0.1 ¢.c. perfusion liquid: 0,45 ¢.c. lodine solution; reduction 0,08250/,. 0,1 e.e. urine of right kidney : 0,48 c.c. lodine solution; reduction 9,090/,. 0,1 ec. urine of left kidney: 0,46 c.c. Iodine solution ; reduction 0,0850/,. Retained by right kidney : 0,0825—0,09 = 0. Retained by left kidney: 0,0825—0,085 = O. Experiment 26 (Aug 20). The same perfusion liquid as in exp. 25. 0,1 c.c. urine of right kidney: 0,45 c.c. lodine solution ; reduction 0,0825°/». 0,1 e.c. urine of left kidney : 0,44 c.c. lodine solution ; reduction 0,08°/,. Retained by right kidney: 6,0825—0,0825 = 0. Retained by left kidney : 0,0825—0,08 — 0,0025°,,,. Thus also in this experiment no mannose was retained. Experiment 27 (Aug. 21). 0,1 c.c. perfusion liquid: 0,35 c.e. lodine solution ; reduction 0,0575° '. 0,1 ec. urine of right kidney: 0,37 ¢.c. Iodine solution ; reduction 0.0625°/,. 0,1 c.c. urine of left kidney: 0.35 cc. lodine solution ; reduction 0,0575/,. 0,1 cc. urine of right kidney: 0,35 cc. Iodine solution ; reduction 0,0575°/). In this exp. also no mannose is retained. Experiment 28 (Aug. 21). The same perfusion liquid as in exp. 27. 0,1 c.c. urine of right kidney: 0,36 c.c. Iodine solution; reduction 0,06°/). 0,1 e.c. urine of left kidney: 0,36 c.c. Iodine solution; reduction 0,06°/,. Retained by each kidney 0,0575—0,06 = 0. In this experiment the Kidneys have also passed all the mannose. From these experiments we may draw the conclusion that the glomerulus epithelium is totally permeable to mannose. We shall now report some experiments made with mutually isomeric disaccharides: saccharose, maltose and lactose. a) Exse Hinscupens: Zeitschr. f. physiol. Chemie, 100, (1918). 557 These experiments seemed of importance because, amongst other reasons, their molecules are larger tban those of the monosaccharides already considered. If the permeability was in connection with this size it would probably appear that the glomerulus epithelium was impermeable to the dissacharides. E. Saccharose. To estimate the quantity of saccharose in the perfusion liquid and in the urine, 0.1 cc. of the liquid was heated to 37° C. during 1?/, hours with 0.15 ce. hydrochloric acid 1:1 and the reduction of the solution thus obtained determined. Experiment 29 (July 17). The perfusion liquid contains 0,1°/, cane sugar. After inversion it causes a reduction of 0,1275°/,. 0,1 e.c. urine of right kidney after inversion causes 0,13°/, reduction. 0,1 c.c. urine of left kidney after inversion causes 0,1325°/) reduction. Retained by right kidney; 0,1275—0 13 = 0. Retained by left kidney: 0.1275—0,1325 = 0. Result: all the cane sugar has passed through. Experiment 30 (July 17). The same perfusion liquid as in exp. 29. Reduction 0,1 ¢.c. urine of right kidney after inversion 0,1125°/). Reduction 0,1 e.c urine of left kidney after inversion 0,0959/,. Retained by right kidney : 0,2275—0,1125 = 0,015" 9. Retained by left kidney : 0,1275—0,095 = 0,032°/). Here follows a table in which several experiments are summarised. There can be no doubt that the glomerulus epithelium has retained saccharose, either as such or in the form of glucose which has then been formed through the splitting of saccharose in the glomerulus epithelium, while the laevulose has passed into the urine. This latter alternative is of no great probability. F. Maltose. Experiment 35 (July 16). In the suitable perfusion liquid with 0,285° » NaHCOsg, 0,15°/, maltose is dissolved. 0,1 c.c. perfusion liquid: reduction 0,0825"/). 0,1 c.c. urine of right kidney: reduction 0,07°/). 0,1 e.c. urine of left kidney: reduction 0,07°/,. Retained by each kidney : 0,0825—0,07 = 0,0125"/,, thus hardly any at all. Experiment 36 (July 16). The same perfusion liquid as in exp. 35. 0,1 c.c. urine of right kidney: reduction 0,0875°/). 0,1 e.c. urine of left kidney: reduction 0,0625°/5. Retained by right kidney: 0,0825—0,0675 = 0,015°/,. Retained by left kidney : 0,0825—0,0625 = 0,02’/,. 558 Retention capability of kidney for saccharose. Experiments of September 17—23 1918. Reduction perfusion liquid Reduction urine of | REISER containing + 0.1 0/, saccharose right kidney 5 we ¥ ‘tee ee he Fite eee eee right ; ; : after «| Previous after ; : Previous to inversion , ; : ; kidney inversion ‚version | inversion Reduction urine of inversion | | | left kidney Retained if oe ee € by Previous. left to ha kidney inversion | | | Frog A 0.01 % 0.13 % 0.0325 0/, 0.09 : : | 0.08 0.05 Frog A 0.015 0.125 0.02 0.0925 0.0325 : P 0.04 0.0925 0.0325 Frog A 0.025 0.14 0.03 0.1025 0.0325 : : < 0.0875 0.0525 ’ 5 ; 0.09 0.05 Experiment 37 (Sept. 26). The perfusion liquid contains 0,15°/, maltose. O,1 ce. perfusion liquid: reduction 0,095/,. 0,1 ce. urine of right kidney: reduction 0,105°/,. 0,1 ce. urine of left kidney: reduction 0,11°/,. Retained by each kidney: nothing. Experiment 38 (Sept. 26). The same perfusion liquid as in exp. 37. 0,1 c.c. urine of right kidney : reduction 0,1°/ . 0,1 e.c. urine of left kidney: reduction 0,105"/, Retained by each kidney: nothing. Experiment 39 (Sept. 26). The same perfusion liquid as in exps. 37 and 38. 0,1 cc. urine of right kidney: reduction 0,0625°/,. 0,1 c.c. urine of left kidney: reduction 0,06250/,. Retained by each kidney: 0,095—0,0625 = 0,0325°/). Experiment 40 (Sept. 27). 0,1 cc. perfusion liquid: reduction 0,08°/,. 0,1 c.c. urine of right kidney: reduction 0,065°/. 0,1 c.c. urine of left kidney: reduction 0,0625/,. Retained by right kidney : 0,08—0,065 = 0,015°/,. Retained by left kidney : 0,08—0,0625 = 0,0157°/,. | 0, 0.043 °/, 0.025 °/, 0.0825 %/, 0.0475 0 0.0775 0.0525 0.02 0.0925 0.0325 0.03 0.0925 0.0325 0.03 0.10 ‚0.04 0.0825 0.0575 It is clear from all these experiments that the quantity of maltose retained is in any case extremely small, notwithstanding the fact that 559 this dissaccharide is built up from two molecules of glucose, and that the glomerulus epithelium is permeable to glucose to a very slight degree. G. Lactose and a mixture of glucose and lactose. Experiment 41 (July 3). 0,2°/; Lactose is dissolved in the perfusion liquid. 0,1 c.e. of perfusion liqdid: reduction 0,14°/. 0,1 c.c. urine of right kidney : reduction 0,1375°/. 0,1 ec. urine of left kidney: reduction 0,14%/5. Retained by right kidney: 0,14—0,1375 = 0,0025°/,. Retained by left kidney: 0,14—0,14 = 0. Experiment 42 (July 3). 0,1 e.e perfusion liquid: reduction 0,11°/,. 0,1 c.c. urine of right kidney: reduction 0,11°/). 0,1 ¢.c. urine of left kidney: reduction 0,1075"/). Retained by right kidney: 0,11—0.11 = 0. Retained by left kidney: 0,11—0,1075 = 0,0025°/,. Here again the kidney has retained no, or very little, lactose. Experiment 43 (July 3). The same perfusion liquid as in exp. 42. 0,1 c.c. urine of right kidney : reduction 0,1075°/ . 0,1 c.c. urine of left kidney: reduction 0,10750/,. Retained by each kidney: 0,11—0,1075 = 0,0025°/,. Here again neither of the kidneys retained lactose. Experiment 44 (Juli 4). 0.1 e.e. perfusion liquid: reduction 0,140/, 0,1 c.c. urine of right kidney: reduction 0.1450°/). 0,1 ¢.c. urine of left kidney: reduction 0,13250/,. Retained by right kidney : 0,14—0,1450 — O. Retained by left kidney: 0,14—0,1325 = 0,00750/,. Lactose has thus been retained by neither of the kidneys. From this we may draw the conclusion that the glomerulus epithelium allows the lactose to pass altogether, although it is built up from glucose and galactose, of which compounds the former is retained to a large degree, and galactose as well, although less. From a_ theoretical as well as from a clinical point of view it seemed of importance to investigate how the kidney would behave towards a mixture of lactose and glucose. . From these experiments we learn that the retention of glucose by the kidney is not influenced by lactose. The latter is passed while glucose is retained to the same degree as when there was no lactose present. This result is in accordance with the use which for a 560 Retention capability of kidney for a mixture of 0.1 0/, glucose and 0.1 0/, sacch. lactis (lactose). Experiments of September 24—26. Reduction urine Retained Reduction urine Retained of by of by right kidney right kidney left kidney left kidney Reduction perfusion liquid Frog a 21ST OGEN ons B oee Frog B O0 40-0925 | 0.065 „| pigar 10:00 0.065 , Frog C __„ 0.09 0.0675 „ 0.0925 | 0.065 , Frog D - 0.1 ED ven” — | — Frog A 0.16 0% 0.096 0.064 , 0.104 | 0.056 „ Frog B „ 0.10 dine. Gide | 0.056 , Frog C , 0.104 0.056 , 0.004 | 0.066 , Frog D , 0.096 0.064 , 0.10 0.060 , considerable time has been made of lactose for clinical use to estimate the validity of kidneys and which is founded on the consideration that the healthy kidney easily passes lactose. We thank Mr. R. Roerink for his able assistance during this research. Summary and Conclusion. 1. The fact, now again affirmed, that, when we pass a RINGER’s solution to which glucose has been added, through the kidney, this is retained by the glomerulus membrane, while salts, which are also crystalloids, are allowed to pass, has raised the question to what this contrast must be attributed. 2. In the first place we can think of the circumstance that glucose possesses a so much larger molecule, which could then impede the passage. If this hypothesis be correct then the disaccharides such as saccharose, maltose and lactose, which possess a still larger molecule (C,,H,,0,,) than glucose (C,H,,O,), would certainly not pass through. Experiments have however proved that the glomerulus epithelium is permeable to a large degree to these sugars, even to raffinose (C,,H,,0,,). The permeability to lactose is perfect. 561 3. Where the cause for the retention of glucose cannot be ascribed to the size of its molecule, we are bound to consider its structure or configuration. There is the more occasion for doing this because its isomerics, laevulose and mannose, are allowed to pass altogether and galactose to a large degree, as has been proved by the experiments under discussion. 4. Glucose therfore occupies a unique position among the mono- saccharides in regard to the glomerulus membrane. In other words the glomerulus membrane can distinguish glucose from the other monosaccharides in a manner that reminds of the relation of sugars and ferments, in connection with which Emin Fiscuir used the well- known simile of a lock and key. In any case these experiments are again a new ulustration of the doctrine of stereotsometrics, but now not as has thus far been the case, through facts of chemical but of physiological nature, belonging to permeability. 5. Not without theoretical and clinical importance seems the fact that the capability for retention of glucose is not modified when glucose and laevulose are simultaneously present in the perfusion liquid. The two sugars are simply separated as by a filter: the glucose remains behind, the laevulose is passed. This also appears to be the case with a mixture of lactose and glucose: the lactose passes completely into the urine and the glucose is retained by the glomerulus epithelium to the same degree as when there was no lactose present. Groningen, September 1918. Physiological Laboratory. Chemistry. — “On the Significance of the Volta-Ejfect in Measure- ments of Blectromotive Equilibria’. By Prof. A. Smirs and J. M. Bisvorr. (Communicated by Prof. P. ZrrMAN). (Communicated in the meeting of Sept 28, 1918). Introduction. Many physicists are of opinion that the Volta effect amounts to only a few milli volts, and that the electromotive force of an elec- trical cell resides practically exclusively in the potential differences metal-electrolyte, so that these alone need be taken into consideration ; there are even those who think that the Volta-effect is theoretically zero. It is chiefly the German school that assumes that the Volta- effect may be neglected with respect to the potential difference metal-electrolyte. On the other hand a great number of investigators think they have found that the Volta-effect can constitute 4, # and even a still larger fraction of the electromotive force of a cell, and can accordingly amount to as much as 1 Volt. The opinions concerning the value of the Volta-effect are therefore greatly divided, which is owing to the great difficulties which attend the determination of the Volta-effect. | In the application of the recent views about the electromotive equilibria!) to the Volta-effect it appears, that even though this effect should be small for metals in the state of internal equilibrium, it must become great for phenomena of polarisation, so that we may certainly not neglect the Volta-effect for these cases. 2. The variation of the Volta-ejfect on polarization and passwation. The following equation holds for the potential difference of a metal in active state with respect to an electrolyte: 0,058 K'” (MS )active log =: Li My and thus we get for the passive state: 0,058 Kage (lc reese - log = i (Mz ) 4 M active. =S TT Am passive, — — from which follows. 1) Zeitschr, f. physik. Chemie 88, 743 (1914). » es; 5 90, 723 (1915). 563 0,058 ] (MS )active = og —— 4 : (MES Voaaite For the Volta-effect holds the relation: Ou,—Om, Bait on from which follows in the same way as we have derived this for the potential difference metal-liquid, that the following equation holds for the Volta-effect : A M active —L — AM passive—L a Am-—Y, = K'(04 Am —M, = 9,058 log Km) (Ou) so that: (Ou activ A Myactire— Ma a BVM pastels = 0,058 log (OM active (4 M passive When we call: (MG? Vaciive =n (Mis ) passive then also (9.1, active ; EEE (Oan, )passi ve It follows from this that when on polarisation or passivation the change of the potential difference metal-electrolyte is: 0,058 log n that of the Volta-effect amounts to: 0,058 log n hence v times the value. What we measure is the sum of these two changes: vp + 1 YP 0,058 log n. Hence the part ran of this total change is due to the Volta- Yv effect. This is, therefore, 4 for a uni-valent metal, and 2 for a bi- valent one etc. When it is now borne in mind that on passivation through anodic solution the potential difference metal-electrolyte, as it is found by measurement, can change by an amount of 1 or 2 Volt. (e.g. 1.7 Volt, is found for iron), it follows from this that aecording to these considerations also the Volta-effect is subjected to a great change *). 1) It must still be pointed out that the case iron is certainly more intricate than the case considered above, because iron contains ions of different valency. 37 Proceedings Royal Acad. Amsterdam. Vol. XXL. 564 This throws, indeed, a peculiar light on the Volta-effect, for it now appears clearly that when really the Volta-efject for metals in the state of internal equilibrium should be zero or very small, which comes to the same thing as equality or nearly equality of the mole- cular thermodynamic potential of the electrons in those metals, this would be a special characteristic property according to the theory of electrons for metals in the state of internal equilibrium. But apart from the value of the Volta-effect, in the logical develop- went of the given theory of the electromotive equilibrium, the Volta-effect cannot be neglected. *) 1) In this we will also point out that when the new views about the rôle of the electrons and the electromotive equilibrium are consistently applied to an arbitrary electrical circuit, we arrive at the result that everywhere where a potential difference occurs, a reaction takes place on passage of the current, in which the change in free energy of the reaction proceeding at that place, determines the value of the potential difference. It is known that according to the given theory at the places of contact metal-electrolyte this reaction consists: 1. of the splitting up of metal ions into ions and electrons; resp. of the formation of metal atoms from the electrically charged dissociation products mentioned here and 2 of the transition of ions and electrons from the metal phase into the electrolyte or vice versa. In this transition from one phase into the other the ions take a prepon- derant part. When we consider the contact metal-metal, the just-mentioned transition consists to by far the greater part of the displacement of electrons from the metal M, to the metal Mp. This view differs accordingly from that which is particularly met with in German handbooks, in which the opinion is embraced that the reactions during passage of the current exclusively take place at the places of contact metal-electrolyte. It is clear that on application of the relation: dA E=A — T — ar to the transition of electricity from one metal into the other we may only conclude to: OA A - d1 when in the process mentioned here no change of the thermo-dynamic energy takes place (E = 0). LEBLANG does so in his handbook p. 227 (1914). and thus comes to the con- clusion that the Volta-effect must be small, because = is small. According to the theory of electrons the difference in solubility heat of the electrons in the two metals will, however, have to be taken into account. In the isotherm transition of electricity between the two metals the free energy of the electrons will change, and be converted into electrical energy in case of a reversible process, the occurring change of the bound energy becoming manifest through the latent heat, which heat is the heat of PeLrtreR. . | dA It is this quantity, the heat of Perrier which, is represented by Dar: and not the Volta-effect. 565 3. The experimental electrical potential. When we measure the potential difference metal-electrolyte, we do so by the aid of an auxiliary electrode, e.g. a calomel or hydrogen electrode, in other words, we then make an electrical cireuit, which is closed during the measurement. The electro-motive force of this circuit, in which we suppose the diffusion potential between L, and Z, annulled, then becomes: EE Au — 4u,—1, — Òam Ms in which Ay, is a Volta-effect. When in this we put Ay, 7, = zero (N. hydrogen-electrode), we get: E= Ay,-1,— 4u,-my,, in which we sball call # the experimental electrical potential. This expression, therefore, always contains the Volta-effect, and until this quantity is known, and until we introduce further sup- positions about its value, we can of course not determine the differ- ence Ay, alone in this way, and consequently not the saturation concentration of the metal-ions éither (which quantity is equivalent with Nernst’s “Lösungstension’””), which was calculated from: aed Ku” E=— — ln — EW eer ae tE) in which yy, has been neglected. Nor can we draw conclusions about the order of the saturation concentrations of the metal-ions from the so-called tension series, until the Volta-effect shall be known. From the fact that the metal M, immersed in a normal solution of one of its salts appears on measurement to be more negative than the metal M,, a 1 norm.-solution of a corresponding salt, we conclude namely that: immersed in Am,—z, is more negative than Ay, 7, but strictly speaking this conclusion is not permissible, because the measurement only says that Ay», — ym, is more strongly nega- tive than Ay, . We demonstrated that the equations for the exp. elec. potential in the form in which they contain the saturation-concentration of the ions or the ““Lösungstension’”’, have the drawback that the un- known Volta-effect occurs in them. It, is entirely different with the new already before given equation, in which the solubility product of the metal or the solubility quotient of the metalloid occurs, and the same thing may be said of the electron equation. ') 1) Zeitschr. f. physik. Chemie 92, 1 (1916). 37* 566 Thus the following equation was namely derived: BT, (Mya) BT, 2) Maa Man rie abn hae Cares 1 1 (2) ANT MD —Ay,— 1, = we measure, however, E= Amt Da Sige a and the Volta-effect being : Ki RONDE Mi —-M3 — fe F it follows from this that: RT (hy) BR Oy) ——In ——ln —. yf Ly, rf Lu, ES (4) in which the Volta-effect has been eliminated. This equation enables us, therefore, to find the ratio of the solubility- products from the electromotive forces. In the practical application the hydrogen electrode may be taken for the metal J/,, and for Ly, the value may be substituted which had already been given before, viz. 10?*~—48"); in this case we get: (Mr’) 0,058 = ieee 2g) 8, » M When equation (4) is compared with the expression for 4, which contains the saturation-concentrations of the metal-ions found by substituting the values of Ay, and Ay, 7, given by equation (1) in equation (3), which gives: KT “Wee ed ge i= = ala he pF RES) ol ae E Dag. 16 i, oe the great advantage which equation (4) resp. (5) have over equation (6) is very apparent, for the latter equation contains a still unknown Volta-effect. While the construction of a series of the potential differences metal-electrolyte is not yet possible, on account of our ignorance of the Volta-effect, equation (5) enables us to draw up a series for the solubility-products of the metals, and from such a relation there may be found a series for the solubility-quotients of the metalloids, as has been done already *). The determination and order of this series is of course the same as that of the socalled tension-series, which gives the order in which 1) Zeitschr. f. physik. Chemie 92, 1 (1916). *) Smits and LoBry pe Bruyn. Verslag Kon. Ak. 26, 270. 567 the metals oust each other from equally concentrated solution. We demonstrated, however, that in a theory which assumes the existence of Volta-effects, the quantities derived from the electromotive equilibria should not be considered as the saturation-concentrations of the metal- ions or as the “Lösungstensionen”’. The electron equation gives a relation that agrees with equation (3), viz. : BR Gelk, Alt es Amr —À enge 1 de M,—L, Mo—Ly F n ( 0) F ; so that RE lo EE (7) F (07) Of course the Volta-effect does not occur in this relation either, so that when a definite value is assumed for the electron conc. (47,) in the electrolyte in which the hydrogen electrode is found, the electron-concentration (7), which belongs to the other electromotive equilibrium, can be found. Equations (4) and (7) are, therefore, not approximate, but perfectly rigorous, which also appears from the consideration that equation (7), (MD) immediately from the condition for electron equilibrium between the liquids ZE, and Z,. The derivation of this electromotive force as sum of the potential differences occurring in the circuit was accord- ingly only followed here in order to keep in agreement with the derivation given before. from which (4) is found by substitution of for (A,), follows 4. As appears from a previous communication *), the experimental elec. potential of a Ni-electrode, immersed in an electrolyte through which hydrogen was led, was found equal to the hydrogen potential. In order to account for this fact that the electromotive force / of the closed circuit nickel-electrode-electrolyte-hydrogen-electrode is zero according to our considerations we must show, as follows directly from equation (7), in what way the electron concentration of the nickel equilibrium has become equal to that of the hydrogen equilibrium. This phenomenon is already explained in a very simple way ’*) by the assumption that the nickel-electrode is, at least superficially, 1) Smits and LoBry pe BRUIJN loc. cit. 2) Smits and Losry pe Bruyn loc. cit. 568 disturbed, which disturbance for a very inert metal will go so far till the electron concentration of the nickel equilibrium in the electrolyte has conformed to the electron concentration of the hydrogen thee brium, and has become almost equal to it. For the limiting case of an ideal inert metal this equality will become perfect, as is also required by / becoming zero in this case. In a former discussion of these phenomena the neglect of the Volta-effect led on the other hand to the conclusion that the electro- motive force of the circuit, in case of equal electron-concentrations in the electrolyte of the two metal equilibria, would be zero only in approximation. It will have become clear through what precedes that while up to now the Volta-effect has been neglected in the A, X-figure, it is better to draw the experimental potential / (e.g. with respect to hydrogen) as ordinate instead of the potential difference metal-liquid ; then the thus obtained / X-figures are perfectly rigorous. They then indicate that for the three-phase equilibrium the two electrodes possess the same experimental potential; the difference between their potential differences with the electrolyte then being equal to the Volta-effect between the two metals. In this point the following equation then holds: 0.058 (1) 0,058 (4,7) ie te ig or so that from the ratio of the solubility-products the situation of the coexisting electrolyte immediately follows. Though this has been shown already before, this circumstance is once more pointed out here, to make clear that equation (6) is of no use to us here, for this equation gives for the three-phase equilibrium : __ 0,058 log Een Ks ibe +s Kae ae EL Se ARR Bata! wars) M,—Mz P, (Mi) p, (45) In this form the equation contains, however, the unknown Volta- effect and the unknown saturation-concentrations, so that a calcu- lation as above is not possible. Summary. By application of the more recent views on the electromotive 569 equilibria to the Volta-effect the result was obtained that on polari- sation and passivation the change in the Volta-effect must be great according to this theory, and that in these phenomena the Volta- effect would even constitute the greatest part of the total change in the electromotive force. [t was further pointed out that if the Volta-effects between metals in which the state of internal equili- brium prevails, should be really very small, on which the opinions are still divided, but which has, indeed, become probable by the experiments made of late, this would have to be considered accord- ing to the theory as a very characteristic property for metals in the state of internal equilibrium. As in principle the Volta-effect at any rate in the given theory of the electromotive equilibria may not be neglected, it was here taken into account. The quantities that are found on measurement of potential differences metal-electrolyte, and which are here called experimental electrical potentials, always contain an unknown Volta- effect, which is the cause that from the said potentials the saturation concentrations of the metal ions. (resp. the “Lösungstensionen’’) cannot be calculated. The newly derived relations, in which the solubility product of the metal, the so/wbilty quotient of the non-metal resp. the electron concentration in the electrolyte occurs, do not contain the Volta- effect, however, and enable us to draw up a solubility-product-sertes of metals, and a solubility-quotient-series of metalloids, as has indeed already been done, from which conclusions can be drawn about the chemical and electrochemical behaviour of the elements. Amsterdam, (reneral and Anorg. Chem. Laboratory Sept. 1918. of the University. Zoology. — “Androgenic origin of Horns and Antlers.” — By Prof. J. F. vaN BEMMELEN. (Communicated in the meeting of September 29, 1918). In his excellent work: “die Säugetiere”, Max Weber gives as his opinion about the origin of the cephalic armament of numerous Ungulates, that horns and antlers originally started in both sexes as defensive weapons against enemies, but later on more and more came to be used as instruments of offence by the males in their fights for the females, and so either have grown an exclusive attribute of the male sex, or at least have developed much more strongly than in the female sex. In this instance therefore WeBeER evidently shares the opinion pronounced by TanpiER and Gross in their paper: Die biologischen Grundlagen der sekundären Geschlechtscharaktere, where they say: “All secondary sexual features were originally specific features, properties therefore, characteristic of a certain species, even of a whole order of Vertebrates, without their primarily having any connection with the genital sphere.” In their commentary on this proposition they remark: ‘Hitherto in the morphology of the sexual characteristics too little attention has been paid to the question, how much of them is peculiar not for the sex, but for the species.” As a special example they cite the case of the horns of Cavicornia, “which do not constitute a sexual characteristic in themselves, but only in their shape, which differs for males and females, whereas on the contrary it is identical in masculine and feminine castrates.” “The same is the case with the hairiness in man. We have been able to show that such an eminently secondary sexual feature of the male sex, as the beard, is also found in old castrates, but there in form and extension resembles that of old women.” According to TanpLeR and Gross the question should not be formulated: “Is an organ a secondary sexual feature’, but: “How much in the development of an organ is specific, how much sexual.” Though this assertion might be granted, yet I believe to be justified in Opposing to it another view: viz. that the evolution has been just the reverse; the cephalic armament arising in males as a means of attack in their duels for the females, and afterwards passing to 4 571 the latter in a more or less reduced form and only in some of the species (according to the well-known rules of monosexual trans- gressional heredity), the result being that the horns now could be used in both sexes as means of defence. As arguments in favour of this opinion may be cited: 1. In Deer the antlers are absent in all females except those of the Reindeer, and precisely in this species a useful application of the horns, not connected with any sexual function, and alike for both male and female may be conceived, viz. the digging up of food from under the snow, though some authorities (e.g. BREHM) deny this function. 2. Among Antilopes, besides genera in which both sexes are horned, others occur in which only the male possesses these attributes, and in the majority of cases the horns of the females are smaller than those of the males. The latter moreover show a tendency to hypertrophic growth, just as is the case with deer, leading to unwieldy size or to sundry strange shapes (e.g. screwlike contortions) which seem to stand in direct contradiction with the requirements of practical use. 3. The same is the case with Cattle, Sheep and Goats, as shown by the four-horned goat, or the excessive development of the horns in the carbon and other buffaloes. 4. Even in Giraffes, whose minute pedicles with their small os cornu may probably be considered as rudiments of formerly better differentiated antlers (compare Ocapia and Sivatherium) the males possess higher and stronger hornstumps than the females, and more- over the unpaired nasal knob. In Ocapia on the other hand the horns are primarily absent in the female, and the same was probably the case during their whole life in those of Siva-, Hellado- and Samotherium. 5. The more original kinds of Ruminants: the Tragulidae and Camelidae, are destitute of horns, and so were the oldest and primitive extinct Artiodactyla (Pantolestidae, Anoplotheridae) as well as all Nonruminantia amongst them. The oldest fossil deer likewise did not yet possess antlers, as the muskdeer does up to this day, though the miocene Palaeomeryx according to RÜriMEYER and ScHLOssER was already provided with them. Perhaps this might be considered as an indicatiou that the tendency to the formation of antlers arose independently in the tribe of the Cervidae, but also as the mani- festation of a far older hereditary inclination to the production of bony frontal appendages of the male. This tendency then must have been in abeyance in the tribe of the Artiodactyla in general ; in Deer 572 it probably reappeared, and from thence continually increased in potency and complication. ¢ This conception according to my view is strengthened by the fact, that also among the Suidae a tendency to the formation of bony protuberances on the dorsal side of the skull undoubtedly occurs and appears in stronger manifestation in the male sex than in the female, as is shown by the monstrous skull of the male African wart-hog. 6. In Protoceratinae likewise the skull of the male only was provided at its upper side with a complete set of paired bony excrescences. 7. According to Marsu both sexes of the Dinoceratidae possessed these bony protuberances (and the large dagger-like tusks besides), yet in the male they grew unto a larger size than in the females. We might conclude from this that the tendency to the production of bony knobs on the skull is even older than the separation of Ungulates into Artio- and Perissodactyles. 8. The annual shedding of the antlers and their regeneration in Cervinae is apparently connected with the rut. The same appears to be the case in Antilocapra. 9. The abovementioned bony processes on the head of Giraffidae, (sensu latiori) Suidae, Protoceratinae and Dinoceratidae, cannot rea- sonably be considered as really practical weapons, as they are far too cumbrous and hypertrophic for that. Neither ean this be the case with the antlers of most Deer or the horns of numerous Antilopes, Cattle, Sheep, and Goats. On the other hand they wear to a very high degree the character of sexual attributes, in their exuberance, unpractical build, curious complication, obviousness and variability. 10. In the first (primitive) members of the Ruminantia antlers and horns apparently arose at a relatively late stage, though this may be further removed in the geological past than is generally supposed. In any case the appearance or return of this feature is younger than the remaining peculiarities of Artiodactyla. | As I consulted the literature on the subject, | found that in considering horns and antlers as sexual attributes, I had come to a similar conclusion as the well-known popular author on questions of evolution in Zoology, Börscue, has set forth in his Tierbuch IV, der Hirsch.- Yet on the first cause of the origin of frontal appendages our opinions disagree, for BörscuE sees in the excrescences on the roof of the skull of so many Ruminants nothing more than originally purely ornamental attributes, and ascribes their birth to a periodical exuberance of energy of growth, manifesting 573 itself in exostoses of the bones of the cranial roof, especially and at last exclusively of the frontal bones. In his opinion the perio- dicity of this surplus of growth-energy keeps time with the rut of the males, but its presence should be ascribed to the regression of another differential feature of the male sex, viz. the ensiform tusks, as they still occur in Suidae, Tragulidae and the unhorned muskdeer. The strongest expression of this opinion is given by BO6tscnE in the words (p. 88) “The pedicle is no weapon.” On p. 89 he continues: “As we saw, the idea “weapon” cannot be applied without reserve to the beam, although it may occasionally be used as such. Far exceeding that application and evidently its real nature, this beam is an ornamental product, a somatical arabesk, abstract from all usefulness, rhythmic in structure, with an inherent connection with the erotic side of life. The pedicle, in principle a product of the skull like the beam, cannot by any means be considered as a weapon, at the same time however it does not want erotic connections.” Against these views | think objections may be raised in two respects. In the first place there is no plausible reason, why the origin of horns and antlers should not be connected with single combats between males belonging to the same species, in which the more primitive mode of fighting with tusks (as still found among hogs) was gradually replaced by knocking of the foreheads against each other. The question, whether this new custom was the immediate or the indirect cause of the exostotic hypertrophic process (Lamarekism versus Darwinism) may be passed over in silence here as in all similar cases. Nor do I want to deny that exuberant growth, in cooperation with periodical sexual maturity, exerted an important influence on their development, as it still does every time the antlers are shed and renewed: we need only remember how pro- foundly this renewal is disturbed by every injury to the male sexual glands. In the same way, BérscuE’s verdict: “The pedicle is no weapon”, seems to me to be liable to serious doubt. Already in itself, the comparison of the long pedicle in the Muntjac-deer with the shorter ones of the remaining Cervidae leads to the conception, that the pedicle should be considered as an organ in a state of regression. To the same consideration leads a survey of the extinct Deer: in the middle-miocene Palaeomeryx-species no separation exists between pedicle and beam, they only show a long bony ontgrowth of their frontal bones, slightly forked at its top. This excrescence therefore might be considered as a pedicle of extraordinary length. In the 574 first fossil deer showing a separate pedicle, the latter is very long, like that of the existing Muntjac. In accordance with this observation is the fact that in Sivatherinae no rose can be detected near the base of their gigantic and ramified antlers, which therefore as a whole might be considered as pedicles. Starting from the fact that in recent Giraffes the small hornstumps permanently retain their covering of hairy skin, the same may have occurred in their extinct allies: the Sivatherinae, and perhaps likewise in the first antlered ancestors of Deer. The separation of the latter’s antlers into pedicle and beam, combined with the phenomena of yearly shedding and regeneration, the ‘‘rubbing” of the “velvet”, in short the entire process of the renovation of the antlers, so incon- venient and dangerous for the stag, might then have developed from a similar primitive condition as that in Sivatherinae, where the frontal bony outgrowths, clad with hairy skin, must gradually have increased in size and complication. If one applies to the latter the designation ‘‘pedicle’, it follows that for them just the contrary might be true of what is implied in Börscne's assertion: viz. the pedicle would have originated as a weapon and only lost this fune- tion in the Giraffes proper. In the second place I cannot see sufficient reason for accepting such an intimate and strict connection between the regression of the tusks in the upper jaw of the stag and the progression of their antlers, as necessarily follows from the supposition that a surplus of growing energy should pass from those tusks to the frontal bones. Against this hypothesis it may be objected that the male Proto- ceratinae as well as the Dinoceratidae were provided with powerful tusks, largely protruding from their mouths, and yet had a whole range of paired and single bony knobs and projections on the roof of their skulls, somewhat like those still found in the (male) wart- hogs. Among deer the male Muntjac still possesses strong tusks projecting downward and outward out of the mouth from under its upper lip, and yet carries well-developed, though simple antlers. There is moreover little reason for the assumption that in Cavi- cornia the same course of events should have taken place as in Cervicornia, viz. a regression of large tusks, going hand in hand with an increased growth and a higher complication of frontal appendages, and yet the origin of these excrescences may be attri- buted to similar causes in all Horn- and Antlerbearing Ungulates. On the other hand there is nothing incomprehensible in the fact that the upper tusks of Deer should have been reduced, as soon as they were no longer used as weapons, because the male Deer got accus- 575 tomed to a new mode of fighting which made them acquire antlers. This vicariating development can be understood, without taking refuge to such an intricate correlation between upper tusks and antlers, as Börscnr does, where he speaks of a surplus of energy of growth, set free by retardation in the development of tusks, and manifesting itself in hypertrophic excrescences on the frontal bones. When we proceed in this course of thought, the question unvolun- tarily avises, if the far higher development of tusks in the male sex of so many species of mammals might not be considered as a support for the abovementioned hypothesis about the androgenic origin of frontal appendages. The difference between the two pheno- mena lies especially in the fact, that with canines modified to tusks or incisors prolonged into darts, only their stronger development and differentiation need be ascribed to influences of sexual, especi- ally male nature, while for horns and antlers also their first appear- ance had to be traced to this same cause. But this does not exclude that the tusks of Elephants and Cetacea, the canines of so many Apes, Carnivores and Ungulates ete. find their most plausible expla- nation in the assumption, that in so far as they are larger than the other teeth and also differ in shape and position from the incisors and molars, they may be considered as an acquisition of the male sex, which afterwards passed to the female, but in reduced pro- portion, and so to a certain extent again lost its monosexual character. Especially the growth far over the limits of practical fitness may be adduced as an argument for this hypothesis; we. should only remember the tusks of the Mammoth curled up in a complete circle, the gigantie canines of the male walrus or the usually unilateral dart of the male Narwhal. In this connection I should like to move the question whether the uncouth tusks of the extinct Sabretoothed tiger (Macbairodus), might not have formed a special attribute of the male sex, as beyond doubt they were far from practical in defence as well as offence. With cer- tainty this is the case with the Walrus. Also the upper canines of the Babirussa, which perforate the upper lip, and are curled up dorsally and backwards, give us a good example of hypertrophic growth far beyond the limits of real usefulness. Groningen, September 1918. Chemistry. — “On the estimation of the geraniol content of citro- nella oil’. By Dr. A. W. K. pr Jone. (Communicated by Prof. vaN ROMBURGH). | (Communicated in the meeting of September 29, 1918). The chemists of the firm of Scaimmen. & Co. have a method for the estimation of geraniol in citronella oil which in the “Bericht” of this firm of October 1899, 20 and also in that of October 1912, 39, is described as follows: “Etwa 2g Phthalsäureanhydrid und 2 g des zu untersuchenden Oels werden mit 2 eem Benzol zwei Stunden in einem Kolben, wie er zu Acetylierungen benutzt wird, auf dem Wasserbad erwärmt, dann erkalten gelassen und mit 60 eem wassriger Halbnormal-Kalilauge 10 Minuten geschüttelt. Der Kolben ist hierbei mit einem eingeschliffenen Glasstopfen verschlossen. Nach dieser Zeit ist alles Anhydrid in neutrales phthalsaures Kali und der saure Geraniolester in sein Kalisalz übergeführt worden. Nun wird das überschüssige Alkali mit Halbnormal-Schwefelsäure zurücktitriert. Ziebt man dann von der Menge Alkali, die der eingewogenen Phthalsäure entspricht, die fiir den Versuch verbrauchte Menge ab, so erfahrt man, wieviel Alkali dem an Phthalsäure gegangenen Geraniol aqui- valent ist, woraus der Prozentgehalt an Geraniol zu berechnen ist”. This method is at the outset subject to suspicion, since it is based on the assumption that geraniol is quantitatively esterified by phtha- lie anhydride, whereas this is not even the case with acetic anhy- dride (98.5 p.c. was found to be esterified). For the following experiments a very pure pbthalic anhydride was prepared; 1 grm. was neutralised by 135 c.c. '/,, n KOH, where- as 135.1 ¢.c. was calculated for the pure substance. The citronellal used was isolated from citronella oil by means of the bisulphite compound. The sp. gr. at 26° was 0.8526 ; [«]p=-+10°21'. It was faintly acidic; 1 grm. was neutralized by 0.2. cc. */,,n KOH. The geraniol was isolated from Palmarosa oil by means of the calcium chloride method. The sp. gr. at 26° was 0.8752. To the mixture of anhydride, citronellal or geraniol 4 c.c. of benzene was added. After heating, the flask was cooled ‘rapidly, so 577 that the crystals which separated remained small and dissolved readily in $ n KOH. The phenolphtalein was added as powder. Duration of heating: 2 hours. Temperature _ uantit | p Q y Quantity of cc. N/10 KOH of the of phthalic « . : citronellal — ze _ Difference water bath anhydride _ d | | in degrees C. used | kan | found | calculated | | owe | È, | 85 | 1.2600 grm. |0.4450 grm. | 169.9 | 170.1 0.2 85 0.9480 „ 0.5120 „ KA Ge || 127.95 0.35 | | At this temperature citronellal is not attacked by the anhydride, or scarcely so. | tity of iol Temperature . - Quantity | Quantity of aant eecranie Duration er ; found of the of of phthalic | geraniol water bath i anhydride used 5 heating in degrees C. grm. grm. grm. 0/ 99 / 2 hours 2.1740 0.4645 0.4158 | 89.5 88 een 2.1810 0.4960 | 0.4575 | 92.2 84 grees 2.1300 0.4860 0.4481 92.2 TT sera. _ 2.1160 0.4950 0.4527 | 91.5 88 B Dake Lebsadn.| 1 DAAD: [CNRS DT 82 3 1.6930 lee OCAS | 0.4173 | 91.4 ” This showed clearly that the esterification of the geraniol had not been complete. The mixture does not boil, however, and I thought that this might possibly be the cause of the shortage. Hence a few further estimations were made in which the heating was done with a small flame. Duration of heating: 2 hours. Quantity of phthalic Quantity of Quantity of geraniol found anhydride used geraniol used ee in grm. in grm. | grm. 0/5 | | 2.0540 0.8890 | 0.8193 92.2 2.0495 | 0.4970 | 0.4497 | 90.5 578 Nor did an increase in the benzene from + c.c. to 8 c.c. change the result. Found: 90.6 °/,. With this way of heating the citronellal is also attacked more extensively. Quantity of phthalic Quantity of | Number of c.c. N/10 KOH anhydride used citronellal used SSS in grm. | in grm. found calculated | difference 0.9490 | 0.5950 126.5 128.1 1.6 1.2850 | 0.5050 | 1.15 8173.45 2.3 Evidently the scientific investigators of the firm of ScatmmeL & Co only worked with mixtures of geraniol, citronellal and limonene. As the following estimations show, results are obtained with mixtures — of geraniol and citronellal which differ but little from the true values. ‘mmm antit | Temperature panes Y | Quantity | Quantity of | Quantity of geraniol ‚of phthalic AEN BE | found of the ; of geraniol citronellal | anhydride | | water bath ei used used ee ee : u 4 wet | in degrees C. term in grm. | in grm. | grm. | 0/0 88 | 2.4883 0.5610 | 1.3602 0.5480 | 97.7 82 2.1090 1.0115 | 0.8120 1.0255 101.4 | 86 | 2.3600 1.3745 0.7560 1.3883 101.0 That the presence of the citronellal should lead to better esterifi- cation of the geraniol is very improbable; since in the previous experiments about 92°/, of the geraniol was esterified, the amount of citronellal esterified in the last three estimations would be 0.032 grm., 0.095 grm. and 0.123 grm. respectively. It would appear from this that the amount of citronellal esterified increases with that of the geraniol. The cause for the esterification of citronellal in the presence of geraniol must be sought in the formation of the acid phthalie ester of geraniol. Phthalic acid itself has little effect because it is only slightly soluble in benzene. It is well known that citronellal is very sensitive to acids, being con- verted by them into isopulegol. In acetylating citronellal without sodium acetate the same pheno- menon is observable when working with mixtures of acetic acid and acetic anhydride. ‘ 579 An indirect method was employed corresponding to the way in which the geraniol was estimated by phthalie anhydride. The appa- ratus consisted of a small flask with long neck to which a ground in U-tube was attached. Into the flask there was always weighed 2 ee. of citronellal and 2 ce. of the acetic anhydride mixture. 5 ee. of '/, n KOH were placed in the U-tube, which was fitted with a soda-lime tube. tity of cit llal Acetic anhydride content of the oua see kas Sg an ; esterified in °/, | mixture | Mean 9/0 : tT A | A | B | | 95.0 28.3 32.0 ) 30.2 15.9 51.9 55.8 | 53.9 53.6 70.0 68.8 | 69.4 Sree | 59.4 | -— 59.4 15.25 | 43.7 41.8 | 42.5 Acetic acid of 97.2 0/, 30.9 30.9 30.9 When the duration of heating was increased from 2 hours to 3 hours, the amount of citronellal esterified was also increased. On using 95°/, acetic anhydride, 40.1 °/, and with 52.9 a7 anhydride, 76.7 °/, of the citronellal was esterified. Clearly the presence of acetic acid in the mixture favours the esterification of the citronellal. It might be concluded, that the citronellal which is not esterified, is nevertheless transformed in some other way, for instance into a terpene or similar body. In order to investigate this point larger quantities of the various products were prepared in the manner in which the estimation of the so called total geraniol content is carried out (Bericht of Scrimmer & Co April 1910, 155). A portion of each product was examined by means of phthalic anhydride for the presence of alcohols, of another portion the saponification number was determined in the ordinary way and a third portion was acetylated by the indirect method, by heating 2 c.c. of the product with 2 ¢c.c. of acetic anhydride of 95 °/, and 0.2 germ. of sodium acetate for 3 hours. : 38 Proceedings Royal Acad. Amsterdam. Vol. XXI. 580 pe es Acetic anhydride Number of cc. of | © lated : 2 Saponification | /o se ine content of the mixture | N/10 phthalic acid BA ‚_according to 0/, esterified | indirect method | | 95.0 | 1.6 124.31) | 67.7 53.6 = 219.0 13.4 Alive 1.0 | 180.3 19.8 15.25 — | 129.0 28.4 Acetic acid of 97.2 % 1.3 | 98.0 | 38.6 Altogether the following amounts of. citronellal were therefore acetylated. a 0 at 37.7 + 74.6 = 112.3 53.6 72.0 +160 == 88.0 31.2 57.3 + 22.9 — 80.2 15.25 39.2 + 31.6 —=- 70.8 Acetic acid of 97.2 °/, 29.1 + 41.6— 70.7 The original use of 95°/, anhydride therefore leads to partial formation of a diacetate, while the other acetic anhydride mixtures do not yield 100°/,, which would indicate that these cause, in addition, the formation of hydrocarbons terpenes. The presence of geraniol, like that of acetic acid, leads to a better esterification of the citronellal, as was the case in the estimations with phthalic anhydride. While according to the indirect method with acetic anhydride of 95°/, in two hours only 30.2°/, of citronnellal was esterified, mixtures with geraniol gave the following result: Quantity | Quantity of | Quantity of geraniol citronellal « esterified grm. grm. | grm. 0.4940 1.1365 0.9478 0.8275 0.9465 1.3195 1.3390 0.8260 1.8740 Assuming that in the mixture 98.5°/, of the geraniol is esterified, 1) This higher figure is most probably due to increased absorption of water, when working on a large scale. 581 as was found in the experiment with the pure substance, we can calculate that the following quantities of citronellal were esterified : 40.6, 53.3 and 67.2. It further follows that no complete esterification is possible without the use of sodium acetate. Even by increasing the duration of heating to LO hours only 93.3 °/, of the geraniol-citronellal mixture (1 ee. to 4 ¢.c.) was esterified. Finally some estimations were made in which for every 2 c.c. of citronellal 0.2 grm. of sodium acetate (previously melted) was employed. | ; k Citronellal esterified Acetic anhydride in 9 content of the mixture | 7 | Mean 0 . 3 | /o A B | Be | 95.0 95.0 95.3 95.1 88.3 93.5 — | 93.5 15.9 | 90.0 | 92.4 91.2 52.9 be Sala Lu) 10 aad 31.2 | 54.5 | — | 54.5 15.25 - | 40.2 aal od A02 | Acetic acid of 97.2 0/, | 30.3 31.4 | 30.8 Hence the presence of sodium acetate increased the quantity of acetylated citronellal in those mixtures which contained 53—95 °/, of anhydride, and did not affect the others. The rise of tempe- rature due to the addition of the sodium acetate is not the cause of the improved acetylation in the former mixtures, since heating the mixture in a sealed tube without sodium acetate to the same temperature (about 149°) did not result in better esterification. The curve shows, however, that when sodium acetate is used, the presence of acetic acid is harmful, whereas, in the absence of sodium acetate, the acid has a favourable effect up to a certain concentration. It follows therefore that in using sodium acetate we induce a different reaction from that which occurs in the absence of this salt. Buitenzorg, 27 May 1918. 38* Mathematics. — “Observations on the expansion of a function in a series of factorials.” I. By Dr. H. B. A. BockKWINKEN. (Communicated by Prof. H. A. LoreNtz). (Communicated in the meeting of September 29, 1918). 5. We now consider another example of Nrersen’s theorem, not belonging to the cases mentioned under N°. 4 of the remarks made in the preceding paragraph. We choose VS where 6 is a number between 0 and 227, not equal to one of these numbers. For this function we have = = atte 2 Se; the first of these equations resulting from the fact that ¢= 1 is an ordinary point of the function. It is further easily found that the nth derivative of g(t) satisfies the equation pile n(n) (Ltr (LAP! oa I" (d4-n) LP (d-+n)\ eo —t } eer. Gij 1—t The modulus of the expression — is given by the relation et ges | 1—t | 2t(1— cos 6) ei) a ae SS err 1—t+V1—2t cos 640? and it is not very difficult to see that it increases monotonously from the value 0 to 1, if ¢ decreases from 1 to 0. We divide the interval (0,1) of ¢ into two parts, (0, ») and (», 1), where » is a number given by p= mi (Ole Weert, os ne EE so that » depends on n and approaches to zero as a limit when 7 becomes. indefinitely large. The positive number d, is at our disposal and will be fixed immediately. The maximum value of the modulus (22) then differs from unity by a quantity greater than kn} if ¢ lies in the second interval, & being a certain positive number, which is independent of n and f; thus we have in this interval 583 n | 1—¢t BY | ken? eit —¢ ed so that the left-hand member of (21) for these values of ¢ approaches uniformly to zero for n= (the factor n P(n): P'(d + n) is only equivalent to n'~? and does therefore not affect this statement). The integral 1 if nI*(n) 1—t Ala ete MEN +n) eid t ei —t vy consequently has zero for its limit if increases indefinitely, however small the value of d may be fixed. For the interval (0, ») we have, independently of ¢ and n, nI(n) ( =) (1—t)?—! kn F'(n) T(d-+n) \eit —t eb _¢ T'(0+n) < kala, where & is again a positive number not depending on » and tf. *) Thus, considering (23), it follows y f nI(n) ( 1—t i (1 — tje! | ae See T'(d+n) \et9—+t eit —t 0 ee. kn ed) We therefore need only choose d, less than J, to see that also the integral over the interval (O,v) is zero for n =o. Thus the whole remainder (11) is zero for n—o, if only R(x) >0, te, since A== —o and A’ — 0, if R(«) >2’ and R(«) >a. For these values of a the integral < kn'* ip d ad can therefore be expanded into a series of factorials; and the theorem of NrieiseN holds in this case. Again we take the example rn ea vit zb hik if Sith <2 sa e4—¢ (1—t) OZ p< | Here 4’ = 0, on account of the first term, and À == u, on account 1) We shall always, in future, denote by k a finite positive number, without always meaning the same number by this letter. This will not cause any am- biguity, because the exact value of k is of no importance in our reasonings. For the sake of clearness, however, we shall often mention the quantities on which k does not depend. ye 584 of the second. If y(t) were equal to the second term only, the integral (1) could be expanded into a series of factorials for A (wv) > u only, and this series would be absolutely converging for these values of z. Thus the whole function may also be represented by ‘such a series for R(x) >u, ie. for Ra) >4 and Ref >7, but the con- vergence is, on account of the first term, only conditional for | 2 R(e) >0, decreases with 1/n as the n-th power of a number less than 1, which causes that, in the integral (11), only an interval has to be considered which, in a proper manner, ap- proaches to zero as 7 becomes infinite, so that the value of & (2) for which expansion is possible can be depressed by unity. This suggests the idea that something of the kind might occur as a rule, if p (t) has t—=1 for an ordinary point. The truth of this presumption is proved by the following investigation. We again divide the interval (0,1) of ¢ into two partial intervals, with the point {== r as a common end-point, which is to approach ultimately to zero as n becomes indefinitely large; and we assume, as in the preceding paragraph, for » the value (23). Consider the circle, with centre » and passing through two fixed points C and C' lying on the circumference of the circle of convergence (0,1) of y(t), symmetrically with regard to the axis of real quantities, and in the interior of an are DA D' of the latter circle, which does not contain a singular point of g(t), D and D' being also conjugate points, whereas A is the point with the affix ¢—1. Then, from and after some value of n the value of » will be so small that the circle with centre » does not contain any singular point of ¢ (£)in its interior and on its circumference; and at all points of the latter between the radii O D and OD’, including an arc ZB KE! of it (B being the point on that are with argument zero), the modulus of y(t) will remain under a finite quantity A, independent of n and t. As regards points of the supplementary arc EF E’ of circle »), F being the point opposite to B, we may remark that g (t) there has a modulus no greater than pl"), 585 y(t) means the natural majorant of ¢ (d), and 2" the distance of the points D and £. We further remark that the radius of the circle (p) is. greater than 1 vp, say 1 —r + yr’. It is evident that the numbers v' and v" both approach to zero together with », but that their ratios to the latter number remain finite and different from zero. At a point P of the interval (», 1) we have, according to a well- known proposition | p(t) | M | Te cle (ee if M is the greater of the numbers K and p (1 —v"). Instead of this inequality we may write 1 — tyr lpt 1 n—1 een 1—t+r 1—t 1 n! —tty' < lv” (Ltr tp tl Mvo! acme, With regard to p (1 — v") the following remarks may be made. If, in the equivalence-equation or, since for Oz t= 1, i (24) lim a, — n” the quantity A! is no less than — 1, we have, according to the proposition of Cesarò, for any fixed d > 0 lim WHS g(1—v") = 0, v'—0 and hence, in virtue of the remark made above on the relation between rv” and », . lim vy’ +1+¢ g(1—v") =) y==0 and further lim »'+14+7 x M= 0 v=0 since, as a matter of course, the expression KX rv’? has, too, zero for its limit. Thus we may write for (24), in connection with the assumption (23) and the finite, not disappearing ratio between » and v' (1—z)"—lepni(t) knk T(n—1) | ~(14-An%s—1)n a kenker", 586 where A is again a positive number not depending on » and 4. Henee, corresponding to any fixed positive quantity ¢ chosen arbi- trarily small, there is an integral number MN, such that the left-band member of the latter inequality is less in value than ¢, for every value of ¢ in the interval (rl), if only n> N. For these values of n we have therefore 1 1 | | (oe ae he | (1—t)R@ dt « Pen) ; < he if R(«)>—1. For any such value of a, i.e. a fortiori for Ra) > 4’, since 4’ was supposed greater than —1, the part of the integral (11) taken over the interval (r,1) has zero for its limit for n=. For the integration over the remaining interval (O,v) we apply the mode of treatment of $ 3 and the inequality (17). According to the latter there is, corresponding to any fixed d and e, chosen as small as we please, an integer N such that we have wniformly in the interval (0,1), and hence in (O,v), ger Tnet N. | TA dte) | For the interval (0,r) it follows from this that, for n >> N gieken < heul Ra I(x -+-n) | thus v | CME a+n—t1 | | f* 6 ey, en dt| << ken—Ra) +44 I(a+n) | 0 If now Rr) > +’, we can have chosen the numbers d and oj so small that R(x) is also greater than 2 + d-+ 9,, and in this case we infer from the latter inequality that the integral over the interval (Or), too, has zero as a limit for n =o, if Ri) >A’. Thus the theorem of NierseN has been proved, in case t=0 is an ordinary point of the function p(t), If a funetion g(t) has the point t= 1 for its only singular point on the circumference of the circle of convergence (0,1), and if, moreover, it satisfies the conditions of HADAMARD, i.e. if it is con- tinuous and “a éeart fini’ on that cireumference, or if a certain derivative of negative order — has this property, then we always have == ae and the theorem of NrerseN has ceased having anything particular. 587 Again it may happen that y(t) can be divided into the sum of two functions 7,(t) and g(t), the first of which is regular at ¢= 1 and the second of which has the latter point as its only singularity on the circumference of the circle (0,1). If, then, the number 4’ for ¢,(t) is equal to 4,’ and that for p‚(t) to 4,’ and if 4,’ >4,’, so that for the whole function g(¢) the number A’ is equal to 2,’, the integral (1) can be expanded in a conditionally converging series of factorials for pet bee Mee atl if A’ = 4,’ a,’ +1. If, in this case, p‚(t) has the properties of Hapa- MARD, then 4,’-+1—4,=4, and the proposition of NikLsEN is valid, which, now, really has a particular meaning. 7. The following proposition is, as a corollary, included in the theorem of the preceding paragraph : ad . , Ld e e ’ . If the coefficients a, of a function y(t), defined by a power-series ie) Dl p(t) = NE ant” et 0 which has the circle (0.1) as its domain of convergence, are, for n=, equivalent to a power nm}! of n, and if the series 90 bree (25) — yaw, . . . . . . . . « is divergent for O<9 <1, the point t= 1 is a singularity of f(t). For if t=1 is an ordinary point of q(t, the series (6), which, except for the factor I(x), is equal to n a jj a RNL gB) _ (ee +n+1) ( is convergent for (re) > 2’ and the convergence of (25) can be derived from it. For we may write n! ay n! an _ De +n+l) PPT Petn tl) Pn Ie If we choose w such that 2’ < R(wv) <2’ +4, the series formed by the first factor, if „ takes all values from zero to infinite, con- verges, as we have already seen, whereas the series, composed of the terms obtained by taking the first finite differences, with regard to n, of the second factor, converges absolutely; and it is a well- 588 known truth that the convergence of (25) is a consequence of these two facts. The same thing would hold with regard to the series wo ay ~ y(n) lim p(n) — n*H5, and lim & g(n) — n*HP-1. n= 0 n= oe Therefore, in the statement of the above theorem such a series may be chosen as well. We further remark that 4', which was hitherto supposed to be greater than — 1, may also be /ess than the latter number: the theorem of NrerseN, in the particular case demon- strated in § 6, keeps its validity for those values of 2', though we should have to apply our reasonings to an integral of the form (8) (in a footnote of § 1) in the latter case. By substituting ¢—d'e'¥ in the power-series for p(t) we obtain the more general theorem: If the coefficients an of a power-series in the letter t are equi- valent ton” for n=, the function — (t) represented by that series has, on the circumference of its circle of convergence (being the circle (0,1), singularities at all points where the series ae (00 <1 diverges. We may add that this theorem already holds, if only the upper limit of the coefficients a, is, in the sense of equation (14), equivalent to n for m= 20. Finitely we observe that the reverse of the proposition does not hold: if the series (25) converges, the point ¢—1 need not be an ordinary point. To make this clear we need only think of the case that the coefficients «, differ from zero only for values of » lying at a certain distance from each other; it may happen then that the series (25) converges absolutely, but the function g (f) has its whole circle of convergence as a singular line. 8. As already remarked, we doubt of the general validity of NieLseN’s theorem, though we are not in a position to furnish a case of the non-validity. It is our opinion that, if 2'< 24< 4'+1, there will be cases in which the integral (1) cannot, for all values of R (wv) > 2, be expanded into a series of factorials. On the other hand we can prove that such an expansion is not possible for any value of Rw) <4, which is a thing not immediately evident if 2 lies between 2’ and 4 +1. 5 1) If R(@) 6, and to diverge for 0 <,, and, consequently, the series (26) to converge for R(x) > 4+ 0,, and to diverge for Bw) <4 + 4, then the integral (1) will, at any case, not admit an expansion into a series of factorials for any value of Aa) <4 +0, We now shall prove that for any positive d, taken as small as we please, lim (Lt) HH? g(t) = 0°) | so that <2’ 40,; by this the required proof will have been established. For the sake of brevity we write 4460,+ d=. Consider the derivative of negative order —« of ¢ (t), which according to the definition of Riemann ®), is given by Dre g(t) = Ta ml (t—u)*—! ¢(u)du = t*y(t), then y(t) is a function regular at ¢=0O with the same circle of convergence (0,1) as g(t) has; its expansion into a power-series is nia,t” a(t) == en T( fo sen 1) £ e 5 © . 5 (27) From this formula it may be derived that tp (t) remains finite for t= 1, in virtue of the initial hypothesis. Conversely we have gt) = Deftep(t)|- First, let in ee alte Then we may choose J so small that also «<1 and write gt) = Dt Di tw(t)]*) = De Late p(t) + | = t | 1 xan ig (28) ap eer (ets 2 [eap(uyne KO | Now w (uw) is, in the range 0 2 wu < 1, finite and thus less than a certain number g. Hence 1) Or for negative values of A’ a by, lim (1 — PHA p (n) (t) = 0, if mis such that A’-+ 6, + 50. 2) See among others Bort, Legons sur les séries a termes positifs, p. 75. 3) Properly speaking it should be D.D~—1, but this operation, in the present case, is equal to D*-!.D, since the subject of the operation is zero for t= 0. 590 t | RJ | Ld [woow tu) “du << of ult — u)—* du ES | 0 or, substituting — tv 1 1 | Wluju* Ht u) du fe sa —v)—* dv, Ie e 0 so that the integral in the left-hand member of this inequality remains finite for all values of ¢ in the closed interval (0,1). Further we divide the second integral on the right-hand side of (28) as follows, supposing ¢ > 3, : t ; t—(1—f) t foon du == | +f “0 ‘0 i—(1—t) To the first of these two integrals we apply the second mean value theorem, which is allowed, because the expression u? (t—w)—*% increases monotonously in the interval in question. We obtain i—(i—t) ep’ (u)u(t—u)—* da = (2t-—1)+(1 —t) 4 [y(2t— 1) — y(8)] . where 8 is a number in the interval (O, 2—1). This part of the integral, as ¢ (t) remains within finite limits, is therefore for ¢—= 1 at most equivalent to (J—7d)-*%. In order to infer the same thing with regard to the second integral, we make use of the fact that lami (l— 0)" PGD, WEEST, Ba aes). Se es Cee i=1 We shall prove this at once; it should not be thought that it is a consequence of the proposition mentioned in a footnote of § 1: it follows solely from the convergence of the series (27) for ¢= 1. . If we assume, for a moment, the formula to be true, we have for the whole interval O 0 lim (1 —t)’+4+? g(t) = 0 tel and thus, as we proposed to show Reden Secondly, let 4 + /, ‘lie between the integers p — 1 and p; we may choose d so small that the same holds for A + 0, + dze. We write id ne ENE (Oe) so that OST be Lat rin SBB: ania « (8%) In this case we have the following reduction g(t) = DL DP [ew Pmt! Amt plet) T(a'+m) Owing to (31) we may, as in the former case, using here the inequality (29) for n =m, prove that the expression Di [tnd wim( is at most equivalent to (1 —d—@+"—V and thus p(t), as m is no greater than p, is of an order no higher than that of (1 —t)—‘ +p—}, that is, according to (30), of the order (1 — 4”. Thus the required result is obtained completely. P = D'-1| P(e'+p) Sn 0 9. We now give a proof of the proposition used in the preceding paragrapb. It may be stated as follows: If the expansion in a power-series of a function ¢p (t) converges at the point t=1 of the circle of convergence (0,1), we have for all positive integral values of n lim (1—t» g™()=0. (82) i=I This proposition, of course, ceases to have a particular meaning, 1) Properly speaking it should be Dy Dz’—1, but this comes to the same thing as Dz’-1 Dp, because t-(y—1)-times the subject of operation is zero for ¢=0. 592 if t—=1 is not a singular point of ¢ (), but if it is, the proposition is not a matter of course. Since, if the coefficients of the power-series in question are complex, the two series formed separately by means of the real and of the imaginary parts of those coefficients must both converge, we may without loss of generality suppose the coefficients to be real quantities. We then consider, together with the function q(t) Sa, + Ot dee Peel the function (t = gee GEN ee a ae a Re where n i == hd dj, 0 Since s,, as n becomes indefinitely large, approaches to a definite limit s, the series (33) behaves, so far as regards its terms for large values of n, as the power-series of the function $ it and according to the reasoning of Crsaro we have not only but also 5 "(n . et = — ii E Org | EP ae ee Further, from n-fold differentiation of the identity p(t = (1—t) f(t) we obtain the new one (ayy yp (1 pn fO-D) n! ni (n— bya The limit of the right-hand side of the latter equation for ¢= 1, is, by (34), equal to zero for all positive integral n-values, and the required formula (32) has thus been proved. By substituting ¢= ¢’ e? we obtain: Zf the expansion in a power- series of a function y(t) converges at the point t= e+ of its circle of convergence (0,1), then, for all positive integral values of n and for real values of t', we have lim (1—t'jn peterr) = 0 I= Physics. - “Deduction of the third virial coefficient for material points (eventually for rigid spheres), which evert central forces on each other’. By Prof. W. H. Kresom and Mrs. C. NORDSTRÖM- VAN Leguwen. (Communication N°. 3a from the Laboratory of Physies and Physical Chemistry of the Veterinary College at Utrecht). (Communicated by Prof. H. KAMERLINGH ONNrs). (Communicated in the meeting of September 29, 1918). § 1. Introduction. This paper contains a continuation of the research, started in Supplement N°. 24a of the communications from the physical laboratory at Leiden (These Proceed., June 1912), in which the existing data on the virial coefficients B and C in KAMERLINGH Onnus’') empirical equation of state *) eer 4 Bo ME SAD At A Ne LG agate ahr ier er NE SE it | were compared with formulae, which can be derived for those virial coefficients starting from different hypotheses on the structure and mutual actions of the molecules. In the preceding®) papers on this subject especially the second virial coefficient B was considered. In the present communication the third virial coefficient C will be treated. To that purpose there will be derived in § 2 a general expression for this third virial- coefficient for material points (eventually rigid spheres) which exert central forces on each other. In § 3 C is calculated for the case that the molecules (rigid spheres) do not attract each other. The thus obtainea value is at the same time the first term of a series of ascending powers of 7’—!, in which C can be developed for the case that the attraction between the molecules varies with the distance according to a law 7 + (potential energy proportional to r—9). In Comm. N°. 36 the development will be given for the next two terms of this series for the laws of force r—5 and r~®, These calculations had already been started, when the importance 1)"H. KAMERLINGH ONNES. Comm. Leiden N°. 71; These Proceed. June 1901. Comm. N°. 74; Arch. Néerl. (2) 6 (1901), p. 874. 2) Comp. H. KAMERLINGH Onnes and W. H. Krersom. Die Zustandsgleichung. Math. Enz. V 10, Leiden Suppl. N°. 23 (1912), § 36. 3) W. H. Keesom. Comm. Leiden Suppl. N°. 245 (These Proc. June 1912), 25 (Sept. 1912), 26 (Oct. 1912), 39a (Sept. 1915), 395 (Oct. 1915), W. H. Kersom and Miss C. van LEEUWEN, Leiden Suppl. N°. 39¢ (March 1916). 594 of the theoretical investigation of C was increased by a remark of Horst'), who concluded from the investigations on the compres- sibility, that for substances as water and ammonia the molecules of which possess an electric moment such as that of a doublet, C is negative in the considered temperature interval, while for normal substances C is found positive. A calculation of C for molecules whose attraction is equivalent to that of a doublet at their centre and a comparison of this result e.g. with that for molecules like those considered in this communication or with that for quadruplets will have to prove whether really the deviating behaviour of C for the substances mentioned must be ascribed to the molecule possessing an electric moment such as that of a doublet. The calculation of C for doublets and quadruplets must however be postponed till later. § 2. Deduction of a general expression for the third virial coefficient for material points which evert central forces on each other. This deduction is analogous to that of the second virial coefficient for this case given in Suppl. Leiden N°. 246§5, which paper may be referred to with respect to the method and the notations already used there. Definition of the macro-complexion. Generalizing (23) le, the n,, molecules in dv,dw, will now be distinguished in: Nita Single molecules (with no other molecule within their sphere of action), np, Molecules belonging to pairs with a mutual distance between r, and r, + dr, N15, Molecules belonging to pairs with a mutual distance between r, and 7, + dr, ete. N11e43,. Molecules belonging to sets of three, the mutual distances of which are resp. lying between 7, and r, + dr, 7, andr, + dr, r, and r, + dr, etc. (2) We define the group macro-complexion?) by giving the numbers a Without attending to any individuality of the molecules. The individual macro-complexion’) is defined as follows: Mita definitely indicated molecules are single, ') G. Horst, Leiden Suppl. N°. 417. These Proceed. Jan. 1917, Host points there also to the importance of the knowledge of the behaviour of C for the deter- mination of the molecular weight from the gas density, comp. also H. KAMERLINGH ONNEs and W. H. Kersom. Die Zustandsgleichung. Leiden Suppl. N°. 23 §§ 77—80. *) Comp. Leiden Suppl. N°. 24a § 2 (These Proceed. June 1912). 595 N11c123 definitely indicated molecules belong to sets of three, the mutual distances of which are 7, (dr,), 7, (dr), 7s (dr‚), etc. Here we have not yet fixed in what way these 211-193 molecules are divided into sets of three, and which two molecules of each set of three have the distance 7, (dr,) and which that r, (dr,). The number of individual macro-complexions contained in the group macro-complexion is: n! ae NTE (3) Niyal M116! . ss itil a. The micro-complexion is defined in the way mentioned Suppl. Lei- den N°. 245 § 5. Number of micro-complexions in the individual macro-complexion : In dv, n, molecules are to be placed. First we place the 7;, single molecules. There are at the disposal of the 1st molecule : % places Ind ak x l 8 | qe | pe ‚ where et ANN At Ee te de Cd has been written for the volume of the sphere of action, 4 jlaces : dv, \ Peene AS) 3rd molecule: x (here an amount 3 has been subtracted from 6 because of the occurrence of a certain number of cases in which the spheres of action of the molecules 1 and 2 partially overlap each other); bog tn dv, Thus the distribution of the 74, molecules gives for the number in question the factor: in b abeel: bep b—(n1a—2)3) x eet 2 yin EEN Pied — nn di (5) Calculation of 8: Call the hatched spherical segment s (7,,). Probability that molecule 2 has a distance +h molecule: places, etc. 7. wai 2: dar”, drin de, From this we find: or ‘dar a iy, = - — 8(r,,)- eon ee ere iC) 39 Proceedings Royal Acad. Amsterdam. Vol. XXI. 596 As An 8(7", 9) — 3 i airs Her ab we get b? om A Dn er ST BH (8) Then we place the 71,1: molecules: At the disposal of the first molecule are: Ln where a term with 8 has been omitted, being negligible for the order of magnitude wanted; of the 2"d molecule: here we must take into consideration that of the space at the disposal of this molecule in the mean a certain part is occupied by one of the vq molecules; doing this we find for the number of places at the disposal of the 2"d molecule: ar, (v?—/,, 97, er im mj = | places ; * places, 1 of the 3 molecule: x | b 1— (nia +2) ra | places, etc. Thus the placing of all the 225, molecules (comp. Suppl. Leiden N°. 240 § 5) gives us the factor: ! b b Peay 5 pues 2: } nia med, RDA dr 161 dv, ae (5) a (7) Ni b1 N1b1 4nr,?dr,\ ? ur (ve — 7 )t 2 dv, dv, Now we are going to place the 71123 definitely fixed molecules, belonging to sets of three, with the mutual distances r, (/7,), 7, (dr), 10123! fn (dr In ways these molecules can be combined to such N1c123 JS sets, where it has been taken into consideration that 3 fixed molecules can form such a set in 3! different ways viz. 1st with distance from molecule 1 to molecule 2—~7,, distance 2 to 3 —=r,, distance 1to3=7r,; 2rd distance 1 to 2—=r,, distance 2 to 3 —r,, distance dto St ete, 597 Now we ealculate the number of places at the disposal of a definite set of three, going only until the required order of magnitude: molecule 1: x places Axr,?*dr, molecule 2: x - 7 ” dv, molecule 3: at the disposal of this molecule is the space described by the hatched parallelogram in revolving about the line J, 2. dr,dr, Contents :——— . 2ah,. sin y The number of places in question is hot x. 20 23 dr „dr, therefore ——--——--—_ where the relation kh, = 7,7, sn 7 Fig. 2. has been used. The 74. molecules thus give the factor: 7 Ty eo 10123 N1c123/ (2 rl dr, dr, dr ) a Nie AN Nn do,’ dr, drsdra Nic123 ! : 3 where the sign of multiplication has to be extended over all possible COMDMAONS 7,075 25. The number of micro-complexions in the group macro-complexion, W, is now obtained by multiplication of (3) by the product of the factors (5), (7), and (8), taken for each of the volume elements dv,, dv,, ete. From this we derive (8) In W = — nysa IN Aire: « «« — Nidi In midis. + «== Nyterz3 Inn 11e123 al b > 1 >> a eten ie dv dv, dv v, dv dv, a ie sng | MLB NY) ae eg el In iv -— — + dv dv, dv dr ry 2 \ - \ nisi, 429,*dr, Nib Na / 9 a ar, (de 1" | + ”) 2 _ de, 2 dv, +2 5 E 710193 IN Nici23 — F M1c123 + dv dr\dredrs eo | 3 dv,” N1c123 2d! rrr dr ‚dr, ‚dr, ej ‘ 39* 598 The state of equilibrium: As in Suppl. Leiden N°. 24b § 5: — p(r,) may represent the potential energy with respect to the mutual forces for a pair of molecules with the mutual distance 7,, where for r, Sn) For a set of three molecules the total potential energy is: OG. Fa 7s) = PO) Ee where we assume, that the attraction between two ‘molecules is not changed by the presence of a third molecule. The condition for the energy is then: w= EEn un LL a G)—F SZ mees P(r, 7,75) = const. (11) dv dw dv dr dv drydradrg For the state of equilibrium we then find after some reductions e.g. for the distribution of the molecules into single molecules, into molecules belonging to pairs, into molecules belonging to sets of three, with the required degree of approximation: n n? men) —_— — p+ EPE Rs), | v v n? n NE = — |p | — (bP—2P?— R) , » (12) v v n° de v? where passing from summations to integrals: B fer Anr’dr, 0 EY 13 R = fete r?) 4n?r* dr, ae 0 S ow {fe P(r, El Va, r3) 247 2 T,7.15 dr,dr,dr,. e e Deducing further the entropy, then the free energy wy and from this the pressure, we finally find for the second virial coefficient: B= tnlb=D, 2° Zend ee analogous with Suppl. Leiden N°. 246 equation (40), and for the third : C= 1n? (48 6°—36P43P?438R—28) . . - . (15) § 3. Development for rigid spheres without attraction. As first 599 case for the development of C according to (15) and (13) we put perce anos WOE cps. on me « (16) POE Fors Se ee ee es (12) with which case we have to do when we consider rigid spheres with diameter « which do not attract each other. We must then obtain the same result found for the first time in another way by JAGER *) as a second volume correction in vaN DER Waars’ equation of state. Notwithstanding this the calculation based upon the general expression (15) we have derived above, will be discussed here rather in details, because the term obtained here may be considered as the first of the development of C in ascending powers of / for g(r) =ecr—4¥. The calculation of the following in the next commu- nication may then be discussed more shortly. As also for r >t g(r) = 0, the quantity + must evidently vanish from the result. We easily find that for g(r) according to (16) and (17): P, = $x (t*—0') R, = Hart — 2170 | 7 20° where the quantities P and FR have been marked with an index 1 in order to indicate that these values are the first terms in the development in ascending powers of / for y(r) = cr 4. (18) v/e For the calculation of S, =i (il 2407 rror, dridr,dr, it may be be useful that 22 ff *“6, b. rm—t 20, so that the vase 7, < 25 need not be considered here. Let us consider in details case a: 1) G. Jager. Wien Sitz-Ber. [2a] 105 (1896), p. 15, 97. For further literature see H. KAMERLINGH ONNES and W. H. Kersom. Die Zustandsgleichung § 404. 600 In Fig. 3, where A and 5 are the centres of the molecules \ 1 rand 2, “so “that ADS the hatched spherical segment represents the space disposable for molecule 3. The volume of. this segment is: Fig. 3. § nrd rider 110. Multiplication by 12 ar,’dr, and integration between tT + 6 and 2r gives for the contribution to $,: 2 n Gh {177° —30r'°6— 3140" +20r'0? 4 381754 610° — 0}. b. Now not the whole sphe- rical segment CDFE of Fig. 3 is at the disposal of molecule 3, but only the part of it that EG is not overlapped by the sphere of distance of molecule 2 (sphere ] BG), which part has been F represented hatched in fig. 4*). Fig. 4. After integration with respect to 7, between + and Tr + o this gives: ‚SOr'o 4 38146" —20r'0' Ir? ot + 616° + 30%. Dan. UT = as De Oo. This part too must again be subdivided, this time into: a SP OE b Zon u. The space disposable for molecule 3 (the hatehed part in Fig. 5) is now bounded by: sphere AS with radius 7, (as r‚, <7,), plane CDG (as r, . Instead of (17) of the preceding communication we now put: 4 q ie fore 1 Dep EE See and develop the exponential factor in P, Rk, and S according to (13) according to ascending powers of h, indicating the succeeding terms by indices. In the same way C splits up into: GC + C,H On tn eae, ee C’, has been calculated already in Comm. N°. 3a $ 3, see equ. (20). With a view to C, we must calculate: Sr eff The domain of integration is divided in the same way as in § 3 Pen neten wed = +. ar +. 2 rs rs De LE En ar, dr dr r? 1 1) The equations have been numbered in continuation of those of Comm. N°. 3a. 603 of Comm. N°. 3a. For the domain sub 1, for which 7, > 7 and thus y (7,) = 0, the last term in the brackets of S, must be omitted. As to the integrations with respect to r, and 7, in the domain 16 we must still discern between: tt, >to and Be Tir a in the same way in 2a between: a: nnen and B: ne>> In the following table the different integration limits have been collected for the succeeding integrations with respect to 7,,7,, and 7, : mo aR 1 2 £9 : Se b 2a a = SS SS Ce Sa aa b ay | hen a B a | 8 HENS. | en | sees a = Fa.) Pie kala ON Eton wor Keet OV ArS r lp tes Ec | T2 a T ry—o,T | ae lee CiTG, Bi a r;— oO 0, ry 2 2 2 | | ri TO, QT | Et ZOT he 0 20 Finally we find: 1 Oeren T ’ == ne | 8 — — Pr — (11 + 6m 2) 60° + 16—+ Bet ln “|. . (23) oO T Further we have 1 | P‚, = Ar he { —— — | T 7 (24) Rohe dart In— —Art +40 0 Using the value of P, found in (18) we find: Grind A ee ee Pet. ve (20) C, follows from ee 12 11° he te Be, + zin 4 as face +-- oe + s dr,dr,dr,, : rt ar Pie Ps ee hs tae al 604 where for domain 1 the 3", 5, and 6!’ terms between the brackets must be omitted. Performing the calculation in the same way as for C we find: ee 1 66 See. e= dl od hesta en ee foe ae ra) a (26) „0 Ao 18 t a Tt PMT ame ee while by Pacts 1 1 Pi,=ith'e ota (27) 223,23 Reig | 5 Rr We AME AE so that finally C. == ln? 2722 RE 30 LE Jp gn’, Wh7c? (24 ln 2 — Tee af an (28) 30° 36 T 12 Tr ] ; Remarkable is that the radius t of the sphere of action does no longer occur in C,. Developing the logarithmic terms in (28) according : 0 to ascending powers of — we obtain © C,=tn'.n°*h edi 2 — 2453) — 18 5 Bie geass (rt: so that evidently the attractive forces excited by the molecules on each other over mutual distances greater than a certain distance 1, furnish a contribution to C,, the ratio of which to the total term is of the same order of magnitude as that in which the forces decrease at an increase of the mutual distance from 5 to r. If now we put t= o, we find, adding (20), (25) and (29) and introducing the potential energy at contact ot C= nt ($1 67)*{1—3 (19 —24 In 2) hv tig (3840 In 2 — 2458) (hv)? … or C = Ben" (4 1 0")? {1 — 1,418 Av + 1,566 (Av)?...}. (80) 1 As a age kp being the well known constant of PuLanck, this bp gives us the first term in the development of C according to ascending powers of 7’—!, 605 § 3. Development for rigid spheres with attraction proportional with r—®. Performing the calculation as in § 2, but now with c P= BOE BEER et 1 aL) r we find: C‚= — in’. x*heo 32 12 Ci, gn? ah? c? (48339 — 40In 2) — ——_- — — — | ot ty Dem fat de (32) 16 2 Go? 8 40. +t-6G —— ——+ §—— —-—_ — — In dot. " rlr) 5" T The radius of the sphere of action neither occurs here in C,, ij while the development of C, according to ascending powers of — B T induces the same remark as was made in $ 2 concerning the contri- bution furnished by the attractive forces for greater mutual distances. For t= we obtain: C = gyn? (kao) — Shw + (8832 —48ln (ho). | a rates (575) C = yn? ($ 70")? {1 —1,2hv + 0,809 (hv)? … 2 § 4. In order to represent the dependency of C on the tempe- rature more in details and to compare this for the two laws of force discussed in this paper, we shall introduce a temperature characteristic of each gas as a reduction temperature’). As the series for B found Suppl. Leiden N°. 246 § 3 eq. (42) converges still sufficiently rapidly at the Boyrr-point, this point suggests itself as reduction temperature. For p(r)=er * equation (42) Suppl. Leiden N°. 245 becomes: = B {18 hv — B (ho)? — 4 (HON a} 007084) where B, represents the value of B for =O, viz. for 77 = oo. From (34) we find for the Boyre-point for which B=0: (hv) = 0,3228, ’ Writing now tf) for = where 75 is the BoyLE-temperature, so B that fp, represents the reduced temperature with respect to the BorLe-point as reduction-temperature, we find for (34) | oan N —3 B= B, {1—0,9669 tro, — 0,0812 ty, — 0,0019 tay} - (85) 1) Comp. H. KAMERLINGH Onnes and W. H. Kersom. Die Zustandsgleichung. Leiden Suppl. N°. 23 § 385. 606 while (30) becomes: C=C,{1—0,457 B) + 0,168 CB) + Bi oe b. For p(r)=er ® we De BZB hot — gi oy — ae el a ee (hv) BR = 0.6070 B=B,, {1—0,9105 tp) — 0.0789 tp, — 0.0093 tj — 0.0010 rr) …}. (38) C=C, 110,728 rl + 0,298 “G2 aa} eee) én In idle and C have been represented as sitions of tip) according to the above formulae, while B, resp. C,, have been taken = 1. as ar Po) Zz 4 6 3 10 wn rd 16 Fig. 1. We see that within the represented range of temperature C is positive. C decreases with rising temperature, in a considerably less degree however than B. How the behaviour will be towards lower temperatures it is impossible to say before more terms of the developments into series will have been calculated. When following terms do not change this result, the terms here calculated would cause a minimum of C, *) after which C would rise again, so that it is not improbable that C remains positive up to a domain of considerably lower temperatures than those to which fig. 1 refers. While for the two laws of force considered the curves of B are nearly the same above the BoyLr-point, the coefficient C decreases with the temperature for ¢g=5 much more rapidly than for q = 4, at least in the beginning. Beira Comp. the representation of the values of C for helium in H. KAMERLINGH Onnes and W. H. Kersom, Die Zustandsgleichung. Leiden. Suppl. N°. 23, § 38, Fig. 15. Mathematics. — “On the number of degrees of freedom of the geodetically moving system and the enclosing euclidian space with the least possible number of dimensions”. By Prof. J. A. SCHOUTEN. (Communicated by Prof. J. CARDINAAL). (Communicated in the meeting of May 25, 1918). Suppose & to be a non-special curve in a finite part \, of a general space of 7 dimensions, containing no singular points and where only one geodetic line exists between two arbitrary points. Assuming in a point Q on & a system of mutually independent directions, we can move this system geodetically along k. This geodetic moving can be geometrically defined in the following n(n + 1) 2 dimensions, without changing its linear element. There exists in this space a space Y, developable on a euclidian space of n dimensions, tangent to X, in &. The geodetically moving directions will now coincide at any moment with the directions moving parallel to themselves in the euclidian space Y,. It appears analytically that the known covariant differential of a directed quantity e.q. a vector is a common differential judged from a geodetically moving system of directions. Hence if v is a vector stationary with respect to this system, V satisfies the differential-equation : dv = 0; way. X, can always be placed in a euclidian space of or in co-ordinates : ; LY | dv? + ) jer dx = 0, and this equation then gives the analytical definition of the notion geodetic moving. ‘) A geodetic line is characterised by the property that its linear element forms at every point the same angles with a system moving geodetically along the line. 1) The covariant notations in this paper are the customary ones, but the contra- variant characteristic numbers of the linear element dx are written contravariant agreeing to G. HessenBera, but contrary to G. Ricct and T. Levi Civita. For the invariant notations, the here used direct analysis, cf. ‘Ueber die direkte Analysis der neueren Relativitätstheorie”, a paper presented to the “Koninkl. Akademie v. W.” together with this note. (Verh. Vol. 12 N’. 6). 608 Starting from a point Oa system of directions is now geodetically moved along a closed curve. On returning to O the system will generally appear to have rotated. Dependent on the choice of the ala curve it is generally possible to obtain in this manner oo ‘ positions of the system. If this number is for one point and hence for every point of the area a’, we call NV the number of degrees of freedom of the geodetically moving system. Now the following theorem exists: I. The. number of dimensions of the euclidian space, in which a given space X, may be placed, without changing its linear element, is at most equal to the number of degrees of freedoin belonging to the geodetically moving system increased with n. We will prove this theorem. If the number of degrees of freedom _n(n—1) a . is smaller than ital there will remain invariant 7 mutually per- fectly perpendicular directions of p,, p‚, Pp, dimensions p,+....+p,=n (by direction of two dimensions or 2-direction we mean a plane direction, etc). The number of possibilities exactly corresponds to the number of manners, in which n can be written as the sum of whole positive numbers. We imagine the 7 invariant directions marked once for all in OQ. The system then may be brought in every point of X,, always by geodetically moving. The invariant p;-direction,7 =1....,7, will then define at every point a pj-direction, and it is the question whether these directions will compose a system nb ° of o ’-curved spaces P; of p; dimensions. This is a PrAFFIAN problem in a general space. We select a definite invariant direction, say the p;-direction, and for convenience, sake we shall write p for pj. If wenow define the p-direction belonging to this direction at every point by the simple p-vectors, “™ pV =V,...V,, which all pass into one another by geodetically moving and likewise the perfectly perpendicular (n—-p)-direction to this, ZEN by .W=W,...W,, gp, then: d pv = 0 5 d gw =0, / henee: Vv=0., 7 gw d. It is worth mentioning, that the vectors Vu, k=—=1,...,p, do not pass into each other by geodetically moving and hence dv; == 0. The same holds good for w,;, /—1,....,q. If now the linear element 609 be dx, the usual formulation of a problem as ander consideration, is as follows. *) Given the pn functions of z!,...., 2": vig’) SEEN Geen elen pati: 1 (the contravariant characteristic numbers of the vectors v), and the qn functions: UGE Wi Bee splines sels Sash AL (Eg (the covariant characteristic numbers of the vectors w), satisfying the relation : Lee Ee A / Vi wij = 0 so Hie equivalent to Vk - Wi = 0, we ask, when the system of the total differential-equations | HEEE Aat idem | | | rf | | Ue Te aye | = 0 | aft | Riches ep equivalent to ENV re Ns Vd is perfectly integrable. If r and 8 are two vectors, lying in the p-vector ,v, and con- sequently satisfying the relations: Me cat ar a we grees QP Tia A Seg which is. equivalent to: 1,...,2 5 oe A dr MV a ee A REE eee i i but being otherwise arbitrary, the conditions of integrability are, as known: loen (Ow, Ow, B> — a dz” dar These equations are generally covariant’) and are equivalent to: Jee = 0; game its, fee ate q. ras? ygaw=0; RK 1) Cf. e.g. E. von Weger, Vorlesungen über das Prarr’sche Problem, pages 93 and f.!. ; Seg Og 2) Owing to the circumstance that the expression aw ee generally covariant. 610 Now it follows from this mode of notation that they may be replaced by the invariant equation : Ps BA lye NEL w)z=ras?t Gag, or, as follows from the preceding, still more simpliùed without making use of two auxiliary vectors: vive gl.) But this equation is identically true, 7—,w being a multiple- sum of isomers of 7,w, and vy Ww being zero. As the plane tangent-spaces of p; dimensions, in the various points of the spaces P; have pj-directions, which by means of geo- detic motion pass into each other and in the invariant p)-direction in QO, but never in any other direction, two spaces P; can therefore never intersect. A geodetic line in X,, which has a linear element in common with a definite space PP, is apparently altogether contained in that space and in that space it is geodetic too. Hence two different spaces P; can never be tangent to one another. Therefore we call the spaces P; parallel ones. As any geodetic line having two points in common with a / space, falls completely within that space, which will be proved later on, we call a P; space geodetic. The r obtained systems of parallel geodetic spaces /,,..., P, are at every point of X, perfectly perpendicular to each other. We shall first contemplate the case r= 2, p, =p, p,=q. The parameter-spaces of n—1 dimensions of the primitive variables a", ... , a, are placed thus that each of them contains o”?~! spaces /, those of wt,...,a” likewise with regard to the spaces Q. At every point we place the mutually perfectly perpendicular p- resp. g-vectors pv and gw. The measure-vectors e’,, x= 1,...,p are then situated in „v and for the measure-vectors e’,, u =p 1,...,‚n the same holds good with regard to ,w. Because e’, Le’, we have , , it ee on a ee Pan ce rig pp he TAM hence the quadratic form ds* may be written thus: re ey PA coe: de —= L gy, derde 5 | gy dak dex’ %,) Hy Now may be demonstrated, that g,, is independent of art! , ar, and likewise g,, is independent of «’,...,a”. It is always possible to choose a scalar & as function of «',..., a”, so that: 1) This equation can also be obtained very easily by means of the direct analysis used here. Another form of the same equation is: 611 wke, . at Ep Consequently : RE 5 ss 5 resting? F . ; ; VEE. &p (VR) (€,.. 0. Gp) ee Vlaer) te. be Ae... ed By complete transvection with: Cp - . +. EH en Cie - €, €, all the terms except the (x + 1)-th give zero, hence: (5 ) tn Aoi ep Ge | ay, = 0 Ox” uv pees cased Now: Ds Bia Bur and 0 xv ay Ay == Ay Ay = SS Ay Ay Ow, ke u oe thus Ogu» Ju = Aux ay 4. Gy, Ay, = 0. Òa* Hence the linear element in the Q spaces is independent of w,...,eP; the corresponding property of the P spaces relative IER LN. vn is similarly demonstrated. This property can-also be expressed in the following manner. U. Lf in a general space X,, is placed a system of w"~? parallel geodetical spaces of p dimensions P, having perfectly perpendicular to it a system of or similar spaces Q of n—p dimensions, a figure in a definite P-space will be congruently projected by the Q-spaces on all the other P-spaces. For p=1 this is the well-known property that the distance of two definite Q-spaces measured along the P-lines is constant. So we can here introduce in this case for primitive variable «' the curve length measured along these lines from a definite space, the spaces remain- ing parameter-spaces. Hence the linear element may then be expressed in the following way: 2,...5n ds? =de! + & gu dat dx’ by in which the g,, do not depend on 2’ .’). As, however, a quadratic differential form in n—1 variables can 1) This formula has already been derived by T. Levi Civira. Nozione di parallelismo in una varieta qualunque e consequente specificazione geometrica della curvatura Riemanniana. Rend. di Pal. 42 (17). 40 Proceedings Royal Acad. Amsterdam. Vol. XXI, 612 n (n—1) always be written as a sum of oS quadrates of complete diffe- a, — + 1 similar qua- rentials, ds? can be reduced to a sum of drates. Hence the space X, can be ee in this case in a eucli- dian space of — 44 cata —l) (n—2) 2 dimensions. As the aan of degrees of freedom of the geodetic moving system amounts exactly to | (n—1) (n—2) ae the required proof has been furnished. If we now return to the case r=2, p,=p, q,=q, the number il => of degrees of freedom is PP 5 ) ae q(9 aS, The quadratic form 2, breaks up into two forms, which may be written as a sum of 1 Yr a? resp. P(P Per” quadrates. Therefore the space X, can be placed within a euclidian space of ppl), glgt-1) p(p—)) | q (9-1) rr a, Sr cose dimensions, and here again the required proof has been furnished. The case q > 2 may be reduced to the preceding one. For this purpose the differential form is divided into two. The spaces of one of the systems, say P,, then again contain themselves at least two perfectly perpendicular systems of parallel geodetic spaces. Then the second part of the differential form is once more divided ete. If owing to the existence of the P;-system the division of the differential form is: 2 2 2 2 2 2 ds* =a ? dx =a, 2 dx + ay 2 dx, in which a, and a, are the ideal radices of the two parts of the fundamentaltensor a’, the differential equation of a geodetic line will be: dx dx dx dx ded == = = 0. a (: ae (a, Ge tale oe a, compossing itself only of the measure vectors e,,... , ep ’ ? and a, only of Opti, > 7 5, O;, we haves 613 dx dx d( a. )%=0 ; ala jn =o, from which the property is inferred : HI. Zn the proposition made in formulating LI the projection of a geodetic line by means of Q spaces on a P space, or vice versa, is, as far as existent, a geodetic one itself. If two points A and B are situated in a P space P,, the pro- jections of these points on all spaces coincide. Hence the projections on a Q space of the line AB geodetic in X, passes twice through the same point, being at the same time geodetic in Q, which is only possible when that projection has degenerated into a point. But then the geodetic line Ab must be situated altogether in a 7 space, e.g. in the present case in /,. Hence any geodetic line, having two points in common with a P-space, is entirely contained in that space. 40* Mechanics. — ,, Bemerkungen über die Beziehungen des Dr SirTER’schen Koordinatensystems B zu der allgemeinen Welt konstanter positiver Kriimmung.” By Prof. Fertx Krein. (Communicated in the meeting of September 29, 1918). 1. Ich gehe zunächst von dem sphärischem Falle aus und setze: dys sk, ds? = + (d5? + dr? + d&* — dv’ + dw’), wo das —+ oder — Zeichen gelten soll, je nachdem wir einen „Räumlichen” oder „Zeitlichen” Vector messen. Der Uebergang zum elliptischem Falle erfolgt am einfachsten, indem wir setzen: RS Ry RS Rv c= NSS ie ‚UZ. wo w w w 2. Dann will ich von vorn herein betonen, dan die so definierte Welt bei 10° homogenen linearen Substitutionen der §, 7, 5, v, w — also bei oo'® Kollineationen der x, y, z, u (welche 2° + y' + 27 — —u?+ k?=0 in sich überführen) — in sich übergeht. 3. Den Uébergang zum Dr Sitter’schen ds? mache ich nun so, dass ich setze: § = Rs 9 cos p, 4 = £ sin sin p cos w, C= Rsindsing sin W, ae v = Reos B sinh R c w= R eos 3 cosh —, ; R (unter sinh, cosh gewöhnliche hyperbolische Funktionen verstanden). 4. Weiter wird also woraus 615 Dabei sind w + v=0, w—v = 0 irgend zwei Tangential-ebenen, welche man durch den Koordinatenanfangspunkt an dem Gebilde dn Hert + w*? =0 legen kann; ¢ ist der in geeigneter Maass- einheit gemessene ,,projektive Winkel” im Büschel A(w + v) + uw — v) =0. = 5. In Uebereinstimmung damit wird {== — oo, bezw. t=+o, wenn w+uv=—O0 oder w—v=—O0; verschwinden beide Ausdrücke, so wird ¢ unbestimmt'). ¢ ist nur in dem Ausschnitte der vier- =a) }] : be wd hte ete dimensionalen Welt reell, fiir den — — positiv is; jenseits wird es WV umagindr. 6. ds? behält seine Form bei of Transformationen, nämlich wenn man a) & 4, § beliebig orthogonal substituiert, 6) t durch ¢ + const. ersetzt. Mit Riicksicht auf 2 gibt es also in der Welt (1) oo° ‚pr Sirrer’schen Uhren”. 7. Da ist es nun sehr amüsant sich auszumalen, wie sich zwei Beobachter unterhalten würden, die mit zwei verschiedenen DE SiTTER- schen Uhren ausgestattet sind. Ereignisse, die für den einen in der Ewigkeit liegen, sind für den anderen zugänglich, und umgekehrt, ja der eine erlebt Ereignisse, welche der andere für imaginar halt. 8. Dies alles sind zunächst selbstverständlich nur mathematische Spekulationen, ob der Physiker sie aufnehmen will steht auf einem anderem Blatte. 1) Dies ist aber nur eine Singularität der Koordinate /, nicht der zu Grunde liegenden Mannigfaltigheit (1). Physics. — “On friction in connexion with Brownian movement’. By Dr. O. Postma. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of September 29, 1918). § 1. The different deductions of the mean displacement of a suspended particle caused by the Brownian movement in a certain time, can be divided into two groups. In the deductions of the first kind the forces acting on the particle are divided into accelerating and retarding forces, in those of the second kind this difference is not made. To the deductions of the first group belongs that of EINSTEIN, the second one of von SMOLUCHOWSKI, that of LANGEVIN and that of DE Haas——Lorentz (by the method of Einstein and Hopr). To those of the second group belongs the first deduction of von SMOLUCHOWSKI, that of Van per Waars and SNETHLAGK and that of SNETHLAGE (given in her dissertation) *) *). In the theories first mentioned a retarding force (friction) has been assumed proportional to the radius of the particle, which is consi- dered as a sphere and has the velocity v. The applied formula W=6rlav is deduced in hydrodynamics on the supposition that at the surface of the particle the fluid has the same velocity as this. All theories of the first kind come to the result that the mean square of the displacement (A*) is inversely proportional to tand proportional to a. The theories of the second kind give the result: A? proportional to t and inversely proportional to a? and to the density @ *). It is however not to the resolution of the forces into accelerating 1) See: G. L. pe Haas — Lorentz “Die Brownsche Bewegung”’, die Wissenschaft 1913, J. D. van per Waars Jr. and Miss A. Sneruuace: “The theory of the Brownian movement”. These Proceedings Vol. XVIII, p. 1822 and A. SNETHLAGE: ‘ Molecalar-kinetie phenomena in gases, especially tle Brownian movement”. Amsterdam 1917. *) Recently Van per Waats Jr. gave still a theory in which, though in a some- what different form, the two kinds of forces are found back. (These Proc. XX p. 1254. ?) The first result gives the best agreement with the investigations of SNETHLAGE. 617 and retarding ones that these different results must be ascribed. This is evident from considerations of von SMo.ucHowskI, who, though assuming such a resistance proportional to wv, obtained the second results by simply taking this resistance proportial to a? and g. The reason of the difference lies in the assumptions on the sliding at the surface. Without sliding the resistance is found proportional to a, and not to a? and g. As long as the velocities of the mole- cules may be considered as independent of those of the particle, the assumption W proportional to ga?v is the more natural one. The number of collisions is then proportional to the surface of the particle and the density, while the total force is also proportional to v by the added opposite influences of the forces on the fore- and back-side, each of which is proportional to the square of the relative velocity. This resistance has been calculated by CUNNINGHAM ‘), xMm who found: $a*un (M + (M 4-m)h’ When however the velocity of the molecules depends on that of the particle the relative velocity v can again depend on a, so that the dependency on a becomes different from that in the preceding formula. It is therefore evidently not true that the assumption of a resisting foree, represented by w=pv should be connected with the supposition that there is no sliding. It is the last assumption that is of importance here. Van DER Waats and SneraLacGé have objections to this assump- tion. When the molecules of the fluid participate in the motion of the particle, the velocities .of the molecule are no longer independent of the velocity of the particle, which should however be the case in the here existing state of statistic equilibrium. This participation of the motion would e.g. be possible in the case of the fall of a particle under the influence of gravitation, but not here in that of the BROWNIAN movement. Here it may be remarked however, that just by the resolution of the forces into accelerating and retarding ones an analogous case has been created. The accelerating forces take the place of gravitation. The action of the molecules on the particle is resolved into an acci- dental, irregular one, which may be regarded as the cause of the motion and a regular, resisting one which represents the friction. When we consider a long time there is statistical equilibrium. There is however a continual transformation of energy of the yy E.. CUNNINGHAM: “On the Velocity of Steady Fall of Spherical Particles through Fluid Medium”. Proc. Roy. Soc. Ser. A. Vol. 83 p. 357. 1910. 618 molecules by the irregular, accelerating forces into molar energy of the particle, and of this again by the resisting forces into molecular energy. Also the molecular velocities are as a whole independent: of those of the particle, but the molecules can be divided into a large majority the mean velocity of which differs too little from that of the particle, and into smaller complexes whose velocities differ too much from that. The first group joins the motion of the particle, the second causes the accelerating forces. These forces, however, are not only caused by accidental deviations of the velocity, but also by accidental deviations of density. The relative dimensions of the molecules, the particles and the free path agree with the above considerations. While the order of magnitude of the molecules is 107, we generally find for that of the particles about 107°, the free path in the fluid being of the order 10~®. We thus can represent the dimensions of the molecules and the particles on a seale by 1 mm and 1 dm, while then the free path would beeome 1 em. As a rough approxi- mation we may assume that in a fluid 10* molecules are acting at the same time on the particle (of the same order of magnitude as the number that can cover the whole surface) and in a gas 10°. The collisions which keep the particle in motion are therefore due to accidentally arising complexes of these molecules. § 2. Now we find for the mean distance travelled by a particle ie! a | at hi ek EINSTEIN’s formula 4? = — — — .t, when we take for the frictional N: 3xba resistance JI’ = 62¢ar. This value has been deduced by Srokes on the assumption that the fluid is incompressible and that there is not any sliding at the surface, while the velocity may not be too great. It is however questionable, whether it is a allowed to assume this absence of sliding. Cunnincuam has tried the supposition that the velocity of the fluid at the surface of the particle is fv, so that the relative velocity of the particle with respect to the adjoining fluid would be v_—/v. Stokes’ formula would then give for the resistance W = 6na kv. CUNNINGHAM now calculated this & in the following way. As has been mentioned above he found for the resistance by a. 8 aMm purely kinetic considerations the value — a?vn ine En 3 (M 4 m)h v represents the relative velocity. This would now become ‚where 619 8 am peek 7m (when at the same time m is neglected with u ; respect to J/). He now equalizes this value to the value 6a$akv derived from phenomenological considerations, and thus finds an equation from which & can be calculated. In my opinion, however, there are several objections to this way of calculation. The assumptions on which the deductions of the two expressions are based are not the same; the results refer therefore to different cases and need not be equal. In his kinetic considera- tions CUNNINGHAM supposed namely, that the colliding molecules possess Maxwell's partition of velocities, viz. in the gas or the fluid there is no internal friction (this depends on the deviations from Maxwer’s partition), while moreover colliding molecules exert only normal forces on the particles viz. there is no external friction. In the case of Stokes’ formula on the contrary there is an inter- nal friction and the fluid exerts also lateral forces on the body. In my opinion it is therefore impossible by equalization of the two results to obtain a relation which has any significance. Moreover there is still the inner contradiction in the kinetic deduction between a uniform transformation of velocity fv and the assumption of a purely elastic collision. For this reason no great importance may I =! be ascribed to the final result: W = 618av (1 Le 1,63 — ) : a Instead of assuming a uniform motion of the fluid with the sphere to an amount dv it might be preferable to suppose the normal velocity of the fluid to be equal to that of the particle, the tangential one being different and a friction existing proportional to the relative tangential velocity. The resistance of a sphere under this assumption has been calcu- lated by Lamp and by Basser'). The first of these uses in his deduction the dissipation function (with the aid of the property: in a fluid the energy transformed into heat = the work of the forces necessary to entertain the motion); the other one follows a direct method by calculating the pressure at the surface. The two results however 4 BY eat) : 5) ; 1+ 3— (+5 1) See: Horace LAMB “A Treatise on the Math. Theory of the Motion of Fluids”. Cambr. 1879 p. 230 (not reprinted in the later issues) and A. B. Basser, “A Treatise on Hydrodynamics” Cambr. 1888 Il p. 270. do not agree exactly. Lams finds: W = 6 agav 620 2 pene | and Basset: W = 62Sav ae , where f is the coefficient of external Ge eres Ba wai a friction. The second expression is also equal to: 67 av ri 1 43 Co $ Ba so that the difference between the results is: 62269 ————-———— A mistake suggests itself in the indirect method and seeking for it we are led by the remark that the two values become equal when g=0 and when 8B=o viz. when there is no friction and when there is no sliding. In both cases the friction does no work: in the first case because the force becomes zero, in the second because the way becomes zero. Probably we shall therefore have to add the work of the friction at the surface to the heat, calculated from the dissipation- Ld / function with the formula {| Fda dy dz, where # = —2 §(a+6-+c)? + 25 (at + 6? + c? + 2f? + 2g? + 2h?)'). This proves indeed to give the right result. At a point determined by the angle @ the force of friction is: 25 3 val 14+ — 7 A B — +va' Ba U=B eres + v} sind’), where A= and B= ae 14+3- 1 gE 4 pa 4 Ba 3 Su 28a i Pe A Bok so that V=s— u sind. This friction is exerted on the surface bee ar Ba 2aa'sin9 dd, while the relative velocity, opposite to the force, is 3 Ou Eoin. a: : 1) Notation of LAMB. ?) See Lamp p. 230 and Basset p. 270. 621 We thus find for the numerical value of the work of the friction per second: f Zana’ sin 7 dB aaa sin J | = | 27ra* sin? 9 paar di} = ar ee Gee el = 62% Cv*a — at a, When to this term the heat calculated from the dissipation-function is added, the force, necessary to entertain the motion, and therefore also the friction in question, is just increased by the missing term. With the aid of this last formula for the resistance we find for the mean deviation of the particles: 25 en tbe Ba tae ‘Baca Sis) pa § 3. That, when the above mentioned work of the resistance is taken into consideration, the dissipation function can give the resistance to which the particle is submitted, may be made evident in the following way. „We think the incompressible fluid enclosed by a surface 5S, part of which is formed by the surface of a body of arbitrary form, which moves through the fluid. Now we consider the kinetic energy » vse oe if [fer de dydz (the molar energy only, not the heat) and deduce from the equation of motion an energy equation which indicates how the kinetic energy changes with the time. We then tind '): dT Ar Ln} = | [fous Yv + Zw) du dy dz — sf { eq? (lu + mu }-nw) dS + +f (Xu 3 Vet Zin) dS | [Fda dy de. yee Here X, Y, Z represent the components of the external force acting on the fluid, pro unit of mass, X,, Y,, Z the pressure components 1) See Basset Il le. p. 252 and HermHorrz “Wissenschaftl. Abhandlungen’”’ Lpzg. 1882 p. 225. 622 pro unit of surface acting at the surface of the fluid, / the dissi- pation function. The first term on the right hand side represents thus the increase of the kinetic energy by the action of external forces, the second term the increase by the flow inwards of the fluid, the third term the increase by the action of the pressure forces, the fourth the decrease by the transformation of the kinetic energy by the inner resistance. This last integral is always positive, the other ones can be positive or negative. When we suppose the surface S to be quite formed by solid bodies, at the boundaries of which the friction is neglected except along the moving body, the second term falls away, while the third term has to be taken over this body only. When further the state is stationary, equation (1) becomes: offerts schil Fdzdydz. (2) For the moving body we have in this case the equation: =(X'u+ Y'v+ Z'w) dt [Aer aS dt =D Gn The summation has to be extended over the points, where the external forces (\’ ete.) act, the integration over the surface. This notation expresses that the forces acting on the surface layer are equal and opposite to those acting there on the fluid. In the case of sliding however the components of velocity are not equal as far as they refer to the body, on one hand, and to the fluid on the other hand. Calling the components in the fluid w, ete. those in the body wi ete. then u‚= u + u, ete. (4), when uw, is written for the component of the relative velocity. Now equation (2) becomes :- {| (X,u,+ Yer Zw) dS ~{{fr da dy dz = 0 and equation (3): = (X'u + Y'v+ Z'w) — [fom + Y,v/4 Zwi) dS = 0. Summation gives us: =(X'ut Y'v+Z'w) + + J [ew + Feed + 2, (or—uiplas —[[[F aa dy de = 0 so, using (4): 2 (X'u+ Y'ut Z'w) =|{ fF dx dy dz (vet Yv, +Z,w,) dS. 623 As the normal forces at the surface are perpendicular to the relative velocity they do not contribute to the last integral; this term represents therefore the work of the friction. The result is thus: the work done per unit of time by the external forces that act on the body is equal to the heat generated in the fluid increased by the absolute value of the work of the external friction '). Taking now as the body a sphere with the velocity v, which is . kept in a state of uniform motion by a force K in the centre, the work per unit of time is Av. This A being equal to the resistance W that has to be overcome, it is also equal to Wv =|) be de dy dz + the work done by the friction along the surface. From this formula we can calculate W. 1) “Absolute” value, as A, and w;, referring both to the fluid are oppositely directed, so that finally the form becomes positive. The result says therefore only : external work = total generated heat. It must be remarked, that RAYLEYGH, who first introduced the dissipation function, did not mean the above mentioned F’ but the total generated heat. (Proc. Lond. Math. Soc. 1873 p. 363). oh Pal sbvaiyidn seat. bont ale de ar: wiser 0 et rant, Odpet bak 9 here ied ‘ : EENDEN ate CDI ae ee: adt miel Seber a. init Ze et nt) alten tH) hee as NW’ nl bidet Krtek ed ka his ziee held Ln ee hens rs th sinbags ) MFEER * Hernen Ay oiiat ‘tt tah oe NAS Me wy Bah be hah Ae SEU Aid Ar ze ni tote sie TSE, 16 hel # bere lr Se TTPO vravi 1 Anes Bd i iat Bir 4 út my Bn oneal Jt Amt REE Thiele en Ar Sete U gh a ge ! {rin 9 7G) Oa) used) ide, anh ej as Guz tite E 2 eh dh (DUE AAE AGL; Ba “31 win} hoste ‘ ish silke) orate a Jn lata wm HIE 2) ob ae alter, Ae Sips Gass B ily I bonte We th enn NE wer alt AN RE. vi Kata Feild Geh oe út: $ it HERREN “te i Veil Pegs ie EEN DE ‘gary pe 5 Ge A Beter, To A Apes Oa oltaell: wf beau tens hs E $0 6 as A SNEL AVT JRA MOREL si i! faa betaing ait ; o KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN TE AMSTERDAM. PROCEEDINGS VOLUME XXI Now S: President: Prof. H. A. LORENTZ. Secretary: Prof. P. ZEEMAN. (Translated from: ‘Verslag van de gewone vergaderingen der Wis- en Natuurkundige Afdeeling,” Vol. XXVI and XXVII). CONTENTS. D. J. HULSHOFF POL: “Our equilibrium-organ”. (Communicated by Prof C. WINKLER), p. 626. D. J. HULSHOFF POL: “Cerebellar ataxia as disturbance of the equilibrium-sensation”. (Communi- cated by Prof. C. WINKLER), p. 637. J. J. VAN LAAR: “On the Heat of Dissociation of Di-atomic Gases in Connection with the Increased Valency-Attractions p/A of the Free Atoms”. (Communicated by Prof. H. A. LORENTZ), p. 644. P. G. CATH: “On the measurement of very low temperatures. XXIX. Vapour-pressures of oxygen and nitrogen for obtaining fixed points on the temperature-scale below 0° C”. (Communicated by Prof. H. KAMERLINGH ONNES), p. 656. F.E. C. SCHEFFER: “On Phenyl Carbaminic Acid and its Homologues”. (Communicated by Prof. BOESEKEN), p. 664. - J. J. HAAK and R. SISSINGH: “Experimental Inquiry into the Nature of the Surface-Layers in the Reflection by Mercury, and into the Difference in the Optical Behaviour of Liquid and Solid Mercury”. (Communicated by Prof. H. A. LORENTZ), p. 678. F. M. JAEGER and WILLIAM THOMAS B. SC.: “Investigations on PASTEUR’s Principle concerning the Relation between Molecular and Crystallonomical Dissymmetry: VIII. On the spontaneous Fission of racemic Potassium-Cobalti-Oxalate into its optically-active -Antipodes”, p. 693. L. E. J. BROUWER: “Ueber eineindeutige, stetige Transformationen von Flächen in sich”. (Sechste Mitteilung), p. 707. EuG. DuBOIS: “The Significance of the Size of the Neurone and its Parts”. (Communicated by Prof. H. ZWAARDEMAKER), p. 711. A. PANNEKOEK: “Expansion of a cosmic gassphere, the new stars and the Cepheids”. (Communi- cated by Prof. W. DE SITTER), p. 730. J. D. VAN DER WAALS JR.: “On the Theory of the Friction of Liquids. (Communicated by Prof. J. D. VAN DER WAALS), p. 743. A. B. DROOGLEEVER FORTUYN: “On two Nerves of Vertebrates agreeing in Structure with the Nerves of Invertebrates”. (Communicated by Prof. J. BOEKE), p. 756. Pd zp N. G. W. H. BEEGER: “Ueber die Teilkörper des Kreiskörpers K (7 IN (Zweiter Teil), p. 758, (Dritter Teil), p. 774. (Communicated by Prof. W. KAPTEYN). . Proceedings Royal Acad. Amsterdam. Vol. X XI. Physiology. — “Our equilibrium-organ”. By Dr. D. J. Hersnorr Por. (Communicated by Prof. C. Winker.) (Communicated in the meeting of November 24, 1917). When studying the functions of the cerebellum *) | always halted before the difficulty of interpretation of the observed phenomena. It is not easy to deduce whether they are dependent on the organ in itself, or resulted from the interruption of tracts in the cerebel- lum, whieh took origin in other parts of the central nervous system. Therefore it seemed to me desirable, before continuing my inves- tigations on the cerebellum, to trace in the first place the connection between those influences which are lying outside of this organ and the cerebellum itself. As experiments on this subject have often been made, it was clear that repetition of the former investigations would not bring a nearer solution of this problem. Therefore I resolved not to start by experimenting on animals, but to examine the suffering people and especially to pay full attention to ataxia. As this phenomenon is often observed by disturbances of the central nervous system and as it is known, that according to the illness, also the decomposition of the movement can show a different type, [ thought, that perhaps it could be possible by putting together the different types to get a more distinct insight into the nosology of the cerebellum. Now it is in general accepted that cerebellar ataxia is caused by a disturbance in efferent paths, and thus could be a motor dis- turbance, but as this kind of conducting fibres is not well imaginable without afferent tracts, one may accept that these too, interrupted in the cerebellum, will show disturbance in movement. j As my former investigations were exclusively restricted to the motor funetions, I thought it now wise to draw attention to the afferent tracts. 1) Cerebellar ataxia. Phsych. Neurol. bladen 1909 N°. 4. Cerebellar functions: in correlation to their localisation. Psych.-Neurol. bl. POLS NOx: 627 Of the sensory and sensorial stimuli which reach the cerebellum along centripetally conducting paths and which are well-known to us, we may mention in the first place the functions of the deep sensation and of the sense for the muscle tone and the equilibrinm. The first pass with the posterior nerve-roots into spinal cord and run partially unerossed, with the Column of Clark as mid-station, to the tractus spino-cerebellaris dorsalis (FovitLE-FLEcusic), while another part, with the area nuclei intermedii as mid-station, run also for more than the greater part uncrossed to the tractus spino-cerebel- laris ventralis (Gower), while a smaller part goes to the same column of the crossed side. This bundle is therefore partially composed of crossed, partially of uncrossed fibres. The tracts of Frecusie and Gower lie in a long but narrow strip at the lateral edge of the spinal cord and run centripetally. The tract of FLecusig goes through the restiforme body and the inferior brachium conjunctivum towards the vermis of the cerebel- lum, without coming in contact with the dentate nucleus. The tract of Gower does not pass into the restiforme body after having reached the medulla oblongata, but runs on in longitudinal direction. On the level of the nervus trigeminus it bends round in latero-dorsal direction and passes along the brachium conjunctivum cerebelli into the vermis superior and the nuclei tecti cerebelli. From the posterior columns of the spinal cord however, there are along other paths also tracts connected with the cerebellum, e.g., through the nuclei of Goïr, and Burpacn, along the fibrae arcuatae externae and anteriores to the restiforme body and from here to the cerebellum. As the influence of these latter fibres is far from known, J will leave them in the further discussion out of the question. Now the experiments of MarBurG and Brine ') have taught, that the lesion of the spino-cerebellar tracts provokes a very serious disturbance of the statotonus. Partial or total destruction of these bundles from the entrance'in the spinal-cord to the cerebellum, will therefore show disturbance of the equilibrium. If the connection between the tracts of Fiscnsig and Gower with the cerebellum are well-known, this is less the case with those between the vestibular organ and the cerebellum. LANGELAAN®) writes that the end-arborisation in the oblongata of 1) EDINGER. Zeitschr. f. Nerv. Heilk. V. 45, 1912 p. 308. 2) J. W. LANGELAAN. Bouw van het centrale zenuwstelsel. Amsterdam. VERSLUYS. _ 1910. 41* 628 the vestibular nerve is T-shaped, of which the ascending fibres con- tinue into the cerebellum. They unite into bundles, between which is found gray matter, belonging to the nucleus of Deirrgs. These bundles form the greater part of the corpus juxta-restiforme and pass with the fibres of the corpus restiforme into the cerebellum, where they end in the dentate nucleus and in the nucleus tecti. JELGERSMA *) too points out (p. 217) the fibres of the nervus vesti- bularis as thick bundles running through and along the braechium conjunctivum inferius towards the cerebellum, being everywhere visible as distinct bundles. Winker is of a different opinion and thinks from sections, which he possesses, he can make out that the nervus vestibularis does not stand in direct connection with the cerebellum, but that the fibres all end in the corpus juxta-restiforme, around cells of the nucleus of Derrers, the nucleus triangularis, of the proper nucleus of the radix descendens N. VIII. Cells of middle size, lying in the regions of these nuclei, carry the impulses through the curpus juxta-restiforme towards the cerebellum. It does not receive direct nerve-roots of the N. VIII. In this case therefore the connection has to take place by means of an interjacent link. As to the physiological function, the investigators of this region are almost of the same opinion, that the vestibular organ will be an organ for the muscle tone and for the equilibrium. Its great importance for our equilibrium has gradually and regu- larly come to the foreground, even so, that Gotz made a sixth organ of it. *) We may say, therefore, recapitulating in short the above mentioned that from the spinal cord as well as from the vestibular organ, strong tracts run to the cerebellum and that interruption of these disturbs the equilibrium. In my investigations 1 thought I was allowed to start from the standpoint, even although from both organs paths go to the cere- bellum, yet the difference between the stimuli which they conduct, is so great, that it could perhaps show me the way in tbe intricate mass of the cerebellar phenomena. Thus, as the spinal path, taking its origin in the peripheral nervous system, enters the spinal cord through the posterior nerve roots and ia. as the thick posterior fibres of the roots send their collateral fibres to the column of Crark and the area nuclei intermedii, I thought I could best study the disturbance in the equilibrium when 1) G. JELGERSMA. De functie van het cerebellum. Psych. en Neur. bl. 1915. 2) H, ZWAARDEMAKER. Physiologie. De Erven F. Bonn. Haarlem 1915. bl. 286. 629 the centripetal impulses along the tractus spino-cerebellares were fallen away, as happens in cases, affecting the posterior funiculi, e.g. in tabes dorsalis. For disturbances of the vestibular organ | had best limit myself to those cases, in which the equilibrium organ had lost its functions, inter alia after scarlatina, cerebro-spinal meningitis ete. In working out these investigations | tried in the first place to devise a scheme, which could serve as well in lesions of the posterior funiculi, as in those of the vestibular organ and in altera- tions of the cerebellum. I thought I had found one in letting the patients perform walking- tests, which were registered on paper. For this purpose a line was drawn in the middle of large pieces of paper, on which the patients walked after blacking their footsoles. When in this paper | leave the results, which | found in cere- bellar alterations out of account, and when I limit myself to those which are correlated to lesions of the posterior funiculi and the vestibular organ, | find what follows: 1. If a patient, who suffers from tabes in a rather far advanced state, walks on the paper, then the reproduction of figure 1‘) appears : The patient tries to fulfil the task of walking on the line (foot print 1—2 and 3), but sways to and fro, as in the RomBerG syndrome and he is obliged to put down the right foot lateralwards, (4). Still worse the swaying becomes in the following right footstep (6) when he replaces the foot three times to keep his equilibrium. The deviation in the line of equilibrium is most distinct with the footsteps 8 to 11, which he had to put down close to each other and during which it was impossible to him to remain on the line. He therefore leaves off trying it and walks on rather well along a broad gait-path. 2. When that which is ordered sub I, is repeated, but with eyes shut, then the deviations of the gaitpath are still more districtly visible. 3. If one lets the patient repeat the same walking exercises as sub 1 and 2, but allows him at the same time to touch our hands ’), then one sees a gaitpath nearly as normal (fig. 2). 1) The cross on the photo indicates the moment when | thickened the contours of the foot prints with ink, because otherwise the footsteps do not come out well enough on the photograph. *) While walking the patient, who is standing on the middle of the paper, stretches his arms to the left and to the right and lays his hands on the dorsal plane of those, which are tended to him from the side. The persons, who help to Tabes no contact-sensation. Tabes contact-sensation. 631 ! ; ; ; : i / ) | | | , 2’ If 5 ate . Er | dE | k ES \ b ke kf ; . A | iP Fs í h 3 m, LYON | | eps bv Hil: Jesirtir | dl } a { Fig. 3. Fig. 47 Vestibular affection no Vestibular affection contact sensation. contact-sensation. 632 4. If one reproduces of a patient, suffering from a vestibular intlec- tion, a gait-path, then distinct deviations will be visible (fig. 3). 5. If we repeat the experiment, but with eyes shut, then the deviations sub 4 are more distinetly visible. 6. If we allow the vestibular suffering patient to touch our hands, then one gets a very important amelioration, even a nearly abolished ataxia (fig. 4). If the symptoms which these patients show are put together, then we have three types: a. distinct deviation while walking with open eyes, while the hands do not find contact. b. increasing of these deviations, when the eyes are shut. : c. gait nearly normal, at least important improvement of it, when the hands find contact. If one wants to comprehend these three differences well, then it is necessary to bring to the foreground that our movement equili- brium, as it were, is principally regulated by the eyes, the vestibular organ and the equilibrium sensation *) of the trunk and the lower limbs. Of these three factors the eyes are the least important, which is easily tested by the fact that a person can walk very well with his eyes shut, but directly shows disturbances, when the vestibular organ or the equilibrium sensation are suffering. This can also be explained, because the last two factors give proprioceptive stimuli, according to the particular conception of SHERRINGTON, which do not affect consciousness, while the eyes convey exteroceptive stimuli, with regard to the gait. We could describe it best in this way, that the equilibrium is governed by the vestibular organ, also by the equilibrium sensation of the trunk and the limbs, while the eyes only regulate the intended direction of the movements. Therefore when a patient suffering from tabes walks, he does not wholly dispose of the three above mentioned factors, but he walks, if we are allowed to express it thus, by his eyes, by the vestibular organ and the rests of the equilibrium sensation of the trunk and the legs. The result is, that the movements become uncertain. If such a person shuts his eyes, then the exteroceptive stimuli, moreover, do this and who walk along the edge of the paper, are asked not to support the patient, but to give way as it were in vertical direction to the movements, which the patient, makes. Their hands therefore have to balance too. The patient has no support, but only contact sensation with the persons who walk alongside of him, and by which his equilibrium sensation can orientate itself. 1) Equilibrium sensation has to be interpreted as an independent subdivision of . that, which till now is brought together in conception of “deep-sensation”’. 633 fall out, because the eyes cannot give direction to the intended movements and therefore it is comprehensible that the uncertainty of the gait augments. However on the other hand it can be said that as the equilibrium in particular is only regulated by proprioceptive stimuli, disturbances in gait will oceur, when a part of those fall out, but the distur- bances will be partially improved by the exteroceptive stimuli, which by means of the eyes can convey their stimuli. _ The above mentioned is known, but it is most important, that when to such a tabes patient, either with opened, or shut eyes, contact is given through persons walking alongside of him, the gait greatly improves, even so, that the ataxia nearly altogether disappears. As such a patient is not supported, but as he has only contact with persons walking alongside of him, I think it may be assumed that the equitibrium sensation of the upper limbs is put into action. The equilibrium-sensation orientates itself along this new path, brings in this way new afferent proprio-ceptive impulses towards the central nervous system and therefore it can better control the move- ments of the lower limbs. The equilibrium sensation of the arms, in the tabes patient, thus takes over the function of the equilibrium sensation of the Jower limbs and trunk (this only of course when the tabes is present in the caudal part of the spinal cord), which for a great part has disappeared. Therefore we can say the following: a. a tabes patient walks by his eyes; by the vestibular organ and the rests of the equilibrium sensation of the trunk and the legs. 6. if such a patient has contact-sensation with persons walking alongside of him, then he moreover walks by the equilibrium sen- sation of the arms. In case a he lacks afferent equilibrium impulses on a great scale and therefore he walks atactic, in case / the amount of these equi- librium-impulses is very considerably augmented and the ataxia there- fore is improved, nay it even has entirely or almost disappeared. If we have a patient suffering from a vestibular affection, then we see on the whole the same effect. In this sort of patients the central nervous system too receives the exteroceptive impulses by way of the eyes, and moreover the proprioceptive ones of the trunk and the legs, through the posterior roots of the spinal cord, but none or only partially from the proprio- ceptive ones of the vestibular organ. The result of this is, that the amount of equilibrium impulses is not sufficient, therefore the patient walks atactic. 634 For these cases too it is important, that when the patients, by means of their arms, receive equilibrium contact through persons walking at their side, then the ataxia either very importantly im- proves or it entirely is abolished. Here too we see, that the equilibrium contact of the arms replaces totally, or for a very large amount the proprioceptive impulses from the vestibular organ. Therefore we can say for this case: a: a vestibular patient walks by his eyes, by the equilibrium sensation of the trunk and the legs, and the afferent-proprioceptive stimuli from the vestibular organ, which is left to him. 6: if such a patient has equilibrinm contact with persons leading him, then he moreover walks by the equilibrium sensation of the arms. In case a he lacks afferent equilibrium stimuli and the patient walks atactie. In case 5 the ataxia totally or partially disappears, because his lack is supplied. Now the peculiarity of the results found is, in tabes as well as in vestibular affection, that the equilibrium sensation of the arms can compensate the equilibrium sensation of the trunk and legs as well — as the tinpulses from our vestibular organ. On account of this the question arises whether it is possible, that in our equilibrium different organs can replace each other. This question deserves to be answered in the affirmative to a certain extent. If e.g. we close the eyes of a person, who is then asked to walk straight on, there will be many, who deviate to the right or to the left. The reason of it will depend among other things on the fact, that the proprioceptive equilibrium stimuli, which arise from both the halves of the body, are not of the same strength; the result is that one half predominates and that the gait will not be totally straight. If we place, however, at a distance a person, who counts, then the blindfolded person will be able, guided by the sound, to walk straight on towards the counting person. The extero-ceptive stimuli, which pass from the ears towards the cerebrum, complete the others, through which the straight gait is made possible. The sense of hearing comes to the aid of the equilibrium sensation. It is also well-known that the eye sense can give direction to our movements. It is comprehensible that as these two senses are already able to give assistance under normal circumstances to the equilibrium sensation, they can help the suffering person in yet higher degree after practice. It is also a well-known fact that e.g. the ataxia in tabes patients, who can still walk straight with their eyes opened, comes to the 635 foreground, when the same movements are performed with eyes shut, or when they are walking in the dark and cannot make use of their eyes. This help may be rather sufficient in light cases of tabes, but it will not be possible to totally improve the disturbance, if the illness has become of a rather serious nature. If we give to this kind of patients contact-sensation by the arms, there will yet be an important improvement. (Fig. 2). . Consequently from this follows as is moreover near at hand, that the equilibrium sensation of the arms, being of the same sort as of the trunk and limbs, compensates in reality, while the other senses can only correct to a certain degree. : I have pointed out in the preceding pages, that one finds the same facts back in lesions of the vestibular organ. Here too the ataxia improves, when the patient uses his eyes (Fig. 3), but here too one finds, that when the eyes cannot sufficiently correct any more, the atavia totally or nearly totally vanishes, when through the arms equilibrium-sensation is obtained with the surrounding world. Here too we find, that the equilibrium sensation of the arms acts totally or nearly totally compensatory. (Fig. 4). The question arises how we can explain this. 1 think that I may except as easy to comprehend, that the pro- prio-ceptiwve stimuli of the equilibrium sensation of the trunk and the limbs and those of the vestibular organ are to be considered of the same sort. Referring to our equilibrium, no difference should be made between the afferent-proprioceptive stimuli, which from the vestibular organ are conducted to the central nervous system and those which come there from the trunk and the limbs. It is a large system of equili- brium fibres that is spread over our whole body and its aim is to regulate the equilibrium. The sixth sense, the one for our equilibrium, has therefore not only to be looked for in the vestibular organ, but it is, as I explained, spread over the whole of our body. The vestibular organ is but a part of it. Now probably one might ask, why does that organ form a whole while in the other equilibrium paths very little independency is found. The reason for this, according to my view, has to be found in the extraordinary relation of the head, in comparison to the rest of the body. To make this clear, one has to keep in mind, that the equilibrium sensation of the different parts of our body is not everywhere the same, or otherwise expressed, is no teverywhere equivalent. Kg. the equilibrium sensation for the trunk, whieh can only move 636 moderately, is but little developed. For the lower limbs it must be already higher, because the movements which they perform are already much more complicated. In yet higher degree this is the case with our arms. So it is known, that as soon as our equilibrium gets in any danger, we do not only immediately put our arms into function, but that we trust even more to our arms, which are weaker concerning our muscle strength than to our much more muscular legs. If the equilibrium sensation of our arms has already reached a high degree, this will be yet more so with our head, which is above all designed to bring to our knowledge our attitude in space. It speaks for itself that without this knowledge no equilibrium is possible. Moreover, with exception of the lower jaw, the different parts of the head are not linked together by joints, but they are tightly grown together. That this puts the head into an extraordinary relation is comprehensible. The trunk and the limbs are in opposition to the head composed of movable parts, which are joined together by tendons and sinews. All the changes in attitude, therefore also those which are of importance for our equilibrium, come to our consciousness. This is not the case with our head. By means of the neck and all that is connected with it, it can, however, fix its own posture with relation to our body, but it will not be of any use for the determination of the equilibrium. The equilibrium organ of the head has, on account of what was above reported, to be not only much more highly developed, but it must be differently composed from the equilibrium paths for the trunk and the limbs. Therefore too the vestibular organ is built up in a different way. As the joints in our head, necessary for the equili- brium, are all missing, it is most probable, that the statoliths through movements during the changes of attitude stimulate the equilibrium fibres aud thus put into action the necessary afferent- proprioceptive impulses. The central nervous system is immediately warned of any danger that threatens our equilibrium and can take the necessary steps against it. CON GL US TONS: a. The sixth equilibrium sense is not placed in the vestibular organ only, but has its tracts spread over the whole body. | hb. The vestibular organ is, as far as it refers to our equilibrium only to be considered as a part of the equilibrium sense. c. The different parts of this sense can compensate each other reciprocally. Physiology. — “Cerebellar ataxia as disturbance of the equilibrium: sensation.” By Dr. D. J. Hursnorr Por. (Communicated by Prof. C. WINKLER.) (Communicated in the meeting of Jan. 26, 1918). In a former paper’) I explained, that the ataxia which a patient, suffering from tabes, shows, while walking, is nearly quite abolished when he has by means of his hands contact-sensation with the persons walking to his right as well as to his left side. the reason for this improved walking has to be looked for in the fact, that as long as the affliction resides in the lower part of the spinal cord, it enables him to make use of the equilibrium- sensation of the upper limbs (afferent-propriaceptive stimuli), and in this way he can orientate himself better in space. If this latter happens to be, then all the efferent impulses, neces- sary for the regulated movements, can run down along the motor paths to the trunk and the lower limbs, through which the ataxia becomes abolished, respectively ameliorated. Ll expressed myself thus, that where an ordinary person walks by his eyes and the equilibrium-sensation of the vestibular apparatus, the trunk and the lower limbs, a tabes-patient does it by the equilibrium-sensation of the vestibular apparatus and the rests of that which is still left in the trunk and lower limbs. If such a person therefore has contact-sensation with people walking alongside of him, he moreover walks by the equilibrium- sensation of the upper limbs. If one examines, not a tabes-patient, but a sufferer from the vestibular organ, then, as Ll wrote, such a patient will walk by his eyes, the equilibrium sensation of the trunk and legs and the rests of the equilibrium impulses which are obtained through the vestibular apparatus. If one therefore gives to such a patient contact sensation through persons walking alongside of him, then it also appears that he is enabled to walk normally again or at least nearly normally. As the ataxia, which both these patients show, are both almost totally abolished through the same influenee, viz. contact sensation 1) Our equilibrium-organ. These Proceedings p. 626. 638 by means of the upper limbs, I thought I had to accept the involved afferent-proprioceptive impulses in those cases as being of the same sort. The vestibular apparatus therefore has to be considered, as far as our walking function is concerned, as a’ modified and higher de- veloped organ for the equilibrium-sensation of the head. akc) a a a , : BG JS ‘i vi R : Bae rs si } ' or a | ee J a | L b Es an | > al eed) wy WJ IK h | Jt + : tate IS ce et A z } Lk > | of | J R N | Ap i dj | ® ae i | : eN f ¢ R \ rad % Ks Zout open Ly Olle id se del ‘ / =) Y ypcirund ? era. AL ouge i verband $ 5 ¢ Ke alles viv e: L / dus nuit gee 9 f 4, n 639 It would take me too long in this paper to enter more in particular into this hypothesis; I refer the interested reader to the original communication. The one thing to which I will draw atten- tion, is, that according to this view, considering our equilibrium, there should be made no principal difference between the afferent- proprioceptive stimuli which are conducted from the cerebellar tracts of the lateral column and those from the vestibular apparatus to the cerebellum. All these stimuli are related to the equilibrium- sensation and therefore regulate our gait. These different afferent tracts thus form a whole and they are to be considered as a sub- division of the same equilibrium system. Now it is important to trace, whether they change, and if so, what alterations these afferent stimuli undergo, when they arrive in the cerebellum. If, investigating this, one makes patients, suffering from e.g. cerebellar tumors, perform the same walking-tests, as 1 did with sufferers from tabes or vestibular affliction, then the results prove to be totally the same. For instance I will report the following case, in which during life the diagnosis was made of a tumor, which was located at the left side, where it pressed as well on the cerebellum as on the nervus octavus. During the operation and also post-mortem the diagnosis could be confirmed. If one had the patient walk totally unsupported with open eyes, then the part of her gait was obtained as reproduced in fig. 1. The ataxia is distinetly visible. If her eyes were shut, then the path of her gait became as in fig. 2. When comparing the two the ataxia proves to be considerably augmented. This is comprehensible, because in the latter case the afferent-extero-ceptive stimuli from the eyes are missing. The differ- ence between these two paths of gait results from the influence of the eyes on the performed movements. It is important to point out once more (see preceding paper) that, when by abnormalities of the static organ a second organ comes to his aid, the latter only can partially replace the deficiencies of the former, if these are of rather severe nature. This is distinetly visible in fig. 1, because notwithstanding the aid of the eyes, the ataxia however is far from gone. If we now let such a patient walk quite free with eyes open, but giving her contact sensation through persons next to her, then although also the neck-equilibrium-impulses are shut out by means of a bandage, the gait path in fig. 3 is reproduced, in which the 640 ataxia proves to be totally gone. The equilibrium sensation which is augmented through the contact-sensation of the arms is enabled to compensate all that is missing. What the eyes could not do, the afferent-proprioceptive stimuli of the upper limbs could do. They compensated, thus abolishing the ataxia. Gn ea | | jd | ne , cae ae 4 A 2) 5 ate ») | ij R | | | | Se i. | Jm | ie Vrouw wand i ? | 3 rar Neal : Doge hn ver band Steel onze hand L hals mn ver band | | of / | aleut Op onze handen oy 1} 641 As one could oppose against this important amelioration that the result had been due for the greater part to the help of the eyes, I repeated the test in precisely the same way, but now with eyes closed. The produced gait path is found in fig. 4. Now too no sign of ataxia is seen. The only deviation is, that the patient walks to the right instead of walking straight on. As this deviation takes place gradually and the gait-path remains straight, the circus gait must be excluded; it seems to me, that in this case one cannot speak of a deviation to the right. It is most probable that the patient did not stand straight in front of the line, but somewhat in oblique direction and therefore walked in that way. The 4 fig. proves that the contact sensation through the arms is yet able to abolish the ataxia, even if the patient misses the afferent-extero-ceptive impulses from the eyes. One thus sees here exactly the same phenomenon as with the ataxia in tabes and in vestibular affliction. Therefore it lies at band to assume that cerebellar sensory ataxia arises when in the cerebellum the equilibrium paths are being interrupted which from the spinal cord and the nervus octavus pass into it. This would confirm JELGrRsMA’s') view (pg. 217): “The supposition that the cerebellum is a central place of innervation for both these organs °), is therefore probable.” Should my view be right, then cerebellar sensory ataxia will occur, when the process of the disease arises in those parts of the cere- bellum, in which the equilibrium-paths pass, In connection with the above a few questions arise. The first is whether the cerebellum exclusively dominates the equilibrium. This question is answered in negative sense by many investigators, because e.g. experiments on animals have taught that experimentally obtained cerebellar ataxia can totally pass away after a certain time, which proves that the afferent equilibrium impulses can arrive at the cerebrum also along other paths. The second question is whether the cerebellar ataxia always shows the same image. Should 1, as regards the experiments on animals, confine myself to _the well-known investigations of Lucianr*), then he too made a 1) JELGERSMA, G. The function of the cerebellum. Psych. Neur. bl. 1915. 2) Meant are: tonus- and equilibrium organ and the deep sensation. 3) Luctani, L. Das Kleinhirn. Georg. Thieme. Leipzig 1893. 42 Proceedings Royal Acad. Amsterdam. Vol. XXI. 642 difference between the ordinary atactive movements and those which were indicated by him as “dysmetria’. When one wishes to see these differences reproduced distinctly I refer to my photographs in my communication on “cerebellar ataxia”. Plate VIII, X to XIV’). In man, where the phenomenon cannot be localized as sure as in animal-experiment, a conclusion is drawn with more difficulty. Yet I will quote some writers, who showed that the deviations in gait do not always give the same aspect. ; JELGERSMA (le. p. 227) e.g. writes that the occurrence of the cerebellar ataxia is due to the fact that the trunk sways over the legs: “a gait, which greatly resembles the walking of drunken men.” He only describes one type of deviation of the gait. OPPENHEIM, however, differentiates in his hand-book two forms of cerebellar ataxia, 1. “auf (Schwindel und) Gleichgewichtsstörung beruhende, “die grosse Aehnlichkeit mit der Gangweise des Betrunkenen “zeigt,” and 2. “eine auf Bewegungsataxie beruhende. Patient “geht breitbeinig und stampfend, aber ohne dass ein übermässiges “Schleudern eintritt.... Eine scharfe Unterscheidung dieser Gehstörung “von der spinalataktischen ist wohl nur möglich, wenn sich die unter “de beschriebene Abart mit ibr verbindet.” OppennEmM therefore thinks the cerebella ataxia e.g. also dependent on the spinal cord. Dúeúrine on the other hand writes in his work “Traité de Pathologie générale” 1901, on page 648 “%. Ataxie labyrinthique. Les affections “de l’oreille interne produisent quelquefois des troubles de la marche “et de leéquilibre, qui ressemblent jusqu’a un certain point aux “disordres de l’ataxie cerebelleuse.”’ DÉGÉRINB points out in this case the connection of the cerebellar ataxia with the phenomena which are found in vestibular disturbances. Now the question arises to interpret the difference in the atactic aspect. In my former communication | explained that even although the equilibrium-impulses, originating from the vestibular organ may be considered as to belong to the same which come from the spinal cord, yet there exists a great difference in their results, owing to the higher development of that organ. The consequence of this is, that according to the afferent equilibrium path suffering more in the one case than in the other, the complex of atactic symptoms will also appear differently. If the spinal tracts have suffered most, then the type as described 1) Psych. Neur. Bl. 1909 N°, 4. 643 by Opprennem will be found in general, viz. his second form of cerebellar ataxia. Should the vestibular tract be hurt, then the description of DÉGÉRINR comes right. If there is an interruption of both the paths or if the disturbance of the paths from the right and from the left side commences more or less simultaneously, then perhaps the gait of the drunken man will become more prominent. If the results of the animal experiment are compared with those found in man, then we may say that the cerebellar ataxia does not always show the same aspect and that this can be explained by the fact, that the cerebellum possesses more than one afferent tract, whose interruption causes disturbance in its course and that according to the suffering of the one or the other or more paths, the aspect will change. Tbe third question is to explain in cerebellar disturbance the fact how the equilibrium sensation of the upper limbs can totally compensate the ataxia, because one may accept that its equilibrium paths, just as those of the lower limbs, pass into the cerebellum and therefore will also be broken off by the process of the illness. I must acknowledge that I cannot very well give an explanation of this fact, if not the possibility should be accepted of a better connection of the equilibrium sensation of the arms (apart from the cerebellum) with the cerebrum, than is the case with the legs. Later experiments will have to give a decision on this point. CONCLUSION. |. By interruption in the cerebellum of afferent cerebellar tracts, originating from the vestibular organ and the tractus spino-cerebellares, ataxia appears. 2. According as these tracts suffer more or less, whether alone, or together, the aspect of the cerebellar ataxia will present a different type. 42* Physics. — “On the Heat of Dissociation of Di-atomic Gases in Connection with the Increased Valency-Attractions WA of the Free Atoms’. By Dr. J. J. van Laar. (Communicated by Prof. H. A. LorENTz). (Communicated in the meeting of October 27, 1918). Introduction. 1. In a series of Treatises in these Proceedings’) on the additivity of the values of 5 and Wa of the equation of state, and on the fundamental values of these quantities for different elements in connection with the periodic system, I determined the following values of 6.10° andp/a.10°, expressed in so-called “normal” units. The values of 6;.10° (per Gr. atom), determined up to now, are found collected in Table I. TABLE I. (Values of bx. 105). H = 59 | | || He = 105 (34 ; 14) | | Li = 145 | C= 100 N=8 | O=70 | F=55 || Ne=76 (75) (60) (80). | Na = 270 | Si = 155 P= 144) $=125 Cl=110 Ar = 144 | Fe 115 K = 480 | Ti=180 | | | Cu = 10, | Ge=210 | As= 195) Se= 180|Br = 165 || Kr — 177 Rb = 580 | Ash | | Ag = 150 | Sn. = 265 |Sb = 250|Te=235| 1= 220 || X= 228 Cs = 710 || Ce = 290 Au=150 Hg=150 Pb=320 Bi=305 Nt = 277 | A 375 | | | | Th = 400 | It will be remembered that these fundamental values present a great regularity. Starting from the carbon group the decrease in every 1) These Proceed. 18, 1220 and 1235; 19, 2, 287 and 295 (1916); 20, 138, 492 and 505 (1917). Cf. also Journ. de Chim. Phys. 14, 3 (1916), and Zeitschr. f. anorg. u. allg Chem. 104, 56—156 (1918). 645 horizontal row of the periodic system towards the right is 15 units, in every vertical row the increase downward 55 units. The elements of the valence-less Helium group evidently fit in with the elements of the preceding Halogen group in a natural way. In this Helium group the values of bx are directly calculated from those of 7% and px; also the value 59 for H from that of H, ’). The other values have been calculated from the compounds of the different elements (for Cl the value 115 seems to be more satisfactory than 110). From 77, and pj; we can directly calculate for N, P, O, Cl from N,, P,O, and Cl,: 6 = 86, 135, 71 and 113 to 125 (according as for Cl, the data of Dewar or the more recent ones of PELLATON are used). All the values of 6 can now be built up additively by the aid of these fundamental values for the most divergent compounds. In this also the “condensed” values 34 and 14 are valid for H; for C, N, and O the condensed values 75, resp. 60 and 50. The rules holding for this are found in the two cited principal papers of 1916 in These Proceedings and in the Journ. de Ch. Ph. No exceptions have been found to this additive law, those that still existed for a few organic substances (e.g. the amines) have now all disappeared, owing to the later determinations of Tr, and p‚ by BerrHouD *) at Neuchatel, undertaken expressly in connection with these deviations. The critical pressures determined earlier by Vincent and CHApPpuIs appeared to be all faulty to a high degree *). It is to be foreseen that this will also be the case with other earlier determinations. We will also state that the values of 6 for H, Li, Na, K, Rb and Cs are to each other in the ratio '/,:1'/,:2'/,:4:5:6. The values of 5 found are entirely independent of the state in which the atom is: whether as free atom e.g. in metallic tin, or as part of a molecule as in SnCl,. It will presently appear that this is not the case with the fundamental values of Va. 2. For the values of 10?\’a, per Gr. atom, again expressed in ‘normal’ units, were found up to now the values recorded in table 1) Of course the latest data were used for this; for H, and Ne the values found very recently by K. Onnes, CROMMELIN and Carn. *) Journ. de Chim. Phys. 15, 3 (1917). 3) Thus py, appeared to be =55,5 for NH .C,H;, whereas V. and Ch. found 66; for NH(C,H;), the value 36.6 was found, against 40 by V. and Ch.; etc. Also for C,H;Cl BertHoup found 45,2, whereas V. and Ch. found 49. Errors therefore from 10 to 200/,! 646 II; the values printed in bold type refer to the increased values of the free atoms, the others to those which are found in compounds : hence the rest-values. TABLE II. (Values of Wax. 102). | || Hee | He=08 (3,2; 1,6) | Li=? \C=3,1 N=29 | O=27| F=29 | Ne=2,l 23 3 | Na = Si P.= 6,45), sS:=-Gs. | ACI Sn Ar=52 27 | 34 | Kf iti =? 33 || 35 | | Ge=7,| As = | Se 0 ler ae Kr = 6,9 Pe 3G a] Rb=? |Zr=? 35 37 | Sn =?| Sb=89 | Te=9 1=88 X=9,1 | KE) 33 | Gs | Ce =? 38 | | 39 | Hg = 10,7 Pb=? Bi= 11 |_Nt=115 | | si] | | ea = tak ol | | 41 | | Not from the fundamental values of a can the values of a of all possible compounds be built up additively — but those of Wa from the fundamental values of Wa given in the above table. Here too no important deviations were found; those for the amines have again disappeared for the greater part through the later determinations of Berrnoup (loe. cit). The values of Waz for the noble gases (calculated directly from those of 7 and p;) fit in again perfectly with the values holding for every horizontal row. [t may be said of them that (starting from the carbon group) the values of Waz in compounds are about the same for every horizontal row of the periodic system, and can be represented in approximation by the whole numbers 1. 3. 5.7.9. 11 from the 0. to the 5. row (inclusive). The values of a, calculated for N, P, O, and Cl directly from N,, P,, O, and Cl, are resp. 2,6, 6,4, 2,6 and 5,4 to 5,8 (Dewar or PELLATON). 647 For H in eompounds only the values 3,2 and chiefly 1,6) hold. The value 1,1 calculated directly from 7}, and ps of H, is only found for H, itself. 8. However — what is self-evidently not found for 6 — it should be taken into aceount that for the fundamental values of Wa the indicated attractions can be entirely or partly destroyed by an interception of the rays of attraction. Thus the value of Va of a central atom surrounded on all sides by atoms or atom groups ewer. Omni, Cali, ele, ol Mm Piel, on and Gein Snil, and GeCl,, N and P in NH, and PH, — is found = 0 every where, so that these enclosed atoms do not exert any attractive action towards the outside *). But as soon as the C-atom gets partly free again, by double bindings e.g. as in C,H, (ouly imagine the position of the C-atom stereo-chemical), the value of Way rises immediately from 0 to 1,55, hence half the fundamental value 3,1. And when the C-atom gets entirely free, for triple bindings e.g. as in C,H,, the full value 3,1 is accordingly at once found. This rule holds everywhere. An interesting example is the zso- amylene. Here there are three singly bound and two doubly bound C-atoms. Hence the value 3 x 0 +4 2 « 1,55 +10 X 1,6 = 19,1 is calculated for ag. From 7, and pz 19,2 was found. Indeed, an exceedingly remarkable confirmation of the rule. The same thing holds for benzene and naphtalene. But for the singly bound C-atoms in the substitution groups the old value 0 is again valid. Thus for Toluoene = C,H,.CH, the value of Waz is = —6 « 1,55-+1xX0-+8 < 1,6 = 22,1, while 22,2 is found; for 0-Xylene = C,H,(CH,), we calculate 6 X 1,55 +2 X 0410 XK 1,6= = 25,3, quite identical with the value 25,3 found from 7%, and pr. We might add numerous others to these examples, but we refer for this to the earlier principal papers. We have now reached the main question: what happens when the atoms no longer oecur in compounds, as N in N,, Cl in Cl,, ete. — but ean oecur entirely free, as for the metals, or for the free atoms Cl, Br and | in Cl,, Br,, I,, which get decomposed at 1) Accordingly in consequence of this (see the table) we were not able to give the values of Va, for Si, Ti, Ge and Sn in compounds, though compounds of them are known, of which 7 and ps have been determined. But it is exactly in these compounds (SiCl,, GeCl, etc.) that the attraction of the central atom Si, Ti, Ge, Sn is eliminated. 648 high temperatures, or for H, N and O in the gases H,, N,, O,, which also dissociate at very high temperatures ? Then — and this is perhaps the most remarkable of our results, which is fully contirmed by what follows presently with respect to the heats of dissociation — the values of Wa rise at once to the greatly increased values from 30 to 40 (instead of from 1 to 11). These increased values are therefore the real valency-attractions, whereas the values found in compounds represent only the so-called rest-values: what still remains for action outside, after the chief valencies are saturate, and have therefore become inactive towards the outside. That for the elements of the helium group only the ordinary rest values are found in spite of their atomistic behaviour is of course owing to this that these elements are valence-less. For arsenic only a partially increased value was found, pointing to a dissociation of As, at 7}, to an amount of 20°/,, whereas phosphorus appeared to be still perfectly normal =P, at 7%. For Se and Te we found amounts of dissociation (always at 7%) of 30°/, resp. 80°/,. In the halogen group only a very slight degree of dissociation (5 and 10°/,) was observed for Br, and 1,. As the atoms occur more and more as free atoms, the metal character asserts itself more and more in a group: As—Sb— Bi; Se— Te; ete. Besides the rise from 0 to 1.55 and 3.1 for carbon — according as the C-atom is entirely or only partially shadowed by surrounding atoms or atom groups, to which it is bound — another rise of Va takes place, up to 32, the ten-fold value, when the C-atom has also got released from these bindings, and can occur quite independent as atomistic carbon. Hence the enormously high value of the critical temperature, viz. 6500 abs., which would only have amounted to 120° abs. in the case of C,, as is easy to calculate. The carbon would have become comparable with N,, O, ete., whereas it is now on a line with a metal that is exceedingly difficult to melt. | For Tellurium the peculiarity is still found that the normal rest- attraction 9, which is among others found for TeCl, ‘), has already 1) That for TeCl, the central Te-atom does not exert an intercepting influence like C in CCl, Ge in GeCl,, etc. is owing to the configuration of the molecule. While for CCl, the four chlorine atoms are regularly situated in the space round the C-atom (in the direction of the four angular points of a tetrahedron), it should be supposed that for TeCl, the Cl atoms lie in one plane round the central Te- atom. Nor do we find a complete intercepting influence for PClg and POC, AsH; and AsCl;, SbH3 and SbCl, which for these substances is probably owing to the comparatively greater extension of the central atom. For SbCl; we find even again the full rest value 9 (Cf. the cited papers in these Proceedings). 649 risen to 13 for TeCl, — where two valencies have been liberated. For entirely free metallic Te a will rise still further, namely to about 30. What consequences this behaviour of molecules and atoms with regard to the attraction exercised by them can have for the properties of many substances (volatility, surface tension, etc), has been set forth at length by me elsewhere. *) 4. The heat of dissociation of di-atomic gases. After these introductory remarks we may now proceed to the real subject of this Paper, viz. the calculation of the heats of dissociation Q from the values of the increased valency attractions VA, which were mentioned above, and which are recorded in table If (the values printed in bold type under the elements), for so far as they are now known. Let us take hydrogen as an example. For the internal energy of the not dissociated gas H, we may write: Ey, = (Z,—Q,) Een ~ she CH, TE eee («) in which £, represents the so-called constant: of Energy of the unbound H-atoms, hence H,—Q, that of the bound H,-atoms in H,. Hence Q, is the absorbed (internal) heat of dissociation (in units of energy eg.) in the dissociation H,-—»2H, at 77=0O — ie. the chemical energy (at Z =0), which is liberated in the formation of 1 Gr. mol. H, from the separate H-atoms). For the dissociated H + H we have evidently : 4A mE ert stee Be ewe va" ease RR) While, namely, for H, the quantity a represented the ordinary rest attraction (per Gr. mol.) between the molecules, A now represents the increased attraction of valency between the separate, now free atoms per Gr. atom, hence 44 per Gr. mol. = 2 Gr. atoms. In the zdeal gasstate, where v is very great, we shall simply have: Ha =(£,—,) tent; 2Ha= Ey + 2enT, and as these two quantities of energy will just differ the total (internal) heat of dissociation Q, we have then: Q = 2Eyq—Epn,=Q,+ (cu—ew,)T, (© the well-known expression for Q in the function of 7, when cy represents the limiting value of the specific heat at constant (large) 1) Cf. Chemisch Weekblad, Sept. 14, 1918, NO. 37 (p. 1124—1137). 650 volume of 1 Gr. atom H, and cy, the same quantity for H,. As CH, is= 5, and cy = 3 (in Gr. cal), Q= Q, + T may be written in the case of H, > 2H for Q. When, however, the volume becomes smaller and smaller, and the a A Sony é quantities —and— larger and larger, at last a (fictitious) volume will v v arise, in which the difference of the two energies has become = 0, in consequence of the fact that with respect to the internal energy it will then have become quite indifferent whether the atoms are separately present in that small space, or combined to molecules — i.e. when also the energies of translation do not differ, hence at 7’= 0. EE) NID CIES OOOO Hs H+ H For the difference Q' = 2 — H'y, we have in this case: 44 a be =O, ——+—4+ Cen—eg)T,’ - See M4 vy in which wv, represents the above mentioned small volume, which we shall have to define more closely. Now it follows immediately from (d) at 7'=0, in which case Q' must be = 0, that 4A a Oo se eel He letalsotet. dier ee v, v, aud this is the simple relation between the heat of dissociation Q, at 7'—=0 and the two attractions A and a, which we have sought. We must now determine the small volume v,. This will evidently be of the order of the limiting volume, which the molecules themselves (see the above figure) occupy in the natural state in unconstrained condition, i.e. the volume expressed by 6, — and not e.g. the smaller volume 6, in the liquid state, where the molecules will be compressed in consequence of the smaller space, and which therefore denotes a constrained, and no free, no natural condition *). Now 4, is about = by, so that we may put: v, — bg = yb, ) In this it has also been supposed that by is not = 4m, when m denotes the real volume of the molecules, but simply =m itself. According to recent views the latter supposition is theoretically at least as well justified as the former by = 4m, which refers specially to collisions of mathematical spheres, and not of real 651 in which y(>>1) will not be far from unity. When we put 44 = na, (1) becomes: (n—l)a, (n—-1) ea; Goa ee a ybr in which we put a,, the attraction in the small volume v,, = ak. In this e(©>1) will differ somewhat more from unity than y, so that 27 e will be >y. For ax: bp we may now write gilt To in which y is only little smaller than 1, so that we finally get, when yÀ :e —= @ is put: 27 mn nd rd et oe ee ig TE) which enables us to calculate Q,, when A and a, and in consequence of which also and 7% are known. But as we do not know the values of A for H,, N,, O,, but as we on the other hand do know those of Q, in approximation, we shall follow the reversed course, and calculate n from Q,. We can then see whether the values of A calculated in this way are really of the expected order of magnitude, compared with the already known values of WA for elements of the periodic system lying near (cf. Table II). We then get: 4 Q | a se . ae hence: = ee AA —= 1 = ee ; «( + 57 6 ze) when 2, the value in Gr. cal., is put for R, which renders it possible to leave also Q, expressed in Gr. cal. The value of a refers to 1 Gr.mol. H,. When therefore a' represents the value of one gram-atom (these values are recorded in table II), then a = 4a’, and we get finally : VAp=Var XV1I+ 4/0 S/T, -- - + . (2) when we take A and a’ both at the critical temperature. Accordingly these two quantities now duly refer to 1 Gr.atom. Under the root sign, however, Q, refers to 1 Gr.mol. according to the derivation of the formula. As e will always be >y4, O= yA: € will be somewhat smaller than unity, also when we take into account that R is not molecules (which can differ considerably from the spherical shape, e.g. the elongated molecules of the hydro-carbons). Moreover, when the real size of the molecules is calculated by another method, we also find values that are in concordance with bk (— by), calculated from Tx and ph, and not with 1/4, bk. Cf. among others my Article in These Proc. of Oct. 1914 (Vol. 17), especially p. 883 and the Note on the same page. 652 exactly = 2, but somewhat smaller. The values of WA, calculated from (2) will, therefore, be sooner foo large than too small, when we shall take for the present @=1 in what follows. 5. Calculation of the values of |} 4; from Q,. Let us begin with hydrogen. Isnarpy (1915) determined Q, = 95000 Gr. cal., whereas LANGMUIR (well-known method) gave 132000 Gr. cal. in 1912. But the latter found from 70000 to 80000 Gr. cal. in 1914. We are therefore not very far from the truth, when for H, we put the value of Q, at round 90000 Gr.cal. With 7% = 33,18 (K. Onnes, CROMMELIN, Catu, 1917) Q,: 7; becomes therefore = 2700, hence ‘/,,Q,: Tr == 402. And as Va',=—1,1 (see table II), we get (all the values of a’, and (Ax are still to be multiplied by 10-2): / Ap = 1,1 X 1/403 = 2,08 = 22, which is in very good harmony with what we have found for Li (23) ~ and Na (27) (see table II). For nitrogen Briner') calculates Q, = at least 150000 Gr. cal. from a single value given by Lanemuir. This value of BRINER's seems too high to me considering the values for H, and for O,, and in my opinion 130000 Gr. eal. will be nearer the truth. When we calculate 7 Ar from. the two values” of” Q,, .we get */,, Q,: Ti= Mn an . 126 ) —‘/ an = 176 to 153. And Va; being = 130000 ye a 032 ; — 2,6, we get: | VA = 2,6 X VI = 34,6 or 82,3 =35 or 82. As in connection with C = 32 also a value in the neighbourhood of 32 can be expected for N, the result is also here a confirmation of our formula (1) or (la), the more so when we bear in mind that the result may possibly be slightly too high in consequence of our having put 6= |. . For Oxygen SmeeL*) has found Q, = 160000 Gr.cal. Hence ‘/,,Q,: Tr here becomes = ‘/,, < (160000 : 154.25) = */,, X 1040 = — 154. With War — 2,6 we find further: Ar = 2:6. 0 Isse A which is again in excellent agreement. In the second place we shall examine the Halogens. First of all Chlorine. Pier®) found the value 113000 Gr.cal. for 1) Journal de Chem. Phys. 12, 119 (1914) and 13, 219 and 465 (1915). 2) Zeitschr. f. physik. Chem. 87, 642 (1914). Cf. also Bruner, l.c. 3) Ibid. 62, 385 (1908). 653 Q,. According to PELLaton the critical temperature is 144° C = 417°1 abs., so that ‘/,, Q,: Tx becomes — ‘/,, x 271 = 40,2. Hence with War=5,6 we shall get about (from the critical data of Dewar would namely follow 5,43, from those of PeLLaton 5,75) Ap =O | 41,2 —'30)0 = 36. Though this seems somewhat too high to us, as from 32 to 30 may be expected, the order of magnitude is yet again in agreement with what was found for it for A with other elements. Possibly Q, is somewhat too high, or also VWa%< 5,6; and perhaps in this case — where az: a, will be further from unity than for H,, N, and O,, which have so much lower critical temperatures — 6 will also be so much smaller than 1, that the found value 35 will have to be lowered to + 32. With regard to Bromine Perman and Atkinson’) found for Q, the value 57000 Gr.eal, With 77 3022,2.C = 475,3 abs: we have of ROE T=") lar X 99 = 14,7. We calculate with a,=7: ay ae ee el a value that is very plausible. In conclusion Zodine. Stark and Bonrrsrrin (1910) found for it Q, = 35500 Gr.cal., so that with 7; —= 512°C = 785°, 1 abs. we iti). Pato, =O. fom. Orly. With. ar = 9 ‘this gives : accordingly : AO i 1 Nor is this value, though somewhat small, at all impossible. 6. Conclusion. It has, indeed, appeared very clearly from the above, that the heats of dissociation Q,, on the decomposition of the mole- cules H,, N,, O,, Cl,, etc. into their atoms, are perfectly accounted for by the increased valence attractions |)’ A of the separate atoms found by us in earlier papers. By means of (1) or (1°) we are henceforth able to calculate Q,, when Wa and A are known or reversely to compute V A according te (2), when Q, is known. In GrBBs-PraNckK’s well-known formula for the dissociation of gases *) 1) Ibid. 33, 215 ( (1900). 2) Of course this formula has nothing to do with the so-called theorem of heat of Nernst, as many pupils and followers of this scientist erroneously think (cf. many articles in many periodics). The formula was already given in nuce by GiBBs in 1878, and was later frequently elaborated by Pranck (1887), v. p. Waars (1891), myself (1892), Dunem (1893) and many others for different cases. [Cf. 654 Q ¢ wee bog hk = Sia _ > E € -|- 1) | log T + (v,) logp + 2 (v,C,), (3) in which K represents the constant of dissociation in the relation k,” Kor ERS ] ax I 1—2 4x” ee or di-atomic gases, where 4, = ——, k, = ‘ DE ot 6 RE die eS, will be therefore = K, when « represents the so-called degree of dissociation and #, ete. the so-called molecular concentrations), we cannot only theoretically get nearer to the constants C, etc. (the so-called chemical constants of the components), and so also to (nC) = C — this has been of late done by Lorentz, PrANCK, Sackur, Terropr and others in virtue of considerations of probability in connection with the so-called theorem of heat of Nernst and PrancK’s theory of quanta — but we can also calculate the heats of reaction Q, for 7=0. Up to now we had to be satisfied with determining Q,, just as C= X(r,(,), experimentally from a few values of wv, but now we should be able to calculate the value of log K at given temperature and pressure accurately for every gas reaction, as soon as only the values of the chemical constants and of VA and Va are accurately known for every element separately. This must henceforth be the task of physicists: to get to know these values completely. They are essential for the knowledge of the behaviour of the chemical substances reacting on each other. When we are further acquainted with all the values of 5 for the different elements, then 7% and px are known of every simple or compound substance, hence also their further thermical behaviour. among others my Lehrbuch der math. Chemie (Bartu, 1901), p. 1—13, 25 — 28, and the “Sechs Vortrdge’ (Vtewee, 1906), p. 64 et seq. These latter appeared originally in the Chem. Weekbl. 1905]. Nernst has only said something about the constant &(»,C,;) = C — which is in connection with the constants of entropy — in reference with his theorem. This enabled him namely to bring the said quantities C, (the “chemical constants’’) in connection with the constants of the equations of the vapour pressure at very low temperature. But all this has of course nothing to do with formula (3) itself, which is quite independent of the theorem of heat. The latter says only something concerning the approach to O of entropy, specific heat etc. in condensed systems, in connection also with PrancK’s theory of quanta. When formula (3) (p constant) is differentiated with respect to 7, we get: dlogK\ — Q , =iv,(4/r+))} dT ‚Tart yh d log K Q aoe oe ( AES RT” Q= const. — Qo ar EG + BT, hence ios const. = Oo + X(4 4) T, in perfect agreement with (c) of 8 4. As regards the values of 6, and of a and \/A, I have already started an examination of them, and [ hope I shall be able to continue this work. Whether after all a substance as H, or Na at ordinary tempera- tures and pressures will occur in the form of molecules, e.g. H,, or in that of free atoms, as Na, depends entirely on the values of C and Q, (hence on Ja and WA). The greater Q, will be and the smaller C, the smaller will be the value of A, hence of « — the sooner therefore the substance under consideration will oeeur in the state of molecules, and not in that of atoms. I will still point out that the coefficient of log 7,viz. =|» (7 al 1)]. is erroneously stereotypically put = 1,75 by Nernst, Porrrzer and many pupils of Nernst. For 2HI— H, + 1, this coefficient will be =——27/,+1x'"/, +1 7/,=0 (as is, indeed, sufficiently known), hence not =1,75. For H, ~ 2H it will be = —1 X ?/, + 2 ‘°/,=—1,5, which again is not = 1,75! The same thing applies to many other reactions. It will, therefore, be advisable to determine the value of the coefficient under consideration separately for every reaction. The same thing applies to the constant C. It will not do to assume the chemical constants C, all in the neighbourhood of 3; these will, indeed, also no doubt be different according to whether we have to do with a mon-atomie or a di-atomic substance, which theory indeed confirms. Many values of (/, are inaccurate because they have been calcu- lated from experiments by means of a formula with faulty coefficients (1,75 and wrong values of C); it will, therefore, deserve recommen- dation to calculate for a reaction not only the values of Q,, but also those of the coefficient in question and of the constant C= Xv, C)) from the experiments themselves. Only in this way is it possible to obtain accurate experimental values of Q,. Since in the computations of $ 5 we could not always reckon with perfectly reliable values of Q,, the values found for WA are of course not perfectly accurate. Also in connection with our putting 6 =1, which will also not be perfectly true, the agreement between the found values of A, and those which we could expect (see table Il) in virtue of the values found already (in an entirely independent way) may be considered very remarkable indeed. La Tour pres de Vevey. August 1918. Physics. — “On the measurement of very low temperatures. XXIX. Vapour-pressures of oxygen and nitrogen for obtaining fixed points on the temperature-scale below O° C.” By P. G. Carn. Communication N°. 152d from the Physical Laboratory at Leiden. (Communicated by Prof. H. KAMERLINGH ONNES). (Communicated in the meeting of Oct. 26, 1918). }) § 1. Introduction. The measurements of Comm. Nes 152a and 6 (1917) conducted in conjunction with Prof. KAMERLINGH Onnes (these Proc. XX pp. 991, 1155, 1160) and those of Comm. N°. 152c carried out with Prof. KAMERLINGR OnNes and J. M. BurGrrs (these Proc. XX p. 1663) offered the welcome opportunity of making some determi- nations of vapour-pressures of oxygen and nitrogen using the apparatus and measuring instruments which had been fitted up and used for obtaining and measuring constant temperatures. KAMERLINGH ONNEs and Braak (1908) (Comm. N°.107a, these Proc. XI (1) p. 333) had added to the available fixed points on the temperature-scale the boiling point of oxygen and a couple of lower points, determined under sufficient guarantees as to accuracy. The determinations mentioned above were intended to supplement the latter and might serve to obtain an independent calibration of the oxygen-*) and nitrogen vapour-pressure thermometers, founded directly *) on the temperature-scale of K AMERLINGH ONNes and his collaborators. Prof. KAMERLINGH ONNES was good enough to invite me to avail 1) This paper was originally presented in the meeting of June 1917; in being prepared for the press it underwent some slight modifications on points of secon- dary importance. 2) The oxygen vapour-pressure thermometer is often used in the form given to it by Srock and Nietsen (Ber. d. D. Chem. Ges. 2 (1906) p. 2066) and is then usually simply called Srocx’s thermometer. For accurate measurements it is necessary to resort to more suitable forms of the instrument, such as the one described in this paper. The scale given by Srock and Nietsen is in need of very considerable corrections (Comp. G. Horst, Comm. N°. 148a; these Proc. XVIII (1) p. 829). 8) Von Siemens (vid. G. Horst Comm. N°. 1484) determined the tempe- ratures at which he observed the vapour-pressures by means of a platinum- thermometer, whose readings were reduced to those of Pét;’ of KAMERLINGH ONNES and his collaborators, chiefly by means of four calibration-points. in such a manner that the scales were made to coincide at the boiling point of oxygen as determined by KAMERLINGH Onnes and BRAAK. 657 myself of this opportunity. I am happy to express my sincere thanks to him on this occasion for his assistance | 09 ko in the investigation. *) | $ 2. Details of the determinations. The temperatures were measured with the helium thermometer described in Comm. N°.152a and were reduced in the same way as on that occasion to a scale which approaches the absolute scale as closely as possible (Avo- gadro-scale of helium according to a future communication by Carn and KamerLincH ONNES). The vapour-pressure apparatus agreed in its main features with model A used by ‘KAMERLINGH ONNES and Braak (Comm. N°. 107a), in particular I adhered to the use of the copper tube the purpose of which is to prevent a lower temperature existing anywhere in the apparatus but in the bulb where the liquetied gas collects, at which spot the temperature of the bath is measured. As in the previous experiments mentioned use was made of the advantage offered by the apparatus of allowing different quantities of gas to be condensed in the bulb. As regards the manometer the modification was adopted which Horst introduced in his vapour-pressure measurements of methyl chloride (Comm. N°. 1446 Sept. 1913): the moveable tube connected to the fixed tube by a rubber tube is replaced (see fig. 1) by a fixed manometric tube*) to which a mercury vessel is attached by means of a rubber tube. This fixed tube can be exhausted or it Part of the present paper is embodied in P. G. Carr, Dissertation Leiden 1917, which appeared a few months after the presentation of this communication (see also nole 1). | *) The letters are for the greater part the same as in fig. A Comm N°. 1074, to which we may refer for the further description; a more complete explanation than that given above seems unnecessary. 5) The protection from temperature-changes was obtained by packing in wool. The temperature of the mercury column was determined by means of thermometers suspended beside the tube at the upper and lower ends of the mercury column, 43 Proceedings Royal Acad. Amsterdam. Vol. XXI. 658 can be. connected to a space of constant pressure and a barometer through the stopcock &,. The ground joint ¢ may be used for ex- haustion, for drying with the aid of a moisture-catcher placed in liquid air and for filling. The oxygen was prepared from potassium-permanganate in an apparatus which could be evacuated by means of a Toepler-pump and dried by freezing with liquid air. The gas was condensed in a small bulb which was attached to the apparatus and cooled in liquid air, whence it was conveyed to the vapour-pressure apparatus by means of the air-pump '). The nitrogen was prepared by Dr. LemKEs under supervision of Prof. van ITALLiv, to whom I express my sincere thanks. § 3. Results. The results for oxygen were as follows: | TABLE |. Vapour-pressures of oxygen. p Pp P LE | 4 in int. c.m.2) in int. atm. calculated O—C = | 1 | — 182.62 C. 90.47K. 78.663 1.0350 1.0351 —0.00deg. Il 182.88 90.21 716.560 1.0074 1.0003 + 0.02 mi 183.22 89.87 13.867 0.9719 0.9725 |+ 0.01 IV 183.91 89.18 68.776 0.9050 0.9053 0.00 v 186.91 86.18 49.330 0.6491 0.6491 0.00 VI 192.01 81.08 21.319 0.3463 | 0.343 « 0.00 _ VII 195.50 | 77.59 16.215” | 0.2138 | 0.2137 | 0.00. | VIII __ 201.38 | 71.71 6.401 _0.08423 0.08398 — 0.02 IX 204.52 | 68.51 3.611 0.04152 0.04753 _ 0.00 X 210.72 62.37 0.959 0.01262 0.01286 +0.08 1) One third of the distillate was used for rinsing out the apparatus, the second third for the filling. *) In order to prevent possible misunderstanding it is mentioned here that the pressure is given in “international centimeters” i.e. in the 76th part of an inter- national atmosphere or in other words as the height of a mercury-column measured not at Leiden itself (local cm.), but at a place where one international atmosphere exactly agrees with 76 cms mercury at 0° C., ie. where g = gBur. int. : 1.0003322, the value which is taken as gnorm. and at present is to be put at 980.615. (At Leiden g = 981.276 and 1 int. atm. = 75.9488 cms). For the definition of inter- national atmosphere Comp. Leiden Comm. Suppl. 23. Mathem. Ene. V 10. Einh. a. When in these Communications about observations at Leiden an atmosphere pure and simple is mentioned, the international atmosphere is always meant. 659 The last two columns show, that the observations are very well represented by an beu of the form lgp == Be: B+ CT (p in int. atm.) with A = —419.31, b=5.2365, C= —0.00648, kindly calculated for me by Professor VrerscHarreLT. According to the equation the normal boiling point of oxygen would be ES KE 0 = — 182°.96 C., whereas the observations in the immediate neighbourhood of the boiling point themselves give: PE Kr 6 = — 182°.95 C. This boiling point was found ‘to be 7 = 90°.11 K.') by | TABLE II. Vapour- pressures of oxygen. ‚ Comparison with KAMERLINGH ONNES and BRAAK and with v. SIEMENS (corr). | En ae Tr | | (in atm.) | p (in atm.) | p (in atm.) | Ra | AT ‚(K.O.andB.)| (v. S. corr.) | (CATH) - eke B.—C.) (v.S.—C.) 90.69 1.0618 | 1.0583 | —0.038deg. 90.47 1.036... ). 1.0351, —0.01 deg. 90.21 1.010 ‚1.003 |, 0.00 ... 90.11 "| 1.0002 0.9075. | —0.03 | | 89.87 er 0.9742 nh 0.0725. | 0.01 89.18 0.9059 0.9053 | ET RA 86.56 0.6792 | 0.6777 —0.02 | 86.18 0.6500 | 0.6491 —0.01 83.66 0.4819 0.4808 | —0.02 81.08 | 0.343 | 0.343 0.00 71.59 | 0.2187 | -; 0.2137. | 0.00 71.71 0.08390 0.08308 6,00. |] BAG: peveg | 004695 (0.04753. | 40.05 | | 62.37 0.01272 | 0.01286 | 40.04 1) Comm. N°. 107a gives 7 — Tooc = — 182°.986C. In the computation, however, the pressure-coefficient of hydrogen at 1100 mm. freezing point pressure is taken at 0.0036627, whereas, if we put Tooc = 273°.09 K. and adopt the corrections to the absolute scale given by KAMERLINGH ONNES and BRAAK, the correct value is 0.0036628, which leads to —182.98 and 7'=90°.11 K, 43* 660 KAMERLINGH Onnes and Braak and 7’=90°.12 K by Henning. t). In Table II the results of KAMERLINGH Onnes and BRAAK and those of v. Siemens, the latter after the reduction of his temperature- measurements to those of KAMERLINGH ONNEs and his collaborators has been corrected according to Host’), are compared with the formula which represents my observations *). The results show that except for the lowest points Horst’s reduction of v. Stemens’ observations has been very successful. The results for nitrogen are given in Table III. TABLE III. Vapour-pressures of nitrogen. | No. 3 P P PN | (in Eed (in atm.) | (calc.) | (O—C) 1 — 188.88 C. 84.21 K.| 159.11 | 2.0942 | 2.0948 0.00 deg. ll | 190.66 82.43 133.44 1.7558 1.7539 0.00 | UI 193.91 79.18 93.86 1.2350 1.2382 0.02 IV | 195.84 71.25 75.19 0.9893 0.9904: 0.01 ai es (2 sane | 56.13 0.7385 0.7366 — 0.02 bekur | tapis: | Ssi 36.16 0.4758 0.4756 0.00 VII 204.69 68.40 22.837 | 0.30049 0.30031 0.00 | VIII | 208.58 64.51 12.090 | 0.15908 0.15882 | — 0.01 en Beato are | arg | FK gale at 59.95 4.695 0.06178 | X. 1 2215.20' «| 57789 2.1882 0.03792 | 1) Ann. d. Phys. (4) 40 (1913) p. 635. The theoretical correction to the readings on the scale of the hydrogen-thermometer given by BertHeLot which HENNING applies, does not differ appreciably in this region of low temperatures from the experimental correction found by KAMERLINGH ONNES and BRAAK. All the same it is doubtful whether Hennixa’s value may be looked upon as a final confirmation of the boiling point found by KAMERLINGH ONNES and BRAAK, since his measure- ment contains an uncertainty to which attention was drawn in note 1 to p. 998 of these Proc. XX (2) (Comm. NO. 1524). 3) In this correction Horsr puts the temperatures at the higher points 0.02 and at the lowest point 0.01 of a degree higher than KAMERLINGH ONNES and BRAAK, which is due to the fact that in the reduction he uses his own formula of interpolation instead of the observations themselves. This explains the difference of the deviations at the top of the two columns for coinciding points, although v. SrEMENs in his determinations takes those of KAMERLINGH ONNES and BRAAK as his starting point. 5) As observed in Comm. N°. 107a, there is a systematic difference of 0.10 degree at the boiling point and 0.13 degree at the lower points between the observations 661 The observations I to VIII refer to liquid nitrogen, the last two to solid nitrogen. The last two columns show that for liquid nitrogen the relation between vapour-pressure and temperature can be success- fully represented by an equation of the same form as for oxygen, with A= — 334.64, B= 4.6969, C' = — 0.00476 (pin int. atm); according to this equation the normal boiling point of nitrogen is P= (05.81 1K. 0 = — 195°.78 C. Fiscuer and ALT‘) give — 195.67 on the hydrogen scale, on the basis of « = 0.0036625. With « = 0.0036627 the result changes to —195°.66, whence 6 = — 195°.61 C. With the smaller dimensions of their thermometer a smaller accuracy was to be expected. They give their accuracy as being + .05 of a degree *). | TABLE IV. Vapour-pressures of nitrogen. Comparison with v. SIEMENS (corr.) and with HAMBURGER and Horst (corr). x | p in atm. | pin atm. pin atm. | pul hes swede | (v.S. corr.) | (H. en H.) | (Carn) | (v.S—C) (HHC) | | | 8421K.| 2.00 2.0948 —0.02 deg. | | 82.43 1.756 1.71539 0.00 | 80.93" 1.4983 1.4894 —0.06 deg. | 79.18 1.236 1.2382") 0.0100. «| | | zes | 1.156 © 1.1561 1. 1460 —0.03 | 17.25 0.9895 0.9904 0.01 74.83 0.7331 0.7366 0.04 72.15* 0.5222 0.5143 | ~0.06 11.54 | 0.4706 _ 0.4756 0.09 | 69.34 0.3484 0.3439 0.028 68.40 0.2952 0.30081 0.10 | 64.51 | 0.1546 0.15882 0.14 | 50.95 | 0.058 | 0.06178 «0.17 67.89 | 0.03618 | 0.03792 | 0.19 ! by TRAVERS, SENTER and JAqgurRop (London Phil. Trans. A. 200 (1902)) and those of KAMERLINGH ONNES and BRAAK, the latter being lower. The former are therefore now only of importance historically. 1) Ann. d. Phys. (4) 9 (1902) p. 1149. *) The uncertainty in their determination seems to have been larger, for as 662 The two measurements for solid nitrogen give the relation 358.73 MOND a + 4.7769 ; as holding near the triple-point; this combined with the equation for liquid nitrogen gives for the triple point (i.e. the point of inter- section. of the two vapour-pressure curves). T = 63°.23 K, p = 0.1269 int. atm. — 9.64 int. cm. whereas v. Stemmns, account being taken of Horst’s corrections, found 7 == Gor soo ae p= 9.35 cm. In Table IV v. Siemens’ results — after correction according to Horst of the reduction to the scale of KamMerLINGH ONNES c.s. — and those of Hampurcer and Horst to which I have applied a small cor- rection’) are compared with the equation which represents my observations. Finally I have tried to connect my results to CRoMMELIN’s measure- ments between the boiling point and the critical point’). The latter may be represented satisfactorily *) by the equation *) regards my measurements it seems improbable that the uncertainty in the deter- mination of the temperature, independently of systematical errors, has been larger than about 0.01 of a degree tef. table III), and — in view of the recent deter- minations with the instrument, which will be published in the next communi- cations — the systematical error of the Leiden gasthermometer (apart from the effect of errors in the expansion of the glass and in the constant for the capil- lary depression of the mercury) also may b* put at about 0.01 of a degree. With regard to the differences in Table IV it should be borne in mind, that the observations of v. Siemens as well as those of HAMBURGER and Horst are based on the readings of a platinum thermometer, which has not been compared directly with the gasthermometer. They cannot serve therefore to estimate the systematical error in the temperature-determination. [This note has been somewhat modified in the translation). 1) HAMBURGER and Horst (these Proc. XVII 1 (1915) p 872) obtained their results by reducing the readings of their platinum-thermometer to the vapour- pressures as found by KAMERLINGH ONNES and BRAAK (comp. note 2 page 660). As the authors also give the temperatures at the oxygen-pressures observed by them and as at 78°.42 K. their temperature is 0.05 degrees higher than my results for oxygen which are at present the only direct measurements of sufficient accuracy in that region, I have applied a correction of that amount to their temperatures. These are marked with an asterisk. 2) Comm. NO. 145d, these Proc. XVII (2) p. 959. 3) We leave out of account the reading at 81°.21 K., which GRomMELIN himself marked as inaccurate by placing it in square brackets. 4) The agreement with this formula containing two terms of the expansion of Tlogp in powers of T is not inferior to what is obtained by means of three terms of the expansion in !/7 as given by CROMMELIN. The latter formula, however, _663 T log p= A+ B(T—T,) + C(T—T;," where A= 190.86, B= 3.9649, C=0.00100 for p in int. atm. This equation is again due to the kindness of Prof. VeRSCHAFFELT. Table V gives the comparison between observation and calculation. For 7’= 81°.87 K. the equation gives p = 1.6363 (CROMMELIN), whereas p = 1.6584 (Carn) is found from the equation which represents my observations. The TABLE V. Vapour-pressures of nitrogen (CROMMELIN). | bi p (in atm.) p calc. Il. | AT(O—CIl) | 125.96 K. | 33.49 | 32.728 | —0.09 deg.! 124.24 | 30.364 30.219 0.00 | 120.98 | 25.889 25.892 | 0.06 117.62 21.820 21.906 0.04 111.78 15.949 | 15.993 |-—0.01 | 99.51 7.3105 7.3652 q08 93.91 4.8218 4.8467 — 0.06 | | 90.62 3.7248 8104870 egi62 tS 86.21 2.5067 2.5016 difference between these two values (corresponding to A Tc = .10 of a degree) is so large, that for the present it seems impossible to combine the observations above the boiling point with those below it in one and the same equation. satisfies the condition of giving the correct critical pressure at the critical point. The present formula gives p,, = 32.728 atm. instead of the observed value pj = 33.49. Chemistry. — “On Phenyl Carbamime Acid and its Homologues’’. By Prof. F. E.C. Scuzrrer. (Communicated by Prof. Bérsexkun.) (Communicated in the meeting of September 29, 1918). 1. Introduction. It was observed by Drirrr in 1887 that under high pressure and at temperatures lower than room temperature aniline with carbonic acid can react under formation of a solid compound which consists of equal molecular quantities of aniline and carbonic acid *). It may besides be inferred from his paper that unmixing takes place at ordinary temperature. Some years ago Dr. J. J. Porak carried out a number of experiments with the same system of substances in the organic chemical Laboratory of the Amsterdam University *); he too succeeded in ascertaining the existence of a compound, and the result of his analysis pointed to the same composition as was given by Dirre. It further appeared in his researches that the compound melts on being heated in the presence of a liquid and a vapour rich in carbonic acid, before the meniscus liquid-vapour disappears, with formation of a second liquid layer; this suggested the thought to me that the system aniline-carbonie acid would present an analogy in its behaviour with Bakuuis RoozeBoom’s gas-hydrate systems and with the system sulphuretted hydrogen-water, the phenomena of which I have fully described in These Proceedings *). It will appear from the below-mentioned observations that the compound, which in my opinion is to be considered as a carbaminic acid, gives rise in the P-T diagram to the appearance of a quadruple point, where solid compound, two liquid layers, and gas coexist, and that the three-phase lines which intersect in this quadruple point, can be determined with sufficient accuracy. This system. also furnishes a new application of the quadruple point rule, drawn up by me in 1912 *), which was described by SCcHREINEMAKERS in the Zeitschrift für physikalische Chemie almost at the same time ®). 1) Compt. rend. 105. 612. (1887). 2) Not published. The results of his research have been kindly put at my disposal by Dr. PoLak, for which I gladly express my indebtedness to him here. 3) These Proc. 18. 829 (1910/11) and 14 195 (1911/12). 4) These Proc. 15 389 (1919/18). 5) Zeitschr. f. physik. Chem, 82. 59 (1913). 665 I further extended this investigation to the three toluidines, and [| have succeeded in determining the limits of existence of three compounds. I have determined the composition of two of these compounds by analysis; the composition of the third had already 20 Fig. I. been found by Dirrs. The great analogy between the three systems will probably justify the conclusion that these compounds are the three isomer tolyl carbaminic acids. 2. The system aniline-carbonic acid; the phenyl carbamine acid. From “anilin purissimum” of Merck coloured red by contact with the air a middle fraction was separated by fractionation ; a slight quantity of this was put into the Cailletet tube by the aid of a long glass capillary. As a test tube I used a tube with a widened upper end of the same shape as in my researches on the systems ether-water and hexane-water ’). For the filling with carbonic acid the test tube was connected with a ground piece to an apparatus consisting of a generating apparatus for carbonic acid, which was obtained from diluted sulphurie acid and sodium bicarbonate, and was dried with phosphorus pentoxide, 1) These Proc. 15. 380 (1912/13), 2) These Proc. 16. 404 (1213 14). 666 an apparatus for a high vacuum, consisting of a vessel with cocoa-nut carbon and a GerssLeR tube, a tube for condensation of the carbonic acid (by the aid of liquid air) and a vessel of about 1, liter capacity as carbonic acid reservoir’). As it was exclusively my purpose in these experiments to determine three-phase pressures, a determination of the concentration of the mixtures used was superfluous. For the first observation | used a mixture with great excess of carbonic acid. When the test tube had been screwed on to the pressure cylinder after the filling, and the mixture had been heated to the ordinary temperature, it appeared that the solid substance could be kept at ordinary temperature only under high pressure. When the available volume was so small that there was only a small quantity of gas present, then a three-phase equilibrium of compound by the side of a thinly liquid layer (rich in carbonie acid) and gas occurred at the ordinary temperature at a pressure of about 50 atmospheres. When the pressure was diminished, the liquid vanished with violent boiling, and solid remained by the side of gas. Below 30 atmospheres the solid substance decomposed with formation of a liquid rich in aniline, a strong generation of gas being perceptible in this layer. Accordingly the solid compound is decomposed into liquid and gas on decrease of pressure. It is clear that the pressure at whjeh this decomposition just sets in indicates the three-phase coexistence of compound, liquid rich in aniline, and gas. On increasing enlargement of the volume there remains coexistence of liquid by the side of gas. At the ordinary temperature the existence of phenyl carbaminic acid is, therefore, only possible at pressures above about 30 atmospheres. Hence in perfect analogy with the gas hydrates the dissociation tensions of this compound are three phase equilibria. This applies also to the determinations which Drrrw carried out by observation of the pressure at which gas begins to form from the crystals, or of the pressure at which this generation ceases, which are of course theoretically the same, but practically different according to Dirrr *). Dittr ascribes the latter to inaccuracies of the temperature deter- mination, in my opinion the slowness of the transformation S > L + G is undoubtedly responsible for this. In my former researches on the system sulphuretted hydrogen-water I have also been able to observe such a slow transformation °). In order to be able to determine the three-phase pressures 1) Cf. also These Proc. 18. 880 (1910/11). 2) loc. cit. 5) loc. cit. 667 accurately, I have adjusted a wide cylindrical vessel narrowed at the lower end round the test tube, in which vessel alcohol was stirred by means of vertically moving leaden plates; the heating took place electrically; cooling was effected by introduction of solid carbonic acid. In the observations with small volume it now appeared that the maximum temperature at which the compound can exist by the side of gas, is about 18°. The quadruple point lies at this temperature; the pressure is about 52 atmospheres. In this point there is intersection of the three-phase lines S (compound), L, (liquid rich in carbonic acid), and G (gas), which is stable at temperatures below the quadruple point, and cannot be prolonged above the HE WE ee quadruple point, S + L, +G, which exhibits a value of —- which a rapidly increases with the temperature in the neighbourhood of the quadruple point, L, + L, +G, which indicates stable equilibria above the quadruple point, but can also be easily determined below the quadruple point; then these equilibria are, however, metastable with respect to the solid phase. The fourth three-phase line S+L,+L, rapidly moves from the quadruple point to higher pressure. The situation of the three-phase lines is indicated in fig. 1 by the letters given above; the quantitative data have been collected in table 1; they have been obtained with two mixtures; one contained a great, the second a small excess of carbonic acid; the determinations carried out with the two mixtures, are in good agreement. When the figure is consulted the phenomena deseribed in the beginning of this paragraph will be clear. As long as gas is present, the solid compound can only exist for pressures which are higher than the three-phase line SL,G. Dirrn’s determinations, which are indicated both in the figure and in the table by the symbol D, appear to depart perceptibly from mine; only in the neighbourhood of O° do the observations agree fairly well. It makes the impression that Dirre has determined the points where solid substance is formed on increase of pressure, and that the pressures have been found much too high through the retardation of the transformation L GS, though Drrre mentions that he observed the pressures at which the generation of gas ceased. In this respect the phenomena are again in perfect analogy with the system sulphuretted hydrogen- water, where CaiLuerwr and Borvet’s observations present analogous deviations with mine.’) Dirre does not lay claim, however, to great accuracy for his observations; he states that his determinations give 1) These Proc. 13. 833, fig. 2 and table on p. 834. (1910/11). 668 only rough values, but that they may yet give an impression of the way in which “aniline carbonate” dissociates. TABLE 1. LG(CO.) Eis SL,G 7 P T P T P ot | 0 odd 8.7 41.8 0 6 D 5.0 39.0 9.85 42.8 0.95 1.4 10.1 44.6 10.0 43.2 2 9D 15.0 50.2 10.9 43.9 5.0 10.9 20.0 56.6 11.5 44.8 5 17D 25.0 63.4 11.85 45.1 7 28 D 30.0 71.4 13.25 46.6 8.1 15.0 31.1 72.9 13.95 47.4 9.7 17.5 a Tat 16.0 49.7 11.6 21.7 wae 16.95 50.8 13.5 26-71 ry il P BOO hi 51.8 15.5 34.6 adden ¥ 19.7 54.1 15.9 36.4 —0.6 Al Bieke 1007 55.4 17.1 43.8 +2.2 35.7 | 215 56.3 17.6 48.3 41 38.1 25.15 61.1 Piss an 7.4 EUS ele 603 62.7 Stile 9.0 fee 80525 ralen = wai 10.8 site ati lle 300 aen 12.8 46.5 | 35.8 71.1 18.0 | 52 13.4 47.3 37.15 79.5 182074 = lee RO 14.8 48,6). ae eee On een Quadruple point 18.09 52.0 atm. 16.95 50.9 When we pursue the three-phase line L,L,G towards higher temperature, the fluidity of the upper layer becomes greater and greater, and at 37° the critical phenomenon presents itself; the critical end- point lies 6° and about 7 atmospheres higher than the critical point of carbonic acid. We further derive from the figure that the three- 669 phase tension L,L,G begins to depart more from the carbonic acid tension with increasing temperature; at the quadruple point the deviation amounts almost to 2 atmospheres; at the critical point of carbonic acid to about 3'/, atmospheres. It follows from the already mentioned quadruple rule, which 1 formulated before as follows: The region that does not possess metastable prolongations of threephase lines in the P.-T.-projection, is that of coezistences of phases of consecutive concentration '), that the region between SL,G and SL,L, satisfies the above mentioned condition. In this region, which besides by the two mentioned three-phase equilibria is also bounded by SL,G and L,L,G resp. by SL,L, and SL,G, the coexistences occur of the two phases which the adjoining three-phase equilibria have in common, hence in this case S + L,, L,+G, and S+L,. As these coexistences according to the rule mentioned must refer to phases which succeed each other in con- centration, the succession is GL,SL,; the concentration of the com- pound lies, therefore, between that of the two coexisting liquid layers. Hence the transformation SL, + L, takes place on the three-phase line SL,L, in the neighbourhood of the quadruple point. 3. In order to get acquainted with the concentration of the compound Dr. Porak has caused a weighed quantity of aniline to act on an excess of carbonic acid in a fused-to tube at the ordinary temperature. After the compound had been formed the tube was opened again at — 80°, and placed in a bath of about — 60°; after half an hour the tube was again fused to, and weighed after having been heated to the ordinary temperature. This analysis yielded the concentration C,H,NH,.1.01 CO, I have carried out three analyses in a way that differs but little from that described here; the method of investigation was the same as that which | have described in my second paper on the system sulphuretted hydrogen-water*); the excess of carbonic acid was sucked off at — 80° by means of a waterjet pump. For the quantity of carbonic acid in gramme-molecules which combines with one mol. of aniline, was found successively 0.98, 0.99, and 0.98. The compound consists, therefore, of an’ equal number of molecules of aniline and carbonic acid. d. The system o-toluidin-carbonic acid; the o-tolyl-carbaminic acid. Also in the system o-toluidine-carbonic acid I have been able (o as- certain the formation of a compound; the quadruple point lies here, 1) loc. cit. 4) loc. cit. however, at lower temperature. As the inquiry into the equilibria LL a Fig. 2, at low temperatures is attended with experimental difficulties, | have rested satisfied with the determination of the quadruple point and of the three-phase line L,L,G with the critical end point. The found three-phase pressures are recorded in table 2 and indicated in fig. 2 Fe TABLE 2. L,L,G ik P T P oO ear | 19-4, ele BI Hide Bb ae OEE 3.9 36.8 BATE Ne GOA Kete RT 29.6 67.1 BiB, dia 42E 34.5 | 14.9 18 | 44.7 38.1 80.8 15.6 | 49.0 Quadruple point — 7,5° 27.5 atm. by crosses. At temperatures below the quadruple point a compound again occurs, which has also already been observed by Dirrr, and which according to him consists of equal molecular quantities of o-toluidine and carbonic acid. I myself have not determined the concentration of this eompound; the application of the above de- scribed method of analysis is accompanied with pretty great diffi- culties at the low temperatures. The possibility that the solid substance should be pure o-toluidine is excluded, because the quadruple point lies at higher temperature than the melting-point of the pure substance. (Cf. table 5). 5. The system m-toluidine-carbonic acid, the m-tolyl carbaminic-acid. There occurs a quadruple point SL,L,G in the system m-toluidine- carbonic acid at a temperature which lies between that of o-toluidine- carbonic acid and that of aniline-carbonic acid. The three-phase lines have again the same relative situation as was described above. The observations referring to the three-phase lines L,L,G, SL,G and 51.,G are recorded in table 3 and indicated in figure 2 by triangles, In this figure are also found the vapour tensions of pure carbonic acid (see table 1). 6. The analysis of the compound according to the method TABLE 3. L1G | SL,G | SL,G . | A ek P IJ | P 1 P DN 7.5 40.5 LEG 10.4 — 3.4 30.8 10.0 43.1 dengerink dies Late 33.4 13.2 46.4 2.9 23.1 + 1.9 35.3 15.5 49.2 5.0 31.5 2.9 36.2 18.0 52.1 4.7 31.7 20.9 55.7 5.7 38.6 23.9 59.5 ve rol Ga _colbeiiiet a 21.35 64.2 30.3 68.5 33.7 13.8 | Quadruple point 6.39 39.2 atm. 35.85 17.4 672 . mentioned in § 3 yielded varying values in contrast to those of aniline and of p-toluidine, as will appear in § 8. For the quantity of carbonic acid which combines with one molecule of m-toluidline, was found successively 0.76, 0.79, 0.85, 0.86, 0.88, 0.88, 0.93, 0.89, 0.86, 0.92, and 0.89 mol. These values for the carbonic acid content are most probably all too small. The cause of this deviation is in my opinion the following: Liquid carbonic acid and m-toluidine are little miscible. Accordingly the formation of the compound on cooling takes chiefly place on the boundary of the two layers. In consequence of this a partition of solid substance is continually formed, separating the two layers. Part of the toluidine can, therefore, be withdrawn from the action of carbonic acid. It is clear that after the excess of carbonic acid has been sucked off, the quantity of bound carbonic acid is found too small. In order to render the formation as complete as possible, the tubes were kept in ice for several days; in the successive determinations this period increases from 2 to 10 days. It appears, therefore, that the time has not much influence on the result of the analysis. In the last determination the tube was cooled for 7 hours with ice and salt (—15 to —20°); it also yields too low a result. The supposition that the m-toluidine should be impure, appeared erroneous, as the correct value of 65° was found for the melting-point of the acete compound. | think I am justified in concluding from the above-mentioned determinations that the compound likewise consists of equal molecular quantities of toluidine and carbonic acid. 7. The system p-toluidine-carbonic acid ; the p-tolyl carbaminie acid. In the P-T-diagram the system p-toluidine-carbonic acid yields a three-phase line L,L,G, which deviates little from that of the said systems. (Cf. fig. 3). The quadruple point SL,L,G lies here at higher temperature; the four three-phase lines which intersect in this quadruple point, are indicated in fig. 3; the quantitative data in table 4. The relative situation of the four phases is the same here as in the preceding systems. The stable part of the three-phase line SL,G terminates at lower temperature in a second quadruple point SSpL,G (Sp is solid p-toluidine). In the preceding systems the corre- sponding quadruple point lies at lower temperature and pressure; in -this system the temperature of the two quadruple points differs little from the critical temperature of carbonic acid. The three-phase lines SL,G and SplL,G are easy to determine, when we heat at constant pressure and read the temperature at which liquid is formed. Without further examination we may state about the three-phase 673 line SSpL, that the slope will be steep. Of three of the three-phase lines that pass through the quadruple point SSgl,G the situation Fig. 3. could, therefore, easily be given. The fourth three-phase line SSpG was, however, difficult to find. The quadruple point rule mentioned, however, gave me an indication where it was to be found. There exist two possibilities for the situation of this fourth three-phase line, which are represented by fig. 4a and 6. The three-phase line Fig. 4a. Fig. 45. SSpG must namely lie between the metastable prolongations of SL,G and Spgl,G (fig. 4a) or between those of SL,G and SSpl, (fig. 45). Other situations are impossible, because else two-phase coexistences 44 Proceedings Royal Acad. Amsterdam. Vol. XXI. would occur with an angle larger than 180°; I have set forth in the mentioned paper that this is impossible. When with the aid of the quadruple point rule we examine what succession of the phases would appear according to fig. 4a, it appears that no metastable prolongations occur between SSgL, and SL,G, that the two-phase coexistences in this region are: G + 5,:S + 1, and Li, + Sg and that the succession of the phases is given by GSL,Sp. In an analogous way it would follow from fig. 46 that the order of the phases would be SGL,Sy. This order indicates diminishing carbonic acid content, because Sg represents ‘solid p-toluidine. That the compound S would be richer in carbonic acid than the gas phase, whieh practically consists of pure carbonic acid, is excluded ; p-toluidine has a very slight vapour tension (b.pt. 200°) at this temperature, and the content of p-toluidine in the vapour is, therefore, very small. The only possibility is, therefore, given by fig. 4a. I have, therefore, TABLE 4. SL,G | L,L,G SL,G T P T P 7. P A 33.9 32.8 72.0 30.2 47.7 45 4 39.1 33.9 73.9 30.8 52.5 9.6 43.6 35.05 15.1 31.1 57.3 13.1 47.6 37.75 80.4 31.3 62.2 17.5 52.7 ECT ee 65.0 19.8 55.5 Seb mon ann 23.3 50.8 a - ds sela = 25.6 62.4 = = 27.0 64.3 21.5 24.2 adden 29.7 67.7 | 23.3 21.6 30.3 43.0 30.8 69.0 25.8 33.4 32.0 38.2 |. 2739 38.8 34.0 32.5 | 36.2 26.8 de Sete ea ee paral . i Sf Quadruple point SL,L,G 31.5° 10 atm. SSpL,G 29.7° 44 atm. tried to find the required three-phase equilibrium in the region between 675 the metastable prolongations of SL,G and SgL,G. The determinations were difficult; the only way to find the equilibria was by examining whether rise of pressure or descent of pressure takes place at constant temperature after some lapse of time. It is clear that below the three-phase line SSpgG (fig. da) the two-phase coexistence Sg + G is found, and above it S+G and 5 + Sp, because S with regard to its concentration lies between G and Sp, and the coexistence of the two solid substances will extend towards higher pressure *). Hence the transformation Sg + G 25 occurs on the three-phase line. The upper arrow indicates the conversion on diminution of volume, the lower one on expansion..When at a definite temperature a fall of the pressure takes place, we are above SS, G; when the pressure increases, we are below SS; G. In this way an upper and a lower limit was found, which were no further apart than one atmosphere, sometimes some tenths of an atmosphere. The slowness of the transformation rendered this method of working necessary ; the lower limit was found to yield values which were better reproducible than the upper one. The explanation of this is in my opinion to be found in the fact that the transformation S— Sg + G takes place more easily than the opposite one. This is self-evident, as the action of G on Sg can exclusively take place on the boundary of the two phases, and formation of a phase S can stop the action. Accordingly the values of the lower limit are recorded in table 4; besides, the upper limit often differs no more than a fraction of an atmosphere from the lower one, as has been said. The relative situation of the three-phase lines is actually that which was predicted with the aid of the quadruple point rule. 8. The analysis of the compound did not present any difficulties. The results of the analyses were resp. 1.00, 0.97, and 0.99 mol. CO, to 1 mol. p-toluidine. Hence the compound contains equal molecular quantities of the two components. 9. Summary of the results. The four examined systems yield pretty well coinciding three- phase lines L,L,G. The critical end-points lie close together. The great difference. between the systems consists only. in the situation of the quadruple points. In table 5 the four systems are arranged 1) This can also be immediately derived from the relative situation of the three- phase lines in fig. 4a. 44* according to ascending quadruple point temperatures. The order in temperature is the same as that in pressure. TABLE 5. Quadruple points SL,L,G. o-toluidine m-toluidine aniline p-toluidine Melting points. m-toluidine | o-toluidine aniline p-toluidine — 30 | —15 —T 44 | This is indeed also necessary on account of the coincidence of the L,L,G lines. The order of the melting-points of the pure components deviates from this only in so far that m-toluidine and o-toluidine have exchanged places. I have determined the four melting-points given in table 5 myself. That of m-toluidine was not known, as far as I could find out; that of o-toluidine agrees with the observation by KNOEVENAGEL !). For the melting-point of aniline we find —8° given; my value lies somewhat higher; Timmermans’ value lies again higher than mine ’). I think, however, that I may conclude from the small melting-range presented by my preparation, that the substance was pure. We find 45° for the melting-point of p-toluidine in the handbooks; my value is lower, and agrees with Hurert’s very carefully executed determi- nation (43, 9°) ®). The compounds that occur in these systems, contain the components in the ratio 1: 1. I think, therefore, that I have to consider them as carbaminie acids. These compounds were still unknown, only Ditrr has evidently observed two of them in his experiments. It is clear from the limits of stability of the compounds, why they have not been found; at ordinary temperature only two of these compounds are possible (phenyl- and p-tolylearbaminie acid). The first decomposes directly into liquid and gas, the second into solid p-toluidine and gas, when the tubes are opened. 1) Ber. 40. 517. (1907). KNOrVENAGEL finds besides the melting-point of —15.5° another belonging to a metastable modification (—21°). TrmMERMANs’ determination (—24.4°) may refer to this metastable modification. 4) TIMMERMANS. Bull. Soc. Chim. Belg. 27. 334. (1914). 3) Hurerr. Zeitschr. physik. Chem. 28. 650. (1899). 677 At low temperature they can all exist at ordinary pressure, but the formation will be hampered by the afore-said reason that the compound can put a stop to the action of the two phases (gas and solid) on each other by separation. It is worthy of note that evidently through the action of aromatic amines on carbonic acids free acids are formed in contrast with ammoniae and the aliphatic amines, which form salts. Of the said carbaminic acids a few salts are known. When it is tried to obtain the free acids by double conversion with acid, they split up into carbonic acid and the free amines; only at high pressure or low temperature could the free carbaminic acids be formed. Yet it is possible, and even probable that these acids, though they are durable at the ordinary temperature only under increased pressure, occur in the liquids L,. The assumption that for certain reactions the carbaminic acids can act as intermediate product, is therefore certainly not to be deemed impossible. It appears from the P-T-diagrams that the quadruple point of p-tolylearbaminie acid lies higher than that of phenylearbaminic acid. Possibly the quadruple point of one of the xylylearbaminic acids lies at still higher temperature. The as-o-xylidine certainly invites to further investigation, the quadruple point will probably have shifted here to higher temperature, as the melting-point lies higher than that of p-toluidine. It is possible that for this system the quadruple point has already disappeared; we should then pass to another type of binary systems; in this case the behaviour will become analogous to that of sulphuretted hydrogen-ammoniae, the particulars of which J deseribed on an earlier occasion. ') When we think the quadruple point gradually removed to higher temperature, it will disappear when it coincides with the critical end-point. A gradual change is not to be realized, as the change in constitution takes place discontinuously. It is here, however, possible that by suitable choice of the homologues the displacement takes place in small leaps, and the transition of the type presented by these systems into that of sulphuretted hydrogen-ammoniac appears very clearly. In his thesis for the doctorate BücuNer already pointed out the existence of such a transition for systems without compound. I shall describe the pbenomena which present themselves in these trans- formations in a later treatise. Delft, August 30% 1918. Technical University. 1) Thesis for the Doctorate (1909). Zeitschr. physik. Chemie. 71. 214 and 671. 1910). Physics. — “Experimental Inquiry into the Nature of the Surface- Layers in the Reflection by Mercury, and into the Difference in the Optical Behaviour of Liquid and Solid Mercury”. By Dr. J. J. Haak and Prof. R. Sissinen. (Communicated by Prof. H. A. Lorentz). (Communicated in the meeting of September 29, 1918). 1. Introduction. Since in 1850 L. Lorenz’) advanced the suppo- sition, that the elliptical polarisation on reflection by transparent bodies is the consequence of a gradual transition between the two adjoining media and elaborated this view theoretically, the influence of these surface layers has been more than once theoretically inves- tigated, both in case of the reflection by transparent bodies and by metals ®). There have, however, been made only few experimental investigations into the nature of these surface-layers and our know- ledge of it is confined to more or less plausible suppositions. Great influence is always assigned to the grinding and polishing and also to the grinding and polishing material itself, with which these ope- rations are made. An-investigation by RayLeen shows, however, that it is not yet possible to state in what way this influence arises *). Besides, an influence of the condensed gas layers has often been supposed and examined *). Up to now however, attempts to demon- strate the influence of a condensed gas layer have not yet succeeded. 2. Purpose of the research. In order to obtain the optical con- stants of a metal, quite independent of the grinding and _ polishing and the material used for this purpose, mercury was chosen for this investigation. Both the liquid mercury and a mirror of solid mercury can be examined. It becomes then also evident, whether at the transition from liquid to solid mercury the optical constants are subjected toa modification. In the investigation of liquid mercury the impression, 1) L. Lorenz, Pogg. Ann., 111, 460, 1860; 114, 238, 1861. 2) C. A. VAN RIJN VAN ALKEMADE, Thesis for the doctorate, Leiden, 1882; Wied. Ann., 20, 22, 1883. P. Drupr, Wied. Ann., 43, 126, 1891; R. CG. Mac Laurin, Proc. Roy. Soc., (A), 76, 49, 1905. 3) RAYLEIGH, Proc. Roy. Inst., 16, 563, 1901. 4) J. J. Spepeck, Pogg. Ann., 20, 35, 1830; P. GLAN, Wied. Ann., 11, 464, 1880; R. SisstncH, Thesis for the doctorate, Leiden, 1885; Arch. Néerl., 20, 171, 1886. 679 however, gained ground, that the layers of air, which are condensed on the surface, exert an appreciable influence on the elliptical polarisation at the reflection, so that in the first place this influence has been more closely examined. 3. The used monochromator. The determination of the optical constants took place in an entirely analogous way, as has been de- scribed by one of us *). For the investigation a goniometer has been used, the graduated circle of which can be placed vertical. Before the goniometer: there is a monochromator of a very simple structure, which was constructed from material, present in the laboratory. The monochromator consists of a collimator with an aperture 1 : 6, a flint- glass prism of Sreinnein with an angle of refraction of 60°, and a second collimator, which will be referred to in future as the colli- mator of the goniometer. The illumination takes place by means of an are-lamp of 18 Amperes. A lens of 7 dioptries forms an image of the crater on the slit of the collimator of the monochromator. Behind the prism a lens of 11 dioptries forms a spectrum on the collimator of the goniometer. The axes of the two collimators are placed horizontal, the slits and the edge of the refracting angle ofthe prism vertical. Care has always been taken, that the image of the crater and the spectrum fall on the middle of the collimator slits. A silvered glass mirror is adjusted. to the collimator of the goniometer; it can revolve round an horizontal axis and throws a monochromatic, cylin- drical beam of light af the required angle of incidence on the mirror in the middle of the goniometer. The wave-length of the incident rays lies between 5790 and 5990 Angstrém-units. During the observations the invariability of this colour-sifting of the incident beam of light is repeat- edly examined. The fringes in the Basinger compensator are always uncoloured. Monochromator and goniometer are mounted on a firm foundation, erected free from the floor in the room, in which the observations have been made. The three levelling-screws of the legs of the goniometer stand each on two thick pieces of india-rubber, in order to prevent as much as possible the influence of vibrations on the liquid mercury surface. 4. The goniometer. For the adjustment and the centring of the parts of the goniometer we refer to the investigation of one of us ’). Only a few points are briefly mentioned here. 1) R. Srssineu, loe. cit. 2) R. SISSINGH, loc. cit. 680 The following expedient proved very convenient in the mutual adjustment of the parts of the goniometer. The sledge on which the silver mirror stands, which served for these adjustments, was fastened to a socket, which fits over a conical pivot in the middle of the goniometer circle. This pivot can be levelled with three adjusting screws. The mirror with the socket can in this way easily be placed on the goniometer and can be removed from it’). As polarizer a nicol is used with pretty large oblique end-planes, which only gives a small deviation to the rays of light, which pass through it. This amounts to 2’.5. For the adjustment of the compensator we refer again to the investigation of one of us’). t The polarisation planes of each of the two compensator wedges are placed in the required position, i.e. parallel to the plane of incidence and normal to it *). To bring the movable wedge in the required position, we make use of the images of the collimator slit, which are formed in the eye-piece behind the goniometer. There are formed three pairs of images. The images of each pair coincide, if the planes of the wedges are parallel. The principal positions of the nicols, in which their planes of polarisation are parallel to the plane of incidence or normal to it, have been determined both in the vertical and in the horizontal position of the goniometer circle. The azimuth of the polarizer is called 0, when the light, that the polarizer trans- mits, vibrates normal to the plane of incidence, that of the analyzer, when the direction of vibration of the transmitted light is pallallel to the plane of incidence. We obtained successively in the horizontal and vertical position of the goniometer circle: mean Polarizer in azimuth 0° 81°57’, 81°43’; 81°50’ Analyzer „ ” O°. «86°S3’; … 86°35"; OBB E AE Considering the inevitable errors of observation, the agreement may be called satisfactory. 1) For fuller details compare J. J. Haak, Thesis for the doctorate, Amsterdam, 1918. 2) R. Sissineu, loc. cit. 3) It is noteworthy, that it is supposed both in the investigation of R. Hennie, Gott. Nachr., 18, 365, 1887, as in that of P. Drupe, Wien. Ann., 34, 489, 1888; 86, 532, 1889; 39, 481, 1890, that the principal sections of the wedges are normal to each other and only the angle between the principal section of one of the wedges and the plane of incidence is determined and taken into account. Cf. SISsINGH in Bosscua’s Textbook of physics, Light, II, p. 555, note 2. 4) All the experiments have been made by J. J. Haax. Compare for further details Mr. Haak’s thesis for the doctorate, Amsterdam, 1918. 651 The positions of the movable compensator wedge, in which the difference of phase for the narrow beam of light, which passes between the threads before the fixed wedge, amounts successively to — },0, + 4, are: F 63.86 ; 49.52; 35,10: As the displacements of the wedge are always reduced to those between 49.52 and 63.86, a displacement of 14.34 mm. corresponds with a phas-edifference of 4 or zr. *) The angle of incidence on the mercury surface is derived from the position, in which the line of sight of the telescope of the goni- ometer is horizontal. This position is found halfway between the posi- tions, in which that line of sight runs successively parallel to that of an incident beam of light and of the corresponding beam of light, which is reflected by a liquid surface in the centre of the goniometer. These positions are 81°36’ and 104°24’, so that the axis of the telescope of the goniometer runs horizontal in the position 81°36’ + (104°24’— —81°36’): 2 = 93°0’. As reflecting surface is used that of thick machine oil, because this gives rise to a pure image of the slit, which is not the case with mercury. *) The goniometer circle is vertical, when the axis of the incident beam coincides with that of the tube in which later the polarizer is placed and after reflection with the axis of the telescope. 5. Preliminary investigation of the optical influence of a condensed layer of air. When, as was already communicated in § 2, it became evident from the observations, that the layer of air condensed on the mercury surface was optically active, the angles of principal incidence Z/ and the principal azimuth /7 were determined on the mirror of pure mercury, immediately after the formation. This took place by the determination of the phase-difference and the restored azimuth for two angles on both sides of the angle of principal incidence from extensive series of observations. From these values those of / and H were obtained by interpolation. *) It was found that J = 79°18’, H = 35°45’. This determination could be made within three hours. On an earlier occasion we obtained / = 78°23’, H = 36°18’, in which the mercury surface was exposed to the air for some days, 1) The phase-differences are given as phase-retardations with respect to the light vector in the reflected hight, perpendicular to the plane of incidence. *) For the mercury surface the compensator fringes remain however straight, only slightly less sharply defined. The accuracy of the adjustment of this fringe is somewhat less than with «a solid surface, viz. 0.03 instead of 0.02. 8) R. Sissineu, loc. cit. 682 but protected against contact with dust, before the observations took place. In both cases the mercury had been distilled in vacuum. With mercury that had been purified by being shaken with potassium hydroxide and nitric acid and had then been dried, we found in the same way some days after the formation of the mercury surface : : = 78%14/5.43\ dhe Gra aye which is in satisfactory agreement with the second determination. The sign of the difference between the values of / and #7 in the two first determinations is the same as is to be derived from the theoretical research by Drupe. A surface layer greatly diminisbes the angle of principal incidence and enlarges the principal azimuth but little’). The following observations show the influence of the layer of air very clearly. The pure mercury is in a well cleaned receptacle of LeysoLtp with plane parallel side walls, as is very often used for light- filters. The observations have been made at an angle of incidence of 78°, with the analyzer at an azimuth of 45°”). Adjustment of the Compensator Polarizer 56.46 *) 43°33’ Two days later, during which time the mercury was protected from dust, 56.71 43°47’ After the mercury in the receptacle had been shaken and a new surface had been formed: 56.44 43°33’ After one day 56.64 43°44’ After shaking 56.42 43°34’ In a following set of observations, in which a pure surface was obtained by means of the method of overflowing of RÖNTGEN ‘), whieh has also been used by Ray.eieH, the observations yielded : Adjustment of the Compensator Polarizer 56.43 43°44’ After two days 56.74 43°41’ 1) P. DrupE, Wied. Ann., 36, 532, 865, 1889; 43, 126, 1891. : *) At this azimuth the error in the determination of the phase-difference intro- duced by the metallic reflexion and of the azimuth of the restored plane polarisation, which henceforth will be referred to as restored azimuth, is a minimum. Cf. R. StssinGu, Thesis for the doctorate, p. 57; Arch. Néerl., 20, p. 138. 3) These and all the further values are every lime the mean of four readings. 4) R6NTGEN, Wied. Ann., 46, 152, 1892; RayLeiaH, Phil. Mag. (5), 30, 398, 1890. 683 After overflowing 56.45 43°37’ After one day 56.68 43°44’ After overflowing 56.45 43°50’ The polarizing action of the side walls of the glass receptacles not having been examined, these values have no absolute value, but they represent the change by the adsorbed air layer no doubt very accurately. 6. Change of the thickness and the influence of the layer with the time. There is no doubt but the thickness of the adsorbed layer of air and so also its optical influence increases with the time. In order to examine this influence the adjustments must be effected in a short time. Those of the polarizer have been omitted, because it appears from § 5, that the influence of the layer of air on the restored azimuth is very small and about of the order of magnitude of the errors of observation. In order to be able to execute the observations in a short time, only observations at one angle of incidence, viz. the angle 79°46’, which is very near the angle of principal incidence, were made, the analyzer also always being set in the same quadrant. The determination of the phase-difference, that arises at metallic reflection between the components vibrating perpendicular to and in the plane of incidence, took place only by annulling this phase difference ‘). The shifting of the movable compensator wedge was therefore exclusively from 49.52 — 63.86 (see § 4). In this way the errors in consequence of the deviation of the light in the polarizer and the inaccurate position of the planes of polarization of the compensator wedges continue to exist, but their influence on the slight change in the phase-difference, that is to be determined, may be considered as of the second order of magnitude. Care should, however, be particularly taken, that the incident beam of light consists always of the same part of the spectrum and keeps the same direction, i.e. always falls on the middle of the slit of the collimator of the goniometer. A slight shifting of the spectrum, which was not even so much as the height of the spectrum, already ‘ modified the compensator-reading by 0.06. This is to be ascribed to the change in the angle of incidence. In the observations the mercury was placed in a shallow iron dish, attached to the bottom of a bronze cylinder. Two side-tubes, closed by plane parallel glass plates, the axes of which lie in a same meridian plane 1) SISSINGH, Thesis for the doctorate, p. 79: Arch. Néerl., 20, 196, 1886. 684 of the cylinder, enable us to make the light strike the mereury at the required angle of incidence. The mercury is conveyed into the dish by a tube in the upper surface of the cylinder. This bronze cylinder is attached to the middle of the goniometer circle. A vertical sledge makes it possible to place the mercury surface so, that the axis of the goniometer lies in it. By means of an horizontal sledge the dish may be placed so, that the axes of the incident and the reflected beam of light pass through the middle of the glass windows. The cylinder can be exhausted and filled with air, that has been dried with caleium-chloride, sulphuric acid, potassium-hydroxide and phosphor-pentoxide. When the position of the movable prism of the compensator, immediately after the formation of the mercury surface is called c,, that ¢ seconds later c, and the final position c,, a very plausible supposition on the increase of the thickness of the layer of air and its optical influence leads to the differential equation : d(c—c,) di. o? == hie —E) So that Ct, == (ee) U — e—kt), In this /% will be proportional to the pressure of the air. This supposition is confirmed by series of observations, made at a pressure of one and of half an atmosphere, which are graphically represented in fig. 1. There have been traced six curves for different values of 4. It appears clearly from the traced lines within what limits the value of & lies for the two series of observations. As value of c,—c, has been taken 0.25, viz. the change of the readings in dry air after 24 hours. When the cylinder is exhausted, no change in the compen- sator-readings can be demonstrated even after 8 hours. Compare the line..., which indicates the observations in the exhausted cylinder. Here follow the means of three series of observations in dry air for a pressure of an atmosphere. Time Angle of Incidence 79°46’ 0 Cicer C—C, > hour C = 56.863 0,023 sne 56.89 0.05 14 hours 56.92 0.08 Zar 56.918 0.078 TL 56.94 0.10 ie 56.965 0.125 Bh, 56.975 0.135 A, 56.972 0.132 685 44 hours 56.97 0.13 ieee 56.992 0.152 by ee 56.99 0.15 Gone 57.00 0.16 Ghia 57.01 0.17 ath, 57.013 0.173 Akties, 57.03 0.19 Se 57.03 0:19 id A=25.25e 25E AN ee Ad 225.250 9b pina 4 ESE Lee a eer Pe Rene As tele A=25_25e "St AA len En L Fe iad ag ad Ne hy BEE AT Tae 4 dd A=25-250 °°7° ‘ de Pt ‘ Eee AE aka ee EEE alie lake uae 4 mis ; time i in ei ona 5 8. hours Fig. 1. When it is borne in mind, that the error in the readings of the compensator amounts to 0.03, the agreement of the observations with the curves traced may be considered as very satisfactory. As according to § 4 a displacement of 14.34 of the movable compensator wedge corresponds to a difference of phase of 2 in circular measure, c,—c, denotes a difference of phase of 0.25: 28.68 = 0.0087. In the following way the change in the angle of principal incidence /, corresponding with this, could be obtained. 686 The measurements on pure mercury, immediately after the formation of the mirror, gave: Angle of Incidence Compensator Angle of Principal Incidence 78°38’ 56.24 19197 Observations made at the same angle of incidence on the mirror with an adsorbed layer of air yielded: 78°38’ 56.90 16°19": It follows from this, that if the compensator-reading in the neigh- bourhood of the angle of principal incidence diminishes by 0.66, the angle of principal incidence increases by 59’, so that it follows from the observed value c,—c, = 0.25, that the adsorbed layer of air decreases the angle of principal incidence by 227.5. It is to be doubted very much whether the change, which has been observed for crystals on natural cleavage surfaces, shortly after the splitting, in the phase difference between the components of the reflected light, which vibrate normal to and in the plane of incidence, should be attributed to layers of air. Drupr found this change for fresh cleavages surfaces of antimony glance, calespar, and roek-salt. *) For rock-salt the cause is not to be looked for in a layer of water on the hygroscopical crystal. The observed change in the phase difference is greatest for antimony glance at the angle of principal incidence. According as the optical axis of the crystal is parallel to the angle of incidence or normal to it, this change amounts successively to 0.01 and 0.06. This value is many times greater than has been observed for mercury. Besides the greater part of the change has already taken place in 2 hours and the retardation in the increment of the ellipticity on standing is much more considerable than for mercury. DrupE considers fresh cleavage-planes as unsaturate and thinks that also a greater conden- sation of the gas layers would have to result from this. Experiment should decide, however, whether gas layers play a part also here. Fresh cleavage-planes of lead glance do not exhibit a change in the elliptical polarisation with the time. 7. On the changes in the phase-difference obtained in a previous investigation. It appears from the observations mentioned in § 5, that changes of nearly 1° in the angle of principal incidence caused by surface-lay ers were observed. Hence the question is, what gives rise to these greater changes? Very probably these occur in consequence of liquid layers, which are enveloped by the dust falling on the mercury. When 1) Drupe, Wied. Ann., 34, 489, 1888; 36, 532, 1889. 687 some dust is swept up from the floor and some of it is strewn over the mercury, the displacement of the compensator fringe can immediately be observed without any adjustment. No liquid is, however, to be observed on the mercury either with the naked eye or with a telescope. When the dust is taken from a place, where oil has been spilt, the compensator-fringe assumes a tortuous form and liquid streaks are to be observed on the surface by means of the telescope. The sinuous compensator-fringe indicates, that not everywhere an equal quantity of liquid is spread over the mercury. That in this case really a liquid is spread over the mercury surface, is also in agreement with the fact, that the mercury surface is smoother, so that an image may be observed in the telescope of the thread, stretched across the centre of the slit of the collimator of the goniometer. This does not succeed with a clean mercury surface in consequence of the vibrations caused by the traffic in the streets. 8. On the values of 1 and H for mercury without surface layer. It follows from § 6 that for a pressure of air of one atmosphere, the curve for £=0.79 best represents the observations, so that the compensator reading successively increases by 0.045, 0.08, and 0.11 in 1, 2, and 3 hours or on an average by 0.04 an hour. As the determinations of / and H, mentioned in $ 5, took up three hours, / has been diminished by 6’ on an average in this time, the etl azimuth MZ has, however, remained unchanged. The change in A lies namely within the errors of observation. From this follows, that for mercury without adsorbed layer of air: f= 1921 H = 35°45’ We subjoin the values obtained by other investigators: BREWSTER *) 78°27’ 34°46’ QUINCKE °) (afters 33°47’ Des Couprrs *) Lon 37 33°30’ Drupw *) 79°34’ 35°43’ MEYER *) 78°23’ gs pe Merse *) 79°22’ 3627 In this the following points are noteworthy. BREWSTER gives 26°0’ as restored azimuth after two reflections under the angle of principal incidence. From this the above given value of the Spaerne azimuth ) BREWSTER, Phil. Trans., 287, 1830. 2) QuINcKE, Pogg. Ann., 142, 202, 1871. 3) Des Coupres, Thesis for the doctorate, Berlin, 1887. 4) Drupe, Wied. Ann., 39, 511, 1890. 5) Meyer, Ann. d. Phys., 81, 1017, 1913. 6) MrÊsr, Gott. Nachr., 530, 1913. 688 has been calculated. It cannot be inferred from BREWSTER’s records, whether his observations have been made on a free mercury surface or on mercury against glass. In the latter case scratches in the glass may be responsible for the too low value of the principal azimuth. Also the too low value of the principal azimuth determined by Quincke should be attributed to scratches’). Drs Couprrs’ observations have been made on a free surface. The deviations seem to be owing to inaccuracies in the observations. Drupre obtains a clean mercury surface by the aid of two funnels. No further particulars aré given about this. The method is probably similar to RöNrGEN’s method of overflowing. The observations were made within two hours after the formation of the mercury mirror. QuinckE and Meyer used a mercury surface against glass. A surface layer between mercury and glass has undoubtedly caused the too low value of the angle of principal incidence. Meerse also made observations on mercury against glass and demonstrates the existence of surface layers, that are then present. The values of / and H given here, have been calculated from his observations. In how far a condensed layer of air has also exerted an influence cannot be ascertained. From the values determined by us and those of Drupr, Meerse, and Meyer, whose value for J, which is certainly too low, has not been taken into account, the following values of the optical constants for mercury may be assumed as the most probable: Poste 2a" =S 35°43"; § 9. The thickness of the adsorbed layer of air. Both Van Run VAN ALKEMADE (see $ 1) and Drupr have given equations, from which the thickness of the surface layer may be derived. According to Drupr®) the change in the phase-difference between the compo- nents of the reflected light, parallel to and normal to the plane of incidence, brought about by the surface layer is: : L Ke Bag. = An CosepSin?g (a— Cos’) ie. je dl 2 (a—Cos*g)* +a’, n? In this p is the angle of incidence, L the thickness of the surface layer, n the index of refraction in this layer at the distance / of the reflecting metal surface: Cos 4 H Sin 4 H AZ {UE Sin? [tg? I Sin? Ltg? I. 1!) J. J. HAAK, Thesis for the doctorate, Amsterdam, 1918, p. 47. 2) Drupr, Wied. Ann., 39. 481, 1890. 689 The index of refraction n of the surface layer will increase on approaching the reflecting metal surface. When for n a mean value is taken, viz. that between 7,;, and the value for the greatly condensed air, immediately adjoining the metal and when with QuinckE’) it is supposed, that the density of this is equal to that of the mercury, n = 4.048 is found for this by the aid of the relation: (n—1):d= constant *). Hence the mean value of ” is (4.048 + 1.003): 2 —= 2.52. As A'—A, the phase difference for the absorbed layer of air, amounts to 0.0087 (see $ 6) and g—79°46’ (see § 6), we find with the values of / and H mentioned in $ 8? Bat b pu This value is in agreement with that for the transition layer liquid-vapour, for which BAKKER *) gives J—2 uu. When in the same way the thickness of a layer of oil (n =1.5, d= 0.9) is calculated, which according to $ 5 can modify the angle of principal incidence by 1°, the compensator-reading by. 0.50 and more, we find, introducing the value 0.50, 2=83 uu. This is in accordance with RarYrereH’s and FiscuEr’s*) determinations. RAYLEIGH found, namely, for the thickness of the thinnest layer of oil, that stops the movements of the camphor particles on water, 2uu. Frscner found for liquid layers, which spread over mercury, thicknesses smaller than Suu. As the adsorbed layer of air of a thickness of 1.6 uu changes the compensator-reading by 0.25 and the mean error in the reading amounts to 0.02, a layer of a thickness of 0.13 uu can still be demonstrated in this way by this optical method. Such a layer is of the thiekness of a molecule. It is not possible to prove the existence of such thin layers by the aid of the capillary phenomena. It is not possible to remove the once adsorbed layer of air by means of a very far exhausted vacuum, as the mercury airpump of GAEDE can bring about. After eight hours’ pumping no displacement of the compensator reading could be demonstrated °). 1) QuincKke, Pogg. Ann., 108, 326, 1859. *) L. Lorenz—H. A. Lorentz’s formula cannot be age here, as ”? would Agen i ie? 8 nt? In As be maj + then 5 must be > 1, if „1? wet become negative. Let n, = 5.108, so that the ; B ; d is to be positive. In the case considered here rh 10 condition is not fulfilled. 3) G. BAKKER, Z. f. Phys. Chem., 91, 571, 1916. 4) RAyLriGH, Phil. Mag., (5), 30, 396, 1890; Fiscuer, Wied. Ann., 68, 436, 1899. 5) This result is in agreement -with other experiments, which show with how 45 Proceedings Royal Acad. Amsterdam. Vol. XXI. 690 10. Testing of the obtained results by means of a mercury mirror got by distillation of pure mercury. According to § 6 the phase- difference was measured with the compensator by annulling this phase-difference, in order to determine the influence exerted by the adsorbed layer of air. From the values given in $4 it appears, that the lowest compensator-reading corresponds with the smallest phase- difference. As according to Drupr every surface layer diminishes the angle of principal incidence, hence increases the phase-difference for every angle of incidence, the mercury surface is the better, i.e. less contaminated by surface layers, as the compensator-reading is smaller. For an angle, somewhat greater than /, this smallest reading was 56.84 *). Mercury, distilled in vacuum, which was conveyed into a dish after being filtered through a paper funnel, yields the reading 56.84. This dish had been placed free in the air and was not sur- rounded by a case (see § 6), so that the rays of light need not pass through glass windows. Mercury purified by being shaken with potassium hydroxide and nitric acid, but not distilled in vacuum, yields the reading 56.85. The difference with the preceding value falls within the errors of observation. Mercury, conveyed into the dish through a drawn-out glass tube, yields 57.00, if the tube has not been very well cleaned. For a thorough cleaning heating to a dull red glow is generally sufficient. Touched by a piece of cloth, which is not clean, the mercury gives the adjustment 57.20. When breathed upon, the mercury yields a reading increased by 0.04 or 0.05. In order to prove by another way, that 56.84 is the reading for pure mercury without surface layer, pure mereury was distilled in vacuum into the iron dish, which is situated in the bronze cylinder. It was previously ascertained, that the glass windows — carefully cooled glass plates of a thickness of 3 mm. — did not modify the compensator-reading, even though the cylinder was exhausted of air. For this purpose an iron mirror was placed in the dish and the compensator-reading was observed before and after the ex- haustion. The pure mercury was heated ina glass globe. The vapour was condensed in the spiral windings of a glass cooler and received in a glass bottle, from which it flows out into the iron vessel through a glass tube with drawn-out point. All the junctions of this apparatus consist of sealed glass. The air-tight connection of the glass tube with great a force these adsorbed layers of air are attached to the surfaces. Cf. among others Voter, Wied. Ann., 19, 39, 1884. Likewise it is in agreement with the fact, that the layer of air cannot be removed by means of carbon powder, which has been heated just before to ared glow. Cf. among others SissinGu, Thesis for the doctorate, p. 162; Arch. Néerl. 20, 228, 1886. 1) The values recorded here are again the nieans of four readings. 691 drawn-out point, which passes through the short tube in the upper wall of the bronze cylinder, with this short tube was effected by means of an airpump tube and a little collodion. A GArpe mercury airpump exhausts the space. The compensator-adjustment was 56.84, hence exactly the same as that which prevails for mercury without adhering layer of air, according to the observations communicated in this §. 11. Lnquiry into a difference in the optical constants of liquid and solid mercury. At first the air in the bronze cylinder was very carefully dried. The cooling of the cylinder and the dish with mercury in it took place by putting a mixture of solid carbonic acid and ether on a tin plate screwed on to the bottom of the cylindrical case. By means of ebonite as heat-insulator the conduction of heat from the metallic parts of the goniometer to the cylinder is prevented. In order to prevent the cooling of the glass windows in the side tubes, because no water vapour from the air may settle on them, these windows are cemented to ebonite tubes, which are screwed on to the side tubes of the bronze cylinder. The air-tight closure was effected by means of very tough Ramsay-grease. As it appeared that in spite of this precaution some water-vapour deposits on the glasses, a current of dry air was blown along them by means of a GARDE box pump, which quite remedied this evil. During the cooling the following phenomena are observed. When during the cooling the compensator is adjusted for the dark eompen- sator-fringe, this winds, while the mercury is still liquid and becomes less dark, after which it disappears altogether. At last the fringe, which has then become black again, jumps back to its original position. When the telescope, which is placed behind the analyzer, is then adjusted on the mercury, ice crystals appear to float on the mercury. From this it is evident, that the explanation of the observed phenomenon is the following. In spite of the careful drying the air contains traces of water vapour, which are deposited on the mercury - surface during the cooling and spread over it as a liquid *). The compensator-fringe is probably sinuous, when the water layer consists of incgherent patches and is very thin. When the water forms a coherent layer and this layer has become so thick, that the reflection takes place on water, the fringe disappears, as at the chosen angle of incidence of 79° 46’ water does not perceptibly polarize the light elliptically on reflection. As soon as the tempera- 1) These phenomena are not observed on an iron mirror during the cooling, so that the condensed water vapour does not seem to spread over this mirror. 45* 692 ture falls below O°, the water freezes and contracts to a few ice erystals, which le spread with large interstices on the mercury. Then we have again a mercury surface and the normal adjustment of the compensator-fringe. In the meantime the mercury is still liquid. Attempts to prevent this deposition of the water vapour by careful drying of the air proved unsuccessful. This was done thoroughly however by exhausting the bronze cylinder. When an iron mirror is placed in the cylinder, it appears, that the glasses on the side-tubes do not become bi-refringent when the case is exhausted, but that they do so through the one-sided cooling during the fall of the temperature of the bronze cylinder by solid carbonic acid and ether. Then the adjustment of the compensator-fringe changes by the constant amount 0.08. During the cooling of the mercury in vacuum no sudden change in the position of the compensator-fringe is observed at the moment of the freezing. The reading of the compensator-fringe diminishes gradually, till this change reaches an amount of 0.08, It appears from this that only the slight double refraction of the glasses plays a part here and that neither during the freezing of the mercury, nor during the further cooling down to —80° the phase-difference, hence also the angle of principal incidence of the mercury changes perceptibly. At first the position of the polarizer presented a change during the freezing, which could amount to as much as 3°. This, however, must be attributed to the influence of wrinkles, which make their appearance with the freezing in consequence of inevitable vibrations caused by the traffic in the streets or by tram cars. This is in agreement with the observations of Fizeau, Drupe, and Haak *) on the diminution of the restored azimuth through grooves or scratches in various directions on the reflecting surface. In order to prevent these wrinkles as much as possible, the iron dish is filled brimful with mercury and then the temperature is slowly lowered. When the traffic in the streets is not too great, it is then sometimes possible to get such a smooth mercury mirror, that a pure image of the collimator slit can be observed. In this case the adjustment of the polarizer does not change. It appears from this, that on freezing and cooling — ofthesolid mercury to — 80° neither the angle of principal incidence nor the principal azimuth are subjected to any change. Optically liquid and solid mercury behave in the same way’), 1) Fizeau, Ann. de Chim. et de Phys., 3, 373, 1861; Drupe, Wied. Ann., 39, 497, 1890; J. J. Haak, Thesis for the doctorate, Amsterdam, 1918. 2) Besides it follows from this that LuMMER and SorGe’s supposition (Ann. der Phys., 31, 325, 1910), according to which internal tensions would give rise to the elliptical polarisation, cannot be valid for solid mercury. Chemistry. — “Investigations on Pasteur’s Principle concerning the Relation between Molecular and Crystallonomical Dissymmetry : VIII. On the spontaneous Fission of racemic Potassium- Cobalti-Oxalate into its optically-active Antipodes.” By Prof. F. M. Jagger and WriutaAm Txomas. B. Sc. (Communicated in the meeting of Nov. 30, 1918). § 1. As a continuation of the fissions, accomplished up to this date, of the racemic complex triovalates of potassium and the triva- lent metals: chromium’), rhodium”), and iridium *), — it appeared desirable to make-an attempt to separate the analogously built potasstum-cobalti-oxalate: K,{Co(C,O,),} + 34H,O, into its optically- active components for the purpose of a comparison of their rotatory dispersion and crystal-forms. The series of the complex oxalates investigated, would then be really complete. The proposed separation into its antipodes was, however, hindered till now by a number of difficulties of various kinds, partially caused by the salt being not very resistant towards an increase of temperature, and on the other hand by its particular solubility-relations, when combined with active bases; moreover a troublesome circumstance was its sensitiveness to light-radiation, this causing a rapid decom- position of these salts in solution, under formation of a pale pink, hardly soluble precipitate, — a reaction, the study of which is now started in our laboratory. The racemic salt: ,{Co(C,0,),} + 34 H,O has been studied by Coraux *). It has, like the corresponding salts of the other metals, triclinic symmetry, but it is „ot isomorphous with them, as follows already from the deviating content of water of erystallisation : while the iridium- and rhodium-salts contain 44 molecules of water, the corresponding chromi-salt has three, the cobalti-salt 34 molecules of it. The salt was prepared in the following way in greater quantities. A mixture of 25 grammes of cobalti-carbonate, 250 eem of a satu- 1) A. Werner, Ber. d. d. Chem. Ges. 45. 3061. (1912). *) A. WERNER, Ber. d. d. Chem. Ges. 47. 1954. (1914); F. M. JarGer, Proceed. Ak. v. Wet. Amsterdam, 20. 263. (1917). 3) F. M. JAEGER, Proceed. Kon. Akad. v. Wet. Amsterdam, 20. 273. (1917); 21. 203. (1918). *) H. Copaux, ‘Bull. de la Soc Min. 29. 75. (1906); Ann. de Chim. et Phys. (8). 6. 508. (1905). 694 rated solution of putassium-ovalate, and 230 ccm of a saturated solution of oxalic acid, was heated on the waterbath under continuous stirring, till all the carbonate had entered into solution. The liquid thus obtained was cooled to 40° C, and then 30 grammes of finely pulverised lead-peroxide: PbO, were added. After some time 50 cem of a 50 °/, solution of acetic acid were slowly added to the rigorously cooled solution under continuous stirring. Then the liquid was filtered and precipitated by 400 cem of 97°/, alcohol; the green precipitate was sucked off, and several times washed with absolute alcohol. In this way 80 grammes of the dark green potassium-cobalti-ovalate were readily collected. § 2. The silver-, and the bariwm-salts being both only slightly soluble, we used the potassium-salt itself for the preparation of the corresponding strychnine-compound, thus avoiding the troublesome use of large volumes of solution; this is of advantage, because also at lower temperatures the solutions are partially decomposed under development of carbondioxide. The strychnine-salt *) is for the greater part precipitated, if the calculated amount of strychnine-sulphate is added to the solution of the potassiwm-salt, and only so much cold water is consequently added as to dissolve the precipitate formed. All these experiments were executed in a dark room, where the solution is left standing in an open vessel for several weeks, at a temperature of about 16° C; the fractions successively deposed from the mother-liquor are collected separately. The crystals obtained are treated, in the same way as described on former occasions, with an excess of potasstum-iodide, the strychnine- iodide is sucked off, and the filtrate precipitated by means of 97°/, alcohol. The salt obtained is purified by repeated erystallisations from a small quantity of water. The first fractions of the strychnine-salt in this way gave crystals of the faevogyratory antipode, containing 1 molecule of water of crystallisation. The determination of the water-content cannot be made at 120° C, because of the decomposibility of the substance; it was therefore made by passing a current of dry air at 20° C over the finely powdered substance during a very long time, and a loss of weight corresponding to 0,8 molecules of water was finally observed. With respect to the light-absorption by the dark green solutions, 1) Originally the separation of the racemic salt was tried by the aid of cincho- nine, but without success. Afterwards we repeated these experiments under some- what varied conditions, but they gave no positive results either. , 695 it was found that in layers of 20 cm., a solution of 0,41 percentages by weight showed a pronounced absorption-band in the yellow and blue part of the spectrum between the wavelengths of 5510 and 6520 4.U. For concentrations of 0,82°/, and 1°/, no light was transmitted; but the solution just mentioned allowed the light to come through between 4850 and 5515 A.U. With a solution of 0.31°/, these limits were: 4770 to 5670 A.U., and 6480 A.U.; with one of 0.27°/,: 4720 to 5750, and 6450 A.U.; ete. Determinations corresponding to wave-lengths within these limits can only be made with extremely diluted solutions, and the incertitude of the readings caused thereby may: explain the deviations of the values obtained in the case of the laevo-, and dextrogyratory components, in so far as these values are observed in the immediate vicinity of the deep minimum in the dispersion-curve. But notwithstanding this incerti- tude, the characteristic slope of the dispersion-curve is in all cases fixed with full certainty. For the salt from the first fractions, we found values of the rotation in good agreement with each other, which are suited to elucidate the strange form of the dispersion-curve (fig. 1) immediately : Molecular olalun LL Degrees y 13000° 12009° 11000° 10000° 9000" all 8000: 1000° (bl 6000° “atl. > Upp, ssilt! sooo“ | S* 8 Yolo 8 4000" [SS Ny 3000°) S 2000" N 1000") | ij 0 1000° De: 4900 5000 5100 5200 5300 5400 5500 5600 S700 5800 5900 6000 6100 6200 6300 6400 - 6900 2000°| K 3000" 4600° Ne s000"| = Vavelenthin ‘100 4800 4800 $000 SIOD 5200 $300 5400 $500 5600 5700 5800 $900 6000 6100 6200 6300 6400 — 7000 Cngstrim - Units . Fig. 1. Molecular Rotatory Dispersion of Laevogyratory Potassium Cobalti-Oxalate (+ 1 H,0). 696 MOLECULAR ROTATION OF LAEVOGYRATORY POTASSIUM-COBALTI-OXALATE | | (+ 1 H20). Wave-length in Angström-Units: | a 4730 — 3913° 4780 4031 4870 4196 4945 4399 5020 4619 5105 4916 5180 5123 5260 5487 5340 5900 5420 6387 5515 7086 5610 7805 5700 8682 5800 | 9708 eek | 11327 poze | 12508 6140 — 8506 6260 | 4 263 6380 | 5391 6520 | 4126 6660 | 1799 6800 | + 160 This curve therefore appears to possess the peculiarity, that the rotation at first rapidly increases for greater wave-lengths, but decreases then very steeply in the vicinity of the absorption-band, to assume the opposite algebraic sign at circa 6260 A.U. The right- handed rotation now reached, shows a minimum at about 6400 A.U., and decreases at first rapidly, afterwards more slowly, so that the 697 curve approaches more and more the axis of the zero-value. The dispersion has therefore, properly speaking, only a really “normal” character between 6240 A.U., and 6400 A.U. The maximum of the laevogyratory rotation is situated at about 6000 A.U. On comparing the magnitude of the rotations for corresponding wave-lengths, in the cases of the complex owalates of chromium, cobaltum, rhodium, and iridium, — the influence of the specific nature of the central metallic atom on the whole character, as well as on the absolute values of the rotation, is immediately evident. The figures 2 and 3 will show this clearly; in fig. 2 the curves of the cobalti-, rhodium- and iridiwm-salts are drawn, in fig. 3 those of the chromium-, and cobalti-salts. While the complex rhodium-, and itridium-ovalates show an analogous dispersion, the cobalti-salt seems to have a Moteedar Kelalion tt hyp. d)- usg, ge JN Js WEE 4800 4900 5000 5100 5200530) 5400 5500 5600 S700 5800 5980 ‘ <_— ALY IL th np a | Muse hinglh om 4100 4800 4400 5000 S100 5200 5300 5400 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6100 6800 6900 7000 / Ing Hh Vit My “ls ; Fig. 2. 698 Meluular Yolalu WW 4/1 AUS © 26000" de 24000 22000" 20000" 8 JE | Ë “Aa ail [© Usvxtion 4800 5000 5200 5400 5600 | 5800 6000 JE 17 -hinglh “uw 4600 4800 sooo 5200 S00 5600 5800 6000 6200 6400 6600 (Ygs/irvm Uh Meliular Hellum U bigger if beregoed vv Llasnumdbvrmian rll 0H 0) and Selassium Clair A Jaah (#140) Fig.3. deviating position amongst them. ') However, this salt shows a complete analogy in this respect with the corresponding chromium-salt, as may be easily seen from the measurements made in this laboratory by Mr. P. J. Beeker with the potasstum-chromt-ovalate: K, {Cr(C,0,)33 + 3H,0, which was separated into its antipodes after Werner's method ®). On comparing the rotation of this salt with that of the cobalti-salt at corresponding wave-lengths, it may be seen that the rotations of the chromium-salt are, up to 5640 A.U., and above 1) According to an investigation of G. Bruuar (Bull. de la Soc. Chim. (4), 17, 226 (1915), there is also a maximum in the dispersion-curve of the complex iridium-oxalate at short wave-lengths (about 4930 A.U.). It therefore seems probable, tbat the character of the anomalous dispersion curves is really very analogous in all these cases, however with a considerable difference in the positions of the’maxima. 2) A. Werner, Ber. d. d. Chem. Ges. 45. 3061 (1912). It is difficult to obtain good crystals of this compound which, moreover, rapidly autoracemises in solution ; therefore crystallographic measurements could till now not be made in any way. 699 6280 A.U., considerably greater than those of the cobalti-salt; while between 5640 and 6280 A.U. the reverse occurs, and the two salts even show an opposite rotation as a consequence of the shift of their maximum and minimum; this shift is, in its turn, intimately connected with the very different situation of the absorption-bands : for the chromium-salt has a very broad band in the orange, yellow, partially in the green and the violet. It must, moreover, be remarked, that our former experience has made it clear, that the triethylene- diamine-chromi-salts show in general only about half the rotations of the triethylenediamine-cobalti-salts for corresponding wave-lengths. From this it is evident how great and unexpected an influence the presence of basic groups, or that of acid radicles, has, on the magni- tude of the rotation when they are dissymmetrically arranged round the central metal-atoms of such complex salts. § 8. As was already mentioned before, the racemic cobalti-salt has already been investigated by Coraux, who described it as a triclinic substance, but not isomorpbous with the corresponding rhodiwim-salt. As will be soon demonstrated, Copaux most probably obtained his erystals from solutions evaporated in the darkness and at low tem- peratures; for even in diffuse day-light the solution is decomposed with development of carbondioxide and precipitation of cobalto-oxalate, — a decomposition which is quite analogous to the photochemical decomposition of the corresponding ferric-salt. It must be remarked that in this photochemical reaction, blue light decomposes the solutions much more rapidly, than red or green light, — in full agreement with Drapgr’s law. But it is a remarkable fact that yellow and orange light, which is absorbed also to a considerable amount, has scarcely any stronger influence than the only slightly active green or red rays. In our experiments the saturated solutions were evaporated in a dark space, the temperature of which differed only slightly from 0° C. In this case we really obtained triclinic-pinacoidal crystals of a dark green, almost black colour, and showing in most cases curved faces and rudimentary forms; accurate measurements were therefore very difficult. The angular values obtained really differ not inconsi- derably from those published by Coraux, at least within some zones; but the identity of his crystals and ours need not be doubted in any way, as may appear from the following values: Angular values: Observed: Calculated: JAEGER: COPAUX: mie = (110) (MOPS 60° 37 60° 36’ — c:m — (001): (110) = 79 22 19 26 de 700 q:m = (011):(110) = 65.510 65 1428 — aw = (100): (010) = 89 2 88 42 88° 40’ c:q = (001):(011) = 33 10 32488 — b:c = (010):(001) = 88 ei - 88 39 BS). 262 etc. The specifie gravity of these crystals, which commonly only showed the combination-forms: m = {110}, u = {110}, e= {001}, q = {011}, w = {112}, and a—{100} and b= {010} very narrow, — was at 15° C. determined at: dj = 1,877; the molecular volume is there- fore: 268,14. An analogy of form with the corresponding rhodium-, and tridium- salts is not present; neither is this the case with the monoclinic chromium-salt, which possesses 3 /7,0. *). These facts prove in every case undeniably, that at temperatures in the neighbourhood of 0° C., the saturated solutions deposit crystals of the racemic compound. But the stranger therefore appeared to us originally the behaviour of solutions evaporating at room- temperature. For from an also inactive solution, which during the summermonths was slowly evaporated in a dark room at tempera- tures only slightly differing from 18° C., dark green, almost black needles were obtained, which even on superficial examination appeared to differ appreciably from the triclinic racemate. Crystal- measurements taught us, that they had trigonal symmetry, and that their form was identical with that of the laevogyratory antipode. A erop of small crystals of this crystallisation-product, dissolved in water, did however not show any appreciable rotation. Suspicion immediately arose, that the racemic salt might have been split under these circumstances into its antipodes spontaneously, and that no trace of rotation could be detected according to our way of investigation, only because the solution deposes the crystals of the two antipodes besides each other in about equal number, so that a crop of several crystals, which by the lack of hemihedral faces cannot be discerned from each other, contains in general almost an equal number of dextro- and laevogyratory individuals, when collected from the solution at random. It is evident that such a mixture will not exhibit any appreciable optical activity. If this suspicion were true, the optical activity must appear immediately, if only a single crystal at the same time were dissolved. Indeed, experience proved, 1) The parametres of the cobalti-salt are: u:b:c=0,5963: 1 : 0,6590 ; a = 91942’; B = 101923’; y = 88°22’. The chromium-salt is monoclinic, with: a:b:¢=1,0060: 1: 1,3989; @ = 86°90’. For the rhodium- and iridium-salts, cf. these Proceedings, 20, p. 270, (1917); and 21, 214, (1918). 701 that the first erystal thus investigated, showed the full activity of the devtrogyratory component, which then had not yet been obtained in the pure state by our fission-experiments; the following numbers can convincingly demonstrate the fact mentioned: MOLECULAR ROTATION OF THE DEXTROGYRATORY POTASSIUM-COBALTI- OXALATE (+ 1 H20). | Wave-length in ANGSTRÖM- Molecular Rotation Units: | in Degrees: 4730 | + 3876° 4780 __ 4009 4870 4167 4945 4428 | 5020 4689 | 5105 4923 | 5180 | 5106 5260 5553 5340 6013 5420 6416 5510 7023 5610 | 7916 5700 | 8703 | | 5800 | 9764 | | 5910 | 11365 | 6020 . 12812 | 6140 | + 8269 | 6260 | Ns (I: | eN, 6380 | 5468 6520 | 4317 6660 | 1678 6800 526 6940 | — 198 | | The minimum in the dispersion-curve is here somewhat steeper 702 than our former measurements with the laevogyratory component indicate, — a deviation probably caused by the uncertainty of the readings in the interval of the absorption-band. But the correspondence of the curves cannot be doubted any longer. Continued study of the single crystals deposited from the solution, taught, that besides the lefthanded crystals, also those of the laevogyratory component occur. It is commonly quite impossible to recognise the two kinds of crystals from each other by their ontward appearance, and thus to select them, because the facets of the right- or lefthanded trape- zohedra or trigonal bipyramids are commonly lacking, so that the aspect of the crystals is in both cases quite the same. It cannot be doubted therefore any longer, that we have found here a first instance of a fission into optically active antipodes of such complex metallic compounds, by spontaneous crystallisation ; for the case of potassiwm-rhodium-oxalate formerly indicated by WERNER as an instance of this kind, can no longer be considered as such, as was some time ago proved by us.) For the purpose of justifying this view, it was necessary to deter- mine the temperature of transition of the racemic compound into its antipodes as accurately as possible. This was done in two ways: by means of the dilatometrical method, and by the study of the Sodibily Uv 100 a $f, Jemperature: 0° 3° Gi: i lo. ~~ Wee 21° Fig. 4. 1) KF. M. JAncer. Proceed. Kon. Akad. v. Wet. Amsterdam, 20. 264, 265. (1917). solubilities of the salts at different temperatures. For we have already formerly drawn attention’) to the fact, that below and above the transition-temperature, the metastable forms must have, as in all such cases, the greater solubility; and this was exactly one of the arguments used by us to reject WeRNER’s conclusions about the occurrence of a spontaneous fission in the case of the potassium- rhodium-ovalate. Indeed, our experiments fully confirmed this view: we were able to demonstrate, that below 14° C. the solubility of the inactive form is really smaller than that of the optically-active antipodes, whereas above 14° C. the reverse was the case. Thus 100 grammes of water at 0° C. e.g. appeared to dissolve 3450 grammes of the racemic salt, at 14° C. 36,81 grammes; ete. On the other hand, 100 grammes of water at 20° C. dissolved 37.40 grammes of the laevogyratory salt, at 22° C. 37,6 grammes; etc. The fig. 4 shows, that the transitionpoint to be determined, without appreciable error, may be fixed as 13,2° C.; this temperature, at which the reaction: 2 rac. LK{CoC, Oo} + 34H,0] TER fd KCo(C, 004) + es LO KCH ON ISLA, ONE DAD takes place, is therefore a mendmum-temperature for the existence of the optically-active salts. The dilatometrical experiments were rather difficult, because of the tendency of the compound to decompose, when its solution is kept at somewhat higher temperatures for a long time, and because of the inevitable retardation-phenomena. Notwithstanding this, we were able to prove a sharp discontinuity of the volume- temperature-curve, at a temperature between 12° and 16° C. That such retardation-phenomena really oecur, cannot be doubted; even in solution, the active salt is transformed just below the transition- temperature into the racemic one, with considerable slowness. Thus we found, that at 12° C, the dilute solution of the laevogy- ratory antipode lost in one day about half, in two days two thirds, ‘in three days almost five sixths, and in four days about nine tenths of its original optical activity, while at the said temperature the optical antipodes beyond all doubt are already metastable with respect to the racemic salt. Crystallisation-experiments made in a thermostate at 22° C gave results in full agreement with our conclusions: the solutions deposited 1) A. WERNER, Ber. d. d. Chem. Ges. 47. 1954 (1914). 704 always the trigonal needles of the active components besides each other, but at 0° C we got only the triclinic crystals of the racemate. Therefore complete proof has been given now, that a fission of the potassium-cobalti-ovalate by spontaneous crystallisation into its optically active components really occurs at temperatures above 13°,2 C. § 4. Crystallographical research taught us, that both the optically active components occur in commonly not distinguishable crystals of trigonal-trapezohedrical symmetry, which are completely tsomor- phous with those of the optically-active rhodium-, and dridium-salts. They have the appearance (fig. 5 and 6) of a prismatic forms of more or less extension; the dertrogyratory component hitherto always presented the rhombohedron-like shape of fig. 6. Trigonal-trapezohedrical. a:c =1:0,8968 (Bravais); a = 100°27’ (MILLER). Cc’ Laevogyratory Potassium-Cobalti-Oxalate (+ 1H,0) Fig 5. Forms observed: R= {1011} [100], large and very brilliant; ° ec = {0001} [111], always present, but subordinate ; m = {1010} (21 1], commonly predominant with the lefthanded crystals, and in the case of the dextrogyratory individuals small, but well developed and yielding sharp images; 7 = {0111} [221], and s = {0221} [111], always present, small, but very lustrous; ¢ = {2021} ip it |; rather large and yielding good reflections. Hemihedral combination-forms were hardly ever 705 observed; only once a crystal of the laevogyratory antipode presented the righthanded trigonal bipyramid x = {2241} [715] as an extremely narrow truncation of the.edge A: m. From this it may be concluded, that in this case too, the substance manifests only a very weak tendency to present hemihedral forms, unregarded the enormous optical activity, which these salts exhibit in aqueous solution; and, moreover, that also in the case of the complex ovalates, the same morphological relation between the cvbalti- and the rhodiwm-salts appears to exist, as between the corresponding cobalti- and rhodium- triethylenediamine-nitrates ©, in so far, as the oppositely rotating cobalti- and rhodium-salts, which are separated from the less soluble compounds with optically-active bases (strychnine) or acids (tartaric acid), yet exhibit hemihedral forms of the same algebraic sign: Dextrogyratory Potasstum-Cobalti-Ozalate (+ 1H, 0). Fig 6. | Rotation of the salt separated from the less soluble compound : Algebraic sign of the hemihedral forms present: Substance: | Triethylenediamine-Cobalti- Nitrate. (Chloro-tartrate): d. + sphenoid. Triethylenediamine-Rhodium-Nitrate. | (Chloro-tartrate):1.| __ —+- sphenoid. | Potassium-Cobalti-Oxalate. (Strychnine salt) : 1. + bipyramid. Potassium-Rhodium-Oxalate. (Strychnine-salt): d. + bipyramid. | Potassium-Iridium-Oxalate. (Strychnine-salt): d. + bipyramid. At the same time it is evident that the cobalti- and rhodium-salts 1) F. M. JAEGER, Proceed. Kon. Akad. v. Wet. Amsterdam, 20. 258, 261 (1917). 46 Proceedings Royal Acad. Amsterdam. Vol. XXI. 706 set free from the less soluble compounds with optically active bases or acids, in the case of triethylenediamine-derivatives exhibit just the opposite rotatory power, as is observed in the case of the complex triowalates; a fact, which sustains the view, according to which it is the basic or acid nature of the radicles placed round the central metallic atom, rather than the special nature of the latter, which in the first instance determines the direction of the rotation. Angular Values: Observed: Calculated : ce: R = (0001): (1011) = # 46°95 = m:t = (1010): (0221) = 25 43 25° 461,’ m:m = (1010):(0110) = 60, 0 60 0 R:R = (1011):(1101) = id Was) WO Tt. t2 c:r = (0001): (0111) = 46 8 46 5 r:s = (0111):(0221) = 18”, 0 18 8h s:m — (0221):(0110) = 25 55 25 4615 s:R = (0221): (1011) = 51 13 515 R:t = (Of11): (0221) = 17 58 188 R:m= (1011): (1010) = 43 55 43 55 x:R = (2241): (0111) = 18 13 8 9 No distinct cleavage was observed. The specific gravity of these crystals was at 15° C. determined at: do = 1,8893; the molecular volume is therefore: 242,57, and the topical parameters become: 7: w = 7,4676 : 6,6971 ; or 7’ = 6,3789. The values of 4 of the Co-, Rho-, and /r-compounds appear to decrease therefore continuously with increasing atomic weight of the metallic atom, while w reaches a minimum in the case of the Rho-salt. Groningen, November 1918. Laboratory for Lnorganic and Physical Chemistry of the University. Mathematics. — ‘Ueber eimeindeutige, stetige Transformationen von Flüchen in sich” (sechste Mitteilung')). By Prof. L. E.J. BROUWER. (Communicated in the meeting of November 30, 1918). § 1. In einem in 1912 in den Göttinger Nachrichten (S. 603—606 *)) in Auszug abgedruckten Briefe an R. Fricke habe ich (S. 605, Fussnote®)) kurz skizziert, wie das von Hurwitz herrührende analy- tische Theorem, dass birationale Transformationen einer Riemannschen Fiche vom Geschlechte p> 1 in sich unmöglich ein vollstiündiges kanonisches Schnittsystem der Fliche in ein üquiwalentes kanonisches Schnitisystem iiberfiihren können, mittels der Analysis Situs bewiesen werden kann, wobei sich seine Gültigkeit herausstellt für alle perio- dischen, eineindeutigen und stetigen Transformationen. Die damalige Andeutung wird im folgenden näher präzisiert und gerechtfertigt werden. Sei O die gegebene zweiseitige Fläche, / die Menge der bei der n-periodischen, eineindeutigen und stetigen, die Ränder invariant lassenden Transformation ¢ von QO invarianten Punkte. Wir nehmen an, dass jedes von / in O bestimmte Gebiet von ¢ in sich trans- formiert wird (was, wenn ¢ die Indikatrix von 0 invariant lässt, stets der Fall ist), und unterziehen die Wirkung von ¢ auf eines dieser Gebiete, welches wir mit w bezeichnen werden, einer näheren Betrachtung. Dabei ziehen wir im Falle, dass ¢ die Indikatrix von O umkehrt, jeden eventuellen für ¢ nicht invarianten Rand von w 1) Vgl. diese Proceedings, XI, S. 788; XII, S. 286; XIII, S. 767; XIV,S. 300; XV, S. 352. *) Das daselbst S. 604 auf künftige Publikationen von P. Korse (der Neujahr 1912 im Besitze einer Abschrift meines Briefes an R. FRICKE war) hinweisende Zitat ist nach der Erledigung der Korrekturen von einer mir unbekannten Hand, ohne meine Mitwirkung oder Vorkenntnis eingefügt worden; die beziigliclien Noten sind mir erst nach ihrem Erscheinen bekannt geworden. In engem Zusammenhang mit dem Inhalte meines (Anfang Marz 1912 gedruckten) Briefes an R. Fricke stehen die Karlsruher Verhandlungen über automorphe Funktionen vom Jahre 1911; der über dieselben erstattete Bericht (Jahresber. d. D. M. V. XXI), ist, ebenso wie die in Gött. Nachr. 1912 erschienene KorBE'sche Mitteilung über den Kontinuitätsbeweis, im Sommer 1912 gedruckt worden. Das in diesem Berichte enthaltene Referat über den Vortrag von P. KoeBe (insbesondere die auf S. 162 befindliche Anmerkung !)) ist insofern irreführend, dass im wirklichen KoeBE'schen Vortrage, nach der ihm vorangegangenen, S. 156—157 wieder- gegebenen kurzen Diskussion mit mir, vom Kontinuitätsbeweise überhaupt nicht wieder die Rede gewesen ist. 46* 708 in einen Punkt zusammen und fiigen diesen Punkt zu w hinzu. Sei R die Menge der übrig bleibenden (für ¢ invarianten) Ränder yr, von w und seien (a,,0,), (a,, 5,), (as, 6,),.... die Rückkehrschnitt- paare einer vollstdindigen kanonischen Zerschneidung von w. Wir nehmen an, dass innerhalb jedes dieser Riickkehrschnittpaare ein a, so gewählt werden kann, dass nicht nur a, selbst, sondern auch seine beiden Seiten von ¢ungeachtet der Rander äquivalent abgebildet werden (was, wenn ¢ jeden Riickkehrschnitt der kanonischen Zer- schneidung ungeachtet der Réander äquivalent abbildet und die Indikatrix von O umkehrt, stets der Fall ist). Weiter schliessen wir den Fall aus, dass entweder kein Rüekkehrschnittpaar (a,b) und höechstens zwei Ränder 7,, oder kein Rand r, und nur ein einziges Rückkehrschnittpaar (a, b,) existiert. $ 2. Im Falle, dass in w wenigstens zwei Rückkehrschnittpaare (a,b) existieren, verstehen wir unter 2 diejenige (kein Rückkehr- schnittpaar mehr aufweisende) Schottkysche Ceberlagerungsfläche von w, welche die a, als blättertrennende Ufer besitzt, unter L die Menge derjenigen Ränder /, von 2, welche durch eine unendliche Zahl von Ueberschreitungen der a, auf w erzeugt werden, unter As, al... die Ueberlagerungsbilder von a, auf 2, unter a’, das durch ¢ auf w bestimmte Bild von a,, unter ,a',, ,a',,... die Ueber- lagerungsbilder von a’, auf 2. Wenn wir jedem „a, dasjenige ja’, zuordnen, dessen Umlaufkoeffizienten zwischen den /, die gleichen absoluten Werte besitzen, wie die entsprechenden Umlaufkoeffizienten von ,a,, so ist in Anschluss daran eine durch die Ueberlagerung von @ über w in ¢ übergehende, eineindeutige und stetige Trans- formation ¢’ von @ in sich bestimmt, welche, ebenso wie ¢, n-periodisch sein muss und jeden Rand /, invariant lässt. Ob alle Ränder von @ für t invariant sind, lassen wir dahingestellt, ziehen aber jeden eventuellen für ¢ nicht invarianten Rand in einen Punkt zusammen und fügen diesen Punkt zu 2 hinzu. Im Falle, dass in w nur ein einziges Riickkehrschnittpaar (a, 0) existiert, verstehen wir unter @ diejenige (kein Riickkehrschnittpaar mehr auf weisende) Schottkysche Ueberlagerungsflache von w, welche a als blättertrennendes Ufer besitzt, unter /, und /, diejenigen Rander von @, welche durch eine unendliche Zahl von Ueberschreitungen von a auf w erzeugt werden, unter ,r,,7,.... die Ueberlagerungs- bilder auf 2 eines (für ¢ der Annahme gemäss invarianten) Randes r von w. Alsdann existiert eine durch die Ueberlagerung von 2 über w in t übergehende, eineindeutige und stetige Transformation t’ von @ in sich, welche ‚r (mithin auch ,7, ,”,....) invariant lässt, 709 so dass sie, ebenso wie ¢, n-periodisch sein muss. Ob alle Ränder von @ für ¢ invariant sind, lassen wir wieder dahingestellt und ziehen jeden eventuellen für ¢’ nicht invarianten Rand in einen Punkt zusammen, den wir zu @ hinzufügen. Im Falle, dass in @ kein Rüekkehrschnittpaar (a,, b,) existiert, verstehen wir unter 2 die Fläche w selbst und unter ¢’ die Trans- formation ¢ selbst. In jedem der drei Fälle besitzt die Flache2 wenigstens drei Ränder und ist sie eineindeutiges stetiges Bild eines Teilgebietes der Kugel, während ¢' eine n-periodische, eineindeutige und stetige Transfor- mation von @ in sich darstellt, welche jeden Rand, aber keinen Punkt invariant lässt. Wir dürfen annehmen, dass von den Rändern von 8 wenigstens einer isohert ist. Im entgegengesetzten Falle können wir nämlich in 2 ein Gebiet g bestimmen, zu dessen Grenze kein Grenzpunkt von Rändern von 2 gehört, und welches, ebenso wie seine Komple- mentärmenge, Ränder von @ enthält. Die Vereinigung von g und seinen Bildern für t/,#*,.... #"—! bildet eine Fläche, welche, ebenso wie @, wenigstens drei Ränder besitzt und von 7?’ mit invarianten Rändern und ohne invariante Punkte in sich transformiert wird, überdies aber einen isolierten Rand besitzt. § 3. Seien P,, P,,-...P,—1 die Punkte, in welche ein in 2 will- kürlich gewählter Punkt P von 1?’,¢’?,....¢’"-! der Reihe nach übergeführt wird. Ein P und P, verbindender stetiger Kurvenbogen jp bildet mit seinen von (?’,¢’?,....¢’"—' bestimmten Bildern eine geschlossene stetige Kurve /p. Die Menge Zi’ der Ränder r’, von lässt sich in solcher Weise in eine endliche Zahl > 3 von voneinander isolierten Teilmengen R’, (kp), R’, (kp),.... R' (kp) zerlegen, dass der Umlaufkoeffizient von kp zwischen einem zu R’, (kp) und einem zu PR’, (kp) gehörigen Rand modulo » gleich einem durch vp und 2 bestimmten, für »=A fortfallenden Wert c (kp) 20 und < n ist. Diese R’, (kp) behalten ihre Brauchbarkeit und die zugehörigen cj (kp) ihre Gültigkeit bei, wenn fiir festes P der Bogen jp diskon- tinuierlich geändert wird. Mittels gleichzeitiger stetiger Variierung von P und jp sieht man weiter ein, dass auch durch Aenderung des Punktes P die Rolle der R’, (kp) und c,; (kp) nicht gestört wird. Indem wir aber P in hinreichender Nahe von &’;, und jp in pas- sender Weise wählen, können wir dafür sorgen, dass der Umlauf- koeffizient von kp zwischen einem zu &’,(kP) und einem zu R'kp) gehörigen Rand, für » und à beide von A verschieden, fortfällt und 710 hieraus folgern wir unmittelbar, dass c; (kp) = 0 für jedes v, jedes A, jedes P und jedes jp. § 4. Wir bezeichnen einen willkürlich gewählten isolierten Rand von @ mit 0, die Menge der übrigen Ränder von £ mit A’, und betrachten die sich um f&’ aperiodisch herumwindende Ueberlage- rungsfläche S von $2. Diese Fläche S ist eineindeutiges und stetiges Bild der Cartesischen Ebene; ihr Rand A” enthält einen aus @ hervorgegangenen Teil FA," und einen aus A’, hervorgegangenen Teil R,"; diese beiden Teile sind voneinander isoliert. Sei P ein Punkt von 2 in der Umgebung von g, P, sein von ¢’ bestimmtes Bild. Wir verbinden P und P, in der Umgebung von o durch einen solchen stetigen Kurvenbogen jp, welche mit seinen von ?’,t’?,....¢’"—! bestimmten Bildern eine geschlossene stetige Kurve kp erzeugt, deren Umlaufkoeffizient zwischen zwei willkürlichen Rändern von @ fortfällt. Die Möglichkeit, einen derartigen Kurven- bogen jp herzustellen, folgt aus §3. Sei P,, der Anfangspunkt, figs der Endpunkt eines Ueberlagerungsbildes von jp auf S, so existiert eine durch die Ueberlagerung von S über 2 in ¢ übergehende, ein- eindeutige und stetige Transformation #” von S in sich, welche P,, in Pi, überführt. Alsdann lässt die Transformation #” den Punkt P,, invariant, so dass tf’, ebenso wie tf, n-periodisch sein muss. Hiermit sind wir aber zu einem Widerspruch gelangt, weil eine periodische, eineindeutige und stetige Transformation der Cartesischen Ebene in sich ohne invariaute Punkte nicht existieren kann. § 5. Im Falle, dass ¢ die Indikatrix von QO invariant lässt und von einer vollständigen kanonischen Zerschneidung von O jeden Riickkehrschnitt samt seinen beiden Seiten ungeachtet der Ränder äquivalent abbildet, besitzt ¢ dieselbe Eigenschaft in bezug auf w (was unmittelbar wie folgt eingesehen werden kann: Sei s ein Riickkehr- schnitt von w, der w nicht zerlegt, so entspricht einer stetigen Variierung von s in O, wenn die Ränder von w je in einen zu w hinzuzufügenden Punkt zusammengezogen werden, eine stetige Vari- ierung von s in w). Wäre nun w samt seiner Grenze nicht identisch mit O, so besässe w einen durch eine zusammenhingende perfekte Menge von fiir t invarianten Punkten abgeschlossenen Rand und würde auf Grund davon die in den $$ 3 und 4 hinsichtlich der Transformation # von @ angestellte, auf einen Widerspruch führende Ueberlegung auch im Falle, dass in w kein Riickkehrschnittpaar (as, b) und nur zwet Rédnder existieren, in Kraft bleiben. Mithin ist in diesem Falle w samt seiner Grenze mit O identisch und O entweder eine Kugel, oder ein Zylinder, oder eine Cartesische Hbene, oder ein Torus. Physiology. — “The Siynificance of the Size of the Neurone and its Parts.’ By Prof. Eve. Dusois. (Communicated by Prof. H. ZW AARDEMAKER.) (Communicated in the meeting of October 26, 1918). The existence of definite relations of quantity of the neurone and its parts to the weight of the body is no longer open to doubt. *) For homoneurie species of mammals (species with the same organi- zation of nervous system), whose body weights are to each other in the ratio of P:1, the volume of homologous neurones — as the volume (or the weight) of the brain — varies proportionally to P?5%; the volume of their central part, the cell body proportionally to 205, It may be assumed that the ideal values are P055-- and 0277, These relations can best be verified by a study of the peripheral nerve fiber. As this constitutes by far the greater part of the neurone to which it belongs, also the volume of the homologous peripheral nerve fibers varies pretty accurately proportionally to P°5% for homo- neurie species. And given that the length of the nerve fiber, for perfectly uniform homoneuric species, must vary proportionally to the longitudinal dimension of the animal, i.e. to P88, the conclusion follows naturally that both, the area of the section and the length of the homologous peripheral nerves of homoneurie species, which in reality — for physiological reasons — cannot be perfectly uniform, varies about proportionally to the longitudinal dimension of the animal. For perfect uniformity, the area of the section would have to increase proportionally to P°-6%- and the volume proportionally to P, when the nerve length increases proportionally to P°%%- This now, is physiologically impossible, as may appear in what follows. The available numerical data, considered physiologically, really lead to the conclusion that the length and the area of the section of a nerve increase uniformly, i.e. both proportionally to P02, hence in the same ratio as the cell-body becomes more voluminous. This holds both for the neurones with peripheral nerve fibers, which conduct the influxions centripetally, the sensitive nerve fibers, and for the neurones with peripheral nerve fibers, through which influxions are 1) These Proceedings. Vol. XX. (1918), p. 1828—1337. 712 transmitted from the center outward, the motor nerve fibers chiefly. Thus the volume of homologous neurones increases proportionally to 056, Also for the “volume of homologous neurones in the brain (certainly to be compared, if not morphologically, yet as functional units with the peripheral neurones), the same proportionality may be assumed as for the neurones with.peripheral nerve fibers, as appears from the equal relations of quantity holding for brain-weight. Direct data concerning the relative areas of the sections of homologous nerve fibers, are, however, at our disposal only to a very limited degree as yet. Most of them refer to the eye, more particu- larly to nerve ends, where the retinal area marks the total area of the receptive terminations of a very important group of centripetally conducting nerves. Many of these data have been furnished to us by Laricqte, in collaboration with Laver and Warrrtor *) through measurements of the diameter of the eye in a number of Mammals, Birds, Reptiles, and Amphibians; from these we can derive the relative size of the retina in approximation. Though at first (1908) LAPICQUE, in virtue of these measurements, assumed that the diameters of the eye vary pretty nearly proportionally to the power '/, of the body weight, he later on (1910) considered as more accurate pro- portionality the power '/, of the body weight. According to the first estimation of Laricqur the retinal area would have to vary about proportionally to P28 or P°?5, on the other hand proportio- nally to P27 or P2857 according to his last estimation. The data as such allow, indeed, only estimation of the general result. Some examples may suffice here, derived in the first place from the tables of Laricqur and his collaborators. For the exponent of relation in question, which I shall denote by 7, in the formula P\ro O ae pa log O—log o and Vo = ——————_, log P—ingp where P and p denote the body weights, and O and o the retinal areas (more accurately here the areas of section of the eye-ball proportional to these in approximation) I find the following values. In comparison of the area of section of the eye, computed from the diameter of the eye, of Equus caballus and Antelope (dorcas ?) 0.2643, of Canis lupus and Canis vulpes 0.2668, of Felis pardus and 1) These Proceedings Vol. XX. (1918), p. 1337, Note 1). 713 Felis domestica 0.2390, of Mustela putorius and Mustela erminea 0.3004; as a mean of these three pair of Carnivores 0.2687. Of the Reptiles examined by WarerLor the gigantic Lizard Vara- nus niloticus, with a body weight of 7500 grams and a diameter of the eye of 5.8 mm., compared with the small Gecko Hemidac- tylus Brooki (8 indiv.), which weighs only 4.9 grams and has a diameter of the eye of 4.25 mm. on an average, gives the value 0.2942 for r,. Comparing the same Varan or Monitor with a Green Lizard, Lacerta viridis (examined by Lapicqur) of a weight of 16.8 grams and a diameter of the eye of 5.8 mm., I find #, = 0,2517. The mean of these two values is 0.2730. Among Amphibians I mention (again examined by Lapicqur) Rana fusca with a body weight of 53.0 grams and a diameter of the eye of 6.6 mm., and Hyla arborea, with a body weight of 4.8 grams and a diameter of 4.6 mm. They give 7, = 0.3006. Two species of Fishes according to WerckKer '), Cyprinus carpio and Gobio fluviatilis (2 indiv.), of resp. 1817.3 grams and 42.2 grams body weight, and 1550 and 238 milligrams weight of the eye (from which here the sectional area is calculated) give r, = 0.3290. Records of retinal areas of some adult species of aquatic mam- mals, accurately computed from direct measurements, of which some differ very much in the size of their bodies, may be found in Purrer’s extensive treatise *). The most homoneurie species, Phoca barbata and Phoca vitulina, whose body lengths were in a ratio of 3 : 1.75 to each other, had retinal areas of 2543 and 1980 square millimeters. Assuming uni- formity of these animals, we find, for a ratio of weight of only a little more than 5:1, r, = 0.2972. Of two Toothed Whales, which are, indeed, not so closely akin, but with much greater difference of weight, Hyperoodon rostratus and Phocaena communis, the ratio of length was 6:1, and the areas of the retina 5000 and 1225 square millimeters. Assuming again uniformity, we find for the ratio of the body weights 216:1 and then 7, = 0.2617. As however the body of Hyperoodon is somewhat slenderer than Phocaena, the real ratio of weight must have been somewhat smaller; the real value of 7, was, therefore, somewhat higher. 1) H. Wetckern—A. Branpt, Gewichtswerthe der Körperorgane bei dem Menschen und den Thieren. Archiv fiir Anthropologie, Bd. 28 (1902), p. 60. 3) A. Pürrer, Die Augen der Wassersäugetiere. Zoologische Jahrbiicher, Abtei- lung fiir Anatomie und Ontogenie der Tiere. Jena 1903, p. 167, 174, 198, 209, 239, 243, 272, 280. The calculations of the areas of the retinae, not immediately comparable, were made by A. Leren. 714 Comparing the gigantic Balaenoptera physalus, which is. likewise somewhat slenderer than Phocaena communis, with the latter, of still more distant relationship, but which it exceeds 5323 times in weight ‘aleulated according to the length, we find 7, = 0.2610 for a ratio of the retinal areas of 11500 : 1225 square millimeters. This value, too, would certainly become somewhat larger, if the real weight could have been taken into account, instead of what has been found from the length. The deviations of the found exponents of the value 0.277.. are not very important if it is considered that : 1. The compared species are not all perfectly homoneuric, — . 2. also the specimens are not always typical for their species, 3. the retinal area can only be calculated from the diameter of the eye (in one case the weight of the eye) in approximation, 4. in other cases the body weight was not directly determined. In virtue of these and many other data, considered in the light of their physiological significance, | think | may assume that the area of the retina really varies on an average proportionally to 1.28 or more accurately 025 for homoneuric species of Vertebrates, and that the same proportionality is of general application for the area of the section of the homologous nerve fibers. About these relations for individuals within a species hardly any direct data are available. Lapicqur') states that for Canis familiaris the diameter of the eye only varies from 20 to 23 mm., whereas the body weight varies from 5 to 40 kilograms, i.e. about as the fifteenth power root of this weight. For 7, we find here 0.1344, which means the individual ocular exponent of relation has de- creased in the same ratio (taking the degree of accuracy of these measurements into account), with respect to what is observed between different species, as the encephalie exponent of relation. That the found value lies nearer half 0.28 than half 0.22 may be remarked in passing. Measurements of the thickness of homologous nerve fibers in individuals of different weight of one species, to form an opinion of the ratio considered here, are entirely wanting, but I think I may deem it probable that the area of the section of the peripheral nerve fibers remains the same, and that on the other hand the white nerve fibers of the brain vary both in length and in section proportionally to Pot (or POH), which is in connection with the absence of the A 1) L. Lapicue, La grandeur relative de l'oeil et l'appréciation du poids encé- phalique. Comptes rendus des séances de l'Académie des Sciences. Paris 1908 (2). Tame 147, p. 210. 715 nodes of Ranvier at those nerve fibers in the brain, and with the fact that the number of nodes at the peripheral nerve fibers remains the same for large and small individuals of the same species, whereas between different species this number varies proportionally to the length of the peripheral nerve fiber. ') The meaning of the term P?°?8 must be looked for in the physio- logical function of the nerve fiber and the retina, namely the con- duction of the impulsions or influxions. Likewise of the same rela- tion between the body weight and the volume of the ganglion cell, which volume must undoubtedly be in relation with the quantity of the impulsions which it ean receive and emit. Görnrin (in 1907) was the first to pronounce the idea that the rate of conduction of the nerve influxion in an axis cylinder seems to be in a definite relation to the diameter of the axis cylinder and to the thickness of the medullary sheath, when he tried to apply WitiiaAmM THomson’s cable formulae to the white nerve fiber’). Ten years later GOrTHLIN justly considers the data enabling us to judge about this relation, deplorably few*). He reminds (in 1917) of the fact that according to the exceedingly important researches by Crauvrav in 1878, the conduction in the motor nerve fibers of the larynx of the Horse is about 8 times more rapid than in the motor nerve fibers of the oesophagus of this animal *), and he brings this difference in rate of conduction of the “influx nerveux” in connection with the very different thickness of the nerve fibers in question’). In the very thin and at the same time non-medullated fibers of the splenic nerve of the Ox the rate of conduction, according 1) A. E. Boycor, On the Number of Nodes of Ranvier in Different Stages of the Growth of Nerve Fibres in the Frog. Journal of Physiology. Vol. 30 (1904). London, p. 370—380. At this place we find also a comparison of Cavia porcellus and Mus musculus. 2) G. F. GörHrin, Experimentella Undersökningar af Ledningens Natur i den Hvita Nervsubstansen. Uppsala 1907, p. 120 seq. 5) Relation entre le fonctionnement et la structure des éléments nerveux. Upsala 1917, p. 15. 4) A. CHAUVEAU, Vitesse de propagation des excitations dans les nerfs moteurs des muscles rouges de faisceaux striés, soustraits 4 l’empire dela volonté. Comptes rendus de l'Académie des Sciences. Tome 87. Paris 1878, p. 288—242. There, p. 138—142, also: Vitesse de propagation des excitations dans les nerfs moteurs des muscles de la vie animale, chez les animaux mammifères. 5) A. VAN GEHUCHTEN and M. MoLHAnt, (Contribution a l'étude anatomique du nerf pneumogastrique chez l'Homme. Le Névraxe, Vol. 13. Louvain 1912, p. 96) for the Rabbit. 716 to A. Fiscuer') is only */,, of that in those slow oesophagus nerves of the Horse. GörnriN adds to this (p. 16) that only by assuming the influxion to be conducted in the same way in the nerve fiber as electricity in a cable, it is to be understood why the velocity of conduction varies according to the dimensions of the fibers. With the thickening of the medullary sheath, which serves as “relative isolator’, the capacity of the cable evidently diminishes, and with the enlargement of the area of section of the “conducting” axis cylinder, the resistance of the cable diminishes. In fact then they equally enhance the conductivity of the nerve fiber, and thus it becomes comprehensible that (as DonaLpson and Hoke and also others found) the mean area of section of the medullary sheath in all vertebrates remains equal to that of the axis cylinder, which it envelops. However, GOTHLIN again justly points this out, the nerve fiber should by no means be imagined as an equally passive con- ductor as e.g. a telegraph wire. On the contrary, many circumstances render it necessary to assume that in all long nerve fibers which are rapid conductors, the influxion is regenerated in some way or other during its conduction, and thus compensates for the losses of energy during the propagation in an ever enlarged space’). Crauvrav evidently supposed" a relation between the greater or smaller rate of conduction of the ‘excitations’ in the nerves, and the voluntary or involuntary character of the movements they excite. It is pretty firmly established now that we have to think here of more tangible causes, Our knowledge took an important step forward by CaRLSON's researches (in 1904 and 1906)*). He demonstrated that for Reptiles (Snakes), Amphibians (Frog), Fishes (the Californian Hagfish Bdello- stoma), Cephalopods (Octopus, Loligo), Gastropods (Slug Limax, Ariolimax, Sea hare Pleurobranchaea) and Crustaceans (Spider Crab, Lobster, Limulus) there exists proportionality between the rate of propagation of the impulses in the motor nerve and the contraction 1) A. Fiscner, Ein Beitrag zur Kenntnis des Ablaufs der Erregungsvorgange im marklosen Warmbliiternerven. Giessen 1911. ef. Göthlin, p. 15. 2) Cf: J. B. Jounsron, On the Significance of the Caliber of the Parts of the Neurone in Vertebrates. Journal of Comparative Neurology and Psychology. Vol. 18. Philadelphia 1908, p. 609—618. 8) A. J. Cartson, The Rate of the Nervous Impulse in the Spinal Cord and in the Vagus and the Hypoglossal Nerves of the Californian Hagfish (Bdellostoma Dombeyi). American Journal of Physiology. Vol. X. Boston 1904, p. 401—418. ——, Further evidence of the direct relation between the Rate of Conduction in a Motor Nerve and the Rapidity of Contraction in the Muscle. Ibid. Vol. XV. Boston 1906, p. 186—143. 4 717 time of the muscle. In every animal “the swifter the action of the muscle, the greater the rate of propagation of the impulse in the motor nerve supplying the muscle”. All the nerves are, so to say, tuned to the muscle they supply. This relation having been established, LarrcQqur and LEGENDRE ') examined which anatomic characteristic of the nerve fibers answers to those physiological properties. They found for the common Frog (Rana esculenta) that the thickness of the nerve fibers regularly increases with the rate of conduction of the influxion, to be measured by the rapidity of contraction in the muscle. It may be derived from the values communicated by them that the rapidity in question in the nerve fiber varies in geometrical ratio with the area of the section of the nerve fiber. They also found with regard to the Rabbit, that among others the nerve fibers for the rapid adductor magnus muscle are thicker than those for the semitendinosus, which is a slow muscle. These data make it highly probable that the principle holds good universally: ‘Les fibres nerveuses sont d’autant plus rapides qu’elles sont plus grosses.” Lapicqur demonstrated further that the movements of different Amphibians are the quicker or the slower as the rate of propa- gation of influxions in the nerves of the hind-legs is greater or smaller.*) For the slow Common Toad (Bufo vulgaris) the rate of conduction in the nerve for the musculus gastrocnemius is only about half so great as for the common Frog (Rana esculenta), which jumps quickly and far. The Frog-toad (Pelobates fuscus), far exceeds the Toads proper in the rapidity of its movements, and has, in agreement with this, equally rapid nerves in its hind-legs as the Frog. But also the Green Toad (Bufo viridis), which is more agile and quicker than the Common Toad, which makes comparatively big jumps, and swims and climbs well, is on a par with the Frog as far as the rate of conduction in the nerves is concerned. Though the Walking Toad (Bufo calamita) does not, indeed, jump like the Frog, it runs almost as fast as a mouse (hence its other name cursor), it swims nimbly and rapidly, climbs better than any other Toad 1) L. Lapicqve et R. LEGENDRE, Relation entre le diamétre des fibres nerveuses et leur rapidité fonctionnelle. Comptes rendus de l'Académie des Sciences. Paris 1913 (2). Tome 157, p. 1163—1166. Also: La rapidité fonctionnelle des fibres nerveuses mesurée par la chronaxie et son substratum anatomique. Bulletin du Muséum d’histoire naturelle. Année 1914, NO, 4, Paris 1914, p. 248—252. 2) Louis Lapicqur, Rapidité nerveuse des membres postérieurs chez divers Batraciens anoures. Bulletin du Muséum d’histoire naturelle, Année 1914, NO. 6, p. 363—366. 718 and easily digs holes, (casting out the earth with its hind-legs); it has about the same rate of conduction in the nerve as the Frog. The same thing is the case with the Tree-Frog (Hyla arborea), which pounces upon winged insects as ifs prey. Now, for a slow American Toad (Bufo lentiginosus), which has the same size as our Common Toad, the mean diameter of the fibers of the nervus ischiadicus is, according to DonaLpson and Hoke, 11.2 micra, as against 14.7 for a frog specimen of about the same weight, belonging to Rana virescens. In a common American Lizard (Sceloporus undulatus) of a body weight of 8.2 grams, the mean diameter of the nerve fibers in the plexus brachialis was 98 micra, on the other hand only 6.2 micra in the Horned Toad (Phrynosoma cornutum), which is at least six times heavier, and owes its name to the fact that it moves more like a toad than as the proverbially quick lizard, to the family of which it belongs. For an equally great rapidity as the said Lizard the nerve fiber of the so much heavier Horned Toad would have to double its diameter (to 12.7 micra). The American Turtle Chrysemys marginata, though probably weighing scarcely less than a kilogram, i.e. certainly as much as a hundred times more than the said small Lizard, has nerve fibers of a mean diameter of no more than 12.4 micra, in its plexus brachialis.') It would have to amount to 18.7 micra for equal rapidity as that Lizard. “G6THIIN also points out that Shrimps of the genera Crangon, Palaemon and others, which are among the quickest animal species, possess thick nerve fibers provided with medullary sheaths. *) L. and M. Laricqur found the greatest rapidity of nerves and muscles of all the Invertebrates which they examined, in the tail of Palaemon. *) ArcockK was the first to inquire into the possible influence of the size of the body on the rate of propagation of the influxion in nerves, by experiments on the nervus ischiadicus of the Frog, and, externally, in the nervus medianus of Man. *) He finds for Man, as 1) Cf. these diameters of nerve fibers in H.H. Donatpson and G. W. Hoke, On the Areas of the Axis Cylinder and Medullary Sheath as seen in Cross Sections of the Spinal Nerves in Vertebrates. Journal of Comparative Neurology and Psychology. Vol. 15, Philadelphia 1905, p. 9—11. 3) G. F. Géruuin, Die doppelbrechenden Eigenschaften des Nervengewebes. Kungl. Svenska Vetenskapsakademiens Handlingar, Ny Följd, Bd. 51 (1915), p. 84. 3) Louis et MARCELLE Lapicqur, Quelques chronaxies chez les Mollusques et Crustacés marins. Comptes rendus de la Société de Biologie. Année 1910. Vol. 2 (69). Paris 1910, p. 280. 4) N. H. Arcock, On the Rapidity of the Nervous Impulse in Tall and Short Individuals. Proceed. Roy. Society, Vol. 72 (1903). London 1904, p. 414—418. 719 well as for the Frog, that in all the examined individuals, of one species and in the same nerve, the rate (per unity of length) is the same, hence independent of the size of the body. Small differences in the values of the rate, viz. of on an average 67.5 m. per second for two men of a mean height of 1887 mm. and on an average 65.9 m. for two men of a mean height of 1721 mm., are neglected, evidently considered as experimental errors. The mean body weights of these men may be put at 85 and 67 kg. according to Hassine’s tables, *) and by means of this an exponent of relation for the velocity of impulsion of 0.1007 can be calculated, to which, however, in itself, not much significance should be ascribed. Of greater importance, for the study of the influence of the size of the body on the rate of propagation of impulsions in homologous nerves are Münnicu’s experiments. ?) The rate (66 m. in the nervus medianus) found by him for Man is in good concordance with the mean rate according to Arcock and the latest determination by HermHorrz and Baxr (64.56 m., in 1870), which speaks, indeed, for the reliability of the method. The greatest importance for our subject have, however, the determinations of the rapidity in the nervus ischiadicus of some mammals, viz. three dogs of different sizes and breeds, two cats, and a rabbit. Minnicu justly lays stress on the fact that the reliability of the results of his experiments on animals must be greater than those of experiments on Man, where the nerve cannot be laid bare. Besides, it is of importance for a judgment of the influence of the dimension of the nerve on the rate of conduction of the nerve influxions that among M@énnicu’s dogs a large and a small specimen was examined, the body weights of which inter se differed more than six times as much as those of the tall and the short individuals examined by Ancock. Ménnicn found not inconsiderable deviations between a large and a small dog, with which the breed had nothing to do, and a particularly important deviation in a representative of the so remarkable breed of Dachs- hunds. Also the rapidities found for cats are of great importance. Though Münnien himself, not realizing that the found numerical differences could have any meaning, leaves undecided whether they rest on different rates of conduction or are only caused by the defectiveness of the method, it has now been raised above doubt, 1) H. Vrerorpt, Anatomische, physiologische und physische Daten und Tabellen. 3 Aufl. 1906, p. 589. 2) FERDINAND Miinnicu, Ueber die Leitungsgeschwindigkeit im motorischen Nerven bei Warmblütern. Zeitschrift für Biologie. Bd. 66. München und Berlin 1916, p. 1—22. 720 that they have a real meaning. In this connection the much smaller deviation found by Atcock between tall and short men, does not even seem to be entire/y devoid of interest, now that it appears to run parallel with that between the dogs of a much more conside- rable difference of weight, and these results are in accordance with what was forced on our attention about the significance of the dimensions of the nerves. For a dog, “brauner Bastard von der Grosse eines Foxterriers,” which must have weighed about 7 or 8 kg., Ménnicu finds a rate of 61 m. per second, equal to that for the Rabbit, which can only have had about one fourth of this body weight. Hence the rapidity for the Rabbit is comparatively great, and we accordingly find for this burrowing Rodent thicker nerve fibers than for the Dog. As mean diameter in the plexus brachialis DonaLpson and Hoke found 13.3 micra for a Rabbit and 11.6 micra for a Shepherd’s Dog (probably 8 times heavier). A definite ratio between the rapidity and the area of the section can, however, not be derived from this, because different nerves and different breeds of dogs are compared here. For his largest cat Minnicn found a rate of 81 meters per second, for another specimen, of which he feels it necessary to state that it is adult, and for which he experimented on a shorter length of nerve, which specimen will therefore have been smaller, the rate was 75 meters. As even the largest cat can hardly have reached half the weight of a middle-sized Foxterrier, the rapidity is certainly relatively very large for the Cat. Undoubtedly the considerable thickness of the nerve fibers in the Cat is in connection with this, on an average in the plexus brachialis a diameter of 16 micra, as against 11.6 micra for a dog 5 times heavier according to DoNALDSON and Hokr.') Besides, the Cat is also distinguished by particularly large ganglion cells (a peculiarity of all the Felis species). The muscle .fibers of the extremities, too, seem to be particularly thick. Already Cavazzant had found that the cells of the cervical and of the lambal ganglia spinalia are particularly large, and equal to the corresponding cells of dogs which are 5 times heavier*). Also Levi found the cells of the fifth cervical ganglion spinale in the cat much larger than in an about equally heavy dog (diameter 81 as against 65.6 micra). They only reached an almost equal diameter (79.7 micra)*) in a dog of 7 times the weight (of 23 kg.). According to 1) Loc. cit. 2) E. CAVAZZANI, Sur les ganglions spinaux. Archives Italiennes de Biologie. Tome 28. Turin 1897, p. 52. Shepherd’s Dogs, Pointers, and adult cat. | 8) G. Levi, 1 gangli cerebrospinali Supplemento al Vol. VII dell’ Archivio 721 Levi the Cat has also larger ganglion cells in the columnae anteriores of the intumescentia lumbalis of the spinal cord than the Fox, and of all the Mammals examined by him, among which the Ox, the largest pyramid cells in the cerebral cortex *). It has appeared, from particularly comparable measurements of Harpesrty, that the mean diameter of homologous cells of the intumescentia cervicalis in the Cat is not much smaller than in the Foxhound, which is 6 times heavier (53.5 micra as against 58.7 micra) *). The muscle fibers in the rectus femoris were thicker in the Cat than in any other Mam- mal, that Levi examined, with the exception only of the Horse; while the mean diameter in the Cat amounted to 55 micra, he found it only 36.2 miera in a dog of about twice the weight (of 6.3 kg.) *). It appears, therefore, very clearly that the quick muscles of the Cat’s leg receive rapid impulsions from large ganglion cells through thick nerve fibers. Something similar holds for the mole-footed Dachshund. Still somewhat more rapidly than in the nervus ischiadicus of a 9 times heavier butcher’s dog “von der Grösse einer Deutschen Dogge”’ did the impulsions propagate in this nerve of a dachshund (at a rate of 88 meters a second as against 85 meters for the butcher’s dog). From the rate of conduction (given by Ménnicn) of the influxion of 61 meters a second in the nervus ischiadicus of the mongrel dog of the size of a foxterrier and 85 meters a second for the butcher’s dog, likewise of a mixed breed, which had the size of a German Boarhound, and must therefore have had about 8 times the weight of the smaller dog of the same species, an interindividual exponent of relation for the rate at which the influxions are conducted in the nervus ischiadicus of 0,1595 can be calculated. This is in satisfactory agreement with the exponent 0.1344 derived above from a record by Lapicque for the area of section of the eye-ball within the species Canis familiaris, and the deviations from the value 0.28 are so great, as to justify for both cases the conclusion that here the same relation of the homoneuric species is not valid, but the other relation, the interindividual, which is expressed by an exponent of half the value of that of the species. It does, however, not follow from this that the area of the section of the nerve fiber varies in the same ratio as that of the retinal area, for as the Italiano di Anatomia e di Embriologia. Firenze 1908, p. 177, and: Studi sulla gran- dezza delle cellule. I. Ibid. Vol. V. Firenze 1906, p. 332. 1) Ibid. (1906), p. 334 and 337. Cf. also: K. BRODMANN, Vergleichende Lokali- sationslehre der Grosshirnrinde. Leipzig 1909, p. 81. 2) Journal of Comp. Neurology. Philadelphia 1902, Vol. 12, NO, 2. p. 160. 3) Loe. cit., (1906), p. 327. 47 Proceedings Royal Acad. Amsterdam, Vol. XXI. 722 length of the nerve fiber varies, from one individual to another within a species, without change of the number of the nodes of Ranvigr, where the resistance for the nerve influxion must be much larger than in the internodia, the resistance in the larger animal becomes slighter in inverse ratio to the length of the nerve fiber. I imagine that in this way the nerve fiber, from individual to individual within a species, does not actually become thicker, only longer. This is different for homoneuric species; there the number of internodia, hence also of the nodes, increases in direct ratio to the length of the nerve fiber, which does not bring about a change in the rapidity of the influxion (per unit of length); this is, therefore, directly dependent on the area of the section. When the propagation of the influxions in the nerve fiber is compared with the motion of electricity in a circuit, in which the resistance (the reciprocal value of the conductivity) is in inverse ratio to the section, and in direct ratio to the length, mach becomes clear in the relations of the size in the nervous system that would otherwise remain unaccountable. Then the ganglion cell may, in a certain sense, be compared with an electric condensator or storage battery, which is charged and discharged. In the first place it becomes clear that only on comparison of homoneuric species the regular quantitative relations of the neurone and its parts to the body weight are found. For the exponent of relation for the volume of the largest ganglion cells of the columnae anteriores in the intumescentia cervicalis the value 0,2387 is found between Horse and Mouse, according to Harpersty’s records, the value 0.3931, on the other hand, between Cat and Mouse; the expo- nent is only 0.0851 between Horse and Cat. A Cat that had the weight of the Horse, would have a ganglion cell for the motor nerve fiber of the claw muscles 2.763 times as voluminous as the cell belonging to the longest motor nerve fiber of the Horse. Harprsty points out (p. 166) that the American Hoary Bat (Atalapha cinerea Pal.), which is considerably smaller than the House Mouse, possesses somewhat larger cell bodies in the columnae anteriores of the intumescentia cervicalis than the latter, which he brings in connection with the innervation of the wings. OBERSTEINER also sees a connection between this great functional importance of the fore- legs as wings in the Bats and the fact that, while for most Mammals the cells of the intumescentia lumbalis are appreciably larger than those in the intumescentia cervicalis, the reverse is true for Bats *). 1) IER OBERSTEINER, Bemerkungen zur Bedeutung der wechselnden Grösse von Nervenzellen. (Del Volume Jubilare in onore L. Brancut. Catania 1913), p. 4. 723 From the much older records of Kaiser it may be derived that the . exponent of relation for the volume of the largest ganglion cells of the cervical medulla of the much larger Bat Plecotus auritus is only 0.1568 in comparison with the Mole *). This Bat has, accord- ingly, a much larger ganglion cell than the Mole, in ratio of equal body weight. OBERSTRINER points out that Man and the Orang-outang have strikingly small cells in their cervical medulla ’). According to Harpesty’s measurements homologous cells of the cervical intumescentia of Man are actually somewhat smaller than in the Foxhound, which he exceeds four times in body weight ®, and not much larger than those of the Cat, which he exceeds more than twenty times in body weight. As well for Macacus, Cynoce- phalus, Ateles as for Man, the nerve fibers of the plexus brachialis are thin, in comparison with those of other Mammals, the size of the body being taken into consideration. Evidently the slight quickness of the muscles, but comparative delicacy of the muscle fibers for the hand and the fingers, as neces- sary factor for their finely regulated collaboration, may be imputed to the slight thickness of the nerve fibers and the comparatively small volume of the ganglion cells, with which they are connected. The dimensions of the peripheral “conductors” and of the central “accumulators”, connected with them, can become of great importance for the quantity of the brain. Where they are to supply muscles which are both strong and quick, the nerve fibers become particularly thick, the cell-bodies large, and thus the neurones voluminous, to whieh also voluminous brain neurones must answer. Of the problems of the Elephant having a more voluminous brain than the Anthro- poid Apes, to which it was thought that an extraordinary mental height had to be attributed for this reason, of the Ateles and Cebus, equally high above the Anthropoids with respect to their quantity of brain, of the aquatic Mammals, among which the Mysticetes possess comparatively larger brain than the Dogs, of the Denticetes, which like the Seals can almost be put on a line with the Anthro- *) O. Karser, Die Funktionen der Ganglienzellen des Halsmarkes. Haag 1891, p. 63. *) L. ce. p. 4. The same property may be assumed for the Gorilla, whose spinal cord was examined by W. Warpeyer (Das Gorilla-Riickenmark. Abhandlungen der Kön. Preus. Akademie der Wissenschaften Berlin. Jahr 1888. Physikalisch- mathematische Classe. Abt. lll, S.1—147). Cf. Harpesry, l.c., p. 168. 3) Cavazzani (1. c., p. 52 and 53) had already found that the ganglion cells in Man are smaller than in Shepherd’s Dogs and Pointers, smaller even than in the Cat. 47* 724 poids, in this respect’), — facts that have led to speculations about uncommonly great intelligence of tle Whales — of the compara- tively highly cepbalised Crocodiles, and of the class of the Fishes, which in general are not inferior to the Reptiles, as far as the quantity of brain is concerned, of all these problems the solution is now obvious *). The proboscis of the Elephant, which plays a pro- - minent part in the animal’s life, is not only strong and agile, but also provided with very voluminous muscles. That of the Asiatic Elephant measures 2 meters for a length of the body of 3°/, meters. Quick strong muscles move the long tail of the American Apes mentioned, which muscles are of still greater service to these animals than those of their hands and feet. But also their arms and legs are very long in comparison with the body, especially for Ateles, which owes its popular name of Spider-Monkey to this, and they consequently contain a large mass of muscles. The Whales have an exceedingly quick and strong motor apparatus in their long body for the tail, which admirably like a ship’s screw propels the gigantic 1) Judging from not yet full-grown animals, some investigators have assumed somewhat too great cephalisation. Thus the Seal is not full-grown with 12!/, kg. body weight (Louis Lapicqur, Sur le poids encéphalique des Mammifères amphi- bies. Bulletin du Muséum d'histoire naturelle. 1912, N°. 1 p. 2). An adult female, examined many years ago by E. H. Werrer (Ueber den Bau des Seehunds, Phoca vitulina. Verhandlungen der Kön. Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math.-Phys. Classe. Jahrgang 1850, p. 108), however somewhat emaciated in captivity, weighed as much as 43.11 kg. The animal possessed 266.5 grams of brain, from which a coefficient of cephalisation k = 0.6766 can be calculated. Of a female Otaria californiana, which had lived in the Amsterdam Zoological Gardens since 1902, and was already fully adult at the time, the body weight was 74 kg., the weight of the brain 374.5 grams, at its death in Nov. 1913, from which k = 0.7025 may be calculated. The weight of the body of this animal, too, must, indeed, have been somewhat higher in its healthy state. Accordingly the calculated coefficients of cephalisation are decidedly somewhat higher than normal for both Pinnipeds. As regards the Whales, we may refer to the weight of nen sulfurea, 22.5 meters long, viz. 63 Am. tons, i.e. 64000 kg. (according to determinations by Lucas inaccessible to me), which is mentioned in Vol. 12, p. 503 of the fourth edition (1915)of BreHm’s Tierleben. The brain, freshly removed, of a Balaenoptera musculus, 18.8 m. (= 60 Norwegian feet) long, weighed 6709 grams, according to G. A. GurpBeErG (Forhandlingar i Videnskabs Selskabet i Christiania. Aar 1885. Christiania 1886, p. 128). Assuming uniformity with the other Balaenoptera species, a body weight of 87385 kg. is found for the latter, and in connection with this a cephalisation coefficient of 0.3841. 2) The significance of the influence of the organs of hearing, touch and other organs of sense on the quanbiy of the brain of these Vertebrates is, however, not slight either. 725 mass with great velocity, and skilfully steers it. The flipper-like hind- limbs of the Seals (Phoca), which form a kind of tail, move the body along through pretty intricate but nimble movements; in other cases (Otaria) very large pectoral fins render such services to the animal, the feet, directed backwards here too, acting more like a helm, worked by powerful muscles. For the Fishes it is again the tail which derives its propelling force from the muscles of the hind part of the body. The Crocodiles have a strong propelling tail. For all these vertebrate aquatic mammals the locomotor muscles must not only be quick, but also particularity voluminous and strong, because of the great density of the medium, in which the movements take place; hence they must consist of thick, but also of numerous muscle fibers. Hence very voluminous and at the same time very numerous neurones. In fact LrGENDRE demonstrated that the nerve fibers of the medulla and the roots of the Dolphin (Delphinus delphis L) are among others much thicker than in Man, the Stag, the Dog, the Rabbit, and the Mouse. He partly accounts for the high cephali- sation of this aquatic Mammal by the thickness of its nerve fibers *). Though 1 do not enter more fully into the discussion of the mutual relation of the dimensions of muscle- and nerve fibers here, | will, however, point out that between, homoneuric species the variations of the dimensions of the neurones and their parts, in function of the body weight, entirely account for the variations of the weight of the brain, in funetion of the body weight; hence that the number of the neurones (differently from what | thought possible at first) remains the same. This must also apply to the sarco-neurones, hence to the muscle fibers. That the cell-body of the neurone must become more voluminous, for homoneuric species, in ratio to the arithmetical longitudinal dimension of the body, is a consequence of the already long known dynamie proportion that the weight (the mass) of similar animals aug- ments according to the cube of the homologous longitudinal dimen- sions, the muscular force, on the other hand, proportional to the area of the section of homologous muscles or the square of the homologous longitudinal dimensions. Already ninety years ago STRAUS-DÜRCKHEIM *) set forth clearly that thus, for similar animals, consequently having the same organization, but of different sizes, the time of every movement must be nearly proportionate to the longitudinal dimen- 1) R. LEGENDRE, Notes sur le système nerveux central d'un Dauphin (Delphinus delphis). Bulletin du Muséum d’histoire naturelle, 1912, N°, 1, p. 6—7. Pl. I. 2) H. Srravus-Diincknem, Considérations générales sur l'anatomie comparée des animaux articulés. Paris 1828, p. 189 et seq. 726 sions, although the rate of the locomotion is equal. For this, increase of the accumulated influxion in the cell-body of the neurone is required, and hence of its volume, in the ratio of about PP or P33... In about the same ratio the lengths of the nerve fibers of homoneurie animal species necessarily increase, these lengths being nearly proportionate to the longitudinal dimensions of the animals. At the same time, it may be assumed, as a matter of course, that there is a tendency for the homologous nerve fiber to increase with the least possible deformation. Without any deformation its area of section should increase in the ratio of ?°??-, while its length increases in the ratio of PO. On the otber hand, the mode of propagation of nervous influxions being similar in homoneuric species of animals, the only possibility to obtain this similarity, i.e. physiological homoneury, under the given conditions, is that the area of section of the nerve fiber should increase proportionally with its length. Therefore the area of the section of the fiber has become somewhat greater and its length less than for uniformity of the homologous nerve fibers, both in the ratio of P55... Hence the ratio P°?77--: 1, as was discussed in the outset. P0217. — Po333.. P0055. — P0222. y¢ P0055. — P0277... Thus the accumulated influxion, the time and the rate of propagation increase, as they ought to do, in the same ratio. Through this way of viewing the matter, which starts from the relations of quantity existing in the nervous system, many impon- derabilia, such as the mental height, the intelligence, and not seldom also the degree of brain evolution may be eliminated from the foreground of our considerations of the relation between structure and functions of the nervous system, which can only promote the fruitfulness of researches in this region, among others and particu- larly with regard to the cortex of the brain. A striking proof for this is furnished by Orro Maysr’s researches on the density in which the cells occur in the cortex of the brain of Apes').- He namely determined, for seven genera, the mean number of cells per 0.01 cubic millimeter, in 10 fields (according to BRODMANN), throughout the whole thickness of the cortex. His results induced him to point out emphatically that the densities by no means run parallel with the order of the examined animals in the zoological system, their degree of brain organization or their intel- 1) Orro Mayer, Mikrometrische Untersuchungen über die Zelldichtigkeit der Grosshirnrinde bei den Affen. Journal fiir Psychologie und Neurologie. Band 19. Leipzig 1912, p. 233—251. 2 Tafeln. 127 ligence. A result which strongly reminds of the absence of systematical order in the “relative brain weights”. ] have, therefore, treated the densities in a similar way as the brain weights, namely by considering them in connection with the weights of the body. For this purpose calculating the mean densities of the whole cortex (over those ten important fields), from Marer’s records, I find a meart density of cell of 1765,4 for the Chimpanzee (Anthropropithecus troglodytes), of 3160 for the Gibbon (Hylobates syndactylus), of 3580.9 for the Capuchin-Monkey (Cebus capucinus), of 3603.4 for the Saimiri or Squirrel-Monkey (Chrysothrix seiurea), and of 3448.1 for the Marmoset (Hapale jacchus), which mean values no more show systematic order than the densities in the separate fields, from which they are computed. ‘This disorder is replaced by regularity, as soon as the size of the animals is taken into account. The ratio of the density of cells between the Chimpanzee and the Siamang is as 1:1.79. The Chimpanzee has 8 times the weight of this Gibbon-species, and now it is very remarkable that, between these homoneuric species, according to the proportionality demon- strated in my former paper, homologous cells must be more voluminous to a ratio of 8°28 — 1.79 for the Chimpanzee than for this Gibbon. Between these homoneuric species the density of cells is, therefore, accurately in inverse ratio with the volume of every cell. In other words: for a Gibbon that bad the size of a Chimpanzee, the density of cells in the cortex of the cerebrum would be equal to that of the Chimpanzee. It may, therefore, be assumed that these Anthropoids of equal cephalisation (equal quantity of the brain in function of the body weight) are also equal in the organisation of their cortex of the brain. The cells of these two species must be uniform, both as far as the dendrites and the other interstitium is concerned, and with regard to the cell-body. When Cebus is compared with Hapale, which he exceeds 6 times in body weight (these weights are on an average 1300 and 215 grams; the brain weight of Cebus is 125, that of Hapale is about 8 grams), the cortical density of cells for Cebus is found 1.7 times greater than for a Hapale of equal body weight. In about the same ratio, viz. 1.8: 1 the cephalisation is, however, also greater for Cebus than for Hapale, a very fair agreement when it is borne in mind that different specimens were compared. Compared with Chrysothrix (Saimiri), (whose body weight amounts to about 400 grams, and which possesses 24 grams of brain) Cebus has 1.5 times higher cephalisation, and a Saimiri of equal body weight as Cebus would 728 have almost 1.4 times greater density of cells. For these allied American Monkeys the density of cells, in function of the body weight, is, therefore, proportionate to the cephalisation: the greater the quantity of brain, in function of the body weight, the greater the density of cells. A Cebus, however, of the same body weight as Hylobates syn- dactylus, would have only two thirds of the density of cells of this long- armed ape, though the latter has three fifths of the weight of the brain of the large Cebus-species in question. Here the density of cells is about in inverse ratio to the cephalisation. Of special interest is the comparison of the density of the cells of the Chimpanzee with that of Man, both having about the same body weight. HAMMARBERG ') determined the density of cells, in normal human brains, through the whole thickness of the cortex, per 0.001 cubic millimeter, i.e. '/,, of Mayrr’s unit of volume, in different cortical aveas, some of which may be compared with the areas according to BRODMANN. Thus in the lobus oecipitalis the area striata or field 17 of Bropmann. Calculated to the same unit of volume as that of Mayer we find here 386 cells, whereas Mayer’s chimpanzee has 2888, i.e. 7.5 times as many. Thus 152 in the area gigantopy- ramidalis or BropMann’s field 4, as against a density for that chim- panzee of 1172, i.e. 7.7 times as many. In the area frontalis agranularis or field 6 the density for Man is only 111, as against 1136 or 10.2 times as much for the chimpanzee examined by Marer ’). As, according to BRODMANN’s *) measurements, the entire surface of the cortex in the Chimpanzee is about a third of that in Man, the regio 1) GC. HAMMARBERG, Studien über Klinik und Pathologie der Idiotie, nebst Unter- suchungen tiber die normale Anatomie der Hirnrinde. Upsala 1895. *) In the part of the gyrus frontalis superior, which belongs to the area frontalis agranularis or BRODMANN’s sixth field | have estimated the density of cells in the deepest half of the third layer. and likewise in the deepest half of the fourth layer of the part of the lobus occipitalis, which corresponds to the area striata or BRODMANN's seventeenth field, from drawings (Table 1, Fig. 2 and Table II, Fig. 4); namely according to the ratio of the other densities drawn and also calculated by HAMMARBERG. Thus was also evaluated the density of cells of the insignificant first layer in the part of the gyrus centralis anterior, which belongs to the area gigantopyramidalis or BRoDMANN’s fourth field. HAMMARBERG Calculated the number of cells per unity of volume from 10 suc- cessive sections, each of a thickness of 0.01 mm., or 5 sections of a thickness of 0.02 mm., Mayer on the other hand from only a single section of a thickness of 0.01 mm. As also parts of cells are counted, the number must be slightly greater according to the latter method. Judging by HAMMARBERG's drawings the difference ean, however, not be considerable. 5) K. BRODMANN, Neue Forschungsergebnisse der Grosshirnrindenanatomie, mit 729 praecentalis (the fourth and the sixth field) of the Chimpanzee about two thirds of that in Man, the area striata about equal to that of Man, it must be assumed, according to these determinations, that the absolute number of the nerve cells in the cerebral cortex of Man is much smaller than in the Chimpanzee; greatest is the value of the ratio in the area striata (7, against 6 in the regio praecentralis). The human brain being, however, certainly more complicated func- tionally, (which is generally called highly developed), it would appear that not the number of neurones, but the multiplication of the contiguities of their dendritic processes, which principally constitute the interstitium, corresponds to more complicated (higher) functions. besonderer Berücksichtigung anthropologischer Fragen. Gesellsch. Deutscher Natur- forscher und Aerzte. Verhandlungen 1913. Leipzig 1913, p 9, 22 and 25. The brain weight of the examined chimpanzee amounted only to 295 grams; the average in full-grown state may be estimated at about 400 grams. The ratio of the surface of the cortex to that of Man may, therefore, become somewhat more favourable. With greater brain weight the density of cells in the cortex of the chimpanzee would have become proportionally less, hence the number of cells would have remained the same. The cortex of Man is, indeed, somewhat thicker, in consequence of which the actual values of the ratio must be proportionately smaller than those calculated. Astronomy. — ‘“Lupansion of a cosmic gassphere, the new stars and the Cepheids’. By Dr. A. PANNEKOEK. (Communicated by Prof. W. DE SITTER). (Communicated in the meeting of October 1918). The new stars which are most fully known in their changes of intensity are of two distinet types. The sudden quick flaming up is common to both; but they differ in their further changes of light. To the one class belong the two brightest Novae of this century, Nova Persei 1901 and Nova Aquilae 1918, as well as Nova Coronae 1866. Immediately after attaining the greatest brilliance the light begins to decrease quickly; then the diminution becomes slower, while a periodicity sets in. In the other class, of which Nova Aurigae 1592 is the best known example — and to which Tycho’s star Nova Cassiopeiae 1572 belongs — the star retains its brilliance for a long time, fluctuates irregularly, and finally loses its brilliance rather rapidly. These two types of change of light show a certain correspondence with the two types of light- maximum, long and short, which are observed, alternately in the same star, in the Antalgol stars such as SS Cygni. Whether this analogy is more than an accidental correspondence, or that a real relationship exists, cannot yet be ascertained. In connection with the appearance of Nova Aurigae SEELIGER has given an explanation which fits the phenomena of this type very well; when a star enters a nebulous mass, thereby being brought to a high temperature, as long as it flies through denser and thinner parts, its temperature will fluctuate up and down. This theory fits the other type less well. Here there is obviously an enormous rise of temperature, caused by a momentary event, of which all the further processes are merely the consequences. The cause from which this sudden heating arises need not be discussed here. We only put the question of what may be deduced concerning the further events, from simple hypotheses. A cosmic body, suddenly brought to such a high temperature, will not be in equilibrium. It will expand adiabatically, and as a consequence it 131 will cool down. Usually the loss of heat by radiation is given as the principal cause of the cooling of a star; but the cooling from adiabatic expansion is of much more importance. In a first approxi- mation, therefore, radiation may be neglected. The force of gravity is also left out of aceount, which to some extent diminishes the force of expansion; this may be done the more legitimately as it is to a greater or less extent compensated by the radiation-pressure. We assume that all changes take place homocentrically. A volume-element at a distance 7, from the centre is found by expansion after a time ¢ at a distance r=r, + A. We must then have the relation A volume-element #,*dr,dw shifts to the distance r and becomes dr dw. By this the density changes according to: Uy ase 0, B dr es dr, As the change takes place adiabatically, pg +* remains constant, or @ = Const. X p°/,, therefore: / ah Poli air dp 7 (pilt\ dpi kde el aay 0, dr” The index 0 specities the conditions at the time 0, which thus remain a function of r. We shall indicate by the index 00 the condition for {— 0 and r=0O, at the centre therefore; putting (2)"= ae Por Our. |) =y, “=a —-"=8 Po Ooo Poo Qo ah 7 p, dy we find Tok WERL Then « is a constant for this gas ball, of the dimension L°7'-?: it is according to p=o AT (H = gas-constant) proportional to the temperature at the centre, and has the physical meaning of the square of the speed of propagation of isothermic disturbances of equilibrium at the centre. 3 is a number without dimensions, which at the centre is —1, a function of 7, which gives the course of a from the centre outwards in the initial condition. The equation of motion now becomes: where y is determined by p\i o \7/s pe \ 4s y=(|—)}) =| ~ |] =/ — : Po Qo , ar r\? dr PJ an, y gives the change of the temperature. By «the formulae (1) and (2) the change of A with the time is determined, when the quantity 8, which determines the original con- dition, is known as a function of 7. We may, for instance, assume a density distribution such as Emprn has calculated for a gaseous sphere in equilibrium, but supposing a much higher temperature at each point than belongs to an equilibrium form of this kind. The original conditions must be such that 4 continually increases, owing to the strong force of expansion which is working all the time towards the outside in consequence of the high temperature. This will cause the temperature of each layer to fall in a ratio which is given by the quantity y, and in consequence the luminosity will decrease. The most external coldest layers, which absorb the light of the central parts, will move towards us with great rapidity; this evplains why in all new stars, as soon as the light begins to decrease, the dark absorption-lines are displaced strongly towards the violet — a phenomenon which it has been attempted in vain to explain by a rapid approach of the whole star, or by differences of pressure. Even when the initial conditions are simple, the equations (1) and (2) are difficult to integrate. An attempt to find the course of the change by mechanical quadrature failed through the fact that or 2 Ti Ab (2) y= nk … dy small variations in y come out greatly increased in aa? and there- r fore also in the 4 that is found and the subsequent values for y, so that each step gives an increasing inaccuracy, which, after inte- gration through a few units of time, makes the results quite un- reliable. On this account we have not succeeded in explaining the periodic variation in brightness — which both in Nova Persei and Nova Aquilae began to appear after the star had decreased 4 classes of magnitude — by special initial conditions. On the other hand the general mean course of the process may be calculated. The question may be asked: is it possible for the 733 whole mass of the star to expand completely uniformly and what are the initial conditions required in this case? We must then have r==r,f(t, where f is independent of r. It follows that | ih ii £ = fs P=(*) = ae ae a (5) zj Oo Po Q, Ei Q, 1 dp 1 dp, ay, Sn og. dr-- 0, adr, dt? © at hence the equation of motion (1) becomes ld ee a Q, dr, For ‘= 0 this gives 7 1 dp Toe er aoe 9 A7, The quantity /,” must be a constant; a special distribution of density and pressure along the radius for ¢=0O must therefore be sought, in which Ld SA eN ER a ro Oy dr, (The dimension of A is ZY; / is then determined by f fil — A? Yamal ï L 4 ; é 5 . : ; (4) By integration of this equation f is found as a function of the time Ie dt The constant is determined by the condition that for t=O and f=1, whew the expansion begins, its velocity is 0, hence df pane 2 bf; ( ) =a hi) af? (5) =— 2 A? f®s +. Const. dt << = A tes: Bedek En 5 Abd ey df 5 Vi fh This form can be integrated by partial integration in the form “of a series. Putting ham — we have a 8 al aA Gtod:10 fa" dage We irt laar —… | + const. 1 Vl 17.8 ~] Ge = 2.4.10 2.4 (1-.«)? - (1-«#)*'— a + const. —= 2a le (1—x)'/s 13 — (1-2) — 9.15 0.15.21 whence 2. Aiel 2 VWEAM TS hell elk Eid j | LS ae | + +e or ‚ (5) 24 2.4.10 —V5fV1—f-% == t= PS a ie a 7 st) ETA hk TET | The former series may be used when w is small, or f very large, the latter holds for small /, « being near unity. The additive con- stant in the second series must be equal to O, since for ¢= 0 and f are both equal to 1. For the first series the constant cannot be determined by this condition: it was found by computing the value of At for one and the same value of w from both series. By means of these series the time-function At was calculated for a number of values of the expansion-factor 7. The temperature is connected with / by the relation | ie ON da Since the temperature changes according to this law throughout the entire star, we are entitled to assume that the same law holds for the effective temperature; the radiation per unit area then changes proportionally to 74, i.e. to fs. The surface itself changes as f*, and thus the luminosity as f—"s, hence log L — log L, = — **/, log f. or expressed in terms of classes of magnitude Mi El LOU Fee hod re ey ee NN The following table contains for a number of values of f the corresponding At and m — m, if At m—m, 1.0 0.0 0.0 AA 0.455 0.290 1.2 0.655 0.554 1.3 0.815 0.798 1.4 0.956 1.023 1.5 1.086 1.233 1.6 1.207 1.429 157 1.322 1.613 1.8 1.433 1.787 1.9 1.541 1.951 2.0 1.645 2.107 / At Mm — M, 2.5 2.137 2.786 2.135 3. 2.599 3.340 3.479 4.214 5. 4.327 4.893 10 8.397 7.000 20 16.307 9.107') The intensity thus diminishes slowly at first and then faster and faster, but the velocity soon reaches a maximum, when the star has fallen rather more than 1 magnitude below the original intensity. The velocity of decrease then becomes slower once more and finally approaches a logarithmic curve. The slow decrease in the beginning is not observed in the new stars, as the process of blazing up has not yet worked out then. Both Nova Persei and Nova Aquilae had their maximum one day after they had reached the first magnitude, and Nova Persei one day before that had already attained the 3 magnitude. As the starting-point {== 0 we must not therefore take the moment of maximum luminosity, but one or two days earlier. Then follows a rapid decrease which, however, soon becomes slower and is then accompanied’ by periodical variations. On comparing the observed light-eurve and the one here calculated they are found not to agree during the further course of the change; the mean observed intensity decreases much more slowly than according to the above calculation. Evidently other influences are at work here, lying outside the simple theory here given. It is therefore only for the first period of rapid decrease of luminosity that agreement may be looked for. For Nova Persei we shall take 0,0 as the ideal maximum inten- sity, a little higher than the greatest brightness observed, because for it the final stage of the blazing up overlaps the beginning of the expansion, and for the ideal starting-point the 21st of February. The following values of At are then found from the values of m—m, on the smoothed observational light-curve. Date m obs. Al t quotient A Febr. 25 1.00 0,97 4d 0.24 2,8.10—6 cme | meno 1.35 6 0.22 Bb Mrch 1 2.07 1.62 8 0.20 > ae 1) From 1.0 to 2.5 equation 5) has been used, from 2.5 to 20 equation 5a. 736 Date m obs. At t quotient A Mreh 3 2.42 1.85 10 0.18 dee RT Zlin Se AEO 12 OAT7 Pp ae ET 3.02 2.30 14 0.16 10, „0 3.27 2.53 16 0.16 18e lk 3.48 2.71 18 0.15 | ha „13 3.65 2.88 20 0.14 17 The diminution of the quotient shows that those influences which Jater on retard the decrease to a higher degree than the theory requires, begin to manifest themselves even in the first stage. However that may be, the order of magnitude of A as found here cannot but be correct, and from it conclusions may be drawn as to the constitution of the Novae. When all quantities are expressed in the absolute system, ¢ is measured in seconds; taking 0.21 as a mean value of the quotient in the above table we have A = 0,21: 86400 == 23,5710. For Nova Aquilae about the same value is found. Il. The distribution of pressure and temperature for ¢=0, which is required for a uniform expansion, and the dependence of A on this distribution are determined by equation (3) in which we shall now leave out the indices 0: l dp _ 5 rgdr or d = — A’ro dr or p=t+ At (orde. R where R is the radius of the external surface, where p=0O. If a definite density-distribution is assumed as existing at the moment of flaring up, the latter equation determines the pressure as a function of r, and therefore also the temperature: At ; Dies mde et | ee a ee mijer 0) x For the density the values have been assumed which EMDEN 727 e has calculated for the equilibrium-forms of spherical cosmic gaseous masses (for n= 2'/,, £ = 1.4); the integration has been performed by mechanical quadrature. The integration-intervals were taken four times smaller than the unit of 7, as used by EMDEN; expressed in our unit the radius of the external surface is 21.67. The result of the integration was as follows: r log @ 21°67? BEET ret tel e Q, R® Q ev Qo Vite R R 0 0.00000 22,5274 22.53 2 9.95523 20,6746 22.92 + 9.82604 16.0941 24.02 6 9.62544 10.6138 25.62 8 9.36944 6.3898 27.30 10 9.06954 3.3527 28:56 12 8.72945 1.5525 28.94 14 8.34375 0.6148 26.61 16 7.88832 0.1922 24.84 18 7.29284 0.038928 20.04 20 6.32566 0.003815 15.12 The integral I is proportional to the temperature. The result there- fore shows, that the uniform expansion requires a distribution of temperature which differs very little from an even temperature throughout the mass. If the original process is not a rise of temperature at the surface by friction in a nebulous mass, but if through some catastrophe the entire mass becomes hot throughout, an approximately equal temperature through the whole mass might be expected and in that case, as was here shown, an approximately uniform expansion would take place. Now for gaat 21.67 ee (2 r dr ii. neen Q o, KR R we have | IR? A? [R?AN TF BV CTE 21,67*.8,3,107 if « is the molecular weight of the gas of which the star consists. Substituting the value of A found above, the mean temperature (taking / = 25) becomes: _ 29 X 6.25,1012 ~ 91,67? X 8,3.107 ‘y’ Riu =410-AUR u. . . . (9) 48 Proceedings Royal Acad. Amsterdam. Vol. XXL. 738 With 7’=10* degrees this gives: R'u = 2,5.10** hence for u = 1 (dissoc. HT) u = 50 (metals) hho Rotor 2,8: xl 08 = 23 times thesun 8,5 times the sun and with 7'== 10° degrees Rp =:2,5,107* „hence Ai oa 7 ee = 71 times thesun 10 times the sun Considering that at this high degree of heat the mass will be highly dissociated, the first values are probably nearer the truth than those corresponding to u == 50. It shows that a Nova at the moment of greatest brightness is a body much more gigantic than the sun, not only in luminosity but also in radius and volume. The theory, that a new star arises when a dark body of the size of our sun, i.e. an ordinary cooled-down dwarf star, suddenly rises to | a colossal temperature, is in contradiction with the above calcula- tions: for R=7T7 X 10", the radius of our sun, with u = 50 and A as observed, the temperature would only rise to 1000°; for T=10*° A would be 10 times larger, that is: the time in which the star loses its light would be 10 times smaller. This result is in accordance with the value of O”.O11 for the parallax of Nova Persei, derived by KarrerN from the supposition that the nebulous rings which were photographed half a year later arose from reflected star-light. This leads to a luminosity 10000 times that of the sun; since the intensity of the surface-radiation was not much different from what it is in an ordinary white star — HerTZsPRUNG found a similar distribution of light in the spectrum of Nova Aquilae as in « Aquilae') — the radius of the Nova must have been 30 to 50 times the radius of the sun. Supposing our interpretation of the dark lines which always ac- company the bright lines on the violet side being correct, this also leads us to a high value of A. The velocity with which the outer- most particles move towards us is R¢/,,. At the moment when the light has fallen by two magnitudes, we have df /aae == 1, hence Räflg= AR=25 X 10-* Rk. For R= the radius of the sun this would become 1.7 km. per second. On the other hand the observed displacement of. the dark lines was as much as would correspond to 700 km. per second. The real velocity must have been smaller, however, since the absorption-line is partly effaced by the broad adjoining emission-line; on the assumption that the velocity may have been about 100 or 200 km./sec. A is found equal to 60 or 1) Astonomische Nachrichten Bd. 207. Nr. 4950. 739 120 times the radius of the sun, therefore again a value of the same order of magnitude. The Novae in the first stage of their brightness thus possess the characteristics of the giant-stars; in order that their mass may not become too exceptionally large, their density must be small even before the expansion. The relation found here between 7’, R, and A cannot teach us anything on this point, as it does not contain the density. A further indication for a small density may be found, however, in the fact that after a decrease of 4 magnitudes the spectrum at the minima of the light-variations more and more approached the character of a nebula-spectrum, and after another few months the star had become a nebula. At this stage the density has become so small that the visible emission is derived from the whole body ineluding even the hindmost layers and still gives but a feeble surface-brightness ; the fact that this condition sets in, when the expansion factor has become something like 10 or 20, proves that the original density must also have been far below unity. Ill. The original equation of motion (1) may also be written in such a form that it does not contain any dimensiOns. Let us put r= ys KS ye en eet ETO} where y is a linear measure, d a length of time and s, x, and z are numerical values. The equations then become de 7d dy dy dz? Goer ae eae : s\? ds \7/s : Bs alent =(()a) 0 So | where 8 is a function of the coördinate s,. The function 8 and the constants «, y, d which determine the special constitution and size of the star are united in the one coéfficient B. The law of change of w with z is solely dependent on this coéfficient, and is the same for all bodies with the same B. Equations (11) determine all possible movements — progressive, irregular or periodical, which may occur in a cosmic gaseous mass, in so far as they are a function of + only and as gravitation may be left out of account. Without calculating these movements themselves, a relation of similarity may be derived from the formulae which establishes a connection between the changes in different stars. If for different 48* (11) 740 cosmic gas-spheres the distribution of 8 along the radius is the same, the expression 4 t 2 zjn 5) Y Yr 2 must be the same function for them, provided B i.e. a8-= for them ; is the same number. If for each of them a suitable time- and distance- scale is assumed, the motions and variations expressed on this scale are for all these bodies identical. Assuming that a periodical solution of the equations (11) exists in which the particles move radially to and fro and the density perio- dically becomes adiabatically larger and smaller, this condition ot motion will be valid for all such bodies provided the periods of the variations are expressed in d as unit and the dimensions of the bodies in terms of 7 We must then have the relation ora aad Loe yn Now «= HT, (at the centre), therefore proportional to the tempe- rature at the centre. Calling P the period of the variations and A the radius of the gas-sphere, this gives: BRS Ba! at : Ie If we may assume, that similar bodies of this kind have the same temperature, the brightness becomes proportional to R*, ie. to Zan Otherwise the temperature will still depend on some power of hk. and we have the more general relation | P? — Lr or 2 log P = Const. + n log L or 2 log P = Const. — 0,4n X M if J/ represents the absolute magnitude. A relation of that kind was found by Miss Lravirr for the variable stars of the d Cephei-type in the small Magelhanic-cloud *). For 25 stars with periods from 1.25 to 127 days she found, that the period increased with the magnitude in such a manner that the logarithm of the period changed by 0.48 per magnitude-class. The Cepheids are giant-stars, to which our suppositions are in so far applicable, that gravity, small in itself by the small density, must moreover for the greater part be neutralized by the radiation- ) Harvard Circular Nr. 173. 741 pressure. They are all nearly of the same spectral type, lence their temperature cannot differ much. The relation which has been found to hold for them between period and intensity may therefore be explained in a simple manner by assuming that the variation of light arises from a_ pulsation of the gaseous sphere; not, as is often assumed, a pulsating deformation, but a pulsating expansion and contraction. Hereby the absorbing layers at the front of the star will alternately move away from us and towards us, hence in the spectrum a periodical displacement will take ‘place. This displacement has usually been taken as indicating an orbital movement and for this reason the Cepheids are admitted amongst the spectroscopic double stars. Still amongst these they occupy a very exceptional position. Calculating the mass from the elements of the orbit, very much smaller values are found for the Cepheids than for other spectroscopic double stars, although their volume is much larger than that of the sun. Although an extre- mely small density is not altogether impossible a priori, still in the relatively small radial velocity an indication may be seen for the assumption, that a different explanation must be given here than for ordinary spectroscopic double stars. But the question arises: is it possible that from an expansion and contraction a radial velocity arises of such a value as the experi- ments give — of several times ten kilometers per second ? The luminosity of d Cepbei and 4 Aquilae was found by Abams from the spectrum to be 60 times that of the sun; for a mean Cepheid with a period of 6.6 days HertzsPruNG derived from the proper motions 600 times the luminosity of the sun. Assuming on the ground of the accordance as to spectral type and colour an equal radiating power per unit surface, these results give a radius equal to 8 and 24 times respectively that of the sun. Representing the maximum expansion and contraction by the factor f—=1 + Af, the maximum radial velocity will be 22 Of. R ~ 86400P, where P is the period in days and R the radius. In kilometres R is 8 or 24 « 7 « 10°. Taking for P 6 days, this gives. = la eas 7 EN EEA 82 resp. 2464 / KM. 4,3 X 10° Since these Cepheids fluctuate rather less than 1 magnitude visu- ally and rather over 1 magnitude in photographic intensity, we shall assume one magnitude for the variation in complete radiation; 742 therefore log L varies by the amount 0.20 above and below the mean. If the radius changes as the number f, the density changes as f°, the temperature as f°/, and the radiation as f7*/,; from log L=+0.20 it then follows that log f = + 0.04, hence f fluctuates between 1,1 and 0,9. In the expression for V we must therefore take 0,1 for Af and the maximum radial velocity becomes 8 or 25 kilometers per second. This value has to be somewhat lowered, since spectographically the mean velocity of the entire front surface is measured, of which only the central parts have the velocity which we have here calculated. But even then the value found: agrees sufficiently with the measured velocities (10 to 20 kilometers _per second) to admit the explanation of the light variation and the variation in radial velocity on the ground of contraction and expansion. There are some other objections to this explanation. The one is the same objection which also holds against the explanation through an orbital movement viz. that the maximum intensity coincides with the highest velocity towards us. The other objection lies in the coefficient 0.48 found by Miss Leavirr. If for these Cepheids equality of spectral class and thus of emissive power and of 7 may be assumed, the brightness becomes proportional to the surface, which gives P? — L. or log P = Const. — 0,2 M. In this case therefore the coéfticient should be 0.2, whereas Miss Leavitt finds a much larger change of the period or a much smaller change of the brightness. It is therefore difficult to explain the deviation by means of a dependence of the temperature 7’ on the linear dimension R; for in that case 7’ would have to be smaller, the larger the star. Possibly an explanation may be found by assuming, that the mass of the Cepheids is actually small, and therefore the density very low, so low, that the rays emitted from one side of the star may penetrate the complete body without being completely absorbed. If a glowing gas-sphere is so rare, that we observe the emission even from the hindmost layers without any diminution, the total light from the sphere will no longer be pro- portional to its surface, but to its mass, therefore be the same for two bodies of equal mass and different dimension. Intermediate conditions are conceivable in which the total light will then be proportional to a lower power of R, say to the first power. In the latter case the coëffficient of M in the formula for log P would become about 0.40. Physics. — “On the Theory of the Friction of Liquids’. By Prof. J. D. van ver Waats Jr. (Communicated by Prof. J. D. van DeR WaAAats). (Communicated in the meeting of November 30, 1918). § 1. /ntroduction. The theory of the friction of gases has been made the subject of numerous researches, the theory of friction of liquids on tbe other hand has met with but scant attention. Yet it is clear that the explanation given to account for the friction of gases — viz. that it is brought about in consequence of this that molecules diffusing from one gas layer to another, at the same time transport an amount of momentum from one layer to another — cannot equally apply to the friction of liquids. For the friction of gases increases at higher temperature. For liquids:on the other hand the viscosity becomes slighter at higher temperature. Such a beha- viour cannot be accounted for with ‘friction by means of transport.” Maxwell calculated that on the supposition of “friction by means of transport” the coefficient of friction 1 should be proportional to VT if we assume that the molecules are perfectly rigid spheres, which do not attract each other. Other assumptions concerning the nature of the molecules (repulsion in inverse ratio with the fifth power of the distance, Maxwerr, or mutually attracting rigid spheres, SUTHERLAND and REINGANUM) lead to a still more rapid increase of y with 7. Nor can the thermal expansion of the liquids explain the sign of the coefficient of temperature of 7. For gases 1 appears to be independent of the volume. For liquids the expansion will promote an increase of 4 with 7, and not a decrease. This has been shown experimentally (except for water, where the reverse takes place), and it is also easy to understand that this is to be expected for friction by means of transport, at least for not associating or dissociating liquids. The expression derived by MaxweE.v: == ot ee NG TT, POR) in which g represents the density, / the mean length of path, s the mean velocity, m the mass, and o the diameter of a molecule, will namely have to be corrected for liquids, to: 744 + eee o—b 1 RET nr B > ee (la) which quantity increases with v. Other well-known corrections have been left out of consideration. Accordingly for liquids we shall not prinecipally have to think of transport of momentum by the diffusing molecules, but we shall have to explain the friction by forces which the molecules exert on each other. If at an arbitrary moment we could suddenly check the motion of the molecules, and if we could arrest them in the position which they occupied at that moment, the friction by means of transport would at the same moment be destroyed, so that we should not have any means to study the friction in those resting molecules. The case. is different for “friction through molecular forces’. At least when we think the molecular forces independent of the velo- city, the frictional forces would continue to exist also after the immobilisation of the molecules. They would be a consequence of the grouping of the molecules in space. Lt is now the question: of what nature are the molecular forces and what is the grouping of the molecules, which gives rise to the existence of the tensor of tension as we meet with it for the friction of liquids. The following three answers might be given to this question: l. Friction through inpact forces or through an instantaneous transfer of momentum. We might assume that the forces that the molecules exert on each other at an impact would furnish the explanation of viscosity. Let us consider the simple case of a liquid in which the current only moves in a single direction, which has been chosen as w-direction of a cartesian system of coordinates, this velocity (2) being a linear function of z, hence: sae, We ee aa Then the layers with greater z will move towards the righthand side with vegard to the underlying layers, if the system of axes is orientated in space in the usual way. A consequence will be that the line connecting the centres of two colliding molecules, which I shall call the central line, will be found more often in the second quadrant of the we-surface than in the first. When the system of coordinates is turned over an angle of 45°/,, so that the + z-axis moves towards the + z-axis, and when the new axes are called w’ and 2’, the pressure that the molecules exert on each other will be greater in os / the <2’ direction than in the «’ direction. It is evident that this agrees with the value of the tensor of tension in this case. 745 Instead, however, of the caleulation of the forees appearing in case of collision, the friction through this cause can also be calcu- lated by means of the momentum that at impact is momentaneously conveyed from the centre of one of the colliding molecules to that of the other. This method of calculation seems simpler and will be carried out in $ 2. Il. Friction for double points. Formation of streaks. We might also: assume the molecules to be electrical or magnetical double points. When they were orientated with regard to each other quite arbi- trarily, they would equally frequently repel as attract each other, so that the mean force would be zero. Through the couples which they exert on each other, they will, however, turn so that attraction prevails. When we now assume that molecules that approach each other, are still little orientated, whereas this is the case to a higher degree with molecules that have moved past each other, and recede again from each other, the molecules whose central line lies in the v-direction will be more orientated on an average than those for which it lies in the 2’ direction, so that a traction in the a’ direction will result, greater than in the 2'-direction, which can again account for the tensor of tension. When we consider more than two molecules whose centres lie on the same line in the w-direction, the couples they exert on each other, will strengthen each other, which can give rise to the forma- tion of a kind of streaks, which still more promotes the friction. It is difficult to compute the accurate amount of this orientation of the molecule axes; it will be different according as one thinks the rotations of the molecules determined by classical mechanics or by the laws of the theory of quanta. Besides there is no occasion in the experimental data to assume that this case actually presents itself. I shall, therefore, not attempt to calculate the friction according to this hypothesis, though possibly it plays a decisive part in the friction of exceedingly viscous liquids, which present themselves as bi-refringent in case of friction, as likewise in the glassy state. HI. Friction in consequence of formation of groups. Finally we can assume the molecules to combine to groups in consequence of their mutual attraction. In liquids at rest these groups will possess spherical symmetry on an average. When, however, a liquid is in a motion for which w= az, these spherical groups will be elongated to ellipsoids. This variation of shape will now again give rise to a greater traction in the «'-direetion. This cause of friction will probably chiefly make itself felt in the neighbourhood of the critical point. In § 4 and following paragraphs | will make an attempt 746 : to calculate the amount of the friction which is to be ascribed to this cause. § 2. Friction in consequence of impact forces. For an accurate ‘alculation of the friction through this cause the accurate knowledge of the distribution of the velocities would be required. I shall, however, confine myself here to an approximate method of calculation of about the same nature as the method of calculation of the “friction by means of transport” for gases by Maxwerr in his papers in the Phil. Mag. in 1860. I shall, namely, assume that the distribution of the velocities of the molecules the centres of which lie in a definite layer z=z, is found by compounding the velocity of the current of the liquid in that layer with a thermal motion for which the unmodified partition of velocities of Maxwrir is thought to hold. The error that we make on this supposition will probably be smaller for liquids than for gases. The free length of path is namely very small here, and the supposition departs little from Maxwe.1’s supposition that the molecules have the velocity of current of the layer in which they have collided last. Even when JrANs’ correction is taken into account for the persistence of the velocities, we shall have to assign a velocity to the molecules corresponding with the velocity of current of a layer which is only a small fraction of 5 removed from the layer in which their centre is situated. 1 shall disregard this small fraction. When we now consider a definite horizontal layer, for which we choose z—0O, an instantaneous transfer of momentum through this layer takes place at every collison for which the centres of the colliding molecules lie on different sides of this layer. At every impact an instantaneous transfer from above downwards takes place and one in opposite direction. These two quantities are equal and of Opposite signs. Hence we may also take into account double the amount of the transfer from above downwards. We shall now first consider the collisions for which the centre of molecule I lies between the planes z==z, and z=2z,+dz, (0 >> z, >-— cosy), the central line’) forming an angle between y and y + dy with the z-axis, and lying in a plane forming an angle between 8 and 3 + dè with the wz-plane. Further the components of velocity of molecule I will lie between w, and uw, + du,, v, and v, + dv, and w, and w, + dw,, those of molecule II lying between w, and wu, + du, ete. The chance that such components of velocity occur is represented for the two molecules respectively by 1) Counted in the direction of molecule | towards I, 747 1 1 —= bu—azioi td UU, Wi, ine al La d= i avm a «A and i ] — — $(ug—aze)® Hats? Ue AU w —_—¢ zt My ded Va erve rs in which z, == z, + 0 cosy. Hence the number of the collisions in question per second and — per surface unity of the layer is: 1 nm? Su ay PE vy? wi (te — a, —aaz 087)? oa wsl P . atcha ag RS UR *" cos 6? sin y dy dpdz . (3) Jt in which ” denotes the number of molecules per cm?., v, the relative velocity of molecule II with respect to molecule I, and u the angle between the direction of », and the central line, so that: vr COs u = (w,—U,) sin y cos B + (v,—v,) sin y sin B + (w, - w,) cos y At each of these collisions the «component of the quantity of motion, which is instantaneously transferred from above downwards is: Wats COs pirapmuy COS BA. rus. ce Fee on oe PAA) The condition that really transfer of momentum through the chosen plane is to take place is: 2 2, > 0 ot ¥ < Bg eos = ( Hence 7 is found by multiplication of (8) by (4), and then by integration with respect to: aes ~ y between 0 and Bg cos —— 0) B ” 0 LE) 2 wv ld Ms 0 erg + Zi ” a 0 ” 0 We have then still to multiply the expression by 2 for the transfer from below upwards. We must, however, still pay attention to something else. In the limits set above collisions have been taken into account which are impossible in reality. Only those combinations of values of the independent variables can occur, for which u is obtuse, hence cos u <0. It is simpler to introduce the condition that v.cosu <0. This condition can be introduced in the way of DiricaLer pAn by multiplieation by few PY dip. a gp hen 748 Which integral is 1 for —p + p. Now if we put p=s and ¢=y,cosu+s, and if we make s to increase indefinitely, the integral appears to become 1 for v‚ cos <0 and 0 for », cos uw > 0. Thus we finally find for the foree which the liquid above the plane z =O exerts per surface unity on that below it: 2n?0?m e= an 1 —qz,)2 2 re) ie “ey )2 ej Seq a Cu 2)? Hr Her + (ug—az — a6 cos 7) + vy + ws} ' pi x fic u‚)sinycos 8 } (v,-v,)siny sin B + (w,-w,) cosy}? >< Xe ‚(5a)') sun sg En LE N ty (ug) sun 7 COS BH rg ey ) sin 7 sin B4-(wWe—w) ) cos Hs PE Et, rl i a ll 7 Sin; z 1 € ww = ZA p en u, u, X sm’ y cos Bdydy dpd — ...d — dz, a it: < oe a — (twa bie § (ug—uy) sin 7 cos BH (va —v)) sin 7 sin 3+ (we—w,) cos ys xe 12. fg) Fig) (ua Î 2 t 2 1X} (55) sin s(p — sin? y cos y cos B dy dB dep du, ... dw, dz, Pp ~ x If we substitute in this u, + bipsiny cos B= &, u, — hep sin y cos B= &, v, + kipsmysn3=—y, v, — kep esin y sin B = 1, (6) w+ } up eos y —— w,— kep cos y El, we get: en ON UT | HELE) siny cos8-+(n4,-n,) siny sin B+ (54-6, )eosy-Hep}? | X (E, HE up sin y cos B) Xe Gi?) —"b P+ es & hts ps (,£9e) ye X sin? y cos y cos B dp dy dp d§, ... d&, dz, On integration with respect to §,...$, terms containing odd powers of £,...6, vanish, so that the only terms left are those with HIE + §,7) sin? ycos? 3 + (1, +2,7)sin?ysin?B + (Cj? + ¢,?) cos*y | kepsinycosp + + 26,7 tg sin y cos B — 4 up? sin y cos B). These terms do not change when §,* is substituted for 5%, 1),*, 4,°,6,7, and £,°, so that + (3&7 tp —4 ep’) sin y cos 3 may be written for the sum of the remaining terms. After execution of the integra- tions we find: 9 . ol . N= — nome if (B-p°) X ear Hits & sin sep X xX sin’ y cosy cos*BdpdydBdz, . . . . . . (5d) Let us now replace et by cos gs-+csings, and execute the integration with respect to ,‚ bearing in mind that we seek the value of the integral for /im.s =o. Then the term with sin sp cos sp vanishes, and in the term with sim? sp we may replace this expres- sion by its mean value 4. Thus we find: +a — fee X sins p XK (8— yg?) dg = V2a — & 2 fora ap 75 0 Bg cos (—2,) ‘ Pe ay do 1 Leip eee an eee sin? y cos y dy = F[sn* x], =f (1l—2;*) Q 0 2\2 _— fan dz, 15 dd ik + Vn n° ot ma ; (5e) — pani =e me be Hay See re 5 tn. eers zee 5 e 15 In the calculation of the number of collisions we have, however, up to now disregarded the influence of the mutual attraction of the molecules and of their dimension in the direction of the velocity. If for this we introduce the usual corrections, we find: „atv o' ma ares E ee EN 15 v—b a in which ¢« represents the difference between the amount of potential energy that the molecules in the liquid possess on an average, and the amount which they possess at the moment of a collision. G. JAcer') and M. Brrrrovin *) had already derived expressions for the friction of liquids; JÄcer considers exclusively “friction in consequence of impact forces’, whereas BRILLOUIN takes besides these also the friction by means of transport into consideration. The method of calculation differs somewhat from that followed above. The results at which they arrive, are in somewhat modified notation : es OSs JÄGER N= ER 6 == Pe A Vv B : D+ : RILLOUIN %j — 508 {| 4 — one : 2(D—o0) In this @ represents the density, s the mean velocity of the mole- cules, and D the mean distance of a pair of adjacent molecules. « and gs are two unknown constants, which will not differ much from 1, and which have been introduced, because all kinds of approximations have been introduced into the calculation, which renders the numerical coefficients not entirely certain. The first term of BRILLOUIN’s formula refers to transport, the second to impact forces. It seems to me that BriLLoviN should also have corrected the first term for the ‘thickness’ of the molecules. In his train of 1) G. Jager, Wiener Sitzungsber. CH, p. 255, Anno 1893. 2) M. Brititouin, Lecons sur la Viscosité des Liquides et des Gaz. Paris. GAUTHIER- Vittars 1907. 751 thought this might have been done by multiplication by a factor 0 = That he failed to do so deprives his test of the experimental data of much of its value, in my opinion. § 8. Test of the formulae for liquids not too near the critical point. Let us call the “coefficient of friction by means of transport” 7,, that through forces of collision %,, and that in consequence of formation of groups 7,. For liquids not too near the critical point we shall disregard 2,. We have further: Ve b? iy pa uri OY in which ec is a numerical coefficient of moderate value. We may no doubt consider this quantity as large compared with 1, so that we shall also neglect 1, When we do so we notice first of all that for constant volume 7 according to the formula must inerease with 7’ proportional to V7. There are only few substances for which the experimental . 00 data are available, required to verify whether the sign of bale: OT» : dr really positive. It is clear that always Fr has been measured, and P dn not ae Ether and Benzene are the only substances for which I 07 0» have found records for — 1), so that Ek can be found according Òpr 07’, to the formula 1 ov 1 On 1 òn 1 On Op 1 On 1 07 ve ar, ndr, = y OD OP» me Se ae 1 OP, ia dv - Vv Op We find: for ether for benzene 1 On / == 0,01075 °) — 0,01853 4) No dT, 1) Been De water, which will most likely also behave abnormally in this respect, and for CO, in the neighbourhood of the critical point, which observations will be discussed later on. , Le Or 8 *) These values, like those of - 57 for other substances given below have Ny B 752 1 oO = 000078. 4 0,00098*) Yo OP, 1 dv ==". .0,001585-%) 0,0011763 *) v or, 1 je — — 0,000139 *) — 0,0000783 °) v Op» This yields for ether Vd Ze 001075 4 0,0088 — — 0,00195 7, OT: and for benzene 1 07 — ze = — 0,01853 + 0,01462 = — 0,00391 1. 07, Theoretically we should find according to equation (5): AEN. wrm BEY 09 Accordingly there is not even agreement in the sign of —~—. When ib ben we, however, take into consideration that the eee fan is the difference of two values which are each about five times the value of the amount sought, and that they are very inaccurately known, fie LSS Dee. . Ret : it is not excluded that 7,07, is in reality negative. Even in the Oly | dv value of > app An error of 4°), is by no means excluded, and the v Opr 1-03 error in the determination of —— 5 ‘will without doubt be many Lao ce 1 -Ov times larger than that in — ae On the other hand it is of course v PT a (64° in which THorre and Ropeer, Phil. Trans. Royal Soc. of London 185 p. 397, A. 1894 comprise their observations. 1) According to WarBure and Sacus. Ann. d. Phys. u. Chem. 22 p. 521. A. 1884. The pressure is expressed in kg. per cm?. 2) According to Amacar 1893, extrapolated for 0° and 1 atm. from the values given in the Recueil de constantes physiques. 3) According to Kopp. 1847. Borrowed from the ‘Recueil etc”. 4) According to Sucmopskr 1910, extrapolated for 0° and 1 atm. from the values given in the “Recueil etc.” 5) According to Rösroen 1891, extrapolated for ge and 1 atm. from the values given in the “Recueil etc.’ been found by differentiation of the empirical formulae of the form: 7 = 753 fdr also possible that even if formula (5) is valid, the value of — —~ must Yo v be negative, in consequence of the factor e RT, or because o, hence also 6 depend on the temperature. The experimental data are not sufficiently accurate to decide this question. More satisfactory results are furnished by another test, which can be applied on a more extensive scale. It consists in this that we Oy compare the experimental values of — : " OT, ") with the values follow- ing from equation (5). For this purpose we write: POs, kouN | Òn Ov HOT, H{\Olo. droll, and in this we put: a GES ed lòn_ 1 (eres AN =) (» jk #) Nar ie eeh Oey.) vR We shall neglect p by the side of i and roughly assume v pa : 8 a 1 Sa and Pk = 57 pe WE then get: 1 07 1 8\° R° Ty? ET TR (=) oe 1 v is the volume per gram-molecule, hence v =m —, so that we 4 finally find to test: 1 07 BAG 3° R* 9’?T;' (1 Ov ; TRT TT ee 1 Ov Borrowing the values of 9, pr, Tr and — Sn from the “Recueil Vv p etc”, we find: (See Table p. 754). The agreement is on the whole as satisfactory as could be expected in view of the many approximations. Generally the experimental value is somewhat smaller than the theoretical one, for ether more than for other substances, benzene and orthoxylene deviating in the opposite sense. For acetic acid and for the alcohols the agreement is much less than for the normal substances. 1) See note 1 on pag. 751. Proceedings Royal Acad. Amsterdam. Vol. XXI. t si ih ues a = as OG oO 1 OT, exp Lf} OT, theor. Pentane 0.01019 0.01269 5.486 4.863 Isopentane 1081 1171 5.774 4.838 Hexane | 1123 1354 6716 5.166 Heptane 1214 1353 7.119 5.456 Octane 1394 1574 9.808 51 Chloroform 1149 1066 | 6.515 4.317 Ether 1075 1463 ; 6.578 4.731 Benzolene 1853 1382 9.770 4.559 Toluolene 1462 1524 ne) 4.867 Orthoxylene 1700 1385 10.871 © 5.170 Metaxylene 1418 1478 1.913 5.223 Paraxylene ') 1472 1414 6.716 5.187 Acetic acid ') 1826. 2607 4.713 4.382 Methyl alcohol 1634 1988 4.527 3.749 Ethyl ö 2086 1250 10.273 4.046 Propyl pe 2887 0970 36. 103 4.421 So far we have tested the temperature-coefficients of 4. We can also test equation (5) directly, namely by for instance calculating 6 from it, and by comparing the values obtained thus with the values of o calculated in another way. When we again omit EE RI the factor e RT and when we put $7 0° Sn num- Pk ber of molecules per gram-molecule), substituting again ——— for a P TS . a 7 v—b, and neglecting p by the side of —, we find: v 1 oO 1) For these substances the values for — 5e also for 0° C. have been calculated 1 p from the empirical formula of RopGER and THORPE, though they are solid at this temperature. C 758 640 /a TN: mp’ a (8) BT Rhee ORE The values thus calculated for 5 are recorded in the table on p. 754. With these the values calculated from the critical quantities: gh TE Nis DE AD Ae AS MATIE l6a Apz. have been compared. N = 6,08 X 10° (SOMMERFELD). They are represented by o' and recorded in the last column of the table. It appears that equation (8) gives values that are in perfect concordance with those of equation (9) as far as order of magnitude is concerned. It is noteworthy that the values for o' differ little inter se, those for 5 presenting much greater differences between each other. The alcohols show again great deviations. 49% Anatomy. — “On two Nerves of Vertebrates agreeing in Structure with the Nerves of Invertebrates.’ By Dr. A. B. DROOGLEEVER Fortuyn. (Communicated by Prof. J. Bork). (Communicated in the meeting of November 30, 1918). As a well-known fact the olfactory cells in the mucous membrane of the nose of Vertebrates are “Sinnesnervenzellen” or ‘conducting sense-cells” as | propose to call them, unless they have received already another English name. (The word “sensory nerve-cell’ may then be reserved to “sensibele Ganglienzelle”). They are sense-cells which are not surrounded by nerve-fibres, but whose cell-body directly passes into a process with all the characteristics of a nerve-fibre. These nerve-fibres, the fila olfactoria, constitute the nervus olfactorius. So the olfactory nerve deviates in its structure from all the other nerves of the vertebrated animals. The truth of this remark may already be deduced from the fact that in Vertebrates besides in the olfactory mucous membrane conductive sense-cells are only found in the retina (rod- and cone-cells) and perhaps in the pineal organ. So all the nerves of the Vertebrates with exception of the olfactory nerve are devoid of nervous processes of conducting sense-cells. On the contrary the majority of the nerves of invertebrated animals do contain processes of conductive sense-cells, which in these animals are always spread about the whole body in all kinds of sense-organs. Often in Invertebrates nerves are composed exclusively of fascicles of neurites of conducting sense-cells as is the case in the olfactory nerve of the Vertebrates. From this I conclude that we have to look upon the olfactory nerve of the Vertebrates as a nerve constructed in a way which is often met with in Invertebrates, and nowhere else in vertebrated animals. Another striking difference between the nerves of vertebrated and invertebrated animals is this that in the Invertebrates ganglion-cells are generally dispersed along the whole course of the nerve either separately or in groups, ganglia. In the nerves of Vertebrates, however, they are totally wanting or they are accumulated in very few ganglia (spinal ganglia or those of the nerves of the brain). Now if one reads (e.g. in the review on this subject by Prof. van WisHE in these Proceedings, Vol. XXVI, 1918) that also in the nervus 757 terminalis of the Vertebrates ganglion-cells are scattered along the whole course of the nerve, then one is compelled to grant to the terminal nerve the character of a nerve of an invertebrated animal. Nevertheless this type of nerve is obviously another than that of the olfactory nerve. In connection with the preceding remarks I should like to point to the fact that Amphioxus, whose nervous system is generally compared with that of vertebrated animals, possesses in its sensory nerves the nerves of a true Invertebrate. All or most of them contain processes of conducting sense-cells as these cells are scattered about the whole body, and moreover most of the sensory nerves are accompanied by dispersed ganglion-cells. Therefore spinal ganglia are lacking. Leiden, Anatomical Cabinet. =) Mathematics. — “Ueber die Teilkörper des Kreiskörpers he DAA (Zweiter Teil). Von Dr. N. G. W.H. Brreerr. (Communicated by Prof. W. KaPTEYN). (Communicated in the meeting of October 26, 1918). Wir beweisen zunächst einige Hülfssätze über die Function F und das Karaktersymbol. Hiilfssatz 1. (oF acme 5) eri jh Qkrni — ai bn ue Ley >> E eh — Uk Beweis: il ak(lk—n)i 2knm i E 2 eae h_j —n ou h == —— | * — | € Ll lh ril [h — |b Ik —l [py Jou — 2 = — = <= e l [h pee Uh To Jeet 2kri | _ 2 —-(— 1) fle =| Hiilfssatz Il: Es sei / die höchste Potenz von / die in w aufgeht. Dann ist n+ Uh Ju n le Beweis: Es sei n + [lp (mod I") dann ist n=! (mod lW") wegen h ch Es sei nun n= rm (mod?) also auch (mod lk—"') dann ist gin! == pn! (mod [h—h') 159 also m == n' (mod lh—K'—1 (l—1)) oder n' = m' HAAL (1—1)v. Hieraus ergibt sich: / Jen ui 2m ui on Ee (L—1)ut n+lh —h' Ju RL TEC B + a a Enk id (ll) — el (Ll) eme d Se Der letzte Factor ist = 1 da w teilbar ist durch 7?’. Hülfssatz U1. Es sei /” die höchste Potenz von / die auf u teilbar ist, sO ist: eni th : ne i Jen wenn die höchste Potenz von /, die auf £ teilbar ist, Plu ke /lh' —bu oh 7) : == F\e! wenn die letzt genannte höchste ioe Potenz = /"' ist. Beweis: Es folgt aus Hülfssatz II: Qhri h euri jh 2lenni En L—1[, Bilan ye ka —1 ih S| buy sn : h n h n+ h I ¢ I ) = 3 | | eb = 3 | e ! na! Ih JA | —eheni akin) h “ Et hl |r = a h' n+l h i= l —2kri 2kri h! h eel iN aal ) 2kri eri as h 1—e! ) F\e ! ) — 0 Wenn nun die héchste Potenz von /, welche in & aufgeht, kleiner ist als 7", so ergibt sich hieraus “=O. Nehmen wir nun weiter an dasz & teilbar ist durch eine höhere Potenz von /,/*+°, e > 0, dann ist : ek A . ll ele Ps jh a 5 n u h 2l F\e ! =| pn De A i | also ist nm heh nme gh 1 =e Ee wegen & = Ute k' 760 also: 2kri he B pe, [rt ole lr) end n=1 s—0 Ks sei n + slh—h'—e = ys! (mod l4) und n=r" (mod lA) dann ist n =r’ (mod lh—h'—c) n=" (mod lh—l'—c) also rs == ' (mod lh—h'—c) s =n' mod g (lh-H'-e) und san +ngp(i-' 2): a Hieraus folgt: ’ ee h—h'—c_ PN Ta) So n 8 mm {ber u oi cH ) =i oe jie =a jh ag moh Lh (wo uly’) 27i n'bu Ubu! EA == 271 et 7 as le v_bu! 271 2 und schliesslich a Uri ji =) a b 2her nt pete Dn h a PRE nk (6) Fe ! — ee EST go = ee n=1 Ë s=0 Wenn nun in der letzten Summe zwei Werte von vs (mod l°) mit einander congruent sind, so ergeben diese Werte zwei gleiche Glieder dieser Summe. Nehmen wir an dasz: a Us, = vs, (mod le) Us, == Vs, + Wlt, und dasz weiter s' =n + vs, ple) und 8’, = 1' + vs, p(lh—h'—e) dann ist auch edi te, ele, =) sn’ + vs, pl HE) + wleg (lh) ss’, + wlep(ll'—e). Weiter ist: n+ sl W-em=rs! und n+ s,lh—h'—c — psa! (mod Ih) 1 ler! _— und auch ist also: n-+ s, Jh—h'—c == psa! + wly ui We; dk == ê == pq! _ pullf el ) nn = (n +s, Uh mij, wl gil haze ) und wegen — pl NA) = 1 (mod Uik), folgt hieraus n + s li—h'—e == Nn J sl —h'—e (mod lh—h') also (mod le) §, = 8, Sehliesslich bemerken wir noch dasz die Werte von v,, die zu zwei verschiedenen Gliedern der Summe aus (6) gehören, nicht Ss gleich sind; denn, nehmen wir an dasz die zwei Werte von », die zu s, und s, gehören, gleich sind, so folgt aus (5) s, = s,, und also würde n+ slk Wer! en n+ a,lh—h'—e = ps! also se = slk ke (mod Ih) = s, (mod U +) “A Und dies ist nicht méglich da s in der Summe = nur laüft bis s= l*t+c_I [h'+ec . Hieraus folgt dasz —— RRT X die Summe aller /-ten Hinheits- wurzeln. Diese letzte Summe ist aber, wie bekannt, gleich Null, so dasz sich ergibt = —=0 und # = 0. s Der Beweis des zweiten Teiles gestaltet sich wie folgt: 2Qhri Qknzi eni =a ae B/L! | —bu ki you > h n h n I ene | be a WA U jh 4 h n l da k/l*’ nicht teilbar ist durch /. Hieraus ergibt sich weiter 2kri ankjit mt Th! bu Hin OW earner CD a l n l 762 Nun durehlaüft nk//” zugleich mit » ein reducirtes Restsystem (mod m), folglich ist: ( a 8 i uni h ALT enn n et Tr hd ——— { Je l Hiilfssatz IV. — 2d eed me e I= F gli — Wh ear jh == 5 al'—! wo {” die höchste Potenz von / ist, die in w aufgeht, und F aus der Function # erhalten wird, wenn in letzterer « durch —vw ersetzt wird. Qn Ist w=} al'—', a gerade, so ist Fed — Yo jat? Beweis: Ks ist ees Har bu ane pry k bu en ene 5 | A's 5 | „dt n=1 Uh EN wenn & nicht teilbar ist durch /. Hieraus geht hervor: Qn h Qrknt —| Fie! == Eur el. Uh n=1 lh 2x ki Wir multipliciren mit e!" und nehmen die Summe über alle Werte von &, die nicht durch / teilbar sind. Es ergibt sich SPE Eus jh et ee F' (5) ce es i | DS n=1 lh} k Wir teilen die letzte Summe in drei Partialsammen : 1°. Die Summe über alle Werte von » für welche n+ 1 teilbar ist durch: Zr ™. 2°. Die Summe über alle Werte von 7 für welche n-+ 1 teilbar ist durch /*-*'-! und nicht durch eine héhere Potenz von J. 3°. Die Summe über alle Werte von » für welche » + 1 durch eine niedrigere Potenz von / teilbar ist, inclusiv die nullte-Potenz. Bei dieser Einteilung haben wir keine Zahl n vergessen. Wir betrachten nun jede dieser drei Summen. 1°. In diesem Falle ist jedes Glied von 3 gleich Eins, also == p(l”). Die hierher gehörigen Zahlen n +1 sind: k Uh, ik, 763 Für die Zahlen n ergibt sich hieraus n=—l Rv, vn =1,2,.....- [A also: Qrbun't n bu h ao el”) n =r" (mod l'), Dann ist aber — 1 =r" (mod U) und also: n' = 0 (mod bg(lk-*')) Folglich : Bl we pap Po, bu ort 5 : n h ; i | ik pl ) —— (—1)o« In der Summe 1°. finden sich /* Glieder vor. Infolge dessen ist der Wert dieser Summe: Dee U U) 2°. In diesem Falle ist: 2nk(n1)t dk — einer /-ten Einheitswurzel. gl’ Die hierher gehörige Summe XZ ist also gleich a die Summe q ke tree de | p (!/) der primitiven Lten Einheitswurzeln, d.h. gleich — ETI p ( . n hu . . . Weiter ist A — einer /-ten Einheitswurzel von + 1 wegen n= —1+/-"—-1y, wo v, alle Werte annimmt. welche < +1 sind und nicht teilbar durch /. Das sind g(*+) Werte. n {ee Auf gleiche Weise wie unter 1°. zeigt man dasz H eine /-te Einheitswurzel ist aus 1 oder — 1 je nachdem bu’ gerade oder ungerade ist, wo w= "wu. Jede primitive l-te Einheitswurzel tritt … 9 (U) in der Summe Y —~ X aut. pl) n Gemiisz diesen Bemerkungen findet man leicht dasz der Wert der hierher gehörigen Partialsumme. > ist: Ch ht ee) bar = 5 Ce) — [h'. [h-1 (— Ie’, 7 (0 pb) 3° Die Summe E kan man verteilen in einige Male die Summe k 764 der primitiven /*~*'—-te Einheitswurzeln, wo c > 1. Daher ist auch x= — 0. n Nach einer kleinen Rechnung findet man nun den ersten Teil des Hülfssatzes. Beweis des zweiten Teils: 2d a n |eabl F'\e = = | — 2 = ace al) 3 a = S(—1 )indne lo = bk (—1)te Poe n — wo 7 den kleinsten positiven Rest bezeichnet von 7! (mod -/) ni ; =i... ESD F ie ie So & +... Non ist Peri) =r,also t==1 t=! 2nr,t I-41 fis a el a (— 1)! e ! fi Die letzte Summe verteilt man in zwei Teile, nl. für ¢ gerade und ¢ ungerade. Nun ist 1, immer quadratischer Rest, und 724 quadratischer Nichtrest. Man findet leicht dasz die Summe eine sogenannte Gaus’sche ist, deren Wert [Yo q'/al—1)? Es ergibt sich nun aus (3) dasz die Klassenanzahl H (welche im 1 Ersten Teil dieses Aufsatzes durch A angedeutet ist) — Mal dem x Producte über u = 1, 2,....a/*-'—1 von (4) gleich ist. Die weitere Herleitung der Formel für die Klassenanzahl, fällt nun sehr ver- schieden aus, je nachdem der Relativgrad 5 gerade oder ungerade ist. Es ist ersichtlich dasz im ersteren Falle der Teilkörper reell ist, und im lezteren imaginar. 1°. Die Klassenanzahl fiir reelle Teilkörper (b gerade). Aus Hülfssatz I ergibt sich kri — kri neen) Weiter ist: arki ior onl kji > af EE = ae s ) = k=1 h —22rki h —2rki —1 h ) 1 h =' > PRe dt — = kF\e ! ij k= | 2nki 2rki hi EI Pr En inZ F\e! J— BokF \e! h Qrki A 2nki U —1 h en | Er 2 > de En = in ) k=1 k=1 Es ist aber: 2nki Qnknt if = wh 7 —1 jh —1 n bu h Se a NSS hele k=) | met We [h 2nkni Ae = oS “gr U nl lh k=1 I iM SS — | ed | © Also ist: und: oder j Qnki (==)! En: h f= a Ne U ) log Ak pita Tijk Jel wo: kri krt ; kri okri Att ik 7 Ta Ti 4 . 4 Ann ve £, Ao ny é l I= l Das Product zerlegen wir in verschiedene andere Producte: 1. Ueber alle Werte von w die nicht durch 7 teilbar sind, 9. Ueber alle Werte von w die durch / teilbar sind aber nicht durch ( schliesslich über alle Werte von u die durch /'-! teilbar sind. Fassen wir nun ins Auge die Werte von w welche teilbar sind durch W und nicht durch MF! 766 Wegen Hülfssatz II] finden wir 2nki Qt po ar kT ub Pe (, i" ) log Ap if i n) = cai log Ax k=1 ‘ k Lk ; wo, in der Summe rechter Hand, & alle Zahlen durchläuft < /, welche teilbar sind durch / aber nicht durch #1. Diese Summe veränderen wir in eine Summe welche sich über alle Werte von k erstreckt, die nicht durch / teilbar sind. Es sei £ = /".k’, so ist kri 2k ri kri 2nki alt + p=") h eis he he ; PAs i, Ae Le IE l eons WS l ) zt ( Qn (Ry (fal yi ) . \l—e kl kri aki mea"); n(x lF'—1) ph—hl); | h aah EE SNE ee are BAN oe l ee PR l Hieraus folgt: Al: == Ans’, A, hi! pt a Ar ohh eee Ar yy Es ist also: eee ee | = ae og Az, re ih (log Aj! +. log Ars” + Ar (ffm) und wegen Hülfssatz IIT: el ]—bu k' aL ji—h! |—bu > | log Ap + = || log Ay, hh te Eee Be ie En) fig ye Die urspriingliche Summe dehnt sich aus über (/—1)/*—"'—1 Werte wie man leicht findet. Diese Summe ist nun ersetzt durch /’ andere und kann daher aufgefasst werden als eine Summe iiber 1, ((—A) WA = (ld) Zahlen. Und diese Anzahl ist gleich der Anzahl der Zahlen welche < /* sind und nicht teilbar durch /. Es sind weiter nicht zwei dieser Zahlen (mod /*) mit einander congruent; denn wäre z. b. ki +. U = fk A lA" (mod If) so wiirde sich ergeben:. 1 =j (mod It’) und dies ist unméglich da # und j beide < /*' sind. 767 Schliesslich ist also: > kj -- ta ‘ s kobe , 4 —— A: oe — Oq « ae Rate k LU tee wo, in der Summe rechter Hand, die Zahl k& alle Werte durchläuft < l* welche nicht durch / teilbar sind. Wir haben nun gefunden: py pe pate Rene hee x¢ LA(al —1) w Ug Uh kk \-—bu SA S| log Ap. dd ols Ih Das erste Product erstreckt sich über alle Werte von w die auf der vorletzten Seite unter 1° genannt sind u.s.w. Fassen wir nun insbesondere das Product ins Auge, welches läuft über alle Werte von uw welche teilbar sind dureh /' und nicht durch /'+!. Wir nehmen die Factoren zu zweien und gebrauchen Hülfssatz IV. Dies ist möglich wegen ON le bu n vat! zj u) H =|; und da w und al——wu beide zugieich durch /# teilbar sind. Man findet dann fiir jedes Factorenpaar: +. Die Factorenzahl wird gleich der halben Anzahl der Zahlen welche < al’! sind, und die durch U” teilbar sind. Also 4((—1) al? Das Product, welches wir ins Auge gefasst haben, hat daher den Wert BHE Dalt —2 Fish =O. Im Falle h’=h—1 gibt es a—l Factoren, denn es gibt a—1 Zahlen 5 ae ek A +4 l thal’—l ta = 768 und fiir die Klassenanzahl findet man h—1 ay" | DE eae H= - = ih AL. : — Il B, | log A TWEE: u. =l ia ge xl Für jeden Körper gilt’): 1 wi D| x =) 2ni+raeR In unserem Falle ist 7 = 2, denn der Körper ist reell und es sind daher +1 die einzigen Einheitswurzeln welche sich im Körper vorfinden. +, = 0, r, = al’—!. Die Discriminante D ist bestimmt in Theorem 5. A ist der Regulator. Man findet, indem man dieses alles substituirt, dasz die Potenz von / verschwindet und alk—1—] lh a [7 Lee Er Hi log Ax u=l k=1 lh ; py rn ge ih ee gal —Ì Das Product is gleich einer Determinante, welche man findet durch ein bekanntes Theorem der Norm einer Algebraischen Zahl*): 2arbuindk ot Bl lof Ay = = LUT) log Az, k LU wo & alle Zahlen durchläuft << /* und die nicht durch /teilbar sind. Es nimmt also ind alle Werte an von 1 bis p (/"). Setzen wir also ind k =1, so ist 1’ =k(mod /). Die letzte Summe nimmt daher folgende Form an: (it) 2abu ti U) 2aruti PO Ra p ne = el CI tog Ay = = gal” ' log At iik t=) al! Zal bal! en En et as J- = i= tal’ +] t= (6 —1)al2?—14-1 he ie 2auti al 2aruti h— h=1 rn el ' log Atal ....4 2 eal log Atol! t=) i=1 el 2auti en 4 = ree, pI q = rt = pirat ee i pt+(b—l)al p= 1) H. Seite 229. 8) Barrzer. Th. u. Anw. der Determinanten. S. 98. 769 Es sei jetzt: ri hl ar (et Ht pt(b—1)al ) ep Pare ET eee — qo ; gmt eft s g ‘ (ie ise he gras (balk 5) So ist: (wo g eine ganze Zahl bedeutet) MENE 2auti ML zie al 2mu(t—1)2 Hees bo A Se U log A i! sil Jil il log Bu log A, Shoots log Agy 2a | ali (1+2+4+ ....+ al’!—1) | log Agye-i log A, .... log Aalt-1_1 log A, + ..... + log Ap ‚log A, log As clog A; Man kan diese Determinante auf folgende Weise umformen. Von den Elementen einer jeden Colonne werden die entsprechenden Elemente der vorangehenden Colonne subtrahirt, wober man mit der vorletzten anfängt, und schiesslieh von den Elementen der ersten Colonne die entsprechenden Elemente der letzten Colonne subtrahirt. Jetzt addirt man die entsprechenden Elemente einer jeden Zeile zu depen der letzten. Alle Elemente dieser letzten Zeile verschwinden dann, ausgenommen das letzte, welches folgenden Wert annimmt: log A, + log A,+....+ log Apt. Dies ist auch der Factor der sich im Nenner des Bruches vor der Determinante findet. Er fallt also hinweg. Die Elemente der Deter- minante haben nun folgende Form erhalten: Tt (pt + rte H.H rt ODE) — (ril 4 |) ik By’ St ne i og € 5 t—1 SALEEN En EA anh ae Pe ET er Oe) Man findet leicht dasz der Exponent von e gleich (1p re rd +p (6—1) aih—l) (ri—rt—}) Ti d wird, also eine gerade ganze Zahl ist. Hieraus folgt: En AEN 0g. = log eee Stas A == log Et 1 ee gia ee te (ieee) 50 Proceedings Royal Acad. Amsterdam. Vol. XXI. 119 Die Factoren 2, welche sich vor dem Producte aus (7) vorfinden, bringen wir in die Elemente der Determinante. Diese werden dann loge, anstatt loge” wie sie bis jetzt waren. Nachdem man noch Zeilen verwechselt hat, findet man die ganze, im Theorem darge- stelte Form. 2. Die Klassenanzahl für imaginire Teilkirper (b ungerade). Der Hülfssatz I ergibt in diesem Falle Zkt Qhai (Toen) Wir zerlegen das Product, welehes von (4) genommen werden musz, in zwei Factoren: 1. das Product über alle geraden Werte von uw; 2. das Product uber alle ungeraden Werte von u. Das erstgenannte Product kann man auf dieselbe Art umformen wie im Falle dasz 4 gerade ist, wegen kari Qk AF) rl) Für dieses Product findet man daher = arg U 1 rami rab Weel bet 25 u gerade ih k==1 Fassen wir nun weiter das zweite Product ins Auge. Hs ist, ebenso wie früher schon gefunden ist: und weiter: ik kai ee 2(lh—k) mi = He In ) bey Ape & B Je A = a Sik =| LE 2kai ese 2kai pn: RETE | sa ie ==. ar AR AC log Ap = — = Fe log Ax, k=1 k=1 Hieraus ergibt sich dasz diese Summe verschwindet. Wir finden daher fiir das Product: ze ale mia! zld = + at rk BAR l k=1 u ungerade ; 1 Um die Zahl — zu bestimmen bemerken wir dasz der Körper x 171 imaginär ist. Und da es ein Garois’scher Körper ist, sind alle con- jugirte Körper imaginär, dh. 7, =0, r, = Sal’. Wiederum ist w—2 (als 6= 1 ist) Wir finden, nachdem die Werte substituirt sind : 2 aki 2 whi ri: I ae ne zerk. Û ) nterle® ling d 2h uungerade l u gerade k — ——— - hes RS SS FE: Tr enn eee 8 = Sapir ral! R (8) hi Die Factoren der beiden Producten werden auf dieselbe Art umgeformt wie für 6 gerade. Im ersten Product erscheinen dann die Factoren JES —bu SE k AR | wo wu teilbar ist durch / und nicht durch +! und die Summe erstreckt sich über alle Werte von & die teilbar sind durch und nicht durch MH. Die Summe S formen wir auf folgende Weise um: kli as Ke" —bu |; Ke |b! ~j—bu k —Jh1 >» ns Een h—h! EAN en kl + eet En h! 1—bu k (”—1) [h—h al 7 pt )- hk! | —$bu She MN oe Em =| d | Uebrigens ist ef yh —bu hie da es die Summe aller (/—1)/'—’’-1-ten Einheitswurzeln ist, multi- cirt mit einer ganzen Zahl. Weiter sind nicht zwei der Zahlen 4//*'—: l/—! (mod (*) mit ein- ander congruent. Keine dieser Zahlen ist teilbar durch / und ihre Anzahl ist gleich y(/"). Man findet also k/l* —bu 7 & J—=bu = == ser niger wo in der letzten Aan die Zahl & sich erstreckt uber alle Werte < /* welche nicht durch / teilbar sind. Nach Umformung aller Factoren findet man sodann, wenn man bemerkt dasa gerade ist und dasz das erste Product 3 a/’—! Factoren enthalt und das zweite 4 a//—!—1: Il pou Ul by @ly#) Ts i | Wisa DBs: ate H log Ak a uungerade k=1 l _ ugersdekt LY] ROPE gl Pe al! R 772 Die im ersten Factor auftretenden Summen kan man auf folgende Weise umformen : 2a bu ind ki Ue iy en Sn H bee Boe Le Oe Es sei indk=t so ist k=rt=r; (mod l') wo 7, den kleinsten positiven Rest bedeutet von 7! (mod /*). Dann wird Pl -1) 2nuti Ee » ali rt ll Wenn man diese Summe verteilt in andere die sich erstrecken von “= 1 bis «== al", so -ergrbt- sick: 2 uti al*—! ia ijf al ‘ ‘ arta 2 "t+jalb-t Es ist leicht ersichtlich dasz die Summe & für jeden Wert von d j teilbar ist durch //, Die Summe des zweiten Productes lässt sich wie folgt umformen : 2auiindk Inl fe You — = = H log Ay = Ze al? log Ay = kl 5 , ! 1) 2auti a" 2ruti ARTE al! == e log A w= 2e log A,= el = 3) $ Ry yal? eA rere 2 aruti ral! 2arutt 2 ht || aay) ME 35 = 2 pul log A; + 2 el log Ag zal! i=1 t—4al’—141 iI =1 (weil w gerade ist) 2aut (== Auf gleiche Weise, wie im Falle dasz 6 gerade ist, kann man diese Summe in eine Determinante verwandlen. Wenn man dann die Factoren 2 des Nenners aus (8) in die Elemente der Deter- minante hinein bringt, so findet man die im Theorem aufgegebene Determinante. Man musz nun noch die Potenzen von —1 beachten, und von 7, die enstehen bei der Benutzung des Hülfssatzes IV. 773 Wenn wir schliesslich noch ein System reeller Grundeinheiten nehmen, so ist log n(®) = RIES, (3) a Wir müszen also in A substituiren (,(4;) = 2/og 4) wonach die Potenz von 2 im Nenner verschwindet. ') H. Seite 215. 2m Mathematics. — “Ueber die Teilkörper des Kreiskörpers ip Von (Dritter Teil) ). By Dr. N. G. W. H. Brererr. (Communicated by Prof. W. Kapreyn). (Communicated in the meeting of November 1918). Der Ausdruck fiir die Klassenanzahl der primären Teilkörper hat die Form eines Productes zweier Brüche, wenn 5 eine ungerade Zahl ist. Ebenso wie es der Fall ist mit der Klassenanzahl des Kreis- körpers der /'-ten Einheitswurzeln selbst, sind auch hier die zwei Brüche ganze Zahlen. Ich will dies hier beweisen, und nenne dabei die Briiche den ersten, resp. zweiten, Factor der Klassenanzahl. Zuerst beweise ich einen Hiilfssatz. Hiilfssatz 1. Jedes System von Grundeinheiten eines primären Teilkörpers, in Bezug auf welches der Kreiskörper selbst den Relativgrad 25 hat, ist auch ein System von Grundeinheiten des primären Teilkörpers, in Bezug auf welches der Kreiskörper selbst den Relativgrad 5 hat, wenn 6 eine ungerade Zahl ist. Beweis : Wir nennen den ersten Teilkörper & und den zweiten Á. Ks ist dann & ein Teilkörper von K. Der Grad von K ist ce; der von & ist dann Je und be = g (l*—!). Wir zeigen erst dasz & in K zu der Substitution siske, wo s==(Z: Zr), gehört. (r eine primitive Wurzel von /*). Die, den Körper A erzeugende, Zahl ist: ne ree „be Also ist c rc c+/,be bed-lfgbe A LR Nach einer kleinen Rechnung ergibt sich: „ilse (26—1)'/gc rac r ZZ +...-2 Die, den Körper & erzeugende, Zahl ist: Tt Mec 2. Use EE a? A shbe ng = Z _r(2b—1) Ya ae ee aA 1) Fortsetzung von ,,Proceedings’’ XXI Seite 454 und Seite 758. 775 Man sieht also dasz Nk = NK + she nK woraus sich ergibt dasz die Zahl 9; durch die Substitution s’/22 nicht geändert wird. Es ist daher & der Teilkörper von A, der zu dieser Substitution gehört. Man folgert nun aus (1) dasz s'/teng conjugirt complex ist mit 7K da p'labe = —1 (mod Py, ; Es sei jetzt H eine Einheit von XK, dann ist also s'h’°H conjugirt complex mit £, und daher nee | — veen den absoluten Betrag 1 besitzen. Man weisz dasz eine solche Einheit immer eine Einheitswurzel ist). Im Körper A bestehen aber nur die Einheitswurzeln + 1. Man schlieszt hieraus ist also eine Einheit, die selbst, und deren Conjugirte sämtlich EEN (2) shoe E mnd e . . . . Ld . . 5 Wenn nun WE — shtek wäre, so würde sich ergeben dasz der reelle Teil von F gleich Null wäre, und also £ das Product einer reellen Einheit mit der Zahl # H? würde also reell sein. Setzen wir #2? =—=e, KH= Ye. Die Relativdifferente der Zahi £ in Bezug auf den Körper & ist daher 2)’, da die relativ conjugirte Zahl von We gleich — We ist. Die Relativdifferente der Zahl £ ist also nicht teilbar durch /. Die Relativdifferente des Körpers K in Bezug auf % kan daher auch nicht teilbar sein durch /. Der Körper K ist relativ eyelisch in Bezug auf /. Ich benutze darum einen Satz’) der relativ cyclischen Körper des Relativgrades 2 : Es sei { ein in / aufgehendes Primideal des Körpers 4, dann ist | nicht teilbar auf die Relativdifferente von A. Man schlieszt hieraus dasz die Primzahl / in K nicht teilbar ist durch das Quadrat eines Primideals. Nun gilt aber in & die Zerlegung L = I've und in K: t==1,¢ wo l,= (1-2) © wie sich aus Satz 4 ergibt. Es ist weiter in K: CIS Hieraus folgt aber «2, und 7 ist in A wohl durch das Quadrat eines Primideals teilbar. 1) Hl. Satz 48, %) H. Satz 98. 776 Man schlieszt hieraus dasz die Gleichheit E= —shbe E nicht bestehen kann. Dann musz aber E == sabe K, HK ist daher eine Einheit des Körpers- 4. Satz 1. Der zweite Factor der Klassenanzahl von K stellt die Klassen- anzahl des Körpers # dar, und ist also eine ganze Zahl. Beweis: Aus Satz 6 (S. 461) ergibt sich dasz die A der beiden Körper einander gleich sind.) Aus dem Hülfssatz folgt dasz auch Bed Satz 2. Der erste Factor der Klassenanzahl von A’ ist eine ganze Zahl. Beweis: Ks ist leicht ersichtlich dasz PET Tepe Ts. HE PE (bie teilbar ist dureh /*. Man braucht daher nur noch zu zeigen dasz uti WSee (ry vie Hes HFD) visi teilbar ist durch 2'2¢-!. Die Summe teilen wir in zwei Partialsummen: IJse c a en >>; ‘=I t=1/4c-++1 Fiir die letzte kann man schreiben: "ge 2ru(t!/yc)t 7 — ie e Pein ri43 ac 45 es == Piode) t=1 Ks ist aber: rijn = nbn = — rn hbe == — rit npifybe (mod U) und daher: Pen = Ur pntpi abe. Nach einer kleinen Rechnung findet man dasz die in der letzten Summe auftretende Form zwischen () gleich Dalet ripe +. + M4(6—1)e) 1) Im Satz 6 musz, im Falle 6 ungerade, vor den ersten Factor der Klassen- anzahl, (— 1)'¢ gesetzt werden. Vor der Determinante soll nicht (—1)'/s(¢—2\\a—4) stehen, sondern (—1)'/s(c—4\(e— 6), on ‘wt a ist. Da w ungerade ist, ergibt sich nun für die im Producte auf- tretende Summe: yi, emule 1), 2ruti ge ge == a € . Bet (outre «4 Fiber) +26. Be € t=1 Es ist also, nachdem man die letztere dieser beiden Summen berechnet hat: —2rui Qrutt (it Bere et Papet en re beie) = =1 i Qrutt c —=2ib.lk Ee © (rid... Hrib-e) i=! Met Riicksicht auf die Beziehung Sisto (lee )=2 u findet man dasz das Product, welches im Zähler des ersten Factors der Klassenanzahl auftritt, teilbar ist durch 2/2! Man kann den Beweis dieses Satzes auch erbringen auf die näm- liche Art wie Kummer und Kronecker bewiesen haben dasz der erste Factor der Klassenanzahl eines jeden Kreiskörpers (der m-ten Einheitswurzeln) eine ganze Zahl ist, oder ein Bruch mit Nenner gleich 2. *) Verzeichnis einiger Werte des ersten Factors der Klassenanzahl, | | drop eh b | erster Factor. Nes ANP | | ‘4 1 3 1 7 2 3 | 11 1 5 | | 13 ee an 1 19 1 3 1 23 | 1 11 3 29 | | 1 1 31 1 15 3 at ed 9 | 37 | 3 1 47 1 23 5 ~ 1) Monatsber. Berlin 1863 und Crelle’s Journ. Band 40. 778 Satz 3. Der erste Factor der Klassenanzahl des Teilkörpers, für welchen die Zahl 5 ungerade ist, ist ein Teiler der Klassenanzahl des Kreis- körpers (der /-ten Einheitswurzeln) selbst. Beweis: Der erste Factor der Klassenanzahl des Kreiskörpers selbst, ist: *) 2aruti et) en ke et en u =" Da 9—-1RYap—1) wo, zur Abkiirzung gy den Wert der Function ¢(/") vorstellt, und das Product sich erstreckt über alle Zahlen 1, 3, .... g—1. Man weisz dasz der Bruch einer ganzen Zahl gleich ist. Die im Zähler auftretende Summe bringen wir in die folgende Form 2nuti Zult +e)i 2au(t + (b— Dei Cc zie oe mte Piet --- +e gp Ti4+(b—l)e i=1 Fiir die Werte BG Se Tse ee | hat diese Summe denselben Wert wie die Summe welche auftritt im Zähler des ersten Factors der Klassenanzahl des Teilkörpers des Relativgrades 6. Der letztgenannte Zähler ist daher ein Teiler des Zählers des Kreiskörpers selbst. Der Quotient der beiden ersten Factoren der Klassenanzahlen ist nun 2Zauti OENE Ee B mee en Zap tac [AC fap—"ac—1) tt wo w alle ungeraden Zahlen 1, 3,.... gy—1 durehlauft die nicht durch b teilbar sind. Es ist weiter: 2arut 2aruti 2au(t+ 1) SOON JANE os pO ie (re p -1) Ze p M= = € - (rri-r{41) ist =| und rr =0 (mod I). Auch ist: 2 2aui lp ed —— a (re 1 wt? ies} Hire P21) > = 5 2u rie 4 1 DH. Satz 141. 779 wo, im ersten Product, « alle Werte 1,3,....g—1 durchläuft die nicht dureh 5 teilbar sind; im zweiten Product aber, durchläuft « alle Zahlen 1,3,....g—1. Der Zähler des letzten Bruches ist teilbar durch /*; der Nenner nicht. Man kann die primitive Wurzel 7 so wählen dasz der Zähler nicht teilbar ist durch /*—'. Hieraus ergibt sich dasz der Zähler aus (3) durch die Potenz von / teilbar ist, die sich im Nenner von (3) vorfindet. Auf dieselbe Art wie beim Beweise des vorigen Satzes kann man beweisen dasz der Zähler von (3) teilbar is durch die Potenz von 2 die sich im Nenner vorfindet. Auf ganz gleiche Art kann man den Satz beweisen: Wenn ein Teilkörper & Teilkörper ist eines Teilkörpers A, so ist der erste Factor der Klassenanzahl von / ein Teiler des ersten Factors von K. (Der Relativgrad 6 ist für beide Körper also ungerade anzunehmen). iv ta. vC, ONE a? Me ith se Ik Cu aie ‘ad Are Ne a a Beit waka wat "Tobit s, vi otk LONT de Ser iia ; rant aa A i st Mrs ate aib Je site } tte? zi Î ‘ ; ee) je “it ih ae a Nobalier (Bi #07) peo: he “uur esus MOTTO: BONS Oeekvaaes ial Ate ht, AT É ste oils Gait We difoh 18) (Bh mortsel dist eh ith joie via ? ata Noir: gade”. ae sing * ! « i. “i Ce ian “er « kk ot tig ee Rat Sine aaa date outs ane fat oe A FeO Teta Ief es Bayo Ke by ids 7 dl HSP sab Ale This A UO esta its hak | Shai ie hadde heg zoe wid. hn aa iat ca EN % € a ao Dt ie A | KD = a ae 7 % k iP - d 4 id 3 ‘ ke = Aue hd ¢ D . ar KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -- PROCEEDINGS OF THE SECTION OF SCIENCES VOLUME XXI — (IST PART) — oes Th JOHANNES MULLER :—: AMSTERDAM : APRIL 1919 s . x EEV | DRUKKERW HOLLAND © AMSTERDAM ie) oe ‘ SI a Ly Ta ° M DE an ba tah: teh a i hone ARO de ia Ve Dien) ee ees TAT