FOR THE PEOPLE FOR EDVCATION FOR SCIENCE
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KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN -- TE AMSTERDAM -:-
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PROGEEDINGS OFTHE eee LION MOE SCIENCES
VOLUME XXI
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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°—<an, hence 1 1 a, approaches w x — « 1 n° == oo. Now 2,’ lies between 1—n? = +1 Nn
at high temperatures and (O0) at low temperatures, so that at high temperatures 0 will lie between + O° and 90°, and at low temperatures between (90°) and 90°. And the limiting value of p, is near oo, ie. 7, near O, so that the available range of temperature is very large in this case. That a, now becomes infinite, is not astonishing, for to obtain a finite value, F(r) should decrease much more rapidly with » than is the case on our assumption (8) — viz. in inverse ratio to 7’. This assumption, however, only holds for not too large values of a: s.
§ XVII. Calculation of (a,),.
Now we must carry out the second integration in (7*). This applies, therefore, to all the molecules that can come in collision, as Ó now remains smaller than the limiting angle @,. It should be carried out in two stages, viz. from «(—cosé)=p to c=a,, and from «=—1(6=—0) to «=p. -For in the general integral with respect to 7 (see $ ioe viz.
en Vp? ra’ (pe — cos’ cos 0)
p? — cos? 6 = p?—a’? will be positive in the first case, negative on the other hand in the second case. Accordingly the first stage gives rise to a Bgty, the second to a log. The first stage, integrated with respect to_r, yields:
1 23 2/2 3 pra (p—a')\" El ct aVp— = g q a? (p?—2’) i: ae 1 sv == as Penge ——— — Bg tg es | aV p? — x? Vp at V p?—a? “
because p(t “= 5 is ==,’ (see § XVI). Hence we have
a’?
Vat —wx, ry — tf = ; “ia oie rr? or Py tg Ze 4g fi EN "Vp? —? =]
ee ; 1 : 1) Botg fer namely = Boig = Bg cosn = 4m — n, hence Bgg =
= 4 nnn.
6
x p 1 The first integral yields > (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, <x dlog = fe Vp’ Wo el / X d log, 0
p seeing that
Var a v?—p? BE log Va : == log V (ir TE): + yi) Er k . p
while from (#?—yp’): (v’—2a,’) = y’ follows 2? = (p?—y’«,*) : (1—y’)
e
and w*—a,? = (p?—a,’) : (1—-y*). Now log Y= = — bgtghyp y, so that we find with bgtghy=y:
1
ae . cos hw wdyp=k wdy,
V p*—2, ef v : I 1+ costh bes *_ ty*h wp tg 99:k 1s ”
a’ 9” En because —— +5 ee y can be substituted for cos°h Wp — a
& a x
XK suvhw, with sin?h w = cos*h p—1, and s? : (a?—s?) is = k? (see § XVII). For 2,7: p> 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. <wXx}. <4. Then (n=1—d) : n3(1--n)’ log (1: n°) becomes 2 (1—n) in (13°), so that (a,), will approach wo X lo : : V1—’¢
If on the other hand » —0 (a large with respect to s), then (a), approaches w xp in (139, whereas this quantity will approach
infinite >< (/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 While p remains:
>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.
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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 intervalO<s<V 38a. _ While the following discussions will be based on the fundamental equations (5), (6), (7), we suppose u, w,7, 7, 75, Tú to be functions of s. As s is the distance from the centre of symmetry expressed in natural units we obtain, attending to the meaning of the quantity u (see I § 3)
ds == die? leskaart sat (38)
As (37) gives by differentiation
s? Gi SS (5 — 1) be aN ee Meen A ord MRL (39) a
we find for u
rn EE ae PIN B el
ae That w is negative for s< a, does not cause any trouble, as the fundamental tensor depends on u? only. Now we must introduce in equation (6) the expressions (37) and (40) for r and w. Introducing to begin with the expressions on the left hand-side only we obtain
"ll a er : hee or re 3 a =O =S Ta 5 ie 5 = (dla)
Introducing the expressions on the right-hand side too we find for 7,* as a function of s
NN ee A Le a? —1 3a? The formulae derived here hold evidently only inside the material system of which the outer boundary may be indicated by s=S.
In order that the space occupied by the system may cross itself at any point we must have because of (87). S>V38a.
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<cs<ST% is
negative, which though somewhat startling is not at all absurd.
Further formula (41) indicates that di becomes infinite for s = 3 a. Within a finite extension there is however only a finite mass of matter, which follows from the fact that "Tí is everywhere finite according to (41).
The equations (40) and (41) for « and Ts involve together with (37) that the fundamental equation (6) is satisfied. Now we must still determine w, 7. and 75 as functions of s, so that also the equations (5) and (7) are satisfied. As (5), (6) and (7) form the complete set of field equations for a stationary gravitation field, we
79
may choose for one of the quantities w, 77. and 75 an arbitrary
single-valued function of s. When also the expressions for 7, w and 4 . .
7’; are introduced, the equations (5) and (7) determine now the two
quantities. All these possible material systems give — if only the : é 3 Db dns f . 5 distribution of 74 is the same — a three-dimensional space of the
same curvature, because the formulae (25)/ perfectly determine the space-components of the fundamental tensor (when p= 1). The curvature of the four-dimensional space-time continuum on the contrary depends also on the distribution of 7’, which quantity according to (5) influences w. The following simple assumptions might e. g. be made to obtain a definite system : w — constant, 7 = 0 or T= Tí (normal pressure in all directions). Performing the integration of (15) and (7), we might choose the integration constants in such a way, that at the boundary s== 9 w takes the value that holds there for the outer field, for, as was proved in $ 1, 2 changes continuously into a surface of discontinuity.
The purpose of our investigation being reached no further calcu- lations will be added. We have shown that Einsteins theory of gravitation really admits such a distribution of the stress-energy-tensor T, that the (three-dimensional) space crosses itself at a certain point. We can also prove without difficulty, that systems can exist in which the space filled with the matter runs out into a narrow neck.
It is still of some importance to investigate the action of the electric forces within the space which is just dilated and afterwards again contracted. We might e.g. investigute the state, when, with constant Ti, 7”, T'5 for the non electro-magnetic matter, a point charge was placed at the centre of symmetry. The gravitation field will evidently change. We have not only to calculate this field, but also the laws of the equilibrium and the motion of other electric (point)-changes in the new electric field. Here we must treat the matter as perfectly permeable. These indications may however suffice, which show already that Einstein's theory opens wide possibilities to explain the state in the inside of an atom.
Chemistry. — ‘Determination of the Configuration of cis-trans isomeric substances’. By Prof. J. BöÖRSEKEN and Cur. VAN Loon.
(Communicated in the meeting of May 25, 1918).
1. The appearance in a number of isomers of unsaturated and cyclic compounds, has undoubtedly been a momentous incitement to the acceptance of van ’T Horr’s hypothesis on the carbon atom, which is supposed to lie in the centre of its valencies.
The permanence of the optical activity at moderate temperatures necessitated the attribution of a rather great stability to these valencies. Obviously in cyclic molecules the same rigidity had to be accepted, and also around the double bond.
Apart from objections that may rise against the substance of this
supposition — objections connected with hypotheses on the internal structure of the atom — it must be granted that a very elegant
interpretation has been given of the existence of the aforesaid isomers.
In fact, only very seldom cis-trans isomerism could not be observed when it was to be expected according to this theory, while on the other hand, if identical radicles are united to one of the unsaturated atoms or members of the ring, and therefore no isomers are to be anticipated, they were indeed not to be found.
2. This interpretation of the existence of cis-trans isomers is little to be doubted; however, it is much more difficult to determine which isomer has the cis- and which the trans-configuration.
By a happy coincidence the classical case of cis-trans isomerism, viz. that of maleic and fumaric acid, has offered, also in this respect, the greatest certainty.
GOOR ROOG GS
| hee: BEG OOR H-—C—CO0H maleic acid fumaric acid
The cause is evident: the configuration determination here is based almost exclusively on properties that may be deduced from the molecules themselves.
The determination namely of the configuration of geometrical isomers takes place along different lines.
a
81
First of all the configuration may be deduced from properties that are to be anticipated in consequence of the reciprocal influence of the groups in the molecule; this is the surest way, if enough (critically examined) data for comparison be available.
Among these properties are: dissociation constants of acids, for- mation of anhydride, resolution into optical antipodes, etc.
The formation of complex compounds is to be included in so far as the valencies, which are the bearers of cis-trans isomerism, are not attacked.
This is seldom easily established, since the structure of complex compounds is uncertain. Probably we are authorized to use this method in judging the cis-trans isomerism of cyclic glycols by the influence on the conductivity of boric acid,
| because the boric acid-radicle is united to the oxygen atoms —C—0.. and not to the carbon atoms, which determine the isomerism.
|
| _BOH It is doubtful if it may be applied to cis-trans non-saturated —C—0O.” acids; if the formation of complex metallic compounds must | be represented by the following formula:
H H—C—R HO—C—R | + HgO = | H—C—COOH H—C—CO
att Hg —O
or anyhow, if it is to be supposed that the double bond is attacked, we can rely upon this method as little as on all others, by which the bearers of the isomerism are affected (see below).
Now the contrast between the two groupings H and COOH, which are bound to the very simple skeleton of maleic and fumaric acid, is exceptional; besides, the two carboxyl groups may enhance each other’s acidity or react with one another under anhydride formation. Also a mutual repulsion of the COOH groups is to be anticipated, from which could be deduced that fumarie acid is more stable than maleic acid.
The particularly simple structure of these acids, by which the carboxyl groups take the leading position, has rendered the above- mentioued considerations so successful in determining the configurations.
As soon as this reciprocal influence fails or the structure becomes much less simple, we have no longer any certainty.
We will examine a- and zso-crotonic acid in this respect:
i. The dissociation constants are: a-crotonic acid 2.10~°.
WSO- _ „ js 36 102:
By comparing propionie with acetic acid it could be deduced that a methyl group weakens the acidity, so that in «-crotonie acid the methyl group must lie on the same side as the acid group; in fact acrylic acid (t= 5,6.10-5) is dissociated in a higher degree than
6
Proceedings Royal Acad. Amsterdam. Vol. XXI.
82
both erotonie acids, and dimethyl-aerylie acid (+ 7 .10~—®, according to preliminary determinations of Mr. P. E. VrrKape, Dissertation Delft 1915, 2rd Note, p. 66) is weaker still. Also citraconic acid is much weaker than maleic acid.
Still one ought to be cautious, for {so-butyric acid is somewhat stronger than propionic acid (1.44 to 1,31. 105) and the dimethyl succinie acids are much stronger than succinic acid (1,9 and 1,3 to IND ee):
2. Formation of an anhydride is not possible and therefore cannot help us. ;
3. Formation of complex compounds: BuLMann has shown that maleic, citraconic and alfo-cinnamic acid form salts with mercuric acetate, which are soluble in sodium hydroxide, and from which the original acids could not be regenerated by elimination of mercury ; in this case p-hydroxy-acids were formed and in consequence B. surmises that the salts were complex mercury salts of these hydroxy- acids, which are formed thus;
H—C—COOH HO—CH—COOH || | > | 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
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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.
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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,,
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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.
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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.
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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