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KONINKLIJKE AKADEMIE
VAN WETENSCHAPPEN

-- TE AMSTERDAM =:-

A

PROCEEDINGS OF THE
SECTION: OP SCIENCES

VOLUME xx —
=} (135) PART) =
BENE AB oe

JOHANNES MULLER :—: AMSTERDAM
: MARCH 1918

cae i 7 ih x ae 4 Rider de i

eee :. tbs vt Se
“ ee Po E a 4
| EN, i 7 Ab (J £ wee cy He
i BET A0 bass say ee AE es Ak
WRU MAIL ENA + Me
nr Aes YROT EUG ARAL SVR a

: § i C06
; ‘ log > me | H
" ke i 6 ‘ a pr 2 i
te
4 kr ie
"ae Fi 4 rt 7 .
> i 5 if

dye ace ay 4

sa PF (Translated from: Verslagen van de Gewone Vergaderingen der Wis- en, rk
Matuurkdndige Afdeeling Di. XXIV, XXV and XXVI. ze

i

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+
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Proceedings N°.
Ns
Ned
NS

N°

CONTENTS.

Se

à4 1

629

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS

VOLUME XX

NEE

President: Prof. H. A. LORENTZ.
Secretary: Prof. P. ZEEMAN.

(Translated from: “Verslag van de gewone vergaderingen der Wis— en
Natuurkundige Afdeeling,” Vol. XXIV, XXV and XXVI).

CONTENTS.

H. A. LORENTZ: “On EINSTEIN’s Theory of gravitation”. Part III. p. 2, Ibid. Part IV, p. 20.

J. P. TREUB: “On the Saponification of Fats.” I. (Communicated by Prof. P. ZEEMAN), p. 35.

P. VAN ROMBURGH and J. M. VAN DER ZANDEN: “On Polymers of Methylchavicol.” (Preliminary
communication), p. 64.

C. F. VAN DUIN: “Action of Organo-magnesium Compounds on, and Reduction of Cineol.” (Commu-
nicated by Prof. P. VAN ROMBURGH), p. 66.

Miss M. A. VAN HERWERDEN: “On the nature and the significance of volutin in Yeast-cells.” (Com-
municated by Prof. C. A. PEKELHARING), p. 70.

H. I. WATERMAN: “Influence of different compounds on the destruction of monosaccharids by
sodiumhydroxide and on the inversion of sucrose by hydrochloric acid. Constitution-formula
of g-amino-acids and of betain”. (Communicated by Prof. J. BOESEKEN), p. 88.

TH. DE DONDER: “Sur les équations différentielles du champ gravifique”. (Communicated by Prof.
H. A. LORENTZ), p. 97.

B. P. HAALMEIJER: “On Elementary Surfaces of the third order.” (First eoninranicaltba. (Communi-
cated by Prof. L. E. J. BROUWER), p. 101.

W. VAN BEMMELEN and J. BOEREMA: “The semi-diurnal horizontal oscillation of the free atmosphere
up to 10 km. above sea level deduced from pilot balloon observations at Batavia”. (Communi-

cated by Dr. J. P. VAN DER STOK), p 119. (With one plate).

Proceedings Royal Acad. Amsterdam. Vol. XX.

Physics. — “On Einsteins Theory of gravitation”. WI By Prof.
H. A. Lorentz.

(Communicated in the meeting of April 1916.) ')

§ 32. In the two preceding papers’) we have tried so far as
possible to present the fundamental principles of the new gravitation
theory in a simple form.

We shall now show how Einstein's differential equations for the
gravitation field can be derived from Haminton’s principle. In this
connexion we shall also have to consider the energy, the stresses,
momenta and energy-currents in that field.

We shall again introduce the quantities g,, formerly used and we
shall also use the “inverse” system of quantities for which we shall
now write ge. It is found useful to introduce besides these the
quantities

gab — y= ge,
Differential coefficients of all these variables with respect to the
coordinates will be represented by the indices belonging to these
latter, e.g.

Ògas 0° Gab

Jab,p a 1 Jab‚pg — ef

We shall use CHRISTOFFEL’s symbols

ab
, EZS 4 (Jac, b —~ Ybe, a TE Jab, c)

and RreMANN’s symbol
(tk, lm) =F (Jim, kl + Qkl, im — Gil, km — Ykm, il) +

Sab) ge im|[kl il) [km

+ zoe TL JL ILS
Gin = (hl) oF (tks lm) tl es ee
G = = (Mgr Gin os >.

This latter quantity is a measure for the curvature of the field-
figure. The principal function of the gravitation field is

Further we put

1) Published September 1916, a revision having been found desirable.
2) See Proceedings Vol. XIX, p. 1841 and 1354.

3

1
— ds ,
mak

GEE
In the integral dS, the element of the field-figure, is expressed in
x-units. The integration has to be extended over the domain within a
certain closed surface 6; « is a positive constant.

where

§ 33. When we pass from the system of coordinates w,,....a, to
another, the value ofG proves to remain unaltered; it is a scalar quantity.
This may be verified by first proving that the quantities (ik, lm)
form a covariant tensor of the fourth order). Next, (gf!) being a
contravariant tensor of the second order’), we can deduce from (40)
that (Gin) is a covariant tensor of the same order’). According to
(41) G is then a scalar. The same is true‘) for Qd S.

We remark that gia = gar’) and Jas,fe = Jatef. We shall suppose
Q to be written in such a way that its form is not altered by
interchanging goa and gay Or Jasfe ANd Gases. If originally this condi-
tion is not fulfilled it is easy to pass to a “symmetrical” form of
this kind.

It is clear that Q may also be expressed in the quantities g@ and
their first and second derivatives and in the same way in the ges
and first and second derivatives of these quantities.

If the necessary substitutions are executed with due care, these
new forms of Q will also be symmetrical.

§ 34. We shall first express the quantity Q in the gq,’s and their

1) This means that the transformation formulae for these quantities have the form

(tk,lm)' = & (abee) paiPbk Pcl Pem (ab, ce)
See for the notations used here and for some others to be used later on my
communication in Zittingsverslag Akad. Amsterdam 28 (1915), p. 1073 (translated
in Proceedings Amsterdam 19 (1916), p. 751). In referring to the equations and
the articles of this paper I shall add the indication 1915.
2) Namely:
gh = Pe (ab) Nak 741.9%.
The symbol (g/) denotes the complex of all the quantities gk,
3) Namely :
Guan ze (ab) Pai Pom Gap.
4) On account of the relation
ig! as Ag dS.
5) Similarly:
got — geb, gea — gv
1%

+

derivatives and we shall determine the variation it undergoes by
arbitrarily chosen variations dg; these latter being continuous functions
of the coordinates. We have ae

S9ab,e —

0
JQ = Zab) a dgab + Za)

Jak ee
By means of the equations
0 ò
dgabef = 3 Jade And dropje = ~— Ogab
day Ox,

this may be decomposed into two parts
dQ = dQ. IIP AT OO NSS EN

namely

dQ 0Q di
b,Q= EO 5 Or a DD aan, ter 0)

0 0
d‚Q == (a a (ae a = A oer = (ge tat TE

(9 0Q )
= Edge ee Gee datg i) ok ee

The last equation shows that

‘[4Qas=0 JA

if the variations dgq, and their first derivatives vanish at the boundary
of the domain of integration.

§ 35. Equations of the same form may also be found if Q is
expressed in one of the two other ways mentioned in § 38. If e.g.
we work with the quantities ge we shall tind

(dQ) = (4,2) + (4, Q),
where (d,Q) and (d,Q) are directly found from (43) and (44) by
replacing gat, Jabes Yabef, Say ANd Pyare ete. by get, gebe, etc. If the
variations chosen in the two cases correspond to each other we
shall have of course
(SQ) = dQ,
Moreover we can show that the equalities
(dd 0 (d,Q) = 0,0,

exist separately. *)

1) Suppose that at the boundary of the domain of integration 3ga =O and
dgab.e = 0. Then we have also 3gab =O and dJgab‚e =O, so that

[0 dS = 0, faaas=—e

and from

5

The decomposition of dQ into two parts is therefore the same,
whether we use gq, g™ or 9%.

It is further of importance that when the system of coordinates
is changed, not only dQdS is an invariant, but that this is also
the case with d,QdS and d,QdS separately. *)

We have therefore

ta Se
Vig Ving
§ 36. For the calculation of d,Q we shall suppose Q to be

expressed in the quantities 3” and their derivatives. Therefore
(comp. (43))

(46)

Oe =e (aD ap Oder ty en ee Pot alee)
if we put
dQ 0, 0Q wre 0Q
M= Dn = (e) BE Open + (ef) mv
_ Now we can show that the quantities M‚, are exactly the
quantities G,», defined by (40). To this effect we may use the
following considerations.

i
We know that 6
Y

ve) is a contravariant tensor of the second
nd

{wo dS = [dQdS
we infer
NI Q)dS = {d,Qd58.

As this must hold for every choice of the variations dgad (by which choice the
variations 3q¢ are determined too) we must have at each point of the field-figure
(J,Q) =4,Q

2) This may be made clear by a reasoning similar to that used in the preceding note.
We again suppose dgab and 8gav‚e to be zero at the boundary of the domain of
integration. Then 39’ab and dg‘ab,e vanish too at the boundary, so that

fees =0 : [4008 = 0.
f vaas = f vaas

we may therefore conclude that

f vaas = f 4,008.

As this must hold for arbitrarily chosen variations dgab we have the equation

J, QdS' = d,QUS.

From

ke

1
order. From this we can deduce that (a) is also such a

=
tensor.

Writing for it et we find according to (46) and (47) that
S (ab) May 8%
is a scalar for every choice of (é%).
This involves that (M‚‚) is a covariant tensor of the second order
and as the same is true for (Gas) we must prove the equation
MG ay
only for one special choice of coordinates.

§ 37. Now this choice can be made in such a way that at the
point P of the field-figureg,, =9,. = Jun ——1,9.=+1,90—0
for a=|=6 and that moreover all first derivatives gane vanish. If
then the values gas at a point Q near P are developed in series
of ascending powers of the differences of coordinates za (Q) — wa (P)
the terms directly following the constant ones will be of the second
order. It is with these terms that we are concerned in the calcula-
tion both of Ms and of Ga for the point P. As in the results
the coefficients of these terms occur to the first power only, it is
sufficient to show that each of the above mentioned terms separately
contributes the same value to Ms and to Gas.

From these considerations we may conclude that

6.Q = S(ab) Gado. to . oR

Expressions containing instead of dg” either the variations dg”
or dgab might be derived from this by using the relations between
the different variations. Of these we shall only mention the formula

il gu
Sige = —— dg — —~— Zed) geadged ww . (49)
V—g VG

§ 38. In connexion with what precedes we here insert a con-
sideration the purpose of which will be evident later on. Let the
infinitely small quantity § be an arbitrarily chosen continuous func-
tion of the coordinates and let the variations dgq, be defined by
the condition that at some point P the quantities gq, have after the
change the values which existed before the change at the point Q,
to which P is shifted when w, is diminished by §, while the three
other coordinates are left constant. Then we have

gab —= — YJab,h §
and similar formulae for the variations dg”,

7

If for d,Q and d,Q the expressions (48) and (44) are taken, the

equation
TE A REE ee)
is an identity for every choice of the variations.

It will likewise be so in the special ease considered and we shall
also come to an identity if in (50) the terms with the derivatives
of & are omitted while those with § itself are preserved.

When this is done 0Q reduces to

dQ
Oak
and, taking into consideration (44) and (48), we find after division
by §
dQ ò / 0Q ò / dQ
Re vn (EN > yrs (er Pa =
Oay, ee Oa (ao 7 ) et) Ove an ee 7)
6 (0 0
— = (abef) Q Jab‚ht == — = (ab)Gargt*. (51)
Òz, Owe Ògas, ef.
In the second term of (44) we have interchanged here the indices
e and f.
If for shortness’ sake we put, for e == h

js 0 0 0 0
sj GD Jab,h + = (abt Ee Jab,fh — = (abt ls a ras

Jab, ef

and for e=h

0
8; = —Q+ (ab) sn gab,h + &(abf)
Jab,h

Ògab, fh de
0 0
= Sat) ze (arr) gas a eas eee
ef \Ògav,nf
we may write
08),
J) St = — E (ab) Gas ge! beet), MIE GET
ie

The set of quantities g, will be called the complex & and the set
of the four quantities which stand on the left hand side of (54) in
the cases h—1, 2, 3, 4, the diwergency of the complex. *) It will
be denoted by div 8 and each of the four quantities separately by
div, 8. 3

The equation therefore becomes

divj8 = — Z(ab)Ga gbh . . . … « « (55)

1) Einstein uses the word “divergency” in a somewhat different sense. It seemed
desirable however to have a name for the left hand side of (54) and it was diffi-
cult to find a better one,

8

If we take other coordinates the right hand side of this equation
is transformed according to a formula which can be found easily.
Hence we can also write down the transformation formula for the
left hand side. It is as follows

div';8' = p= (m)pmpditms — DI + 2p (abe)patc 9° Gay. . (56)
4. $

§ 39. We shall now consider a second complex s,, the com-

ponents of which are defined by
$e, = — GE (a)g” gan + 2= (a)9% Gar . a. Cae
Taking also the divergency of this complex we find that the
difference
div',8'9 — p X (m)pmhk divnëo
has just the value which we can deduce from (56) for the corre-
sponding difference
div! 8'— pX(m)pmpdivys
It is thus seen that ;
div',8'—div';,8'9 = pX(Mm)pmn(Lirns — divm8o)
and that we have therefore
WO SS DU BY eC he ole (te
for all systems of coordinates as soon as this is the case for one
system.

Now a direct calculation starting from (52), (53) and (57) teaches
us that the terms with the highest derivatives of the quantities
Jab, (viz. those of the third order) are the same in div, 8 and div, 8,.
Further it is evident that in the system of coordinates introduced in
§ 37 these terms with the third derivatives are the only ones. This
proves the general validity of’ equation (58). It is especially to be
noticed that if $ and 8, are determined by (52), (53) and (57) and
if the function defined in § 32 is taken for G, the relation is an
identity.

§ 40. We shall now derive the differential equations for the
gravitation field, first for the case of an electromagnetic system. *)
For the part of the principal function belonging to it we write

fuas,

where L is defined by (35) (1915). From L we can derive the
stresses, the momenta, the energy-current and the energy of the

1) This has also been done by pe DonpeER, Zittingsversiag Akad. Amsterdam,
25 (1916), p. 153.

9

electromagnetic system; for this purpose we must use the equations
(45) and (46) (1915) or in Einstein's notation, which we shall follow
here, *)

feo —L+ Ory, Wig vase re
and ior bt
zb in (a) yy, fa Ceo ieee ENA EK

dig
The set of quantities £,/ might be called the stress-energy-complex
(comp. $ 38). As for a change of the system of coordinates the
transformation formulae for © are similar to those by which tensors
are defined, we can also speak of the stress-energy-tensor. We have
namely

1 l
——_—_ FT ZE > (kl ke Nps
Par, ee (Al) p 4

$ 41. The equations for the gravitation field are now obtained
(comp. $$ 18 and 14, 1915) from the condition that

Ì
4, { Las + 5. feas=o DD

for all variations dg,, which vanish at the boundary of the field
of integration together with their first derivatives. The index in
the first term indicates that in the variation of L the quantities War
must be kept constant.

If we suppose L to be expressed in the quantities g@ and if (42),
(45) and (48) are taken into consideration, we find from (61) that
at each point of the field-figure

òL
= (ab) &

If now in the first term we put

1
) dg + — S(ab) Gad =0 . . (62)
p : 2x

1) The notations & Vr and bi, (see (27), (29) and § 11, 1915), will however

be preserved though they do not correspond to those of Erster. As to
formulae (59) and (60) it is to be understood that if p and q are two of the
numbers 1, 2, 3, 4, p/ and q’ denote the other two in such a way that the order
p q pq is obtained from | 2 3 4 by an even number of permutations of two
ciphers.

If 2, 2, 3, XZ, are replaced by x, y, 2, ¢ and if for the stresses the usual
notations Xx, Xy, etc, are used (so that e.g. for a surface element de perpendicular
to the axis of x, Xz is the first component of the force per unit of surface which
the part of the system situated on the positive side of dz exerts on the opposite
part) then %!= Xa, T° = Xy, etc. Further —3,*, —*, — Tt are the components
of the momentum per unit of volume and %!, 34, {4° the components of Ee
energy-current. Finally is the energy per unit of volume.

10

OL ze
EE WT 3. |) eee ee

and if for dg the value (49) is substituted, this term becomes
x 2 (ab) Tas 39? — # 2 (abed) 9” gea Tar 59°,
or if in the latter summation a,b is interchanged with c, d and if
the quantity
TEA Ta > I DR
is introduced,
4 2 (ab) (Tas — 4$ gas T) 69”.
Finally, putting equal to zero the coefficient of each dz we
find from (62) the differential equation required
Gop = — 2 (Lap Fab IN ge a ee
This is of the same form as Einstein's field equations, but to see
that the formulae really correspond to each other it remains to
show that the quantities 7,, and £/ defined by (63), (59) and (60)
are connected by EiNsrEiN’s formulae

ENV GT So ae

We must have therefore

OL
2 2 (a) gee (ele —L+ (A) Wie We sn -- - 16m

==
and for b==c
OL
2 S(a) gab (; ‘= D(a) Wis dae. nl EEN
ge Wade

$ 42. This can be tested in the following way. The function L
(comp. $ 9, 1915) is a homogeneous quadratic function of the wWes’s
and when differentiated with respect to these variables it gives the

quantities was. It may therefore also be regarded as a homogeneous

quadratic function of the was. From (35), (29) and (32)'), 1915 we
find therefore

L = 4£V—g & (pqrs) (97995 — 99° 9?*) Wg Wrs - - + (69)

Now we can also differentiate with respect to the gts; while not

the wss but the quantities Wah are kept constant, and we have e.g.

aby 5 (DL
(os) =~ (ae)

According to (69) one part of the latter differential coefficient is

1) The quantities y,, in that equation are the same as those which are now
denoted by get.

11

obtained by differentiating the factor W—g only and the other part
by keeping this factor constant.
For the calculation of the first of these parts we can use the

relation
kn Meten en vei ete „orda et jo de AND

and for the second part we find
EV gE (19) 94 Wap Weg:
If (32) 1915 is used (67) and (68) finally become
= (9) Weg Weg + 2 (a) Warder = 2L,

ei

= (9) Weg Wig + Zou Wa ZO.
a — bi

These equations are really fulfilled. This is evident from : woa = 0,
Obi can Wia = — Was; besides, the meaning of Wis
($ 14, 1915) and equation (35) 1915 must be taken into consideration.

$ 43. In nearly the same way we can treat the gravitation field
of a system of incoherent material points; here the quantities wa
and wa (§§ 4 and 5, 1915) play a similar part as Was and Was in
what precedes. To consider a more general case we can suppose
“molecular forces” to act between the material points (which we
assume to be equal to each other); in such a way that in ordinary
mechanics we should ascribe to the system a potential energy
depending on the density only. Conforming to this we shall add
to the Lagrangian function L (§ 4, 1915) a term which is some
function of the density of the matter at the point P of the field-
figure, such as that density is when by a transformation the matter
at that point has been brought to rest. This can also be expressed
as follows. Let do be an infinitely small three-dimensional extension
expressed in natural units, which at the point P is perpendicular to
the world-line passing through that point, and edo the number of
points where do intersects world-lines. The contribution of an element
of the field-figure to the principal function will then be found by
multiplying the magnitude of that element expressed in natural units
by a function of 9. Further calculation teaches us that the term
to be added to L must have the form

; Ns f
tf = = . . . . . . . 71

12

where P is given by (15) 1915. As the Lagrangian function defined
by (41) 1915 equally falls under this form and also the sum of this
function and the new term, the expression (71) may be regarded as the
total function L. The function p may be left indeterminate. If now
with this function the calculations of §§ 5 and 6, 1915 are repeated,
we find the components of the stress-energy-tensor of the matter.

The equations for the gravitation field again take the form (65).
Tur is defined by an equation of the form (63), where on the left
hand side we must differentiate while the w,’s are kept constant.
Relation (66) can again be verified without diffleulty.

We shall not, however, dwell upon this, as the following consider-
ations are more general and apply e.g. also to systems of material
points that are anisotropic as regards the configuration and the
molecular actions.

§ 44. At any point P of the field-figure the Lagrangian function
L will evidently be determined by the course and the mutual
situation of the world-lines of the material points in the neighbour-
hood of P. This leads to the assumption that for constant Jas’S the
variation dL is a homogeneous linear function of the virtual dis-
placements dx, of the material points and of the differential coefficients

0daq
Ons ;

these last quantities evidently determining the deformation of an
infinitesimal part of the figure formed by the world-lines *).
The calculation becomes most simple if we put

OEL) isug Soma sa ee anes
and for constant gas’s
| | òde,
Oes
Considerations corresponding exactly to those mentioned in $$ 4

—6, 1915, now lead to the equations of motion and to the follow-
ing expressions for the components of the stress-energy-tensor

Se es ag ee ee ee

0H = > (a) Ua Òza += (ab) V? (73)

and for b=—c
=S ae Pe 2 ee

C

The differential equations again take the form (65) if the quantities
Ty, are defined by

1) In the cases considered in § 43, 5L can indeed be represented in this way.

in the differentiation on the left hand side the coordinates of the
material points are kept constant. To show that 7; and ©? satisfy
equation (66) we must now show that

N SEN ann OL
—L-—V—g ie =22@9(5 =)
g x
and for DIG
ETEN OL
—VY—g V =2 5 or ( )
c dg ee po

If here the value (72) is substituted for L and if (70) is taken
into account, these equations say that for all values of 6 and c we
must have

Oro) + PF oe O10 Dig NE)

Now this relation immediately follows from a condition, to which
L must be subjected at any rate, viz. that LdS is a scalar quantity.
This involves that in a definite case we must find for H always
the same value whatever be the choice of coordinates.

§ 45. Let us suppose that instead of only one coordinate x, a
new one 2,’ has been introduced, which differs infinitely little from
Te, With the restriction that if

Gi she HS Ge
the term § depends on the coordinate 2; only and is zero at the
point in question of the field-figure. The quantities g then take
other values and in the new system of coordinates the world-lines
of the material points will have a slightly changed course.

By each of these circumstances separately H would change, but
all together must leave it unaltered. As to the first change we
remark that, according to the transformation formula for g, the

variation dg vanishes when the two indices are different from c, while

05.
dgee == 2geb nS.
Ox,
and for a=—=c
‘ is
dg* = dgee = geb Oe

Oxy

The change of H due to these variations is

0&, 0H
ab
Ov, (a) 9 (or):

2

14

Further, in the new system of coordinates the figure formed by .
the world-lines differs from that figure in the old system by the
variation 02,—= & which is a function of zs only. Therefore accord-
ing to (73) the second variation of H is

ys %
x Oxy

By putting equal to zero the sum of this expression and the

preceding one we obtain (76).

§ 46. We have thus deduced for some cases the equations of
the gravitation field from the variation theorem. Probably this can
also be done for thermodynamic systems, if the Lagrangian function
is properly chosen in connexion with the thermodynamic functions,
entropy and free energy. But as soon as we are concerned with
irreversible phenomena, when e.g. the energy-current consists in a
conduction of heat, the variation principle cannot be applied. We
shall then be obliged to take Einstein's field-equations as our point
of departure, unless, considering the motions of the individual atoms
or molecules, we succeed in treating these by means of the gene-
ralized principle of HAMILTON.

§47. Finally we shall consider the stresses, the energy ete. which
belong to the gravitation field itself. The results will be the same
for all the systems treated above, but we shall confine ourselves to
the case of §§44 and 45. We suppose certain external forces A, t
act on the material points, though we Een see that strictly croak
this is not allowed.

For any displacements dza of the matter and variations of the
gravitation field we first have the equation which summarizes what
we found above

dL + 10 + (a) Kadee = Vv = (a) Ure +
x
Oty ter Oren
+ (ab) ee Veda Dab V7) 02,

OL
+ aw (% sz) ot + 5 0 + 5, J,Q + (a) Kadea.

In virtue of the equations of motion of the matter, the terms
with dz, cancel each other on the right hand side and similarly,
on account of the equations of the gravitation field, the terms with
dg and 0, Q. Thus we can write’)

1) To make the notation agree with that of § 38 b has been replaced by. €.

15

On i Le
Z(a)Kadra = — IL + Za) Vg Va daa) aaa (dQ - d,Q). (77)

Let us now suppose that only the coordinate #, undergoes an
infinitely small change, which has the same value at all points of
the field-figure. Let at the same time the system of values Jab be
shifted everywhere in the direction of 2, over the distance day.
The left hand side of the equation then becomes Kidz, and we
have on the right hand side

oh es dQ = — oe Sx}.
Oxy, Oa,

After dividing the equation by dz, we may thus, according to

(74) and (75), write

0

En
— Ze) = — dvs.
Òz,

By the same division we obtain from dQ—d,(Q the expression
occurring on the left hand side of (51), which we have repre-
sented by

0872 Sa fe
ae: — WIG;

=(¢)

where the complex 8 is defined by (52) and (53). If therefore we
introduce a new complex t which differs from & only by the factor

= 0 that
1 e
A ee 78)
we find
Hey tp, din Pee oe. (09)

The form of this equation leads us to consider t as the stress-
energy-complex of the gravitation field, just as © is the stress-energy-
tensor for the matter. We need not further explain that for the
case Ky, =O the four equations contained in (79) express the
conservation of momentum and of energy for the total system, matter
and gravitation field taken. together.

§48. To learn something about the nature of the stress-energy-
complex ¢ we shall consider the stationary gravitation field caused
by a quantity of matter without motion and distributed symmetri-
cally around a point O. In this problem it is convenient to introduce
for the three space coordinates w,, x,, 2, (v7, will represent the time)
“polar” coordinates. By «2, we shall therefore denote a quantity r

16

which is a measure for the “distance” to the centre. As to a, and
x, we shall put 2#,=cos3,2,—=g, after first having introduced
polar coordinates 9, p (in such a way that the rectangular coor-
dinates are 7 cos 0,7 sin 9 cos p‚r sin 9 sing). lt can be proved that,
because of the symmetry about the centre, ga, —=0 for a =/=},
while we’may put for the quantities gaq
U

i Wiid ae 95, = A) Ii — Is, = wv, (80)
where uw, v, w are certain functions of 7. Differentiations of these
functions will be represented by accents. We now find that of the complex
t only the components t°, t,° and t‚* are different from zero. The
expressions found for them may be further simplified by properly
choosing 7. If the distance to the centre O is measured by the
time the light requires to be propagated from O to the point in
question, we have w=v. One then finds

1 u'? 8 uv? wo!
Te + 2u Br oF SM

EK v
4 1 5 u wui!
fs = oS Hh RENEE aller Sy ’ ODN ar cage a (81)
1 u? Kd
t.4 = — | — 20 — 2u + — |.
2x 2u v |

§ 49. We must assume that in the gravitation fields really existing
the quantities gap have values differing very little from those which
belong to a field without gravitation. In this latter we should have

7", 5 = 1. :
and thus we put now
ur (LE): n=,
where the quantities « and » which depend on r are infinitely small,
say of the first order, and their derivatives too. Neglecting quantities
of the second order we find from (81)

1
a oe (2 + 2u + Oru' + 27?u" + 1r’v"),
x
3 1 '
ts ss lice Mine Hh + rv’),

1
Aen = (2u — 2 + Óru' + 27r?u" + rr").

For our degree of approximation we may suppose that of the
quantities 7%, only 7’,, differs from 0. If we put

a Ore

Pasig gt e A TE
a quantity which depends on # and which we shall assume to be
zero outside a certain sphere, we find from the field equations

r

gee i ae ; .
z c BE rcp reede + f raar).
0 0 0 ao

r r

1
== J mear tf reae| ‘
(fs
0 Go
We thus obtain

r r :
1 3 l
CS tf voor „fear RO en a SO
x Pp
2 0
oe B

EPEN ne eee at (BA)

u=%

DP =o

$ 50. If first we leave aside the first term of t,', which would
also exist if no attracting matter were present, it is remarkable
that the gravitation constant x does not occur in the stress t,', nor
in the energy f,‘; the same would have been found if we had
used -other coordinates. This constitutes an important difference
between EiNsTEIN’s theory and other theories in which attracting or
repulsing forces are reduced to “field actions”. The pulsating spheres
of ByserKNus e.g. are subjected to forces which, for a given motion,
are proportional to the density of the fluid in which they are imbedded;
and the changes of pressure and the energy in that fluid are likewise
proportional to this density. In this case we shall therefore ascribe
to the stress-energy-complex values proportional to the intensity of
the actions which we want to explain. In ErnsreiN’s theory such a
proportionality does not exist. The value of ¢,‘ is of the same order

of magnitude as Z,‘ in the matter. To our degree of approximation

we find namely from (82) 3,‘ —= 7’.

§ 51. If we had not worked with polar coordinates but with
rectangular coordinates we should have had to put for the field
without gravitation g,, = Qs, = 9s; = —1.9,, =1, gas = 0 for a == b.
Then we should have found zero for all the components of the complex.
In the system of coordinates used above we found for the field

i :
without gravitation t,!—W—,; this is due to the complex ¢ being no
x

tensor. If it were, the quantities t‚/ would be zero in every system

of coordinates if they had that value in one system.
2
Proceedings Royal Acad. Amsterdam. Vol. XX.

18

It is also remarkable that in real cases the first term in (83)
can be much larger than the following ones. If we consider e.g.
a point P outside the attracting sphere, we can prove that the
ratio of the first term to the third is of the same order as the
ratio of the square of the velocity of light to the square of the
velocity with which a material point can describe a circular orbit
passing through P.

The following must also be noticed. In the system of polar coor-
dinates used above there will exist in the field without gravitation

1
the stress t,1—-—. If a stress of this magnitude were produced by
x

means of actions which give rise to a stress-energy-tensor, the passage
to rectangular coordinates would give us a stress which becomes
infinite at the point O. In those coordinates we should namely have
sin 1

pa
1
ger

§ 52. Evidently it would be more satisfactory if we could ascribe |
a stress-energy-tensor to the gravitation field. Now this can really

be done. Indeed, the quantities 8), determined by (57) form a tensor
and according to (58), (79) may be replaced by

KK, == dh Ee OOR to hos a ee EN
if t, is defined by a relation similar to (78), viz.
e bee
thh—=— 80. (86)

Equation (85) shows that, just as well as t°“, we may consider the
quantities te, as the stresses ete. in the gravitation field. This way
of interpretation is very simple. With a view to (41) we can namely
derive from the equations for the gravitation field (65)

; bs
and
1
ha Be (Gar 5 Jab@).
Further we find from (66)
= aes G2 (a)3 Gah — ea zi (a) 3° Gah

2x x

and from (57) and (86)

6; See ERE
At every point of the field-figure the components of the stress-
energy-tensor of the gravitation field would therefore be equal to

19

the corresponding quantities for the matter or the electro-magnetic
system with the opposite sign. It is obvious that by this the condi-
tion of the conservation of momentum and energy for the whole
system would be immediately fulfilled. It was in fact this circum-
stance that made me think of the tensor f, = — ©. The way in
which 8, was introduced in $$ 38 and 39 has only been chosen in
order to lay stress on (58) being an identity, so that equation (85)
is but another form of (79).

At first sight the relations (87) and the conception to which they
have led, may look somewhat startling. According toit we should have
to imagine that behind the directly observable world with its stresses,
energy etc. there is hidden the gravitation field with stresses, energy
ete. that are everywhere equal and opposite to the former; evidently
this is in agreement with the interchange of momentum and energy
which accompanies the action of gravitation. On the way of a light-
beam e.g. there would be everywhere in the gravitation field an
energy current equal and opposite to the one existing in the beam. If
we remember that this hidden energy-current can be fully described
mathematically by the quantities gq, and that only the interchange
just mentioned makes it perceptible to us, this mode of viewing
the phenomena does not seem unacceptable. At all events we are
forcibly led to it if we want to preserve the advantage of a stress-
energy-tensor also for the gravitation field. It can namely be shown
that a tensor which is transformed in the same way as the tensor
t, defined by (57) and (86) and which in every system of coor-
dinates has the same divergency as the latter, must coincide with t,.

Finally we may remark that (78), (86), (58), (87) give

one dwt=dwt, — — dwf,
so that we have, both from (79) and from (85), A, = 0.

The question is this, that, so long as the gravitation field is con-
sidered as given, we may introduce “external” forces, but that in
the equations for the gravitation field itself we must also take into
consideration the stress-energy-tensor of the system by which those
forces are exerted.

2%

Physics. — “On Einstein's Theory of gravitation.” IV. By Prof. H.
A. LORENTZ.

(Communicated in the meeting of October 28, 1916).

§ 53. The expressions for the stress-energy-components of the
gravitation field found in the preceding paper call for some further
remarks. If by dj,’ we denote a quantity having the value 1 for
e=h and being 0 for e=—h, those expressions can be written in
the form (comp. equations (52) and (78))

1 0
tae = — jd Q + 2 (ab a
2x Odab

9 ( 2Q
— Z(abf)—— Jah, h
dae Ogab, of
They contain the first and second derivatives of the quantities gap.

ErnsTEIN on the contrary has given values for the stress-energy-
components which contain the first derivatives only and which
therefore are in many respects much more fit for application.

It will now be shown how we can also find formulae without
second derivatives, if we start from (88).

= (abf d
oy Gad Is

(88)

§ 54. For this purpose we shall eee the complex u defined by

dQ
e—__ e b Ee 89
Uy, oe Sh Q— = (a Dan Carole ( )
and we shall seek its divergency.
We have
ou sE, dQ dQ
(div u), = & (e On linge eee Ox os ae b, fe aired ‚)I
or 2
1 OR
j , = — —— ENE ee eee = eee 90
(div u); On dz, (90)
if we put

0
RQ 3 (at) (aprts) i aoe

Now Q=—V= 9 G can be divided into two parts, the first of
-which Q, contains differential coefficients of the quantities gq, of the
first order only, while the second Q, is a homogeneous linear function

21

of the second derivatives of those quantities. This latter involves
that, if we replace (91) by

„ 0Q 0
Rt QS ery (5 el )— ='(abjfe) — ~( a ) seu
Ògas Oare Ògab, fe

the second and the third term es each other. Thus

R= Q = = otf) zn (a -) o,s - 2 + + (92)
Jab, fe
If now we detine a complex v by the equation
a RP Seep Orin CR a ee ed CE) |
2x
we have —
af, 1 OR
OEM <> see, aps Fett is Sen AEN
If finally we put
t=—t+uy,

we infer from (90) and (94)
NN AAL GE
and from (88), (89), (93) and (92)
1 0
es en ei at Q,+ Se ult SN ee I ee Garen ) t—
Jab. fh

Oud, h Òzy,

> (ab = = (ab ge a —) 96
win 2 (5 re) daa + (bf) (5) amas} (08)

and for e=|=h

a | dQ
a ab OG
= Sat k= = 2 ( Nae (geen) b‚f

— 2 (abf) — ar) bal (87)
0x \ 0gub,ef nie DE Soa

Formula (95) shows that the quantities tje can be taken just as
well as the expressions (88) for the stress-energy-components and we
see from (96) and (97) that these new expressions contain only the
first derivatives of the coefficients 9,5; they are homogeneous quadratic
functions of these differential coefficients.

This becomes clear when we remember that Q, is a function of
this kind and that only Q, contributes something to the second
term of (96) and the first of (97); further that the derivatives of Q
occurring in the following terms contain only the quantities gas and
not their derivatives.

!
j=
2x

§ 55. EINSTeEINs stress-energy-components have a form widely
different from that of the above mentioned ones. They are

22

1 1
PES = dje = (abef) g®® Tach Dope — — & (abe) ge? Vac’ Mon’,
x x

where for the sake of simplicity it has been assumed that V—g—1.
Further we have

ab ab
ea se See |e
If now our formulae (96) and (97) are likewise simplified by the
assumption Aiel (so that Q becomes equal to G), we may

expect that t’ will become identical with fp. This is really so in
the case gus = 0 for a=/=6; by which it seems very probable that
the agreement will exist in general.

In the preceding paper it was shown already that the stress-
energy-components ;¢ do not form a “tensor”, but what was called
a “complex”. The same may be said of the quantities t’;¢ defined
by (96) and (97) and of the expressions given by Einstein. If we
want a stress-energy-tensor, there are only left the quantities téo,
defined by (86) and (57), the values of which are always equal and
opposite to the corresponding stress-energy-components %,° for the
matter or the electromagnetic field.

It must be noticed that the four equations

Oes
Nkk Sa

always express the same relations, whether we choose ten, te, t/ze
or (pn as stress-energy-components 2%), of the gravitation field.
If however in a definite case we want to use the equations in order
to calculate how the momentum and the energy of the matter and
the electromagnetic field change by the gravitational actions, it is
best to use t’%, or t%),, just because these quantities are homo-
geneous quadratic functions of the derivatives gn, ;
Experience namely teaches us that the gravitation fields occurring
in nature may be regarded as feeble, in this sense that the values
of the gqo’s are little different from those which might be assumed
if no gravitation field existed. For these latter values, which will be
called the “normal” ones, we may write in orthogonal coordinates
In = 92 = Ia = — 1, Au =O", gab = 9, for a==b. (98)
In a first approximation, which most times will be sufficient, the
deviations of the values of the g,»’s from these normal ones may
be taken proportional to the gravitation constant x. This factor

also appears in the differential coefficients g,5,-; hence, according to
the character of the functions t’,¢ mentioned above (and on account

«

23

1
of the factor zs in (96) and (97)) these functions become proportional

to x, so that in a feeble gravitation field they have low values.

§ 56. Because of the complicated form of equations (96) and (97),
we shall confine ourselves to the calculation for some cases of t’,*,
ie. of the energy per unit of volume. This calculation is considerably
simplified if we consider stationary fields only. Then all differential
coefficients with respect to «, vanish, so that we have according
to (96)

1
pe == an

0 0
Tei Q, + = (abfe) de (s—--) ons| . . (99)

We shall work out the calculation, first for a field without gravita-
tion and secondly for the case of an attracting spherical body in
which the matter is distributed symmetrically round the centre.

If there is no gravitation field we may take for the quantities
Jab the “normal” values. For the case of orthogonal coordinates these
are given by (98). When we want to use the polar coordinates
introduced into § 48 we have the corresponding formulae

r?

Zn = — ri (1 —e®), == == 1, = 60
In Ten Je m(L— 2,”), Jas NN
dar Oy» LOPrd ib.
If, using polar coordinates, we have to do with an attracting sphere
and if we take its centre as origin, we may put ©

I = Inde) = TV I, =v, (101)
1

where wu, v, w are functions of r. The g,s’s which belong to an
orthogonal system of coordinates may be expressed in the same
functions.

These gq,’s are

Ui, (U
Oe — — Uv], ete.
7123

re \?

Ius = Ia, = Ia, = 9, Lin
The “etc.” means that for g,.,9;,; we have similar expressions
as for g,, and for g,,,93, similar ones as for 9,
§ 57. In order to deduce the differential equations determining
u, Vv, w we may arbitrarily use rectangular or polar coordinates ;
the latter however are here to be preferred, If differentiations

.

24

with respect to 7 are indicated by accents, we have according
to (40) and (101)

aes 1 i u uv ww
Or NE SCP erk

u ue uw! |
Opel 5-24), E

u 13 fet le | " 12
pe u u uv vw w w
Ks itis Wy PRO RAET)
u 2u? 2uv Avw 2w 4w?
G uw ow w" wi?

Quy 4e? 2 | dow’ |
Gis for a is b.
So we have found the left hand sides of the field equations (65).
Before considering these equations more closely we shall introduce
the simplification that the g,s’s are very little different from the
normal values (100). For these latter we have

2

i Po) See Or to EN
and therefore we now put
eer (ld) ap ce Sl sek Cea (ke) See
The quantities 2, u, », which depend on 7, will be regarded as
infinitely small of the first order and in the field equations we
shall neglect quantities of second and higher orders.
Then we may write for G,, ete.

1 an re ' |
G,, award (4 + 2ra' + $r°A" — w — bru’ + rw),
eval
Oe = (l—a,’)(a al rh! aes gr?" ica see Sr Br =F trv’),

1
G,=—¢ & ie ae ’)
Je

On the right hand-sides of the field equations (65) we may take
for gas the normal value; moreover we shall take for 77, and 7
the values which hold for a system of incoherent material points.
We may do so if we assume no other internal stresses but those
caused by the mutual attractions; these stresses may be neglected
in the present approximation.

As we supposed the attracting matter to be at rest we have
according to (10),..(16).and (15) (1915) wv, = w, = w, = 0; wi =e,
Us Sa, SON = ce Pleas :

In the notations we are now using we have further, according
to (23) (1915),

25

UhWe
Teg
so that of the stress-energy-components of the matter only one is
different from zero, namely ;

>
B

?

Tei

4 x
Further (66) involves that, also of the quantities 7,5, only one,
namely 7, is not equal to zero. As we may put V—g =c7r"’, we

have namely

Finally we are led to the three differential equations

Ae brrr terp Sk wey.) (109)
Bande rde bet pS — Fa eee (105)
DO ager er er (OR)

It may be remarked that odz,de,de, represents the “mass” present
in the element of volume dz,dz,dx,. Because of the meaning of
a@,,2,,0, (§ 48) the mass in the shell between spheres with radii
r andr + dr is found when odz,dz,dx, is integrated with respect to
a, between the limits —4 and +1 and with respect to «, between
O and 2a. As @ depends on r only, this latter mass becomes Anodr,
so that o is connected with the “density” in the ordinary sense of the

word, which will be called 9, by the equation

o=—r 0.

The differential equations also hold outside the sphere if @ is put
equal to zero. We can first imagine vy to change gradually to 0
near the surface and then treat the abrupt change as a limiting case.

In all the preceding considerations we have tacitly supposed the
second derivatives of the quantities g to have everywhere finite values.
Therefore v and v' will be continuous at the surface, even in the
case of an abrupt change.

§ 58. Equation (106) gives

: À

vi fom, Ek err DT)
"9

where the integration constant is determined by the consideration that
for r =O all the quantities g,, and their derivatives must be finite,
so that for » =O the product 7°»P’ must be zero. As it is natural to

suppose that at an infinite distance pv vanishes, we find further

26

dr
Pek rj RD
Oe! 10

The quantities 2 and u on the contrary are not completely determined
by the differential equations. If namely equations (105) and (106) are.
added to (104) after having been multiplied by — } and + 4 respecti-
vely, we find En

Ah Ar OC Se an er
and it is clear that (104) and (105) are satisfied as soon as this is
the case with this condition (109) and with (106). So we have only to
attend to (108) and (109). The indefiniteness remaining in A and u is
inevitable on account of the covariancy of the field equations. It does
not give rise to any difficulties.

Equation (107) teaches us that near the centre

Er :
Di Rog?

if 9, is the density at the centre, whereas from (108) we find a
finite value for v itself. This confirms what has been said above
about the values at the centre. We shall assume that at that point
4,u and their derivatives have likewise finite values. Moreover we
suppose (and this agrees with (109) that 2, u, 2’ and w' are
continuous at the surface of the sphere.

If a is the radius of the sphere we find from (108) for an exter-

nal point
a

x
i odr.
r

0
Without contradicting (109) we may assume that at a great

1
distance from the centre Zand u are likewise proportional to — , so that 4’
Us
1
and u’ decrease proportionally to —.
V

§ 59. We can now continue the calculation of t',* (§ 56).
Substituting (101) in (99) and using polar coordinates we find

14 1 Ee ; u? a uw!
et mek a == Ns
: 2x v u? uw

whence by substituting (102) we derive fora field without gravitation
C

x

This equation shows that, working with polar coordinates, we

40

27 ;

should have to ascribe a certain negative value of the energy to a
field without gravitation, in such a way (comp. § 57) that the
energy in the shell between the spheres described round the origin
with radii 7 and 7+ dr becomes

Arc

— — dr.
x

The density of the energy in the ordinary sense of the word
would be inversely proportional to 7’, so that it would become
infinite at the centre.

It is hardly necessary to remark that, using rectangular coor-
dinates we find a value zero for the same case of a field without
gravitation. The normal values of gq, are then constants and their

derivatives vanish.

§ 60. Using rectangular coordinates we shall now indicate the
form of t*‚ for the field of a spherical body, with the approximation
specified in § 57. Thus we put

(110)

a,? |

9, = — (Ll + A+ — (u), etc, |

TT,

Irs =z (A), ete. |
he

Inu In =I = 9 0 Ius H (Ll +).
By (109) and (110) we find) ;

1) Of the laborious calculation it may be remarked here only that it is convenient
to write the values (110) in the form

apne

where x and 3 are infinitesimal functions of 7. We then find

da Oy Oa
—120 (= “) + BO ie ze +

0*3 0°38 0°38 2
bk ander Òz der? — (ae ==) ||

(a, i, k = 1, 2, 8)

which reduces to (111) if the relations between 7, 6 and A, u, viz.

1 U 1 "
tee Pah ’ wetn =d

ees

c
2x

and the equality 2° = v/ involved in (109) are taken into consideration.

28

' 6 t 1 1 ! t
(=F eren Om +2 || (EL)

Thus we see (comp. $ 58) that at a distance from the attracting
1

sphere t‚* decreases proportionally to — Further it is to be noticed
Ui

that on account of the indefiniteness pointed out in § 58, there
remains some uncertainty as to the distribution of the energy over
the space, but that nevertheless the total energy of the gravitation field

Edn fr dr

0
has a definite value.
Indeed, by the integration the last term of (111) vanishes. After
multiplication by r? this term becomes namely

d
(A—p) + 2r (A — U) (A — u) = = fr (4 — #1.

The integral of this expression is 0 because (comp. §§ 57 and 58)
r(4—u)* is continuous at the surface of the sphere and vanishes
both for r=0O and for r=o.

We have thus
zE adr, ES SEINEN
x

where the value (107) can be substituted for »'. If e.g. the density
@ is everywhere the same all over the sphere, we have at an internal
point

vy =txor

and at an external point

a
xO —-
a

I

From this we tind
E == merg a.
$ 61. The general equation (99) found for t't, can be transformed
in a simple. as We have namely

i 0 0 dQ
= (abje) —— aa ~ (5 = aus f=— Gop. — hee mg) —

0Q |
So (abfe) en Jab, fe

ab, fe

and we may write —Q, ($ 54) Me the last term. Hence

29

EN " ay 0Q
t' ml Ct Elaa). ee)

where we must give the values 1, 2, 3 to e and f.
The gravitation energy lying within a closed surface consists therefore
of two parts, the first of which is

1 »
B= — 5 [Qade, de, de Ed AID

while the second can be represented by surface integrals. If namely
Ji» Jas Je Are the direction constants of the normal drawn outward

1 00)
Be (abe ~—— Gah fQGedO6 . .-. . (1
2 2% 4 7 Wase en ( Ms)

In the case of the infinitely feeble gravitation field represented
by 2,u,r ($ 57) both expressions £, and £, contain quantities of
the first order, but it can easily be verified that these cancel each
other in the sum, so that, as we knew already, the total energy
is of the second order.

From Q=V—gG and the equations of $ 32 we find namely

ze Weg (29% gfe—gbf gre—gvf gb), . . . (116)
so that we can write
Al —= :
E, os „fv > (abfe) (29%? gfe—agPh gee —gf gbe) Jab, f Je do.

The factor gas,¢ is of the first order. Thus, if we confine ourselves
to that order, we may take for all the other quantities these normal
values. Many of these are zero and we find

c
BE, = — Dn = (ae) | 9% (gaae —Jae,a)gedo. » . ~ (117)

Here we must take a=1,2,3,4; e=1, 2,3, while we remark
that for a=e the expression between brackets vanishes. For a = 4

Ov
the integral becomes f: ek do, which after summation with respect
Ve

to e gives
Ov
dn
n representing the normal to the surface. If a and e differ from
each other, while neither of them is equal to 4, we can deduce
from (110) and (109)

p OUST se de eter aen (018)

Ov

Jaa,e — Jae,a — —

30

Each value of e occurring twice, ie. combined with the two
values different from e which a can take, we have in addition to (118)

— i ue do
On :
so that (117) becomes
REO’ Ov
rn

we have for every closed surface that does not surround the sphere
E,=0, but for every surface that does

a

Bte f od et eg ley ae her
0
As to ZE, we remark that substituting (65) in (41) and taking
into consideration (64) we find,

C= eT OH eV 9 T OOS ~ ee
From this we conclude that ZE, is zero if there is no matter
inside the surface o. In order to determine KE, in the opposite
case, we remember that G is independent of the choice of coordinates.
To calculate this quantity we may therefore use the value of 7’
indicated in $ 56, which is sufficient to calculate EL, as far as the

terms of the first order. We have therefore

4

x
a
Ga ae
B
and if, using further on rectangular coordinates, we take for Vi
the normal value c,

Cx
Miis
qe

From this we find by substitution in (114) for the case of the
closed surface 6 surrounding the sphere

a

Ei, =—2ne | ear.

0
This equation together with (119) shows that in (113) when
integrated over the whole space the terms of the first order really —
cancel each other. In order to calculate those of the second order

ot

and thus to derive the result (112) from (113), we should have to
determine the quantity 7’ (comp. 120)), accurately to the order x.
The surface integrals in (115) too would have to be considered
more closely. We shall not however dwell upon this.

§ 62. From the expression for t'‚* given in (113) and the value

E Tan E, zi vijf
derived from it, it can be inferred that, though t' is no tensor, we yet may
change a good deal in the system of coordinates in which the pheno-
mena are described, without altering the value of the total energy.
Let us suppose e.g. that 2, is left unchanged but that, instead of the
rectangular coordinates 2,, x,, 2, hitherto used, other quantities
a',, ©, ©, are introduced, which are some continuous function of
Bis Las Los withthe restriction that z'; = 2,, Za, 2', = #, outside
a certain closed surface surrounding the attracting matter at a
sufficient distance. If we use these new coordinates, we shall have
to introduce other quantities g',5 instead of gas. As however outside
the closed surface the quantities gags and their derivatives do not
change, the value of /#, will approach the same limit as when we
used the coordinates 2,, 2,, #,, if the surface o for which it is calculated
expands indefinitely. The value which we find for Z, after the
transformation of coordinates will also be the same as before. Indeed,
if dr is an element of volume expressed in 2,, z,, #,-units and dr’ the
same element expressed in z',, 2',, z's-units, while Q' represents the
new value of Q, we have

Ode = Qdr':

It is clear that the total energy will also remain unchanged if
ete, differ from «,,,,2, at all points, provided only that these
differences decrease so rapidly with increasing distance from the
attracting body, that they bave no influence on the limit of the
expression (115).

The result which we have now found admits of another inter-
pretation. In the mode of description which we first followed (using
Bi, Vo, Lo), @*) and gas are certain functions of 2,, 2,,,; in the new
one 0’, Jas are certain other functions of 2’,, 2’, 2',. If now, without
leaving the system of coordinates x,,2,,7,, we ascribe to the density
and to the gravitation potentials values which depend on 2,, 2,, «,
in the same way as @’, g'as depended on 2’,, 2’,, x, just now, we
shall obtain a new system (consisting of the attracting body and
the gravitation field) which is different from the original system

1) By @ we mean here what was denoted by @ in § 56.

32

because other functions of the coordinates occur in it, but which never-
theless no observation will be able to discern from it, the indefiniteness
which is a necessary consequence of the covariancy of the field
equations, again presenting itself.

What has been said shows that the total gravitation energy
in this new system will have the same value as in the
original one, as has been found already in § 60 with the restrictions
then introduced.

$ 63. If t were a tensor, we should have for all substitutions
the transformation formulae given at the end of § 40. In reality
this is not the case now, but from (96) and (97) we ean still
deduce that those formulae hold for linear substitutions. They
may likewise be applied to the stress-energy-components of the
matter or of an electromagnetic system. Hence, if &,/ represents the
total stress-energy-components, i.e. quantities in which the corres-
ponding components for the gravitation field, the matter and the
electromagnetic field are taken together, we have for any linear
transformation

—— 21,6 = TE (kl) pre zE. (121
ya tt= pew (121)

We shall apply this to the case of a relativity transformation,
which can be represented by the equations

! —___ id ou! — id sf —— id ail — ,
a Sst a DEE ne Aa PE a ti 7 z,, (122)
with the relation

at Ee a CA
In doing so we shall assume that the system, when described in
the rectangular coordinates «,, z,,r, and with respect to the time a,,
is in a stationary state and at rest. i
Then we derive from (97) ')

1) We have 914 = Jos = 934 = 0, while all the other quantities gap are independent
of x, Thus we can say that the quantities gad and gab, are equal to zero when
among their indices the number 4 occurs an odd number of times. The same may

be said of gab, ga4,c, en (according to (116)), as sas) and also of pro-

Ògas, cd Òzz \ O9ab,cd /

ducts of two or more of such quantities. As in the last two terms of (97) the

indices ad, b and f occur twice, these terms will vanish when only one of the
indices e and h has the value 4.

As to the first term of (97) we remark that, according to the formulae of § 32,

each of the indices a, 6 and e occurs only once in the differential coefficient of

) with respect to gab,e,-while other indices are repeated. As to the number of
Pp g Pp

33

sit Sat ae; eS fe a,
which means that in the system (#,, 2,, 7, «,) there are neither
momenta nor energy currents in the gravitation field.
We may assume the same for the matter, so that we have for
the total stress-energy-components in the system (2,, a,, 2,, w‚)

had NN vj NEEN
oe

a NS a a 0.

Let us now consider especially the components ©“, £',1 and 2’,
in the system (v’,, v',, 2’,,a’,). For these we find from (121) and (122)
x ab ab | es
Bei REL abe abe (124)
C c

Gh ee ae ite Pak ge Sith RS, aa de (125)

It is thus seen in the first place that between the momentum in
the direction of 2, (—$’,‘) and the energy-current in that direction
(f',1) there exists the relation

Pall] Ja rel
oS a

well known from the theory of relativity.
Further we have for the total energy in the system (2’,, 2’,, 2’,, 2',)

E fe de, de aa’,

where the integration has to be performed for a definite value of
the time «/‚. On account of (122) we may write for this
! 1 te 4
hee 8!‘ de, dz, dz,,
where we have to keep in view a definite value of the time z,.
If the value (125) is substituted here and if we take into con-
sideration that, the state being stationary in the system (7,, x, 7, v,),

ae de, dz, dz, = 0

Bak,

if His the energy ascribed to the system in the coordinates (w‚, #,, «,, #).
By integration of the first of the expressions (124) we find in thie
same way for the total momentum in the direction of «,

we have

b
Gas.
C

_ times which e, Jh and the other indices occur we can therefore say the same of
the first term of (97) as of the other terms. The first term also is therefore zero,
if no more than one of the two indices e and h has the value 4.

That #4e vanishes for e=|=4 is seen immediately.

Proceedings Royal Acad. Amsterdam. Vol. XX.

34

$ 64. Equations (122) show that in the coordinates (a',, x',, «',, «'‚)

sce Olan
the system has a velocity of translation — in the direction of «’,.

a
If this velocity is denoted by v, we have according to (123)
1
OSS SS .
v?
pae i
C
If therefore we put
E
M= rar
C
we find
Me* M
EE eee oe

ak WAE
c €

When the system moves as a whole we may therefore ascribe
to it an energy and a momentum which depend on the velocity of
translation in the way known from the theory of relativity. The
quantity M, to which the energy of the gravitation field also con-
tributes a certain part, may be called the “mass” of the system.
From what has been said in $62 it follows that within certain
limits it depends on the way in which the system and the gravita-
tion field are described.

It must be remarked however that, if for the gravitation field we
had chosen the stress-energy-tensor t, ($ 52), the total energy of the
system even when in motion would be zero. The same would be
true of the total momentum and we should have to put M= 0.

At first sight it may seem strange that we may arbitrarily ascribe
‘to the moving system the momentum determined by (126) or a momen-
tum 0; one might be inclined to think that, when a definite system
of coordinates has been chosen, the momentum must have a definite
value, which might be determined by an experiment in which the
system is brought to rest by “external” forces. We must remember
however (comp. § 52) that in the theory of gravitation we may
introduce no “external” forces without considering also the material
system S' in which they originate. This system S’ together with
the system S with which we were originally concerned, will form
an entity, in which there is a gravitation field, part of which is
due to S' (and a part also to the simultaneous existence of S and S’),
There is no doubt that we may apply the above considerations to
the total system (S,S') without being led into contradiction with
any observation.

Chemistry. — “On the Saponification of Fats’ .1. By Dr. J. P. Truus.
(Communicated by Prof. P. Zerman).

(Communicated in the meeting of December 21, 1916).

INTRODUCTION.

§ 1. The saponification of esters of glycerine has been first expe-
rimentally studied by Gerrer. *) He determined the velocity of sapo-
nification of the three acetines in diluted acid solution, by titration
of the split off acetic acid, and came to the result, that the ratio
of the velocity constants of the reactions: triacetine — diacetine >
monoacetine — glycerine is as 3:2:1, from which follows that the
estergroups are all saponified with the same velocity, and that the
velocity of saponification of a certain estergroup is independent of
a neighbouring group being saponified or not.

ABEL”) advanced against this that good constants are likewise
found when it is assumed that the saponification leads directly from
triglyceride to glycerine, and that therefore Gerrer’s measurements
of velocity do not prove anything.

This is clear, as we arrive at the same equations of velocity in
the two different cases as Aspen *) has proved in another paper for
the general case of a reaction in 7 stages.

However with his measurements of velocity Geiten has not proved
that the saponification of triacetine proceeds in stages, but only that
if it proceeds in stages the velocity constants of the three stages
must be in the ratio of 3:2:1.‘) This result on the contrary shows
the impossibility to decide whether the process goes by stages or
not from measurements of the velocity of the splitting off of fatty
acid alone”)

GeitEL proved that the acid saponification of glycerine esters actually
takes place in stages by demonstrating that rancid fats contain more

1) Z. f. pr. Chem. (2) 55 429 (1897), 57 113 (1898).

2) UrzeRr u. Kuimont, Chemie der Fette 244 (1906).

3) Z. f. phys. Chem. 56 558 (1906).

4) J. Meyer has proved (Z. f. Electrochem. 18 485 (1907)), that this ratio
only holds in approximation. For 18° C. the following ratio seems to hold more
accurately : 3.10 : 2.00 : 1.14, for 25° C.: 3.06 : 2.00 : 1.25.

5) Cf. also § 12.

3%

36
bound glycerine than agrees with an immediate splitting up into
glycerine and three molecules of fatty acid. He could therefore
assume analogous behaviour for the acetines.

Jur. Meyer *) has mathematically examined the course of the sapo-
nification in acid solution of esters of bivalent acids or alcohols, and
has brought this into equation in a very lucid form. It is evident
from his formulae that when the first stage passes twice as quickly
as the second, the whole saponification becomes seemingly mono-
molecular. Measurements of velocity carried out by him for the acid
saponification of the glycol acetates and of esters of different symme-
trically built bi-basie acids, confirm this fully.

Also to J. Mryer’s conclusions the objection might be made that
a simply monomolecular saponification explains his results equally
well. J. Meyrr has, however, also determined the velocities of sapo-
nification of the methyl esters of the asymmetrical camphoric acid.
Of the dimethyl camphorate one ester group now appeared to be
much more quickly split off than another. Hence the velocity
constants in the saponification of di- and mono-ester are not in the
ratio of 2:1, so that here the results of the measurements of velocity
lead us to conclude directly to the process in stages of the reaction.
The assumption that also in the saponification of glycol esters ete.
the reaction takes place in stages, is then perfectly justified. Besides
Jur. Mryrr’s experiments support Gerrer’s view that the acetines
in acid solution are saponified stagewise.

THE SAPONIFICATION IN EMULSION.

§ 2. Both Gerrer’s papers and those of Jur. Mever treat the
saponification in solution. In the saponification of fat, however,
always more or less fine emulsions of fat and an aqueous solution
are worked with, and it is, therefore, now the question in what
way the reaction takes place in this case.

In the first place it is the question: Where does the reaction take |
place? There are, namely, three possibilities:

1. Reaction takes place in the water phase.

2. Reaction takes place in the fat phase.

3. Reaction takes place on the boundary of the two phases.

Let us consider each of these possibilities separately.

1. The reaction takes place in the water phase.

In this case the velocity with which the triglyceride is converted,
is determined by the number of molecules dissolved in the water

ly Z, f. phys. Chem. 66 81 (1909).

|
| |

37

phase. Now follows immediately another question, viz. may in this
case an equation of velocity be applied which holds good for a
solution? As Nernst ') has observed an equation of velocity holding
for a homogeneous system leads to an entirely wrong conclusion in
a heterogeneous system, when the velocity of reaction is dependent
on the velocities of diffusion. This will always be the case where
the velocity of reaction is great with respect to the velocity of
diffusion.

When, however, on the contrary in a heterogeneous system the
concentration equilibrium sets in rapidly, and the reaction proceeds
comparatively slowly, the influence of the diffusion velocity is only

slight, and can become quite imperceptible. This now is generally

the case when both phases are liquid When asubstance A dissolved
in a solvent B is shaken with a solvent C, which does not mix
with 5, only a very short time is required to establish the equili-
brium between the two solutions.

H. Go.pscumipt’) has determined the velocity of saponification
of ethyl acetate dissolved in benzine and shaken with about nor-
mal hydrochloric acid. Assuming that the reaction takes place in
the aqueous solution he represented the velocity of saponification by
the equation:

En eee as eh aaa ea” a TRE Phe al gee CAR

in which v, = volume of the aqueous solution, v, = volume of the
benzolie solution, C= constant of partition of ethyl acetate between
water and benzene. On the whole the reaction velocity appeared to
be well represented by this equation.

Towards the end, the reaction in the opposite sense had to be
taken into account.

It appears from ‘this that when the velocity of reaction is not too
great, the equations of velocity which hold in a homogeneous system
may be applied in a heterogeneous system, consisting of two liquid
phases.

Let us now return to the saponification of fat, and Jet us imagine
the case that a triglyceride is saponified with diluted sulphuric
acid according to the Twrrcurur process, in whieh fat and aqueous
solution is held in emulsion by blowing in of steam, after addition
of about '/, °/, TwitcHELL reagent. GoLipscumipt’s formula may cer-
tainly not be used in this case for quantitative determinations. For

1) Z. f. phys. Chem. 47 55 (1904).
*) Z. f. phys. Chem. 31 235 (1899),

38

Nernst’s law of partition cannot be applied here unreservedly, as
the fat phase consists chiefly of triglyceride at the beginning of
the saponification and chiefly of.fatty acid at the end. The “constant”
of partition C' can, therefore, not be constant in this case. We can,
however, draw a conclusion from equation (1) as to the probability
or improbability of the supposition that the saponification takes place
in the water phase. For at any rate there appears from it that if
the said supposition is valid, the extent of the surface of contact
between fat particles and water particles plays no part. But then
the action of the TwrrcHELL reagent must chiefly rest on this that
it causes an increase of C, in other words, increases the solubility
of the fat in the water phase. This is in itself very well possible,
but seeing that the saponification without reagent practically does
not take place, and obtains an efficient velocity on addition of not
quite half a percentage to the emulsion, it is very improbable indeed,
that increase of the solubility of the fat in the water phase should
be the cause of it.

As will appear in $ 4, the action of the TwitcHELL reagent can
be quite plausibly accounted for by the supposition that the saponi-
fication takes place on the boundary of fat and water.

There is, however, another phenomenon that points to this. It
appears namely, that, when triglycerides which contain little or no
free fatty acid, are saponified, the reaction velocity is very small at
first, then it increases and reaches a maximum. WEGSCHEIDER ‘),
who assumes the reaction in the aqueous solution, wants to explain
this by taking the concentration of the triglyceride in the water-
phase constant. The increase of the reaction velocity would then be
caused by the presence of lower glycerides in the waterphase. On
this assumption WEGSCHEIDER comes to the following equation of
velocity for the splitting off of fatty acid:

da

OH MOE Gib AC ten = ee
dt

In this C represents the not changing concentration of the tri-
glyceride in the aqueous solution, / is a constant of velocity.

Equation (2) would really be able to explain the increase of the
velocity of saponification, it it could be applied to the saponification
of fat in this form. Now it is clear that (2) can only hold for the
saponification by means of bases, as only in this case the fat phase
which is in contact with the aqueous solution, consists practically
exclusively of triglycerides, because of which the concentration of

1) Kais. Ak. d. Wissensch. Wien 116, II b. 1325 (1907).

Bn

ee _ -

39

the triglyceride in the waterphase may be taken constant. As will
appear in $ 19 the saponification in alcalic surroundings takes place
however practically directly from triglyceride to glycerine + fatty
acid. Then the second term of the second member of (2) disappears,
and we should have a veiocity of saponification which does not
change with the time. The facts, however, are different.

If the reaction takes place on the boundary of the two phases,
the increase of reaction velocity is at once apparent. At the beginning
we have namely a not very intimate emulsion of lye and triglyce-
ride. As the saponification advances, the soap concentration in the
waterphase increases, the surface tension between fat- and water-
phase accordingly decreases; hence the emulsion becomes more
intimate, and the surface where the reaction can take place, be-
comes greater.

After what precedes we may, therefore, put aside the first possi-
bility as very improbable.

§ 3. 2. The reaction takes place in the fat phase.

This supposition is still less tenable, as a reaction which is cata-
lytically accelerated by H° or OH’ ions, is very improbable in not
aqueous surroundings. :

§ 4. So the last possibility remains, namely :

3. The reaction takes place on the boundary of the two phases.

In the saponification in acid solution the velocity is a function of
the number of collisions in the unity of time between an ester
molecule and an H’ ion. For a given concentration and a definite
temperature this number of collisions is fixed and therefore the
velocity constant also.

If, however, as in the TwitcHELL process we have an emulsion
of fat- and water particles, which move through one another in fine
division, and if the reaction takes place on the boundary of the two
phases, the velocity will be a function of the extent of the surface
where the collisions can take place, i.e. of the fineness of the
emulsion; hence the velocity constant will not be definite at a
given temperature.

In the TwrrcuuLL process the accelerating influence of the reagent
must chiefly, if not entirely, be found in the enlargement of the
surface of contact between fat- and waterphase, in other words in
the decrease of the surface tension between fat and water. That
actually this surface tension is considerably decreased by traces of

40

reagent can be easily shown with Donnan’s pipette. ') (see § 7).

It may seem arbitrary that where it appears that already traces
of TwitcHELL reagent considerably decrease the surface tension
between fat and water, it has been assumed in § 2, that those traces
cannot practically influence the solubility of the fat in water. Yet
this is by no means the case. In order to increase the solubility of
fat in water sufficiently a solvent for triglyceride would have to be
added to the waterphase, which mixes with water. Further the
waterphase would have to exhibit a certain (pretty considerable)
concentration of this solvent throughout its volume. For a substance,
however, which lowers the surface tension between fat and water
this need not be the case. For the action of a substance to lower
the surface tension is accompanied with adsorption at the surface
common to the two phases, in consequence of which such a sub-
stance, though if calculated over the whole mass, it is present only
in traces, can occur in pretty considerable concentration at the
common surface. It is exactly this surface layer that counteracts
the tendency. of two colliding drops to join to one whole. ’)

The same considerations are also valid for the saponification in
alealic surroundings. Here the soap formed in the saponification acts
so as to lower the surface tension between fat and water.

We arrive therefore at the conclusion that in the saponification
in emulsion the reaction practically takes entirely place on the boun-
dary of the fat and the water phase. We may then apply the
equations of velocity holding in solution, when we take the fact
into account that the constant of velocity depends on the fineness
of the emulsion.

§5. Measurements of velocity have been carried out by M. Nicrovx *),
who studied the saponification of cottonseed oil by the aid of the
ferment found in ricinus seed. He found for:

1 a
ms A log

ad

a good constant especially at low temperature (15°). From this it
appears that in this case the fineness of the emulsion does not appre-
ciably change during the saponification and that the ratio of the
saponification velocities of the three glycerides is as 3:2:1 or as
Are 0057)

1) Z. f. phys. Chem. 31, 42 (1899).

2) Donnan loc. cit.

3) Saponification des corps gras (1906).
4) See § 12.

41

As Nicioux states') that the quantity of glycerine split off after
a certain time corresponds to the split off quantity of fatty acid,
the latter ratio must be correct. In the experiments of M. Nictovux
triglyceride seems to have split off practically directly into fatty acid
and glycerine.

In the saponification by means of bases the fineness of the emul-
sion does certainly not remain constant. For here the soap that is
formed gives rise to a lowering of the surface tension between fat
and aqueous solution, hence the fineness of the emulsion will increase
during the saponification. The same thing holds, at least for the
beginning of the reaction, for the autoclave saponification with zine
oxide and likewise for the saponification with lime. Nor does the
fineness of the emulsion remain the same in the course of the
TWwITcHELL process. As can be shown with the aid of Donnan’s
pipette the surface tension between e.g. linseed oil fatty acid and
water is smaller than between linseed oil and water. Here too the
surface of contact: between fat and water phase will therefore
become larger in the course of the reaction.

It is clear that in these cases measurements of velocity are of
little use. The constant of velocity will always present a course,
and then there is no criterion whether the equations of velocity that
have been drawn up, are correct or not. We shall have to adopt
another course here.

When we draw up equations of velocity for the splitting off of
fatty acid in the saponification of triglyceride, and when there occurs
in them only one constant 4, which is dependent on the extent of
the surface of contact of fat- and waterphase, and which there-
fore from the beginning of the saponification may be considered
really constant only during a small period Af, we arrive after inte-
gration of the drawn up equations between the limits 0 and At, at
a relation between the number of molecules of fatty acid (z) split
off after the time At, and & and At,. Let this function be:

. ae OE ee eo. a (8)

For stagewise saponification a second equation denotes : the number
of molecules of glycerine (s) split off after the time Af, as function
of k and At. Let this function be:

Bale AE ee eat)

If we now can eliminate £ Xx At, from the two ae (3) and
(4), we find a relation:

EEN Sten EEND £5)

Le le. 5.

42

which indicates the relation between the number of molecules of
glycerine and fatty acid split off after the time At.

Let us now imagine that after the time Az, has elapsed, the con-
stant of velocity £ changes into 4’, and let us now consider a
following period At. At the beginning of this period the following
equation holds:

z=f(k X At)
s == p(k XxX At,)

The same values z and s could, however, have been obtained

with the constant of velocity 4’ in a certain period Af',, so that:
k

Ma SS kl 5 At, e . e . . . . e (6)

At the beginning of this period we have, therefore, also:
SE NL)
P= (EI)
but then is after the lapse of the time t,:
rf x (At, + Ar)
and
spk KA = Aaah.

From these last equations h' X (Af, + At‚) can be eliminated in
the same way as kX At, from (3) and (4), whicb proves, there-
fore, that (5) also holds after At, has passed.

Since the same reasoning may be extended over the whole sapo-
nification, it appears that when the number of molecules of split off
fatty acid in the saponification of fat can be represented by :

Zj (k XC)
and the number of molecules of split off glycerine by :
$= hx 0;

in which equations k varies with the time, we must be able to
derive a function:

wW (z,s) = 0
by elimination of £ Xt, the form of which does not change during
the saponification, and which is independent of the change of &.

Since in the saponification of fat both split off glycerine and free
fatty acid can be determined separately, we have a means in this
to examine the mechanism of the reaction.

It may still be pointed out here that in the change of & with
the time must also be included the decrease of concentration of the
lye taking place in the saponification in alealic surroundings. We
shall, therefore, have to arrive at analogous equations for acid and
alealic saponification.

at i ta
|

43

§ 6. Before proceeding to the derivation of an equation y(z,s) = 0,
we must first discuss the question what is to be expected in the
saponification of fats in which different fatty acids are present.

The natural fats are, namely, mixtures of different triglycerides
and in a molecule of triglyceride there are often found two, some-
times three different groups of fatty acid. Now it is first of all
conceivable that e.g. the oleic acid group is more easily separated
from a molecule of oleo-dipalmitine than a palmitinie acid group.
Secondly, however, the possibility exists e.g. for a mixture of
trioleine and tripalmitine that the surface tension of one of these
glvcerides in contact with the water surroundings with which the
saponification is carried out, is lower than that of the other. The
consequence of this would be that the triglyceride, which has
the lowest surface tension in contact with the water phase, was
adsorbed at the common surface, and was consequently more rapidly
saponified.

Of this, however, nothing has ever appeared.

It has been shown by Trum *) that in the saponification with
bases as well as when palm oil and olive oil become rancid, the
iodine value of the split off fatty acids agrees with that of the fatty
acids that are still combined to glycerine.

STIEPEL®) finds for autoclaved tallow fatty acids that the still
combined fatty acids exhibit a somewhat higher iodine value than
the split off ones; for the autoclavation of cocoanut oil and palm
kernel oil he arrives, however, at the conclusion that the split off
and the combined fatty acids have the same composition. STIEPEL
finds a corroboration of this’) in the fact that on distillation of
partially saponified cocoanut and palm kernel oil the distillate
presents the same acid values as that on second distillation of the
fat mass that had first remained behind in the kettle, after this
mass had been saponified anew, and now entirely.

It follows from this that a difference in saponifiability between
ester groups of different fatty acids may in general be neglected,
and further that the surface tensions of the glycerides occurring in
the fats examined by Trum and Srizprx in contact with the saponi-
fying surroundings can be only little divergent.

CoNNsTEIN, Hoyer and WARTENBERG *) have found that the fermen-
tative saponification with the ricinus seed ferment proceeds more

1) Z. f. angew. Chem. 3 482 (1890).

2) Seifens. Ztg. 831 937, 965, 986, 1006, 1026 (1904), 36, 788, (1909).
5) Seifens. Ztg. 35 1359 (1908).

4, Ber. 35. 3988. (1902).

44

slowly as the molecular weight of the combined acids is lower. It
is therefore not excluded that also when glycerides of fatty acids of
different molecular weight occur side by side, they will present here
a specifie saponification velocity. Experiments of the same nature
as have been made by Tuum and Srieerr, are not mentioned by
CONNSTEIN C.S.

§ 7. In order to be able to form an opinion about the surface
tensions of fats in contact with different media,
I made the following experiments:

With the aid of Donnan’s pipette the number
of drops were counted which, when a definite
volume of different triglycerides flowed from a
thin walled capillary, mounted up in water and
some solutions, all this at 100° C. The arrange-
ment of the experiments follows from fig. 1.
The pipette with the tube C which contained
the aqueous solution, was heated to 100° in a
large glass beaker with water. During the outflow
of the fat the beaker was not heated to avoid
shaking through the boiling of the water. The
filling took place by the fat being sucked up
into the bulb 4. The mouth of the capillary
was then cleaned as well as possible with a
cloth, and the tube C was fastened with a rubber
stopper. The whole was now carefully heated in
the beaker till the fat just appeared from the
capillary mouth, then the solution (always a
same quantity) was poured into C' along the wall.
When the whole had reached the temperature of
100°, then the level of the fat was reduced to-a by opening of the
cock, and then the number of drops was counted which mounted

Fig. 1.

in the solution in C during the fall of the level of the fat from.

a to 6. The diameter of the capillary mouth was about 1 mm.
Trilaurine flowed out from a to 5 in about 4 minutes.

During the outflow of the fat every drop remains hanging at the
capillary mouth till the upward pressure exceeds the tension of the
surface. The greater, therefore, this surface tension is, the fewer
drops will get detached when a definite volume of the fat substance
flows out, and the smaller the number of drops that will mount in
the aqueous liquid. When we disregard the difference in specific
weight of the different triglycerides, the tension of the surface of

|

7
’

45

contact is roughly inversely proportional to the number of drops.
„It it clear that the results obtained with the described apparatus
give an indication only in rough approximation about the ratio of
the surface tensions of different glycerides in contact with aqueous
solutions. To determine this ratio quantitatively more accurate
measurements. are necessary than can be carried out with the Donnan
pipette. It will, however, appear in § 19 et seq. that the data
obtained in the described way, can qualitatively entirely account
for the phenomena that present themselves in the saponification of
fats.
The obtained results are recorded in table I.

TABLE 1.
1 2 3 4 ab ey ae, | 8 | 9 | 10
Saponification 5 Number of drops mounting
value is at 100°C. in:
is CB) S ~ | NO
3 3 3 lef) [ee [ZES
g 5 3 i gs =o es 5E
= z BRON ek care oe. |e leu SEG,
= 2 = 2 3 [SES | Sat | 28 | sem.
En Di i rn hi S |Ghe|Be| 35
0 2 sk IS 15
S Si he 28
op) os SS
oa — Sf
Trilaurine 0.0 | 264.0 | 264.3 | 0.0 | 43.7 | 10 | 44 53 | 190
Tripalmitine | <0.2 208.9 | 208.3 0.8 62.0 12 45 | (110) | (870)
Tristearine | <0.2 | 189.2 | 190.0| 0.0/68.7| 13 | 50 | (85)| -—
Olive oil 1.43 | — | 191.0] 83.9] 22:5] 11 | 50 | (100)| (340)
Linseed.oil | <0.2 | — | 191.3 182.3 20.6) 15 | 48 \ (70)| (210)

Trilaurine was obtained by recrystallisation of Tangkallak fat
from alcohol, then from ether, tripalmitine by recrystallisation of
Chinese vegetable tallow from benzene, then washing of the obtained
product with alcohol, and again recrystallisation from ether. Tristearine
by recrystallisation of catalytically hardened linseed oil from benzene,
then also washing with alcohol and recrystallisation from ether. The
olive oil used was oil sold for consumption from French origin.
The linseed oil had been freed from free fatty acid as well as
possible by treatment with lye. From the constants recorded in

46

columns 2—6 the purity of the examined triglycerides appears
sufficiently.

It now appears from columns 7 and 8 that the surface tensions
of the examined triglycerides in contact with water and a 1 °/,
solution of TwitcHELL reagent diverge but little ater se, and it is
clearly visible that in the presence of reagent the drops get sooner
detached, the surface tension between fat and water phase has
therefore decreased.

In alealic surroundings greater divergencies were found between
the different triglycerides inter se (see columns 9 and 10). It is,
however, very much the question if they are essential. In triglycerides
which cannot, like trilaurine, be purified by recrystallisation from
alcohol, it is exceedingly difficult to remove the last traces of free
fatty acid. In alealic surroundings these traces cause a lowering of
the surface tension, and give moreover rise to irregular moistening
of the capillary mouth, which is the cause that often great deviations
are found in repeated determinations. The values which are little
reliable for this reason, have been placed between ( ). The lowest
number of drops (rounded off to tens) that was found on repeated
determination, has always been given. These values are of importance
in so far, that they show clearly the influence of the molecular
weight of the soap which is dissolved in the water.

As appears from Trum and StTIEPEL’s observations and from the
results with Donnan’s pipette described in this $, no difference need
in general be made in the derivation of an equation w (z,s) = 0
between natural fats and simple triglycerides. A function derived
on this supposition must, however first be tested by different fats
in the fermentative saponification, before further conclusions are
drawn from it.

DERIVATION OF AN EQUATION w (z, s) = 0.

§ 8. As the reaction takes place on the boundary of fat and
water phase, the velocity with which each of the stages of saponi-
fication proceeds, will be governed by the surface tension of tri-,
di- and monogiycerides against saponifying surroundings. For ife. g.
the surface tension of the diglyceride against the water phase is
smaller than that of the tri-glyceride, the diglyceride will directly
after its formation be adsorbed at the surface of contact, and there-
fore reach a greater concentration in the surface layer than when
no adsorption took place. The consequence of this will be that an
estergroup of a molecule of diglyceride has on an average a greater

47

chance to be saponified than an estergroup of a molecule of tri-
glyceride.

To be able to form an opinion about the surface tensions of tri-,
di-, and monoglycerides against saponifying surroundings the behaviour
of the laurines was examined by the aid of the apparatus described
in the preceding §. The results are found in table 2.

TABLE
1 2 3 4 5 | 6 | 7
Saponification Number of drops flowing
value out at 100° C. into:
| = x
| SO = 5 oe
a. on ras
Fat subst ae: ba ES =m
at substance See. E 5 250. 2-0:
5 5 = = SAL | SEE
2 © = | og ox
= [Ly = = are | Oo DE
pe | Se | 28
eee
Trilaurine 264.0 | 264.3 | 46° 10 44 53
Dilaurine 246.3 | 246.3 | 54 1855 42 410
Monolaurine 205.0 | 204.4 | 62.8 — — =
90 Trilaurine + 10 Dilaurine — — — 12 44 90
90 a + 10 Monolaurine — — — 80 280 flows !
Laurinic acid 280.7 | 280.4 | 43-711) | 38 70 af

The laurinie acid was prepared by saponification of trilaurine,
obtained from Tangkallak fat, followed by distillation in vacuo. The
lower laurines were obtained by esterification of laurinic acid with
excess of glycerine at about 200° in the way indicated by van Erpik
TareME’). To purify the dilaurine it was first recrystallized from
alcohol (to remove monolaurine), then from benzene (to remove
trilaurine). The monolaurine was first recrystallized from petroleum-
ether (to remove di- and trilaurine), then from alcohol (to remove
monolauryldiglycerine). All the glycerides were perfectly free from
oleic acid and free fatty acid. For the rest the constants mentioned
in columns 2—-4 sufficiently express the purity of the substances used.

The number of drops of monolaurine rising in aqueous solutions
could not be determined, as a skin is formed on the boundary of

1) Solidifying point.
2) Thesis for the doctorate. Delft (1911).

48

monolaurine and water, so that there is no question of “drops’’.
To be able to form in spite of this an opinion about the surface
tension of monolaurine against aqueous solutions, the number of
drops that mounted of a mixture of 90 °/, trilaurine and 10 °/, mono-
laurine, has been given in table 2, while for a comparison the thus
obtained values of dilaurine have been given.

From table 2 the following conclusions can be drawn:

1. In acid and neutral surroundings the surface tensions of tri-
and dilaurine in contact with the saponifying medium differ little,
that of monolaurine is much less. We must therefore expect that
in case of saponification in non alcalice surroundings the monoglyce-
ride will be adsorbed at the boundary of fat- and water phase, and
_will, therefore, be saponified with a velocity greater than that with
which it has been formed.

2. In alealic surroundings both the surface tensions of di- and of
monolaurine in contact with the saponifying medium are much
smaller than that of trilaurine. Both di- and monoglycerides will,
therefore, be absorbed here at the boundary layer; hence they are
saponified with velocities greater than that with which they have
been formed.

As appears from what precedes the increase of concentration of
the lower glycerides at the surface of contact between fat and water
phase must be taken into account in the derivation of an equation
w(z, s) = 0. We now put:

p resp. q = the number of times that the concentration of the digly-
ceride, resp. monoglyceride at the boundary layer is greater in conse-
quence of the adsorption than if no adsorption had taken place, and
we put p and q both constant.

This assumption is an approximation, because the adsorption is
not proportional to the total concentration of the adsorbed sub-
stance. ') This approximation will be the closer as the concentration
of the lower glycerides will vary between a narrower margin during
the saponification.

$ 9. Since the difference in velocity of saponification between
esters of primary and secondary alcohols is only slight, and the
isomeric di- and monoglycerides can therefore be considered here as
equivalent with close approximation (which also follows from Gerrer’s
and J. Meyers results), we come to the following scheme for the
saponification of fat:

1) See FREUNDLICH, Z. f. phys. Chem. 57, 385 (1907).

Pe ee hie

ado mate

NELE Fe Hi
ee
k p-k qek

(r) (x) (y) (s)

In this A represents a molecule of triglyceride, which can be
converted into a molecule of diglyceride (6, C, or D) in three ways
with a constant of velocity 4. A molecule of diglyceride, e.g. B can
give a molecule of monoglyceride (W or F) in two ways with a
constant of velocity p.k, while finally every molecule of monogly-
ceride forms glycerine (H) with a velocity constant q.k. The number
of molecules of A present after a time ¢ will be expressed by r,
the number of molecules of B, C, and D each by w, of EZ, F, and
G each by y, and of H by s.

It is clear, that when there is no difference in saponifiability of
the different ester groups, the concentrations of B, C, and D, and
of EZ, F, and G are equal inter se at any moment.

Let us now suppose a molecules of triglyceride A to be present
at the beginning of the reaction.

The velocity of saponification of A is now denoted by the equation:

dr
eee at Tar Roan an eet oes a (LA)
When we integrate (7) and consider that for ¢=0, r = a, then
follows:
aa Ag re TE tense EB)
The variation of the number of molecules B (hence also of C

and D) is represented by:

dx
es ee 0 Ee Ak wi ha wD)
When the value of 7 from (8) is substituted in (9), then follows:
du
apt Bits Bad on Oe teed Een ene, 3 (10)
This equation can be solved by putting:

in which, therefore, an arbitrary value can be given e.g. to m; n
is then fixed. f

4
Proceedings Royal Acad. Amsterdam. Vol. XX.

As

La —
dt dt Fat
(10) passes into:

dn dm
m.— +n \— + 2pk.m) —=a,k.e kt oy, (12)
dt dt
Let us now take m so that
dm
Ok ah
ae

On integration of this last equation we then find for m:
ie its Cale. Beas | ie ee oe
Introducing this value into (12), we get:

e—2pkt , a —a.k.e.—akt,
dt
from which by integration:
gp oH + CO. Wie * = ia ea
Now follows from equations (11), (13), and (14):
J — et pa seas oe ER GER Cage ee sat Se
Bearing in mind that for _=O also «=O, we find for the
integration constant C:
a
ME
2p—3
through which (15) passes into:
00
dennen «tg fe, oe 6 EN

The change of the number of molecules £ (hence also of # and
G) is represented by:
dy

— = 2nk. e—gk. nn Tear aa pee J
Er pk.a—gk.y (17)

If the value of « from (16) is substituted in (17), then follows:

dy 2pk
utreg eek. EC

This differential equation can be solved in the same way as (10).
We then find:

1) A constant of integration can of course be omitted here

Di
e—3kt e—2pkt e—qkt
‘ |@p—8Xq—3) (@p—8)(q—2p) | @—8)(q—2p)4

At last the number of molecules of split off glycerine (s) can be
calculated from:

y= a.ap (19)

ds
Oe SL
or from:
s—a— r— d(x+y).
We then find:
feel ns ois 2g SM ES SO
(2p-3)(q-8) _(2p-5)(g-2p) (7-8)(q- 2p)
The number of molecules of split off fatty acid is, as appears
from the scheme at the beginning of this $:

zr 3. 2y + 3s.

. (20)

Su!

When we substitute in this the equations (16), (19), and (20), we get:

ut BRENG) e—3kt EE e—2pkt — UVR tat

(2p-8)(q-8) (2p-3)(q-2p) (q-8)(q-2p)

Now we can eliminate £.t from the formulae (20) and (21) for
definite values of p and g, which gives usa relation between s and z.
It is however, more practicable to substitute two other quantities
for s and z.

The total number of molecules of glycerine is a. If we now call
that part of the total quantity of molecules of glycerine that is
split off g, then

zie,

s

j= —-.
a

The total number of molecules of fatty acid is 3a. If we now call
that part of the total quantity of molecules of fatty acid that is
split off 7, then:

Tk aoe
Now follows from the equations (20) and (21):
GE {-— dE et Pe, Ep Ek op — e—qkt (22)
(2p-3)(q-3) (2p-3)(q-2p) (9-8)(q-2p)
and
gn EY pe BP ek (23)
(2p-3)(q-3) (2p-3)(q-2p) (q-8)(q-2p)

If now p and q are successively given different values, we obtain
4*

52

equations for g and 7, from which & . ¢ are more or less easy to
eliminate in accordance with the values assumed for p and gq.

§ 10. Before proceeding to substitute for p and q numerical
values, we point out that the equations (22) and (23) do not
allow us to substitute the following values:

p="*/s + q=8 and q = 2p.

It is easy to see that in the derivation of (22) and (23) operations
have been performed, which it is not allowed to execute with the
above mentioned values of p and g. If we yet wish to introduce
these values, we must proceed as in § 9, and substitute the assumed
values for p and q from the very first. We then get transcendental
equations for the function :

wp (Tg) = 0.

§ 11. Let us now first define the limits between which all the
curves represented by the functions w(7Z’,g) = 0, for different values
of p and q, must lie. It is clear that the extreme values, which

1
p and q can have, are @ and —.
ee)

Let us first put p=o and g=o. The physical meaning of
this is, that the increase of concentration of the lower glycerides
at the surface of contact in consequence of the adsorption is so
great that their velocity of saponfication compared with that of the
triglyceride, is oo.

The equations (22) and (23) pass in this case into:

gie a er
TSA ek SS a

from which
RS

This result can of course at once be understood. In fig. 2, where
g and T are both given in percentages, A represents this limit.

The equation g = T will be more fully discussed in § 19.

l
Let us now put y=—. The physical meaning of this would be
oO

Le
that the monoglyceride reaches only a concentration of — in conse-
ee) g

quence of negative adsorption in the boundary layer. This limit
has only mathematical signification. (See Fig. 2 p. 53.)
The equations (22) and (23) pass in this case after finite time,
into: |
JOE ie te Ae)

53

EN

Be
oa
a
al
fal
ES
a
|
Ps
a
B
a
iS
kai
a
ie
Z

\

100T

(28)

i.e. in finite time no glycerine is split off, the reaction goes no
further than monoglyceride. In finite time 7 approaches °/,.

1
If ¢ =o, (22) and (23) passes for g = — into:
co

Md be SON eo weary ee ten Aa)
TER Ao. Ye a 2 (50)

from which:
Gea eI ay Ree arene. (SL)

I
For g=— we find therefore the two boundary lines
le ©}

g=0 and T=*/; + !/a9
both are indicated by B in fig. 2.
All the curves y(7,g) =O which are possible for different values
of p and q, must therefore lie within the limits:
g=T, 9=0, and T*/, + "hg

54

§ 12. Let us now put p=1, g=1. The physical meaning of
this is that there is no question of adsorption, because the surface
tensions of all the glycerides in contact with the saponifying sur-
roundings are the same. The velocity of saponification of an ester
group is now independent of whether or no a neighbouring group
is saponified.

For this case we find from (22) and (23):

g=(l—e—*ty? . (32)
T=1—e-# (33)

from which:
EO, Fe ae ot a ree ae

This curve is indicated by C in fig. 2. It touches the boundary
lines: Boat g= T == 0 and atg= hel

Equation (34) is valid for the saponification in solution. From
Gueirer’s*) and J. Meyer’s*) measurements follows that it holds for
the saponification of triacetine, at least with close approximation. If
it is possible to measure the split off quantity of glycerine in this

1) loc. cit.

TABLE 8.
Pa DE | 40> | 5 | 6 | 7 | 8
Value of 100 7 for
Se a) pt p= eo eal a= p=1 |: p=3 | pH=e
| q—2 q=4 Cat ge 7 =o) ge
(q = 23)

0 0 0 0 0 0 0 0
10 | 46.42 39.03 34.201; 125,00: 2266) 4} 47 20 10
20 58.48 51.90 45.99 | 37.56 | 34.59 | 28.24 20
30 66.94 61.45 55.20 | 47.25 | 44.73 | 38.26 30
40 73.68 69.27 63317 1) 55.99 «| 5302 el 247075 40
50 79.37 75.98 70.37 | 64.12 | 62.50 | 56.90 50
60 84.43 81.89 11.02 1917839) 70363: -:| 265,82 60
70 88.79 87.17 83:24 S 0-21 ay GAEL 6) 14555 70
80 92.83 91.94 89.15 | 86.34 | 85.88 | 83.15 80
90 96.55 96.22 94.73 | 93.26 | 93.10 | 91.62 90
100 100 100 100 100 100 100 100

55

ease, then the saponification of triacetine in stages can be directly
proved by this way.

When the equations (25) and (33) are compared it appears *) that
measurement of the velocity of fatty acid separation can never be
conclusive with respect to the question whether or no the saponi-
fication of triacetine in acid solution takes place in stages’).

In column 2 of table 3 (See p. 54) we find the values of
100 7 corresponding to the given values of 100g for the case that

p=q=1.

§ 13. Let us now put p = 1, q = 4, Le. in consequence of the adsorp-
tion the concentration of the monoglyceride in the boundary layer
is 4 times as great as it would be if no adsorption had taken place.

For this case equations (22) and (23) become:

g = 1 + 8e-3kt — Ge—2kt _ Be-4kt, . . . . (35)
T= 1 + 3e-3kt — Bet — ¢—Akt — ] — e— kts] — (1—-e—*t)3}. (36
From these two equations e—* can be eliminated by solving the

fourth power equation (35) and substituting the found value in (36).
From (35) we find:

eht, [4-1 16-92 + |/92-4-81 16-92 + 18) (2-2) +*/,(I-9)} |

in which : (37)

3 3
a") - Vol + 1-9}.
In column 4 of table 3 are found the values of 100 7’ calculated
by the aid of these equations.
These equations will be more fully discussed in $ 22. The curve
D of fig. 2 represents the corresponding values of 100 7’ and 100 4
graphically. It touches the boundary line B at y= 7'= 0.

§14.- If we ‘put p— 2, g=2, then:
g = 1 + 8e—3kt__Be—4ht__Ge—2kt . . . … … (35)
T=l He MAU leki(d-ekt) . . . (38)
It appears that the split off quantity of glycerine is a same func-
tion of the time as in the case that p=1, g = 4. This is a general
property. If p= 2m and ¢= 2m, or p=m and q= 4m, in both
cases we have:
8m?

gl - ——___1__ e—3kt ETE Shite Es e—2mkt
(4m — 3)(2m — 3) 4m —3 2m—3

1) In the acid saponification in solution & is invariable.
2 Cf. § 1. The fatty acid splitting off likewise becomes seemingly monomolecular
for p=1/,, q=@ and for p=, g= 2,

56

Hence in order to calculate 7’ for a definite value of g, for the
case p= 2, g=2, we can make use of the equations (37) and
(38). The thus obtained value is found in column 3 of table 3.

§ 15. If we put p=3, g=9, (22) and (23) change into:

9 Sal Se DE AT otter ee ee
Pa (ee ee ee eg ee ee

from which:
SS AG Age GINS | o> fee ee

In column 5 of table 3 are found the values of 100 7 calculated
by the aid of this equation. The curve F of fig. 2 represents the
corresponding values of 100 7’ and 100g for this case graphically.
It touches the boundary line B at g = 7=0O.

With a deviation < 0,3°/, equation (41) holds also for the case
p=1, ¢= 23. This will be more fully discussed in § 21.

§ 16. For p=1, ¢g=o (22) and (23) change into:

gat Bee Tek a (42)
PS eae Seo ee ee

from which
Erg Sag = Fay ook EN

If 7 is solved from this, we find:

T='/, {1 + 49 + 2V1 + 8g. cos (120° — 1/, pt,
in which:
(45)
89° + 20g—1
V (1489)
The values of 100 7’ calculated from this are found in column 6
of table 3.

cos P = —

§ 17. If we put p=3, g=o, (22) and (23) become:

g= 1—2¢—-8kt + pene oe Us ae (46)

PS frege sa (EN

from which:
LSR FRE eee a

In column 7 of table 3 are found the values of 100 7’ calculated
from this. The curve F of fig. 2 represents the corresponding values
of 100 7 and 1009 for this case graphically. It touches the boundary
line B at y= T'==0. Equation (48) will be more fully discussed

in § 20.

57
Testing of the Derived Formulae.

§ 18. Measurements for the purpose of a comparison of the split
off quantity of glycerine with the split off quantity of fatty acid
have been carried out by KerLNERr!), who determined free fatty
acid and combined glycerine’) of partially saponified palmkernel oil
by different methods of saponification.

Let us first examine how 7’ and g are to be found from Krraner’s
observations. For the calculation of the percentage of split off fatty
acid the procedure is always as follows: The acid value is determined
of a sample of the fat (which has first been washed with water
and then dried), this is divided by the acid value of the esterfree
fatty acid, and multiplied by 100. The value obtained (we shall
call this 100 7”) now indicates how much free fatty acid the sample
contains in percentages, but only in approximation what percentage
of the total fatty acid present occurs as free fatty acid (100 7’).
The acid value of the esterfree fatty acid indicates how many mer.
KOH is required to neutralize 1 gramme of this fatty acid. If ofa
sample of partially saponified fat we want to determine what per-
centage of total fatty acid present occurs as free fatty acid, we
must know, not the number of mgr. KOH (a), required to neutralise
the free fatty acid of 1 gramme of fat, but the number of mgr. KOH
(6) required for a quantity of fat which contains the same quantity
of total fatty acid as 1 gramme of esterfree fatty acid. The value of
saponification being a measure for the total fatty acid present,

_ we get:

6 _ saponification value of the esterfree fatty acid
a saponification value of the fat to be examined ©

(49)

It is clear that in consequence of the glycerine content of the
partially saponified fat, always 6 >a. To find, therefore, 7’ from
1’, we multiply by b/a.

To calculate g we multiply the glycerine content of every sample
again by 6/a, and thus find the number of grammes of glycerine present
in a quantity of the sample, which contains 100 grammes of total fatty
acid. If we now also know the glycerine content of the triglyceride,
hence also the quantity of glycerine present in so much triglyceride
as contains 100 grammes of fatty acid, g can be directly determined.

§ 19. Let us now discuss KeLuner’s results.

1) Chemiker Ztg. 33, 453, 661, 993. (1909).
*) According to the oxidationmethod.

58

We calculate from the given acid values and fatty acid contents
(table 4—8) :

Acid value esterfree fatty acid .

Mol. weight __,,

9 >

grammes of glycerine in 100 gr. triglyceride

” 2)

bi a>

triglyceride ,

= 258.0, from this:

= 217-5
= 691.4
13.31

ie „ triglyceride of 100 gr. fatty acid—= 14.08.

For the saponification of palmkernel oil with aqueous KOH
KELLNER now gives the values of columns 1, 2, 3, and 6 of table 4.

TABLE 4,

1 7 3 4 5 6 af

PATER ue fie ere ieee peenig ,
: aponitication| % 9 glyc. | with resp. Og free fatty

Acid value value ‚in the fat, to total | 100 g acid in the fat | 100 7
| fatty acid |

96.3 249 8.26 8.56 | 39.20 31.32 38.67

193.3 253.8 3.41 3.41 15.35 | 74.92 16.16

It appears from columns 5 and 7 that g = 7, hence practically
p=q=~o (ef. § 11 and fig. 2 line A), in other words, there prac-
tically directly takes place splitting up into fatty acid and glycerine.

The values found by KeLuNeEr for the saponification of palm kernel
oil with lime are found in columns 1, 2, 3, and 6 of table 5,

ABIES:
1 | 2 | 3 4 5 6 1,
Is ificatio 4 1 | oe | 0/ free fatty
- aponification | % glyc. | with resp. | 0
Acid value value inthe fat | to total 100 g acid in the fat | 100 7
fatty acid
101.65 248.8 37-86 8.09 42.54 39.39 40.85
169.5 251.0 4.31 4.43 68.54 65.69 67.52

Here too it appears on comparison of columns 5 and 7 that

g=T, and therefore p—=g==oo must be practically valid also
here. It follows therefore from KELLNER’s experiments that in the
saponification of palmkernel oil with aqueous lye, as well as with

59

lime, the triglyceride splits up practically directly into glycerine and
fatty acid.')

This result is in conflict with the results of LewkowitTcu ?), who
in the alealie saponification of tallow and cottonseed oil concluded
to a saponification in measurable stages from the increase of the
acetyl value. It is, indeed, not probable that tallow and cottonseed
oil would have a stagewise saponification in alcalic surroundings,
and palmkernel oil practically not.

As in the saponification of olive oil, tallow and tristearine with
normal KOH R. Fanto’) has found that here too the separated
quantity of glycerine agrees with direct splitting up of the trigly-
ceride into glycerine and fatty acid, in LewkowiTcn’s experiments,
the increase of the acetyl value must be explained by other causes
than the presence of lower glycerides. Marcusson *) has shown that
this is really the case. The increased acetyl value is as well caused
by the fatty acids, as by the fat that has remained unsaponified.
Probably the oxidation of the unsaturate fatty acids plays a part
here, which also explains the irregularity of increasing and decreasing
of Lewkowitcn’s acetyl values.

The results obtained by Fanto and KeriNeEr, perfectly confirm the
conclusion drawn at tbe end of $ 8. In alealie surroundings the
adsorption of the lower glycerides at the surface of contact between
fat and water phase is so great that the chance to collision between
an OH’ ion and a molecule of di- and monoglyceride is practically
oo compared with the chance to collision between an OH’ ion and
a molecule of triglyceride.

§ 20. For the fermentative saponification of palmkernel oil
KeLLNER found the values given in columns 1, 2, 3, and 6 of table 6.
(See p. 60).

On comparison of columns 5 and 7 it appears that here g—/= 7.
In column 8 are recorded the values found for 100 7, when 7’ is
calculated from g by the aid of formula (48), i. e. on the assumption
that p—3 and g=o. (See § 17 and fig. 2 curve F’)).

It appears that the calculated and observed values of 100 7’ agree
sufficiently, especially when we consider that g cannot be determined

1) Kettner draws this conclusion by comparison of the found glycerine content
of the partially saponified fat with that calculated on the assumption of a direct
complete splitting up.

2) Ber. 33, 89 (1900); 36, 175, 3766 (1903); 37, 884 (1904); 39, 4095 (1906).

3) Monatshefte f. Chemie 25 919 (1904).

4) Ber. 39 3466 (1906), 40 2905 (1907).

60

TABLE 6

ms I Gata zi

0 |
Saponifi- % | glyc. with. eee 100 7
Acid value | cation | glycerine |respect to, 100 2 ed dal 100 7 calc. from
value in the fat | total fatty Peder (48)
|

| acid |

66.7 241.5 10.63 | 11.36 | 19.32 | 25.86 21-63: “| 275

18.4 243.3 |. 9.95 | 10.55 | 25.07 | 30.39 | 32.23 | 33.40
84.27 | 243.6 | 9.63 | 10.20 | 27.56 | 32.66 | 34.50 35.87
16.10 | 247.2 | 71,92 | 8.27 41.26 | 44.99 | 46.96 | 48.92
165.15 | 250.8 | 5.38 | 5.53 | 60.72 | 64.01 | 65.85 | 66.45
284.22 | 252.7 | 1.41 | 1.44 | 89.77 | 90.78 | 92.68 | 91.43

| |

more accurately than to about 1 °/, (for smaller values of ga much
greater error is even inevitable).

Of course the conclusion may not be drawn that in the said sapo-
nification p=3 and g=o. Also by assuming other values of p
and q equations can be drawn up (which however in general do
not enable us to express 7’ explicitly as function of 9), which more
or less accord with the values found experimentally '). Accordingly
equation (48) and likewise the equations discussed in the following
$$ must be considered as formulae of approximation, which roughly
give an insight into the relations of the surface tensions of the three
glycerides against the saponifying surroundings. When the relation
between p and g on one side and the surface tensions between
aqueous solutions and the three glycerides on the other side are
quantitatively known, then we shall be able to decide in how far
the here assumed values of p and g are conformable to the truth.

With regard to table 6 it may still be pointed out that the results
obtained by KerLNeR do not agree with what was found by M.
Nrcroux for the fermentative saponification of cottonseed oil. From
Nic.ovx’s values follows a practically direct splitting up into glyce-
rine and fatty acid. (See $ 5). Possibly the difference lies in this
that KeLLNer kept the emulsion in motion by blowing in air, Nicroux
on the other hand brought about the emulsion by stirring, and left
it undisturbed after that.

1) It is the question whether p and q are here only functions of the surface
tensions between glycerides and the aqueous solution, as the enzym is not in
solution according to Nictoux (loc. cit). (See also § 7).

61

In conclusion it may still be pointed out here that the relation
existing between free fatty acid and separated glycerine offers a
chance to throw light on the mechanism of the splitting up of fat
in germinating seeds. It is still an open question whether the reaction
takes place there analogously to the saponification by the aid of the
ferment from ricinus: seed.

$ 21. In the saponification according to the TwitcHELL process
KerLNER found for palmkernel oil the values from columns 1, 2, 3,
and 6 of table 7.

FAB LE-7.

wd
1 2 3 4 Bll 26 KA Hee 8
Oo. 5 ‘
Ds | Saponifi- Oo. glyc. with | fred 9 atty | 1007
Acid value, cation glycerine | respect ta) 100 2 agian, | 100 7 calc. from
value jin the fat, the total the fat | (41)

fatty acid

56.9 241.7 | (11.36) | (12.13) | (13.85) |, 22.05 | 23.54 | (30.79)

| |

91.9 242.0 | 10.322 11200 21.88 35.63 | 37.99 39.48

122.4 244.8 S272 SIS OL 1G 34.73 47.43 | 49.99 51.48
171.5 245.0° jw 5.15 5.34 62.07 68.81 | 71.33 13.38

210.3 252.0 2.87 2.94 79.12 81.54 83.48 85.12

In column 8 are found the values of 100 7’ calculated from g
by the aid of formula (41), which has been derived for the case
that p=3 and q=9, but which with a deviation smaller than
0.3°/, is also valid for the case that p=1, g = 23 (ef. § 15 and
fig. 2 curve E). .

In the first row the found and the calculated values of 100 7’
diverge greatly. This, however, says little, as this great difference
already disappears if the glycerine content of the fat is 12.1°/,
instead of 11.36°/,. When little glycerine has as yet been separated,
a small error in the glycerine content of the concerned sample or
of the triglyceride, on which the calculation of g is based, has a
very great influence on the calculated value of 7. The agreement
between the other values of column 7 and 8 is satisfactory.

As it now appears from the experiments described in § 7 and
$ 8 that p—=1 in general in the TwitcHELL saponification, g must
have a value in this case, which, as appears from the agreement

62

of columns 7 and 8 in table 7, differs little from the value g — 23
at the lower limit. *)

§ 22. The values found by KeEtiner in the autoclave process of
palmkernel oil are found in columns 1. 2, 3 and 6 of table 8.

TABLE S,
1 | 2 3 | 4 | 5 6 7 8
| | |
| | 0 | 100 7
Acid value Saponi- | seated gehn 100 g ree fatty 100 T “30 fou

| value (in the fat | bre ‘in the fat (37)
55 | 242 | (12.16) | (12.96) | (1.95) | 21.30 | 22.71 | (31.24)
(315% 1 229 9.84 | 10.26 27.13 | 50.96 53.12 5 Bf fs
193 251 5.28 | 5.43 | 61.43 | 74.80 16.89 | 71.94
212 aoe 3.75 | 3.044 | 72.23 | 82.17 84.13 | 84.89
218 |, 258 | A ON yeeee | 79.47 | 84.48 | 86.15 | 88.85
229.5 | RT PP Bl Sp LBA | 84.80 88.94 90.09 | 91.86

| | |

In column 8 are found the values of 100 7’ calculated from g
by the aid of the equations (36) and (37), i.e. on the assumption
that p=1 and g= 4 (see $ 13 and fig. 2 curve JD).

In the first row the deviation between the found and the calcu-
lated values of 100 7 is again greatest. Much importance should
not be attached to this here either, as this deviation -already dis-
appears when the glycerine content of the fat is 12.6°/, instead of
12,16°/,. The other values of columns 7 and 8 agree sufficiently.

It is not improbable that also in the autoclave saponification,
where the saponification takes place in feebly acid surroundings,
p=1. Nothing can be said of this, however, with any certainty,
as the influence of zinc soap has not been examined in the experi-
ments of §§ 7 and 8. If really p= 1 also here, g must have a
value which differs little from g = 4 in virtue of the en of
columns 7 and 8 of table 1.

SUMMARY.

It has been shown in this paper that in the saponification in
emulsion the reaction takes chiefly place on the boundary of the

1) A deviation upward has little influence. (See table 3 columns 5 and 6),

63

two phases and that in this case the process of the saponification
is governed by the value of the surface tensions between the gly-
cerides and the saponifying medium.

As velocities of saponification do not give an insight here in the
mechanism of the reaction, because they are influenced by the
variable fineness of the emulsion, equations were derived which
give the relation between separated fatty acid and separated
glycerine.

The equations, in which the increase of concentration of the
lower glycerides at the surface of contact between fat and water-
phase were taken into acount, appeared to be able to account for
the different course of the saponification in different surroundings.

In conclusion I gladly avail myself of the opportunity to express
my thanks to Dr. Geren for the kind interest he has taken in
my work.

Laboratory of the Royal Stearine
Candle Works “Gouda”.
7ouda, November 1916.

Chemistry. — “On Polymers of Methylchavicol’. (Preliminary
communication). By Prof. P. van RoMmBureH and J. M. van
DER ZANDEN.

(Communicated in the meeting of March 31, 1917.)

Some years ago one of us (v. R.) communicated that through
heating of methylechavicol some products of polymerisation were
obtained, viz. one that melts at 98°, and one that melts at 166°,
while moreover still a very slight quantity of a compound melting
above 200° could be separated. Elementary analyses and deter-
minations of the molecular weight of the two first-mentioned products
made it probable that these are dimers of methylehavicol.

The compound melting at 98° gave a dibromide melting at 87°
with bromine.

Since then a greater quantity of these polymers has been prepared,
so that a more extensive investigation was possible, some results
of which will now briefly be communicated by us here.

In the first place it has been tried to augment the not very large
yield of polymers. On change of the duration of the heating and
of the temperature it appeared that a slight improvement of the
yield can only be obtained by prolonging the former.

Besides the crystalline compounds a viscous syrup, which still
contains large quantities of it, is formed. Now we tried to separate
part of it by distillation in vacuo (15 m.m.), and we actually
succeeded in getting a fairly large yield of crystals from the fraction
283°—313°. When the residue in the flask is heated to a higher
temperature (to 350°), a reaction evidently sets in, at least the
distillate becomes thinly liquid and the liquid that was distilled over
goes over at ordinary pressure between 150° and 350°.

By continued fractionated distillation a product boiling between
175° and 178° was obtained, whose smell resembles that of the
methylether of paracresol boiling at 175°.5. Oxidation of the compound >
formed with the mixture according to Kimianr yields an acid, which
after recrystallisation melts between 178° and 181° and does not
give lowering of the melting point when mixed with anisic acid.

If the substance is treated with cone. nitric acid (Sp. gr. 1.5),
light yellow crystals are formed melting at 122°, which do not

65

exhibit lowering of the melting point when mixed with the dinitro
compound obtained through the same treatment from p. cresyl methyl-
ether. The liquid obtained is therefore undoubtedly p. cresyl methyl-
ether. How this substance is formed, has not been explained as yet.

The polymer melting at 166°, dissolved in chloroform, gives with
bromine a bromide erystallising out from petroleum ether in fine
needles, melting at 139° while assuming a black colour. About the
experiments which have been made to determine the structure of
the polymers, we, may communicate what follows.

The product melting at 98° was oxidised with a solution of
potassium permanganate in acetone. At this oxidation there is formed
besides anisic acid, an acid which on recrystallisation separates from
toluol into hard massive crystals. Melting point 113°. A micro-
elementary analysis gave C 69.42 and 69.34°/,. H 8.06 and 7.68°/,.
These results point to a substance of the composition C,,H,,O,
(Theory C 69.22 °/,, H 7.74 °/,).

We found 207 for the molecular weight, by titration, assuming
the acid to be monobasic. Calculated for C,,H,,O, : 208.

Continued oxidation, now with the mixture according to Kmanr,
caused a new acid melting at 138° to be formed from this acid
melting at 113°. We are still occupied with this research.

‘Oxidation of the product melting at 166° yielded only anisic acid.

Utrecht. Org. Chem. Labor. of the University.

Proceedings Royal Acad. Amsterdam. Vol. XX.

Chemistry. — “Action of Organo-magnesium Compounds on, and
Reduction of Cineol.” By C. F. van Dur, chem. docts.

(Communicated by Prof. P. van ROMBURGH).

(Communicated in the meeting of March 31, 1917).

The first communication in the literature about the action of
organo-magnesium compounds on internal oxides is given by BLAIsE *),
who found that on action of RM, Br on ethylene oxide, both a
primary alcohol and the bromic hydrine of glycol are formed. The
formation of these two products led him to assume that the organo-
magnesium compound is added to part of the oxide according to
equation RM,Br + (CH,), O = RCH, — CH,OM,Br (f) and to another
part according to the reaction scheme: RM,Br + (CH,),0 = CH, Br —
— CH,OM,R (II).

GRIGNARD ®), however, rightly pointed out that according to BLatsr’s
reaction scheme II the organo-magnesium compound splits up into
RM, and halogen in its addition; a case which had never yet been
observed. He then proved that in this first of all an oxonium

CH, M,br
compound is formed of the formula | O , which on heating
CH,” \R
CH,OM,br
passes into | through an intramolecular conversion. The
CH,R
appearance of the bromic hydrine of the glycol can then also be
explained by action of HBr (formed at the hydrolysis of the M,Br’)
on oxide that has not reacted and had then split off. The mechanism
of the reaction had been made clear by this. Henry studied the
action of organo-magnesium compounds on isobutylene oxide *), symm.
butylene oxide *), and propylene oxide‘); the first reacts then as the

1) Braise. CG. R. d. l’Acad. des sciences 134, 551 (1902).

2) Gricnarp C. R. d. l’Acad. des sciences 136, 1260 (1903), Bull. Soc. chim.
(3) 29, 944 (1905).

3) Henry C, R. de l'Acad. des sciences 145, 21 (1907); cf. ibid. 145, 154 (1907).

4) Ibid. 145, 406 (1907).

5) Ibid. 145, 453 (1907).

67

isomeric isobutylaldehyde, the second as methylethylketon, the reaction
for propylene oxide proceeding as for ethylene oxide.

Numerous are the investigations with epichloric bydrine; losrrscu *)
got on action of C,H, M,Br a chlorie amylaleohol, and Kune ®)
obtained quantitatively the chloric iodine hydrine with C,H, M,I;
IositscH *), repeating his experiments, now got both the chloric
bromie hydrine and a cbloric amylaleohol. Then Fournrav and
TirreNeau *) found that with aliphatic organo-magnesium bromides
only the chlorie bromic hydrine is formed, while with aromatic
eo compounds chloric propanoles with the constitution

—CH,Cl ari ae oe en
RCH, — ° con arise for which formerly the formula

CH,CLC > CH,OH had been assumed. *)

In conclusion we may mention that Fourneav and TrrrenNrAu ®)
drew the conclusion from their study on the action of organo-
magnesium compounds on mono substituted and asymmetrically
di-substituted ethylene oxides that for the former the general process

AE
ERZ
for the latter a previous isomerisation to the isomeric aldehyde
must be assumed.

When we take the general result of the above mentioned researches
into consideration, we should expect the primary formation of an
addition product on the action of an organo-magnesium compound
on cineol; that this is really so on action of CH,M,I was already
shown by GRIGNARD'), who also observed that on heating of this
addition product a violent reaction sets in. This reaction, however,
consists in more than only in an intra-molecular conversion; the
reaction taking place at about 160° is very violent; the temperature
rises to about 260°, while during violent gas generation a liquid
distills over, which is coloured brown through free iodine. The
reaction is not always equally violent; in connection with this the
quantity of gas that escapes, and the quantity of liquid that distills

ZH
of reaction is: R CH, + R’M,Br > ROK oa CH‚R', whereas

1) losirscH, Journ. russ. phys. chem. Ges. 34, 96. (1902).

2) Krine, Bull Soc. chim. (3), 31, 14 (1904) C. R. d. Acad. des sciences 137,
756 (1903).

3) losrrscH, Journ. russ. phys. chim. Ges. 36, 6 (1904).

4) FoURNEAU and TIFFENEAU, Bull Soc. chim. (4) 1, 1227 (1907).

5) J. D. Rieper, Akt. Ges. D. R. P. 183361; Ch. C. 1907 1, 1607.

6) FouRNEAU et TIFFENEAU, C. R. d. l’Acad. des sciences 145, 437 (1907).

7) GRIGNARD, Bull Soc. chim. (3) 29, 944 (1903).

5*

68

over, varies. The latter is best obtained by adding 52 grammes of
cineol to an etherie suspension of 56 grammes of CH,M,I, obtained
from 8 grammes of Mg and 48 grammes of CH,I; the whole mass
is heated for a few hours and left for 3 or four days.

Then the ether is distilled off, in which but very little cineol
goes over, the residue is heated up to + 160° on a sand bath, a
very wide exit tube is placed on the flask and the reaction proceeds
without further supply of heat. Thus from 30—40 grammes of distillate
are obtained and from 4-—5 L. of gas. There is, however, always
some liquid left behind in the flask; it is obtained by treating the
residue further in the usual way; the thus obtained liquid is the
same as that which distills over. The latter is first washed with a
thiosulphate solution and then fractionated in vacuo, collecting every-
thing together that goes over at 21 mm. to 85°. Then the distillation
is stopped, because then the residue begins to decompose with split-
ting off of iodine. What is distilled over is shaken a few times
with a 50°/, resorcine-solution for the removal of unchanged cineol,
which is present in a fairly large quantity ; then it is dried on
chlorie calcium and finally distilled over metallic sodium. Then a
liquid is obtained of b.pt. 170°—178° at 759 mm., consisting of a
mixture of hydrocarbons C,,H,,. The elementary analysis namely
gave: (burned with lead chromate in a closed tube): C=87, 84°/,;
H =,12.00°/,; (calculated. for. CH,» C= 88.15°/,; H—iieatee
the determination of the physical constants of the fractions obtained
by repeated fractionation :

fraction b.pt. 170°—172°,5; Sp. Graco —= 0.841
nps =d 4679
Mol. Refr. = 44.99
fraction b.pt. 172° 5-—175°: Sp. Grig = 0.846
N D160 = 1.4706
Mol. Refr. = 44.94
fraction b.pt. 175°—178°: Sp. Grice = 0.853
N D160 = 4 Ayo2
Mol. Refr. = 44.95
Calculated for C,,H,, 2 ©: Mol. Refr. = 45.25.

Besides, the fact that in not a single way a crystalline product
could be obtained from any of these fractions points to the presence
of a mixture. I tried the preparation of the tetrabromide according
to Warracn *) in alcohol ether, in sulphuret of carbon, and in

1) WALLACH, Annalen 227, 280 (1885).

69

methyl aleohol, of a nitrosochloride, and of a chloric hydrate, but
could not obtain a crystalline product anywhere. Nor did oxidation
with potassium manganate according to Wagner!) lead to a result;
the only products that could be identified were acetone and oxalic
acid ; hence I must rest satisfied with the communication that with
generation of methane a mixture of hydrocarbons C,,H,, is formed.

With C,H;MgbBr the reaction takes place much less vigorously,
and chiefly unchanged cineol distills over; the diminution of the
specifie gravity, as well as the reduction of an ammoniacal silver oxide
solution pointed, however, to the presence of hydrocarbons ©,,H,,.

The action of C,H,;MgBr proceeds quite differently; it is true,
here too an addition product is formed, but this remains unchanged
on being heated, even up to 200°, so that when treating it we get
back the cineol, besides diphenyl. Perhaps that here an intramole-
cular conversion may be obtained on continued action in the cold,
for then crystals separate from the etheric solution of cineol and
C,H,MgBr ; if these crystals are treated separately, a little (8 grammes
from 42 Gr. C,H,MgBr and 36 Gr. cineol after 3 months’ standing)
of a liquid that goes over between 80° and 90° at 21 mm., which
with phenylisocyanate gave no crystalline urethane. The liquid which
was poured off the crystals, yielded nothing but cineol.

With regard to the reduction of internal oxides according to the
method of SABATIER and SENDERENS, there exists in the literature
only one example, viz. the reduction of cyclohexene oxide to cyclo-
hexanol ®), in which the yield was even quantitative. Besides SABATIER *)
expressed the opinion that all the ethylene oxides could be hydrated
according to his method.

I could not get a reduction for cineol in this way ; when at 170°
this substance with an excess of bydrogen had been led twice
through a tube 40 em. long, filled with pumice nickel, the sp. gr.
had not changed in the least.

These researches, both with cineol and with other oxides, are
being continued.

Utrecht. Org. Chem. Lab. of the University.

1) WaGner, B. B. 23, 2315 (1896).
2) BRUNEL, Ann. Chim. Phys. (8) 6, 237 (1905).
3) SABATIER, Die Katalyse in der organischen Chemie Leipzig 1915, pag. 80.

Bacteriology. — “On the nature and the significance of Volutin
in Yeast-cells”. By Miss M. A. van Harwerpen M.D. (Com-

municated by Prof. C. A. PEKELHARING).
(Communicated in the meeting of April 27, 1917).

At the same time when the improvements in the technique of
microscopical science enabled investigators to demonstrate a nucleus
in a number of so-called non-nucleated unicellular organisms, atten-
tion was called to the presence of basophilic granules in the cell-
plasma of bacteria, hypho- and blastomycetes, algae and protozoa,
which behave towards various reagents in a similar way. Where
there is a cell-nucleus, as is the case with the greater part of these
unicellular organisms, these granuies are disposed apart from it, and
as to the affinity to stains they do not completely correspond with
the chromatin of the nucleus.

Originally Basks looked upon these basophilic granules in bacteria
as spores and described them afterwards as ‘‘Corpuscules metachro-
matiques’”’), a name still in common use in the French literature
up to this day. In Germany they are called ‘‘volutin-granules” since
A. Mrysr’s’) extended investigation of their chemical nature. The
name was borrowed from the Spirillam volutans, in which one of
Mryer’s pupils, Grimme*) found the granules. Now, if we consider
that this Spirillum volutans is only one out of many hundreds of
bacteria, moulds, and algae in which similar corpuscles are found,
the term is, surely, not well chosen. The objection to the term
“metachromatic corpuscles” is that the granules or drops under
consideration do not under all conditions stain metachromatically
with methylene-blue, as also Meyer has remarked, and also that
— as will be seen lower down -— whether or no a change into
a red-violet tint occurs, may, with most granules, depend not only
on the condition of the cultures, but also in a great measure on
the source of the methylene-blue preparation. In this paper L shall,

1) Zeitschr. f. Hygiene. Bd. XX, p. 412.
2) Botanische Zeitung 1904, p. 113.
3) Zentralblatt. f Bakteriol. Bd. XXXII 1902, p. 172.

(a

therefore, adopt the term ‘‘volutin’ as being generally received in
the microbiological literature.

For a detailed morphological description of the volutin-granule
I- refer to the relative literature’). When fixing moulds or yeast-
cells in formol, staining them by methylene-blue and differentiating
them in 1 pere. sulphuric acid, the cell is decolorized, not however
the spots containing the substance called volutin. This substance can
be observed as fine granules scattered all over the cells, again
coalesced to coarse granules, sometimes imbedded in a vacuole,
varying much as to dispersion and quantity. A. Meyer ®, who
realized the importance of subjecting to a close microchemical ex-
amination a substance of such frequent occurrence in the vegetable
kingdom, has discovered other qualities in addition to the typical
colour-reaction just mentioned. If e.g. we treat the methylene-blue
preparation thus obtained with a solution of iodine in potassium
iodide, the blue granules turn black and gradually lose the colour in
> perc. sodiumearbonate. The volutin rapidly dissolves in warm
water, also in the fresh preparation in 5 pere. sulphuric- or hydro-
chlorie acid. The digestibility by pepsin could not be made out,
since 0.2 pere. hydrochloric acid will throw the substance into
solution at bodily temperature. The behaviour of volutin towards
dyes induced Meyer to compare it with that of nucleinic acid pre-
pared from yeast. He found the one to be very similar to the other.
In 1904 Meyer, therefore, advanced the hypothesis that volutin is
a nucleinic-acid compound. This view has also been adopted in the
later literature on the basis of Mryrr’s investigation, although Meyer *)
himself became aware that his reactions, from a chemical point of
view, were not such as to prove his hypothesis, as witnesses his own
pronouncement upon it on page 125, where he positively acknow-
ledges that the question as to volutin being a nucleinic acid compound
has not yet been set at rest. Still, there was more ground, no doubt,
for Muyer’s supposition than for JANssENs and Lipiancs*), who, ten
years before, in their morphological investigation of the nucleus of
yeast-cells, consider basophilic granules in the cell-plasma, which
they suppose to play the part of reserve-material, as nucleo-albumins.

1) CurrrorD DoBeLL. Quart. journal of mier. science, 1908, p. 121.

SWELLENGREBEL. Arch. f. Hygiene Bd. LXX, 1909, p. 380.

Guituermonp. Recherches cytologiques sur les levures etc, These de Paris, 1902.
Arch. f. Protistenkunde, Bd XIX, 1910, p. 298.

Reicnenow. Arbeiten aus dem kaiserl. Gesundheitsamte. Bd. XXXIII, 1910, p. 1.

2) 1c. pit.
5) La Cellule, T, XIV, 1898, p. 203.

72

The reactions, performed by Meyer in 1904, were in the main sub-
stantiated in 1913 by SvmBaLY) and in the same year by LINDE’).
Linpr points out that there is not any method to fix the volutin so
as to prevent it from being ultimately dissolved in water at room-
temperature. Besides Meyer himself also other investigators are
wavering in their opinion about the chemical nature of this sub-
stance. GUILLIERMOND*) for instance says that “though Meyers hypo-
thesis seems highly plausible, “aucune preuve décisive”’ has as yet
been given” (p. 307). We quote from Kont’s*) manual of yeast-
cells (p. 40) the following passage: “Concerning the chemical nature
we are still much in the dark. The assertions brought forward have
not yet reached beyond the hypothetical stage.”

Now, would it not be possible to settle this question with the
aid of a subtler micro-chemical reagent than Meyer and his followers
had at their disposal? If volutin is a nucleinic-acid compound, it
would presumably be dissolved by the enzyme that splits up nu-
cleinic acid, i.e. nuclease. If this could be demonstrated, the sup-
position would be proved. In connection with other experience of
the action of nuclease on the nucleoproteins in the cell, I have also
examined the volutin of various hypho- and blastomycetes and, as
has been said elsewhere, 1 first made use of a nuclease prepared
from the spleen of the ox. These efforts came to nothing, as volutin
dissolves in water at bodily temperature inside the time required
for a nuclease digestion. In glycerin the activity of the nuclease
appeared to fall short. When following another way I came to the
conclusion that Meyers suggestion was perbaps right, as I found
that the volutin was dissolved within a very short time by the
nuclease-action of the living cells themselves, when bronght in con-
tact with coverslip-preparations fixed in alcohol. No such result was
achieved with cells that were killed by formol-vapours and conse-
quently had been deprived of their nuclease action. ’)

The volutin-granules, which along with fat and glycogen often
occupy such a considerable space in the cel-plasma of these unicel-
lular organisms, prompted me to look at them more closely both
morphologically „and physiologically. The following questions arise:
Is there no other way to establish the chemical nature than the

1) Zeitschr. f. allgem. Physiol. Bd XV, 1913, p. 456.

2) Centralbl. f. Bakteriol. Abt. Il. Bd. XX XIX, 1913—14, p. 369.

3) Arch. f. Protistenkunde, Bd. XIX, 1910.

4) Die Hefepilze 1907. Quette und Meyer. Leipzig.

5) For a description of these experiments with Ustilago maydis and a species
of Torula, see Anat. Anzeiger. Bd. XLVII, 1914, p. 312.

73

one just referred to? What is the function of these granules in the
cell-body, does their presence depend on a special composition of
the culture-medium? Do the various enzymic actions result from
their presence? And as to their morphology: is there any connection
between these granules and the development of reserve-materials
such as fat and glycogen? and is there any reason for identifying
them with mitochondria, as some researchers have done?

I have not carried out an extended cytological investigation.
Still, I found that with Torula monosa'), the yeast-cell with which
most of the experiments described lower down have been performed,
the result of mitochondria-staining after Brnpa is positive for the
volutin-granules — another proof that all granules classed as “mito-
chondria” upon the basis of these colour-reactions alone, do not
necessarily agree in composition; for these nucleinic-acid-containing
granules e.g. stain in the same way as the mitochondria in the
liver-cells, which very probably are compounds of phosphatids.
When stained with methylene-blue and subsequently differentiated
with 1 perc. sulphuric acid the volutin-granules do not at all appear
to be metachromatic, as was also found by Meyer. The greater
cells often show a diffusely coloured vacuole, which is stained
violet-red, i.e. metachromatically; the granules often stain deep-
blue as well in formol as in aleohol-fixation.

Nor is the methylene-blue-preparation used Abee to the
degree of metachromasia. With methylene-blue ‘“f-patent” for
example a higher metachromasia was found than with GRÜBLER'’S
methylene-blue for bacteria, the latter being probably purer. With
toluidin-blue on the contrary nearly all granules are stained meta-
chromatically. It is certain, therefore, that metachromasia cannot be
taken as a criterion. This, as I said on page 70, also induced me to
prefer the term ‘‘volutin-granules”. Besides with these anilin-dyes
the granules can also be demonstrated with carbolic fuchsin; again
also by Unna’s staining-method (a mixture of two basic dyes: pyronin
and methyl-green, of which only the former is taken up by the
volutin-granules.)

With Hemennain’s ferric-hematoxylin the volutin-granules are not
or hardly stained when the nucleus of the yeast-cell becomes clearly
visible. In some cases we succeeded in identifying in the living cell,
with dilute neutral-red, occasionally also with methylene-blue, granules
disposed similarly to the volutin. This, however does not occur with

1) For further information regarding Torula monosa we refer to KLuyver
Biochemical Sugar-determinations. (Thesis. Delft 1914, p. 16.)

74

all cells’) nor always in such a quantity as is found in the fixed
preparation after staining with methylene-blue and differentiation in
sulphuric acid. Of course it is not outside the probabilities that,
coincidently with the fixation, compounds are isolated from the living
matter, that contribute to the mass of volutin-granules. Allowance
should always be made for such possibilities in the examination of
fixed substances. It will no longer do, of course, to apply the term
vital staining to the staining of living cells with metbylene-blue +
5 perc. formol, as HENNEBERG does.’)

Fat may be stained in the yeast-cell at the same time with volutin.
We need only apply Sudan III to the preparation treated with
metbylene-blue and sulphuric acid. As other researchers, among whom
HENNEBERG, have also pointed out, the volutin-granules are, as regards
their location, independent of the fat-drops; it can only be altered
passively by the pressure of glycogen- or fat-drops, by which they
may be pushed back to the periphery of the cell. In other respects
their independence also appears from the presence of fat and glycogen
in cultures where volutin is absent.

I here wish to emphasize that moulds as well as various other
kinds of yeast may be cultivated and multiplied without any
volutin being produced in the cell. Earlier researchers were also
acquainted with the fact that in the case of insufficient nutrition
the amount of volutin in the cell decreases, that some of the cells
in the culture may even become completely devoid of volutin.
In 1910 ReicueNow *) has also demonstrated in Haematococcus
pluvialis, that by not adding phosphate to the culture-fluid the
volutin may be made to disappear ultimately, which, however,
results in the death of the haematococci.

In my own culture-experiments with Ustilaginae and Torulaceae
I have likewise watched the effect of the amount of phosphate in
the culture-medium and endeavoured to obtain permanent cultures
completely free from volutin. To accomplish this effect it will not
suffice to omit the phosphate in the agar-cultures because, though
in doing so the majority of the cells will indeed be free from
volutin, still many volutin-bearers will be seen through the microscope
in an alcohol-coverslip- preparation.

1!) It is remarkable that in an agar-malt medium stained rather deep-red with
neutral red, only a very small percentage of the growing cells had, after several
weeks, taken up the dye.

2) Centralblatt f. Bact. Bd. XLV, 1916, s. 50; Wochenschrift f. Brauerei 1915,
NO, 36—42.

SEP page sie

75

I could not possibly detect volutin-granules among the glycogen-
drops in the living cell-plasma. It is necessary, therefore, in watching
the culture-experiments, to fix coverslip-preparations. With Torula
and Ustilago absolute alcohol answered the purpose very well; not
quite so well with Saccharomyces in which, as HeNNeBerG ') has also
pointed out, the cell-body after fixation in alcohol and colouring in
methylene-blue is not sufficiently decolourized by 1°/, sulphuric acid.

The culture medium was prepared by adding to the agar chemic-
ally pure substances; the agar was extracted for one hour with
0.5 °/, acetic acid and subsequently washed out with distilled water.
No tap-water was used’). As nutritive substances we first used
5 pere. glycose (pro analysi), 0.5 perc. peptone Mrrck, 0.05 perc.
Mg SO, (pro analysi) and a trace of KNO, (pro analysi). When
examining this medium after Neumann*) to ascertain whether any
phosphorus be present, this will be found to be the case and is
chiefly due to the peptone. Neither was the albumose, obtained in
the laboratory from the white of a hen’s egg, found to be entirely
free from phosphorus. When using pure glycocoll or asparagin as
N-source, phosphorus is not found anymore macrochemically after
NEUMANN, yet on a microchemical examination the analysis always
brings to the front traces of phosphorus ‘).

On the medium carefully prepared with glycocoll or asparagin
not a single yeast-cell containing volutin is to be found after some
days at the second inoculation from a malt-agar-culture of Torula
monosa. The growth, however, is slighter than in the phosphate-free
cultures with peptone and since in the latter only a few
volutin-containing cells were found in a field of thousands of volutin-
free cells, they could safely be used for most of our experiments.
Also in these almost phosphate-free peptone-cultures the growth, as

1) lie. page 74.

2) It should be noted that Retcuenow found the methylene-blue-sulphuric acid
reaction absent in his experiments with Haematococci, even when simply omitting
the phosphate. In the description of the culture-fluids no mention was even made
of the use of distilled water. Probably the hypho- and blastomycetes studied by
me can seize upon smaller quantities of phosphorous-compounds present in the
culture-fluid than the Haematococci studied by RetcueNnow.

3) After this method the substance to be examined is incinerated with 10 c. c.
of distilled nitric acid s.g 1,4 and strong sulphuric acid aa. For further details
we refer to Horre Seyrer, Hdb d. Physiol. u. Pathol. Chemischen Analyse p. 359.
After incineration and subsequent neutralization phosphoric acid was reacted upon
with ammonium-molybdate.

4) Here part of the nutrient medium was incinerated, the ashes were taken up
in a drop of nitric acid and were put on an object-glass. Reaction was then .per-
formed under the microscope

76

might be expected, is less rapid and less strong than on similar
media with phosphate. I have not succeeded in making a nutrient
medium that was microchemically completely free from P. The
chances, indeed, are that on such a medium there is no increase
of cells at all.

Volutin-free cultures were thus made from Ustilago maydes, from
Torula monosa, from Saccharomyces cerevisiae and from a Lactose-
yeast *) (also a Torula).

When transplanting the cells upon a phosphate-containing medium
they were seen to be loaded again with volutin within afew hours,
in a fluid medium even after a few minutes. This also applies to
cultures that had been free from volutin for 8 or 9 months. That
they have preserved the power to produce volutin is evidenced
whenever an opportunity offers. The volutin-free yeast-cells react
very rapidly on the addition of a small quantity of phosphate; for
instance the addition of 0,04 m.grm KH,PO, to 10 ee. of fluid will
evolve volutin in several cells. Also from purely organic compounds
the phosphoric compound required for volutin-formation may be
obtained in many cases. When using an albumose prepared from
white of egg that contained 0.09 m.grm P. per test-tube, a large number
of the cells were loaded with volutin. Volutin is also developed
rapidly by the addition of nucleinic-acid-sodium to the culture-
medium. As for the amount of fat and glycogen no difference worth
mentioning could be made out, on microscopic examination, between
phosphate-free and phosphate-containing cultures

In a Torula monosa culture upon a phosphate-free medium we
observed in March 1916 a sudden formation of pigment in the
yeast-cells. This could be made out macroscopically as a dark-grey
to black coloration, microscopically from the presence of fine, brown
pigment-granules in the living-cells, which did not disappear upon a
short treatment with alcohol and that could be fixed in Canada-
balsam-preparations. The pigment-production was arrested as soon
as the pigmented cultures were transplanted on a phosphate-containing
medium. It appears, therefore, that it is brought forth only in a
phosphate-free medium. For five months this pigmented culture was
maintained, then outside influences destroyed it. I never again detected
this pigmentation in the same stock of Torula moncsa. „

As appears from the above, the presence of phosphorus-compounds
is essential to the formation of volutin. They alone, and not as
HENNEBERG asserts calcium-salts or ammonium-carbonate or sugar, can

1) Vide Kuuyver, |. c. page 73.

ith

regenerate this substance in volutin-free cells. It is not the calcium-
salts, but the amount of phosphate in the tap-water that constitutes
the causative agent. When HENNEBERG *) reports that in a culture of
beer-yeast, that has lost much of its volutin, a fresh volutin-formation
takes place a short time after an incubation in sugar, notably after
the addition of ammonium-carbonate, this is to be attributed, in my
Opinion, only to some degree of impurity of the preparation used.
It is a fact that commercial glycose or cane-sugar, as 1 myself found
experimentally, will produce an increase or a recovery of the volutin.
In perfectly pure glycose (pro analysi), on the other hand, whether
or not after the addition of ammonium-carbonate (pro analysi), the
volutin is not recovered.

There is, therefore, no longer any room for doubt that volutin
needs phosphorus for its formation. The question now arises: are
there other methods by which its composition can be ascertained ?
When Torula monosa is treated for some time, say one hour, at

nN.
room-temperature with T NaOH, all the volutin is dissolved. By the

same method nucleinic-acid may be prepared from yeast in the
usual way. Does perhaps the volutin furnish the material for it?
This hypothesis has been previously brought forward by Meyer
also, as it seemed improbable to him that the comparatively slight
portion of chromatin in the cell-nucleus — in connection with
his reactions on volutin described above — could furnish all the
nucleinic acid produced from yeast. Is there any foundation for this
hypothesis, to which also other workers (among others ReicHe-
now 7) and Konr ®)) inclined? In order to ascertain this the following
experiments were made.

Twenty-five malt-agar tubes were inoculated with Torula monosa ;

n
after 28 hours the strongly developed cultures were washed in 7

NaOH. The suspension, thus obtained, was well stirred; one hour
later (when all cells were. free from volutin) it was filtered by
suction through paper-pulp, after which 1 ec. 5 perc. H,SO, per
10 cc. filtrate was added. A precipitate was formed that was cen-
trifugalized and washed out again in distilled water. : When it was
analysed this precipitate appeared to contain purin-bodies. It gave
after hydrolysis in the autoclave with dilute sulphuric acid a distinct
xanthin- and adenin-reaction and on further analysis a phosphate-

1) L ce. page 74.
2) 1. c. page 71.
5) |. e. page 72.

78

reaction. According to what has been found in the preparation of
nucleinic acid from Saccharomyces cerevisiae we might expect the
above mentioned precipitate originated by the addition of 5 perc.
H,SO, to contain nucleinic acid (similar reactions I achieved after-
wards also with commercial beer-yeast).

That the nuclease of yeast-cells is of itself capable of splitting up
this nucleinie compound and to set free phosphoric acid may now
clearly be demonstrated. A suspension of the washed precipitate in
1 pere. glyeose and 0,05 perc. MgSO, inoculated with Torula monosa
does not give a phosphate-reaction; but this reaction is obtained
when the yeast-cells are ground with quartz-sand and are then
brought in contact with a suspension of the precipitate in water.
After an incubation at 20° for 48 hours (with addition of a grain
of thymol) a distinct phosphate-reaction with Mg-mixture takes place,
which is absent in the precipitate itself (also after 48 hours)
and is likewise absent in the ground Torula-cells. The impractibility
of demonstrating the nuclease-action in the living cells by the
phosphate-reaction is probably due to the fact that these cells
immediately seize upon the phosphate they have isolated from
the nucleinie acid. By crushing the cells, however, we were
in a position to prove the nuclease-action of Torula monosa.
It appears that this enzymic activity was not lost in the
cultures grown without volutin, for with these also phosphate
could be isolated from the nucleinic acid precipitate after grinding
the cells with quartz-sand. | had ascertained beforehand that among
thousands of cells at the most a single volutin-containing cell was
present in coverslip-preparations of this culture.

Volutin is separated from the yeast-cells by — NaOH while by

the same process a nucleinie acid compound is thrown into solution.

Though all this seems to point to the identity of these two sub-
stances, the decisive proof is furnished only when observing that
phosphate-free cultures, not containing any volutin and extracted with

- NaOH, do not yield a precipitate with 5 perc. H,SO,.
o

This was evidenced by me in the following way. Twenty-five
culture-tubes with volutin-free Torula monosa (at most one granule

of volutin per field of thousands of cells) were treated with =
NaOH, and the filtrate was passed through a filter by suction. With

1 ce. 5 pere. H,SO, per 10 cc. of filtrate at most a trace of turbidity,
never, not even after centrifugalizing, a precipitate was produced.

ah

79

This does not imply that volutin-free yeast does not contain
nucleinic acid, as in working with very large quantities of yeast,
the nucleinie acid will also be isolated from the nuclei, and will be
demonstrable in the filtrate. But it does imply that Torula monosa,
devoid of volutin, produces much less nueleinie acid than the normal
Torula. It is, therefore, quite justifiable to say that the volutin is
answerable for by far the greater part of the nucleinic acid, gene-
rally prepared from yeast.

The fact that volutin-free cultures may be kept alive for months,
moreover that, when transplanted on phosphate-free media, a fresh
growth is developed, and lastly that on a phosphate-containing medium
volutin is formed again directly, shows conclusively that the presence of
this nucleinie acid compound is not essential to the vitality of these
cultures. This does not mean, however that the conditions of life
are not more favourable for the yeast-cells that contain volutin. On
the contrary, with them the growth is more rapid and more abundant,
the size of the cells is often larger. Nor is it to be expected, that
a substance, present in each yeast-cell to such a considerable amount,
has no function to perform in it.

In how far the absence of phosphate in the nutrient medium —
aside from the volutin-production — is answerable for the decrease
in the growth, could never be made out in these experiments.
Theoretically speaking the protoplasm of the yeast-cell may, also
apart from the volutin in consequence of the absence of this salt,
lack a stimulant agent that cannot be replaced by other salts. It
should not be forgotten, of course, that also in these phosphate-
free nutrient media traces of phosphorus were always demonstrable
microchemically, i.e. organic phosphorus-compounds that may be
borne off and assimilated by the cell to render the multiplication of
the nuclei possible. But for this the capacity of the yeast cells to
grow and to multiply would be out of the question, as has been
remarked above. The rare reports in the literature concerning a
growth on a medium entirely free from phosphorus, cannot bear
criticism as far as a careful analysis is concerned.

It stands to reason that, alongside of the efforts to detect the
chemical nature of volutin, observers have always tried to establish
its significance in the life of the yeast-cell. The tirst observers who
studied these basophilic granules in moulds and bacteria naturally
inclined to believe them to be a reserve-material. When upon the
basis of Meyer's investigation it was surmised that volutin was
built up of nucleinic-acid compounds, it was natural that one should
look for some relation with the nucleus. But when nothing was

80

found microscopically that indicated a production of the volutin
granules from the nucleus, it was concluded that volutin-granules
generated in the cell-body and formed a reserve-material from which
the nucleus could draw at will. In this way RricHeNow ') e.g.
represents in his culture-experiments with Haematococcus pluvialis
the colliquation of the larger clumps in the centre (initially on the
side of the nucleus) and a breaking-up into smaller granules at the
periphery during the growth of the cultures.

Of late HeNNEBERG*®) went farther than the researchers who
admitted volutin to have significance only as a reserve-material.
According to him volutin plays an important part in the fermenting
process of Saccharomyces cerevisiae. He even inclines to believe
that the fermenting enzyme itself is to be found in the volutin
and tries to support his view by a series of microscopic experiments
tending to observe the abundance of volutin in the yeast-cell and
its distribution in the cell-body, together with the fermenting power.
He found that the yeast-cell during rest bears some spherical volutin-
granules and that a dissemination into finer granules, rendered
visible at the initial stage of the fermentation, indicates activity.
The lower fermenting power (the carbonic acid production was
taken as an index of the fermenting power) runs parallel with a
diminution of the volutin. In a sugar-solution, as HENNEBERG asserts,
the volutin is recovered. Before discussing HENNEBERG’s results will
return to my Own investigation. ae

It being possible to make volutin-free cultures from moulds
and yeast-cells that could be kept alive for a long space of time I
put the simple question: Is the presence of volutin indispensable
to fermentation? To answer it it was only necessary to use for
the yeast-tests the volutin-free cultures under microscopic control.

First of all we had to find out whether fermentation can take
place without volutin. When this appeared to be actually the case,
the next thing of importance was to obtain some quantitative data
regarding the relationship between the fermentation of a volutin-
containing and of a volutin-free culture of tbe same age, both
cultivated under the same circumstances (except the absence of
phosphate in the culture medium).

For these experiments the rate of fermentation was estimated
partly physically, partly chemically. In the first case the carbonic
acid produced was collected above mercury; in the second, the
reducing power of the remaining non-fermented sugar was determined

1) Le. page 71.
2) Le. page 74.

81

after Benepicr’). It goes without saying that sterilisation was
applied in these experiments. The fermentation-flasks connected with
the mercury tube were kept at a temperature of 23°—25° and the
fermentation took place at the same temperature in the flasks used
for the determination after BENEDICT. -

These series of experiments go to show by either method that a
distinct fermentation takes place in the volutin-free cultures of Torula
monosa, Saccharomyces cerevisiae and Lactose-yeast. The fluid used
was free from phosphate and contained 1 to 24 pere. glycose, 0.05
perc. MgSO,, a trace of KNO,, 0,2 perc.’ peptone or, to make sure
of the absence of any volutin in thousands of yeast-cells, 0,2 perc.
asparagin or glycocoll instead of peptone. Peptone, however, serves
our purpose quite well; especially in comparative fermentation-tests
it was quite immaterial whether over against the richly filled yeast-
cells of the volutin-containing culture an occasional cell containing
a few volutin-granules, emerges in the volutin-free culture. As said
before, peptone yields the advantage of a larger growth. Into the
culture-fluid thus prepared, yeast-cells were inoculated from
a culture-tube. At the commencement as well as at the end
of each experiment the absence of volutin was verified. In the
experiments with asparagin or glycocoll the same substance was
also made use of for the culture of the volutin containing yeast
with the addition of 0,1 pere. KH,PO,. Occasionally also the entire
volutin-free culture was transferred from a tube to a flask of a phos-
phate-free fluid, and the fermentation was compared with that which
was brought about also in a phosphate-free fluid by a nearly equal
quantity of Torula monosa loaded with volutin-granules. In this way
a greater uniformity was obtained in the medium in which the
fermentation took place, than with a phosphate-containing control-
fluid; still, a perfect uniformity was never obtained, since it may
be possible that these volutin-containing cells, rich in nucleinic acid,
as has been said on page 9, of themselves furnish phosphate to
the medium on which they have been planted. I subjoin some of
my experiments and their results:

1 October 1916. Two flasks A and B each with 25 c.c. 2 perc.
glycose, 0,05 perc. MgSO,, 0,1 pere. glycocoll and a trace of KNO,
(consequently phosphate-free). The preparations used are all chemic-
ally pure. To A is added a culture of Torula monosa cultivated on
a phosphate-free medium, (only a few cells with volutin, at most
one in a thousand); to B is added a culture full of volutin-granules,

1) Vide Nagasaki, Zeitschr. fiir physiol. Chem. Bd. 95, p. 61.

Proceedings Royal Acad. Amsterdam, Vol. XX.

82

the culture being grown on a phosphate-containing medium. The
amounts are such as to render the turbidity in the two flasks apparently
equal. Temperature 20°—25°. The carbonic acid output is read
above the level of the mercury. After the lapse of five hours not
yet an appreciable fall of the ‘mercury-column ; after 48 hours with
A a fall of 4 ce, with B 44 ec. (to the level of the mercury-
reservoir) '). At -the termination of the experiment in 4 exclusively
yeast-cells without volutin; in 5 a large part with volutin.

26 January 1917. In flask A 25 ce. 0,5 pere. glycose, 0,05 pere.
MgSO,, 0,5 pere. asparagin and a trace of KNO,, in flask B the
same liquid + 0,1 pere. KH,PO,. In A inoculation with Torula
monosa without volutin from a phosphate-free asparagin medium.
In B Torula monosa that has been incubated from 29 June 1916
to 23 Jan. 1917 with a phosphate-free medium and by transplanting
into malt-agar has again grown rich in volutin. Carbonic-acid pro-
duction read above the level of the mercury. Temperature 20°— 25°.
Fall of the mercury-column with A in 6 hours 2} cc. with 63 c.c..

3 April 1917. A flask of 25 ec. 2,5 perc. glycose, 0,2 perc.
peptone, 0,05 perc. MgSO, and a trace of KNO,. Inoculation with
volutin-free culture of Torula monosa, which had been transplanted
on March 30 on a phosphate-free medium and which originates from
a phosphate-free culture dating from June 19, 1916. After two hours
fall of the mereury-column = 3} c.c. (which means fermentation of
a volutin-free culture of rather more than nine months).

13 March 1917. In flask A and B each 25 c.c. 2,5 perc. glycose,
0,2 perc. peptone, 0,05 perc. MgSO, and a trace of KNO,. A. inocu-
lated with a second inoculation from a volutin-free culture, 2. from
a volutin-rich culture. Apparently eqúal turbidity in the two flasks
at the commencement of the experiment. After one hour both mer-
cury-columns are fallen 1 cc, after 18 hours A = 3¢.c. B= 44 cc.
(down to the mercury level in the reservoir.)

March 1917. Beer-yeast from the brewery, kept under 10 perc.
cane-sugar, in which initially a marked fermentation has taken place
(microscopically coarse volutin-granules; no appreciable contamination

1) The determinations were read from non-graduated tubes, on which the level
of the mercury was marked with anilin pencil. The fall was expressed in centi-
metres; 4 cm. corresponded with a fall of 4 c.c. of mercury. Minute quantitative
determinations cannot be made in this way, which indeed was not aimed at. In
the case of a fall down to the level of the mercury-reservoir, the amount of car-
bonie acid given off may, of course, have been larger than we were able to read.
Therefore, for a more accurate quantitative comparison some determinations were
made after Benepicr (vide p. 83).

83

of the yeast-cells through bacteria or moulds). After three days’
incubation in canesugar, inoculated in flask B with 0,5 perce. glycose,
0,5 perc. peptone, 0,05 perc. MgSO, and a trace of KNO, In flask A,
filled with the same fluid, also beer-yeast from the same source,
which, however, has already twice been transplanted on a phosphate-
free medium, and does not contain any more volutin. After 18 hours
both mercury columns fallen 44 c.c. (to the mercury level in the
reservoir). *) In a subsequent similar experiment both fallen 2 c.c.
after one hour’s fermentation.

Reduction tests after BeNepicr.

July 1915. A 20 cc. culture fluid with glycose, 0,05 perc. MgSO,
and a trace of KCl. B similar fluid + 0,1 pere. KH,PO,.

Before the fermentation determination of reduction = 0,5 glycose.

A inoculated with volutin-free, B with volutin-containing culture
of Torula monosa. After 48 hours’ fermentation

A=0,15 perc. glycose.

B= 0,15 perc. glycose.

March 1917. A 20 e.e. of culture-fluid with glycose, 0,5 perc.
asparagin and a trace of KNO,. B similar fluid + 0,1 pere. KH,PO,.
Before fermentation determination of reduction — 2,25 perc. glycose.

A inoculated with volutin-free culture, B with volutin-containing
culture of Torula monosa. After 48 hours in A still 2.06 perc. glycose.

in B also 2,06 perc. glycose.

March 1917. A 15 c.c. of culture-fluid with glycose, 0,2 perc.
asparagin, 0,05 pere. MgSO, and a trace of KNO,. & similar fluid.
+ 0,1 perc. KH,PO,. A inoculated with volutin-free, B with volutin--
containing culture of Lactose-yeast. Before the fermentation determ-
ination of reduction = 2,04 perc. glycose. After 48 hours in
A still 1.53 perc. glycose.

5 1.44 perc. glycose. *)

It cannot be doubted therefore, that with Torula monosa, Saccha-
romyces cerevisae and Lactose-yeast fermentation takes places in-
dependently of volutin, the same holding perhaps for other
zymase-containing hypho- and blastomycetes. It is not surprising that
in some experiments the fermentation in the phosphate-containing.
control-cultures is somewhat stronger in the same space of time. As
I stated before, the growth of yeast is more intense on a phosphate-
containing medium, so that the number of the cells, participating
in the fermentation-process must be more considerable. We also

1) See preceding note p. 82.

2) | am endebted to Mr. J. W. Besr for the last two determinations.

84

know that the presence of phosphate aids the fermentation. The
experiments cannot be arranged so as to enable the investigator to
study this influence irrespective of the volatin-production. HARDEN
and Youne*) have even rendered it very probable that phosphates
are indispensable to fermentation. According to them the course of
the fermentation is represented by their well-known equation :

U,H,,0,+2 PO,Na,H = C,H,,0,(PO,Na,),+2 CO,+2C,H,OH+ 2 H,0.

C,H,,0,(PO,Na,), + 2 HO = C,H,,0, + 2P0,Na,H.

In connection with this my experiments go to show that (if HARDEN
and Youre are right) traces of phosphorus-compounds in a culture-
fluid, too small to cause the formation of volutin, bring about
these chemical changes. That they are organic and not anorganic
phosphorus-compounds is of no consequence; we need only think
of the nuclease-action, which, as described above, also belongs to
the volutin-free yeast-cells.

Zymase-action therefore does not depend on the presence of
volutin, no more does the nuclease-action; I have been able to prove
the same for katalase-action. Yeast-cells, entirely free from volutin,
suspended in 3 pere. H,O}, evolve immediately an abundance of
oxygen. No gas-formation takes place in H,O, when the yeast-cells
have been previously destroyed by boiling.

HENNEBERG’S totally different results with the zymase-action of
Saccharomyces cerevisiae may be attributed to the fact that such
problems can be solved only when, as was the case in my ex-
periments, the volutin-production can be absolutely precluded, for, as
HENNEBERG himself asserts on page 15, the volutin-contents of the
cells could in his experiments be replenished from the medium in
which they live.

When HeENNEBERG observes regeneration of the volutin in a solution
of sugar in water without the addition of salts, it is not the sugar that
constitutes the active substance, for in glycose pro analysi no fresh
production of volutin takes place, as I found over and over again.
When some months ago commercial glycose was sold in a some-
what impure condition, numerous volutin-containing cells appeared
in all my cultures without addition of pbosphate.

As for HENNEBERG's microscopic findings during the fermentation,
viz. a finer distribution of the volutin-granules in the initial phase
of the fermenting process, the question may be asked whether these
changes may perhaps be connected with other life-symptoms in the

1) Proceedings of the Royal Society Vol. LXXXII p. 361. See also Eurer and
Hammarsten Bioch. Zeitschr. Bd. LXXVI, 1916, p. 314.

85

protoplasm during the transplantation into a fresh culture-fluid (pre-
paration to the division of the cell etc.)

I have myself repeatedly made microscopic preparations in different
phases of the fermentation. In a beer-yeast, kept for two weeks
under 10 perc. cane-sugar, I found along with volutin-free cells
many with one or more coarse volutin-granules, also a few with a
great number of scattered fine granules. When they were incubated
in a phosphate-free fluid, fermentation set in at once. After 20
minutes (1 c.c. fall of the mercury-column) the same picture was
seen under the microscope as before the fermentation. After 1 hour
(fall of the mercury-column 2 c.c.) some cells had begun to germinate.
They were obviously richer in coarse volutin-granules than the other
cells. The bridge between the knob and the mother-cell was also
stained deep blue. Just as in this case the cells in a volutin-containing
culture generally exhibit a picture differing so much the one from
the other that, in my judgment, it is impracticable to point out a
distinct change in connection with the fermentation. True, with the
Torula monosa in a fresh phosphate-containing cultnre-fluid faint-
violet coloured vacuoles, which have generated after a long incuba-
tion in the same malt-agar medium, are often seen to disappear,
but there is nothing whatever that points to a correlation with
fermentation. On the contrary. what goes very much against such
a correlation with the fermentation as such, in the case of Torula
monosa, is the discovery that when these yeast-cells are incubated
with a fluid containing 2.5 perc. lactose instead of 2.5 perc. glycose
(along with peptone, MgSO, and KNO.) exactly the same picture is
presented by the spreading volutin-granules in the coverslip-prepara-
tions, made during some hours. Now it is well-known that Torula
monosa, as the very name indicates, attacks exclusively monose not
lactose. In a similar way I examined at the same time the beer-
yeast-culture mentioned above, which had been incubated under
10 perc. cane-sugar, as well with 2.5 perc. lactose as with 5 perc.
glycose (both with an equal amount of peptone, MgSO, and KNQ,).
Fermentation-experiments taught that, as might be expected, only in
the latter case an evolution of carbonic acid oecurred, since lactose
is not fermented. A convincing difference between the yeast-cells in
these two fluids was not noticeable. True, after 1, 2, and 3 hours
a more diffuse dissemination of the volutin-granules was to be seen
in part of the cells, but only in a very small percentage of the
cells, which was perhaps a little higher in the glycose-containing
fluid. The difference was, however, not such as to point to a close
relation to the fermentation,

86

To ascertain whether volutin is indispensable to the fermenting
process we had to follow up another plan. It then appeared that
the zymase as well as the nuclease and the katalase do not depend
on the presence of this substance. The question might now be asked
whether — thongh the material that exhibits the colour-reactions,
demonstrated by Meyer, is not the mother-substance of the
enzymes — it may be that the principal constituents of the volutin
still remain in the cell after the said reactions have disappeared.
To this question we can say that — as seen on page 80 — in ex-
tracting volutin-containing Torula monosa with dilute alkali we get
a nucleinie acid compound, not to be obtained by the same method
from an approximately equal amount of volutin-free Torula. It
follows then that it is-not only a colour-reaction, which gets lost
through unknown chemical changes in the cell, but undoubtedly a
nueleinie acid compound which the Torula loses in a phosphate-free
medium. We are, therefore, justified in assuming that this nucleinic-
acid compound is not indispensable for the enzymic action. Nor is
it indispensable to sustain the stock, for the Torula monosa maintains
the properties of the stock after a nine months’ incubation in a
phosphate-free medium; also the multiplication continues to proceed
regularly, though at a somewhat slower rate. Such a culture trans-
planted after 9 months upon a phosphate-free medium is still capable
of fermenting (experiment page 85); in a phosphate-containing
medium volutin is formed again directly in the usual way. This
substance, which appeared indeed to be a_nucleinic-acid com-
pound, we can hardly consider to be anything else but a reserve-
material, and most likely one of a peculiar nature. If namely we
bear in mind that most probably phosphates must always be present
for the cells to draw from in the fermenting process, it will be an
advantage to the cell when it can draw this phosphate immediately
from that reservoir of nucleinic-acid in virtue of its nuclease; the
quantity of this phosphate may be ever so small, as after being set
free, it can be utilized again. It is obvious, therefore, that the
presence of volutin in hypho- and blastomycetes may be of great
significance for the fermenting process without however being indis-
pensable to the fermentation, as it has been proved that even the
merest traces of phosphorus, demonstrable only under the microscope,
in a volutin-free culture, suffice to render fermentation possible.

SUMMARY. =

The production of volutin-granules in moulds and in yeast-cells
depends on the presence of anorganic or organic phosphorus-com-
pounds in the culture-medium. In a phosphate-free medium Ustilago

87

maydes, Torula monosa, Saccharomyces cerevisiae and Lactose-yeast
can be cultivated, without volutin being generated in the cells. On
transpiantation into a phosphate-containing medium a fresh formation
of volutin ensues.

With dilute alkali a nucleinic-acid compound is extracted along
with volutin from Torula monosa and Saccharomyces cerevisiae,
which cannot be obtained from an approximately equal quantity of
volutin-free culture. The hypothesis, never proved before, though
indirectly supported, that volutin consists of a.nucleinic-acid compound,
has thus been placed beyond dispute. Nucleinic-acid prepared from
yeast in the usual way originates no doubt chiefly from volutin.

The nucleinic-acid compound dissolved from the volutin-containing
cells, is decomposed by a nuclease formed in the Torula-cells
themselves, in which process the development of phosphoric acid
could be demonstrated. Also the volutin-free cultures of Torula
monosa still contain a nuclease. Other enzymic actions also continue,
e. g. the katalase- and the zymase-action. Contrary to HENNEBERG’s
recent pronouncement that the zymase-action of the yeast depends
on the presence of volutin, that the latter is probably to be con-
sidered. as the enzyme itself, it may be proved that volutin-free
cultures still evolve a very distinct fermentation. Considering the
slower growth in a phosphate-free medium, this fermentation is not
appreciably inferior to the fermenting power of the volutin-cultures
of Torula and Saccharomyces. Even after a sojourn of 9 months

in a phosphate-free environment a volutin-free culture of Torula

monosa is still liable to fermentation.

Volutin is a nucleinic-acid compound which is presumably nothing
but a reserve-material. The presence of this reserve-material
though it is not indispensable to the life and the multiplication of
cells, is no doubt of great moment for their individual develop-
ment. There is reason to believe that, though not being required
for the fermentation, it facilitates the fermenting process by contin-
ually supplying small amounts of phosphate, which can be liberated
from the nucleinic-acid by the nuclease. The relationship of the
volutin to the multiplication of the cells is a subject that must be
left for further investigation.

Upon a phosphate-free medium with Torula monosa a pigmented
variation was once developed with fine brown pigment-granules.
This pigment disappeared after transplantation upon a phosphate-
containing medium but recurred repeatedly in a phosphate-free medium.
Also these pigment-containing cultures had retained the glycose-
splitting enzyme.

Chemistry. — “Influence of different compounds on the destruction
of monosaccharids by sodiumhydroxide and on the inversion
of sucrose by hydrochloric acid. Constitution-formula of
a-amino-acids and of betain”’. By Dr. H. 1. WATERMAN. (Com-
municated by Prof. J. BOgsEKEn).

(Communicated in the meeting of April 27, 1917).

The destruction of monosaccharids, such as glucose, galactose etc,
by basic substances, is accompanied by a decrease of polarisation
of the solution in question whilst at the same time the colour
becomes brown.

The action of hydroxylions is measured by the rapidity of dimi-
nution of polarisation and by the colour-intensity of the solution.
I have noticed that different substances belonging to a series of
compounds which generally have no or only a slightly acidic cha-
racter, for instance amino acetic acid and «-aminopropioniec acid, are
able to neutralize the destructive action on glucose of considerable
concentrations of alkali *).

In order to increase the general importance of my observations
I have repeated the experiments with solutions of galactose instead

of glucose. The results obtained are quite the same as with glucose °

(See table U. The action of hydroxylions on galactose is retarded
too by different “neutral” substances. Whereas the addition of
5 em°. 1,06 normal NaOH-solution in the experiments mentioned
in table I after three hours has lowered polarisation from + 12,4
to + 9,3 and + 9,4, in the presence of 500 milligrams of alanin
the same concentration of alkali has lowered polarisation only to
+ 11,1. After 24 hours and especially after 48 hours the pheno-
menon could be observed much better still. So, after 48 hours,
without the addition of alanin the polarisation had diminished from
12,4 to 4,0 respectively 4,3; in the presence of alanine the polari-
sation had only diminished to 9,7.

The difference in colour-intensity of the solutions examined was
in accordance with these facts. After + 48 hours, in presence of
sodium hydroxide, but without alanine, the colour was brown yellow,
with alanine only pale-yellow.

1) H. 1. Waterman, Chemisch Weekblad 10, 739 (1913); 14, 119 (1917),

Lil

89

From these and other experiments described in previous commu-
nications it follows that the quantity of alkali, which is fixed by
glycine and alanine is very important.

TABLE I. Retarding action of alanine on the destruction of galactose
by alkali,

c | D
40 cm3 of a solution containing + 3,5°/,
galactose !)

aan

0 | 500 milligram

Quantity of alanine added

Number of cm? 1,06.
normal NaOH-solution added

Filled up with H,O to 50 cm

A,B,C and D were at the same time placed in
a thermostat with watermantle (temperature : 33°)

At the beginning
} (temp. of the polarisa-| -++ 12,3 + 11,1 + 11,1 + 12,3
Polarisation | tion-solution: 18,5°) .
d After + 3 hours (temp.
in grades | of the polarisation- | + 12,4 + 9,3 A Bee eile
v solution: 20°)
NTA ARAE + 24 h. (temp.
(length of the | Of the polarisation- | + 12,5 + 5,6 + 5,8 + 9,9
polarisation- solution: 18,5°—19°) itd
tube 2 dm.) |After + 48 h. (temp.
of the polarisation- | -+ 12,4 + 4,0 + 4,3 + 9,7
solution: 18°)
At the beginning | colourless} colourless colourless} colourless
Colour pale pale
After + 3 hours | colourless yellow yellow colourless
of the
Solon After + 24 hours | colourless} yellow yellow colourless
brown- | brown- | scarcely pale
After + 48 hours | colourless yellow yellow yellow

The number of cm’. alkali used for titration of the same quantity
of these amino acids dissolved in water (with phenolphtalein as
indicator) can practically be neglected with regard to the above.

By further experiments it has appeared that glycine and alanine
behave under the circumstances belonging to these researches as
one-basie acid.

'! Before using this solution, it was- boiled for a moment and afterwards cooled.

90

Some of the observations referring to this are united in table II.

TABLE II. Determination of the quantity of alkali fixed by
_ glycine and alanine.

: | A E | ae PE B | 6 H | I | 4

80 cm3 of a solution containing + 5°/ glucose

Number of |

Biv
cM? 1,06 normal) | |
NaOH-solution 0 | 2 3 | 4 5 | 6 10 | 10 10 10

„added. : ; | |
| a 500 500
Added | | | | | milligr. | milligr.
| | | eN, el ‚glycine alanine
Filled up to 100 cms and placed in thermostat with
watermantle Cemperarure 33°)
Shin tar ae | | | a we
SCW | 22 |+ 11,0) | + 10,6 + 9,7) + 10,5
SONS <5) | | |
Beede oel ae Bees
Pe Er 10,9 nl 96+93+83 +100 + 85 + 9,6
5 HOMER Wire, pa | os
Colour 53) 2 So8| a8 | so) 8 EEE | G86 iis 458
lution ES & Ris E52 5 SS 8 | ses | ese | Be
solution aS 3 8 Nef poe 8 | > | BE | 32° nen ane

A solution of glucose in water of fixed concentration was under
comparable circumstances submitted to the action of different quan-
tities of alkali. The number of em? normal NaOH-solution added
varied between O and 10 em?. per 100 cm’. solution.

The most important decrease and the darkest yellow colour occurred
there, where most alkali had been added (experiments G and U).
After about two hours the polarisation of G and I had diminished
from +10,9 to + 8,3 and + 8,5. By the experiments H and J
was proved once more the protective influence of glycine and alanine;
the polarisation was respectively -+ 19,0 and + 9.6.

The polarisation of H lay between that of C and D. From this
it is apparent that 500 milligram glycine has fixed 10—3} = 6,5 cm’.
1,06 N.NaOH = almost 7 em°. N.NaOH. The intensity of colour
was in accordance with this. In an analogous way it was demonstrated
that 500 milligrams of alanine had fixed about 5 >< 1,06 = 5,3 cm°.
N.NaOH.

Glycine and alanine regarded as monobasic acid, 500 milligrams

91

. of these compounds would fix respectively ==.6;/5*@na.. one.

500 ;
nn = 5,6 cm’. N. alkali, so the agreement is sufficient.

On the other hand amino acetic acid and «-aminopropionie acid
behave in hydrochloric acid containing solution as monacidie alkali,
so that these aminoacids slacken the velocity of inversion of sucrose
by hydrochloric acid considerably (Table III).

TABLE Ill. Slackening influence of glycine and alanine on the inversion of sucrose by hydro-
chloric acid.

mgee

130 Gr. sucrose was dissolved in H,O and filled up to 500 cm? (solution R.)

50 cm3 of solution R

| 500 500 500
Added | milligr. | milligr. | milligr.
ts 3 glycine | alanine | phenol
Number of cm3 1,01 |
Normal hydrochloric 0 Pe 4 6 10 10 10 10 10
acid added hale el gd
Filled up with H,O to 100 cm3 and placed in thermostat with
watermantle (temperature 33°)
At the nl ele alen en En
beginning | 1 496 | + 49,4 | 4404 | + 49,5 | +403 | + 49,4 | + 49,1 | + 49,0 | + 49,0
periments al
after |
gore + 49,8 | +-48,5 | + 47,6 | + 46,3 | + 43,9 | 47,2 | +464 | + 43,1 + 43,4
me ours | Aj -
Polarisation | after DS TE — 13,8 | — 16,7 a We = 16,6 7 16,1
5 + 43/92 (temp. | —83 | (temp. (temp. | — — (temp. emp.
in grades | hours | SS | 139) ne el se ett | 145) | 149)
Denner after | SEE |= 2.0 | = 13,9) — 16,1 | — 16,5.) —13,1 | 4,9 | — 16,4 | — 16,4
+3X 24 Sun | (temp. | (temp. | (temp. | (temp. | (temp. | (temp. | (temp. (temp.
(2dm tube) |_hours | 285 | 1795) | 18%5) | 1795) | 170,5) | 189,5) | 18°,5) | 18°) 18°)
after (CS of | “not: | — 14,8 | not not | —147 | —15,2 | not not
+424, 58° determ-| (temp. | determ-| determ- (temp. | (temp. determ- | determ-
hours | 352) ined | 21°,5) | ined mede Pie) 20°) |- med ined
after | ~ a note r16:0 | stief not | —16,0 | —15,9 | not not
+6X 24 5 & determ-| (temp. | determ-| determ-| (temp. | (temp. | determ-, determ-
hours | => ined 189,5) | ined ined | 189,5) 18°) | ined ined

As follows from the survey given in table III the protective
action of glycine and alanine on sucrose corresponds with respec-
tively +.6 cm? and + 5 ecm’. N.HCI.

It is thus proved that under the circumstances of these experi-
ments both aminoacids behave as monacidie basic substances.

92

Phenol has no protective influence; from the results obtained the
opposite would sooner be inferred.

In acidic solution the properties of phenol differ widely from those
of glycine and alanine, in alkalic solution on the contrary phenol
behaves as monobasic acid just as glycine and alanine.

TABLE IV. Influence of phenol on the destruction of glucose by alkali.

| 80 cm3 of a solution containing + 5%) glucose

cm3 1,06 normal NaOH- |
solution added 0 | 3 > | 10 | 10 | 10 | 10
| | | 500 | 1500
quantity of phenol added 0 | 0 | 0 | 0 milligr.| milligr. 0

Filled up with H,0 to 100 cm3. Placed in
thermostat with watermantle (temperature 33°).

At beginning | not

of the experi=/ + 11,1 |+ 10,6 + 10,3 | +9,8 | + 10,2} + 10,9 | determ-
ments | ined

After

|
| |

+ 3!) hours) + |D! | Bic cl ea cS Oa | +85 +106 +6,9

[= |

Temp. of the
Polarisation | Polarisation-| (19°,5) | (19°) | (19°) | (19°) | (19°) | (18°)
| solution | SN | :
in grades | ager | | | A
+ OA hours) +140 | +3,0 | +04 | 40 | 406 | oss Sem

VENTZKE

Temp. of the
(2dm-tube) | polarisation- | (20°) | (20°) | (199,7)
solution | | |

| not | | |
De +08 | 02 | —0,7) 06 | +81 | — 0,6

‚determ-
+2X24h, ined |
Temp. of the
| polarisation- | (118,5)

solution |

|

(179.5) |

| almost |
yellow colour- |
less

| deep

After | colour: | pale
Colour yellow | yellow |

deep
+ 24 hours, less yellow

yellow

500 milligram phenol neutralizes the action of about 5 ecm’.
1,06 N. NaOH-solution = + 5,3 em’. N. NaOH. |

Regarding phenol as monobasic acid 500 milligram corresponds

“th 500
NV
wi 94

=5,3cm’. N.acid. So the agreement is sufficient.

In the above we have made acquaintance with two sensible
methods, which enable us to determine the acidic or basic character

93

of a compound in another way than this has usually happened
up to now.

Besides they give us an important indication about the condition
of the amino acids in watery solutions.

These amphoter electrolytes have apparently a neutralising influence
in two directions. This can only be understood well, when we sup-
pose that the action of the alkalie substances and of the acids on
the destruction of the monosaccharids and on the inversion of
sucrose is caused by molecules or molecule-fractions, which can be
fixed by the amino acid.

From these results we may conclude that with these reactions
under the influence of strong acids the amino acids assume the
character of rather strong basic substances and under influence of
strong alkali they act as rather strong acids.

These two opposite properties of the amphoter substance come
very distinctly to the front, which can best be symbolized by the
supposition of the open chain as well in strong acidic as in alkalic
medium.

When we suppose the ring-constitution in pure water, against
which no decisive difficulty exists because the electrice conductive
power is so small, the above can also be defined as follows:

By strong alkali the carboxyl-side

CH,—NH, CH,— NH, CH, —NH,Cl
| zeide Kost |
CO — OK CO-——O CO — OH

of the ring is opened, by strong acids the ammonium-side.

Now it was interesting to know how betain should behave; this
compound has no doubt in pure water the ring-constitution and it
could be expected that this ring should not open on the two sides as
easily, at least not in an equally strong degree, as this proved the
case with the amino-acids.

Really, the experiments joined in tables Ve and V? show that
betain does not hinder the destructive action of alkali on glucose,
whereas betain acts as monacidic alkali on hydrochloric acid during
the inversion of sucrose. Accordingly in presence of alkali the ring-
formula must be assigned to betain; in presence of strong acids
this compound has an open chain.

CH,—N(CH,), CH,—N(CH,),Cl

| sE

CO —O CO — OH
Betain in neutral and - Betain in strong

alkalic solutions. acidic solutions.

94

TABLE Va. Behaviour of betain in, alkalic solution.

| 40 cm3 of a solution containing + 5°/, glucose

Quantity of betain-hydro- | | | |
cilorie acid added 499 milligr.

|
| |
Number of cm3 1,06 Nörmial| 0 | 25 | 5 | |
NaOH-solution | 5 | |
Filled up to 50 cm3 and placed in
thermostat with watermantle (temperature 33°)

At beginning, | |
EN 419.4, ee ijs Astor, ABe lS
Polarisation | eta
, After | 125) + 97.) + 75 | +102 | + 76
in grades ze iz
Verne. | BRERA oe Wega] 8,0 4,1
+ 6 hours ao 7 | ae , ote ) | ai , — )
(2 dm. tube) Aft 5 | |
arousal Be SN
Ey | hardly hardly |
Colour | After | colourless. pale | „pale
| pale pale
ES | + S hours } | yellow | yellow yellow yellow
: |. After | Sree’ pale = vellow | ie ellow
solution + 24 hours | yellow | y | velo y
See : 499
499 milligram betain-hydroc =
1

almost 3,3 em? normal HCI. If in alkalie medium betain behaves
as a neutral compound, the action of the added 5 cm’. 1,06 N.
NaOH — N. NaOH must be diminished by that of
3,3 cm°.; then 5,3—3,3—2 em?. N. NaOH remains. From the
experiments it follows indeed that 499 milligrams of betain-hydro-
chloric acid and 5 cm*. 1,06 N. NaOH act together as something
less than 2,5 cm’. 1,06 N. NaOH.

Whilst in alkalie solution betain behaves entel: in acidic solution
it acts like one-acidic alkali.

From this results that the betain-hydrochloric acid-complex behaves
as practically neutral.

Hence the inversion of sucrose by hydrochloric acid is accelerated
but little by betain-hydrochloric acid.

95

TABLE V0. Behaviour of betain in acidic solution.

130 Gr. sucrose was dissolved in HO and filled
up to 500 cm3 (R)

50 cm° of liquid R

Quantity of betain- barre ac EN
hydrochloric acid 500 500
added milligf, milligr.
Number of cm3 1,01 |
normal HCI-solution 0 3 5 7 7 10 10 10
added
Filled up with H,0 to. 100 cm’, placed in thermostat
with watermantle (temperature: 33°)
5 ;
eee ng + 49,8 + 49,4 |-++ 49,3\+ 49,2 + 49,3 |+ 49,2 + 49,1 |+ 491
Temperseure | | :
polarization (HAS) |) TSP PLESK | Gs?) | 2182)" | 38%)
liquid | |
After + al
23/, hours + 49,7 | + 46,7 |+- 45,0|+ 42,5) + 41,9 |+ 40,2) + 39,57-+ 39,4
Polarisation
it eg |
in grades nalerieation (199) | (19°) | (199) | (19°) | (19°) | (189)
liquid |

VENTZKE After +
(2dm) tube | 3/2 hours

Remue

of the

polarisation |
liquid

H-49,8 | + 45,5 443,14 39,9] + 39,1 |-+ 36,1) + 35,8 |+ 36,0

(19°) (19°) | (19°) |(18,5°)} (19°) | (199)

not not ; ;
After + determ- determ- — 16,2 — 16,4| — 16,2 — 16,6 — 16,3 |acterm-
i | ined ined ined

en ee
ngladegtion | (179) | (17°) | (179) | (17°) | (179,5)
liquid

«

The summary of the above mentioned results becomes as follows:

1. Amino acetic acid and «@ amino propionic acid retard the
destruction of glucose by sodiumhydroxide.

2. This phenomenon is independent of the presence and the
quality of the monosaccharid, for the destruction of galactose by
sodium-hydroxide is retarded too by the substance mentioned.

3. Amino acetic acid and «-aminopropionic acid bebave in alkalic
medium as acids. By further examination it was demonstrated that
they behaved as about one-basic acid.

4. Just on the other hand these aminoacids in presence of hydro-
chlorie acid behave as monacidic alkali, so that they considerably

96

retard the rapidity of inversion of sucrose by hydrochloric acid.

5. The behaviour of glycine and alanine deserves special attention
because these compounds behave by the usual way of titration as
practically neutral.

The number of cm°. alcali necessary for colouring pink a solution
of glycine or alanine, which contains phenolphtalein, is insignificant
if compared with the quantity of alkali which would be necessary
when both compounds should behave in this case as one-basic acids.
The same holds for phenol (Compare 7°).

6. The destructive influence of sodium-hydroxide on monosac-
charids and the inversion of sucrose by hydrochlorie acid can be
used for the edification of two sensible methods, which enable us
to judge in another way -than was usual up to now, whether a
compound has acidic or basic properties *).

7. Remarkable too is the behaviour of phenol in alkalic solution.
Phenol acts then as about a one-basie acid, whilst this compound
practically has no influence on the inversion of sucrose by hydro-
chlorie acid. :

8. The pure amphoter behaviour of glycine in alkalic and in
acidic solution, together with the behaviour of betain, which com-
pound in alkalie solution is practically neutral and in acidic solution
acts as one-acidic basic substance, make it probable that glycine
as well as alanine possess in alkalic and acidic solution the open
formula of constitution. In entirely neutral solution the ring formula
is sufficient. To betain the ring-formula must be granted in neutral
and alkalie solution, in acidic solution the open constitution-formula.

This research will be continued in different directions in order
to study the acidic and basic character of the substances and at
the same. time to determine how far their usual constitution-formula
corresponds with this character.

Dordrecht, February 1917.

1) These methods can only be used if we know with certainty that the sub-
stance to be examined is not destroyed in acidic or aikalic solution and has for
the rest no disturbing influence.

° i, . . 2 . 4 en
Physics. — “Sur les équations diffbrentielles du champ gravifique.”

By Mr. Tx. pr Donper. (Communicated by Prof. H. A. Lorentz).

(Communicated in the meeting of May 26, 1917).

En étudiant le champ gravifique dans univers stellaire, M. EINSTEIN
a été amené tout récemment'), a adopter l’hypothése que j'avais
faite anterieurement, en la considérant comme nécessaire dans tous
les cas’), a savoir que la courbure totale C de l’espace-temps doit
être nulle. |
D’autre part, M. Ersrein modifie ses equations du champ gravi-
fique: dans chacune d’elles, il introduit.un terme nouveau; je démon-
trerai dans cette note que les équations ainsi corrigées d’ EInstuin
sont identiques aux équations que j'ai données le 12 juin 1916 ®,
à savoir:
k(—g)! rare ah! Gh, Um) = = Yim Vik — 4 Yim Tre) + + (Ql)
dG Pao ee
Dans ces dix équations différentielles, les symboles g;,, représentent
les dix potentiels d'EiNsruin; g est le déterminant symétrique formé
au moyen de ces gi; chacun des g* représente le mineur algébrique
de gx, divisé par g; (ik, lm) est une parenthèse a quatre indices de
CuristorreL ou de Rismann; & est une constante universelle; les
Tp sont 16 fonctions qui dépendent du champ électromagnétique *)
de Maxweti-Lorentz et du mouvement de la matière’) dans le
champ gravifique. La configuration de l'espace-temps est déterminée

1) Sitzungsberichte der Akademie der Wissenschaften, Berlin, 8 février 1917.

2) Zittingsverslag Amsterdam XXV, 1916, p. 156.

Archives du Musée Teyrer, Série 2. T. Ill (voir la fin de ce mémoire).

Dans la suite de cette note, nous désignerons le mémoire précédent sous le
nom de: mémoire (Archives TEYLER).

8) Voir la fin de mon mémoire (Archives TEYLER).

4) Voir €quation (355) de mon mémoire (Archives ‘TEYLER).

5) Voir mémoire de M. ErNsrein, p. 799 (Silz.ber. Akad. Wiss., Berlin, 4 novembre
1915). Remarquons que quand M. Einstein écrit

==)? Eg TZ} Eg Ti,
ed ed

nous employons le symbole Tiz.

Proceedings Royal Acad Amsterdam. Vol. XX.

98

par la forme différentielle quadratique.

as? == = me Vik FE at fe re ER (2)
ik s

Démonstration. M. EisrriN a proposé, dans son récent travail
cité, d’étendre le principe d’Hamivton de la manière suivante: il
annule la variation

J Bea (£41 4 HD dz dy de di. 2 EA

où Lest une fonction qui joue un rôle analogue a celui de L dans
Pétude da champ électromagnétique dépourvu de matière *); où
1=kC (—g}, et où À est une fonction de x,y,z et t, que nous
déterminerons plus loin®); dans le calcul de la variation J, on prend,
comme on sait, de = dy = dz = dt = 0. Les tirets horizontaux qui
surmontent les symboles employés servent a rappeler qu’on a fait
usage des variables g* et de leurs dérivées.
Les 10 équations différentielles des extrémales de (3) sont’):

im

OLE + 14 Mp =0, (4)

im =d

im d d d d* d
Ke Se ES = + = ie els
V dg s ats dg? Te dede diner

Remarquons que

im nt d(—g)* Sit dg
peo EE En CN ef 2
Le ( PI dyim ee, 2 ( 9) dgim
Grâce au déterminant réciproque de g, on trouve aisément que:
dg a
dim == TT (2 = Eim) Y Yim
où ¢;—1, et ej, — 0, quand 7 est différent de m.
Donec:
im = St
Ng HO ed) DE am 6)
Les équations (4) et (5) donnent:
in dL 2 =~ ;
ae dgim Ps = tial lt eN

Par extension *), on aura encore dans le cas d'un champ gravi-

1) Voir équations (324) à (329) de mon mémoire (Archives TEYLER).

2) M. EINSTeIN supposait que A était une constante.

3) Pour les détails, voir le chapitre VIL de mon mémoire (Archives TEYLER).
4) Voir *équation (353) de mon mémoire (Archives TEYLER).

og

fique contenant de la matière:

dl Eim Ae
— Tiss 1 = Ink di . . . ° . . (7)

dgim
On a dautre part lidentité remarquable *):
im

(1 + in) Tk (— gh SS gH (ik, tm) AC Cin - (8)
Heal]

Rapprochons les relations (6), (7) et (8); les 10 équations diffé-
rentielles du champ gravifique prennent la forme:

k(— g) zE gr (ch , lm) — k = 7 C Jim = YIkm ER 4a (C5 Jim (9)
eee ; ley:

Multiplions ces équations (9) par g’”, et sommons par rapport a
iet a m; d'où, en vertu de?):
CI SLT ol gin (ik, lm), res ae ter eh.)
do Ue
on trouve la relation

- CO (oe AE Ty = a(S Ik ze 0 OS)
k

- Rappelons que dans le champ électromagnétique de Maxwerr-
LORENTZ, On a °*)
pd Ai eee | pee MR rae tars Acer A a (12)
k

D’autre part, les 7%, provenant de la matière valent, d'après
EINSTEIN *) :
a a da, da},
Tizk—=(— 9) 0 = gie — —. . . . s . (13)
a ds ds

où 9 désigne la densité de la matière: c'est une fonction de a, y, z,
et ¢. On en déduit que (2):

3 Tee = ge: EEN ce
d'où, en vertu de (11) et (14),
ond
== = — —C base oreo al tos HED
‘ 4 Di GE)

Il résulte de (45) que, pour que 2 soit fonction de x, y, z, ¢
seulement, c'est-à-dire pour que 2 soit indépendant des gj, et de
leurs dérivées, il faut et il suffit que:

SA i A AE HATE EE eo ts (16)

1) Voir la fin de mon mémoire (Archives TEYLER).
2) Dans la formule (6) qui se trouve a la fin de mon mémoire (Archives TYLER),
il y a une erreur typographique: le facteur 1/ a élé omis.
3) Voir la relation (y) à la fin de mon mémoire (Archives TEYLER).
4) Voir p. 799. Sitz.ber. Akad. Wiss. Berlin, 4 novembre 1915.
7

100

Jai done démontré que mon équation complémentaire est nécessaire
et suffisante.

En vertu de (15) et (16), on a done, enfin:
Q
AS SS i eee eee ee 1
; (17)
et les équations du champ gravifique deviennent [(9), (16) et (17)]:

a8 = o we
B 9)? 22 gil (tk ’ Lm) == = Yam Tir — nn 9)? Gimnma =. (18)
k l k

ou encore, en vertu de (14)
k(— gy? Soy oe (ik ’ lm) = a (Jk Eir— + Yim Tk) ; : (19)

ce sont textuellement les équations que nous avons données le 12 juin
1916"); elles entrainent comme conséquence’*) léquation comple-
mentaire
Gs).

Remarquons enfin, que le principe d’HaAMILTON généralisé pourra
s’énoncer comme suit:

Les équations différentielles de tout champ gravifique et électro-
magnétique expriment que, dans un espace-temps euclidien, Pintegrale :

SINE (=a! [ae ay dz dt
est extrémee.

Remarque I. [’hypothese (14), ainsi que nos équations *) (353)
(Archives TeyLer), sont satisfaites dans le cas où Von prendrait
£=160 (—9)*; alors; on Aura Tij (— g)'h ey.

Remarque Il. Si Pon wintroduit pas Phypothese (14), la relation
(11) montre qu'en vertu de C=O, on aura:

nn ? = Tea. dn en ee EUD

En substituant cette valeur de 2 dans les équations (9), on obtient
encore mes équations (19). Pour Vapplication du principe d’ Haminron
(3), on devra dans le second membre de (20) exprimer toutes les
quantités en fonction de a, y,z et ¢; on obtiendra ainsi le 4 attaché
au système considéré. .

1) Voir la dernière page de mon mémoire (Archives TEYLER).
2) Nes conclusions précédentes sont indépendantes de \hypothése (13).

Mathematics. — “On Elementary Surfaces of the third order”.
(First communication). By B. P. HAaLMeter. (Communicated
by Prof. BROUWER).

(Communicated in the meeting of May 26, 1917).

Introduction. The existence of certain numbers of real straight
lines on cubic surfaces is well known. In Math. Ann. 76 C. Jurn
makes a clever attempt to prove the existence of straight lines on
certain surfaces of the third order which are non-analytically defined
and which he calls elementary surfaces. His methods however are
not always convincing and some conditions he puts to his surfaces
seem to be artificial and out of place. The object of this note is to
introduce elementary surfaces of the third order ina natural way and
to prove the existence of at least one straight line on such a surface.
Our starting point is formed by the elementary curves of the third
order which are extensively dealt with by Juer in the Proc. of the
R. Acad. of Denmark, 7* series, t. 41 N°. 2. Besides this we shall
principally use well known theorems of the analysis situs and the
theory of sets of points.

In carrying out the following researches I am indebted for many
suggestions to Prof. lL. E. J. BROUWER, who also has attracted my
attention to this subject. ‘

Definitions and exposition of the problem. An open JORDAN curve,
which, together with the linesegment *) between its endpoints, forms
the boundary of a convex region, is called conver arch. These convex
arches form the building material for the elementary curves. Let a
set of points be composed of a finite number of convex arches, in such
a way that it forms the continuous representation of a circle. To
every point of the circle is to correspond one and only one point
of the set under consideration. Besides, the tangent (touching line,
Stiitze) is to change continuously with the corresponding point of
the circle and lastly the set of points is not to contain linesegments,
but may include entire lines. A closed set of points consisting of
a finite or countably infinite number of these above defined sets
is called elementary curve. Isolated points are admitted though
tangents in the ordinary sense disappear.

1) In the following Zine will be used for straight line.

102

An elementary curve is said to be of the n order, when lines
exist which have mn, but no lines which have more than 7 points
in common with the curve (unless the curve includes the entire line).

In this note we chiefly consider elementary curves of the third
order. Some of the results obtained by Jurr which shall prove most
useful are the following:

The possible forms of elementary curves of the third order are:

1. Oneconnected curve of the third order without double point orcusp.

2. One connected curve of the third order with a cusp (the two
branches arrive at the cusp from different sides of the tangent, cusps
where the two branches meet from the same side cannot exist on
curves of the third order, as a slight change in the position of the
tangent would produce 4 points of intersection).

3. One connected curve of the third order with double point,
(this variety can be considered as composed of a curve of the third
order and one of the second') having only the double point in
common and each forming an angle at that point).

4. One connected curve of the third order and one of the second *)
(that is: oval, boundary of convex region) having no points in common.
5. One connected curve of the third order and isolated point.

6. Straight line and oval).

7. Straight line and isolated point.

8. Three straight lines. |

As points of intersection with a line are counted:

double: ordinary point (that is: internal point of a convex arch)
on the tangent, isolated point on every line, cusp on every line
except the tangent and double point on every line except on either
of the tangents.

triple: point of inflexion on tangent, cusp on tangent and double
point on both tangents.

All other modes of intersection are counted single.

We define as elementary surface of the third order F* any set
of points in the projective R, possessing the two following properties *) :

1) These curves of the second order of course need not have finite breadth, but
can have one or two points in common with the line at infinity. (We always
consider projective space).

2) Ultimately it may be advisable to make this definition less restricting. In order
to admit conical points it will be necessary to extend the first condition and to
make it possible that the surface degenerates both conditions have to be revised.

The ultimate definition must be couched in such terms that no essential altera-
tions are required for defining elementary surfaces of order higher than the third.

103

1. f° is to answer the most general definition of a twodimen-
sional continuum *).

2. Every plane section of J” is an elementary curve of the
third order.

This note is divided into two parts:

In the first part we shall prove: The tangents to plane sections
passing through an arbitrary point A of F*, not situated on a line
of F*, form one plane, which may be called tangent plane to F* in
A Only one exceptional point is possible having the following
character: It is isolated in every plane except the planes through
one line, and in these it is cusp with that line as cuspidal tangent.

In the second part we begin by proving some further theorems
concerning points of /'* not situated on a line of F*. At the end
we assume that no point of a certain plane section is situated on
a line of /*. By showing that this leads to contradictory results,
the existence of at least one straight line on F* is established.

First part. We divide the proposition as foilows:

§ 1. If A is isolated in a plane a, then « is tangent plane to
F* in A or A is exceptional point.

§ 2. Only one exceptional point is possible. |

$ 3. If A is double point in a plane @ and cusp in not more
than one plane, then « is tangent plane. f

$ 4. If A is cusp in one and not more than one plane «a, then «
is tangent plane.

$ 5. If A is cusp in two different planes, then A is exceptional point.

$ 6. Through A passes at least one plane in which A is either
isolated point, double point or cusp.

§ 1. Jf A ús isolated in a plane a then a is tangent plane or A
is exceptional point.

The first thing to be done is to construct a plane in which A is
not isolated. The vicinity of A on F® is the (1,1) continuous
representation of the vicinity of a point in a plane, hence a
sequence of points A,, A,, A,.... of #* can be chosen having A
for sole limiting point. Let a be an arbitrary line through A in «
and §,,8,,8,.... the planes passing through a and A,, A,, A,....
respectively. These planes have at least one limiting plane 8 passing
through « also. In case A is isolated in each of the planes @,, 8,, 3,
it can be shown that A is not isolated in ~. ;

1) Brouwer, Math. Ann. 71, p 97.

104

In a plane in which A is isolated the remaining points belonging
to F° form a connected curve of the third order or a straight line.
This resteurve is a closed set of points (it is the continuous repre-
sentation of a circle), hence A has a finite minimum distance from
it. Let this minimum distance be e‚ in ~,, €, in 8, ete. When a point
B moves along the resteurve in ~, the distance AL changes
continuously from ¢€, to oo, in #, from &, to oo etc. (when a point
A is situated at distances 6, and b, from two points 5, and 5,
belonging to a connected set of points then to every distance 5, such
that 6, > b, >, corresponds at least one point B, of the set such
that Ab, = b,).

The sequence ¢,, ,,&,....has zero for limit. Let d,,d,,d,.... be
a decreasing sequence chosen from it and let the corresponding
planes be represented by 8, 82, Bs, ----

In 85, we choose a point B, of F* such that d, > AB, >d,

2) Bo, 2 > ” ” Bi 339) #93 ” ” d, AD De Jb,

ande. oek B 5" 5 ean ene Uae eee

In Ba, we choose a POMEE rn Ng ee ene ree

and Ar Kae ei eh aes SB Panel een Aline nee
ANU rsr ces Pid Re go a Wagan cae Oe meena ar mete

etc.
B, B,", B"... have a limiting point B, in 8 such that J, 2 AB, = d,
VEER oe Pathe Nag eh eae es js. Be Se 5R > 5 Oy ae
ete.

F* is a closed set of points, hence B, b,, B,... all belong to f°.
Besides d,,d,, d,.... is a decreasing sequence having zero for limit

hence A is limiting point of #° in ~.

We now proceed to construct a finite sphere round A inside of
which /* is entirely situated on one side of the plane a (except the
point A in a). A is isolated in «a, hence with A as centre there
exists in « a finite circle c containing no other points of £*. Let
b be the sphere with A as centre passing through c. The vicinity
of A on F* is the (1,1) continuous representation of the vicinity
of a point in a plane. Let A, be the point corresponding to A.
The correspondence is (1,1) continuous, hence a finite circle c,
round A, can be found in the plane such that all internal points of
c, have corresponding points inside the sphere 5 and a sphere 6’ con-
centric with 5 can be found such that all internal points of 5’
belonging to /’ have corresponding points inside ¢,.

Inside 6' f° lies on only one side of «a for if this were not the
case, a contradiction might be obtained as follows: Two points 5

105

and C of F* are chosen, both internal to 6’ and on different sides
of a. The corresponding points B, and C, are situated inside c, and
can be joined by an open JorDAN curve not passing through A, and
entirely internal to c,. The set of points AK corresponding to this
curve is closed and connected (both these properties are invariants
for (1,1) continuous transformations). A’ is situated entirely inside 4,
contains points on both sides of « but no points of « itself (A is
the only internal point of e belonging to /*), Hence K is composed
of two closed sets of points, one on each side of «, but this is im-
possible, because K is connected.

The above results may be taken together as follows:

Through the line @ passes a plane «, in which A is isolated, and
a plane 8, in which A is not isolated. Besides, inside a sufficiently
small neighbourhood of A the surface /” lies entirely on one side of
a, let us say below a. Hence inside that neighbourhood of 4 the inter-
section of 8 and F® lies entirely below a (always excepting the
point A itself, which is situated on a). Considering the possible forms
of elementary curves of the third order, there remain two possibilities:

1. A is ordinary point in 8 with a as tangent.

2. A is cusp in 8B.

Let A be cusp in 8 with 5 as cuspidal tangent. In no plane through
b can A be isolated, because the two branches meeting at the cusp
in 6 furnish points of F? on both sides of each of these planes
inside every vicinity of A. But above a@ there is a finite neigh-
bourhood of A containing no points of £*, hence in every plane
through 6, A is either cusp or ordinary point with the tangent in
a. We proceed to show that if A is cusp in B it cannot be ordinary
point in two other planes through 6.

Let «,,a,,¢,.... be a sequence of parallel planes each of which
lies above all preceding ones and which have « for limiting plane.
Let the points of intersection of b and a,,a,,a,.... be respectively
B,, B,, B,.... If the sequence is started high enough every plane
a,,a@,... has a point in common with each of the branches meeting
at the cusp in g. Let these points be B,' and B", B, and B,",
To es Eee CD None, of « these). pointe’ Baseren 28.” can be
isolated in the planes «,,«,.... considering the branches meeting
at the cusp in 8 furnish points on both sides of each of these
planes in every vicinity of B,', B,"....

A sequence of connected sets of points, each having a breadth
>p, has for limit a connected set of points with breadth > p.
From this follows that when n increases the points B, and 5,"

~

106

cannot continue to be situated on odd curves in a@,, for an odd
curve is never entirely internal to a finite region (in other words:
always has infinite breadth), so the limiting set would be a curve
in @ passing through A. But if for larger than some finite value
the points B, and #6," can neither be isolated nor situated on odd
curves, they must lie on even curves. which in this case must be
ovals. Obviously these ovals contract when 7 increases and A is the
sole limiting point. Let y and d be the planes through 5 in which
A is supposed to be ordinary point (with the tangents in a).
Let a, c, and d, be the lines of intersection of «a, and 9, y, d res-
7 pectively. Obviously a, intersects
BE " the oval in plane a, at B, and B’.
De B, is a point of the cuspidal
tangent in 3 and B, and B, are
points of the branches meeting at
the cusp from different sides of
the tangent, so on line a,, B, is
situated between 5, and B’,
hence B, is internal point of the
oval in «,.-From this follows that
the lines ¢, and d, passing through
B, bave each two points in common
with the oval, one on either side of B,. Let these points be C,’, GC”
and 052 DL
In plane 8 A is cusp with 6 as tangent, but in y and d A is
supposed to be ordinary point with the tangents in @ From this

B, Bu q B
B, 63> Bae
may be made as small as desired, even of the second order with
respect to the distance of the planes @ and a@,. Besides the angles
of a, en and d, are the same for every 7, hence for 7 large
enough, the linesegments CD, and B, B, will have no point in
common and this result contradicts one of the fundamental proper-
ties of ovals.

The following question arises: Is it possible that A is ordinary
point in y and d and cusp in 7, but with a cuspidal tangent not
coinciding with 5? We shall show that the answer must be negative.
The notation of points of intersection etc. is kept the same as above.
In 8 the branches meeting at the cusp would arrive from the same
side of 6, but in 7 and © the branches meeting at A arrive from
different sides of 5. Hence for n large enough the oval in a, would
be such that on the lines c, and d, the point B, is situated between

Wig. ds

follows that by taking m large enough the ratios

107
the points of intersection with the oval, but on the line a, both
points of intersection lie on the same side of B. This means that
B, is at the same time internal and external to the oval, and this
is impossible.

The above results may be taken together as follows: A is supposed
to be isolated in plane «, and 5 is a line through A not situated
in a. Now, if A is ordinary point in two different planes through
b, it cannot be cusp in any other plane through b. But if A is ordinary
point in a plane through 5, the branches meeting at 4 in this plane
furnish points of #° on both sides of every plane through 6 inside
every vicinity of A. Hence in no plane through 6 can A be
isolated. Besides above « there is a finite vicinity of A contain-
ing no points of #* so in no plane can A be double point, point
of inflexion or ordinary point with tangent not situated in «. Hence
when A is supposed to be isolated in @, and 5 is a line through A
not situated in @, the final result may be formulated as follows:
If through b pass two different planes in which A is ordinary point,
then in every plane through b, A ts ordinary point and all the tangents
are situated in @.

Above we found that in 8 the point 4 is either:

1. Ordinary point with a as tangent.

2. Cusp.

Let the first possibility be assumed. We turn the tangent qin the
plane 8 round the point 4 in both directions to the positions « and
a". Provided these rotations be small enough the lines a’ and a’
have each three different points in common with /*'). Hence in no
plane through a’ or a" can A be isolated point, double point or
cusp. Points of inflexion aré also excluded, because one of the branches
meeting at such a point would arrive from above «, but above «
there is a finite neighbourhood of A containing no points of /’*. The
only remaining possibility is that in every plane through a’ or a",
A is ordinary point and the tangents must all be situated in @ because
above « there is a finite neighbourhood of A containing no points of £'*.

Let ce be an arbitrary line through A, not situated in « orp. The

1) Juez, loc. cit. Acad. of Denmark. When points of intersection are counted
as explained, an elementary curve of the third order and an arbitrary line in its
plane have in common either three points or one point. Hence a tangent at an
ordinary point A carries one point more of the curve. Now if this tangent be
turned round A over a sufficiently small angle, A is replaced by two points of
intersection A and B each counting single. But there must be still another point
of intersection, as there are to be three altogether, so the line in its new position
has three different points in common with the curve.

108

two planes passing through c and through a@' and «” respectively
show ordinary points in A. Hence (using the results obtained above)
every plane through c shows an ordinary point in A and all the
tangents are situated in @.

But e is an arbitrary line through A only subjected to the con-
dition not to lie in @ or 8, so it follows that in every plane, except
« and p, A is ordinary point with tangent in «. Besides in 8 A was
assumed to be ordinary point and the tangent was found to lie in a,
hence the only remaining exception is « in which plane A is isolated
and which has now been proved to answer our definition of tangent
plane.

We now assume the second possibility given above:

The point A is isolated in @ and cusp in 8. Let 5 be the cuspidal
tangent. In no plane through 5 can A be ordinary point, for if this
were the case, it might be shown in the same way as above that
A cannot be cusp in fg. Also in no plane through 6 can A be
isolated because 6 has only the point .A in common with £'*. Taking:
into consideration that above « there is a finite vicinity of A
containing no points of #*, the only remaining possibility is that A is
cusp in every plane through 5. 6 must be cuspidal tangent in every
one of these planes because has only the point A in common
with #*. Now a cusp counts double as point of intersection on any
line except the tangent, hence every line through A (=|=6) has one
and only one other point in common with /’*, because in the plane
through that line and 6 the point A is cusp with 5 for tangent.
Thus in a plane through A which does not contain 5, every line
through A has one and only one other point in common with #?,
hence A is isolated in every plane which does not contain 6. Thus
it has been shown that A is exceptional point.

Before proceeding further we shal] just rehearse what has been
done in $ 1: | |

A was assumed to be isolated in plane «. Then a plane 8 was
constructed in which A was not isolated. From the assumed isolation
in « it followed that only two things were possible, namely that A is
ordinary point in 2 with tangent in @ or that A is cusp in B.
Assuming the first possibility we proved that « must be tangent
plane, while the second assumption lead to the conclusion that A
is exceptional point.

§ 2. Only one exceptional point ts possible.
Suppose there could be two: A and B. In a plane through A
and 5 there are a priori four possibilities :

109

1. A and B are both isolated.
2. A and B are both cusps.
3. A is isolated and B is cusp.
4. A is cusp and B is isolated.

But no elementary curve of the third order can have two isolated
points, two cusps or one of each, hence the required contradiction

is obtained.

§ 3. Jf A is double point in a plane a and cusp in not more
than one plane, then a is tangent plane.

The points of /* situated in the plane a form an elementary
curve of the third order A, which has a double point in A. A is
the point of intersection of two convex
arches AK, and K,. Let ¢ be a circle
round A in «, such that all points of
K which are internal to c belong to
K, + K, and besides c must be such
that it has only two points: C and H
in common with A, and only two
points: D and F with K,. All these
conditions can be fulfilled by taking
c small enough.

Now the branches AC, AD, AE
and Af are connected by four sets of points I, I, III and IV,
having no points in common, all belonging to #* and each of
which is entirely situated on one side of a. Respecting these four
sets of points, the JorpaN theorem for threedimensional space ')
leaves only two possibilities.

The first possibility comes to the following: AC and AD are
connected by I, and AD and AZ by II, A# and AF’ by III, and
lastly AF and AC by IV. If the concave side of EC faces F, let
us assume for a minute that III and IV are both situated above a.
Now if a parallel linesegment converges from above towards /’C”
it would end up by having at least two points in common with
both III and IV, and this is impossible. Hence III and IV cannot
lie on the same side of a. If the concave side of DF faces / then
II and III must also be situated on different sides of «. Hence II
and IV lie on the same side of a, but then I 1s certainly situated
on the other side, for suppose all three were on the same side then
a parallel linesegment converging from that side towards PQ would

1) Brouwer, Math. Ann. 71, p. 314.

Le)

finish up by having at least one point in common with each of II
and IV, and at least two with I, and this again cannot be as no
line carries four points of #’*. Hence the final result is that I and
IIl are situated above « and II and IV below « or vice versa.

A representative case of the second possibility is the following:
AC and AK are connected by | above a, AF and AF above or
below a by II, AF and AD below « by HI and lastly A D-and AC
above or below a by IV. If IV be situated below « we choose in
a a point A’ near A and a point D’ near D, such that the line-
segment A’D’ intersects the arch AD at a point near A and at
another point near D. Now a parallel linesegment converging from
below towards A’ D’ would end up by carrying at least two points
of III and two of IV: acontradiction *). Hence the second possibility
left by the Jorpan theorem is excluded and we need only consider
the first. In the following it will be assumed that I and III are
situated above, and II and IV below a.

Obviously the set of points 1+ AC + AD is the (1,1) continuous
representation of a plane region and part of its boundary. Besides,
inside a finite neighbourhood of the point corresponding to A, this
region has the character of a JoRDAN region, because the arches
AC and AD are Jorpan curves, and the same holds for the (1,1)
continuous representations. The same things can be said of I+4+-AD + A A,
UI 4 AE 4 AF and IV + AF + AC.

Lastly we remark that inside a finite neighbourhood of A all points
of F*, not situated in a belong to 1 + 1 + III + IV.

Let 6 be a line in « through A such that the branches FA and
EA arrive at A from different sides of this line. Then the branches
CA and DA will do the same. Let 8 be a plane through 6 (= @).
AC and AD are joined above « by I. [+ AC —+ AD is the
continuous (1,1) representation of a plane region and part of its
boundary. Let I, correspond to I, 4,C, to AC, and A,D, to AD.
Inside a finite neighbourhood of A, the region 1, has the character
of a JORDAN region.

We shall now have to use a property of JoRDAN regions called the
~“Unbewalltheit”.*) For two dimensions it may be formulated as
follows: Let / be a closed JorpAN curve, / the internal and £ the
external region. Two points Q and R of / can always be joined
by an open JorDaN curve belonging entirely to / and by an open

1) By using this last reasoning the first possibility might have been dealt with
in a more simple fashion.
2) Brouwer, Math. Ann. 71, p. 321. ;
SCHOENFLIES, Mengenlehre 2, chapter 5.

ait

JorDAN curve belonging entirely to . Let P be a third point of ./
and c an arbitrary circle round P. Now the “Unbewalltheit” says
that if Q and R are chosen close enough to P, the joining curves
may be kept entirely inside c.

Applying this to our case it follows that points of A,C, and A, D,
can be joined by open JorpaN curves entirely belonging to I,, inside
any vicinity of A,. Hence in the continwous (1,1) representation
AC and AD can be joined by open Jorpan curves entirely situated
on I inside any vicinity of A. Now every one of these curves
has at least one point in common with 8, because AC and AD lie
on different sides of that plane, hence in plane 8 the point A is
limiting point of I, and in the same way can be proved that A is
limiting point of HI in 2. But I and If have no points in common,
hence in @one branch departs from A on | and another on III. I and IIL
are both situated above aso in 3 two branches depart from A above «.

In 8 two branches arrive at A from the same side of 6. Considering
the possible forms of elementary curves of the third order, there
are a priori three possibilities :

1. A is double point in @.

2. A is cusp in f.

3. A is ordinary point in 8 with 6 for tangent.

1. Suppose A is double point in 6. Two branches AP and AQ
arrive in A from above 6, hence two more ASand AR arrive from
below 6 (three from one side and one from the other is impossible
because 6 has, besides A, another point in common with #°)- We
proceed to show that the ‘branches AR and AS are at first both
situated on II or both on IV. Suppose AR and AS were situated
vespectively on II and IV. Then AR and AS could not be connected
below a, because II and IV have no points in common. But AS
would be connected via AC and AF with AP and AQ and AR
would be connected via AD and AH with AP and AQ. From this
follows that AR and AS would only be connected via AP and AQ.
This however leads to a contradiction, because the four branches
meeting at A in p must be connected in an analogous way as those
in «a hence AR and AS are joined by a set of points situated
entirely on one side of 6. Thus it has been shown that AR and AS
are situated either both on II or both on IV, let us assume on II.

The vicinity of A on #* is the (1,1) continuous representation
of the vicinity of a point in a plane. Let A correspond to A,,
AE to A,E,, AD to A,D, II, to II,, AR to A,R, and ASto A,S,.
Inside a finite neighbourhood of A, the region Il, is divided by the

112

open Jorpan curves A,R, and A,S, in three regions having no points
in common. In the vicinity of A, all these regions have the character

ie

Ffg. 3.

of JorpAN regions. We consider the two outside regions, namely
those connecting respectively A,/#, with A,R, and A,S, with A, D,’).
The (1,1) continuous representations of these regions of /’* connect
respectively AH with AR and AS with AD. That this connection
exists inside any neighbourhood of A, again follows from the “Un-
bewalltheit”. Hence in any plane through 5 (fig. 3) such that AZ
and AR are situated on different sides, at least two branches arrive
at A from below «. But we also know that in each of these planes
two branches arrive in A from above « (one on I and the other —
on III), hence the following result has been obtained: When the
plane 8 is turned round 5 (fig. 3) in such a way that the lower half
moves to the left, then in every position as far as « the point A
remains double point.

Let ce be a line in « through A, passing between the branches AH
and AD, and let d be a line in @ through A, separating the branch-
es AR and AS. The plane through c and d is denoted by + (fig. 3).
In y two branches arrive in A from below @, one on II and the
other on IV. The branch situated on I arrives in A from the right

1) A priori it would be possible that A,£, is connected with A,S, and A,R,
with A,D,, but when we consider the representations on F3, this leads to contra-
diction with the JoRDAN theorem for threedimensional space.

113

hand side of AN, because the component region of I] which forms
the direct connection between AF and AS, is situated on the right
hand side of 2. This branch on II cannot have AZ for tangent
because in that case the branch on IV would also have AL for
tangent and cusps where both branches arrive from the same side
of the tangent, are excluded. Hence the branch in y on II forms
at A finite angles with both AM and AZ.

The line c has, besides A, another point in common with #®, and
for this reason can never be tangent at a double: point. Hence the
branch in y situated on IV cannot have AL as tangent, so it must
arrive in A under a finite angle with AL, and it follows that if the
plane « be turned round line 6 in such a way that the right hand
side moves downwards (fig. 3), the point A will at first remain
double point. The above results may be taken together as follows:
a cannot be limiting plane of planes through b in which A is not
double point. But by reversing @ and @ in our reasonings, the same
can be said of plane 8. Hence: if @ be turned round 5 in either
direction, A at first remains double point. In neither direction can
there be a last plane in which A is double point, so either there
is a first in which A is not double point, or all planes through 5
show a double point in A.

In a first plane in which A is not double point, there still arrive
two branches in A from above a (one on I and the other on lil)
hence in such a plane A would be either ordinary point with 6
as tangent or cusp. The case that A is cusp shall be dealt with
sub 2. So at present only two assumptions need be made, namely
that there is a first plane in which A is not double point, but
ordinary point with } for tangent, or that all planes through 5 show
a double point in A. We shall successively show that both these
assumptions lead to contradictions.

Let d be first plane in which A is ordinary point with 6 for
tangent and d,,d,,d,.... a sequence of converging planes (all pas-
sing through 6) in which A is double point. In d a finite neighbour-
hood of A exists containing no points of #'* on one side of the
tangent 6, in this case below b. Considering * is a closed set, this
is only possible when in the converging planes the loop of the curve
(that is the part of the second order) ends up by being situated in
the semiplane of d, which converges towards the lower semiplane
of J. Besides these loops must contract towards A and nothing but
A. Hence for n > some finite number the branches in d, belonging
to the part of the third order depart from A above 6. At first the
concave side of these branches faces 6. Both branches have infinite

8

Proceedings Royal Acad Amsterdam. Vol. XX.

5

114

breadth, hence each has an infinite limiting branch. In the limiting
plane d the branches departing from A at first face 6 with their
conver side (6 is tangent at an ordinary point). But a sequence of
finite concave branches cannot have a convex limiting branch
hence a contradiction is obtained. The possibility might be put
forward that on the converging branches points of inflexion may
converge towards A, but a curve of the third order with double
point has only one point of inflexion *), hence it may be assumed
that only on either the left or the right hand branch points of inflexion
converge towards A and the contradiction remains with regard to
the other branch.

We now proceed to show that not all planes through 6 can
have a double points in A. Again A# and AD are supposed to be
joined by II below @ and AC and AF by IV below a (fig. 3).
AR and AS are situated on II. We found that if @ be turned round
6 in such a way that the right hand side moves downwards, then
at first A remains double point and the branches arriving in A from
below remain situated on IV. In the same way as AC and AF
are connected by IV below a, the branches AF and AS are connected
by a component region of II on the right hand side of B. Taking
in consideration this analogy it is obvious that if 8 be turned round
b in such a way that the lower half moves to the right, then at
first A not only remains double point, but the branches meeting at
A from below a are still situated on II. This may be expressed as
follows: There cannot be a last plane in which the branches are situated
on Ll, and the same can be said of IV.

Let us now consider the set of semiplanes through 5 and situated
below a. If every plane through 5 has a double point in A, then
in each of these semiplanes two branches would arrive in A from
below «a. It was found that if these branches are both situated
on II, then the same holds for the branches in all semiplanes situated
more to the left. In the same way if both branches lie on IV this
is also the case in all semiplanes more to the right. Besides the set
of semiplanes with branches on II cannot have a last element on
the right side and «those with branches on IV cannot have a last
element on the left side. But all semiplanes have two branches
below a, hence the two kinds of semiplanes with branches on II -
and IV respectively must be separated by a semiplane with one
branch on II and one on IV, and this is impossible according to
page 111. Thus the assumption that a// planes through 6 have double
points in A leads to a contradiction.

1) Jue loc. cit. Acad. of Denmark § 5.

115

2. We now come to the second possibility given on page 111,
namely that A is cusp in 2. Again « denotes the plane in which 4
is double point and 6 the line of intersection of « and 3. In the
proposition of § 3 it was assumed that A is cusp in not more than
one plane. Hence if c is a line in «a (—H=b) the point A can never
be cusp in any plane through c. Provided e does not coincide with
one of the tangents in a@ either, the reasoning given sub 1 shows
that A, cannot be double point in any plane through c (except in @).
Considering the possibilities given on page 111 it follows that A must
be ordinary point in every plane through c (except «), with ¢ for
tangent.

Let AF be the cuspidal tangent in 3 (fig. 4). The line c in « we
choose in the same angle of the tangents in A, in which the line

5

Fig. 4.

b is situated. Besides we choose in « a line d through 4A, not being
tangent in A and in # a line e, not coinciding with Af or 6. The
plane through d and e is denoted by d, that through cand Af by y.

The branches meeting at the cusp A in @ arrive from above «
(one on I and the other on [II). We consider a sequence of planes
¥2YpYe¥,--+- turning round AF and converging towards 8. In each
of these planes A is ordinary point with tangent (c,,c,,c, ...) situated
in a. The branches meeting at A in each of these planes arrive
from above « (one on | and the other on III), because the branches

in 8 arrive from above and none of the lines ¢,,¢,,¢, .. . is separated
from 6 by a tangent in A.
Each of the lines ¢,,c,,c,... has, except A, another point in common

with F*. The distance from A to these points cannot tend towards
zero, because if the second point of /* on b is added, they form
8*

116

_a closed set of points to which A does not belong, none of the lines

C,,C,,C,----0 being tangent.
Let ¢,,@,,¢@,... be the lines of intersection of 0 and y,,7,,7,---
respectively. These lines e,, e,,é@,... converge towards e.

In y, a branch departs from A between c, and e,, in y,; between
ce, and e, ete. The distance from A at which these branches can
cross ¢,,C,,C,;-... cannot tend towards zero, hence to make it possible
that in plane ~@ no branch departs from A between bande it is
unavoidable that the branches in the converging planes cross
2, €,,?,--- in points converging towards A. This means that in plane
Ò the iine e would be tangent in A. But considering d does not
coincide with 6 or either of the tangents in «, the plane 0 through
d must show an ordinary point in A with d for tangent. Thus a
contradiction has been obtained.

It has been shown successively that the a priori possibilities 1 and
2 given on page 111 lead to contradictory results. Hence only the
third possibility remains, namely that A is ordinary point in @ with
6 for tangent. But 6 is an arbitrary line in « through A, only
ae to the condition not to coincide with either of the tangents
in A, and 2 is an arbitrary plane through 4, only supposed not to
coincide with «, hence the results obtained so far may be expressed
as follows: In every plane through A which does not coincide with
a and does not contain a tangent in a, the point A is ordinary point
with tangent situated in a.

Thus to complete the proof that « is tangent plane, it only remains
to consider the sections of F* in planes through a tangent at A in «.

In « the point A is point of intersection of two convex arches,
parts of which are indicated by QS and PR in fig. 5. Let a —= DC)
be tangent at A to PR, and let 8 be an arbitrary plane through
1 (=|= a). We assume the senses of curvature of the convex arches
to be as indicated in fig. 5.

In B we choose a line AB (=a) and we consider a sequence of
planes 8,,8,,8,.... all passing through AB and converging towards g,
in such a way that the back part converges towards 8 from the right
hand side (see fig. 5). The line of intersection of « and 8, is denoted
by AC, (a,).

Let the part of #* connecting AP and AS be situated above a
(the other case is treated in a strictly analogous way). In every
plane 8, a branch departs from A above « in the direction AC,
These branches have a limiting set in 8 belonging to the closed set
fF’. Applying the same reasoning given above to show that A cannot
be cusp in any plane, it can be shown that this limiting branch

117

Fig. 5.

in @ departs from A above « in the direction AC. If this branch
formed at A a finite angle with AC, then every line inside this
angle would be tangent at A in every plane except 8, and this is
obviously at variance with the results already obtained. For the com-
plete demonstration it is necessary to know that the linesegment
AC, cannot have points in common with /'*, converging towards A.
Now this is obvious if we remember that when the lines a, converge
towards a, the point A, on AR converges towards A, and that the
point A only counts double on a.

The possibility might be put forward that the branches in the
converging planes pf, have only A as limiting set in 8. Theu how-
ever, it is unavoidable that the converging planes show ovals, con-
tracting from above towards A. Now all these ovals would cross
AB, hence A would be limiting point of #* on AB and the entire
line AB would belong to #*, a possibility excluded at the outset.

Between the branches AP and AS the surface /'* was assumed
to be situated above «, hence the part of #* connecting AS and
AR lies below «. Now if the planes @,, 8,,8,... converge towards
8 from the other side and if we consider the front halves of these
planes (fig. 5), it may be shown in exactly the same way that in
8 a branch departs from A below « in the direction AD.

Taking these results together, it is found that A is point of inflexion
in 8 with a for tangent.

Before passing on to § 4 we shall. prove the following theorem :
It is impossible that a point of intersection A counts double on a

118

line b in some planes through b and single in other planes through b.
To prove this it obviously is sufficient to show that when a point
of intersection A counts double on a line b in a sequence of planes
J,,0,... through 6, converging towards a limiting plane 0, then A
also counts double on b in 0.

Let us imagine two parallel planes, also parallel to 5, situated
close to 6 and on different sides of that line. The lines of intersection
with 0,, 0,....9 are respectively denoted by 5, b,....b and
6," b,"....6". Now if the above proposition were false, then in at
least one of the two planes, for instance the first, there would be
every time fwo points of intersection with 4',, converging together
towards one point of intersection with 6’. This would remain the
same when the plane, parallel to itself, moves towards 5. But then
it is unavoidable that two branches departing from A in 0, which
keep finite breadth, converge towards one single finite branch
departing from A in 0, hence the two sectors of the surface, meeting
at that branch, would be situated on the same side of 0 and this
has been shown to be impossible at the beginning of $ 3.

Meteorology. — “The semi-diurnal horizontal oscillation of the free
atmosphere up to 10 km. above sea level deduced from pilot
balloon observations at Batavia.” By Dr. W. van BEMMELEN

and Dr. J. Bozrrma. (Communicated by Dr. J. P. v. p. Stok).

(Communicated in the meeting of May 26, 1917).

The great regularity of the semi-diurnal variation of the air
pressure in the whole equatorial zone, as well as the constancy of
its amplitude and phase all over the earth prove that the atmos-
sphere as a whole also performs a regular semi-diurnal oscillation.
Above all it was Jur. Hann who brought to light the simple laws
commanding this phenomenon, while Marcuirs proved this pheno-
menon probably to be a phenomenon of resonance by making it evident
that an infinitely thin shell of the atmosphere has a period of oscil-
lation of its own of nearly 12 hours and consequently will resound
to a diurnal disturbance as caused by the sun’s radiation.

From the wind observations on mountain tops in Europe and
North America and also on those in southern British India Hann *)
deduced that this variation of the air pressure is accompanied by a
horizontal wind oscillation possessing an amplitude of some deci-
meters pro sec.

This horizontal atmospheric oscillation may be called an important
geophysical phenomenon. Thus ArtTH. SCHUSTER founded his theory
of the diurnal variation of terrestrial magnetism on the presence of
the above oscillation also in the very upper layers of the atmosphere,
and it might, therefore, be desirable to try and obtain more inform-
ation by observations in the free atmosphere, where disturbances
caused by convection will be of less influence than they must be
on high mountain tops.

Though this has already been done to some extent by means of
cloud observations, no exhaustive nor distinet results could be obtained
in this way.

The only suitable method of observation consists in a series of
pilot balloon observations, which, however, are so complicated that

') Sitzungsber, d, Ak, d. W. in Wien 1908,

120

they can only be applied in those regions where the atmospheric
conditions are quiet enough to retain the series of observations
within practicable limits.

Experience gained by means of pilot balloon observations at
Batavia justified the supposition that here favourable results might
be acquired by such investigation, and therefore during the last
few years we have continued the ascension of pilot-balloons started in
1909. As early as 1912 ascensions at 2 p.m. and 7 p.m. were
added by the first mentioned of us to the ascensions which before
that date usually took place at 8 a.m., in order to gather further
knowledge of the phenomenon of land and sea breezes. The results
obtained induced him to start in 1913 an extensive series of observ-
ations at different hours between 6 a.m. and 6 p.m., more in
particular with a view to study the diurnal and semi-diurnal variation
of the wind. However, at the time he was not able to have nocturnal
ascensions made of balloons carrying lights, like those that first
took place in 1912, but in 1914 we again proceeded to these
nocturnal observations, when the latter of us joined the investigation.

A continuation of these nocturnal observations was checked,
however, by the outbreak of the war in August 1914, so that only
a series of day balloons could be sent up and not before the latter
part of 1915 was it possible to have an extensive series of nocturnal
ascensions made.

From the above it will appear that on account of various circum-
stances, partly not under our control, the ascensions have by no
means been conducted in such a manner as would have been most
suitable to the investigation, i.e. equally distributed over the day
and during the same season.

These drawbacks have partly been neutralized by the following
circumstances. a

For the deduction of the semi-dinrnal variation, which is the most
important and which it is our aim to investigate, it is sufficient to
divide the observations over half a whole-day, as was usually
done before.

The observations were made in periods as little disturbed by
showers as possible, this being even more feasible for half days, or
shorter parts of a day, than for whole-days.

The ascensions took place within the semester May/November,
ie. principally with northern declination of the sun and eastern winds.

Of the greater part of the ascensions the altitude reached by the
balloons has been calculated trigonometrically from double-observation.

The direction and length of the bases were the following :

W. VAN BEMMELEN and J. BOEREMA: “The semi-diurnal horizontal oscillation of the free atmosphere up to 10km above
sea level deduced from pilot balloon-observations at Batavia.”

5 JO

2) fo) 10 O05 OO _05 10 35 JO 05 05 O0 OS

See) sea ast.
: EEE soe Gees ee)
ERS EERE DARE NEBEL
EE INE
SE ee eS...

Ì 4 — et

10 Kan

Sane Jes Si
set
xX

north U.

Amplitudes of the semi-diurnal variation

SQ JT Oo

to

®
el
A
ry ‘GO
©
|

ee | 5
iJ ea Sha Can oa en e <A
o LLLLI | Shs,
XG north 1

Amplitudes of the diurnal variation

Proceedings Royal Acad. Amsterdam. Vol. XX.

121

1912 311° 1860 m.
1914 (night balloons) „ 2

», (day balloons) 348 908 ,,
1915 (night balloons) 296 2640 ,,

During 1913 single-observation only has been applied, and during
the other years the same has been applied mostly for ascensions
between 6 a. m. and 9 a.m., during which hours the lower layers
are little or not disturbed by convection currents.

The material of observation consisted of the direction and velocity
of the wind at various heights and at various hours, derived from
series of consecutive balloon ascensions and the problem remained
to deduce therefrom the diurnal and semi-diurnal oscillations.

To this end it was assumed in the first place that these oscilla-
tions are constant for each height. However, this is certainly not
the case with the diurnal oscillations in the lower layers, because
the latter are influenced by land and sea breezes, the intensity of
which varies from day to day; but the departures from an average
might be classified among all non-periodical variations of the wind.

However, the semidiurnal variation of the East-West-component
might be surmised to possess a great constancy and also the
North-South-component to retain the same sign during at least a
semester.

In the second place the ‘difference method” was applied ; each
time two ascensions, succeeding each other at a short interval, were
joined and by doing so it might be expected that the non-periodical
variations would be eliminated for the greater part; moreover, ascen-
sions with intervals not disturbed by showers were mostly joined
and, as regards the lower layers, with intervals no longer than 24
hours. In case of the upper layers longer intervals were admitted.

Therefore, each couple of ascensions yielded the value of the
following expression for each height and for each of the two com-
ponents (E. W. and N.S.):

ww, (sin t, —sin t,) + y,(cos t, cos t,) + x,(sin2t,— sin2t,) + y,(cos 2t, cos 2t,)

and the values of w,,y¥,,2, andy, were calculated from the whole
for each height and for each of the two components according to
the method of least squares.

When performing the ascensions, which took place by day and
night, and the enormous amount of calculations attending the
deduction of wind components and the making up and solving of
the normal-equations, we have been greatly assisted by the staff of

122

the Batavia Observatory, especially by Observator J. H. Kars, to
all of whom we herewith desire to express our thanks

In order to gather further knowledge of the air currents in the
lower layers, we collected and worked out particulars for each
100 m., up to 1500 m.; upwards of 1500 m. for each 500 m.

The following is a synopsis of the number of cases on which
finally the normal-equations are based. In order to somewhat neutralize
the diminution of those numbers in the higher levels, groups were
formed for the deduction of the coefficients, resulting in the follow-
ing figures:

0.1 km. 406 3.5 km. 302
0.5, 406 Ne 284
0 400 45&5 , 509
Lots 387 55 &6 , 432
ee 357 65 ETL 341
zelde 338 15B&85, 318
Sn 320 Is ee 199

This compensation was also necessary to meet the growing uncer-
tainty of the observation with increasing height and the increase
of the (non-periodical) velocity of the wind.

However, the number of cases of such a combined group may
not be regarded as being equivalent to a non-combined one of the
same number, seeing that the non-periodical velocities of the wind —
here appearing as accidental errors —- are not independent of each
other for successive heights.

It is, however, impossible to ascertain this difference quantitatively,
although it should be remarked that it is principally dependent on
two circumstances.

In the first place on the variability of the wind which the
balloon encounters when ascending. As a rule this variability will be
somewhat less in the higher layers than in the lower ones.

Secondly on the manner of combining the ascensions, i.e. whether”
the differences of the wind have been deduced from the same couple
of ascensions or from various ones. E. g. a balloon sent up at 3 a.m.
is compared up to 6 km. with one sent up at 0 a.m., but higher
up with one sent up the day before at 6 p.m. In this case the
differences for e.g. 4.5 km. and 5 km. will be more dependent on
each other than those for 5.5 km. and 6 km. The letter case will
be more frequent in the upper layers.

123

From the above it will appear that it is impossible to state
this dependency quantitatively to any amount.

The mean errors have been calculated for the groups 4 km.
(6.5—7) km. and (9—11) km. as follows:

4 km. 6.5—7 km. 9—11 km.

North East _ North East. North East
(21) 17 18 20 22 30 32
(1) 30 32 37 40 63 67
(29) 17 18 21 22 32 34
(Ye) 17 18 21 22 30 32

The equality of the mean errors of z,, «‚ and y, and the greater
value of the m. e. of y, are striking; this being a consequence of
the distribution of the ascensions over the day.

The mean errors of @, 4%, 2, and y, are mainly in inverse
proportion to respectively [sin ¢ sin t], [cos t cos t], [sin 2t sin 2)
and [cos 2 cos 2), and the greater part of the couples consisted
of one ascension between the hours of 6—12 or 18 —24 together
with one between the hours of 0—6 or 12—18. For such a com-
bination the values of [sin ¢ sin tl, [sin 2 sin 2t| and [cos 2
cos 2) are indeed about the same, but [cos ¢ cos t] is much smaller,
which may be easily ascertained when making up the limits between
which the values for sin t, cos t, sin 2t and cos 2t are fluctuating.

sin t. cos f. sin 2t. cos 2t.

6b—12" 10 to 0.0 0.0 to—1.0 O0.0over—1.0 to 0.0 —1.0to 1.0

12h—6 h 0.0 to —1.0 —1.0to 0.0 OOover 1.0 to 0.0 1.0 to—1.0
(difference)
absolute value 60 to 2.0 0.0 to 1.0 0.0 to 2.0 0.0 to 2.0

As regards the values determined for y,, these show indeed
— in accordance with the mean errors, which are nearly twice as
large — a greater spreading than those for z,, 7, and y,. The
values determined are shown in the following table and graphic.

The figures in the table (p. 124) show, first: that the influence of land
and sea breezes is distinctly visible in the curves for the diurnal variation
of the N.-component; this influence seems to make itself felt up to
4 k.m. Together with this influence is mixed up the one, which

124

Coefficients of the sinus formula for the diurnal and semidiurnal
wind oscillations at various heights, expressed in centimeters pro sec".

Northcomponent | | Eastcomponent
LEE, 5 4 EN ELK Pa gil okt Vi ry Vas
0.1 km.) —270 | —240 | 65 | 25 || —23 ine eee (oes betes
02°, 1300 108 “eels | eas 3 za Ed
086 12e Ao Parse EMM ee Kn Ee de
OA Le oa 15 Imaal een
05 „ | —193| —65 |° 115 | —40 CL: ee
06 len et 70 | —50 21 lit | —49 | —33
0.7 sb) 2 jes op sij gat 2
68 Pels 23 | —39 || 38 63 | —30 2
09, |—-100; 2 | 5 | —35 ae” en: |) aart SER
rn gs heh arte bs Enne Pab 36 | —31 6
1a ala able eas 68 13 | —46 21
Bt leed ee i ee eN 9 | 19 | —33 39
Ee fs) > A RE TDM 90 | sil alg 4
(B 40 46 | —49 5 | 2 | -23 | —4 | 30
yor 62 43 | —5 Fie (ae es fates pe 25
De Eel 1d 44 16 | —36 | —46 13
Mons ate | is Atel oe 46 ich te ER 5
lp 86 | 47 EL ZL Za aes 38
SA he AE eer | —8 NN Pe en 21
ak bl | 4 | 0 CE 4] =p 43
455 , | 32) —7 eer, be Fa | —95 | —35 | —43 27
55 67,° | 13 |) 6 a) als ese De ate | 43 | —31 23
ee vei 2, —81 =F 1 | a2 | a | —34 2
USS UM Oy pe So 7 Sees en
[es OFS 50 sees

B eed | 137 Je 8 eee

is the result of the intermixing of the layers
(Espy-Köppen effect).
As regards the Kast component the land

of the air by convection

and sea breezes ought

125

to be of small account, the direction of the coastline being mainly
E.-W.; on the other hand the Espy-Köppen effect ought to be fairly
well the same for both components. However, the sea wind blowing
N.-S. exercises its influence on the East West component, in such
a manner that the air above the sea, which is little or not
susceptible to the Espy-Köppen effect, is forced landward, thereby
diminishing its effect above the land in those layers where sea
breezes occur.

The following phases of the diurnal oscillation of the East com-
ponent clearly show the influence of the Espy-Köppen effect.

Phase of the diurnal oscillation of the East component.

© Height Phase | Height Phase
Olkm. | 178° || 0.9 km. | 440
02 , 88 EE te 33
03 , 84 vn 1
iY ge 82 ar ae 12
05 , 85 ‘tele Slee 337
0.6 , 79 14, 346
rs ae 70 Mois BC 321
08 , | 59 (tie A 204

Though in a smaller degree, the curves of the semidiurnal
variation also distinctly show the influence of both phenomena of
land and sea breezes and of the Espy-Képpen effect.

The main reason for this is probably that both phenomena do
not run purely sinus-like, but deviate from it sufficiently to
produce an important semi-diurnal term when applying harmonic
analysis. Indeed both phenomena chiefly originate in insolation to
and radiation from the earth, which do not run purely semidiurnal.

The graphs distinctly show that these influences make them-
selves felt principally below 3 km. and may be neglected above
4 km. Therefore, if we wish to arrive at results for the lower.
layers, not disguised by either of these effects, it will be necessary to
operate far from the land and above the open sea, because here
they are both absent.

Eventually we proceeded to these observations and the last mentioned
of us together with the observator J. H. Kars started a series of ascen-

126

sions from a small coral island in the Java Sea (one most north
of tbe Duizend-Eilanden '); moreover he erected an anemograph
(recording velocity and direction) on the neighbouring Noordwachter
light house (50 m. above sea level). These balloon observations are
still in hand, but of the wind records the results of a few months
are available.

These, however, have shown that there also, i.e. at a distance
of no less than 68 km. from the Sumatra coast, still considerable
land and sea breezes, are found and, seeing that the islet whence
the balloons were sent up is situated respectively 60 km and 70 km
from the Java and Sumatra coasts, the results there obtained will
neither be free from the effect of land and sea breezes *).

We trust to obtain and publish in due course the various results
for land and sea breezes and Espy-Köppen effect, to be deduced
from the foregoing observations, after the necessary reductions have
been completed.

For the present we will deal with the results for the asmospheric
layers above 4 km only.

Then it will at once appear from the graphs that the amplitude
of the diurnal variation must be a minute one for both components,
in any ease too slight to be deduced with any certainty at all from
the results obtained. |

The semi-diurnal variation of the N. component is also a minute
one, however, the scattering of the points of observation is much
less and the curve drawn between these points deserves more
confidence.

On the other hand, the amplitudes for the East component are
much larger, whilst the scattering of the points is also slight.

According to this scattering one would expect mean errors of the

1) The expenses for this investigation have for the greater part been covered
by funds put at the disposal of the Director of the K. Magn et Met. Observato-
rium on the occasion of the dissolution of the Nederl. Ind. Ver. voor Luchtvaart
(Netherlands Indian Association for Aeronautics) with the purpose that these funds
should be utilised for such aerological researches.

2) According to the observations on Noordwacliter during July—November 1916
the amplitudes of the diurnal variation of the E-W and N-S compenents are res-
pectively 83 cm and 54 cm and, therefore are actually in inverse proportion to
the distances to the Sumatra and Java coasts of respectively 68 km and 100 km

54 6
seeing that 88 100° The phases are respectively 249° and 204°, agreeing with
€ 4

‚the phase of the N. component above Batavia at 0.1 km i.e. 222°,

127

amplitudes smaller than those found for 4, 6.5—7 and 9—11 km,
as may appear from the following summary.

Coefficients of the semi-diurnal oscillation, calculated and graphically deduced

(in cm. pro sec.).

N.S. Component E. W. Component
Height §=|_——________ NE En RENE ete ES A
9 J? v2 J2
calc. graph.) , | calc. graph. a | calc. graph. A | calc. graph.) A
4 km. 4 10 | —6 0 0 0; —40 —45 5 43 30 13
Abe 17 OS Pe: 7 Wp =A (sO) top Eke
SS ae ees 2 ee (2 A2 Ii 35 4 23-0003
6.5—7 , —6° —9 3 1 8 |—7| —34 —21 | —7 2 2 0
15-85 , | —27 —9 18 —1 -1 0;—12 —9 | —3 —26 —26 0
9-11 , 9 —4 | 13) —21 —21 0}; 50 30 | 20) —23 —23 0
_ Mean
(absolute values) 10 2 7 3
4 km.) 17 17 18 18
Mean error | 6.5—7 „ 21 21 22 22
| 9-11 , 32 30 34 32

Even if above 7 km the course of the curves may not be quite
reliable and consequently the small deviations as assumed above be
somewhat flattered, this is not the case below 7 km. Thus the fact
that these deviations are so small must partly be explained by the
circumstance that the values arrived at for successive levels are not
independent of one another, because for the greater part they are
based on observations obtained from the same couple of ascensions
for a series of successive heights, and it is especially during un-
disturbed weather that the E-monsoon current exhibits a fairly
amount of homogeneity between + and 11 km.

From the above may further be concluded that the course of the
curves according to the heights possesses a certainty more approach-
ing the above mentioned A’s, but that the curves as a whole
may have a greater error, i.e. that they are drawn either too bigh
or too low on the graph.

Therefore, if their course it can be taken as

may be trusted,

128

fairly certain that, at least as regards the semi-diurnal East com-
ponent, the values for x, and y, above 4 km. respectively increase
and diminish, which comes to this that the phase runs from the
second quadrant to the fourth through the third.

Considering the manner the curves are drawn on the graph, the
following is arrived at for amplitude and phase.

Semidiurnal variation.

North Component East Component

en a oe sae pe eee

Height Ampl. Phase Ampl. Phase

4 km. 10 cm. 0° 54 cm. 144°
5 -% 13» 51 48 5 147
6.3 fa, 108 Sr 152
u aca boe 150 Zet 181
he 10 280 BE el 243
rr Fe are : 20 26 <5 287
1D. 15 Bg * 259 <a 317

As has been mentioned previously, the results obtained for the
lower layers (below 4 km.) are strongly affected by land and sea
breezes and the Espy-Köppen effect and only after tinal reduction
of the observations above the Java Sea we may look forward to a
better insight into the wind movement in these layers.

From the wind records of the Noordwachter lighthouse 50 m.
above sea level, the following particulars have been derived for the
semi-diurnal variation of the East component :

July —November 13 sin (2t—160°)
July—September 23 sin (2t—176°)

This value will probably for the greater part be free from the
Espy-Köppen effect, but not quite from the influence of land and
sea breezes, as the amplitude for the diurnal term amounts to 0.84 m
and undoubtedly it will be accompanied by a semi-diurnal one,
as has been discussed previously for the analogical case at Batavia.

The graphic, however, distinctly indicates the presumable course
of the curves between O and 4 km and this course is opposite to
the one above 4 km.

Height Ampl. Phase
0 km. 28 cm. 180°

Semi-diurnal
pe ons 167

variation
Zn A gene 159

East-Comp.
” 51 ” 151

The surface value of the phase, i.e. 180°, given above, agrees
p 8 8

129:

fairly well with the results for July—September obtained at Noord-
wachter, viz. 176°.

The principal results of the investigation as regards the East
component therefore are:

From O up to 4 km. the amplitude increases and the phase
decreases from about 180°—144°, higher up to 7 km. both return
to the bottom values.

On higher level until 10 the phase seems to increase greatly and
the amplitude to increase again.

When calculating the air displacement (velocity density) we find:

Height Ampl. | Air density |. een
0 km. 28cm: | 1.00 28
ies Sie oth. 0283 | 29
Ben ca 0.75 33
ee a 0.67 34
ae re Oel 33
Avie ABs os ee 055 26
6 ip Ber 0.49 19
ie De 0.44 10
Bd 7 0.40 10
DAR es 03E 10
10 , Zere Id 0:43 13

The foregoing points to slight increase up to 4 km and a
decrease from here upwards. The values above 7 km are too
uncertain to warrant a fresh increase; a gradual decrease is
more likely.

The original decrease of phase for the semi-diurnal oscillation of
the East component in the lower parts of the atmosphere up to an
altitude of 4 km corresponds with that for the semi-diurnal variation
of the air pressure resulting from a comparison of the phases for
Batavia and for the neighbouring Pangerango top.

Oscillation of the wind

85

Air pressure Free atmosph.
Height Ampl. Phase Phase
Batavia 8 m. 1.00 mm. 160° 180°
Pangerango 3025 ,, Ono 5 142° fol?
9

Proceedings Royal Acad. Amsterdam. Vol. XX.

130

However the variation of air pressure on Mount Pangerango will
not be wholly equal to that in the free atmosphere.

The difference in phase for windeomponent and airpressure there-
fore is:

at O km. 20°
PR esos La See

As regards greater heights in the equatorial zone only the obser-
vations on the Misti top (5840 m. lat. — 16°16’)') are available.

Here the phase for air pressure is 100° and for the wind 111°
(presuming that the wind here blows principally from the W.)
making a difference of 11°, i.e. quite in accordance with the value
obtained for the Pangerango.

At the foot of the Misti the semi-diurnal variation of the air
pressure has a phase of 164° (Mollende 24 m) and on the top of 114°,
thus retarding 46° for a difference in height of 5.8 km. With regard
to the Pangerango and Batavia this retardation amounts to 18° for
a difference in height of 3.0 km., making per km. respectively : -

46 18

RS and — — 6

a little difference, seeing the uncertainty in phase for the two
tops. The phase for the Misti has been calculated from one year’s
observations and the one for the Pangerango from three ib but
only from two-hourly observations. 7

If one may go by the value set up for the Misti, one may fix
the phase for airpressure above Java at 6 km. at 112°. This would,
however make a difference with the phase of the windcomponent
at 6 km. of 152°—112°—40°, going by which, we should arrive at:
difference in phase of the semi-diurnal oscillation of wind and air-
pressure

O-kem= 5220?
VA 9
Be AO

and if still higher the phase of the air-pressure continues to decrease,
the difference will increase to 180° at about 10 km.

This result therefore, gives no support to the reliability of our
results as to the renewed increase of phase of the windvariation
above 4 km., in so far that the difference of phase between the .
variations of air-pressure and wind do not remain constant. On the
other hand this support is supplied by the observations of the
Ci-drift made at the Batavia Observatory during 1906—10.

1) Hann, Sitz. Berichte Wien. Bd 118.

131
When calculating the drift components, it was assumed that the
height of the clouds was constant: 11.0 km. for Ci and 10.5 km.

for Ci Cu. *)
For the East component has been found:

Number Number

of cases of cases
6h—7 a.m. 6.1 m.p.s. 114 O—1 p.m. 5.0 m.p.s. 28
7-8 , ie een 74 oe as 405 31
8—9 , ge, 35 23 „ Zen 55
9-10 , ee 36 3—4 , 5.0.5, 30
10 —11 , 5.8) '¥ 28 4—5 , LER 38
11 —12 „ 2.9 » 28 5—6 „ did 140
6-1 , 6. her 34

and consequently for the semi-diurnal term 124 em. sin (2 1+266°). °)

No doubt this result is pretty ‘uncertain, but all the same it rather
distinctly points to a high phase (266°), as did the results of the
balloon observations (10 km. 317°), and it adds weight to the un-
expected result that the phase after originally decreasing up to
4 km., would increase higher up, at first slowly and afterwards
more quickly.

The question now arises, how do the results obtained agree with
the theoretical ones? j

Marcures was the first to proffer a reasonable explanation of the
phenomenon of the semi-diurnal barometrical oscillation, by demon-
strating that the period of oscillation of an infinitely thin atmo-
spheric shell comes “near 12 hours, and consequently will be
excited by the daily disturbance of the temperature. He therefore
proceeds from the oscillation of the temperature and deduces baro-
meter and wind oscillations, but by this process he finds the phases
vastly different to those observed.

This is attributed by Gorp’) to the neglecting of the vertical
velocities, as a consequence of dealing with an infinitely thin atmo-
spheric shell, assumed in order to overcome the otherwise unsurmount-
able mathematical difficulties.

Hence Gorp proceeds on the contrary from the barometer varia-
tion and approximately solves the variations of temperature and
wind (horizontal as well as vertical.)

1) Fieee, Observations Batavia Observatory. Vol. XXX. App. Il.

2) By applying the “difference method” afterwards has been found 42 Sin(2t 263°)
3) Phil. Mag. Vol. 19,

132

In this case it is not necessary for him to proffer an explanation
of the change of the phase with the height, as shown by the variation
of the airpressure.

Of this he only says (p. 37): “This diminution is probably
due in part to the greater resistance to motion near the earth’s
surface; it may be due in part also to a change in the phase of
the semidiurnal temperature variation in the free atmosphere”.

As regards the latter, there are, in effect, only two determinations
available for the free atmosphere, *) i.e. at Lindenberg and at Batavia.
No trustworthy results may be derived from mountain stations.

Phase of the semidiurnal variation.

Lindenberg Batavia
Height - : stor Been E
| Temp. Airpress. | Temp. Airpress.
0 km. | 61° | 136° 63° 160°
EA ay 33 | 22 92 | En
1—2 „ 57 — 146 ze
Sne = | 113 = | 142

At Batavia, therefore, the phases of temperature and pressure
move in opposite direction and at Lindénberg there is no pronounced
sense of change for the temperature.

On p. 39 Gorp writes: “The difference of phase (between tempe-
rature and pressure variation) ought to diminish only slowly with
the height; for latit. 45° this decrease would only be 45° at a height
of 10 km.”.

Lindenberg feebly agrees with the above, but at Batavia the
diminution in difference of phase is too pronounced, the change
in phase (with the height) of the airpressure agreeing indeed, although
to a smaller degree.

Thus Gorp finds (see p. 38, note) from a graphical combination
38° diminution in phase for an increase in height of 2 km.; for
Batavia-Pangerango (difference in height also 3 km.) 18° has been found.

Further from Gorp’s results we find for the North and East
components of the horizontal wind variation at the latitude of
Batavia’s (phase as compared to the one of the pressure variation)

1) See Observatory Batavia. Verhandelingen 4, pg. XXXIX,

133

North comp. East comp.
Ampl. Phase Ampl. Phase
5 cm. 20°? 23 cm. ie

whilst above from the balloon observations bave been deduced 28 cm.
and 20° as being the most probable values for the East component;
a small value being found for amplitude of the North component.

Here, therefore, is a close agreement.

This agreement holds up to a height of 4 km., because the phase of
ihe semi-diurnal variation of the East component follows the one of
the air-pressure, and higher up the amplitude of the North component
also remains considerably smaller than the one of the East component.
E.g. at an average from 4— 5 km. we deduce from our observations:

North component 113 cm. East component 51 em.

51
proportion_— = 4.5, wherens Gorp finds on the surface of the earth

5 and 23 em. i.e. a proportion of 4.6; but as to a reverse in the
change of phase there is nothing to be found in Gorp'’s treatise.

However, are there any indications to explain a similar reverse?

In the first place we looked for them in the vertical motion of
the air, as found by Gorp’s theoretical investigation and have
considered whether, as a consequence of the rising or descending of
air, a returning flow in the upper layers may take place.

According to Gorp the maximum vertical velocity of the semi-
diurnal motion oceurs about the time of the greatest horizontal
velocity viz. an upward motion in the case of Western motion; e.g.
close to the equator at 5 km. he finds an amplitude of the vertical
oscillation of 2 mm. pro sec. If in consequence of this we assume
an average rise of 1 mm. pro sec. occurring over '/, of the
earth’s surface or 10.000 km., then we may also surmise that
this air would again have to flow off through a layer ofa thickness
of say 10 km. This would make an average velocity of 1 m. pro sec.,
Le. actually a velocity of the order required and it may be accepted
as probable that this velocity is in inverse direction to that near
the earth’s surface. Fe

Also, that on account of the ever increasing outflow and inflow
witb increasing height, a gradual increase (up to inversion) of phase
should occur.

Apparently a similar explanation would not seem illogical, but
often such arguments are misleading when we are solving suchlike
problems, so that we will only consider it as a hint in that direction.

Finally we examined the possibility that the inversion of phase

134

might have been occasioned by an accidental distribution of the
errors of our observational results and in reality might not exist.
Let us take it that for the values found the real errors were the

following ones:
Eastcomponent.

| |
| ” | »; |
een | LT PINE See a
ide x Value real | real value real real
calcul. error value calcul. | error value

: | | i
6.5—7 —34 —9 | —2 | 2 —18 | 20 | 142°
ee eee ee 26 —41 | 15 | 124°
Eee she |< he | =d 93 Powe 101°

| |

According to the above the real values for amplitude and phase
ought to be: .

4 km. 59 cm. 133
a ce A ine 148
Wien 5 y 38 „ 144
BEA | Y 20 , 142
8S ae 124

es Cae ies 101

This would do away completely with the inversion and would
carry on the original decrease of phase up to the highest heights
observed, which should be accompanied by a rapid diminution of the
amplitude. 5

If we observe the real errors, it strikes us that, though the ones
for y, are one-sidedly negative, they are of about the same size as
the mean errors.

accepted error mean error
6.5—7 km. —18 18
15-85 „ —41 =
9-11 , —33 32

But, as has been explained above and has been proved by the
graph, the mean error is much larger than the one to be expected.

As regards «x, the real errors accepted for 6.5—7 and 7.5—8.5
km. are small, but the ones for 9—11 km. are much larger than
the mean error, already exaggerated, viz. 52 against 34.

135

This consideration and the support, afforded by the phase pre-
viously arrived at for the Ci-drift, therefore leads to the following
conclusion: ‘Possible but not probable” by which is indicated at
the same time the manner to arrive at a better insight, viz. more
observations. .

The ascensions made on the Java-sea, which supplied a number
of observations up to about 11 km., have partly met the above
requirement, and may conceivably be taken as valid for Batavia.

On the other part it has been decided to commence further ascens-
ions; these are now taking place during the change of the monsoon,
when the velocity of the wind (the non-periodical one) as well as
accidental errors *) will be slight.

SUMMARY.

1. With a purpose of investigating the diurnal and semi-diurnal
oscillation of the motion of the air in the free atmosphere up to
high levels, a great number of pilot balloon ascensions took place
at Batavia during various hours of the whole day.

2. For the first time some series of nocturnal ascensions of bal-
loons carrying lights, were then performed up to great heights,
following the device indicated by C. H. Lei) and as realized by
him in some cases.

3. Afterwards will be published the various results for the lower
layers of the atmosphere up to 4 km.

4. As regards the upper layers up to 10 km., it appears that the
material of observations is insufficient for deducing a diurnal oscillat-
ion, on the other hand that for the semi-diurnal one leads up to
positive results.

5. The amplitude of the semi-diurnal oscillation is larger for the
East than for the North component, the phase of which remains
uncertain.

6. In the lower layers the amplitude and phase seem to be in
accordance with the theoretical results of Gorp (Gorp does not deal
with the upper layers in that respect).

7. Up to 4 km. the phase of the semi-diurnal oscillation of the
East component diminishes, in this respect following the corresponding
phase of the oscillation of the air pressure.

8. Above 4 km. it is obviously probable that the phase again
increases, thus effecting an inversion at a height of about 10 km.

1) Circumstances have prevented till now (May Ist) from doing so.
2) Quart. Journal. R. Meteor. Soc. 1909.

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
| FE ae

PROCEEDINGS

VOLUME XX

NE

President: Prof. H. A. LORENTZ.
Secretary: Prof. P. ZEEMAN.

(Translated from: “Verslag van de gewone vergaderingen der Wis- en
Natuurkundige Afdeeling,” Vol. XXV and XXVI).

CONTENTS.

J. J. VAN LAAR: “On the Critical Temperature and Pressure of Mercury and Phosphorus”. (Com-
municated by Prof. H. A. LORENTZ), p. 138.

J. M. BURGERS: “Adiabatic Invariants of Mechanical Systems.” I, II and III. (Communicated by
Prof. H. A. LORENTZ), p. 149, p. 158 and p. 163.

J. M. BURGERS: “The spectrum of a rotating molecule according to the theory of quanta”. (Com-
municated by Prof. H. A. LORENTZ), p. 170.

H. KAMERLINGH ONNES, C. A CROMMELIN and P. G. CATH: “Isothermals of di-atomic substances
and their binary mixtures. XIX. A preliminary determination of the critical point of hydro-
gen”, p. 178.

H. I. WATERMAN: “Amygdalin as nutriment for Fusarium”. (Communicated by Prof. J. BOESEKEN), p. 185.

H. ZWAARDEMAKER: “On the behaviour of the Uranium-heart towards Electric Stimulation, as
investigated by Mr. M. DEN BOER, p. 189.

N. VOORHOEVE: “Ahypothesis concerning the mutual relation between some hereditary abnormalities
that occur combined’. (Communicated by Prof. I. K. A. WERTHEIM SALOMONSON), p. 194.

P. EHRENFEST: “In what way does it become manifest in the fundamental laws of physics that
space has three dimensions?” (Commumunicated by Prof. H. A. LORENTZ), p. 200.

W. J. H. MOLL and L. S. ORNSTEIN: “Contribution to the research of liquid crystals. II. The
influence of the temperature on the extinction; further experiments upon the influence of the
magnetic field”. (Communicated by Prof. W. H. JULIUS), p. 210.

C. J. Vv. D. HORST: “The Fore-brain of Synbranchidae”. (Communicated by Prof. MAX WEBER), p. 217.

W. DE SITTER: “On the curvature of space”, p. 229.

F. M. JAEGER: “Investigations into PASTEUR’s Principle of the Connection between Molecular and
Crystallonomical Dissymmetry. III. Racemic and Optically Active Complex Salts of Trivalent
Rhodium”, p. 244.

F. M. JAEGER: “Ibid. IV. Racemic and Optically-active Complex Salts of Rhodium-tri-oxalic Acid,” p. 263.

F. M. JAEGER: “Two Crystallised, Isomeric d-Fructose-Penta-acetates”, p. 280.

F. M. JAEGER and R. T. A. MEES: “On Complex Salts of Ferri-Malonic-Acid”, p. 283.

F. M. JAEGER and JUL. KAHN: “On some isomeric, complex cis- and trans-Diethylenediamine-Salts
of Cobaltum, and on Tri-ethylenediamine-Zinc-Chloride”. p. 292.

B. P. HAALMEIJER: “On Elementary Surfaces of the third order.” (Second communication). (Com-

« municated by Prof. L. E. J. BROUWER), p. 304.

E. H. BüCHNER: “The boiling point line of the system: hexane-nitrobenzene.” (Communicated by
Prof. A. F. HOLLEMAN), p. 322.

1. K. A. WERTHEIM SALOMONSON: “Photography of the fundus of the human eye”, p. 326. (With one

plate).

Proceedings Royal Acad. Amsterdam. Vol. XX.

Physics. — “On the Critical Temperature and Pressure of Mercury
and Phosphorus.” By Dr. J. J. van Laar. (Communicated by
Prof. H. A. Lorentz).

(Communicated in the meeting of April 27, 1917).
A. Mercury.

1. In a previous paper’) I calculated the value 1260° abs.
(987° ©.) for the critical temperature of mercury, and 192 atm. for
the critical pressure.

We namely calculated as most probable value from the densities
of the compounds HgCl,, HgBr, and Hgl, (loc. cit. p. 7—8)for bz
for mereury per Gr. atom the value 150.107, and further for Vaz
from the values of 7%, found by Rotisanz (Z. f. ph. Ch. 87, p. 153)
for these compounds, the mean value (likewise per Gr. atom)
11,0.10-2. (loc. cit. p. 8). With these values of Vaz and bz we
further calculate from the formulae

es 8 EN, A
TT : Pk = 35 be
with R—=1:273,1, 4=?'/,, — 0,964, n = 2, the values
Ty == ABD pr — 192 atm.

All this on the supposition that (saturated) mercury vapour at
T, is bimolecular (n= 2); a supposition, to which the course of
the vapour tensions between 0° and 500° C. leads us necessarily.
(Cf. also loc. cit. p. 12—16).

2. Since then the vapour pressure observations of CAILLETET,
Corarprav and Rivikre (1900) have become known to me, which
go up to 880° C. With

Ae:

ule he
log = € hare 1), or f cle log PRI log p ai }
p 1 :

we calculate for f7), the values 3004, 2979, 3233, 3107 (mean 3080)
from the values 8,0, 22,3, 50, 102, 162 atm. at resp. 500°, 600°,
700°, 800°, and 880° C.; always from a corresponding couple of

mf

1) On the Fundamental Values of the Quantities b and Va etc. Il. These Proc.
of March 25, 1916, p. 4 and 9.

139

values of 7 and p. (f between two observations supposed to be
constant). And for f + log pj. the values 4,789, 4,760, 5,022, 4,905 ;
evidently the value 4,880 for f + log py corresponds with the mean
value 3080 found just now for 77%.

The correction quantity 4 is represented (compare also loc. cit.

Be tie pred fc
p. 4) by the expression 4 — Eel En And as y, the coeffi-
cient of direction of the straight connecting line between Dj and D,,
is given by the approximative formula 2y = 1+ 0,04V 7}, we find,
with about W/ 7), = 34,5, for 2y the value 2,38 (which at the same
time indicates the ratio 0,:6,), i.e. 1,19 for y. Therefore 4 beco-
mes = 0,936, i.e. somewhat smaller than we assumed formerly
(0,964, see above).

If now w is the dissociation degree of the Hg,-molecules at 7%,
there are on an average n = 2: (l + 2) atoms available per mole-
cule (the association degree of Hg,), and we have evidently:

, gl Nae ime AE 10— ne 2
Ty = 218 X = X 0,986 x X
150: 1059 oy eee

(11(1—#)+ 362)? . 10-4

(150)? . 10-10
because the quantity Vaz is = 11.10— per Gr. molecule for Hg
molecules bound to Hg,; much higher on the other hand for the
free mercury atoms, viz. 36.10? (ef. IV in These Proc. Vol. XIX,
p. 317, where we found for bismuth, which stands in the same
horizontal row as mercury, 35,6), so that War, becomes on an
average — 11(1—w) + 36 per Gr. atom.

Hence we find:

1
ES 57 0030
(

(11(1 —) + 862)? |
la
log py == 0,1877 + 2 log (11(l—e) + 362) |
with f7), = 3080, f + log px = 4,880 therefore (see above) :
f = 4,692 —2 log (11(1—a) +362),
so that « can be found from

T, = 10,10 X

10,10( )? _ 3080
Rn et £60229 ater We
or from
[4,692 ~ 2 log (11(1—2) +-36.x)] X (11(1—«)+ 365)" Ha) = 305.

If now «=O for 7, i.e. everything bimolecular, this becomes
whe ia; 11:
316 = 305,
10%

140

which, considering the uncertainties in the values of bj, and Vaz
calculated from the halogen compounds, is already very fair agree-
ment. With Yaz = 10,8 we should have found 306 == 305. -

With z=0,1 we should find 393= 305 with Va,=9,7 + 3,6=13,3,
which therefore does not agree at all. And for still higher values
of w the difference would have become greater and greater.

The least dissociation of Hg, to Hg, for 7% is accordingly quite
excluded, so that we must assume that the mercury is perfectly
bimolecular there. The liquid mercury is a fortiori bimolecular, the
saturated mercury vapour on the other hand will be Hg, only at
lower temperatures (where the large volume prevails); at higher
temperatures, however, in consequence of the decreasing volume, it
will become Hg, in ever increasing degree (cf. also loc. cit. p. 3—4).

3. With the somewhat lower value for 4, viz. 4 = 0,936, and
with 6, = 150.105, Va, = 10,8 . 10-2 we now find:

Lp AO 10541166 = 1178" aes. = 9059-C-

It is therefore seen that at 880° Carrerer c.s. have been exceed-
ingly near the critical temperature of mercury, viz. + 900° C.;
it lies possibly scarcely 20° higher than the highest temperature at
which they have still carried out a vapour pressure determination
(880° C.).

Would it not be desirable in virtue of this fact to determine the
critical temperature of mercury at last experimentally? As it was
possible. to RorinJanz to determine those of the mercury halides,
which lie resp. at 976°, 1011° and 1072° abs., it may also be
possible to come to + 1200° abs. (at the melting point of silver,
viz. 960° C., we are already far above the critical temperature of
mercury).

The only scientist that calculated a fairly plausible value for 7%
for Hg, was Happen (Ann. der Ph. (4) 18, 351 (1904), who gave
1370° abs., i. e. only 100° higher than was calculated by me in
1916 (1260° abs). But the values of KöÖNIiGsBERGER (1912), viz.
1270° C. (1543 abs.) and of BeNper (1915), viz. > 1500° C. (>1773°
abs.) are very certainly too high. As f 7), not far below 7% amounts
only to= 3080 (this follows with absolute certainty from the
vapour pressure observations, see above), the value of the factor f
would at high temperatures be only — 2, or even 1,7, instead of
approaching 2,8 (see below), whereas this factor is already about
2,7 or 2,6 between O° and 100° C. — still apart from the fact that

141

then we should find quite improbable values for aj, and 55.
We calculate for pz:

log px = 0,1877 + 2,0668 — 2,2545,

pr = 180 atm.

In order to get somewhat more certainty concerning the values
of Tr, and pj, and also of Vaz and bj, we can still make the
following calculation.

From / 7; = 3080, f + log pp = 4,880 follows namely with f
resp. = 2,5, 2,6 and 2,65:

ye oo ebr hade log pe 2,300 >|: pe = 240
2,60 1185 2,280 191
2,65 1162 2,230 170

From 8n.6,= RT;,:p, follows then with n=2, R—=1: 273
for bx (per Gr. atom) b4= 7): 4370 pz, i.e. br resp. = 117, 142
and 156.105.

With these values for 6, we get then az resp. = 95,2, 111,3,
119,6.10-+, i.e (az: (per Gr. atom) resp. = 9,76, 10,55 and
10,94. 109, from 7% = 151,5 az : bz.

It appears from this most convincingly that — as bj, must lie in
the neighbourhood of 150.105, Yaz in that of 11. 10-? — the
value 1232 abs. is pretty well excluded for 7, and that we have
therefore the choice between from 1185 to 1162. Retaining
- bg =150.10-5, we get Wa, —10,77 .10-?, as we assumed above
(rounded off 10,8), and further:

Bi Pa A abs SINE pd atm.

But it is also possible that 7% becomes somewhat higher, e. g.
1185°, to which corresponds pz = 191 atm. Then the value of Vaz
„would, however, be still lower than that which we calculated from
Hg I,, viz. 10,7 .10—? (loc. cit. p. 8), and that of bj, would become
= 142.10-5, ie. equal to that which was calculated from this
same compound. But these were the very lowest values. Those
which were calculated from Hg Cl, and Hg Br,, were both higher
Woest. p. 6).

1) With regard to the value of a, I have convinced myself that from the com-
pressibility of mercury at 0°, 110°, and 192° C, follows about the same value of
a as was calculated from the mercury halides for az. And with regard to Dz,
even on the supposition that Hg, for liquid mercury has the stoechiometric pro-
perties of mercuro compounds, a value would follow from the densities of Hg,Cls
and Hg,Br. for Hg (per Gr. atom) in Hg, which is only little less than was
found for Hg in the mercuri compounds. (viz. about 140, 10—4 instead of 150, 109),

142

4. For the successive values of vAN DER Waals’ vapour pressure
factor f we now tind between 0° and 886° C. from the following
vapour pressure observations (at lower temperatures mean values)

0° 100° 200° 000 4009 500° 550° 600° 650° 700°
b= 24..10-3. 0,27. AT AS ne LONT LO ate 13,8 © 22,3 934 50

750° 800° 850° 880° C.
72 102 137,5 162 (atm)

the following values.

log P* — 8.7534 5,6903 3,8998 27427 1,9410 1,3497 1,1129
0,9045 0,7213 0,5538 0,3955 0,2442 0,1145 0,0433
4 —1= 3,2930 21421 1,4778 1,0454 0,7415 0.5162 0,4241
0,3425 0,2608 0,2045 0,1457 0,0923 0,0436 0,0165

Fig 266°. . 2664-264." 262°" DiEP 1261 (2362
(min)

2,64 2,67 Zyl 2,71 2,65 2,63 2,62

fi 012- 612 -- 608° ~ 604% <6,03 ~ 16/02 "20604

6,08 615 6,24 6,25 609 6,05 6,04
The value of p at 0° C. has probably been taken still somewhat
too bigh; we assumed 0,00024 for it (Hertz gave 0,00019, v. ».
Praars 0,00047). And especially for the values at the higher
temperatures the slightest error in the vapour pressure will make
itself greatly felt in the caleulated values of f; the same thing
holds with respect to only an minimum error in the calculated
values of pj; and 7}. If we assume e.g. pz — 180 atm. instead of
179 atm., log py becomes 24 units in the last decimal greater, which
would cause the values of f at the highest three temperatures to
rise immediately to 2,67, 2,68, and 2,77 (with nep.log.: 6,15, 6,17,
6,38). Then 54 would get the value 149. 10? instead of 150. 105,
and Vaz would become 10,74 instead of 10,77. But in any case
the course of f is pretty regular; this quantity decreases from about
2,66 at O° C. to 2,61 (the minimum value) at 500° C., after which
it increases again to 2,7 or 2,8 at the critical temperature. The

minimum lies at 7’ = 0,66 7

The value of f at the critical temperature might have been ex-
~ pected higher than 6,4 or 6,5 (nep. log.), since fx is equal to 8y
according to our former considerations, when neither a nor 6 are
functions of the temperature. Now y is about = 1,2, hence 8y would
be = 9,6. But we should bear in mind that exactly in the case of
mercury « would be a temperature function in a high degree. For
only through the predominant influence of the volume does Hg
become Hg, at higher temperatures, whereas if the temperature
influence only could make itself felt, Hg, would dissociate to Hg,,

143

which would cause Va to rise from about 11 to 36. The value of

da : , \
(=) at 7% will, therefore, be very great positive, and this will
;

d "
lower the value of (3) at the critical temperature considerably.
ry’

Henee-the value. of fj, = G Ee) will also be considerably lower than
k

the normal value, In our case the expected value is diminished
from 9,6 to 6,5.

As Tr is now found = 1172° abs. instead of 1260° abs., as I
calculated before, the ratios 7%: 7, and 7%: 7 will also be
somewhat lower. For the former we find 1172: 630 — 1,86, and
for the latter 5,0. So high a value for the ratio 7%: 7), is only
found for He (5,2) and for Bismuth (5,5) of the elements calculated
by us up to now. But we shall soon see (in a following paper),
that 7%: 7%, is also = over 5 for tin, lead and the alkali metals.
A pretty high value of 7%: 7, (i.e. > 1,7) is also found for Argon,
Krypton, Xenon, Niton (1,73—1,79), for the Halogenides (1,75 to 1,72),
for O, (1,71), for P, Sb and Bi (1,75—1,77), but 1,86 was not
reached yet. Among the compounds we mention HCI (1,71), HBr
eS) tl oho ESO ASS: and Se - (1,77) PH. (ly toy,
CS, (1,71), CH, (1,75), H. COH (1,97), while the three mercury
halogenides, examined by RorinJanz, give 1,69 to 1,71.

5. In conclusion I will still point out that 6, = 150 X 105 does
not only ensue from the densities of the mercury halogen compounds
(see § 1), but also from the density of mercury itself. For it follows
from Dewar’s determinations (1902), who found the value 14,882
for the density at 188° C., and those of Marrer, who gave 14,193
for the density at -—39°, that the limiting density D, at about
—250° (below this no appreciable volume diminution takes place)
will amount to 14,46. 200,6 Gr. of mercury then occupy a space
of 200,6 : 14,46 = 13,87 cem., i.e. = 13,87 : 22412 = 61,9 .10-° in
so called normal unities. This is, therefore, 4, = v,. Now according
to one of our formulae bz: 6, = 2y, hence 6, = 61,9: 10 Xx 2,4 =
= 149.10-5, quite identical to the value which we found above
(§ 4) with p=180 atm. We may, therefore, put the value of 0, for
mercury at 150 10-5 with great certainty.

The value of Dj is found from the formula Dz = D, : 21 + y) =
= 14,46: 44 — 3,3.

Recapitulating we probably have for mercury :

144

Ty, == 11722 tabs. (900°C) ae pr = 180 krent dee acd
bj = 149 710 SHS RO 14 0S Ari EE fe= 164
Te: PAHO 86. Sa lee

These values are pretty certain; the critical temperature lies
— taking the vapour pressure observations between 500° and 880°
into consideration — almost as much as 100° lower than I had
calculated in 1916, and only 20° higher than the highest temperature
at which CaiLLeter c.s. have carried out their vapour pressure
determinations.

That it seems at /ower temperatures that the critical temperature
of mercury lies much higher (as among others KÖNIGSBERGER and
BENDER supposed), is owing to this that the expansion of liquid
mercury is abnormally small at those temperatures (0°—300° C.).
But this is to be attributed to still unknown disturbing circumstances
(association e.g.). As little as we may conclude to entirely faulty

critical values from the abnormal expansibility of water in the —

neighbourhood of 4° C. (and still far above it), or of Helium (where
likewise a maximum density was observed), may we do-so for
mercury. We shall come back to this later on.

B. Phosphorus.

1. From the vapour pressures of liquid phosphorus, found by
himself between 169° and 634° (These Proc. of Oct. 17, 1914 and
Jan. 18, 1915; Z. f. ph. Chem. 88 (1914), 91 (1916)), Smits extra-
polated the value 82,2 atm. for the critical pressure, on the assump-
tion of the value 695° C., found by Want for the critical tempera-
ture of Phosphorus. It is easy to see that this value is too low. If
from the vapour pressure formula

Dk n ih " : ry i)

eee == (F — 1) with 7; — 695 + 273,1 = 968,1 we, namely,
he

calculate the values of f at ten different temperatures, we find with

pr = 80, resp. 90 atm.:

4

634° C.
58,6 atm.

t—= 169°,0 210°,0 252°,0 29896 355°,7 40993 | 5049 _ 5509 5039

p= 004°. 0,20: O54> “| (ISS BE (7,36). 1 2324 033 0ON AAD

p 3,3010 2,6021 2,1107 1,7632 1,3143 1,0362 0,5376 0,3846 0,2517 0,1352
Ep - 3,3522 2,6532 2,22185 1,8144 1,3654 10874 0,58875 0,4357 0,3088 0,1863
a —1= 1,1898 1,0039 0,8437 0,6934 0,5396 © 0,4187 \ 0,2458 0,1762 0,1178 0,0672

| PTT = 2050 — 957 92a" As (2,47) 2,19-°\* 24a! Leder leak |

962) ye Gas 263417 20E Sense (200 2,40" 247 Saen en

(min)

i

145

From this it is evident that the first row of values (which
correspond to pp —= 80 atm.) cannot be correct. For then / would
steadily decrease from the value 2,77 at 169° up to the critical
temperature, where the value would even become < 2!, whereas
it is known that f always passes through a minimum at T = about
0,7 or 0,8 7), after which it increases again to Ty. It is easily seen
that also with pz = 82 atm. the decrease has not been checked up
to Tj, and that not until 90 atm. is reached a suitable and possible
course for f is obtained. A further calculation, about which presently
more, has even taught me that the correct value of pj is still
somewhat higher, viz. about 95 atm. at least when we continue
to assume 7%. = 968,1.

That Smits extrapolated a too low value for pj, is owing to this
that he used an invalid formula for this extrapolation; a formula
namely, which is only valid at temperatures that lie far from the
critical temperature — and which can therefore not serve to extra-
polate up to the critical temperature.

For in the well-known relation of CLAPEYRON

dp A
dt Tv
Av = v,—v, can be replaced by v, only at low temperature, disregard-

ing the liquid volume; and only at /ow temperatures v, = RT’: p may
be put, on the assumption that the vapour follows the law of Borin
— so that only then this formula becomes:

d log p 2

Rat Pe
in which A represents the (total) heat of evaporation. In imitation
of so many other authors, who are still of opinion that this last
formula is of general validity, because van ‘tT Horr and others
always used this limiting formula for researches where the above
mentioned conditions are fulfilled, Smrrs assumed that the formula
with dlog p would continue to be valid up to the critical temperature,
when it was only assumed thas 4 decreases linearly with the
temperature up to 7. This now is certainly pretty accurately
fulfilled at dower temperatures, but near 7% 4 suddenly decreases
rapidly and becomes —O at the critical temperature. On Surrs’
assumption of linear decrease, however, 4 would retain a large
finite value still at 7%!

But we need not speak about this any longer, because, as we

observed, the whole formula, the linear decrease of 2 included, holds

146

only for lower temperatures. And this shows at the same time the
incorrectness of the extrapolation carried out by Smits.
At all temperatures, however, vaN DER Waals’ vapour pressure
holds, which may be written in the form:
: ST:

109 aT at, SOP PRN erie ee eg a
in which / is still a temperature function. When we compare this
expression with the integrated formula d log p= ete, on the assump-
tion of 2—= 2, —y RT, i.e. with
(A, s R) + pT log T

0

logp= C— —~ —g log T= C — : b
og p rr he 7 (6)
it appears that in the formula used by Smits, which — we repeat
it — holds only for relatively low values, the constant C will be

=f, + log pr, and that 4,=/,RT;. But though the form of the
last formula shows resemblance with Van per Waats’ formula, the
numerator of the term with ‘4/7 will be in no connection at all
with 4 at higher temperatures, as 2 will approach O at 7%, while
the numerator mentioned remains finite, and is virtually = f 7}, accord-
ing to Van DER Waats’ formula.

2. On the assumption of the quadratic relation
NY ye TT (Tr_—T)
Jedi Sayer: EE PT
for the portion of the vapour pressure curve between the minimum
and the critical temperature, I calculated the values a = 11,71,
B = 26,62, f,= 3,77, pr = 95,3 for the four unknown quantities
a, 8, fr and pj from the four vapour pressure observations at 504°,
550°, 593°, and 634°.

However — neither the values 7% = 968,1, Pk = 95, nor even
with the somewhat lower pressure 90 atm., can satisfy us. It is
namely almost sure that at 695° C., according to the determinations
of the density of Stock, Gipson and Sram (1912), the phosphorus
vapour is still quite normal, i.e. = P,, even at the low pressure of
75 m.m. And this will a fortiori be the case at a pressure of 80
a 90 atm. (ie. at a fotal pressure, internal and external pressure
combined, of f7, Xx 80 or 90 = + 640 or 720 atm.). The same thing
follows also from PREUNER and BROCKMÖLLER’s determinations (Z. f.
ph. Chem. 81, p. 159 (1912)).

From the formula 6, = RT),: 8p, the value 465. 10-3 would now
follow for 6, with 7;,=968,1, pz = 95,3; and with p, = 90 the
value 492. 105. Both most probably too low, as 4 X 140 = 560.105
may be expected.

147

The only way out is that the critical temperature 695° C. deter-
mined by Want is about 20° or 25° too high, and that 675° or
670° C. is perhaps the correct temperature.

When the above given calculation (with f= fe —aetc.) is now
repeated for different values of 7%, we find e.g.

Meene TTP = 926,62 «f= 3,77 | Py 008 | b, = 465
6809 „| 10,31 25,80 3,485 83,6 | 522

| | | |
6709 „| 9,40 | 25,26 | 331 TR | 560

Only the last value of 7}, viz. 670°, gives a plausible value of
bx with the corresponding value of pj == 77,1 atm. But as this value
for phosphorus is not perfectly certain, it is also possible that an
intermediary value, e.g. 675° C., must be assumed. The value of
px would then become about 80 atm., and 5, a little more than
540.105. With a value 2°/, lower, ie. 948,7 abs. instead of
968,1 abs., or 675,6 C., and with pj, 80 atm. we should find the
following values for f for the same ten temperatures as in § 1.

|

Pr. |
En 39010 2602142, 17070. <1,7632 .1,3143 1,0362 0,5376 0,3846 0,2577 0,1352

Tj.
aol — 1,1460 0,9638 0,8068 0,6595 0,5088 03903 « 0,2209 0,1527 0,0954 0,0459
Pees T0 *260 7 261 PBT BBs AEL BEE A2 TON GR

(min)

The value at 409°,3, still determined by Smits with difficulty
between the two series of observations, is rather divergent, as was
to be expected. We find namely 2,65 instead of 2,50 about; but
the other values all form actually one single series, so that the
liquid white phosphorus can be considered with perfect certainty
as the metastable continuation of the liquid red phosphorus below
the triple point at 589°,5 — which’has been proved irrefutably
by Smits.

We have, therefore, with some probability for the critical data
of phosphorus :

Breder abs a= 675° C.3 pp E80 -atm,

At 7; the value of f will then approach 3,4 (with nep. log. it
will approach 7,8). For bp we tind 542.10 ?, ie. 185.10” per

148

Gr.atom'); and for ap= RT, X (27:83) X by the value 0,0665
with 4 = 0,955 (y =1,09); cf. also IV, These Proc. of June 24,
1916, p. 307), so that Vaz, becomes = 25,8, i.e. 6,45 . 10? per Gr.atom.

The value 7,8 found for fj is somewhat lower than would follow
from fr = 8 y, viz. 8,7, or 3,8 with ordinary log.

The minimum. lies: at (TT): Ty = @: 28 = 0193) ie. at 7 =
= 0,81 7, or 183° lower than 7%, hence at 492° C., only slightly
below 504° C. And this minimum value of / will evidently be
= Jr — (a? : 48) = (97,12 :102,1) — 2,45. We found above
the slightly lower value 2,43 with the valne of 80 atm. for pz,
which had been taken somewhat too low (80,7 atm. corresponds
namely with 675°,6).

Fontanivent sur Clarens, March 1917.

1) From the critical data of PH3 would even follow 131 with H = 34 (cf. I,
These Proc. of Jan. 29, 1916, p. 1224). But these data are ere not absolu-
tely accurate either.

Physics. — “Adiabatic Invariants of Mechanical Systems.” 1. By
J. M. Bovrerrs. Supplement N°. 41e to the Communications
from the Physical Laboratory at Leiden. (Communicated by
Prof. H. A. Lorentz).

(Communicated in the meeting of November 25, 1916).

Introduction.

During the past year’) the theory of quanta has made great
progress by the study of a class of mechanical systems which are
characterized by the following property: the integral of action :

WASE; dt
(T’: kinetic energy) separates into a sum of integrals each of which
depends on one of the coordinates only :

"Vk
EN CATO

In general each coordinate can only move up and down within
a certain interval (which is given by roots of the equation /’, = 0) *).

From the formula given for W it follows that the momentum
corresponding to the coordinate qx is equal to:

LE V F(q),

anr = fete. cages ale) aera. Cae

For this class of systems Epstein and other investigators use the
following equation as the principle for the introduction of the quanta :

e

hence :

Ted op == Ni Be dt ne esn UO

1) K. ScrwarzscHiLD : Sitz. Ber. Berl. Akad. 1916, p. 548.

P. Epstetn: Ann. d. Physik 50 (1916) p. 490; 51 (1916) p. 168.

P. DeBije: Gott. Nachr. (1916) p. 142; Phys. Z. S. 17 (1916) p. 507, 512.

A. SoMMERFELD, Ann. d. Physik 51 (1916) p. 1; Phys. Z. S. 17 (1916) p. 491.

2) The radical sign has been written in accordance with the most common cases ;
the function Fx then becomes a rational function.

3) Comp. note 2 p. 150.

150

where during the integration g, moves up and down once between
its limits (nz being an arbitrary integral number).

Now Prof. P. Eurenrest *) has pointed out the great interest for
the theory of quanta of the so-called Adiabatic Invariants, i.e.
quantities the value of which does not change if the system is trans-
formed in an adiabatic way (definition by Eurenrist, Le. and below
§ 1) from one state of motion to another. He has shown that for
rigorously periodic systems the integral of action, extended over a.

1
full period P=~—:
Yr

ns ST
{a 2t=P are
yv

0

does not change its value during an adiabatic variation of the system;
and also that both the quanta-formulae introduced by SOMMERFELD
for the elliptic motion are related to adiabatic invariants. As
Prof. Enrenrest has already remarked it would be very interesting
to inquire whether the above mentioned quantities /; are also adia-
batic invariants. In the following lines I will try to show that this
is the case.

§ 1. General considerations about the adiabatic alteration of a system.

Suppose that the mechanical system under consideration possesses
n degrees of freedom ; the coordinates will be denoted by q,..- qn;
the momenta by p,...p,- H be the Hamiltonian function, expressed
in terms of the g and p. For the present we will only suppose that
no coordinate or momentum can increase indefinitely, but that all
of them will remain between certain limiting values (to be deduced
from the equations of motion) (supposition A). *)

In the function H besides the q and p certain paramaters a occur :
e.g. masses, electric charges, the intensity of a field of force. We
may imagine that during a certain time these parameters are changed
infinitely slowly. A reversible adiabatic variation of the system will

1) P, EHRENFEST, these Proceedings Vol. XIX (1), p. 576, 1917.

2) In the problems treated by Epstrern and others an azimuthal angle p occurs,
which can increase indefinitely. The configuration of the system, however, is periodic
with respect to this coordinate; an increase of p by 27 here takes the place of
the up and down motion of the other coordinates. Apart from this the further
treatment remains substantially the same. (lt is also possible to introduce q = sin?
as a new variable — cf CHARLIER, Die Mechanik des Himmels I p. 112 — in
order to return to the general case).

151

now be defined as consisting of a variation of the a, which is
characterized by the following properties :

(I) The variation is infinitely slow as compared to the motions of
the system; or more precisely: in a time during which every,
coordinate has moved up and down many times between its limiting
values, the a have increased or decreased by an infinitely small
quantity of the first order.

da
(IT) hae approximately a constant.

(III) During the variation the Hamiltonian equations:
dgn OH dpp 0H
one ae Oe ee (4)
remain valid *).
If the motion is transformed from a given state in which the a
and the constants of integration of the equations of motion have
certain values in an adiabatic way to another state, the values of
these integration constants will change. For supposing :
e=S(q PsA?) |

to be an integral, we have during the adiabatic process from (III):
de Oc das
Po tda Bo ee ee

For simplicity it will be assumed that only one parameter is
varied; then the total increase of c ae 1 be:

i de da En
fant ee meee Ea

de ;
where the line over — denotes an appropriately taken mean value’).
t

According to (II) we may take the mean with respect to the time,

ij
whereas on account of supposition (1) 5e may be replaced by the
a

de
value of Sn for the wndisturbed motion.
a

The increase of a function g(c,a) of the integration constants and
the parameters during the adiabatic change is given by the formula:

1) This is for instance always the case if only the a which occur in the
function of forces are varied. — In a system possessing cyclic coordinates the cyclic
momenta may appear as parameters; and the same holds for the cyclic velocities,
if instead of H the function R= H — z= peyel. q eycl. is introduced.

2) Supposition (A) was introduced in order to make possible the définition of a
mean value of this kind. :

152

— 99 0g
— = ees gis ee
dg S50 + gd : (7)

If.for such a function g: dg = 0, it will be defined as an adiabatic

invariant *).

§ 2. If the equations of motion are completely integrated it is
always possible to express the momenta p,... pn as functions of
the g, the « and nm constants of integration a’ ....a"*). In accordance
with what was said in the introduction we shall specialize to
systems where the expression of ps contains only the coordinate
gx (together with the a and a):
; pk = VF RG: ot a) oie ea ee

(Supposition 2).

In connection with supposition (A) of § 1 the functions #%, are
assumed to possess the following properties:

(1) Each equation /% (qx) =90 has (at least) two simple roots
Er and mu; for values of qx between these roots #7 > 0.

(2) At a certain instant gs lies between &, and 17.

It can then be shown that gz remains in this interval, and that
it performs a so-called libration’) °)..... . (Supposition A’).

The following integrals will now be introduced, which will be
called “phase-integrals”

In = (dar pr = | dar VF x (ge) = Ie («era (@)

1) Integrals c=f which are independent of the a are themselves adiabatic
invariants (cf. form. 6). As an instance: in the motion under central forces the
integral of the moment of momentum.

2) This may be accomplished
for instance by the integration of
the partial differential equation of
HAMILTON-J ACOBI.

3) Geometrical interpretation of
this formula: If we draw a q-p-
diagram for the coordinate qz, the
point (q@k,pk) describes a closed
curve, the form of which is in-
dependent of the values of the
other q.

4) Cf. CHARLIER, Die Mechanik
des Himmels, (Leipzig 1902) I,
p. 86, 100.

5) Comp. note 2, page 150.

153

During the integration g; moves up and down once between its
limiting values §; and 17; written explicitly :

Nl:
I=? fan WRR (gp)! eee ae te
Sk
If the system is varied adiabatically, the variation of 4 will

Nk 4 Vk
OL; OT}, ee Jij
di da + > — gray d Tks Bae = Sal ee nt ;
ane 5 qm am
Sk

It is thus necessary to calculate de. Solving the « Jen the equa-
tions (8) we obtain the system of » first integrals:
Lm ET ater ee ER RE A Lg
(One of the @, say a’ is the total energy ; then H' is the Hamin-
TONIAN function).
Hence according to equation (6) we have:

oHm
da” = OG SVs EE ASS OA RE GAD
A Hm OF,
Now the quantities a be expressed by means of the ret
a a

If in equation (11) for the p’s the values (8) are introduced, it
becomes an identity, thus:
oHm xd Om OY F,
tE
da 1 Op, Oa

OR re nie ee nnn VMS

Further we put:
LANE
FR Et EN ee Meg ee (OE)
the determinant of the n° quantities fj, will be called #'; its minors
Firm, From the properties of functional determinants it follows that:
oHm Film

TE erk)
Equations (13) and (15) give:
oHm F lm OV F, °)
iN TEN
a NN (16)

1) At the limits of the interval of integration the integrand WE = 0, hence
it is unnecessary to take account of the variations of these limits.
2) It may be noticed that:
a) of the coordinates only q/ occurs in fin:
b) Fim does not contain q/ but it contains the other gq.

11
Proceedings Royal Acad. Amsterdam. Vol. XX.

154

In order to find the mean of this quantity with respect to the
time, it is necessary to study the properties of periodicity of the
systems under consideration.

§ 3.
We introduce a set of variables ¢,....t, defined by the equations:

Vk
ae Eee Picks EE 5 Gin to as pete eee em

During the motion of the system we have:
dqn OM ls Fh
TE
(Comp. eq. 15).

From this equation it may be inferred that ¢,....4. are constants,
whereas ¢,=?¢—1t,. (The « and the ¢ form a set of canonical inte-
eration constants ')).

All the phases of the mechanical system can be characterized by
the values of the g and p; or by the g and a (cf. eq. 8); or by
the ¢ and a. We will consider the representation on each other of
the following two n-dimensional ‘spaces, obtained by taking the a
constant :

(I) the g-space, limited by the surfaces Ot = Se Gr ne the
t-space.

The representation of these spaces on each other is given by
equations (17). The ¢ are many-valued functions of the g with moduli
of periodicity :

Vk
ome [dae fila Be OE) Eper eN
Sk
(ev)
[wii is the increase of t; if q, moves once up and down between
Er and xz, the other g remaining constant ®)].

Hence the f-space can be divided into period-cells: similarly placed
points of these cells correspond to the same point of the q-space.
The representation of one period-cell on the q-space limited according
to (I) is uniform; on the other hand every point of the g-space is

1) If the integral of action be W, we have:
ow
oe da
*) These integrals obtain a simple meaning if qx is considered as a complex
variable (Cf. Sommerretp, Phys. Zeitschr. 17 (1916) p. 500).

(18)

it ed

155

represented in more than one point of a period-cell, in such a way
that the positive and negative values of p;,—=W fF; are separated.

The determinant of the wi will be denoted by &; it will be
supposed that 2=|—0. Its minors are Q*; we put: ;
Oki

2 is equal to the volume of one period-cell.

In the ¢space the motion of the mechanical system is represented
by a line parallel to the axis of ¢,, which passes through the cells.
If every point of this line is replaced by the corresponding point
in one of the cells, a set of points is obtained in this cell which is
everywhere dense, if no relations of commensurability exist between
the wJ' (relations of the form:

wki —

Sn am

J

the m; being positive or negative integra! numbers '). (Supposition C).
We now replace the mean of a quantity z with respect to the

time, ie. the mean value of z for all the states of the system

represented by a great length of the ¢line by the mean value of

z for all points of one period cell. ?).

1) This theorem is due to SräckeL. It is founded upon theorems given by
JacoBI and Kronecker. Cf. KRONECKER, Werke 3, 1, p. 47.

Remark. We put:

= Sor i.
t

The zj are SCHWARZSCHILD’s ‘‘Winkelkoordinaten” (l.c.; comp. also Epstein,
Ann. d. Phys. 51 (1916) p. 176). If gp moves up and down once between its
limiting values Ex» and yx, while the other q remain constant, only ré increases by 1.

Taking the 7 as a rectangular set of coordinates, the set of period cells becomes
a system of hypercubes, bounded by the surfaces r= integral number, while
the motion of the mechanical system is represented by the line

t) = Wi! t 4+ constant,

The wit are the mean motions.

2) That these methods of calculating the mean value come to the same may
be demonstrated as follows (for the sake of simplicity we limit ourselves to a
system of two degrees of freedom): The quantity 2 may be written as a function
of the “angular variables” +1, 7°, which is periodic with respect to both these
variables with periods equal to 1. If we suppose (which certainly is allowed) that

072

ST exists and is continuous everywhere in the region 0< j <1, this function

Orde?

may be expanded in a double Fourier series in +! and #° (Cf. on multiple
Fourier series for instance Born, Dynamik der Krystallgitter, Anhang (Leipzig,
Teubner 1915) ). Hence :

i ri cos tT! 2
— En eee eo
: oT le A Dee A

156

ey Sie tye
smf fd. EN de (A)

where the integration is extended over the volume of one period cell.
Written as a function of the q:

Pi ee | a fi OE sec la)
ze dg ge tej
Ke. 7 OUA 1/0)
1 Ne ~
=u fe fenn ota hehe "ee ce

During the integration every q moves up and down once between

Hence :

its limits.

oHm
We use this to calculate aa (comp. formula 16):
a

maman

oH™ ae = (5 ve) i

da / Fhe i,
1 OY FL
aS I > oe shed d n=. Fin Ten We . . . 21

By means of the relation :

aE : fan dg dgiti... dgn. Fim = Qin

it can easily be verified that form. (21) reduces to:

“OH OF
=— Soma fd ENE en:
a

Now from form. (10), (12), (14) and (19) we have:

1
A= fare
0

This series converges uniformly and can be integrated term by term. If we
now put: rj = wJl.t + const., and calculate the form:
. o+T
z — Lim je
li ft
to
it is found — as none of the factors rw!!+ sw?! 1s equal to zero — to reduce to:

1
z= A= fa’ HOT 2
0

where:

157

Ff, oHm
dl = tal 2 fin aL + = Wim 5 | of toes Gees)
m a

ESS

oHm
Substituting for Tai its value according to (22), it is immediately
a
found that :
O17, = 0;

hence J; is an adiabatic invariant.

SUMMARY.

If a mechanical system possesses the following properties :

1. every momentum pz; can be expressed as a function of the
corresponding coordinate qj; (supposition 5);

2. the motion of every coordinate q, is a libration (supposition A’);

3. no relations of commensurability exist between the mean motions
wi! of the “angular variables” (supposition C) ;
then the “phase integrals”

are wvariant against an adzabatic disturbance of the system.

Remark. ‘Those cases of degeneration in which supposition C is
not satisfied will be treated separately in a subsequent communication.

Physics. — “Adiabatic Invariants of Mechanical Systems’. Il. By
J. M. Bureers. Supplement N°. 41d to the Communications
from the Physical Laboratory at Leiden. (Communicated by
Prof. H. KAMERLINGH ONNES).

(Communicated in the meeting of December 21, 1916).
Systems between the mean motions of which relations of

commensurability exist.

In the 1st part of this paper’) it was shown that for mechanical
systems, possessing the following properties:
1. each momentum pz can be expressed as a function of the form:

PE V Fr (gk s a’... a", a)
2. the motion of each coordinate is a libration;

the n phase-integrals :
Tj: =| dak

are all of them adiabatic invariants, provided no relations of commen-
surability exist. between the mean motions w/: of the ‘angular
variables” 7. As remarked in the paper quoted, this supposition
was necessary in order that the system might pass consecutively
through all the states which are represented by the points of a
period-cell, so that an integral with respect to the time might be
replaced by an integral over the volume of a period-cell.

In this section we shall consider the case that relations of commen-
surability do exist between the mean motions, and it will be shown
that if the adiabatic disturbances are limited to such as do not violate
these relations, at least certain definite linear combinations of the Zj.
(with integral coefficients) are invariants. If the system is rigorously
periodic, so that the mean motions are all equal, the only combi- _
nation of this character is found to be the sum of all the phase-
integrals (in other words: the integral of action, extended over a
full period of the system), the invariancy of which has already been
demonstrated by EHRENFEST ®).

We shall- describe the motion of the mechanical system in the

1) These Proceedings. p. 149.
2) P. EHRENFEST, ibidem XIX (1) p. 576, 191%.

159

system of coordinates of the angular variables t, which are related
to the canonical variables ¢ by the formulae:
ETL EN
t
In the ¢space the boundary “surfaces” of the period-cells are
given by
= wit; = integral number;
hence the t-space will be divided into ‘‘cubes’” with side —=1. As
f,...¢, are constant during the motion of the system, whereas
Ll =t—t,, the orbit of the system is represented in the r-space by
the straight line: ’
he MO NO LONEN te = oe iro av ae Ae)

In order to simplify our formulae we will assume the ¢ to be
determined in’ such a way that the constants are equal to zero.

We will now suppose that between the mean motions w/: relations
of the form:

SOE CAG) Neen gage ae Re ee ere.)
J
(u=1.-.2 ; the mj being integral numbers).
exist.

If each point of the ¢-line is replaced by the corresponding point
in the first cell, the points thus obtained will not fill up this cell;
they only fill the (mn—)-dimensional regions determined by tbe
equations :

= mij = inteoral number: (fes bor tA eeen ke)
J
Let us consider the region which contains the ¢-line itself; for

this:
ms th =0 (SS RE Fa cos ae ee Ge
J
In this region we may construct a period-lattice in the following
way: the points of the net are the integral solutions of the equations
(5). These solutions can all of them be expressed as linear integral
combinations of a “primitive” set of n— independent solutions:
dr (s=1l...n—}) be a tee (60)

Such a primitive set gives the angles of a primitive period-cell.
In the region (G) defined by equations (5) we shall introduce a
system of n—A coordinates 39%, so that:

fs Sl de ee Tie ee oe OD)

160
Then the period-cells in this region are bounded by the “hyper-
surfaces”
ds = integral number.

In a way analogous to the one used in the general case it can
be shown that the mean value of a function taken for all points of
the f-line may be replaced by the mean value for all points of an
(n—A)-dimensional period-cell in the region G. From this it follows
that the mean of a quantity Z with respect to the time is equal to:

1 1
Z=|...fa9'...d9—.2 RE eae ae
0 0

In the case considered this formula has to be used instead of

Om
eq. (20) of the previous paper in computing the quantities :
a
If now we put:
Yom Erk. Ie SNE es EN
k

we can show that the quantities Y, are invariants for such adiabatic
disturbances as do not violate the relations (3).

Scheme of the calculation.

Making use of the expression obtained in the first part (eq. 23)

GENERE
as the value of nae it is found that:
a

“kh mers
dy, JT}, Fy, OV F
= =n Era | de VE St ffe ty
da k du k: da ZEN da
Eh

The second part of this expression is equal to:

| OVE
af. fer dp Bekom fle VE EERE
a

km

(the quantities re, Wim being constants, they may be taken under the
sign of integration). We will now transform the term of the sum
which bears the index / from the variables ®... 95... 9" tothe
variables: 3... 3! g/95+1,,..9"—4, The Jacobian of this trans-
formation is:

DH. Ds rn) 1

es > Se ae pom EAN
(od eee J .s> ei Ogi =A > Wkm + ry

09s km

Hence (11) ae into :

aj J d}!... d9*—1 dgstt ... dgr- hs dq: _ met gl)

If os changes from O to 1, tl increases by res hence g, describes
l Site
rs full periods’). The expression (13) now becomes:
nl

OVE
Erna f dq. AU esos EGT
1 da

zl

If this is introduced into eq. (10), it is found that: dY,=0; it
has thus been shown that Ys is an znvariant for the adiabatic disturb-
ances considered.

Remarks.

1. It has been pointed out by ScnwarzscHiLD and Epsreix *) that
the total energy a‘ of the system, when expressed as a function of
the Zr, depends only on the linear integral combinations Yx of the Ix;
this is a consequence of the equations (3). From this it follows that
it is always possible to fix the value of the energy by “quantizising”
the adiabatic invariants (i.e. by equating the adiab. inv. to integral
multiples of PLANck’s constant).

2. In the equations: Y,= Sri: {== adiabatic invariant, an
arbitrary primitive system of solutions of eq. (5) may be chosen for
the system of coefficients: rs 7, (¢ =1...n — 4). All such systems
are connected together by linear integral substitutions, the determinant
of which is equal to —+&1.. Hence the same holds for the sets of
m—A independent. Fr: if Y!...Y;—, and Yi... Ya be two
of these sets, we have:

1) This follows from the equation:

dn = fr ER = fim, Lik dirk = = fim, ei. Ee „d9s.
mk mks
2) In the ae of complex values of q/ (cf A. SoMMERFELD, Phys. Zeitschr.
17, p. 500, 1916) the path of integration goes rs times round the branch points
gi= El, Qt = v1 of the function: p= V Figi.
3) K. SCHWARZSCHILD, Sitz. Ber. Berl. Akad. p. 550, 1916.
P. Epstein, Ann. d. Phys. 51 p. 180, 1916.

162

5 S
Yo PN GR

s

and WE = ye. Ye

where the c% like the y{ are integral numbers. If Ys is put equal
to ns. 4, where n, takes all positive and negative integral values,
the Y; also pass through all positive and negative integral multiples
of h. In like manner the same set of values is obtained for the total
energy a’ of the system, whether expressed as a function of the Ys,
or of the Y;.

3. The question arises whether the quantities Y, found above are
the only adiabatic invariants of these systems.

Physics. — ‘Adiabatic Invariants of Mechanical Systems. HI”.
By J. M. Bureers. Supplement N°. 41e to the Communications
from the Physical Laboratory at Leiden. (Communicated by
Prof. H. KAMERLINGH ONNES).

(Communicated in the meeting of January 27, 1917).

In the two preceding papers’) on this subject the question was
investigated as to what quantities possess the property of being
adiabatic invariants for those mechanical systems in which the
variables canbe separated, i.e. where the momenta can be expressed
by formulae of the form:

D= VF: (qx, @... a”, a)

The result obtained was that the “phase-integrals”: J, = f dqx Pk

do not change during an adiabatic disturbance of the system; this
conclusion is closely connected with the quantum formulae as intro-
duced by Epsrrin, DeBrr and SOMMERFELD, who put these integrals equal
to integral multiples of PLanck’s constant. SCHWARZSCHILD’), however,
has put the quantum formulae into another form, which is far more
general. He supposes that by means of certain transformations it is
possible to express the original coordinates and momenta (q,p) as
functions of a new system (Q,P), possessing the following properties:

1. The Q are linear functions of the time;

2. the P are constants; ~

3. the g and p are periodic functions of the Q with a period
2x; hence for instance:

PEO, abd Ore An) = HO Qa):
These variables Q are the so-called “angular variables” (“Winkel-
koordinaten’’). He then introduces the quantum formulae:
27
[ade P= 2 P= ny sh dee COURTS ee ce ys (A).
0
If the character of the system is such that the variables can be

1) These Proceedings p. 149 and 158.
2) K. SCHWARZSCHILD, Sitz. Ber. Berl. Akad. p. 548, 1916.

164

separated, it is always possible to introduce angular variables; in
that case the formulae of ScHwarzscHitp and those of Epstein
coincide). In this paper, however, it will be shown without making
use of the separation of the variables, that — provided certain
conditions mentioned below are fulfilled — it is always possible to
choose the quantities P, in such a way that they are adiabatic
invariants. This is of importance as the possiblity of introducing
angular variables is not limited to those systems.

§ 1. We consider a mechanical system possessing solutions of
the following form: the coordinates and momenta g and p can be
expanded into trigonometric series (multiple Fourier series) proceeding
according to sines and cosines of multiples of 7 variables Q,... Qn:

Siler. cos
Vk = = Am, eM), ee (m, Gs Gale usu monet Qn)

ef)

(1)
Oo kk COs
Pk == = B, eten 4 (m, 9E oes Mn Qn)
aa n (sun
These variables are linear functions of the time:
Ope WEA er hee dal ht cate ea nek Re
we limit ourselves to the case that the mean motions w; are all
incommensurable. €,...&. are n constants of integration; the w; and

the coefficients of the trigonometric series are functions of the
parameters a occurring in the equations of the system (masses,
intensity of a field of foree, &c.) and of n other integration constants
P,...P,, chosen in such a way that together with the Q they form
a system of canonical variables; the transformation of the g and p
into the new variables Q and P is a contact-transformation’).

We suppose that for a given domain of values of the P the
series considered are uniformly convergent, independent of the value
ON

A method of obtaining solutions of this kind is treated in the
last chapter of Wuirrakgr’s Analytical Dynamics (Cambridge 1904 °*)) :
Integration by Trigonometric Series. — If the Hamiltonian function is
a quadratic function of the original variables q and p, the angular
variables Q are immediately related to the normal coordinates or
principal vibrations of the system *); the series then reduce to:

1) Cf. for instance P. S. Epstein, Ann. d. Phys. (4) 51 (1916) pg. 176.

2) Cf. fi. B. T. WurrraKer, Anal. Dynamics, p. 282. (Cambr. 1904).

8) A 2nd edition has appeared in 1917 (note added in the English translation).
4) WHITTAKER, l. c. p. 399.

165

Gk = Yn + = a? cos QG; + EB sin Q; ie ay, ACE
with analogous expressions for the pz.

§ 2. Adiabatic disturbances of the system.

As before we shall assume that during the infinitely slow change
of the parameters the Hamironran equations for the original
coordinates and momenta q and p remain valid (see, however,
below, remark 4, a). In order to investigate how the variables Q
and P behave during such a process, it is simplest to consider into
what expression the differential form:

Pi TING iy pd es ede LS ben ov Re
changes by the transformation from the gq, p to the Q, P'). As
remarked above this transformation is a contact-transformation ;
hence as long as the a are not varied we have:

Epdg=EPdQ HAW... (5)
dw being the complete differential of a function of he, Q An dl
which may also contain the a. During the variation the a are
explicitly given functions of the time; the formula (5) has then to
be replaced by

d
Spdg=XPdQ4+ PF. d+ DW... ... ©
where:
_OW OW 0W da
DW sa d En Pp dt ENE
AQ ae ' Oa dt ) (7)
wos an function: of? .Q. PP sand. a, de — if the P have been
properly chosen — contains the Q only in the form of trigono-
metric functions:
COS
e= = Cm La ee Qed Crys e Hi F, (8)

The proof of this proposition is given in $ 3.
Hence the differential expression (4) changes into:
> PdQ— {H*(Q, Pa) — F.adt+ DW. (9)
H*(Q, P,a) is obtained from A (q, p,a) by replacing the anid p
by their expansions in trigonometric series. Now the characteristic
property of the angular variables is that H* does not contain the Q:
He (P, a) Vou bam ve eee O0
The equations of motion for the Q and P are the canonical
equations derived from a HAMILTONIAN function, which is equal to
the coefficient of dt in the differential expression (9). *)

1) WHITTAKER, |. c. p. 297.

2) In order to simplify the formulae it is assumed that only one parameter a
is varied.

8) WHITTAKER, |. e. p. 407.

166

Hence we have for P:

dE UE ess ba dE mp. Cm. om fe
dt QX n_|eos

If no relations of commensurability exist between the mean motions
w; of the Q; — as is assumed in § 1 — the mean of this expres-
sion with respect to the time is zero: hence during the variational
process P;, remains unchanged’). We have thus proved that the
expressions which are “quantizised” by ScHWARZSCHILD are invariants
for an adiabatic disturbance of the system.

As according to formula (10) the total energy EH = H*(P, a) only
depends on the P and on the parameters, it is always possible to jix
the value of the energy by quantizising the P.*) *)

(m, Q, om, a) | (11) 4

1) The meaning of ©’ is: summation over all + and — values of the m, with
the exception of simultaneous zero values of all the m.
2) This may be formulated more exactly as follows:

da
For the sake of simplicity suppose ap to be constant: then by integrating eq.

(11) term by term (which is allowed on account of the uniform convergence):
. tf k cos lo + It
Oo}, a8 . : En,. -mM, En (mm, Q, sig tate Qn)

Independently of the value of ¢ the value of the term between [ ] always
remains below a finite limit g. Hence:

to

|S Pil < 2a.g
On the other hand:
da=a.T
We thus have:
AE,
LEES —
T=a da

This reasoning also applies to the demonstration given in the Ist part of this
paper (These Proceedings, p. 149).

8) If the original HAMILTONIAN function H(q,p,a) is a quadratic function of the
q and p, H*(P,a) will be found to be of the form:

wy. Py + constant.

: : h
Hence if Px is put equal to mx De the total energy of the system is:
7

h
E = — Say. ng + constant.
Tet
OH*(P,a) .
4) It can be shown that ae is equal to the mean with respect to the

time of the force exerted by the system “in the direction of the parameter a’.

167

$ 3. Proof of formula (8).

In the expression 2pzdqz q and p are replaced by their expan-
sions (1); in differentiating the Q, P, and ¢ are regarded as independ-
ent variables, the parameter a being an explicitly given function
of t. This gives:

| ee
> pr. dgy= = ft. dQ + E fh. AP Hfr dt
‘ 5

sk pk
Ji fe, fs are Fourier series with respect to the Q.

As for a == constant this substitution is a contact transformation,
we must have:

AY

: OW ow
= ft dQ + Eft. dPy = E Pidi + Eon dQ + S—dPy (12)

dQ). OP;
Hence:
ow mea cos
—— = — Py + fk = — Pet y(Pia) + Z'n, | tm, Q, 4 mn Qh)
OQ: 1 0 nr) sin
and:

= | Cos
Wz Gen Pr, si vl) Qk + 2 Din... ph (m, Q, Sean Mn Qn)

Furthermore we have:

ow dy! mr
=e) rn Sag
oa e+e, A +=

In va the Q occur a under sines and cosines; from this it
follows that the coefficient of Q; on the second side of the equation
must be zero, and hence:

cos

(m, Q, asta Mn Qn)

sin

ve = Pr + azo).

As the condition (12) determines the P and Q all but the additive
constants, it is always possible to include the aj(a) in the P. If
we suppose this to be the case, we get:

ye == oy
hence:
cos
PE ae. Ooi hy | : | (m, QF es seni). on or pe RO)
ntsin
It follows that:
F a ow
—— —T Ent

is a Fourier series with respect to the Q, and thus the proposition
has been proved.

168

Remarks.
1. If zj (a) is not made equal to zero by a proper choice of the
additive constant of P,, it will be found that:

P, + aj (a) = adiabatic invariant.

2. In many cases the P/ can immediately be introduced in such
a way that the quantities a,(a@) are zero. As examples we may
mention :

a. systems the HAMILTONIAN function of which can be expanded
according to ascending powers of the g and p, and which are to
be treated by a method given by Warrraker ');

6. systems in which the variables can be separated; the P are
then determined by the formulae:

2a P, = ly = phase-integral corresponding to the coordinate qr =

Nk
= fm . dq”).
Si

3. Suppose the P to be determined as assumed above, so that
W is a periodic function of the Q (form. 13). If the parameters
are not varied:

En SARLAT

Integrating this expression from Q}=0 to Q, = 2a (Qi... Qui
Qi Qn, Pi.. Pn being kept constant), we find:
Qk = 2n t
= pdg = 2x P= adiabatic invariant.
Qk=0
[If the «,(a) have not been included in the 7, it is found that:
Qk = 2
= pdq = 2a (Py + ar) = adiab. inv. according to remark 1].
Qk=0

Epstein has given the quantum formulae in a form which is
equivalent to:

1) WHITTAKER, l.c. p. 398— 408.

2) The constants «x which occur in SCHWARZSCHILD’s formulae (1. c. p. 549, 551;
see also higher up, form. A), and which are determined by the limits of the
phase-space, are probably connected with the quantities x, introduced here; but I
have not been able so far to find a general proof.

169

Qk= 2n
ndgn. be 7)
Ord
and is therefore in agreement with the above.

4. The following points have still to be mentioned:

a. Probably it will be found sufficient that in passing from a=
constant to? a= a given function of the time, the Hamiltonian
equations remain unchanged only if we neglect terms of the 2nd

and higher orders in a. This has yet to be investigated.

6. In the present paper it has been supposed that the mean motions
w; are all incommensurable. The w; are, however, functions of the
parameters. Hence if the a are varied, the w; change too, and their
ratios pass through rational values. It has still to be investigated,
whether this may give rise to difficulties. (This applies also to the
demonstrations given in the preceding papers).

SUM MAR Y:

If a mechanical system of 2 degrees of freedom possesses solutions
which can be expressed by means of. multiple trigonometric series,
proceeding by the sines and cosines of n angular variables, between
the mean motions of which no relations of commensurability exist,
it is possible to determine the canonical momenta corresponding to
these angular variables in such a way that they are adiabatic
invariants for an infinitely slow change of the parameters of the
system. — (The fact that during an adiabatic disturbance the mean
motions change -and that their ratios pass through rational values
has to be further inquired into.)

1) P. S. Epstein, Verh. d. D. Physik. Ges. 18 (1916) p. 411.

Proceedings Royal Acad. Amsterdam. Vol. XX.

Physics. — “The spectrum of a rotating molecule according to the
theory of quanta.’ By J. M. Burexrs. (Communicated by
Prof. H. A. Lorentz).

(Communicated in the meeting of May 26, 1917).

§ 1. Introductzon. :

N. Bserrum has drawn attention to the fact that if a molecule
which carries a vibrating resonator, rotates, this rotation exerts an
influence upon the frequency of the light emitted *). If the frequency
of the resonator is », 7), the angular velocity of the molecule being
w= 2ar', the frequencies: rv, », +2, r, —v’ will be found in -
the light emitted by the system. Molecules of this kind will also
absorb the frequencies »,, v, + »', », —v’ from radiation which falls
upon them. Starting from this principle Byerrum has explained the
structure of the bands which are found in the infra-red absorption
spectra of certain gases. It was assumed that the velocity of rotation
of the molecule is determined by a condition taken from the theory
of quanta, so that w can only have values which are an integral
multiple of a certain quantity w,. In the spectrum of such a gas
a line of the frequency », will be accompanied by a set of equi-
distant satellites, given by the general formula :

Wo
VA el
276

The bands observed in the absorption spectra of water vapour and
other gases actually have a structure that may be described by this
formula *).

On the principles of the theory of quanta, however, one will
be inclined to assume that a given spectral line is not emitted by
a vibrating electron; but that it is emitted when the electron passes
from a certain definite state of motion discontinuously to another

1) N. Bserrum, Nernst-Festschrift p. 93 (1912). — Lord RayLereH was the
first to point out this influence of the rotation (Scientific Papers, IV, p. 17). — Com-
pare also: W. C. MANpersLooT, De breedte van spektraallijnen (Diss. Utrecht 1914).

2) In this paper “frequency” will always denote the number of vibrations per
second.

8) Cf. for instance: EvA von Baur, Verh. Deutsch.Phys. Ges. 15, p. 710, 1150,
(1913); H. Rupens & G. Herrner, Sitz. Ber. Berl. Akad. p. 167, 1916.

171

definite state. If in the first state the energy of the electron is: a’,
in the second state: a’, then according to Borr’s hypothesis the
difference a’—a" will be emitted as light of the frequency :
a'—a"
h

On the other hand the electron can absorb light of the same
frequency if it passes back from the second state to the first.

Now the ‘following question arises: Suppose the electron to move
in the field of a rotating molecule; does the rotation of the molecule
exert an influence of the same kind on the frequency of the light
emitted, as it does in Byrrrum’s theory ? The object of this commu-
nication is to show that following the lines of the theory of quanta,
it is possible to deduce at least for certain rotating systems spectral
formulae which show the same character as the one given by Burrrum.

=

§ 2. General formulae for the motion of an electron in the field
of a rotating molecule.

It will be assumed that the molecule has an unvariable form, and
that it can rotate about an axis fixed in space. The position of the
molecule is determined by the angle of rotation ¢,. In the field of
the molecule an electron moves; its position will be given by polar
coordinates 7, 3, p,‚ (the axis of the polar system of coordinates
coincides with the axis of the molecule).

The potential energy of the system V is a function of the relative
ee is of the electron and the molecule, hence it depends on

, ® and p‚—p, ). If m be the mass of the electron, / the moment
4 inertia of the molecule about the axis of rotation, the Lagrangian
function for the system is:

I= + r7. 9? + 93, sin? D. p,°) + ree — V(r, 9,9,—9,)- - (1)

In this formula we shall put:
P1 — Po = 13 P,—Y, OPTA EEN he Nish Pte (2)
If the momenta corresponding to the coordinates r, 3, w,, w, are
calculated, the Hamiltonian function will be found to be:

©: U (WY, ae)
eel st )+ FE + V(r, 8, W,) . (3)
m

w, being a cyclic coordinate, W, is constant. P, represents the

1) In V ¢,—, necessarily occurs: otherwise the rotation of the molecule cannot
exert any influence upon the motion of the electron. (This applies also to the
theory of RAYLEIGH and BjJeRRUM, cf. W. C. MANDERSLOOT, I. c. II, § 3).

£25

172

total moment of momentum of electron and molecule together; it
determines the rotation of the system as a whole.
If ”, =U, the motion of the electron is determined by the function :

1 Oe LT as
Hy = 5 (Re ahs = ui \+ at Vrs AB): se oy can

2m r? sin? 9 21

It will now be assumed that it is possible to find solutions of
the problem characterized by the Ham. function (4) (in this problem
there is no disturbing influence of the rotation), and that these
solutions are of the following form: the coordinates and momenta
can be expressed as periodic functions (with period 27) of three
variables q,, 9, Jz, Which depend linearly on the time (so-called
“angular variables”)*). If the canonical momenta p,, ps, Ps, COr-
responding to these variables, are introduced *), the original
coordinates and momenta 7, 9, w,, R, O, YW, can be expressed as
functions of 9,4: Je Pr Pa Ps This transformation of the variables
possesses the property of conserving the canonical (HaMILTONIAN)
form of the equations of motion. *)

To find solutions of the problem given by (3) (%,=— 0), it may
be considered as a problem of disturbed motion, and instead of
the original coordinates and momenta the pand q may be introduced
as new variables. The equations of motion of the g and p then are
the HaAMILTONIAN equations, derived from the function A (q, p), which
is obtained if in (3) the original coordinates and momenta are
replaced by their expressions as functions of the g and p. This
function has the form:

YW Ye
K (q, p) =H, — ipa ries
vite eo Ps (PiP>sPs)- as Fett

= AGP Ds Ps) OT T

y }
| (mg, ze MQ, + mat) | (5) 5)
sin

= Fe | Bama (Pp, Pz Ps):

1) Solutions of this kind are — as is known — of frequent use in Astronomy,
especially for the treatment of problems of disturbed motion. In the most common
cases they have the form of trigonometric expansions according to sines and
cosines of combinations of the g. — (The>expression found for J, has a slightly
different form, as this variable can increase indefinitely; for instance J, may be
found to be equal to qz plus a periodic function of q1 2 q3)-

In the theory of quanta K. ScHwarzscHILD was the first to introduce solutions
of that nature (Sitz. Ber. Berl. kad p. 548, 1916) Compare also : J. M. BURGERS,
these Proceedings p. 163.

2) These momenta p, py pz are constant.

3) Comp. WuitTakeR, Anal. Dynamics (Cambridge 1904) p. 297, 396.

4) 5’ denotes a summation over all positive and negative values of the m, with

the exception of simultaneous zero values.

173

(where ¥,, represents the mean value of W, (the moment of
momentum of the electron).

It will be assumed : .

1. that in A, (p, p, p,) all the p occur, in sucha way that between
the three differential quotients 40/3, °40/y,,, %4b/a,, there do not
exist any rational relations;

2. that J is very large, and that the quantity 2/7 is small as
compared to the mean angular velocity of the electron, so that the
second and higher powers of this quantity may be neglected.

It is then very easy to find solutions of the problem considered,
following a method given by DerauNay and Waurrraker *); if these
solutions are restricted to the terms which contain ¥2/7 to the powers
O and 1, they are of the form:

Pr, en cos
as X | tome PA Ps Era | : | (m,Q, si m0, 0) |
Ll haa sin (6)

(iw ar =F

: (Oms) |

Here Q, Q,Q, are new angular variables; P, P, P, are the
canonical momenta, corresponding to them.

The total energy of the system is found to be (to the same
degree of approximation):

ea en
a= A (BEE) —]- Wi (Pa Bey

p, , cos
qq Q, Ree 5 Smymams (P, P, P,). sin

aide
21

ore eu

§ 3. The Quantum formulae.
Following the ideas developed by ScuwarzscHILp *), the quantum
formulae for the system may be introduced as follows :

the quantities P, P, P, ”, are put equal to integral multiples of
h

Qe |
h h h h
sitet Clas) ihn a Ei ree Pitre tg . 2 (8)

The energy, when expressed as a function of the quantum

numbers 7, ”, 7, ”,, becomes:
at, (n‚n‚n,). h h?

pees eee AME en,
2 L ata 8n°I (9)

ia Oe LN Ng

1) Cf. WuirraKer, |. c. p. 404.

2) The three terms of this equation may be interpreted approximately as follows:
A, is the energy of the electron; ¥52/2/ is the energy of rotation of the molecule;
Vo. ¥
the term — aca is related to the CorroLrs-reaction generated by the rotation.

3) K. ScHWARZSCHILD, |. c,

174

If the electron passes from a state of motion characterized by the
numbers n,'n,'7,'n,' to another, characterized by n,"7,"n,'n,", the
energy decreasing from «’ to @'’, according to Bour’s hypothesis the
system emits light of the frequency :

al — a!

i a ere ee

Hence the spectrum lines of the molecule under consideration are

given by the formula:
a,’—a," n,'.a,'—n,'. h
a ae

"
a
4 1 1 ze (n,?—7n,""”)

Rae Fay on I

p= (ty
With the aid of this expression it is possible to show the influence
of the rotation on the spectrum.

§4. Discussion of the spectrum.

The spectrum lines given by formula (11), which are characterized
by 8 numbers, may be grouped in different ways. In order to show
the influence of the rotation of the molecule as clearly as possible
we will consider a definite change n,'n,'n,/— 7," n,"n," (hence the
values of «aa, a," are fixed); then by giving different values
to the numbers 7,',7,", different systems of lines are obtained.

A. First consider the case n,' =," = O (in both states of motion
the rotation of the molecule asa whole is zero); then the frequency is:
a,’ — a,"

Ps = 7

. (12)
B. If „, and n," are equal, and different from zero, the frequency
will be found to be:

! "
‚A — @,

on I . (13)

vy == DV, — Ny,

Hence the original line », appears to be accompanied on both

sides by equidistant satellites, in the same way as in BJERRUM’S
theory. The distance of the satellites is equal to:

if "
A Pek Ie
Qal

In general the value given by (14) is not the same as that given
by Bsgrrum’s theory which is:

. (14)

h

A »r=——_—
E 4 n°]

(14a)

The expressions (14) and (14a) may give the same value if for

175

Instance a, =n, oe for certain systems this may be approximately

the case *).

C. If n,’=—7,", so that the general formula (11) has to be
retained, each line v, appears to possess a double infinite system of
satellites, the distances of which are given by a quadratic formula.
This formula is of the same type as the one given by DustanpREs
and others for the band spectra’). A formula of this kind has been
derived from the theory of quanta for the first time by ScHwarzscuiLD *);
SCHWARZSCHILD has also pointed out that if the moment of inertia /
is calculated from the coefficient of the term of the second degree,
the values obtained are of the proper order of magnitude.

Other groups of lines.

D. If n',n',n', are equal to n",n",n", respectively, so that only
n, changes in the transition from the first state of motion to the
second, a set of lines is obtained, which may be denoted by the
name of “rotation spectrum” :

a, h

ene ee er:

From the order of magnitude of the coefficients it may be inferred
that these lines are to be found in the infra-red (they stretch out
Montan as p= to; A=):

E. Reupens and Herryer *) have observed in the absorption spectrum

(15)

ry = (rn!)

1) The difference between formula (14) and (14a) becomes of importance if it
is desired to calculate the value of the moment of inertia from the distance of

the lines. (In ByJERRUM’s theory 2,31 is sometimes given for Av instead of the
us

a]
value (14a); cf. H. RuBeNs and G. Herrwer, |. c. p. 168).

A more important difference between formula (14) and BjeRRUM's theory is
that the value given by (14) depends on zj’—e}’, and hence on the numbers
M/Na'Na/Ny Nans. This makes the value of Av in general different for different
lines yo, whereas on ByERRUM’s theory Av is independent of vo.

Compare also the example given in 8 5.

2) Cf. H. M. Koren, Das Leuchten der Gase und Dämpfe, (BRAUNscHwWerG, 1913),
p. 214, seq.

8) K. ScHWARZSCHILD, |. c. p. 566. — SCHWARZSCHILD supposes that the
rotation of the molecule and the motion of the electron do not exert any influence
upon each other; the opposite supposition is essential to the theory given above.
This is the cause of the term which is linear in my and #4” being absent in
SCHWARZSCHILD’s formula.

4) H. RuBens and G, Herrner, |. c.

176

of water vapour a system of lines which are related to the series
given by (13) by the formula:

PE EN HRO)

These investigators, who explain the system vr; on ByerRum’s theory
ascribe the lines »v// to the emission or absorption of the rotating
molecules themselves’). The interpretation of the lines on the theory
given above is more difficult, and less general. They may occur in
special cases, if it is possible to tind transitions 7,'7,'”,' —n, n," 7,",
for which a, does not change, while the value of a, changes
by the same amount as is given by the formula (13)*). In general
it will thus not be possible to find a system v/7 corresponding to

each system of lines »,, vy. Compare also the example given in $ 5°).

§ 5. Example.

An illustration of the preceding theory is afforded by considering
a system, which is characterized by the following expression for the
potential energy of the electron in the field of the molecule:

yee =: = ae ss ale ONE eN (17)
r r r* sin” D

(The form of V has been chosen in this way in order to make
possible the integration of the equations of motion by means of the
method of separation of the variables; cf. P. SräckeL, C.R. 116,
p. 485, 1893; 121, p. 489, 1895).

The energy of the permissible motions of the electron, when
expressed as a function of the quantum numbers, is found to be:

da met E° ( 1 8a? ma 82% m.b ) (n, — u). 1?
ne

A? i * Nake i. Oe 8a? 1
i a
in
: An? I el

(in this formula terms of the 2nd and higher orders in a, 6, c have
been neglected).

_ The mean value ¥,, of the moment of momentum of the electron
is equal to:

1) Compare in connection with this: M. Prancx, Ann. d. Phys. 52, p. 491, 1917.

2) In every case changes 1,/22/N3’ —> 7,/"Nq'"n3” must be possible for which only
the sign of the moment of momentum of the electron changes (and hence the
sign of zj); zo then retains the same value. Cf. § 5.

3) It may be remarked that formula (15) can also give a system of equidistant
lines in the infra-red, if n’, = — nv”, (the moment of momentum of the molecule
changes sign, while the direction of rotation of the electron remains the same);
the distance of the lines is equal to: %/Jn.

n, may be positive or negative; this depends on the direction of
motion of the electron.

In the principal part «, of « only the absolute value |n,| occurs.

From (18) the spectral formula may be deduced in the same way
as above. The following groups of lines (corresponding to those
called B and E in § 4) are of special interest:

a). Take: n, = —n,' = — n,; n,' =,'' = n,; the values of n,, n,
change in an arbitrary manner. The frequencies emitted then are:

anime Hf 1 1 | 2n,nyh 2n,n,h
ee ee

If n, has always the same value, different positive and negative
values of n, give a set of equidistant lines, accompanying the line
P,; the distance of two consecutive satellites will be:

2n,-h
Wz .
4x? I
b). Take: n,’ = — n," =—n,; Nn, =N, ==n, (as was done above

under a); while the values of n, and», do not change. In this case:
a (nn nz) == a, (n,n Ni);
hence the frequency emitted becomes:
2n,n,-h
na
The structure of these systems is in some respects analogous to that
of the absorption bands of water vapour ; (I) corresponds to the band
observed near: 2 = 6,26; (ID) to the lines in the far distant infra-red.
It must not be forgotten, however, that this example has been
chosen arbitrarily, so that no great value can be attached to the absolute
magnitude of Av.

SA a Bee es Tk aoe EE

Summary.

An attempt is made to show that it is possible to deduce from
the theory of quanta spectral formulae for rotating systems, which
may explain the structure of the bands, observed in the infra-red
absorption spectra of certain gases.

At the same time a formula has been found which might be
useful in the explanation of the band spectra and which is an
extension of one already given by SCHWARZSCHILD.

In order to simplify the problem it has been assumed that the
system rotates about a fixed axis; hence the question arises whether
it is not possible to give a more general treatment, in which account
is taken of the precessional motion of the molecule.

Physics. — “Jsothermals of di-atomic substances and their binary
mixtures. XIX. A preliminary determination of the critical
point of hydrogen”. By H. KAMERLINGH ONNes, C. A. CROMMELIN
and P. G. Cara (Communication N°. 151c from the Physical
Laboratory at Leiden).

(Communicated in the meeting of June 24, 1916).

1. Introduction. Apparatus and method.

The results so far obtained by different observers in the determi-
nation of the critical data for hydrogen show a wide divergence.
Dewar ') found 7;,—= 29° K, 30° K and 32°K respectively, pz = 15
atm., OrzewskKr’s ®) latest values were 7, = 32°.3 K and pz = 13 atm.,
Berre *) found 7, = 31°.4K and p,=11 atm. It will appear from
this communication that all these results differ very considerably *)
from those which we consider to be the correct ones, viz.
Ty, = 33°.2 K, p,~=12.8 atm. In view of the great difficulty of
the problem this is not to be wondered at.

We ourselves had repeatedly been compelled to defer the accurate
investigation regarding the critical condition of hydrogen, because
we were unable to get over the difficulty of keeping temperatures
such as those, at which the experiments had to be carried out,
sufficiently constant. As soon as this difficulty had been conquered
by the construction of our hydrogen-vapour cryostat’) we were
enabled to avail ourselves of this material improvement in order to
arrive at a first determination of the critical temperature of neon *®)
and also a more accurate measurement of the critical data of hydrogen.

1) J. Dewar. Inaug. Adress. Brit. Ass. Adv. Sc Belfast 1902.

2) K. Oxuszewsk1, Ann. d. Phys. (4) 17 (1905) pg. 986, Ann. de chimie et de
phys. 8 (1906) pg. 198.

8) F Burre, Physik. ZS. 14 (1913) pg. 860.

BuLLe’s results differ a good deal more from ours-than OrszewsKI's much
earlier results.

4) With the exception of Dewar’s pk and ours (Note added in the translation).
In fact the weaker point in Otszewsxi’s work will have been probably the determi-
nation of the temperature.

5) H. KAMERLINGH ONNES, Proc. XIX (2) p. 1049. Comm. N°. 15la.

6) H. KAMERLINGH ONNES, C. A. CROMMELIN and P.G Carr, Proc. XIX (2) p. 1058.
1917. Comm. N°. 1515.

179

On the whole the same apparatus were used for both substances ,
as regards these, we may therefore refer to the communication on
neon just quoted. Only the pressures were measured in a different
way. The readings on the closed manometer M,,, which was used
in the case of neon, do not begin before 20 atm. and the critical pressure
of hydrogen being smaller than that, we had to use the open
standard-gauge of the laboratory ') for the purpose.

The hydrogen which had been purified by distillation was free
from all admixture as shown by the heterogeneous parts of the
isothermals in the pressure-density diagram being straight lines
running parallel to the density-axis (comp. fig. 1). At the same time
we wish our results to be considered as preliminary ones. In the
first place because a larger number of observations than we have
made (comp. fig. 1) will be needed to smooth out certain small
irregularities which our observations still show and thus draw with
sufficient certainty the set of isothermals which have to serve the
purpose of settling in a perfectly satisfactory manner physical
constants of so fundamental a character as the critical data of hydrogen.
But even more because in our opinion it is necessary for definite
determinations to be able to follow the critical phenomena by eye
which we have not been able to do as yet, no more than in the

100 100 dg J00 400 soa 600

Fig. 1.

1) H. KAMERLINGH ONNes, Proc. I p. 213. Comm. NO. 44.

180

case of neon, although we hope before long to be in a position to
carry out this purpose. ')

As regards the method of deducing the critical data from the
observations we have in our Communication on the critical condition
of neon referred to the similar case of hydrogen, which is the
subject of our present communication. In this case we are able to
illustrate our method of procedure by means of a suitable diagram °).

This figure gives the main features of the pressure-density
diagram, drawn during the observations to keep a graphical record
of the results already obtained, in order to guide us, as we went
along, in the choice of the conditions under which the next observa-
tions had to be made.

In representing an observation we used as abscissa the quantity
of gas*) that was at that moment present in the part of the
vapour-pressure apparatus which was at the temperature of the
observation *).

The observations relating to one isothermal are each time indicated
by one and the same of the signs 0,0, « and A. The determination
of the points of beginning and completed condensation had been
preceded by tentative readings in the neighbourhood of the obser-
vations recorded in the diagram. The fact that the heterogeneous

1) Compare, besides the paper already quoted on the hydrogen-vapour cryostat
(151a), particularly H. KAMERLINGH ONNEsS, Proc. XVIII. pg. 507. Comm. N°. 147c,
where a cryostat is discussed working with neon boiling under enhanced pressure
to be used amongst others for this object.

*) The corresponding diagram of neon being somewhat modified owing to
impurities would necessitate an elaborate discussion.

8) The quantity is given in cc. of gas measured in the normal condition.

4) The volume of this part is approximately 1.0! cc. Our apparatus, being a
simple vapour-pressure apparatus, was not suitable either for the accurate measu-
rement of this volume or of the quantity of substance which enters this volume.
In consequence of this the uncertainty as regards the accurate value of the volume
which is at the observational temperature as also the value of the temperature
of certain other parts of the apparatus whose volume cannot be neglected is
such that the accuracy of the density-determinations cannot be guaranteed
beyond a limit of a few percentages.

Owing to the uncertainty of the correction for the “waste” space, the figure
may deviate in a systematic manner from the correct pressure-density diagram to
the same degree: in order to obtain the correct figure the chords of the boundary
curve i. e. the heterogeneous parts of the isothermals will have to be shifted
without any change in length, each by an amount about proportional to the
pressure ; the uncertainty about the part of the volume which has the low obser-
vational temperature causes the value of the unit of density which is assumed
in setting off the density on the axis of abscissae in fig. 1 to be only approxi-
mately known.

181

parts of the isothermals are accurately parallel to the axis of
abscissae proves on the one hand that the hydrogen was absolutely
pure and on the other that the temperature could be kept constant
to Of of a degree *). It proved possible to connect the points which
give the beginning and the completion of the liquefaction by means
of an ordinary parabola. The pressure corresponding to the top of
this parabola was taken as the critical pressure. The temperature
corresponding to the critical pressure thus found was obtained by
extrapolation from the measurements of the vapour-pressure at the
temperatures immediately below the critical temperatures.

Beside the observations just mentioned on the isothermals which
show a heterogeneous part, an isothermal will be found plotted in
the figure which was determined at a slightly higher temperature
and where a similar part parallel to the density-axis (heterogeneous
condition) is no longer found. The tangent to the point of inflexion
is inclined to the axis of densities and this isothermal thus appears
to belong to a temperature higher than the critical temperature. It
js clear, that by means of this figure an upper and a lower limit
may be fixed for the critical pressure and a probable value of the
critical pressure may be established, from which in their turn follow
an upper and a lower limit and a probable value of the critical
temperature. The degree of certainty with which the latter deter-
mination may be effected is fairly satisfactory. For hydrogen the
two limiting values just mentioned as regards the temperature differ
by less than + of a degree. The figure further shows that in the
manner described the temperature itself may be considered to be
determinable with a certainty of .1 of a degree.

§ 2. Results.

In the following table of vapour-pressures the absolute temperature
as well as that on the centigrade scale have been corrected to the
Kervin-scale by means of the table of corrections published on
former occasions *).

The pressure is given in international*) cms mercury and in

1) To make this result possible the distribution of the temperature must also
have been a very uniform one throughout the exverimental space. We are glad
to express once more our thanks to Mr. J. M. Burgers, phil. cand. for his careful
assistance in the regulation of the temperatures.

2) H. KAMERLINGH ONNEs, Proc, X, p. 589. Comm. N°. 1025, and H. KAMERLINGH
Onnes and G. Horst, Proc. XVII, 1, pg. 501. Comm. N°. 1414.

8) In the original paper the pressuresywere given in local cms mercury and in
international atmospheres. [Note added in the translation).

182

international atmospheres, the international atmosphere being taken
equal to a mercury column of 75.9488 ets at Leiden.

TABLE I. Vapour-pressures of hydrogen.

y G6 | PEG (int. cm.) | Pross (int. atm.)
32.02 K OTE 822.7 10.825
32.60 — 240.49 893.2 11.752
32.93 | — 240.16 936.5 12.322

Above 4% the following point was measured:

7 4 p (int. cm.) P (int. atm.)

33.28 K. — 239.81 C. 999.5 13.151

The critical quantities derived from the above figures are as follows :

TAB EIL Critical constants of hydrogen.

T, Op pp (int. cm.) py (int. atm.).

— 23091 C. | 973 12.80

oo
ers 9
En
oo
A

The deviation of BurLe’s result from ours may, as we think, be
partly explained from his having used a resistance-thermometer which
was only calibrated by means of hydrogen- and oxygen-temperatures
and on the other hand perhaps also from the arrangement of his

apparatus. Possibly owing to the rapidity with which he had to
conduct the experiment the flask in which the hydrogen was com-
pressed had not yet assumed the temperature of the resistance-
thermometer when the measurement was made.

§ 3. Estimation of the critical density.
By means of the above value of the critical temperature, of the

183

~

liquid-densities *) and vapour-pressures *) published on former occasions
and finally of the values of the constant B*), it is now possible to
make a calculation of the critical density of hydrogen assuming the
diameter of CarLrerer and Marnias to be a straight line for hydrogen ;
there is all the more reason for the latter supposition as according
to the figure the inclination of the diameter in the neighbourhood
of the critical point cannot deviate much from that at the lower
temperatures where the expansion of the liquid has been determined.
For this purpose, using the vapour-pressures and the value of B,
by means of the relation *) d4 = Eyes
A4 Ag’
ties are calculated at the same temperatures at which the density
of the liquid has been measured. The constant A4, is taken equal
REE EEE
In this manner the ordinates of the diameter are obtained and
by means of these the equation of the diameter. In the following
table ervap and orliq denote the densities of the saturated vapour
and the liquid respectively in gr. per em? and y the ordinate of the
diameter. The two points through which the diameter is laid are
indicated by an asterisk. |

1

P| the vapour densi-

TALBLECESTLE Vapour- and liquid densities of hydrogen.

| | ae Ex
y (R). | W_—R

6. | Orig. | Orvar. Jy (W).
| = 25268 | 07081 | 00135 | 03608 | «= .03604. | + .00004
mar aib | 3 8627 3626 | + I
* — 253.16 7192 | _101 | 3647 3641 0
| —25519 | 7344 | 000064 | 3704 3704 0
255.99 | 421 49 F196 | ee aT !
+7695 |. 7404 | 2. 38 3766 3766 0
— 251.23 1538 | Si | dB oh erde Wee i
— 258.27 7631 Ee ONE es 0

1) H. KAMERLINGH Onnes and C. A. CROMMELIN, Proc. XVI, 1, pg. 245,
Comm. NO. 137a. 3

2, H. KAMERLINGH OnNES and W. H. Kresom, Proc. XVI, 1, pg. 440, Comm.
N°. 137d.

*) H. KAMERLINGH ONNES and W.J. pe Haas, Proc. XV, 1, p. 405, Comm.
NO. 127c. The values of B were smoothed in the manner as given in this paper.

4) BA was found by graphical interpolation; C4 does not come into account.

184

The equation of the diameter becomes
y = — .06453 — .000398 A.
or in the more usual form
y = + .04416 — .000398 7.

Substituting in this equation our value for the critical temperature
6 = — 239.91° C. the critical density is found to be
ork = .0310.
In 1904 Dewar’) published an estimate of the density viz. .033,

calculated from a couple of liquid densities as determined by
himself. *) ;

1) J. Dewar, Proc. R. S. 73 (1904) pg. 251.

*) If the critical density is derived using the quantities dy in the small flask as
read from fig. 1, the weight of a cc. of hydrogen measured under normal
conditions and the volume of the flask as given in note 4 on page 180 the result
is Op, = -033. If the diameter is truly a straight line and thus ¥,, = -081
as given in the text, a comparison of v’ from the figure and v from the diameter
would provide a means of correcting fig. 1 for the systematic deviation (see the
note mentioned) from the true pressure density diagram. It is found that the
direction of the diameter in the neighbourhood of the critical point in the cor-
rected figure still coincides within the Jimits of accuracy with the direction holding
for lower temperatures according to the above table.

Chemistry. — “Amygdalin as nutriment for Fusarium.” By Dr.
H. I. Waterman. (Communicated by Prof. Dr. J. BonseKen).

(Communicated in the meeting of May 26, 1917).

A solution of amygdalin in tapwater which at the same time con-
tained inorganic salts as NH,NO,, KH,PO,, and MgSO, remained in
the laboratory for some time at the ordinary temperature. After 18
days spontaneous infection was observed. On the liquid a white
flocky mass of mycelium had appeared, under which a rose-coloured
underground.

From this mycelium something was transferred to a plate of malt
agar and cultivated at the ordinary temperature. After 24 hours
some growth could already be observed, after 2 x 24 hours a flocky
mycelium had been formed, whilst 24 hours later a very vigorous
development was observed. A white flocky mass of mycelium was
visible then ; the plate had obtained on some spots a yellow and on
other spots a red colour. The red colour was especially concentrated
in those places of the nutrient plate which in transferring had been
jn contact with the platinum-wire.

The following day (after 4 X 24 hours) the whole glass-box was
filled up with white mycelium.

The microscopy of the thus isolated species of mould and especially
the presence of sickle-shaped spores divided into several cells pointed
to Fusarium.

This species of Fusarium developed well on nutrient soils of the
composition: tapwater whether or not coagulated with agar and
containing 2°/, amygdalin, 0,15°/, NH,NO,. 0.15°/, KH,PO,, 0,10°/,
magnesiumsulfate (crystallised). From the means of isolation this
could be expected.

On amygdalin-agar Fusarium developed as white flocky mycelium,
whilst this nutrient plate was for the greater part coloured yellow.

Especially on this nutrient soil the formation of sickle-shaped
spores came to the front.

Some days later the yellow colour had for the greater part become
red, whilst the mycelium had shrivelled. Besides, on malt agar this
shrivelling after a prolonged cultivation was observed too.

The formed red colour of the amygdalin plate did not dissolve

13

Proceedings Royal Acad. Amsterdam. Vol. XX.

186

in boiling water. With hydrochlorie acid the colour became yellow,
with sodium hydroxide, ammonia or soda violet and otherwise. So
the colour acted as indicator.

Furthermore I observed that the infection with Fusartwm occurred
too with solutions of amygdalin, on which Aspergillus niger had
developed first (temperature 34°).

Twelve days after inoculation with Aspergillus niger 1 delivered
the solutions in question containing amygdalin from the mould
layer; the clear solutions were kept at ordinary temperature in
tumblers covered with watch-glasses. Not all amygdalin was used,
a part had remained.

Miss Prof. Dr. Jon. WesrerpijkK was kind enough to determine
the isolated Fusariwm. The found species was Fusarium discolor
var. triseptatum *).

This organism was found first by SHERBAKOFF on rotting potatoes.

It was possible that in my ease the descent would be the same,
because a few years consecutively I had experimented on potatoes
in the same laboratory.

With the isolated species of mould I made almost the same
researches as some time ago with Aspergillus niger. *)

TABLE. TL (Fusarium).

Glucose as exclusive organic Amygdalin as exclusive organic
food. food.

Composition of the culture liquid: 50 cm3 of tapwater, in which dissolved
0.15 /, NHyNOs;, 0.15% 9 KHz PO,, 0.1 % magnesiumsulfate (crystallised),
Ordinary temperature.

A. 2% glucose: 1000 milligr. B. 2% amygdalin: 1000 milligr.
ee
ate Obtained | ee | Obtained
Assimilated | : Number of | Assimilated 5
glucose | ary an cele a days after | = amygdalin | ay oer a
(milligr.) (mgr.). inoculation. | (milligr.). (milligr.).

1000 EN eee 600 | 214
15 | not determined. | 347

1000 297 | 73 | |
1000 | 300 220 | not determined. | 235
| 230 not determined. | 299

1) CG. D. Suerpaxorr, Fusaria of potatoes. Memoir NO, 6. Cornell University
Agricultural Experiment Station, May 1915. p. 239.

2) Amygdalin as nutriment for Aspergillus niger, These Proceedings Vol. XIX,
p. 922 (1917).

187

EAB EE Tia. (Fusarium).
Retarding influence of benzaldehyde and hydrogen cyanide.

Culture liquid: 50 cm3. tapwater, in which dissolved 0.15 0, NH NO3, 0.15%

KH, PO,, 0.1% MgSO4. 7 H,O and 20/, glucose. Ordinary temperature.

No. Added. ei = Â SS

Pik - db EE | EE
2 \1drop of benzaldehyde. | — ? epee yee ee A
3 \3drops of benzaldehyde. | — — = — ee
4 '5 drops of benzaldehyde. | er ee oe. | ==) |
5 | 1 cM3 P Seep kt ets
6 3 CM? P du ETA Sd A
a 5 cM P fo) REE
8 SCMEQ THE

The solution P was prepared as follows: 50 milligr.

Development after ‚Obtained dry

weight of

| mould (Mgr.)

pore eee 21 | 37 days. | after 73 days.

297 2)
324 2)

271 2)
282 2)
238 2)

KCN was

dissolved in distilled water and filled up to 100 cm*. Added 10 em?
of 0,98 X + Normal sulfuric acid. The solution Q was obtained by
adding to 100em* of H,O 10 em’ of 0,98 X +, Normal sulfuric acid.

TABL Ee TEs, (Fusarium).

50 cm3. of tapwater, in which dissolved 0.15%, NH‚;NO3, 0.15 % KH,PO,,

0.1 % MgSO,. 7 H,O. Ordinary temperature.

Development after
No. Added. Se Sa

ee ox ee 6 | 11 days.
1,2 29 glucose ie tet H+
3,4 | 2% glucose +0,04%emulsin | ++ | +4444 |H
5,6 20/, amygdalin HA | 44444444

1,8 | 2% amygdalin-+0.04% emulsin —3 | = 7)

The results obtained with Fusarium were almost quite analogous

with Aspergillus niger.

1) Sweat smell of benzaldehyde could be stated.
2) All glucose was used.
3) The smell of benzaldehyde or (and) HCN was stated.

13*

188

So it was proved:

1. Amygdalin is assimilated by Fusarium whilst young mycelium
is formed at the expense of the assimilated amygdalin. (Table I).

2. Compared with glucose amygdalin is not an inferior nutriment
at least with regard to the dry weight of mould obtained. (Table I).

3. Benzaldehyde and to a small degree HCN hinder the develop-
ment of Fusarium in liquids containing glucose (Table 11%), whilst
the addition of emulsin to liquids containing amygdalin prevents
growth entirely. The same emulsin has practically no retarding
influence on the development of Fusarium in glucose containing
solutions. (Table 116).

Therefore it is impossible that when amygdalin as only source of
carbon is assimilated by Fusarium this glucoside is dissociated to an
important degree into glucose, benzaldehyde and HCN out of the cell.

Dordrecht, May 1917.

Physiology. — “On the behaviour of the Uranium-heart towards
Electric Stimulation, as investigated by Mr. M. pen Boer.
By Prof. Dr. H. ZWAARDEMAKER.

(Communicated in the meeting of May 26, 1917).

When an isolated Kroneckered frog’s heart, after being freed from
circulation-potassium, is fed with a uranium-containing fluid (15 mgrms
of uranyl-nitrate *), 100 mgrms of calcium-chloride, 200 mgrms of
sodium-bicarbonate, 7 grms of sodium-chloride per Litre) it will soon
recover its automatical pulsations and also its mechanic excitability
in a most perfect way. If to such a heart we apply during the
diastole, electric stimuli in the form of opening induction shocks of
moderate force, the usual extrasystoles will be produced when the flow
of uranium begins. Later on when the uranium-circulation has continued,
say, 15 minutes, and when the uranium-salt has penetrated into all
lacunae and cell-walls, a short pause will still be observable after a
stimulus given at the right moment, but ultimately no manner of action
will be seen any more. When the uranium condition of the heart is
complete, the ventricle has become electrically inexcitable. Mechanical
excitability continues, though slightly diminished ’).

If the single induction shocks be replaced by a series of “making”
and “breaking” shocks, strong enough to produce extrasystoles,
something very striking takes place. We procured the said periodic
stimuli by causing the teeth of a rotating disc to be dipped into a
little mercury, the dise being so driven by a small electromotor
that the time of closure of a current (from an accumulator of 2
volts) is equal to the time of opening. In this way we effectuated
500 closures and 500 openings per minute, affording in all 1000
stimuli per minute. When they were sent in some direction or other
through the normally pulsating uranium-heart, a standstill in diastole
ensued in typical cases almost without a latent period. On discon-
tinuing the stimulation the heart resumes its beats suddenly with
normal rhythm and with perfectly regular systoles, perhaps after
a short after-effect.

1) Winter-frogs require 25 mgrms, summer-frogs from 1 to 5; in this case we

took an average quantity.
2) Cf. H. Zwaarpemaker Potassium-ion and automaticity of the Heart. Ned,

Tijdschr. v. Geneesk. 1917, I, p. 1174,

190

A totally different result will be observed when a ventricle,
brought to a complete standstill by removing the potassium, is
acted upon by the electric disc with quite the same foree. Then
the isolated heart will pulsate normally and with a rhythm about
equal to that which occurred before with Ringer-circulation. On
discontinuing the stimulation of the toothed dise the systoles will
also cease.

So there is a marked contrast. Whereas the motionless heart,
freed from circulation-potassium, will for some time resume its
normal contractility through stimulation with the toothed disc, the
normal action of the uranium-heart will cease altogether when
acted upon in the same way and with the same force. In both
cases this holds good as long as the stimulus lasts or only for'a
slightly longer duration and in both cases the tonus remains unaltered.

Under similar circumstances the usual Ringerheart is brought into
a condition of tonicity and undulation.

The alternating current just described was obtained through sudden
closures and openings of a primary current. We also used instead
of it a sinusoidal current of about 1000 complete periods per minute.
It goes without saying that a considerably stronger battery had to
be inserted in the primary circuit with changes of greater slope.
Mostly, however, 8 or 10 volts proved sufficient to achieve similar
results. We made use of platinum-electrodes. *)

The Ringer-heart did not present anything particular while the
alternating current was passing, at the most an increase of tonicity,
which rendered the pulsations somewhat more incomplete.

The motionless heart freed from circulation-potassium, recommenced
its regular beats while the sinusoidal current was passing, its systole-
form, rhythm and tonicity being equal to that of the Ringer-heart.
A latent period and an after-effect were often distinctly manifest,
mostly, however, they were scarcely visible with a stimulus of
about a minute’s duration.

The uranium-heart behaves quite differently. When for instance
the same hearts that during a standstill had been experimented upon
with sinusoidal stimuli, were restored to action by a uranium-containing
circulating fluid, a sudden standstill was seen at making the alter-
nating current, which, when the sinusoidal stimulus was discontinued
after a minute, would as suddenly make room for the normal
rhythm. There is often hardly a latent period or an after-effect.

1) According to S. Rineer (J. of Phys Vol 4 p. 372) an excessive dosis of
potassium or rubidium yields the same effect by faradization as has been described
here for the uranium-heart,

Eel

Still, it sometimes occurs that a latent period of in maximo 1
minute appears and an after-effect that will occasionally persist for
2 or 3 minutes. Then we are in a position to see that, during such
an after-standstill, the mechanic excitability has been maintained.

Whereas the Rincer-heart and the heart, fed with a potassium-
free fluid, generally stand the faint sinusoidal stimuli very well, the
uranium-heart does not. In most cases the minute-stimuli can be
applied only three times even with intervals of five minutes or
more. After that the heart is irrevocably lost (permanent standstill).

So with the sinusoidal current we also observe: a short recurrence
of the pulsations in the motionless heart freed from circulation-
potassium, arrest of the pulsations of the heart beating under the
influence of uranium. Analogous experiments may be made with
the constant current.

A normally beating RincxEr-heart is only slightly affected by weak
currents transmitted with nonpolarizable electrodes. If the force of
the current be raised to about 3 m.A., the tonus is markedly in-
creased while the current passes and a series of rapid undulations
reveal themselves in the constantly contracted heart. After breaking
the current the heart suddenly relaxes and a pause appears that
outwardly bears a resemblance to a compensatory pause after
extrasystole.

A heart that is arrested through deprivation of circulation-potassium
resumes its beats at once under the influence of a constant current
of from 1 to 3 m.A., while it as suddenly comes to a standstill
when the current is broken.

If the heart, either fresh or stimulated in the way described, is
deprived of potassium and is made to pulsate again perfectly by
uranium-containing RINGER’s mixture so long that, with fairly strong
stimuli no more extrasystoles are produced, a gradual transmission
of from '/, to 3 m.A. will bring about a complete standstill with
relaxation. Again normal systoles will recur after breaking the cur-
rent. Only in the case of superexcitation increase of tonicity is noted.
In order to observe the very remarkable arrest of the pulsations
without further by-effect it is, therefore, necessary: 1 to wait till
induetion-shoeks of moderate force are no longer succeeded by extra-
systoles; 2 not to apply stronger currents than are just necessary
for the individual heart.

The direction of the current is of no consequence, the result being
the same whether it flows from base to point or conversely. Frequent
stimuli or such as are too strong or last too long destroy the heart,

192

The pulsations become smaller and less frequent, the mechanic exci-
tability slackens and disappears.

This then is the fourth time that we find a contrast between
the condition produced by potassium-free circulation and the uranium
condition; recovery of pulsation through electrification with the one
and inhibition of the normal systoles with the other.

In a previous paper') I demonstrated an antagonism between
potassium and uranium, viz. that the two elements compensate each
other’s action when acting simultaneously in a circulating fluid.
This antagonism obtained for the heart, the vascular endothelium,
the curarised muscle ®), the kidney *). As for the automaticity of
the heart radium-, respectively mesothorium-radiation (through glass
or mica) also appeared to be antagonisti¢ to uranium. So it appears
after all that electricity in impulsive, sinusoidal periodical or continuous
form also counteracts uranium contained in a circulating fluid.
Electricity annihilates the favourable effect of uranium. On the other
hand electrification acts on the heart fed with a potassium-free
circulating fluid like potassium and like radiation. Potassium-radiation
with g-rays (and y-rays) and prolonged electric stimulation in any
form appear, therefore, in many respects to act physiologically in
the same way.

The physiological actions I have in mind are:

A heart without circulation-potassium resumes its beats.

1. with potassium *)

2. with radiation, so far investigated only in a form in which
negative charges are given to the tissue ’),
3. with electrification.

A normally beating uranium-heart is arrested:
1. with potassium,

2. with radiation °),

1) H. ZWAARDEMAKER. K. Akad. v. Wet. Amst. Proceedings 24. Fel. 1917. Vol.
25. p. 1096.

2) I. GunzBure. Congress April 1917. The Hague.

3) H. |. Hampureer (by word of mouth). Cf. also HAMBURGER and BRINKMAN.
K. Ak. Amst. Proc. 27 Jan. 1917, in which the substitution of potassium by
uranium was demonstrated with regard to the permeability of the kidney.

4) Also with other radio-active elements. H. ZwAARDEMAKER and T. P. FEENSTRA.
K. Akad. Amst. Proc. 30 Sept 1916, Vol. 25, p. 517.

5) H. ZWAARDEMAKER, C..E. BENJAMINS and T. P. FEENSTRA, Ned. Tijdseh: v.
Geneesk. 1916, II, p. 1923.

6) H. ZWAARDEMAKER, Congress. The Hague. April 1917.

193

3. with electrification.

Antagonistic to uranium, provided that the dosage of both factors
be cautiously measured ') are:

1. potassium,
2. radiation,
3. electrification.

For vagus-influences. in all these cases see an article in Arch.
néerl. de physiol. *). Probably they do not play a part in what has
been described in this paper.

1, H. ZWAARDEMAKER, K. Akad. Amst. Proc. March 1917.
®) H. ZwAARDEMAKER and J. W. Lexy Arch. néerl. de physiol. Part I, p.
745, 1917.

Neurobiology. — “A hypothesis concerning the mutual relation
between some hereditary abnormalities that occurr combined”.
By Dr. N. Voornorvr. (Communicated by Prof. Dr. I. K. A.
WERTHEIM SALOMONSON).

(Communicated in the meeting of May 26, 1917).

In the following lines I intend:

1. to explain the grounds on which my hypothesis rests that a
hereditary inferiority of the mesenchyme occurs,

2. to elucidate the significance and the scope of this hypothesis.

I. I was induced to state this hypothesis in consequence of
considerations concerning the results of an examination of three
patients, whilst an examination of the family relations afforded strong
proofs for the validity of this hypothesis.

It appeared, that a father and his two daughters were troubled
with blue sclerotics and brittle bones, a combination which, though
very rare, has already several times been described, and the heredity
of which has been proved.

The father was moreover suffering from haemophily, an abnormality,
which is likewise exceedingly hereditary.

If we account for the anatomical substratum of these abnormalities,
then it appears:

a. that the blue colour of the sclerotics is caused by their
congenital excessive thinness ;

6. that the brittle bones which in my cases proved to be the
consequence of osteopsathyrosis infantilis, is a consequence of an
inferiority of the ossificating elements ;

c. that so much may be stated with certainty concerning haemo-
phily, though its pathogenesis may not yet be perfectly clear, that
it is a consequence of an inferiority of blood resp. bloodvessels.

Consequently there existed in these individuals a hereditary
inferiority of 3 systems of organs: sclerotics, skeleton and blood resp.
bloodvessels. Their hereditary transmission pointed to an endogenous
cause, or in other words, to a defect in design.

195

The abnormalities found here belong in fact to the rarities, and
it was obvious that the combined occurrence of these abnormalities,
which in themselves are already rare, could not be regarded as merely
accidental. Consequently not 3 abnormalities existing beside and
independently of each other, but 3 consequences of one and the same
germinal defect. If this were indeed the case, there should be one
point in the embryonal development from which these three systems
of organs differentiate. And now embryology teaches us that this is
indeed the case, and that they are all three products of the mesen-
chyme. We have now therefore put forward the hypothesis, that
we have here to do with an hereditary inferiority of the mesenchyme.

The foregoing hypothesis is now supported :
A. By the results of a closer examination of the 3 patients.

It appeared namely that in these patients still other products
of the mesenchyme showed abnormalities, which were either
congenital, or consequences of an inferiority revealing itself in abnor-
mally early wastage of the respective organs.

So we find in the father (at the age of 54 years):

on both sides a very strong arcus senilis corneae ;

a rather strong sclerosis of the bloodvessels, though there were no
propitiating causes at work as lues, intoxications (lead, alcohol,
_ tobacco) or nephritis;

a rectangular position of the two auricles with regard to the skull.

In the elder girl the two little fingers were in radial adduction
in the metacarpophalangeal joint.

B. By the examination of the other members of the family.

I could obtain information about 244 members of the family
extending over 5 generations (children that died very young, are
not included). Only the branch of my patient’s father has been
included* in the scheme. Of these 59 children I myself could
interrogate and examine superficially 40. It appeared now that
already from my patient’s grand-parents haemophily occurred in
this family, ‘and that there were several cases of blue sclerotics
and also of haemophily among the 59 individuals whom I included
in the scheme. Moreover there was one case of congenital defect
of the heart, one of split palate with harelip, one of rachischisis.

This is of course the minimum of the existing abnormalities for I
could only make a thorough examination of a few of the 40
individuals whom I. saw myself. Two cases of very severe rickets

196

are likewise indicated in the scheme, in the first place, because
it is possible that someother, perhaps hereditary, abnormality of
the skeleton was present, and in the second place, because perhaps
the occurrence of severe rachitis point to an heriditary inferiority
of the skeleton.

Now it appears that all the congenital defects in this family
relate to products of the mesenchyme. There are however some
mesenchyme-derivations, where about we did not yet speak. These
are the spleen, the lymph-glands and -vessels, the conjunctive
cellular tissue and the involuntary muscles. Were these groups of
organs perhaps not defect in design? In order to find an answer
to this question we must consider the following:

We still know so little about the physiology of the spleen that we
cannot be astonished, if an inferiority of this organ, which we can
miss entirely even without any disturbance worth mentioning, does
not come to expression in a clinical respect. But moreover and
especially, it is very doubtful whether the spleen originates in tbe
mesenchyme.

As to the lymph-glands and -vessels we are struck by the fact
that so much lymph-gland-tuberculosis occurs in this family, whereas
the relation of frequency between tuberculosis of the lymph-glands
on the one hand and that of other organs on the other hand is very
large. Perhaps we may see in this fact a proof of inferiority
expressing itself in a diminished resistance of the lymph-apparatus
against infection with the bacilli of tuberculosis.

With regard to the three latter groups: conjunctive cellular tissue,
ligaments and involuntary muscles it must be boru in mind, that
the physiological signification of this tissues, hidden in the body,
is only of a secondary vital interest. And if we consider now,
that the blue sclerotics are only recognised as such, thanks to
their superficial situation, we need not be astonished that a less
strong design and a decreased power of resistance of these tissues
needs not come to expression in a clinical respect. We still remark
moreover that many women of this family and even men showed
the type of the “habitus atonicus sive asthenicus’; | leave undecided
however, whether this should be considered as a proof of a con-
genital inferiority of the ligaments.

Taking all in all it is not doubtful, but the results of the
scrutinous examination of these three patients and of the other
members of the family form a strong support for the hypothesis
stated above.

There is however more. Whilst the three patients showed us an

197

entirely developed inferiority of the mesenchyme, the examination
of the other members of the family enables us to follow the
development of all the phases of our disease and to observe the
precursors of the outbreak.

Indeed the inferiority of the bloodvessels is not seen exclusively in the
presence of haemopbily, a congenital defect of the heart shows in what
a labile equilibrium the constitution of this organic system is found,
Osteopsathyrosis does not yet occur, but the skeleton already shows
signs here and there of abnormal design, the rachischisis and the
split palate are there to prove it. Blue selerotics, although less
intensive than in my patients are already found in several
individuals, and one of them shows moreover a congenital cornea-
abnormality (embryotoxon).

The catastrophe first take place in the person of my patient and
his daughters; what had been threatening for some generations
is realised: the mesenchyme shows its insufficient design in an
unmistakable inferiority.

II]. The significance of this hypothesis is of for greater general
scope than to explain the origin of these abnormalities in my patients.

In this family we had to do with a hereditary inferiority in
design of one of the 4 great groups into which the embryonal cells
are differentiated in the very first phase of the development of the
germ. It seems to me that it is of great interest for the doctrine of
hereditary abnormalities that it is possible that germinal defects
exist affecting the individual already at so early a stage, and set
their stamp upon him.

Do such inferiorities of the whole mesenchyme or of the greater
part of it occur more? Can they likewise be restricted to the
exclusion of the mesenchyme and of the two other germinal layers to
one germinal layer? Do they likewise occur hereditarily ? Numerous
facts indeed seem to point that way. Innumerable are the questions
rising with regard to this hypothesis. It is really a working hypothesis.
And it seems to me that it is a very important undertaking to
examine, guided by this hypothesis, the combinations of hereditary
abnormalities in connection with other degenerations in one and
the same family. For what is known of these combinations or
correlations is, almost without any exception, limited to the mere
statement of the facts as such, and a dominant idea, if there was
one, in the explanation of these correlations, has not been succesful.

198

GENEALOGICAL TREE.

orbid coerul. (Grandchildren of ancestors)

m
1+ haemophily (Son. of daughter of ancestors) born lame Great-grandchildren
d

=F
et
Group B. | 7
1 hu

elle? aa + epilepsy . Page TA warf of ancestors
83 indiv. 7 cases of tuberculosis B ae i nch-back
- + tuberculosis
Group A. IN a Group B.

+ 58ca 0=9 +62 ca

se =è Ee mee

1/0 40
bo bg ddbdg Boren) 8 pied beh do

| they are indeed of no interest for
the critical examination.

ee 29". 27 25 os ren 27 26 25/9 2019 1544 +4
one Soon after zo (severe rachitis)
birth
$ (severe rachitis)
° 34 = © 53e =@ 506=@ 260=0 b 2 OROFOV OMO
KETEN x CEN ee AERTS TE VN \ / \ 24 died young)
121 bbededogsd oo EEEN seo o 18 GE
Pf 4 4 7 4 ‘ Ip, 4, rj
„tammer ssiiee 9 8 7 6 ilk Le 5 4 2h +1 +l6 + he
COT,
ve Explanation of the signs:
4 individual ages by.me, 8 haemophily. The figures placed under or beside
» n seen > » . ° . . B
5 congenital abnormality. the indiv. indicate the age in years.
d man. 1
tub i
Q woman. uber eos: . Remark. In order not to make the
4 tuberc. of lymphgland. survey too difficult, the details of
a osteopsathyrosis infantilis. = married with Group C have not been indicated;

blue sclerotics.

199

One question be still mentioned here, because I had to put it
already when studying this family:

How far can the degeneration of the mesenchyme proceed in a
family before one or more of the other germinal layers also begin to
degenerate? It is obvious, that precisely families with a defective
mesenchyme are the most suited to give an answer to this question,
in view of the comparatively less important functions of most of
the mesenchyme-organs. .

1 was able to have two members of this family and two of my
patients examined otologically (Prof. Bureer). All four of them,
though they had no complaints worth mentioning, proved to be
suffering from a labyrinth deafness, an hereditary disease par
excellence. But an affection of an ectodermal organ.

On the contrary van DER HorvE and pe Kiryn found in their
patients who suffered from blue sclerotics and brittle bones, otoscle-
rosis (mesenchyme), although they could state with one patient a
combination of otosclerosis and labyrinth deafness. 1 doubted
however immediately, whether the affection of the organ of Corti
in my patients was a primary or a secondary one. For ‘according
to the investigations of Quix and van Lennep the atropby of the
organ of Corti, is most probably caused in some cases of hereditary
labyrinth-deafness (in casu the dancing-mouse) by a primary affection
of the stria vascularis. Now the latter is a product of the mesenchyme.

And so at any rate the possibility is present that the affection
of the labyrinth in this family is a further proof of an inferior
mesenchyme, and the occurrence of this abnormality need not
lead to the conclusion that the inferiority in this family is not
restricted to the mesenchyme.

Physics. — “In what way does it become manifest in the funda-
mental laws of physics that space has three dimensions?”
By Prof. Dr. P. Exrenrust. (Communicated by Prof. Dr.
H. A. Lorentz).

(Communicated in the meeting of May 26, 1917).

Introduction.

“Why has our space just three dimensions?” or in other words:
“By which singular characteristics do geometrics and physics in
R, distinguish themselves from those in the other #,’s?” When put
in this way the questions have perhaps no sense. Surely they are
exposed to justified criticism. For does space “exist”? Is it three-
dimensional? And then the question ‘why’! What is meant by
“physics” of R, or R,?

I will not try to find a better form for these questions. Perhaps
others will succeed in indicating some more singular properties of R,
and then it will become clear to what are the “justified” questions
to which our considerations are fit answers.

”

§ 1. Gravitation and planetary motion.

As to the planetary motion, we shall see, that there is a di
between R, and R, as well as between R, and the higher R,’s with
respect to the stability of the circular trajectories. In R,‚a small disturb-
ance leaves the trajectory finite if the energy is not too great; in R,
on the contrary this is the case for all values of the energy. In B,
for n>3 the planet falls on the attracting centre or flies away
infinitely. In R, for n > 3 there do not eaist motions comparable
with the elliptic motion in R,, — all trajectories have the character of
spirals.

For the attraction under the influence of which a planet circulates i in

‘Mm
the space R,, we put aa ; to this corresponds for n > 2 a potential

energy :
Vine Mm 1

r= {eee tage tk |!

We deduce this law of attraction from the differential equation

of LAPLACE—Porsson. The means: we assume the force to be

201

directed towards the centre and to be a function of 7 only, so that
it can be derived from a potential and we shall apply Gatss’
theorem for the integral of the normal component of the force over
a closed surface (force-current).
The equations of motion thus have the form
dj, Mm vj OV
Er = — x DEE a = — DE: . (zet B
The motion takes place in a plane. In this plane we introduce
polar coordinates. Then the two first integrals can be written down

at once

m- :
OF + 0g) + POSE,
mr? ~ =O.

By elimination of p we find for 7
: ae 2V @
— - — a :
m m mr?

Sepa
pas VAR Bri OF rf)
r

That r may oscillate along the trajectory between positive values, 7
must have real and alternately positive and negative values. Therefore
the quantity from which the root has to be taken must always be
positive, between two values of r for which it is zero. The discus-
sion of the latter cases is to be found in appendix (I). There we
shall also consider the case » = 2 for which (1) has to be replaced by

V=x Mm log r
and (2) therefore by

: i ;
pW or? Br log es ae te (2*)
r
where
2E G
A it AE ale
m m

The result of this discussion is

Motions between two Mole the
n Circular trajectories positive nfinit

values of r tr
ree eed instable | impossible! | possible
Nea a Vi péssible 2 een te aa
° | sable | (moreover closed) | bee

3 Bae an Bele os possible i impossible!

Aiko (not closed) | P ;

14

Proceedings Royal Acad. Amsterdam, Vol. XX.

202

Remarks :

1st. In this connexion we may remind of the following theorem of
BerTRAND '): The trajectories of a material point described under the
influence of a force which is directed towards a fixed centre and
a function of the distance to that centre are only then closed when
the force is proportional to that distance or inversely proportional
to its square.

2nd, It is remarkable that also in a non-euclidic three-dimensional
space the planetary trajectories corresponding to the elliptic ones
prove to be closed, if the changes in the gravitation law and in the
equations of mechanics corresponding to the curvature of the space
are introduced. (Comp. LiEBMANN) ’*).

3rd, We may put the question: what does of Bour’s deduction of
the series in the spectrain R, become, ifm =|= 3. Let us change in this
deduction the law of electric attraction in the same way as that
of gravitation, and like Bonr quantisize the moment of momentum.
From the preceding it it clear that for n > 3 only circular trajectories
ean occur. For n>4 we find infinite series and for n=4 a
singular case which is particularly remarkable with respect to the
theory of quanta. (See appendix IH).

§ 2. Translation——rotation, force—pair of forces, electric
field—magnetic field.

In R, there is dualism between rotation and translation, in so
far as both are defined by three characterizing numbers. This is
closely connected to the fact that the number of planes through the
pairs of axes of coordinates equals the number of axes themselves.

In every other R, these two numbers are not equal. The number
of axes of coordinates is n. Taking two of these at a time we

n(n—1) : n(n —1)
can draw through them RE planes. Evidently BER > nm tor

n(n—1)
n > 3, while n > ae tof 0d “Clee:

for n= 2 we have only one rotation and two translations,

for n= 4 we have 6 rotations and 4 translations.

This corresponds to the dualism which exists only for n= 3
between the three components of the force and the three components
of a pair of forces which together can replace an arbitrary system
of forces.

1) J. Berrranp, Comptes Rendus. T. 77, 1878, p. 849.
3) H. LreBMANN, Nicht-euklidische Geometrie. 2e Aufl. 1912, p. 207.

nne in

203

Starting from the formulae of the theory of relativity we easily
see that also the dualism between the electric and magnetic quantities
is restricted to A,

In R,, the electric field is determined by n components, the magnetic

one by a numbers.

The space-coordinates in the (n + 1)-dimensional “world” will be
denoted by w,....«, and ¢ will be replaced by w, = ict. The electric
and magnetic forces can be deduced from an (n + 1)-fold potential

(corresponding to the four-fold retarded potential in Ry): @,, P,, -- - Pu:
The Saas components of its rotation:
Opn OPK h andk=1,...n
Or, Op == 0
give the magnetic field and the 2 components of the rotation:
Opn IP,
zen hz ns
dx, ows ( ik

the electric field.

$ 3. Integrals of the equation of vibration in R,.
(Generalization of the retarded potentials).

The integrals of the equation :
1 dp
2 Dt —A f= 0,

have the following properties in #,: If at the time t=O we have

everywhere y=:U and 1) except in a small domain y, then

dt

we have at an arbitrary later moment ¢ (if only ¢ is taken large
enough) still everywhere ~ = 0, Et except in a thin layer
between two surfaces (fig. A), which in the limit, when y becomes
small enough, become spherical surfaces with the centre at y.

In R, we have something else: here we have except a disturbance
of equilibrium between two concentric lines round y still an asymp-
totically diminishing disturbance of equilibrium in the whole exten-
sion (III) enclosed by the inner line.

In this respect all Ron41’s behave like R,, all Roy's like R, (see
appendix III).

But among the A2,4,’s Rk, is characterized by a particularity

14*

which becomes evident when the retarded potentials i.e. the integrals
of the differential equation

for R, are compared with those for the higher /2,44’s.

For Ay:
TR {fv Lel
Cc; r
For Zet
1 Ea
c | Ot
=a LIS te tS
For , :

do 070
wage ff ve en, tlie] 1 Loe,
5C,¢ 7 En Ed:
(see appendix IV).
Here C,= 42, C,=22', C,— Ha’; are the areas of spheres
with a radius equal to unity in A, R,, Rk, respectively. The symbol

0 0?
lo], Ee | expresses that the values must be taken at the

205

. ip EN . .
time £— — (the “retarded values”). While in 2, the retarded poten-
c

tials depend on @ only, we see that in &,, R, ete. they are functions
of the differential coefficients of @ with respect to the time too.

It must be remarked here that for high values of 7 (which in
radiation problems are the only ones we are concerned with) the
highest differential coefficient is the most important because here the
lowest power of 7 occurs in the denominator. An electron with sharply
bounded ebarge causes therefore by its motion high singularities.

Appendix.

I. The discussion mentioned in § 1 may be illustrated by fig. 1,
where the dotted lines give the terms Ar’? and Art” as functions
of r, while the full curve represents their sum and the horizontal
line the part C? to be subtracted. In this graphical representation
the condition is that the horizontal line cuts the full curve in
two points between which the line lies below the curve so that the
difference (Ar? + Br4—") — C° is-here positive.

For n = 2 we have added fig. 2 of analogous structure ; the lines
represent: ar? — Br? logr,*) their sum and y’. Then the condition
is satisfied.

2
) —A divided by ae is the energy a planet must have in order to be brought

Jg N= 4

207

II. That the electric attraction gives the centripetal force for
the circular motion is expressed by the equation

mrp = EE fi (A)
Bonr’s condition for the stationary circular paths gives
“it th .
mr EEE
id 27

where t is a whole number.
For the rt circle the energy is therefore

(n—2
2(n—2) )

Mn
eo oe 41?m\ 4 rer n—A4 |
he 2(n—2)

where n > 2.
For &, too we suppose the radiated frequencies to be calculable
from |

EE
Vor =S Tr
‘ h
For n=4 we have a singular case. Equation (A) becomes then

MP 2
m
so that
mr? p ze v/m.

The moment of momentum can thus have only one perfectly defined
value: eVm, so that the coefficient of attraction must be connected
with / if the quantum condition (necessarily with only one value
of r) remains. For n >4 we find

Dir =v, (0* — TE),
where x is a positive fraction in general. Thus we obtain
series in the spectrum which for constant t and increasing 6 contain
lines in the ultraviolet which become more and more distant from
one another.

III. The solution of the equation of vibration for a membrane
ean be derived from that for a three-dimensional body by supposing
in the latter case the disturbances of equilibrium to be in the
beginning independent of one of the rectangular coordinates e.g. of

2
to an infinite distance without velocity, — z divided by en the other hand the

energy required to carry il without velocity to the distance 1 from the centre,

208

z. Then spheres with a radius 7 =ct are continually cutting the
domain of the original»disturbance of equilibrium. Working out
the calculation we find that the number of integrations to be executed
if one of the coordinates does not occur is still the same as when
it occurs.') That is the reason why in A, a disturbance
of equilibrium never vanishes there where it once appeared. In
an analogous way we can pass from a solution for e,44 to one
for Re,. In this way it becomes clear that the continuation of a
disturbance of equilibrium is a common property of all A2,’s.

IV. The easiest way to find these solutions is by means of
KircHHorr’s method.*) A special solution x of the equation without
right-hand side is then used. This x is a function of tand of the dis-
tance 7 to a fixed centre P only so that the equation which is satisfied
by x, becomes in A,

1
Applying the operation La to a special solution of this
par

equation we find a solution of the same equation for + 2 instead
of for n. For odd values of n the special solution is

n—1

=D"? lele).
a(e4 7)
c

where G is an arbitrary function;
fOr n=:

1 0G 1 r
——- oralso =—F (+7)
fm Or tf C

(F an arbitrary function);
for |) == 5: IN

10 r 1 r 1 r
r Or\r c r? c ric c
etc.

Applying GREEN's identity to the required solution w and this x (e.g.
for n=5) in the whole space w outside a small sphere with
radius R round P we find

viz-stor n= dt

1) Comp. e.g. H. A. Lorentz: The theory of electrons. Note 4, p. 233.
2) See e.g. Rayreien, Theory of Sound, Ch, XIV, § 275.

209

lp 22)=
= lh (to) + ill fins

where & represents the area of the sphere and MN its normal
drawn towards w.

Now we must integrate with respect to ¢ from a high negative
value ¢, to a high positive one t, *). For the arbitrary function /’ oceur-
ring in x we take a function which is zero for all values of the
argument except for those very near zero (there we must pass to the
limit) in this way however that for zero the integral of HF over that
small domain has just the value 1. By interchanging the passage to
the limit and the integration *) and by contraction of the sphere the
identity becomes

; hak: 1 r 1 r
—3C,Wp, (i=0) = {af (ff voo el (+ =e P+).
h

or after a partial integration

es
WP, (t=0) = BC. fo

ge
A translation of the point eae then gives the Dn we want.

1) If we want to be accurate the extension, must also be delimited at the
ie
outside. For the largest value of 7 which occurs 4 + — must still be negative..

Only afterwards we pass to the limit of an infinite extension.

2) This interchange which is not further justified will be known to be charac-
teristic of KrrcHHorr’s method. Here we shall simply borrow it from KircHHorr.
If we want to execute the integration rigorously, we shall have to avail our-
selves of a method given by J. Hapamarp: Acta Math. 31 (1908) p. 333;
especially § 22. Comp. for further literature J. Hapamarp, Journ. de Phys. 1906.

Physics. — “Contribution to the research of liquid crystals. II.
The influence of the temperature on the extinction; further
experiments upon the influence of the magnetic field.” By
Dr. W. J. H. Morr and Prof. Dr. L. S. Ornstem. (Commu-
nicated by Prof. Junius).

(Communicated in the meeting of February 24, 1917).

In the further research of the liquid crystals, the results of
which we intend to communicate hereafter, the same method as
described in a former communication was used again. (These
Proc. XIX p. 1815). The method was improved in a few respects
only, the principal change being that a copper-disk with a
central hole of about three millimeter diameter takes the place
of the glass-pieces in the oven. The substance is put between two
object-glasses, which lie on the disk. In this way we get the
advantage that the matter does not come into contact with the
copper, and that it is possible to examine several substances succes-
sively with the same oven. Though the narrow hole in the copper
diminishes the intensity of the image on the thermo-pile it secures
an absolutely homogeneous heating of the very small part of the
matter under observation.

§ 1. The influence of the temperature on the extinction.

The extinction in its dependence on the temperature was measured
in the following way. The matter that has been melted before
between two glasses and congealed afterwards, is put on the oven
the temperature of which is below the melting-point. Then such
a value is given to the heating current, that in the long the
melting to isotropic-liquid will be reached. Then if after some time
the substance has been molten, the current of heating is put off
(or diminished) so that the substance gets liquid crystalline again
and congeales afterwards. In the method described before the
extinction is registered during this rising and falling of temperature.
There were examined p-azoxy-anisol, p-azoxy-phenetol, anisaldazine
and p-azoxy-benzoeacid-aethylester. In fig. 1 and 2 the curves of
extinction are reproduced of two of these substances *).

1) p-azoxy-phenetol and anisaldazine produced melting-curves of thesame charav-
ter as p-azoxy-anisol.

211

P AZOXY -ANISOL

Rising Temperature Falling Temperature
kj

P AZOXY-BENZOEZURE AETHYLESTER

eo!

Rising Temperature Falling Temperature
Fig. 2.

When we consider these curves of extinction, the fact is obvious
that the extinction in the liquid-crystalline condition, proceeding
from melting of the solid phase (“ex-solid”) is different from the
liquid-erystalline condition formed by cooling of the isotropic liquid
(““ex-liquid’””). This different extinction is accompanied by an abso-
lutely different aspect.

These differences are mostly conspicuous in the case of p-azoxy-
benzoe-acid-aethylester. With this substance the ex-solid condition
is milky-opalescent, the ex-liquid grainy-opalescent, and when the
preparation is heated the first condition always changes into the
other at the same temperature. In the curve that transition
appears by a leap-wise increase of the extinction. In the cooling-
branch of the extinction-curve we only found an indication that
at the sudden transition from liquid into liquid-crystal during a
short period the ex-solid state might inconstantly have existed as
instable, however by cooling very quickly we succeeded in obtain-
ing durably the ex-solid condition from the liquid state.

With the three other substances examined, also a very obvious
difference in extinction between the ex-liquid state and the ex-solid
state (vid. fig. 1) shows itself, but contrary to p-azoxy-benzoe-acid
aethylester, with these three substances of both liquid-crystal con-
ditions the ex-liquid one is the most opaque.

Another particularity of the extinction-curves are the different
bag-shaped drops.

The drops at the transition from liquid-crystal into isotropic-liquid
and the reverse are not real, i.e. they have no meaning for theex-
tinction as such. They are caused by the melting (resp. the getting
turbid) not occurring simultaneously in all parts of the substance.

212

By the great difference in index of refraction of the isotropic and
of the crystalline liquid, phenomena of refraction occur at the limit,
and therefore the image of the Nernst-burner is broadened, deformed
or shifted. A temporary weakening of the thermo-current, therefore
a drop of the extinction-curve will be the consequence of this.

In our opinion the slow rise of the extinction-curve when the
liquid-erystalline phase has proceeded from the isotropic-liquid phase
(the continuation of the bag-shaped drop) can be explained by
the fact that at the sudden turbidily a very disorientated state
appears, on which only very slowly the directing influence of the
glass makes itself felt.

Sometimes the extinction-curve of p-azoxy-anisol showed a peculiar
drop at the transition from liquid-erystalline into solid. In fig. 1
this drop is represented. It occurred especially when the preparation
had been examined in a very thin layer and was then strongly
undercooled. At macroscopic examination it appeared to us
that under these conditions greenish-yellow crystals were formed,
which we soon could identify with the meta-stable solid phase
already described by LieHMANN in 1890. *)

Finally from our curves the dependence on the temperature of
the extinction can be read. If we define ourselves to p-azoxy-anisol
(fig. 1) then a strong dependence on the temperature may be stated
in the ex-solid condition, in such a way that at rising temperature
the extinction decreases. Also the reverse effect, increase of the
extinction at falling temperature, could be established with certainty
for the ex-solid condition. In the ex-liquid condition the dependence
on the temperature is very much less evident, without any doubt
it exists however in the same sense as in the case of the ex-solid
state. Already in 1902 Scuenck has performed measurings on this
question with the aid of the spectrophotometer of Guan, and has
only reached a negative result. But he examined the extinction for
yellow light, whereas our method gives the extinction for a mixture
of rays in which ultra-red dominates.

So we thought it useful to examine the dependence on the tem-
perature also in another range of wave-lengths. We chose as such
the photographically active rays. The image of the Nernst-burner
was for that purpose formed instead of on the vertical slit of the
thermopile on the horizontal slit of a photographical registration-

1) Perhaps it is well to remark that we succeeded in establishing the reversible
melting-point of this greenish-yellow phase at 108°. Vorränper as well as SCHENCK |
doubt the validity of an analogeous result of LEHMANN.

213

apparatus. The line thus registrated on the sensitive paper enables
us at once to judge by its breadth and blackness of the extinction of
the substance.

As well with ex-solid as with ex-liquid the temperature appeared
to exercise a plainly perceptible influence on the extinction, but the
direction of the effect is for the photographically active rays just
the reverse as for the ultra-red rays, i.e. for the short waves at
rising temparature the extinction increases. So, as the effect of the
temperature in the case of strongly differing wave-lengths has
a right to a different sign, the apparent contradiction between the
result of ScHEeNcK and that which follows from our extinction-curves
is explained.

§ 2. The influence of a magnetic field on the extinction.

The fact that there appeared to exist two liquid-erystalline states,
made it desirable to extend our research on the influence of mag-
netic field to the second states. Besides the magnetic effect on the
three remaining substances we disposed of, had to be examined. ')

With p-benzoé-acid azoxy aethylester no influence could be esta-
blished, not even with the strongest fields we could excite, (ca. 1100
Gauss). An examination with still stronger magnetic field is being
prepared now. Anisaldazine experiences a strong influence as well |
in ex-solid as in ex-liquid state. With p-azoxy-phenetol the influence is
much weaker, but could still be observed by us with certainty in
both states. The character of the effect is for both substances prin-
cipally the same as for p-azoxy-anisol.

The magnetic effect of p-azoxy-anisol in the ex-solid condition is
represented by the figures 3 and 4. For sake of comparison we

A > E |
Fig. 3. P-azoxy-anisol ex-solid vertical field (1100 Gauss).

1) In our first communication we mentioned the preponderating influence the
nature of the adjacent surface has on the magnetic effect. In order to examine
this influence more closely we have registered the magnetic effect, for substances
of different thickness, put between glass whether or no chemically cleaned, or
enclosed between mica. The differences found, however, were only of quantitative
character.

214

reprint from our first communication the figures 5 and 6, which
represent the effect of equally strong magnetic fields on the
ex-liquid state. *)

A E |

; Fig. 4. P-azoxy-anisol ex-solid horizontal field (1100 Gauss).

A E !
Fig. 5. P-azoxy-anisol ex-liquid vertical field (1100 Gauss)
Cc
D
B
A E |
Fig. 6. P-azoxy-antsol ex-liquid horizontal field (1100 Gauss).

A comparison between figures 3 and 5 and also between figures
+ and 6 shows that the magnetic effect for ex-solid indeed differs
from that for ex-liquid. That however the difference is greatly
quantitative appears when fig. 3 is compared to fig. 7. This figure
too is reproduced from our former communication and represents
the influence of a weak vertical field on the ex-liquid state.

t) For the meaning of these figures and the method of registration we refer to
our first communication.

215

A E |

Fig. 7. P-azoxy-anisol ex-liquid vertical field (300 Gauss).

The different magnetic effect in the two states consequently can be
described as follows: a strong field acts analogeously on ex-solid as
a weak field on ex-liquid.

Finally we shall mention a magnetic effect of a particular kind. A
horizontal field causes lasting clearing up as well in the ex-solid as in
the ex-liquid state a. In our former communication we explained this
diminishing of the extinction by the fact that the particles are directed
to a high degree. Now it seemed to us of importance to examine
how a vertical field would disturb this order. Figures 8 and 9 show
this disorientating influence of a vertical field on ex-liquid and
ex-solid, when the substance has first been exposed to the effect of
a horizontal field.

A E:
Fig 8. P-azoxy-anisol ex-liquid first horizontal field, then vertical field.

A E
Fig. 7. P-azoxy-anisol ex: solid first horizontal, then vertical field.

216

The explication we gave in our first communication of the magnetic
effect, can be transmitted unchanged to the phenomena described
above. The different degree of extinction in ex-liquid and ex-solid
and the different influence thereupon of a magnetic field, we are
inclined to ascribe to the fact that the little parts have a different
directability in the ex-liquid state and in the ex-solid state.

SUMMARY.

The extinction of p-azoxy-anisol, p-azoxy-phenetal, anisaldazine and
p-azoxy-benzoé-acid-aethy lester is examined in its dependence on the
temperature.

It appears that two different liquid-crystalline states exist (“ex-solid”’
and ‘‘ex-liquid”) which possess each a different extinction, and which
undergo in a different degree the influence of a magnetic field.

The coefficient of temperature of the extinction appears to be
negative for ultra-red and positive for ultra-violet.

Utrecht, Februari 1917. Physical Laboratory
Institute for Theoretical Physics.

Zoology. — “The Fore-brain of Synbranchidae’. By Dr. C. J.
v. p. Horst, Amsterdam. (Communicated by Prof. Max Weger).

(Communicated in the meeting of May 26, 1917).

The Synbranchidae are distinguished from all other Teleosts by
a secondary coalescence of the two halves of their fore-brain.

In the rich collection of the Central Institute for Brain Research
at Amsterdam, which includes almost all the suborders of the
Teleostei, 1 found several representantives of other suborders of Teleosts
in which the hemispheres of the fore-brain are pressed together,
but where no coalescence has occurred.

In only one of the three series of Hippocampus in the Institute the two hemis-
pheres partially have grown together in the midline, dorsal from the commissura
anterior. This, however, must be regarded rather as an abnormality in this specimen
caused by the presence of parasites in the brain cavity, whereby the fore-brain
has become totally changed in form.

Of the suborder of Synbranchii, L was able to examine the brains
of Monopterus albus (Zuiew) received from Dr. Sunrer of Batavia,
and of Synbranchus marmoratus Bl, which I obtained from the
Aquarium of the Royal Zoological Society “Natura Artis Magistra”
at Amsterdam. The brains of these fishes were cut in series of
sections 20u thick, treated by the Waicert-PaL method and contra-
stained with paracarmine.

Monopterus and Synbranchus are exactly alike as regards the
formation of the brain, as I have pointed out in a previous paper (2).
The coalescence of the two hemispheres is therefore not an abnorm-
ality here as in the above-mentioned specimen of Hippocampus,
but is a typical characteristic of the family of Synbranchidae, and,
if it also occurred in Amphipnous, even of the whole order of
Synbranchii.

The outer form of the fore-brain.

The fila olfactoria are collected in a short nervus olfactorius which
forms a fairly sharp boundary with the bulb.

As in most of the Teleostei the bulbi olfactorii in Monopterus
and Synbranchus are sedentary; an elongated tractus olfactorius is

not found here.
15

Proceedings Royal Acad. Amsterdam. Vol. XX.

218

Further back the bulbi, which in comparison with the fore-brain
are considerably big, decrease in size according as the fore-brain becomes
larger. On the median side they are separated by a deep groove.
Whereas in most Teleosts with sedentary bulbs, this groove extends
over the dorsal and lateral side of the bulbus, so that the front
point of the fore-brain projects in the ventricular cavity above the

bulbi; this is not the case in the Synbranchidae; the foremost point —

of the telencephalon has already united with the bulb.

On this boundary between the bulb and the fore-brain the
ependyma, which forms the roof of the fore-brain, is attached to
the dorsal and lateral sides of the olfactory bulb, while on the
medial side the place of attachment lies on the bulb rather before
this boundary (see fig. 2). Except for a small fold on the front,
the membranous roof lies flat over the whole fore-brain. To the
ventro-lateral side of the hemispheres in the fissura endorhinalis
(not lateral from it, as in many other Teleosts), the tela choreoidea
is attached and the ependyma passes over into the subventricular
ependyma, which even extends over the hemispheres themselves.

sulcus limitans telencephali

striatum ENDS —

tub /lat.

sulcus palaeopaliio - epistriaricus

Sulcus ypsiliformis

Fig. 1. Monopteus albus. Wax model of the fore-brain Lateral side.

The fissura endorhinalis is very deep owing to the great develop-
ment of the lateral portion of the fore-brain (tuberculum laterale
and tuberculum posterius of SHELDON (6)). As SrHeLpoN has described
of the carp, so also in Monopterus this fissure deviates at the
place of the suleus ypsiliformis rather decidedly in a lateral direction.

lingua
lateralis

219

Thus seen from the ventral side, the fissura. endorhinalis is
shaped like two half ares, which form an obtuse angle with each
other. The anterior of these bounds the tuberculum laterale, the
posterior the tuberculum posterius.

Not only latero-ventrally, but also caudally the tuberculum
posterius is strongly developed like the median portion of the fore-
brain. The caudal portion in these fishes thus covers a greater part
of the thalamus. One consequence of this is that the posterior fold
of the velum transversum points forwards instead of backwards
(fig. 2). Whereas the dorsal sack (pulvinar epiphyseos) usually lies
on the roof ependyma of the fore-brain, here we see just the reverse;
the richly folded dorsal sack is covered by the ependyma of the
fore-brain which bends backwards over it.

sulcus limitans lelencephali

striatum
eptum

"@

epistriatum

velum
transversum

olfactorius

ee
: comm. ant.
nervus opticus

Fig. 2. Monopterus albus. Wax model of the fore-brain Median side.

A short distance behind the middle of the hemispheres, in a rather
frontal position consequently the suleus ypsiliformis of GorsreiN (1)
begins at the place where the fissura endorhinalis forms the afore-
mentioned obtuse angle (fig. 1). The sulcus, very deep at this place,
proceeds at first perpendicularly upwards, but later on bends in a
somewhat caudal direction. As in Cyprinus, according to SHELDON’s
description, the groove then divides. The two grooves then formed
run along the whole dorso-lateral side of the hemispheres. They
constitute the boundary between the lateral part of the fore-brain,

the palaeopallium (tuberculum laterale and tuberculum posterius),
15%

220

and the more dorsal epistriatum. These two grooves together may
thus very justly be termed the sulcus palaeopallio-epistriaticus, as
this has been described by Karpers and THEUNISSEN (5) in Thynnus.

The most posterior portion of this groove is very deep and
narrow; the boundary between the palaeopallium and the epistriatum
can therefore be drawn very sharply there. The direction of this
portion is almost caudo-frontal. Towards the front, the groove
becomes shallower and wider. It then deviates in a ventral direction
and reaches the fissura endorhinalis on the front of the hemispheres,
getting gradually fainter.

Dorsally from this sulens palaeopallio-epistriaticus there lies a
body which I take to be the epistriatum (the primordium hippocampi
of SHELDON). At least the posterior portion, limited by the deep
grooves, corresponds exactly in form to the epistriatum of Gadus,
Silurus and other fishes, as has been described by Karpers. Thus
it shows clearly a lingua lateralis descending in the suleus ypsili-
formis, as well as a lingua posterior projecting backwards. The
latter is especially distinct in Synbranchus (fig. 6). This part too
receives ‘secondary olfactory fibres from the tractus olfactorius
medialis pars lateralis, just as the lateral part of the hemispheres
lying caudally from the suleus ypsilliformis. To this very pronounced
part of the epistriatum an anterior portion joins, which is connected
with it by a narrow strip lying rather deeper, so that one might
say that it is separated from it by a broad and shallow groove.
Viewed through the microscope, these two parts merge invisibly
into each other. I therefore believe that this front portion must
also be considered as a part of the epistriatum. Like the caudal
portion it is closely connected with the palaeopallium. Both receive
secondary olfactory fibres from the tractus olfactorius lateralis. On
the other hand, it is fairly sbarply divided from the striatum, over
which it lies like a hood.

The epistriatum is bounded on the median side by the sulcus
limitans telencephali, which has been described by SHELDON, and
which forms the boundary between the corpus precommissurale
(septum mihi) and the primordium hippocampi of this author
(epistriatum mihi) (figs. 1, 2, 3, 4, 5, 6). This suleus in Monopterus
is very narrow and deep, as is seen in figs. 3, 4, and 5, specially
the posterior portion of the epistriatum (the lingua posterior) is
sharply separated by it from the other parts of the hemispheres. In
Synbranchus the suleus is not so deep, but this is secondary
compared with Monopterus, since, here and there in Synbranchus
a series of ependyma cells is found lying between the epistriatum

221

and the septum at the same place as where in Monopterus the
suleus cuts deep into the hemispheres. From this it is evident that
the suleus limitans telencephali is present at first in Synbranchus
in the same form as in Monopterus, but that later it grows together
in a similar way as in the posterior portion of the central canal
of the spinal cord, where only a septum ependymale remains.
In consequence of this the suleus is only indicated by a very faint
groove in Synbranchus.

In Cyprinus, where SueLpon has described this groove, the sulens
limitans runs entirely on the median side of the hemispheres and
only at its caudal end does it reach the dorsal surface of the fore-
brain then lying on the dorso-median side of the hemispheres. In
most of. the specimens of Teleosts, which I examined as to this,
the groove is found at the same place as in the carp. But in the
Synbranchidae the course of the sulcus limitans is entirely modified
“owing to the enormous development of the septum. This body, in
most Teleosts, covers the entire median wall of the hemispheres
ventrally from the sulcus limitans. Whereas it is comparatively
small at the front of the fore-brain there covering only the ventral
half of the median wall of the cerebrum, it grows out caudally
in a dorsal direction and finally covers the whole median side of
the hemispheres.

In Monopterus the frontal end of the sulcus limitans lies at the
same place as in other Teleosts, about half way up the median
wall of the hemispheres. From here this groove runs slightly
caudally, but then makes a sharp bend and further proceeds in a
dorso-frontal direction to the upper surface of the brain (fig. 2).
Here the groove curves gradually in a caudal direction and then
runs backwards almost parallel to the median line. (Fig. 1).

This course of the sulcus limitans is, as has been said, caused
by the enormous increase of the septum. As in Cyprinus the frontal
termination of this body occupies only the ventral half of the median
side of the hemispheres. But the greater part of it has developed
strongly in a dorsal direction. The whole median side of the
hemispheres and a part of the dorsal side are covered by it.
Moreover it protrudes there somewhat in a frontal direction covering
the striatum, in consequence of which the sulcus limitans is bent
here in a dorso-frontal direction (fig. 2). This dorsal growth also
explains why the suleus limitans cuts so extremely deep into the
fore-brain. This groove also proceeds over the posterior side of the
cerebrum and forms there, caudally from the epistriatum, the boundary
between the septum and the tuberculum posterius,

222

In many Teleosts a large part of the surface of the hemispheres
is formed by the corpus striatum. Kapprrs (4) has described this
in Gadus and Hippoglossus. On the other hand, in Synbranchidae
the corpus striatum is almost completely pushed away from the
surface, owing to the septum growing over it from the median side,
and the epistriatum from the lateral side. Only a small portion of
the striatum remains on the surface, viz. on the dorsal side of the
hemispheres, lateral from the frontal point of the septum.

As was already remarked (pag. 217), the cerebra of Monopterus
and Synbranchus are specially remarkable owing to the two
hemispheres having partially united (fig. 2).

This junction has an important influence on the relation of the
ventricular cavities.

The ventricular cavity dorsal and lateral from the two hemispheres
has been called ventrieulus lateralis by Gorpsrein (1), and the slit
between the hemispheres has been termed ventriculus medianus by
the author. This nomenclature may very suitably be employed here,
now that the two ventricle portions have been separated by the growth.
Ventrally from the coalescence between the hemispheres lies the
ventriculus medianus, for in spite of the growth the median ventricle
remains still clearly visible in these fishes, owing to the fact that the —
hemispheres of the forebrain always deviate slightly from each other
on the ventral side above the lamina terminalis and the commissura
anterior. Behind the commissura anterior the median ventricle is
connected with the recessus praeopticus, while at the frontal pole
of the cerebrum the lateral and median ventricles are continuous.
For the rest, the two ventricles are completely separated, also in the
front portion of the cerebrum. Here the two halves of the fore-brain
lie closely pressed together, each indentation in the one half being
filled out by the other half, and we frequently see a blood-vessel
passing from the one side to the other (fig. 3). Locally too the two
halves have frequently grown together; such a coalescence is rather
larger immediately caudad from the bend in the sulcus limitans
(fig. 2).

The caudal parts of the hemispheres have completely united.
The frontal boundary of this coalescence is not constant. In Mo-
nopterus it runs differently from that in Synbranchus; very probably in
individual cases it will not be constant either, and this is not sur-
prising, considering the local coalescence, which can also be found
in the frontal portion. On the other hand, the boundary behind the
level of the commissura anterior is sharply defined (fig. 2).

Between the two hemispheres, on the dorsal side, is a deep groove,

223

which penetrates to where the hemispheres lie against each other
or are connected with each other. This groove grows shallower and
fainter in a caudal direction and finally disappears altogether (Cf.
figs. 3—6).

The coalescence of the hemispheres is by no means a superficial
one, since it is accompanied by radical changes in the position of
the nuclei and in the course of the fibre tracts. Some of the fibres,
indeed, which in other fishes decussate in the commissura anterior,
here decussate above the ventriculus medianus. The small size of
the ventral commissura anterior of these fishes as compared with
that of other Teleosts, is hereby explained.

I wish to point out that it is a common feature that a part of
a commissure may cross more dorsally if a suitable commissure-bed
is present (c.f. the development of the psalterium in reptiles and of
the corpus callosum in mammals).

The nuclei and tracts in the fore-brain.

The nuclei and tracts of the fore-brain have been frequently
described, and S#HeLDoN in particular has given a most minute and
accurate account of it. [t is therefore not my intention to describe
them all again here, the more so as the position of the nuclei has
already been spoken of in discussing the morphology. I will only
say a few words concerning some fibre-tracts which differ from the
normal type in their course, and concerning the corpus striatum
which has been almost quite pushed away from the surface by the
other portions of the fore-brain (vide supra).

At the frontal part of the telencephalon, the corpus striatum is
seen for a short distance on the medio-dorsal side of the hemispheres
between the septum and the epistriatum (fig. 1). Further caudally the
growth of the septum pushes it quite away from the surface. Its shape
then is oval, in consequence of which in a cross section through this
region the septum appears narrowest in the middle (fig. 3). Further
caudally the striatum becomes broader; it spreads further in a
median direction, dislodging the two septa. This spreading of the
striata goes so far that at the level of the posterior boundary of
the commissura anterior, they grow fogether over the median line,
whereby the septum becomes divided into a dorsal and a ventral
part (fig.4). We can here distinguish a median and two lateral
portions in the striatum.

Further caudally the median connecting portion of the striata is
separated from the lateral parts more or less, This separation is

224

Sulc.lim.tel,

tr. olf.med.
p.dors.

tr. olf. med, gee
p. med.

trolfmed, "OPE:

p. lat.
Fig. 83. Monopterus albus.

sulc.lim tel. striatum

tr. olf. thal.p.dors.

epistriatum

x one vente. lat.

Eo EA tr str. thal.

ling.lat.
2.
epistr.

Sulc.yps. tr. hypoph. olf. med.

comm.olf. internucl.
tr olf. thal. p.ventr.
rec. praeopt.

tr. olf. med. p. lat

tr. str. thal.

n.opticus

Fig. 4. Monopterus,,albus.

225

carried further in Synbranchus than in Monopterus, just as in the
former the whole curious development of the fore-brain has reached
a further stage than in the latter, as was already pointed out in
discussing the sulcus limitans telencephali. The same thing is also
clear in considering the caudal end of the striata. In Monopterus
the striatum is still separated by a part of the septum from the
ventriculus medianus and the recessus praeopticus. The lateral
portions of the striata extend equally far caudally as the median
portion; in the series of sections the striatum is therefore seen to
disappear entirely simultaneously, and at the back it is covered by
the septum and the lateral portions of the cerebrum, which meet here.

In Synbranchus, on the contrary, the striatum in the middle pushes
away the ventral portion of the septum, so that the striatum lies
directly dorsally from the recessus praeopticus (fig. 6). This median
portion of the striata extends further in a caudal direction than the
lateral parts. At the back of the fore-brain the striatum is not

sulc.lim. tel. sulc. yp. p. post.

ot ee striatum

Is.

rec. praeopl. Zena
Vii?

BEZ

Hi)

_ epistriatum
3 ling. post.

tr str. that.

trolf.med. ¢f
p-lat. Lt

tr. hypoth. olf med?

tr. ol F thal.p. vente.

nucl. praeopt.p.m

nucl. pracopt.p.p.

Fig. 5- Monopterus albus.

226

plexus choreoideus

nucl. posthab. os.nucl. nucl. entopeduncularis

intermedius

comm. trans.

ou a nf
tr. praeth.cin. Í
praeth.cin 4 oe nucl. praerot.
tr. praeopt.hab. axe
p. lat. %

comm. trans.

fibrae ansulatae

Fig. 6. Synbranchus marmoratus.

entirely surrounded by other parts of the cerebrum, though the
dorsally situated septum reaches rather further caudally.

Of the fore-brain tracts the tractus olfactorius claims our atten-
tion first. |

The lateral olfactory tract (tractus olfactorius lateralis) is found
in its usual position near the fissura endorhinalis. It sends its fibres
into the lateral olfactory regions, the area olfactoria lateralis of
Karpers and Tueunissen (5). In the level of the suleus ypsiliformis
this olfactory tract has entirely disappeared. lt is my opinion there-
fore that in Synbranchidae only that part of the lateral olfactory
region which lies in front of the suleus ypsiliformis is provided
with olfactory fibres from the tractus olfactorius lateralis. According
to SHELDON in the Cyprinidae the nucleus piriformis and the nucleus
taeniae also receive fibres from the lateral olfactory tract.

In the tractus olfactorius medialis I can distinguish three bundles.
One, non-medullated, connects the septum with the bulb and is
probably the same as SHELDON describes as tractus olfactorius ascen-
dens (running frontally).

227

The course of the very small tractus olfactorius medialis pars
medialis does not differ from that in Cyprinus, while on the other
hand, the thick medullated tractus olfactorius medialis pars lateralis
takes a different course. According to SrneLpon (6), Kapprrs (4),
Go.psTEIN (1), and others, this bundle decussates with the commissura
anterior in at least the large majority of cases among Teleosts. In
Synbranchus and Monopterus, however, nothing is to be seen of
this decussation. The tract here runs somewhat in a lateral and
dorsal direction, on to the commissura anterior, and then penetrates
between the various bundles of the tractus strio-thalamicus (fig. 4).
When slightly lateral from this, i.e. dorsal from the fissura endor-
hinalis, the bundle dissolves into a dense network of fibres which
lie nearly on the boundary of the nucleus piriformis, the striatum
‘and the ventral portion of the septum (fig. 5). The fibres of this
network then spread into the nucleus piriformis and the nucleus
taeniae, which are not clearly distinguishable from each other here,
and further into the caudal part of the epistriatum. This region,
behind the suleus ypsiliformis, is thus provided with olfactory fibres
only by the lateral part of the median olfactory tract.

Very slightly candal from the place where the tractus olfactorius
medialis pars lateralis merges into the aforesaid network, medul-
lated fibres from the nucleus piriformis gather (GoLDsTEHIN’s com-
missura olfactoria internnelearis, described as a non-medullated
bundle by SHELDON under the name of tractus olfactorii mediales
partes laterales). These fibres, forming a considerable bundle, decus-
sate with the most posterior part of the commissura anterior (fig. 4).
It is possible that in this bundle there are still a few decussating
fibres- of the tractus olfactorius medialis pars lateralis; but this I
could not! determine with certainty.

In connection with the coalescence of the two halves of the fore-
brain, the course of a part of the so-called tractus strio-thalamicus
is very remarkable. The majority of tbe fibres which form this
bundle congregate, as in all Teleosts, from nearly every part of the
fore-brain and, after having decussated partly in the commissura
anterior, run medially from the fissura endorhinalis to the mid-
brain. From the most posterior part of the epistriatum (the lingua
posterior), however, a great number of medullated fibres join to
a thick bundle, which decussates somewhat further frontally in the
median striatal portion connecting the two halves of the telencephalon.
(fig. +). After the decussation this bundle runs a short distance
forward in the dorso-lateral portion of the striatum. On the frontal
level of the commissura anterior this bundle bends at a right angle

228

in a ventral direction, and joins the rest of the tractus striothalamicus.

The fore-brain of the Synbranchidae is remarkable, because it
forms, as it were, the final stage in the series of development of
the telencephalon of Ganoids and Teleosts. For, according to SHELDON,
the septum originally forms the ventro-median part of the fore-brain.
In Polypterus, which forms the first stage in this series of develop-
ment, this part still lies at its original place. The septum now
gradually grows on the median side past the striatum, whereby the
striatal portion of the ventricle wall is more and more restricted.
This process going on, in some Teleosts the striatum disappears
altogether from the ventricular wall, and the sulcus limitans
telencephali forms the boundary between the septum and the dorsal
part of the cerebrum, the epistriatum. At the caudal end of the
fore-brain this process is further advanced than at the frontal end;
the sulcus limitans thus lying in front on the median side, at the
back on the dorso-median side of the hemispheres.

Only in the Synbranchidae however, this process goes so far that
the septum reaches the dorsal surface of the cerebrum, whereby
the epistriatum is pushed aside and the sulcus limitans comes to lie
on the dorsal, or even dorso-lateral, surface of the hemispheres.

The coalescence of the hemispheres may also be regarded as a
final stage in the development. In the Ganoids the two hemis-
pheres are far apart and the ventriculus medianus is broad. This
is also the case in primitive Teleosts, such as Salmo. In other
Teleosts the hemispheres approach each other more and more, and
in most Acanthopterygii they lie right against each other; the
ventriculus medianus only being open in the ventral part above the
lamina terminalis and above the commissura anterior. In the
Synbranchidae the hemispheres, at least as regards their caudal half,
have almost entirely coalesced, and of the ventriculus medianus
only a narrow split remains, ventrally from this junction.

LITERATURE CITED.

1. Gotpstein, Kurt. Untersuchungen über das Verderhirn und Zwischenhirn
einiger Knochenfische. Arch. f mikr. Anat. u. Entw. Bd. 66. 1905.

2. Horst, C. J. van per. Die motorischen Kerne und Bahnen in dem Gehirn
der Fische. etc. Tijdschrift d. Ned. Dierkundige Vereeniging. Bd. XVI. 1918.

3. Kappers, C. U. Artins. The structure of the teleostean and selachian brain.
Journ. Comp. Neur. Vol. 16 1906.

4. Karpers, CG. U. Ariins. Die Furchen am Vorderhirn einiger Teleostier.
Anat. Anz. Bd. 40. 1911.

5. Karpers, C. U Arténs und W.F. Tueuntssen. Die Phylogenese des Rhinence-
phalons, des Corpus striatum und der Vorderhirncommissuren. Folia Neurobiologica.
Bd. [. 1808.

6. Srerpon. R. E. The olfactory tracts and centers in Teleosts. Journ. Gomp.
Neur. Vol. 22. 1912.

Astronomy. — “On the curvature of space”. By Prof. W. pr Srrrer.

(Communicated in the meeting of 1917, June 30).

1. In order to make possible an entirely relative conception of
inertia, EINsreiN*) has replaced the original field equations of his
theory by the equations

Gas RG ASS RT the gn Pre den 6

In my last paper’) I have pointed out two different systems of
Ju, which satisfy these equations. The system A is EINSTEIN’s, in
which the whole of space is filled with matter of the average
density @,. In a stationary state, and if all matter is at rest without
any stresses or pressure, then we have 7',,—0O with the exception
of 7, =49,, 9. In the system B this ‘“world-matter’’ does not exist:
we have 9, —0 and consequently all 7’,, —0. The line-element in

‘

the two systems was there found to be

ds* — — R? | dy? + sin? y [dy + sin* wy dd?]}} DS ih (24)
ds* = — R* {dwo* + sin? w [dy* + sin? x (dp? + sin? wdd’)]}. (2B)
In the system A we have:

1

i R : HOEN le Gee Sa ae
and in B:
3
= E ZN nr Ne my

À R? Qo ( B)

In the system Ay, w,% are real angles; in By and 9 are also
real, but w and x are imaginary. If, however, we put
sin wsinysin§ , mn
tan w cos fy = tan, tk.

1) A. Einstein, Kosmologische Betrachtungen zur Allgemeinen -helativitdts-
theorie, Sitzungsber., Berlin 1917 Febr. 8, p. 142.

2) W. pe Sirrer, On the relativity of inertia, these Proceedings, 1917 March 31,
vol. XIX, p. 1217.

In the footnote to page 1220 of that paper it is stated that the four-dimensional
world of the system B can be represented as a hyperboloid of two sheets in a
space of five dimensions, which is projected on a euclidean space of four dimensions
by a “stereographic projection’. This is erroneous. The hyperboloid has only one
sheet. Its projection fills only part of the euclidean space of four dimensions;
the part outside the limiting hyperboloid 1+ o6h?=0O (which is called (a) in the
quoted footnote) is the projection of the conjugated hyperboloid (which is of
two sheets).

230

where i—V —1, then ¢ and y are real and (2B) becomes:

-

r

ds* == dr" — R* sin? = [dy? + sin? yw dd?] + cos? R Gadi? (AE
If in A we also take r= Ry, then (2 A) becomes:
ds = a = [der + sin? wpd) 4+ cdt??. . . .» (4A)

The two systems A and B now differ only in g,,. For the sake
of comparison we add the system C, with

À=0 ‘ OO SEIS ae eer shee (3C)
in which the line-element is
ds? = — dr? — 7" [dy + sin? pdd"] + ed? . . . (4C)

Both A and B become identical with C for R= o.

If in A the origin of coordinates is displaced toa point ,, w,, 9,,
and in B to a time-space point w,, %,, ,. 9,, then the line-element
conserves the forms (2 A) and (2 B) respectively. These can then
again by the same transformations be altered to (4 A) and (4 B).
In A the variable ¢, which takes no part in the transformation,
remains of course the same. In B on the other hand the new
variable ¢ after the transformation is generally not the same as before.

I will put, for both systems A and B

TTS

In the system B this y is not the same as in (2 B), but it is the
angle which was called & above. I will continue to use 7 as an
independent variable, and not z.

2. In the theory of general relativity there is no essential
difference between inertia and gravitation. It will, however, be
convenient to continue to make this difference. A field in which
the line-element can be brought in one of the forms (4 A), (4 B) or
(4 C) with the corresponding condition (8 A), (3 B), or (3 C), will be
called a field of pure inertia, without gravitation. If the g,, deviate
from these values we will say that there is gravitation. This is
produced by matter, which I call “ordinary” or “gravitating”
matter. Its density is o,. In the systems B and C there is no other
matter than this ordinary matter. In the system A the whole of
space is filled with matter, which, in the simple case that the line-
element is represented by (2 A) or (4 A) produces no “gravitation”,
but only ‘inertia’. This matter I have called ‘“‘world-matter’. Its
density is @,. When taken over sufficiently large units of volume

wo

this g, is a constant. Locally however it may be variable: the
world-matter can be condensed to bodies of greater density, or it
can have a smaller density than the average, or be absent altogether.
According to HiNsrriN's view we must assume that a// ordinary
matter (sun, stars, nebulae ete.) consist of condensed world-matter,
and perhaps also that all world-matter is thus condensed.

3. To begin with we will neglect gravitation and consider only
the inertial field. The three-dimensional line-element is in the two
systems A and B:

do? = dr? + RP? sin? = [dy* + sin? pd].

If AR? is positive and finite, this is the line-element of a three-
dimensional space with a constant positive curvature. There are
two forms of this, viz: the space of Riemann’), or spherical space,
and the elliptical space, which has been investigated by Newcoms ®).
In the spherical space all “straight” :i.e. geodetic) lines which start
from one point, intersect again in another point: the ‘‘antipodal
point”, whose distance from the first point, measured along any
of these lines, is 7. In the elliptical space any two straight lines
have only one point in common. In both spaces the straight line is
closed; in the spherical space its total length is 2A, in the ellip-
tical space it is wh. In the spherical space the largest possible
distance between two points is af, in the elliptical space ja A.
Both spaces are finite, thongh unlimited. The volume of the whole
of spherical space is 21°, of elliptical space z° A. For values of
r which are small compared with A, the two spaces differ only inap-
preciably form the euclidean space. |

The existence of the antipodal point, where all rays of light
starting from a point again intersect, and where also, as
will be shown below, the gravitational action of a material point
(however small its mass may be) becomes infinite, certainly is a
drawback of the spherical space, and it will be preferable to assume
the true physical space to be elliptical.

The elliptical space can be projected on euclidean space by the
transformation

fat ian pe ese ee, re eet Ae)

The line-element in the systems A and 4 then becomes

1) Ueber die Hypothesen welche der Geometrie zu Grunde liegen (1854),
2) Elementary theorems relating to geometry of three dimensions and of
uniform positive curvature, CRELLE’s Journal Bd. 83, p. 293 (1877).

232

ds? = — oo Age ae ee os sl Herdt. . (6A)
¢ aE z) Lit a

AR a a r [dyw? + = w dd? | a ijs a _ (6B)
(8 RET AET

For r=o in the system A all g,, become zero, with the excep-
tion of g,,, which remains 1. In the system B g,, also becomes zero.

4. The world-lines of light-vibrations are geodetic lines (ds = 0)
in the four-dimensional time-space. Their projections on the three-
dimensional space are the rays of light. In the system A, with the
coordinates 7,w,0, these light-rays are also geodetic lines of the
three-dimensional space, and the velocity of light is constant. In
the system £/ this is not so. The velocity of light in that system is,
in the radial direction, v= ccos y. It is possible, however, in B to
introduce space-coordinates, measured in which the velocity of light
shall be constant in the radial direction. If the radius-vector in this
new measure is called 4, we have

cos ¥ dh = dr

The integral of this equation is

a] h ; r 7
8 — == sooth Be ee os area ha
nh R an ; (7)

R

In the system A we can, of course, also perform the same trans-
formation. The line-element becomes

h
— dh? — sinh? = [dp? + sin? p dd]

ds? == 7 ad. {SAN
cosh? as
R
i
— dh? — sinh? 7 [dp? -+ sin? ydd?] He dt?
OF ale Svea SEN
h
cosh? —
Jan

The three-dimensional line-element
/
dot a [dep? + sin? yp 19°]

is that of a space of constant negative curvature: the hyperbolical
space or space of LoparscHewsky. When described in the coordinates
of this space, the rays of light in the system B are straight (i.e.
geodetic) lines, and the velocity of light is constant in all directions,

233.

although the system of reference was determined by the condition
that it should be constant in the radial direction.

In this system of reference also all y,, are zero at infinity in the
system B, and in A all g,, soepie Jas. Which remains 1.

To h=o corresponds r=}a A. The whole of elliptical space
is therefore by the kann (7) projected on the whole of
hyperbolical space. For values of 7 exceeding }aR, h becomes
negative. Now w point (—A, ®, 9) is the same as (h,a—y, ax + 9).
The projection of the spherical space therefore fills the hyperbolical
space twice. The same thing is true of the projection, by (5), of
the elliptical and spherical spaces on the euclidian space.

5. Let the sun be placed in the origin of coordinates, and let
the distance from the sun to the earth be a. We still neglect all
gravitation.

In the system A the rays of light are straight lines, when de-
scribed in-the coordinates 7, y, 9, i.e. in the elliptical or spherical
space.

In the system B the same is true for the coordinates A, w, 9
(hypervolical space).

In the system A, consequently to triangles formed by rays of
light, the ordinary formulas of spherical trigonometry are applicable.
The parallax p of a star whose distance from the sun is 7, is thus
given by the formula

a

z
tan p = sin — cot—,
R

R

The square of a/R being negligible, we can write this

a Le a
Ba GO Soni ty TEN SAN oee AEN
In the system B we have similarly, in the coordinates h,w, 9:
hed h
tan p = sinh 2 coth R’
or
a h
D= pe ae ae a Gel 145 RER ej

In the system A we have consequently p= 0 for r=}ahk, ie
for the largest distance which is possible in the elliptical space. If
we admitted still larger distances, which are only possible in the
spherical space, then p would become negative, and for r= aft
we should find p= — 90°.

16

Proceedings Royal Acad. Amsterdam. Vol. XX.

234

In the system B p has a minimum value

aul ©

which it reaches for h=o, ie. r= tak. For values of r exceeding
this distance p increases again, and for »=2R we should find
p=} 90°.

Already in 1900 Scuwarzscui.p') gave a discussion of the possible
curvature of space, starting from the formulae (9A) and (9B). For
the system B we can from the observed parallaxes?) derive a lower
limit for ZJ. Senwarzsemup finds R > 4.10° astronomical units. In
the system A the measured parallaxes cannot give a limit for R.
| In both systems we can, of course, derive such a limit from

distances which have been determined, or estimated, otherwise than
from the measured: parallaxes. These distances must, in the elliptical
space, be smaller than 4a. This undoubtedly leads to a much
higher limit, of the order of 10'° or more.

6. The straight line being closed, we should, at the point of
the heavens 180° from the sun, see an image of the back side of
the sun. This not being the case, practically all the light must be
absorbed on the long “voyage round the universe”. ScHWARZSCHILD
estimates that an absorption of 40 magnitudes would be sufficient *).
If we adopt the result found by SHapiny‘), viz. that the absorption
in intergalactic space is smaller than 0™.01 in a distance of 1000
parsecs, then for an absorption of 40 mags we need a distance of
7.10" astronomical units. In the elliptical space we have thus
Bs 10.

In the system A we can suppose that this absorption is produced

1) Ueber das zuldssige Kriimmungsmaass des Raumes, Vierteljahrsschrift der
Astron, Gesellschaft, Bd. 35. p. 337.

*) The meaning is of course actually measured parallaxes, not parallaxes derived
by the formula p=a/r from a distance which is determined from other sources
(comparison of rad al and transversal velocity, absolute magnitude, etc.). SCHWARZ-
SCHILD assumes that there are certainly stars having a parallax of 0” 05. All
parallaxes measured since then are relative parallaxes, and consequently we must
at the present time still use the same limit.

*) It might be argued that we should not see the back of the actual sun but
of the sun as it was when the light left it. We could thus do without absorption,
if the time taken by light to traverse the distance af exceeded the age of the
sun. With any reasonable estimate of this age, we should thus be led to still
larger values of R.

*) Contributions from the Mount Wilson Solar Observatory Nrs. 115—117.

235

by the world-matter. It is about '/,, of the absorption which Kina *)
used in his calculation of the density of matter in interstellar space.
The density of the world-matter would thus be about '/,, of the
density found by Kine, or e,=#"10-!* in astronomical units.
The corresponding value of A (see art. 8) is R=—= 2:10". The total
absorption in the distance +R would then be only 3.6 magnitudes.
To get the required absorption of 40 magnitudes we must increase
9, and consequently diminish R. We then find 9, =2:°10-”,
R=2:10’. This value of course has practically no weight, as it
is very doubtful whether the considerations by which Kine derived
the density from the coefficient of absorption are applicable to the
world-matter. ;

The whole argument is inapplicable to the system B, since in
this system the light requires an infinite time for the “voyage round
the world’, One half of this time is

IR

1
PS Sd:
v
0

and, since v= c cos x, we find 7’ = o.

7. In the system A g,, is constant, in B g,, diminishes with
increasing rv. Consequently in B the lines is the spectra of very
distant objects must appear displaced towards the red. This dis-
placement by the inertial field is superposed on the displacement
produced by the gravitational field of the stars themselves. It is
well known that the Helium-stars show a systematic displacement
corresponding to a radial velocity of + 4.3 Km/sec. If we assume
that about */, of this is due to the gravitational field of the stars
themselves *), then there remains for the displacement by the inertial
field about 3 Km/sec. We should thus have, at the average distance
of the Helium stars

f= 1— 2.10-5 = cos? —.
3 R

If for this average distance we take 7 = 3-10’ (corresponding
to a parallax of 0"-007 by the formula p = a/r), this gives R = $. 10%. ©
Also for the M-stars, whose average distance is probably the largest
after=that of the Helium-stars, CAMPBELL*®) finds a systematic dis-
placement of the same order. The other stars, whose average dis-

1) Nature, Vol. 95, p. 701 (Aug. 26, 1915).

2)-Cf. pe Sirrer, On Einstein's theory of gravitation and its astronomical

consequences, Monthly notices, Vol. 76, p. 719.
3) Lick Bulletin, Vol. 6, p. 127.

16*

236

tances are smaller, also have a much smaller systematic displacement
towards the red, which can very well be explained by the gravita-
tional field of the stars themselves.

Lately some radial velocities of nebulae‘) have been observed,
which are very large; of the order of 1000 Km/sec. If we take
600 Km/sec., and explain this as a displacement towards the red
produced by the inertial field, we should, with the above value
of PR, find for the distance of these nebulae 7 = 4: 10° = 2000
parsecs. It is probable that the real distance is much larger. ?)

About a systematic displacement towards the red of the spectral
lines of nebulae we can, however, as yet say nothing with certainty.
If in the future it should be proved that very distant objects have
systematically positive apparent radial velocities, this would be an
indication that the system B, and not A, would correspond to the
truth. If such a systematic displacement of spectral lines should be
shown not to exist, this might be interpreted either as pointing to
the system A in preference to B, or as indicating a still larger
value of A in the system J.

8. In the paper which has already repeatedly been quoted,
SCHWARZSCHILD determined the value of A for elliptical space by the
condition that space should be large enough to contain the whole
of our galactic system, the star-density being taken constant and
equal to the value near the sun. This reasoning cannot be applied
to the system A, since the field-equations give a relation between
M and oe, which contradicts ScHWARzsCHILD’s condition.

We have

2
HO a

The volume of the elliptical space is a? k*. The total mass is
therefore a? F* 9,, or

1) NG.C. 4594 PS oe + 1180 km/sec.

SLIPHER +1190 ,
N.G.C. 1068 GSE + 1100 =
Pease -+ 765 je
(Mowe Se oO) as

The nebula in Andromeda however appears to have a considerable negative
velocity, viz. :
WRIGHT — 304 km/sec.
PEASE — 3829 „
SLIPHER — 300 „
2) EpDINGTON (Monthly Notices, Vol. 77, p. 375) estimates 7 > 100000 parsecs.
This, combined with an apparent velocity of + 600 km/sec., would give R > 3.10".

If we take for M/ the mass of our galactic system, which can be
estimated ') at $°10'° (sun = 1), then the last formula gives R = 41,
or only about 1'/, times the distance of Neptune from the sun. This,
of course, is absurd. If we use the other formula we can take for
9, the star-density in the immediate neighbourhood of the sun, which
we estimate at 80 stars per unit of volume of Kapreyn (cube of
10 parsecs side), or o, = 10—'7 in astronomical units. We then find
R=9:10". The total mass then becomes M=7- 10", and con-
sequently the galactic system would only represent an entirely
negligible portion of the total world-matter.

It appears probable for many different reasons that outside our
galactic system there are many more similar systems, whose mutual
distances are large compared with their dimensions. If we take for
the average mutual distance 10° astronomical units, then an elliptical
space with R=9-10" could contain 7-10° galactic systems, of
which of course only a small number are known to us by direct
observation. If, however, they all actually existed, and their average
mass were the same as of our own galaxy, then their combined
mass would be about 2:10'®, and consequently only one three-
thousandth part of the world-matter would be condensed to “ordinary”
matter. It is very well possible to construct a world in which the
whole of the world-matter would, or at least could, be thus condensed.
We must then for @, take the density not within the galactic
system, but the average density over a unit of volume which is
large compared with the mutual distances of the galactic systems.
With the numerical data adopted above, this leads to R= 5-10",
and there would then be more than a billion galactic systems.

All this of course is very vague and hypothetical. Observation
only gives us certainty about the existence of our own galactic
system, and probability about some hundreds more. All beyond this
is extrapolation.

9. We now come to the case that there is gravitation, which is
produced by “ordinary” matter, with the density g,. L will consider
the field produced by a small sphere at the origin of the system
of coordinates, which I will call the “sun”. Its radius is R.

In the system A the world-matter has thus everywhere the
constant density @,, except for values of » which are smaller than

1) Communicated by Prof. KapTEyn,

238

k, 1,e. within the sun. There the density >) is @ =g, 4e. In the
system B, we have eg =g,, and this is zero except for r< R.
The line-element then has the form
ds* = — adr* —b[dp* + sin? w d9?] + fe'dt*,
and in a stationary state a, 6, f are functions of r only. The equations
become somewhat simpler if we introduce
biga m= les Rim iloe 2
If differential coefficients with respect to r are indicated by accents
we find
G,, =m" + 4n"+ bm! (ml) + An! (nl —D,
a

ze Ge =— + Fm + tm (2m + n'— 1),

ar GG, =d" + An (2m' 4 n'—1),
Gis sini G
In order to write down the equations (1) we must know the values
of 7. If all matter is at rest, and if there is no pressure or stress
in it, these are: 7,,=9,,0, all other 7,,=—0. These values I call
1’. If we adopt these, then the equations (1) become, after a simple
reduction

n+ n'(m'+ 3n'—11)=a(uo— 2d) . . . (10)
ma don! (il =a! Saa ob ooh ae
gm im) a. ESE.

It is easily verified that these are satisfied if we take o= 0,
and for g,, we take the values corresponding to one of the forms
(44), (45), or (4C) of the line-element, with the conditions (3A),
(3B), or (BC) respectively. Similarly for (6.4), (6B) and (8A), (8B),
if the accents in (10), (41), (12) denote differential coefficients with
respect to r, or fA respectively. Consequently in the field of pure
inertia we have 7’,,= 7’, ie. by the action of inertia alone there
are produced no pressures or stresses in the world-matter.

') This, of course, is not strictly in accordance with Einstein's hypothesis, by which
the condensation of the world-matter in the sun should be compensated by a
rarefying, or entire absence, of it elsewhere. The mass of the sun however is
extremely small compared with the total mass in a unit of volume of such extent
as must be taken in order to treat the density of the world-matter as constant.
Therefore, if we neglect the compensation, the mass present in the unit of volume
containing the sun is only very little in excess of that present in the other units.
In the real physical world such small deviations from perfect homogeneity must
always be considered as possible, and they must produce only small differences in
the gravitational field.

239

If however the mass of the sun is not neglected, then a stationary
state of equilibrium, with all matter at rest, cannot exist without
internal forces within this matter. The 7, are then different from
FT. If the world-matter is considered as a continuous “fluid”, then
this fluid can only be at rest if there is in it a pressure or stress.
If it is considered as consisting of separated material points then
these cannot be at rest. The difference 7’,,—T7,,° vanishes with g,
for if e=0, both 7, and 7%, are zero. This difference, therefore, is
of the form e.o, € being of the order of the gravitation produced by
the sun. The right-hand-members of the equations (1), and therefore
also of (10), (11), (12) require corrections of the order x.e.0. If
these are neglected, the equations are no longer exact.

10. The mass of the sun being small, the values of a,b, f will
not differ much from those of the inertial tield. We can then, in
the system A, and for the coordinates 7, y, 3, put

a=l+a ; b = R* sin? y (1 + 8) ‘ f=l+y,
and in a first approximation we can neglect the squares and products
of a5 8, y. The equations then became:

2
eae es os Heke atk oa eee Oe
i cot Y U ! | 2a
a+ 2% opal — yy) + Fang... (14
RE „ Cot X
Boose x — ac YH (8 + ¥) =~ = : Teeth)

From (13) we find, remembering that the accents denote differen-
tiations with respect to r= fA. y: :
n
y' sin? y= faxo, sn’ x dr |
0

Outside the sun we have o,=—=0. Thus if we put
R

— Rf a xo, sn? ydr
0
then outside the sun .

from which

a a
AP Ss ae CE == Arade oe eN (16)
Ag R X .

240

For r=3a2 8, ie. for the largest distance which is possible in
the elliptical space, we have thus y= 0. For still larger distances,
which are only possible in the spherical space, y becomes positive,
and finally for == a A we should have g,, =o, however small
the mass of the sun may be, as has already been remarked above
(art. 3).

If now from (14) and (15) we endeavour to determine «a and 8,
we are met by difficulties. It appears that the equations (13), (14), (15)
are contradictory to each other. If we make the combination

(13) 48 SD (15) Ran C)

we find
VHAN = 0,2 nun er oe eee ke CEN
which is absurd. If the equations were exact, they should, in con-
sequence of the invariance, be dependent on each other. They are
however not exact, since on the right-hand-sides terms of the order
of e.xg have been neglected, ¢ being of the order of a, 8, y. In
the world-matter we have’) x9 = xo, = 22, and these corrections
can only be neglected if 4 is also of the order «. This has not been
assumed in the equations (13), (14), (15). If we wish to assume it,
then we must also develop in powers of 4 We can then use the
coordinates r, wy, 9. We put thus
. Ga oe =; tb ae eA Se) a ee ey
The equations, in which now the accents denote differentiations
with respect to r, then become, to the first order

Ww 2 |
Y ied = %Q,,

2 ae
PB eter 8,

B—a+r( + y)=— Art,
which are easily verified to be dependent on each other.
We can thus add an arbitrary condition. If we take e.g.
a= 2B,
then we find, to the first order, outside the sun

a a a
a= Ur He , BEA td 5 y=
r If r

1) Of course, if beside the world-matter there is also “ordinary matter”, 1. e.
if the density of the world matter is not constant, this relation is also only
approximatively true, and requires a correction of the order ), ¢. (See also art. 11).

241

R
where a= fx vy, dr. If a is neglected these are the terms of the
0
first order in the development of (6 A) in powers of à=!/R*,

11. Consider again the equations (10), (11), (12). If these were
exact, they would be dependent on each other. They are, however,
not exact, and consequently they are contradictory. If we make
the combination:

d (12)

2.—-
dr

LZ fen! = Ee (12) = Em! en} (LI) — 10),

we find‘)
VER AME He Anner oe

Consequently the equations are dependent on each other, i.e. a
stationary equilibrium, all matter being at rest without internal forces,
is only possible, when either 9 =O or n’ = 0, ie. g,, = constant.
In the system A g is never zero, since outside the sun e=@,. A
stationary equilibrium is then only possible if g,, is constant, Le.
if no “ordinary” matter exists, for all ordinary matter will, by the
mechanism of the equation (10) or (13) produce a term y in g,,
which is not constant. If ordinary or gravitating matter does exist
then not only in those portions of space which are occupied by it,
but throughout the whole of the world-matter 7, will differ from
T°. We can e.g. consider the world-matter as an adiabatic incom-
pressible fluid. If this is supposed to be at rest, we have

Le = gap VT Ii
where p is the pressure in the world-matter. I then tind

zel)
Re VF

and, to the first order, and for the coordinates 7, yp, #:

cos 2 y +5)
es Rsiny | RB)’

a , 4
x0, = 2A — Itek 1-35 ;

For our sun a/R is of the order of 10.
For y=i2.we have y=0O, and for ,=2 we should find

1) It is easily verified that (18) becomes identical with (17) if all terms of
higher orders than the first are neglected.

242

y=, as in the approximate solution (16), in which p was neglected.
For the planetary motion we must go to the second order. I
find a motion of the perihelion amounting to
; d= SEE tie ae). ee. EN
which is of course entirely negligible on account of the smallness
of 2a’. In my last paper’) it was stated that there is no motion
of the perihelion. In that paper the values 7’,,° were used, i.e. the
pressure p was neglected. The motion (19) can thus be said to be
produced by the pressure of the world-matter on the planet. It will
disappear if we suppose that in the immediate neighbourhood of the
sun the world-matter is absent.

12. In the system JS outside the sun we have g =O, and the
equations are dependent on each other and can be integrated.

Within the sun n’axy, must be of the second order, and conse-
quently »’ must be of the first order. If we put

f = cos" x (t+ 7);

2 y’ tan ¥ .
then n’ =— — tan y + ——, thus —— must be of the first order.
R l+y RK
Since y=—7/R we find that 1/R? must be of the first order, as in
system A.
Developing f in powers of 1/R we find, to the first order
ri
SS le R? =e Ys
In the first approximation we find for y the same value as in
the systems A and C, viz: y= — a/r. Here however we have also
the term —’/r. Thus classical mechanics according to NEWTON’s

law can only be used as a first approximation if this term, and
- consequently also 4—= °*/p: is of the second order. Investigating the
effect of this term on planetary motion, we find a motion of the peri-
belion *) amounting to

3a°
= Ut
2a A?

=

OE bs

1) These Proceedings, Vol. XIX, page 1224.
2) In my last paper (these Proceedings Vol. XIX, p. 1224) I found

sa? ent?
——— nt — ——.
4a R? 2h?
The difference is due to the use of a different system of reference, with a
different time and different radius-vector, in the two cases, the formulas for the

transformation of the space-variables (especially the radius-vector) from one system
to the other depending on the time.

da =

243
From the condition that this shall for the earth not exceeds ay
2" per century we find
R> 10°.
Then */22<C10—'® is actually of the second order compared with

x— 25°10-8. This limit of A is still considerably lower than the
value which was found above from the displacement of the spectral

lines. For the planetary motion — and generally for all mechanical
problems which do not involve very large values of r — we can

therefore in both systems A and 5 neglect the effect of A entirely.

Chemistry. — “Jnvestigations into Pasteur’s Principle of the
Connection between Molecular and Crystallonomical Dissym-
metry: TIE. Racemic and Optically Active Complex Salts of
Trwalent Rhodium. By Prof. Dr. F. M. Jaxerr.

(Communicated in the meeting of June 30, 1917.)

§ 1. In immediate continuation of previous investigations concern-
ing the complex ¢rv-ethylenediamine-salts of trivalent cobaltum 9,
the results of the study of the corresponding complex salts of trivalent
rhodium are communicated in the present paper, namely in so far
as the data already obtained possibly allow a more general discussion
of some problems connected with this subject.

For the preparation of the salts investigated, we started from pure
rhodium-trichloride. Originally Wernegr’s method *) was followed, who
transformed the chloride into sodium-rhodium-chloride: Na,}Rho Clt +
+12H,0, which afterwards was dissolved in gently heated
ethylenediamine-hydrate. It appeared, however, that the troublesome
purification of the complex salt from the adhering sodtumchloride
may be prevented, when the rhodium-trichloride itself is dissolved
immediately into the base mentioned, on gently heating it with the
last. If some few precautions be taken, the required substance is
obtained from an aqueous solution immediately in very beautiful,
colourless crystals. It must, moreover, be remarked in this connection,
that a miscibility between the complex rhodiuwm-salt and sodium-
chloride in the solid state, as supposed by the author mentioned
because of an eventually existing isomorphism of two crystal-species
of cubic symmetry, does in reality not occur. No such isomorphism
is present here, as the complex rhodium-salt has ditrigonal, the
sodium-chloride on the contrary cubic symmetry. Probably only
occlusion or mechanical removal of the one salt by the other may
be the cause of the phenomenon observed. Moreover, it must appear
remarkable, that almost all data concerning the amount of water
of crystallisation present in these rhodium-salts, differ exactly by the
number of half a molecule H,O in comparison with those of the
corresponding cobalti-salts. As the last mentioned salts are, however,

Ly ESM. JAEGER, Proceed. Kon. Akad. Amsterdam, 17. 1217; 18. 49. (1915);

Zeits. f. Kryst u. Miner. 55. 209. (1915).
2) A. Werner Ber. d. d. Chem. Ges. 45. 1228. (1912).

245

in by far the most cases, directly isomorphous with the rhodium-
salts of corresponding constitution, suspicion arises that the content
of crystallisation-water is really the same in the two series of com-
plex salts. The divergence mentioned would then be explained by a
systematical error in the rather difficult determinations of the element
rhodium in these cases.

§ 2. In the following pages the result of the direct measurements
of the racemic salts and of the optically active components are in
the first place dealt with, and the data concerning the rotation-
dispersion of these perfectly colourless salts are subsequently
communicated. Some considerations of a more general character,
regarding the stereometrical configuration of these complex substances
will then be inserted.

The results of the study of some other compounds of this series
will be published later-on.

SODIUM-RHODIUM-CHLORIDE.

{Rho Cl, Na, + 12H, 0.

This salt may be prepared by dissolving the calculated quantity
of rhodium-trichloride in a concentrated solution of sodiwm-chloride,
evaporating the solution on the water-bath, and by crystallisation
at room-temperature. The solution of the black-red, often voluminous
and apparently octahedral crys-
tals, has a beautiful crimson
colour. The crystals are hygros-
copical, and they soon become
dull, when softly heated.

Monoclinic-prismatic.
a: 6: e=1,2034:1:1,4576;
Dr ao coe,

Forms observed: c = {001},
small but lustrous; ordinarily
the facet (001) appears much
broader than (001); 7, = {101},

Fig. .1 r, = {101}, and g= {014}, com-

Sodium-Rhodium-Chloride. monly equally well developed,

and giving good images; /—= {010}, narrow, mostly absent; s = {021},

very narrow, often absent, but sharply reflecting; m — {430}, small,

dull, and often absent. The external shape is apparently octahedral,
and isometrically developed.

246

Angles: Observed:

Gee

r = (001): (101) =* 71932

r‚:rs!=(101):(101) =* 63 8

3 Or MS

: q =(001):(Ol1) =* 50 46
: p=(001):(101)= 45 30
: g=(011):(011)= 78 28
: g=(010):(011)= 39 20
sg = (101) NOLS 63 AT
: q=(101):(Oll)= 77-0
“sE (001): (021) = 67 56
: m= (010): (430) = 52 40

m:m = (430):(430)= 74 40
The erystals are cleavable parallel to {001}.

Calculated:

45920’
18 28
39 14
63 41
Tl 4
67 47'\,
52 49
74 22

$ 3. RacrMic TRI-ETHYLENEDIAMINE-RHODIUM-CHLORIDE.
{Rho (Hine),} Cl, + 8H,O0.

The salt was prepared from the corresponding iodide in solution,
by treating it with freshly precipitated silwer-chloride. The substance
erystallises in big, rectangular-shaped, prismatic individuals, or in
smaller, strongly refracting crystals, which often possess uneven faces
yielding multiple reflexes.

Fig. 2.
Racemic Tri-ethylenediamine-Rhodium-Chloride.

Ditrigonal-scalenohedral.

a2 C= 1 Ob Fa0:

Forms Observed: c = \00013, predominant, often rough, but lustrous;

m = {1010!, well developed, always present, and lustrous; » = {1011},

narrow, often absent; a = {1101}, smaller than 7, and only rarely
present. The crystals are plates parallel to {0001}, or prisms parallel

to the c-axis.

The substance is perfectly isomorphous with the corresponding

247

Co-salt'); WeRNer’s supposition of their being cubic and isomorphous
with NaCl, is erroneous, as has been already said.

Angles: Observed: Calculated:
c:r — (0001) :(1011) =* 37° 51’ 3
m:r —=(1010):(1011) = 52 9 52o 9
m.:m=(1010):(0110) = 60 0 60 0
ce:x =(0001):(1101)= 38 1 37 51

No distinct cleavability was found.

The crystals are uniaxial; their birefringence is negative. The
interference-image is sometimes slightly disturbed, as was observed
also in the case of the Co-salt.

The corresponding bromide was completely isomorphous with the
chloride here described.

§ 4. Racemic Tri-ETHYLENEDIAMINE-CHROMI-CHLORIDE,.
{Or (Hine),} Cl, + 3H, 0.
Beautiful, orange, very transparent and lustrous prismatic erystals,
which commonly only exhibit the forms c and m. Occasionally also

hexagonal plates occur parallel to {0001}; once a form r = {1011}
was observed as a very narrow truncation of the edge c: m.

fad ea
1
an
m | mv

genen

Ig.-ö.
Racemic FOG Ree ee cea
Ditrigonal-scalenohedral.
a:c = 1: 0,6930.
Forms observed: c = 0001}, and m= 11010!, both very lustrous,
while sometimes c, sometimes however m is developed predominantly;
r = 1011}, mostly absent, and in all cases very narrow.

tj F. M. Jagger, Proceed. Kon. Acad. Amsterdam, 18. 50. (1915).

248

Angles: Observed: Calculated:
Cr ==(0001)) ONS SEO Oe oe
rm —=(10Ul) (010) = “5420 51° 20
m:m= (1010): (0110) = 60 0 60 0

No distinet cleavability was observed.

The crystals show a feeble dichroism: on m they are orange-
yellow for vibrations parallel to the c-axis, and lemon-yellow for
such as are perpendicular to that direction. Optically uniaxial, occasion-
ally with disturbed interference-image. The character of birefringence
is negative as it was in the cases of the Co-, and Aho-salts.

§ 5. LARVOGYRATE TRI-ETHYLENEDIAMINE-RHODIUM-
-CHLORO-D-TARTRATE.

Cl
l-Rho (Eine), + 4H,0.
d-C,H,O,

Beautiful, rather big colourless and parallelogram-shaped plates
or short prisms. They are well developed and possess constant
angular values. 7

Monoclinic, and having apparently prismatic symmetry. Of the
facets of o and w commonly only three, e.g. w,, 0, and o,, are present.

a bes 09158" 1:50,6965;
Bld:

Forms observed: m = {110}, well developed, commonly predominant
with two parallel faces and having high lustre; c = {001}, large and
also perfectly reflecting; w= {111}, larger than o = {111}, both
rather dull and giving weak reflections. The crystals are plates parallel
to {110}, with a slight elongation in the direction of the c-axis.

Fig. 4.
]-Tri-ethylenediamine-rhodium-chloro-d-tartrates.

249

Angles : Observed: Calculated:
c:m =(001) : (110) =* 76° 59’ _
m:m—=(110) : (110) =* 82 18 —
C:w =(001):(111)=* 51 53 —
út: = (111): (110)= bt 8 Die: 18
¢:0, = (001): (111)= - 38 ~ 40 38 43
m:o.==(110) : (lll) = 38 19 38 16

m:n =(l1l) : (110) = 86° circa 85 40

Cleavage parallel to e and m.
On {110} the extinction-angle is small, only 5°—10° with respect
to the edge m:m. The plane of the optical axes is {010}.

$ 6. LARVOGYRATE TRI-ETLYLENEDIAMINE-RHODIUM-BROMIDE.
{Rho (Hine),} br, + 2H,O.

The substance was prepared from pure laevogyrate iodide by
heating its solution with freshly precipitated sclver-bromide. After
filtration the solution was concentrated on the water-bath, and
crystallised at room-temperature. Very big, hexagonal thick plates
are deposited from the colourless solution, which are strongly
refracting and give extremely sharp reflexes.

Ditetragonal, _ apparently
bipyramidal, and completely
isomorphous with the corre-
sponding cobalti-salt.

eet: 06530:
Forms observed: m = {110},

ordinarily with two predominant
parallel faces, very lustrous, but

Fig. 5. often uneven; o = {101}, splen-
Laevogyrate Triethylene-diamine- didly reflecting and with well-
Rhodium-Bromide. developed faces. ‘The shape of

the erystals is that of thick plates parallel to faces of the prism ;
elongation parallel to the c-axis. No indication of hemihedrism was
observed in any case, just as was stated in the case of the corre-
sponding cobalti-salts.

Angles; Observed: Calculated:
0:0 = (101): (011) =* 53°49’ 3
o:m=(101):(110)= 63 514 63°51/o’
m:m =(110):(110)= 90 0 90 0
No distinct cleavability was observed.
The crystals are optically uniaxial; a plate perpendicular to the
Le
Proceedings Royal Acad. Amsterdam. Vol. XX.

250

c-axis shows in convergent polarised light the axial image character-
istic of uniaxial crystals without circular polarisation. The character
of the birefringence is negative.

§ 7.

Rotation-Dispersion of Laevogyrate Tri-ethylenediamine-Rhodium-
Bromide.

The solution contained 7,310 grams of the anhydrous salt in 100 grams of
the solution. The length of the tube applied was 20 c.m.

o Wave-length | Specific Rotation [«] : Malev rotation [M]
in ANGSTROM-Units : in Degrees : in Degrees :
| 6780 — 46,66 — 2440
6640 — 48,0 — 2510
6520 — 49,66 — 2597
6380 — 53,08 — 2776
6260 — 56,57 — 2960
6140 | — 59,0 — 3086
6030 — 62,66 nf
5890 — 66,1 — 3457
7500 — 71,07 — 3717
5510 — 76,40 — 3996
5340 _— 81,33 — 4253
5180 . — 85,30 — 4461
5100 rae 87,62 — 4583
4860 | — 91,93 — 4808
4710 — 94,60 — 4947
4310 — 97,20 — 5083 be
4150 | — 91,34 | — 5091

§ 8. Racemic TRI-ETHYLENEDIAMINE-RHODIUM-IODIDE.
{ Rho( Eine),} 1, + 1 H, 0.

Very small, colourless, strongly refracting and well-developed
crystals, which are completely isomorphous with the corresponding
cobalti-salt '). In the last case, however, the symbol {111} was

1) F. M. Jagger, Proceed. 18. 62. (1915).

251

attributed to the form w= {112}, so that the value of 6:c in the
former case is analogous to 4: 2c’ in the present.

Rhombic-bipyramidal.
a:b6:c=0,8541 : 1: 0.8632.

Forms observed: o = {111}, predo- 9,f&
minant and very lustrous; c= 4%
{O01}, also well-developed and giving
splendid reflexes; w= {221}, very
narrow, but exactly measurable ;

g = {041}, small, very lustrous, and Fig. 6.
often absent. Racemic
Tri-ethylenediamine Rhodium- Iodide.
Angles: Observed: Calculated:
c zo = (001) : (111) =* 53° 3’ =
o:0 =(11I) : (111)=* 74 51 =

o:w=(111) : (221)= 16 11 16°20’
w:w= (221) : (221)= 41 33 41 14
0:0 =(111): (111)= 73 55 73 54
o:0 =(111):(111)= 62 43 62 43
o:q =(I11) : (041) = 47 59 48 16
q:w==(04l) : (221) = 60 52 60 56
w:o =(221): (lll) = 70 52 70 48

No distinct cleavage was found.
When considered as a rhombic, but pseudo-tetragonal erystal, c
becomes {100}, and a: 6:c=1,0106 : 1: 1,1708.

§ 9. LAEVOGYRATE. TRI-ETHYLENEDIAMINE-RHODIUM-IODIDE.
{ Rho( Aeine),} 1, + '/, H,O (?).

Small, colourless and only feebly reflecting crystals, which are
however well measurable and possess constant angular values.
Occasionally very lustrous, nicely shaped and regularly developed
little crystals were met with. .

Rhombic; perhaps bisphenoidical, although the geometrical appear-
ance is completely holohedral.

a:b:c=0,8064 : 1 : 0,7380.

Forms observed: c = {001}, well-developed and lustrous ; o = {121},
and w = {121}, almost equally large, and only very rarely w some-
what broader than o. This different size of o and w is the only
indication of the substance eventually being bisphenoidical. Further :

is

252

r= {101}, smaller than s = {201} and a= {100}; t= {021} and
w= {041}, both equally large and greater than-b = {010}; q = {011},
commonly absent, but if present, rather large; m= {110}, absent
in almost every case, always small and receding into the background.

Undoubtedly the crystals are isomorphous with those of the optic-
ally active cobalti-salts, whose expected bisphenoidical symmetry did
not reveal itself in any distinct way either. The rather appreciable
divergence of the values a:4 of the salts, as well as the differences
in shape and combined forms in the two cases, may probably be con-
nected with the eventually existing slight difference in the content
of water of crystallisation, mentioned previously.

Fig. 7 (1). Fig. 7 (2).
Laevogyrate Laevogyrate
Tri-ethylenediamine-Rhodium-lodide. Tri-ethyienediamine Rhodium-lodide.

Angles: Observed: Calculated:
c:t =(001):(021) =* 55°53’ =
é€:07— (001): (121): =""60" 2 —
Fw (021) :.(041) —" 1s 26 15°24!/,’
w:b =(041):(010) = 18 46 18 423),
ein =): (101) = 4221 42 28
ris "= (101) (201 =: AS. 54 18 53
s:a =(201):(100) = 28 45 28 39
0:0 =(121):(121) = 59 52 59 52
-0:0 = (121):(121) = 85 10 85 8
a:0 —(100):(121) = 62 45 62 50
b:o =(010):(121) = 42 40 42 34
tzo —(021):(121) = — 27 101),
w:o = (041):(121) = 57 40 57 291/3
bq = (010): (0l1}: =< 53226 53 34
o:% =(121):(121) = 54 26 54 20

No distinct cleavability could be observed.

The external aspect of the crystals is almost isometrical, with a
slight flattening parallel to {001}, and a distinct elongation parallel
to the direction of the direction of the a-axis.

The optical axes are situated in {001}; the a-axis is first bisectrix.

253

§ 10.
Rotation-dispersion of re NOns Parery Tri-ethylenediamine-rhodium-
iodide.
The solution investigated contained 4,535 grams anhydrous salt in 100 grams
of solution; the length of the tube applied was 20 c.m.
o Wave-length in | Specific rotation [4] Molecular rotation [M]
ANGSTROM-Units : in Degrees: in Degrees:
6840 | — 30,91 — 2051
6660 | — 32,89 — 2183
| 6520 | — 34,98 | — 2322
6380 | — 37,18 | — 2468 |
6260 — 38,50 — 2555
6140 | — 41,14 =2 27100
6030 — 44,11 — 2928
5890 — 45,54 — 3023
5800 — 47,08 — 3125
5700 — 49,06 — 3256
5605 — 50,49 — 3351
5510 — 52,36 — 3475
| 5420 — 54,12 — 3592
| 5340 5a oe — 3665
5260 — 57,20 — 3796
5180 — 58,30 — 3869
5100 — 59,73 — 3964
5020 — 61,05 — 4052 |
4945 ebr — 4110 |
4860 — 63,36 — 4205 |
4780 — 64,46 — 4278 |
| 4710 — 65,34 | — 4337 |
| 4650 — 66,66 — 4424 |
| 4560 261,92 — 4468 |
4480 — 67,87 | — 4505
| 4420 — 68,42 — 4541
4310 | — 68,53 — 4548 |
4260 — 68,64 = = — 4556 |
4150 — 68,82) 5 5 — 4567
4060 | —69,10| 5 — 4585
4010 | — 69,19 — 4592
3940 | — 69,24 | — 4596
3880 | —(60,5) a — (4016)
3820 —(55,2) | = & — (3664) |
3780 | —(61,2)) = 5 — (3398) |
| 3140 —(48,7) | = 3 — (3232) |
= — (3219)

254

§ 11. Racemic TRI-ETHYLENEDIAMINE-RHODIUM-NITRATE.
{ Rho (Eine),} (NO),

This compound is always obtained, when a solution of the racemic
chloride is mixed with a solution of Ag NO,. It erystallises from
an aqueous solution in splendid, colourless, and distinetly hemzmor-
phic crystals, showing a strongly variable aspect, and often distorted
in a most particular way. Some of the forms most often met with
are reproduced in fig. 8 a, 6, and c. For the purpose of immediate
comparison with the corresponding cobalti-salt, which is completely
isomorphous with it, the heteropolar binary axis of the crystals is
also in this case adopted as a-axis. |

To the well-developed pyramid o, which never fails and always
gives good reflexes, the symbol {111} has been attributed, so that
the ratio c:5 is now exactly half the corresponding ratio of the
cobalti-salt studied previously: there the symbol {112} was given to
the form 0.

Fig. 8.
Racemic Tri-cthylenediamine-Rhodium-Nitrate.

Rhombic-pyramidal.
a:b:c=0,7874:1 : 0,5606.

Forms observed: a =$100}, well-developed, occasionally predo-
minant and always very lustrous; a’ = {100}, mostly absent or ex-
tremely narrow, sometimes, however, very large and reflecting
perfectly ; o = {111}, large and lustrous; m= {120}, if present large
and lustrous, but occasionally absent; c = {001}, well-developed and
giving very sharp reflexes; w = §111}, never failing, but always
much smaller than 0; g = {011}, occasionally absent, in most cases,
however, large and lustrous; s = {021}, much narrower than q, but
giving good images: » = {20.01}, small and dull, ordinarily absent;
6b = §010!, always sharply reflecting, and sometimes equally large
as g and c, but occasionally much smaller or totally absent;
p = {120}, extremely small, only rarely present.

255

The colourless crystals always exhibit a distinctly hemimorphie
Shape; the polar axis is the a-axis. The substance possesses a strong
tendency to crystallisation, but the various individuals are in general
smaller than those of the corresponding cobalti-salt.

The external aspect is in most cases rather isometrical, but
occasionally a slight elongation parallel to the direction of the a-axis
could be stated.

Angles: Observed: Calculated:
On On (UIP =De Bi —
Bro (LOOK MEIS" 258 91/z —
Oso (111). (PII) 95:36 950 38
6:0 = (010): (lil)—= 65-24 65 - 274);
ong (GEND SOES . 31-4] 31 501
ERO OO IJ 442510 42 11
c:q = (001): (O11)= 29 10 29 16}/,
g:s = (011): (021)= 19 10 19 0
s:b = (021): (010)= 41 44 41 _ 431),
o:q = (111): (Oll)= 31 50 31 _ 50!
ear OOG Er IDS 42 7 43 Ad
m:m= (120).:- (120) = 115 16 115 10
(ee (LOO) (ION 122 - Paul In 199420
e:r = (001) :(20.01)= 85. 0 84 54
rir —(20.01) :(20.01)= 9 56 (Ont.
o:m= (i111): @20)= 85 50 852755

No distinct cleavage was observed.
On {100} the optical extinction is parallel and perpendicular to
the c-axis. The plane of the optical axes is {010}.

§ 12. LanvoGyratory TRI-ETHYLENEDIAMINE-RHODIUM-NITRATE.
{ Rho ( Eine),} (NO),

This compound was prepared from the laevogyrate zodide by
means of sz/ver-nitrate. It erystallises in beautiful, triangularly-shaped
plates, or, in most cases, in crystals of the form reproduced in fig. 9.

The various individuals are distorted in most capricious ways, a
fact which to no slight degree opposes their investigation.

Rhombic-bisphenoidical.
Di 0,8642 an BE 0,6049.

Forms observed: a —={100} and o ={l11}, almost equally large;
r ={101}, commonly present with only two faces; m—=il10;,

256

small but well reflecting; p={120{, much smaller than m, and
often totally absent. The shape of the crystals is that of plates
parallel to {100}, or to two parallel faces of m. They are completely
tsomorphous with those of the cobalti-salt.

Fig. 9.
Laevogyrate Tri ethylenediamine-Rhodium-Nitrate.

Angles: Observed: Calculated:
a:m = (100): (110) =* 40°50’ =
a: o=(100):(111)=* 59 5 =
a: r=(100):(101)= 55 1. 55° ij!

r= (111): (101) = -26 13 26 211),

> PaO) 20) = EW Ee A

sm = (111): (110) 84 15 84 21

:m =(111):(110)= 4715 47 125

zo =(111):(111I)= 5246 52 42!1/,

op: = (lt) (120) —) 982 4 S182 421,

: p = (100):(120)= 5937 5957

|

TOSS) WOO: . Ss ces

No distinct cleavability could be found.

The plane of the optical axes is {010}; in the corner of the image,
one optical axis is visible under the microscope, when a plate parallel
to m is used.

§ 13.

In the accompanying figure 10 the dispersion-curves of both the
last mentioned salts have been reproduced. Their shape is absolutely
different from that of the corresponding curves of the optically-
active cobalt-salts,

Molecular Rotation in

5800

5600

2200

2000

Degrees

Wave-
length
in Ang-
ström-
Units.

WOO 42 43 44 45 46 47 48 49 SOOO SI 52 53 54 5S 56 57 58 59 6000 61 62 63 64 6&5 66 67 68 69 000

Fig. 10. Molecular Rotation-Dispersion of Laevogyrate Triaethylenediamine-

The liquid contained 3,372 grams of salt in 100 grams of the solution. |

Rhodium-Nitrate and -lodide.

nitrate.

le)

Wave-length in

Specific rotation [2]

Molecular rotation [M]

ANGSTROM-Units : in Degrees: in Degrees:

6840 Bean — 2397

6660 — 55,44 — 2600

6520 — 5650 — 2650 |
6380 — 60,55 — 2840 |
6260 | — 63.55 — 2981

6140 | — 67,99 — 3189

6030 | En | — 3340 |
5890 | SD 15 — 3600 |
5800 | = IGS — 3696 |
5700 | a 8237 — 3863 |
5605 | BST — 4023

5510 | — 89,48 — 4196

5420 | — 91,70 — 4301

5340 | — 93,92 — 4405

5260 | 407,18 — 4558 |
5180 | — 98,66 — 4627 |
5100 | — 100,59 — 4118

4045 — 105,63 — 4955

En eee NE 110,45 — 5180 |
4650 EE (122 — 5357 |
4480 NG — 5500 |
4310 | — 119,85 — 5621 |

258

§ 14. In first instance it may be remarked in this connection,
that the expected hemihedrism of the erystal-forms of the optically
active bromide and iodide investigated, does not manifest itself in
any distinct and convincing way, notwithstanding the enormous value
of the optical rotation of these salts.

This fact is completely analogous to our previous experience in
the case of the corresponding cobalti-salts. It proves once more that
even if Pastrur’s principle be considered as principally correct, the
chemical identity of the dissymmetrically arranged substitutes must
be looked upon as a very unfavourable factor for the eventual mani-
festation of the hemihedrism predicted. These facts seem to sustain
our view previously explained, according to which the dissymme-
trical arrangement as such determines chiefly the size of the optical
rotation of the molecule, while the chemical contrast between the
different substitutes is the predominant factor for the manifestation
of ‘the crystallonomical enantiomorphism.

Moreover, Werner’), on the supposition, that analogously built
dissymmetrical molecules always should combine with the same
optically active radical into compounds showing analogous solubility-
relations, concluded that the laevoyyrate Rho-salts of the kind here
described, and the dewtrogyrate Co-salts would possess the same
stereometrical configuration.

For from the less soluble chloro-d-tartrates of both series, the
oppositely rotating ho-, resp. Co-salts are set free, after the
d-tartaric acid has been removed from them.

The Swiss scientist does not give sufficient and rational proof of
the correctness of his starting hypothesis *). On the contrary: the
solubility of chemical compounds is a constitutive property of so
highly a complicated nature, that there is every reason to doubt
a priori the general correctness of the supposition mentioned above.
Then, however, at the same time the value of Waurner’s
considerations, suggestive as they may be, has become appreciably
diminished, in so far as they concern the specific influence of the
central metal-atom on the direction of the optical rotation of the
molecule.

As a counterpart of the views exposed by him, we therefore
wish here to bring forward the following arguments which, in our
opinion, appear to be founded on a firmer basis.

1) A. Werner, loco cit. Berl. Ber. 45. 1229 (1912). Bull. de la Soc. Chim. (1919),
p. 21; G. Urnsain et A. SÉnÉcuap, Introduction à Etude des Complexes, (1913), p. 174.

°) Some cases are mentioned in his paper: however, there is no certainty that
really no inversion has occurred here during the experiment.

a pen

259

In the preceding paragraphs we were able to demonstrate :

1. that in analogously built, optically-cnactive complex salts of
trivalent cobalt and trivalent rhodiwm, the two metals will replace
each other strictly ¢somorphously. This fact is in full agreement with
what can be expected because of the place these elements occupy
in the eighth group of the periodical system.

2. that in analogously built, optically-active complex salts of
trivalent rhodium and cobalt, this isomorphous mutual substitution
of the central atoms remains. This specific property of the metal-
atoms obviously appears therefore not influenced by the special
dissymmetry of the molecules, in which they are present.

Now we will suppose that the dertro-gyrate, and just in the
same way the laevo-gyrate tri-ethylenediamine-cobalti-chloride is trans-
formed into the corresponding chloro-tartrates by means of silver-
d-tartrate. Of course the erystal-forms of both these compounds
d'd and ld, being no longer each other’s mirror-images, will be
different from each other. For among all properties of chemical
molecules none is certainly so closely connected with their molecular
configuration, as the crystal-form is. With respect to the identical
d-tartrate-radical in the two compounds, it is therefore the two special
configuration of the d’-, resp. l’-tri-ethylenediamine-cobalti-radicals,
which determines the differences of crystal-form in the case of the
two chloro-d-tartrates just mentioned.

If now, while completely preserving the existent stereometrical
arrangement of the radicals round the central Co-atom, we think
this last simply replaced by the Aho-atom which, according to what
is mentioned sub 2°) in the above, will replace it in the way of
a perfect isomorphous element, — ‘then it will be evident that the
two complex Rho-compounds thus obtained will be perfectly iso-
morphous with the two corresponding Co-salts just mentioned, and
more particularly each of them with that Co-salt which possesses
an analogous configuration of its radicals in space. This conclusion
is compelling, quite independent of the other question concerning
the special influence which this substitution eventually may have
on the size and even on the sense of the optical rotation of the
original molecule, or on its solubility. The chloro-d-tartrates of Co-
an Rho-complexes with corresponding configuration therefore will
exhibit perfectly isomorphous crystal-forms, independently of their
specific optical properties or of the differences in their solubilities.

Also the number of molecules of water of crystallisation in the two
isomorphous erystal-species will be exactly the same.

Experience now teaches us that the less soluble éri-ethylene-

260

diamime-cobalti-chloro-tartrate has triclinic-pedial symmetry’), the
less soluble — tri-ethylenediamine-rhodium-chloro-tartrate, however,
monoclinic and perhaps sphenoidical symmetry, without the least
analogy of the parameters existing between them. No analogy
of ‘form whatever can be stated between the two kinds of
chloro-tartrates; and in agreement with this lack of isomorphism,
direct analysis showed that, while the cobalti-salt erystallises with
5 molecules H,0, the less soluble rhodium-salt contains only 4
molecules of water of crystallisation, — the respective data in Wrrnur’s
paper being obviously erroneous. There can be therefore not the least
doubt about the truth of the fact that we have not to deal here
with isomorphous salts of corresponding constitution, but with quite
different substances. In connection with what was said above, we are
therefore compelled to conclude from these facts, that the complex
tri-ethylenediamine-rhodium-ion present as a radical in the corresponding
chloro-d-tartrate, has not the same configuration in space as the radical
occurring in the less soluble cobalti-chloro-d-tartrate, but that it
possesses on the contrary, precisely the antilogous stereometrical
configuration in comparison with it. And because the tri-ethylene-
diamine-rhodium-iodide set free from this chloro-d-tartrate, and all salts
derived from it, appear to be /aevogyratory, it follows from this that
the stereometrical configuration of optically-active triethylenediamine-
cobalti-, and rhodium-ions of the same direction of rotation, must be
the same also, — a fact which a priori might have appeared most
probable.

The d-cobalti-salts must therefore possess the same arrangement of
the radicals round their central metal-atom as the d-rhodium-salts,
and the /-cobalti-salts the same as the /-rhodium-salts.

$ 15. With this conclusion at the same time WeRNER’s supposition
of the strange, rather arbitrarily conjectured specific influence of the
central Zèhv-atom, concerning the total inversion of the direction of
rotation of the original dissymmetrical complex, needs to be given up.
The analogously arranged dissymmetrical complexes containing Cr,
Co, or Rho, must all exhibit the same direction of rotation, and only
the absolute size of it may be different and varying in the way
indicated by Werner. This specific rotation is therefore evidently
determined chiefly by the special configuration in space of the radicals
placed round the central atom, and by the specific dissymmetry of

1) f. M. Jaeger, Proceed. Kon. Acad. Amsterdam, 18, 54, 55. (1915).

261

that arrangement. Only in second instance the mass and the chemical
nature of the central-atom seem to be of influence, and more especial-
ly in so far as concerns the changes of the size of the rotation, when
the one kind of central atom is replaced by another isomorphous
element.

It is worth remarkiug here, that the crystals of the optically-
active tri-ethylenediamine-cobalti-, and -rhodium-nitrate here investiga-
ted, exhibit sphenoids of opposite algebraic signs in the case of active
salts rotating in the same direction: the daevo-gyrate cobalti-salt
manifest the rzght-handed sphenoid, while the /aevo-gyrate rhodium-
nitrate exhibits precisely the /eft-handed form.

On the surface of it, it may seem to be correct to consider this
fact as an argument in favour of Werrner’s view about the anti-
logous configurations of cobalti- and rhodium-salts of the same
rotation-direction. But this conclusion must appear completely un-
justified, as soon as the facts hitherto stated are taken into account,
— scanty as these facts for the rest may be at this moment. The
question: in how far is there any rational connection between the
external appearance of form of a erystal and the stereometrical con-
figuration of its molecules? — seems to be quite unanswerable at
the present moment, because this external appearance of the crystal,
depending on a great number of accidental circumstances during the
process of crystallisation, is a very capricious and variable phenomenon.
It is, for instance, well known, that the A-, (NH,)-, Rb-, and Co-dextro-
bitartrates, all undoubtedly having the same stereometrical configuration
(namely : of d-tartaric acid), may exhibit preferentially the forms {111}
or {111} in a predominant way, if certain salts (e.g. sodium-citrate)
be purposely added to their solutions, or if circumstances during
the crystallisation be arbitrarily varied. In the case of the complex
salts under consideration, which, moreover, appear to vary their
outward appearance to a most intense degree under circumstances
only slightly altered, such arguments, based only on this external
form, can hardly have any value at all for judging the internal
structure of their molecules, unless full certainty is obtained that the
salts compared are deposited under exactly the same circumstances,
as e.g. this may be assumed in cases, where racemoids are separated
by so-called spontaneous crystallisation, the two kinds of crystals here
being deposited simultaneously from the same mother-liquid.

At this moment the only conclusion can be, that the same con-
figuration must be attributed to the deztro-, respectively /aevo-gyrate
complex-salts of cobalt and rhedium, when they exhibit a rotation
of the same direction.

262

On comparing the molecular rotation of the bromides, iodides and
nitrates of the two series:

«

Tri-ethylenediamine- Tri-ethylenediamine- Tri-ethylenediamine-
Cobalti- Bromide : Cobalti-lodide: ~ Cobalti- Nitrate :

LM | n= = 6008"
Tri-ethylenediamine-

Rhodium-Bromide:
[M |p eni | wap sepia OP.) Lic

[M |b ELO [M |p = + 4600°
Tri-ethylenediamine- | Tri-ethylenediamine-
Rhodium-lodide : | Rhodium-Nitrate :
|
{

|
| [M |p = == 36005

we see that the Co-salts, besides exhibiting a much greater rotation-
dispersion, also possess a much greater absolute rotation.

This fact will appear conceivable, if we bear in mind the very
different chemical nature of the two isomorphous central atoms, and
the rather appreciable difference in their atomic weights (59 and
103).

Some experiments on the properties of the analogous /r-salts in
this respect, are planned in this laboratory.

Laboratory for Inorganic and Physical Chemistry
of the University, Groningen, Holland.

Sea

Chemistry. — Investigations into Pasreur’s Principle of the Con-
nection between Molecular and Crystallonomical Dissymmetry :
IV. Racemic and Optically-active Complex Salts of Rhodium-
tri-oralic Acid. By Prof. Dr. F. M. Jarcer.

(Communicated in the meeting of June 30, 1917).

§ 1. In our previous papers we had occasion to draw attention
to the fact that the crystals of the optically active components
which, according to WeRNer’s theory, may be obtained under definite
circumstances from the racemic complex salts of the general type:
{Me X",} Y,, exhibit occasionally the non-superposable hemihedrism
to be expected according to Pastgur’s principle, but that in other
eases of this kind no evidence whatever of this hemihedrism is
detectable by any experimental method at hand.

At the same time attention was drawn to the other fact that the
molecular dissymmetry in cases as these, is by no means caused
by a total absence of symmetry-properties in the molecule, but that
the complex ions of the type mentioned, if once Werner’s theory
be adopted, must possess a configuration of their radicals in space,
possessing the symmetry of the trgonal-trapezohedral class (D,).
From the established fact that the non-superposable hemihedrism
of the erystal-forms could not be stated in several cases where deri-
vatives of the complex tri-ethylenediamine-cobalti-ion {Co(Hine),} :
were studied, we were compelled to conclude that the cause of
this abnormal behaviour must be ascribed to the particular circum-
stance that the radicals placed round the central metal-atom are
chemically identical here. It was remarked, however, that the
expected hemidedrism could be stated without exception in all cases,
where in the salts investigated radicals containing oxygen *) were
present.

It was of interest to study other instances of this kind. Thus such
analogously composed salts were chosen in the first place, as con-
tained the oxygen-bearing radicals immediately linked to the central-
atom in the form of the radicals of bivalent carboxylic acids. Our
choice was finally fixed upon salts derived from the complex

1) F. M. Jagcer, Proceed. Kon. Acad. Amsterdam, 17. 1217. (1915); 18. 49.
(1915); Zeits. f. Kryst. u. Min. 55. 209. (1915).

264

rhodium-tri-oxalic acid, for which the possibility of a fission into
the optically-active components had been proved experimentally ’).
Moreover, a detailed investigation of these salts appeared also desi-
rable from another point of view, because in Werner’s original
paper some facts are mentioned concerning the crystallisation-phe-
nomena of the optically-active components, which a priori must be
considered very improbable, and therefore worth controlling again
by means of new experiments.

For instance, the fact was there brought to the fore, that from
solutions of the racemic compound under favourable conditions crystals
of both the enantiomorphous modifications would be deposited spon-
taneously, which hemihedral crystals of the optically active compo-
nents, if brought at room-temperature into a concentrated solution
of the racemoid, would increase slowly and grow to big individuals
within a few weeks. But, as we found during our investigations,
that the optically active forms are much more soluble under the
same circumstances than the racemic substance is, it seemed highly
probable that an error was made here, because, moreover, the facts
mentioned cannot be right from a theoretical standpoint. The draw-
ings in WERNER’s paper, intended to give an impression of the
crystal-forms obtained, rather point to distorted, and accidentally
non-superposable triclinic crystals of the racemoid being present here,
than to enantiomorphous crystals of true hemihedral symmetry.
Moreover, Werner himself mentions the triclinic symmetry of these
crystals, and therefore the validity of Pasrgur’s principle in his
case cannot yet be considered as proved by the data given in this
paper. Repeatedly we have made attempts in the way indicated by
Werner, to perform a spontaneous fission of the racemic salt; but
the solution of it saturated just above 100° C., first being rapidly
cooled down to 90° C., and subsequently cooled down to room-
temperature, never deposited other crystals than the capriciously
distorted individuals of the triclinic racemoid. The microscopically
small crystals often obtained by very rapid cooling of the hot
solution, appeared to be no crystals of the active forms either; they
were rhombic individuals exhibiting prismatic, domatic and basal
facets, of a new hydrate of the racemic compound, probably at higher
temperatures stable, and containing less water of crystallisation.
These experiments, if varied in several ways, gave unexceptionally
bigger or smaller crystals of the racemic compound. The appreciably
greater solubility of the active forms in comparison with that of

1) A. Werner, Ber. d. d. Chem. Ges. 47. 1954. (1914).

265

the racemic salt at all temperatures between 15° and 100° C.
characterizes the racemoid undoubtedly as the more stable solid
phase within this range of temperature, with respect to the mixture
of these antipodes. The experiment described by Werner can therefore
never lead to a spontaneous fission, and surely it must be quite
impossible, that under these conditions an optically active crystal
should increase, when brought at room-temperature into the saturated
solution of the racemic salt. This may readily be deduced from
Baknuis Roozesoom’s well-known graphical representations *) of the
solubility-relations here prevailing. Indeed, it could be proved on the
contrary by often repeated experiments, that a crystal of one of the
optically-active forms, if brought into a feebly supersaturated or
saturated solution of the racemic salt, ¢mmediately disintegrates and
subsequently disappears completely, and that after some lapse of time,
triclinic erystals of the racemic compound are deposited from the
solution. These last crystals are often rudimentarily developed, so
that occasionally they make the impression of pedia/, unsymmetrical
crystals, which of course must therefore appear non-superposable
with their mirror-images. If dissolved in water, the solutions of these
crystals were in every case optically zmactive, There can be no doubt
whatever therefore about the fact, that the crystals obtained and
reproduced by Werner must have been distorted triclinic crystals
of the racemic salt; it remains, however, doubtful whether the
solutions obtained by him in dissolving these crystals, can really
have been ‘‘optically-active’, unless some optically-active crystals
for inoculation-purposes were previously introduced into the solutions.

We were able, moreover, to demonstrate the remarkable fact that
the crystals of the pure optically active components exhibit exactly
the same symmetry as that previously deduced for the complex ions
of this type themselves. Indeed, they are trigonal-trapezohedral, and
they show forms which externally are quite comparable ‘with the
typical forms of some dextro- or laevogyratory quartz-crystals.

The racemic salt was prepared from freshly precipitated and
washed rhodium-hydroxide obtained from sodium-rhodium-chloride by
means of a dilute sodium-hydrovide-solution at 40° C.; the pure
rhodium-hydrovide was then dissolved in a hot solution of potasstum-
bi-oxalate. The fission into its components, which is a rather tedious
process, was executed by means of the strychnine-salt, from which
afterwards the strychnine was readily eliminated in the form of
its wdide.

1) H. W. Baxuuis RoozeBoom, Zeits. f. phys. Chemie, 28. 494. (1899).
18

Proceedings Royal Acad. Amsterdam. Vol. XX,

266

§ 2. A solution of the active components containing 3,79°/, of
the anhydrous salt, exhibits in a layer of 10 em. an absorption-
spectrum, in which all violet, blue, and green rays are lacking,
while of the yellow light only a small portion is transmitted. With
increasing dilution a small extension of thé spectrum is stated ;
more particularly the yellow and green rays are gradually better
transmitted and become more and more visible. In the case of a
solution of 0,5°', even some greenish-blue rays were visible. In a
layer of 20 cm., the limits between which the light was not appre-
ciably absorbed, appeared to be about as follows :

|
|

Concentration of the solution in Limits of the transmitted light (in

percentages of anhydrous salt: | A.U.) for a layer of 20 ¢.m.:
10.96 | 7000—5800
3.79 | 7000—5700
Zed 7000—5500
1.97 | 1000—5500
1.09 7000 —5100
0.55 7000—4900
0.50 4 7000 — 4800

These data may give an approximate impression of the extension
of the light-transmission for several wave-lengths. In the red part of
the spectrum, the limit is situated at about 7700 A.U.; however it
could not be fixed any more, because of the micrometer-screw of
the monochromator not going so far.

By the study of the seven solutions just mentioned, the rotation
for any of these wave-lengths was measured in a way analogous to
that previously described *). The total behaviour of these orange-red
to orange-yellow coloured solutions is most remarkable. For instance,
if only sufficiently concentrated solutions of the right-handed salt be
investigated (e.g. of 10°/,), and thus only a limited spectral region
be taken into account, the observer would readily come to the con-
clusion, that his salt is laevogyratory. The following data, obtained
within several spectral regions, by means of the seven solutions
mentioned, and which appeared, after controlling, to be sufficiently
exact, may elucidate this strange behaviour’) with the simultaneous

') F. M. Jarcer, Proceed. Kon. Acad. Amsterdam 1%. 1227 (1915).
2) A. WERNER, loc. cit. 1955.

267

aid of fig. 1. The numbers given for the specific rotation [«|, are
the mean values of commonly three or four different values obtained
in various series of measurements; they always relate to a content
of anhydrous salt in 100 weight-parts of the solution :

Rotation-Dispersion of dextrogyrate Potassium-Rhodium-oxalate.
eines De : OS gene roo | Molecular rotation
ANGSTRÖM- Units: in Degrees: | in Degrees: |
4860 1356 + 17240 |
4950 | + 293 | + 14190
5020 | + 253 | + 12250
5100 | + 206 + 9975
| 5180 | 172,1 | + 8335
5260 | 7 | + 6860
5340 | + 114,3 | + 5535 |
5420 | 4. 85 | 4115 |
5510 | + 67,1 | + 3250
5605 | + 46,4 | + 2250
5700 + 30,1 + 1460
5800 | + 16,3 | + 790
5890 | eon | + 300
| 5900 + 5,1 | + 247
5970 | 0 | 0
6030 | BA Lee Nog
6140 | — 11,5 | — 551
6260 | el | er 608
6380 | El | seg CH
6520 | — 23,5 | — 1140
6660 | = 951 | = 1215
6800 | 426 | == 74260
| 6945 | TA | de Ci)

From this it becomes evident that the solutions of the right-handed
salt are only dextrogyratory in reality, if green and yellow light-
18*

268

rays be taken into account; for the red and orange rays however,
these solutions appear to be laevogyrate.

Such solutions are optically-inactive for a wave-length of 5970
A.U. At this wave-length, invariable moreover for solutions of
widely different concentrations, there is no longer any detectable
difference between solutions of the two antipodes.

tc Gotution

360°

780°) SA
170 8
760°) X
750 =
740°
130° AN
120°) A
70°\ 8
700° >
90°} 8
80°
70°
60°
50

Mactive

Ww
6110 6360 6610 6860 Have-Cenglh
a Cagslrom-~
a
MLLS.

5970

Fig. 1.
Specific Rotation-dispersion of right-handed Potassium-Rhodium-Oxalate.
(All molecular rotations are 50 times as great).

It is worth attention, that the solutions do not exhibit for this
wave-length any trace of an absorption-line. Our former supposition
connecting the abnormal rotation-dispersion with the eventual occur-
rence of selective absorption, thus appears no longer justified. As
WERNER points out, the phenomenon is met with in the study of
all complex metal-oxalates hitherto investigated: those of rhodium,
chromium, and cobaltum, exhibit this property in a very pronounced way’):

1) To an investigator studying the heterogeneous equilibria between the racemic
salt and both its antipodes in solution, there could no longer be any difference
between the solutions of the d- and J-component or their mixtures, if he worked
under conditions which enabled him only to use light of a wave-length of a = 5970 A.U,
Indeed, all such solutions would then be found optically inactive. In such circum-

269

Finally it may be remarked, that the optically-active salts have
no appreciable tendency to auto-racemisation. After heating on the
water-bath during some time, no appreciable diminution of the original
rotatory power appears to have occurred. A slight hydrolysation,
however, could be stated in these cases. The salts are slowly
decomposed by the continued action of violet light or by strong
heating, while a black powder (Rho ?), and some rhodium-hydroxide
are set free.

$ 3. I. Racemic Porasstum-Ruopium-Oxaate (+ 45 H,0).
This compound crystallizes in big, ordinarily flattened, garnet-red,

Fig. 2,
Racemic Potassium-Rhodium-Oxalate (+ 41/, H,0).

stances no difference would apparently be any longer present here and in the
well-known case of sodiwm-chlorate; in other words: in this singular point the
descriptive number of components would be diminished with one. However, it
must be insisted, that this is only valid for one definite temperature and one
definite pressure, as |z] is a function of both. For wave-lengths only slightly smaller
or greater than 5970 A.U., the system is again a ternary one, in which the
functions of d- and l-components are reversed.

270

very lustrous, and perfectly transparent crystals. Their shape is
very variable, and some of the most frequent forms are reproduced
in fig. 2 a—c. The external aspect is often highly unsymmetrical
(fig. 2c); in such cases individuals are occasionally met with, which
- may be looked upon as apparently enantiomorphous. However, their
symmetry is most probably holohedral, the present forms therefore
only being special and accidental forms of growth.

The crystals are identical with those studied by Durer’); they
differ from those only in aspect, as Durst’s crystals exhibited the
form «={110} predominant.

Triclinic - pinacoidal.
a:b:e==1;0732 1: 1:0316:

A= 92°45)" a= 98108
B=102 44 B= 104 174
C= 67 243 y= 66 112
Forms observed: m= {110}, very lustrous and predominant ;

u = {110}, smaller, but also well reflecting; 6 = {010}, commonly
a little broader than u; w = {111}, broad and yielding good reflexes;
E = {112}, well developed and very lustrous; p= {111}, commonly
smaller, but rarely also much greater than §, and giving eminent
reflexes; c == {001}, commonly very narrow, occasionally somewhat
broader; o= {111}, broad and beautifully reflecting; «= {111},
small and lustrous, often absent; 4A = {112}, commonly absent,
occasionally very narrow, rarely broad; q = {021}, very small, but
well measurable.

Angles: Observed: Calculated:
m:b =(110): (010)=* 54° 10’ —
Mm: = (110): @10)=* 93. 3 =
mo (MO) (111) = ae 24, =
b:o =(010) : (lll) =* 61 48
mre —=(110): (lll) =* 96 34!/, —
c:b —=(001) : (010) = 87 15 87° 14!/,’
b':p =(010): (110) = 32 45 32 45!
m:p =(110):(111)= 86 18 86 23
mw: =(111): (00l)= 57 501 51 40
m:t —=(110): (112) = T1 33 11-35
c:p =(001): (110) = 95 1 95 19
eee =(110) =(15j= 51520 51 24
o:c =(111): (001) = 42 14 42 14
m:o =(110):(111)= 34 31 34 35

1) H. Durer, Bull. de la Soc. Min. 12. 466. (1889); Cf

Chim. et Phys. (6). 17. 307. (1889).

.: E‚ Lemp, Ann. de

271

Observed: Calculated
b':E =(010) : (112) = 51 25 51 25
a:€ =(111):(112)= 70 8 Wil ial
oe (le) (ilk, 20 +47 20 45
x: =(111) : (110) 30 40 30 40
m:0/ =(110):(111)= 88 59 89 1
b:0 =(010): (lli)= 64 19 64 24
o:p =(111): (168)= 11 35 68 38
usp =(110): (lli)= 27 58 28 4
c’:€ =(001): (112)—= 43 59 43 55
c:p —=(001):(lli)= 57 4 57 10
c:k =(001):(112)—= 27 4 26 56
k:o =(112): (111) = 15 10 15: 18
b:k =(010):(112)= 714 3 11 56
b:q ==(010): (021) = 25 4 25 SiG

c':q = (001): (021) = 67 41", 67 42

No distinct cleavability was found.

The erystals are dichroitie: on {110} for vibrations including an
angle of 60° with the edge m:u they are orange-yellow; for such
perpendicular to those, reddish-orange. On {110} one of the directions
of extinetion includes an angle of 31° with the intersection m : u;
the plane of the optical axes is almost parallel to the edge w:m.

§4. After transformation of the racemic salt into the corresponding
strychnine-salt und fractional crystallisation, the oxalate was resolved
into its antipodes. The d-rhodiwm-salt, namely, combines into a less
soluble strychnine-salt than the /-salt does. By treatment with potassiwm-
iodide, all strychnine may be eliminated as the little soluble strychnine-
todide, and in this way the optically active potassium-salts may be
obtained. It is a tedious task to get a sufficient quantity of the
laevogyrate antipode, because always some strychnine-d-o.xalate is
withdrawn with the /-ovalate; therefore the last mother-liquids always
deposit the racemic salt besides the laevogyrate. Moreover, the
result is also diminished by the hydrolysis of the strychnine-salt
during the concentration of the mother-liquid on the water-bath.

Finally, however, sufficient quantities of both antipodes were obtained.
The erystallographical description of these salts is given in the following
paragraphs.

§ 5. IL. Dextrocyratory Porassitm- RHODIUM-OXALATE (+ 1H,0).

Splendid, sometimes colossal, very lustrous, blood-red and perfectly
transparent crystals. Their external aspect is very variable with the

272

particular circumstances of crystallisation, and as a consequence of
their numerous and most capricious distortions, the right interpretation
of the measurements is often very troublesome. Some of the most
frequently occurring forms are reproduced in fig. 3 a and 6.

Fig. 3.
Dextrogyrate Potassium-Rhodium-Oxalate (+1 H,0).

Trigonal-trapezohedral.
a:c=1:0,8938 (Bravars); « = 100°38’ (MILLER).

Forms observed: '): R = {1011} [100], always present, and exhibiting
large faces; c= {0001}[111], always present too, sometimes very
small, but in most cases rather large; r= {0111}[221], and s=
== 022i (111), rarely failing, well reflecting, but much smaller than
R; t= [2021 [511], often absent, always narrow and dull; m=
= (11010: [244). always present, occasionally with small, mostly with
well-developed faces, and rarely predominant ; 2 = {2241} [715], as
a right-handed, positive, trigonal bipyramid, occasionally absent, but
in several cases with faces almost '/, or '/, of those of R. The
different faces of 2 are in all cases of very different sizes. The
aspect of the crystals is occasionally like that of quartz (fig. 3a),
and appreciably distorted; sometimes 7 is a little broader, so that
the external shape gets a more hevagonal form. No distinct cleavage

1 The symbols between [ | are Muerian symbols, relating to the polar edges
of R as axes of reference.

273

was found. On {1010} occasionally unsymmetrical corrosion-figures,
like trapezia, are observed.

Angles: Observed: Calculated:
R:R =(1101): (1011) =* 76°55’ —
c:R ==(0001):(1011)= 45 59 45°54! /,/
c:r —(0001):(Ol11) = 45 58 45 54!/,
r:s —=(0111):(0221) = 18 25 18 15
s:m =(0221):(0110) = 25 31 25 51
m:R =(0110):(1011)= 68 54% 68 571/3
m:t =(0110):(0221)— 25 43 25 51
s:R =(0221):(1011)= 51 14 51 12
r:R=(1011):(0111)= 42 8 42 51/3
R:m=(1011):(1010)= 44 5 44 53/,
x: m = (2241):(1010) = 40 55, 40 54!4
x:R =(2241):(0111)= 7758 78 17
m:x —(0110):(2241) = 40 55% 40 54!)
x:R =(2241):(1011)= 28 2 28 3
xir =(2241):(0111I)= 28 2 28 3
c:x —=(0001):(2241) = 60 48 60 461/,
x: x = (2241): (2241) = 58 24 58 23

Intergrowths of two crystals occur, with their trigonal axes inter-
secting under right angles, and one individual rotated about it
through 180°.

The crystals are distinctly dichroitie: on {1010} for vibrations
parallel to the direction of the c-axis, they are orange, for vibrations
perpendicular to the first, blood-red.

Plates parallel to {0001} show in convergent polarised light the
interference-image of an uniaxial crystal without circular polarisation :
the bars of the black cross are regularly extended to the centre of
the image. The character of the birefringence is negative for all
colours, and it is strong. For Na-light is n,—=1,6052. ne —1,5804 ;
the birefringence is therefore 0,025.

When the axial image is very much enlarged, it appears, on closer
examination, to be properly biawial, with an extremely small apparent
angle of the axes, and with the axial-plane perpendicular to one of
the edges c:7. Although the rotatory power of the crystals is ob-
viously very weak, it is, however, rather probable that they represent
pseudo-symmetrical intergrowths of lamellae of lower symmetry.

§ 6. III. Larvoeyrate Porassium-RHODIUM-OXALATE (+ 1H, 0).

From the last mother-liquids, in which the more soluble strych-
nine-salt is accumulated, the laevogyrate salt, together with some

274

racemic oxalate, is obtained by means of potassiwm-iodide. First the
less soluble racemic salt crystallizes in the form of fig. 2, sub-
sequently the laevogyrate antipode in splendid, garnet-red, flattened
crystals.

Commonly the external aspect is that of fig. 4a, with a flattening
parallel two opposite faces of the rhombohedron, of which one is
often excavated and uneven. Occasionally also crystals of the aspect

a Fig. 4. b
Laevogyrate Potassium-Rhodium-Oxalate (+ 1 H,0).

of fig. 46 were obtained. Both combinations are also met with in
the case of the dextrogyrate salt, but the antipodes differ in the
occurring of a left-handed trigonal bipyramid y = {4221} [751] in
the /aevogyrate forms, where the dextrogyrate manifested the right-
handed trigonal bipyramid 2 = {2241![715]. For the rest the angular
values are the same as found in the case of the dextrogyrate salt.
A review of some values may convince the reader of this:

Angles: Observed: Calculated:

RzR=(101): (101) =* 7790 —

c: R=(0001):(1011) = 46 1 45°54’
C:r =(0001): (0111) => 45 52 45 54
ris (0111): (0221) = <. 18°-3 18 15
s:m = (0221):(0110) = 26 0 25 51
m:t =(0110):(0221) = 26 1 25 51
y:m = (4221):(1010) = 4058 40 54
y:r = (4221) :(1101)= - 280 28 2

y: R = (4221):(1101) =- 718 8 78 17

275

The geometrical as well as the optical properties, wtth the exception
of the peculiarity just mentioned, are in both cases perfectly agreeing.
There can be no doubt whatever as to the fact that the two optically
active salts erystallise in non-superposable mirror-images, although
the crystals themselves do not show a distinct rotatory power.
Stereographical projections of the dextro- and laevogyrate erystal-
forms are given here in fig. Sa and 55, for the purpose of surveying
the general zonal relations.

jo1o 1010

ON

LESS
INAS
Les

1100

d. 1.
a Fig. 5. b
Stereographical Projection of the Crystal-forms of dextro- and laevo- Potassium-Rhodium-Oxalate.
(+ 1 H20).

From these facts it becomes clearly evident that in the case
investigated Pasteur’s principle appears fully confirmed. Indeed, a
non-superpusable hemihedrism of the erystal-forms can be stated as
inseparably accompanying the enormously strong optical activity of
the solutions of these salts. However, it is worth attention that even
here this hemihedrism only manifests itself by the occurrence of a
single trigonal bipyramid, and never by the presence of any
“trapezohedral” face, as e.g. in the case of quartz; this fact again
may in some way or other be connected also with the lack of
chemical contrast between the dissymmetrically arranged substitutes.

Finally it may be remarked that a solution of these salts after
three days, exposure to the light and even to the sun-light, did not
exhibit any appreciable photochemical decomposition. In aqueous
solution, however, the substance exposed in quartz-vessels to the

276

action of a strong quartz-lamp during some few hours, appeared
decomposed to a detectable degree: metallic rhodium covered the
walls of the quartz-apparatus in the form of a black mirror, while
carbon-dioxide was set free. As the violet and blue rays are almost
completely absorbed by the solutions (see above), the relatively rapid
photochemical destruction of the molecule by rays of short wave-
length may be considered as to be in full accordance with Draprr’s law.

§ 7. If a solution of potassium-rhodium-orvalate be treated with
stlver-nitrate, the silver-salt: {Rho(C,O,),} 4g, is precipitated as a
vividly red compound, which is only little soluble in cold, somewhat
more in hot water, and which erystallises in long needles much
alike the bichromate.

From this szlver-salt, as well as directly from the potassiwm-salts,
by interchange with tri-ethylenediamine-rhodium-halogenides, complex
salts of the type {Rho (Eine),} {Rho (C,O,)};, may be obtained as pale
yellow to orange-yellow crystalline precipitates, which are almost
insoluble in all kinds of solvents. as was to be expected beforehand.

By combination of the racemic and optically-active ions, we have
finally obtained the following nine isomeric salts:

{r-Rho (Kine), } {r-Rho(C,O,),}; {r-Rho (Hine),} {d-Rho (C,O,),};
{r-Rho (Hine),{ {1-Rho (C,O,),3; {d-Rho (Hine),} {r-Rho (C,O,)5};
{/-Rho (Hine),} {r-Rho (C,O,)},; {d-Rho (EFine),} {d-Rho a 0).
{/-Rho (Eine), } {Rho (C,O,)},; {d-Rho (Eine), } { l-Rho (CO) };
and {l-Rho (Eine), } {d- Rho (C,O),}.

$ 8. Furthermore some measurements may be recorded here con-
eerning potasstum-rhodium-malonate: K,{Rho(C,H,0,),} +3H,0, a
new compound obtained in a way analogous to that described for
the corresponding oxalate, and which is now also used in fission-
experiments. Finally the description has been given here also of
potassium-iridium-ovalate: K, {lr (C,O,,}-+ 44 H,O, of which the
fission into its antipodes is now being investigated also in the author’s
laboratory. The available data prove once more plainly the full isomor-
phism between Rho- and /r-derivates of analogous structure.

§ 9. Racemic Porasstum-RHODIuM-MALONATE.
{Rho (C,H, 0,),} K, + 34,0.

This compound was prepared from freshly precipitated and washed
rhodium-hydroayde, by boiling it during a long time with a solution
of potassium-bimalonate, to which some free malonic acid was
added. Complete solution occurs only after heating during a very

277

long time; moreover the salt is more easily hydrolysed than the
corresponding oralate. It contains 9,32 */, Of water, corresponding
to the presence of three molecules of erystallisation-water.

The salt erystallises from an aqueous solution in the form of
thin, hexagonally bordered, orange-red plates, or occasionally in
somewhat thicker crystals, exhibiting however the same combination-
forms. They show appreciable oscillations of their angular values,

Monoelinie-prismatic.
a:0:e=1,2309:1:1,0783 ; 3 = 86°36’.

Fig. 6.
Racemic Pan Malonate (+ 3 H,0).
Forms observed: a = {100}, always predominant and very lustrous ;
0 = {111} and w»=$111}, commonly equally large, occasionally o
much broader than w, and yielding sharp reflexes ; m — {210}, larger
or smaller, but always well reflecting; 6 == {010}, ordinarily narrow,
often absent, sometimes even broader than m. The external habit
is that of plates parallel to {100}, with a slight elongation in the

direction of the c-axis.
Angles: Observed: Calculated:
Bo. = (010): (At) =* 5148" —

a:o =(100):(111)=* 57 30 =
w:a'=(111):(100) =* 60 55 =

a:m=(100):(210)= 31 32 31°34’
m:b =(210):(010)= 58 28 58 26
o:0=(111):(111)= 76 15 16 24
Ga = (111): (111) =" «61 40 61 35
ato==(111):(111)— 79 20 719 44
Beas (010) (111) = 50225 50 8

278

No distinet cleavability has been found. At {100} corrosion-figures
having the shape of isosceles triangles were observed ; their symmetry
is in accordance with that of the monoelinie-prismatie class. The
crystals are only feebly dichroitic. The plane of the optical axes is
{O10}; one optical axis is observable at the border of the optical
field, under appreciable inclination to the plane {100}. The dispersion
is inclined and weak, with o >v. The birefringence has negative
character.

§ 10. Racemic Porassrum-IRIDIUM-OXALATE (+ 44H, 0).

The substance was obtained by dissolving freshly precipitated
widium-hydrowide in oxalic acid. The process goes on extremely
slowly and is accomplished only by heating during about 30 hours
at a reflux-cooler, until the liquid has got a pure yellow colour. It
is then neutralized with potassitum-carbonate and slowly evaporated
at room-temperature’). Besides much potassium-ovalate, orange
coloured crystals of the cridiwm-salt are deposited. They are selected,
purified, and recrystallised several times from aqueous solutions.

The salt is deposited from aqueous solutions in beautiful orange
erystals, which are very lustrous and suited for precise measurements.

Triclinic-pinacoidal.
abe = 10771 1.10405;

A= 93° 224’. Es 98) 3824
B= 101° 36%’. B= 10493
COT? at ye GE ITE

Forms observed: m= {110}, pre-
dominant, much larger than u = {110},
and 6 = {010}; the vertical zône
exhibits occasionally appreciable fluc-
tuations of the angular values, and

Fig. 7. multiple reflexes; c = {001}, smaller
Racemic Potassium-Iridium-Oxalate than uw, but very lustrous; 0 = {111},
(+ 4% H,0). a little larger than c, and w = {111},

much broader than o; all three forms yield very good reflexes;

&— {112}, large and very lustrous. The external form appears
flattened parallel to m; the plates are commonly very thick.

1) Cf. CG. Gratpint, Rend. Acad. Linc. Roma (be). 16. II. 551. (1907); the
crystals were measured by F. Zamponint, (Zeits. f. Kryst. u. Min. 47. 621, (1910),
who also demonstrated their isomorphism with the analogous rhodium-compound
However, his interpretation of the occurring forms is different from the one given
by us.

279

Angles: J Observed: Calculated:
(JAEGER): (ZAMBONINI):

m:b =(110): (010) =* 53°54’ 53° 59’ al
mio == (110): (111) =* 45 33 45 30 =
b:y =(010):(110) =* 32 40 32°37 =
c:» =(001):(110) =* 94 40 94 38 =
c:m = (001): (110) =* 76 40 es En
m:u =(110):(110)= 93 20 93 24 93926’
eto =(110):(1l1) = 96 25 96 18 96 30!/5
c:& =(001):(112)= 43 42 = 43 59
m:o =(110):(111)= 34 38 a 34 20
c:0 =(001):(111)= 42 20 = 42 20
C:m =(001):(111)= 57 46 56(2)46 51 46
m:& =(110):(112)— 78 13 == 17 59
ik =(111):(112)= 70 17 70 16 10 13
uik =(110):(112) = 50 58 50 51 51 45
a:» =(100):(110)= — — 34 47
a:m=(100):(110) = — en 58 39

No distinct cleavability was found.

On all faces oblique extinction. The crystals are strongly dichroitic:
on m yellow and orange, quite analogous to the corresponding
rhodium-compound. |

There can be no doubt about the complete isomorphism of the
crystals in this case with those of the Aho-salt. The corresponding
Co-salt was investigated by Coraux; it is also triclinic-pinacoidal, but
not isomorphous with the two other salts, probably because of a
difference in its content of water of crystallisation. Attempts will
be made to obtain the analogous cobalti-salt.

Afterwards we will communicate in detail about our fission-
experiments of potasstum-rhodium-malonate and of potassium-iridium-
oxalate, as soon as their optical antipodes will have been obtained.

Laboratory for Inorganic and Physical Chemistry
of the University, Groningen, Holland.

Chemistry. — “Two Crystallised, Isomeric d-Fructose- Penta-
acetates’. By Prof. Dr. F. M. JarGeR.

(Communicated in the meeting of June 30, 1917.)

§ 1. Nine years ago‘) the crystallonomical character was described
of a d-Fructose-tetra-acetate (rupt : 132° C.), prepared by Dr. D. H.
Brauns. The compound is monoclinic-sphenoidical, with the parameters:
a:b:¢=1,3463:1:1,5733, and @ = 52°12’. A short time ago Hupson
and Brauns*) succeeded in obtaining two isomeric crystallised
d-Fructose-penta-acetates, whose form-analogy is strikingly evident,
as will be clear from what follows.

Both penta-acetates possess the same cycle of atoms in their
molecules, as is proved by the fact that they both are obtainable
from d-Fructose-tetra-acetate, namely the
a-modification by means of ZnCl, and
acetic acid-anhydride, the g-modification
by means of strong sulphuric acid and
acetic-acid-anhydride.

§ 2. I. a-d-FRUCTOSE-PENTA-ACETATE.

The substance melts at 70° C.; its
specifie rotation at 20°C. is: [alp—
— + 34°,75 in chloroform-solution. Its
taste is a little bitter, and it erystallises
from a mixture of alcohol and chloroform
in prismatic erystals which are repro-
duced in fig. 1.

Rhombic-bisphenoidical. a ae
a:b:c=0,4946 : 1 : 0,3349 Tae

Forms observed: m= {110}, predomi- _ #d-Fructose-pentaracetate.
nant, and 6 — {010}, appreciably smaller. The prism-zône shows
often disturbances, and its angular values may oscillate within
rather wide limits. Furthermore: qg—{011}, large and lustrous;
w = 111}, small and lustrous; s = §021!, smaller and somewhat
duller. The external form is elongated parallel the c-axis.

1) F. M. Jarcer. Proceed. Kon. Acad. Amsterdam, 10. 563. (1908); Zeits. f.

Kryst. u Miner. 45. 539 (1908).
2) C.S. Hupson and D. H. Brauns, Journ. Amer. Chem. Soc. 37. 1283, 2736. (1915).

281

Angles: Mi Observed: Calculated :
m:m=(110) : (110) =* 52038 3
ane Aue): (OEE 31 2 —
m0 €010) (010) = 63 41 63°41’
q:b =(011) : (010)= 71 29 rb
bs (ELI) = (010) — 14.23 14 30
Bom e (LOP == oe 59. 0 52 56
vijl {Mij 32 38 32 42
pe mt = (111) 3 (110) = 68° 42 68 33
Sig MEI ONS De 7 15 18
Sm = (021): (110) == 75. 44 15 43
m:q = (110) : (211) — 81 57 81 541,

Cleavable imperfectly parallel to {110}.
On m rectangular extinction.

$ 3. IL. g-d-FRUCTOSR-PENTA-ACETATE.

This isomeric compound melts at 109° C.; its ee rotation in
chloroform-solution is at 20° C.:falp = —120°,9; in benzene Le)
Be 105° 5.

The crystals here described were obtained from a solution in ether ;

Fig. 2.
2-d-Fructose-penta-acetate.
from benzene the substance crystallises in individuals containing 1
molecule of benzene.

Colourless, well-developed, prismatic crystals, which havea feebly

bitter taste.
Rhombic-bisphenoidical.
a:b6:c=0,4941 : 1: 0,9094,

Forms observed: gq = {011}, very predominant, but yielding bad

reflexes ; 6 = {010}, smaller, much better reflecting than q; 7 = {101},
19
Proceedings Royal Acad. Amsterdam Vol. XX.

282

large and lustrous; w= {111}, small, but giving good reflexes ;
o = 111}, larger than w, but duller ; m= {110}, very small, yielding
sharp images. The habit of the crystals is elongated parallel to the
d-axis.

Angles: Observed: Calculated:
b:q =(010) : (011)=* 47943 =
rir. =(101): (101) =* 57 2 =
g:q =(011): (O11)= 84 34 84°34’
q:r =(011): (101) = 69 16 69 19
rin (08). (110) = 3158 38 1
mig —(110)-: (Oli)= 7236 72 39,
gro —(OM) {Lj — “53 40 53 41
öso=( (ND = ‘i222 72 37
o:m=(l11): (110) = 25 54 25 57
m:o =(110): (lll) = 25 59 25 57
b:o =(010): (111)= 66-39 66 32
b:» —=(010) : (lll) = 66 31 66 32
o:o =(111):(111)= 23 24 23 28
b:m=(010) : (110) = 63 55 63 4213

Perfectly cleavable parallel to {011}.
The plane of the optical axes is {100}; probably the c-axis is first
bissectrix.

§ 4. From these measurements it is obvious that both isomerides
possess the same symmetry, and, — within the limits of experimental
errors, — the same parameter a:b (=0,4944). Such relations are
met with often in the case of polymorphic modifications of the same
substance *). The identity of the eyelie structure in both molecules
compels us to believe that it is this cyclic of atoms common to both,
which determines the value of a: 6. In no cases of polymorphism
where up till now such analogy in the values of one of the para-
meter-ratios was observed, there could be indicated the existence of
a reversible transition between the two modifications. In the case of
a- and ‘B-d-Fructose-penta-acetates too, as Hupson and Braun demon-
strate, no such reversibility seems to exist either (monotropy). It
may seem probable that in cases like this, no ordinary polymorphism
is present, but that in last instance the differences observed between
such modifications are always caused by true chemical isomerism,
in which a great part of the molecule is common to the two modifications.

Laboratory for Inorganic and Physical Chemistry.
of the University. Groningen, Holland.

1) F. M. Jarcer, Zeits. f. Kryst. u. Miner. 40. 131.(1905) : thus in the case of @ and B-
1-3-4 Dinitro diethyl-aniline ; of a- and p-Benzyl-phtalimide (ibid. 40. 371. (1905);
of a- and §-Mannite, (Grorn’s Chem. Kryst. III. 431. (1910); etc.

Chemistry. — “On Complex Salts of Ferri-Malonic-Acid’. By
Prof. Dr. F. M. Jancrer and Dr. R. T. A. Mers.

(Communicated in the meeting of June 30, 1917.)

§ 1. For certain purposes connected with the investigations into
Pasreur’s principle as made in this laboratory, it appeared desirable
to prepare optically-active complex salts of trivalent {ron with the
radicals of divalent carboxylic acids.

Such salts are already known derived from ovzalic acid; some
of them derived from malonic acid were prepared by Scnozz ®),
but in a not very recommendable way. The data given by this
author concerning the content of crystallisation-water in these salts,
do not agree with our numbers, moreover.

There exist in reality several series of hydrates here, as may
become clear in the following paragraphs.

Moreover, analogous complex salts may be obtained, as we found,
from tartronic acid, and from some substituted tartronic acids, as
we shall demonstrate later-on.

§ 2. All attempts made by us with the purpose of resolving
these racemic salts into their optically-active components, never gave
really positive results, neither with derivatives of oxalic, nor with
those of malonic acid.

In the case of the complex malonates the fission was attempted
by means of the strychnine-, brucine-, and cinchonine-salts. These
last gave sirupy, very viscous liquids, which did not erystallise.
The strychnine- and brucine-ferri-malonates crystallise in small,
greenish-yellow crystals, but after fractional crystallisation and eli-
mination of the alcaloids from the salts obtained, no optically active
solutions were obtained. The ¢tri-strychnine-salt whose constitution,
from analysis, could be established to be: {Fe (C,H, 0,),} Str, + 6 H,O,
showed a rotation corresponding about to the amount of strychnine
present in it.

After the base had been removed by means of K/, the potassium-
salt obtained appeared to be optically znactive, probably by very
rapid racemisation. No attempts made with the purpose of prevent-

1) A. Scuotz, Monatshefte f. Chemie 29. 439. (1908).
aS

284

ing this autoracemisation by experimenting in liquids containing
much acetone and by working very fast, gave any better results.
Analogous experience was gathered in the case of the brucine-salts.

In the same way we prepared the #ri-strychnine-ferri-ozalate,
which, from analysis, was seen to have the composition: {Fe (C,O,),}
Str, + 2 H,O, and here also several attempts were made to resolve
it into its antipodes. The result was always negative, and the same
occurred with : Diammonium-strychnine-ferri-ovalate, diammonium-
quinine-, diammonium-cinchonine- and diammonium-mor phine-ferri-
oxalates. Only from the solution of the strychnine-salt a green sub-
stance was obtained, which, bowever, did not show an activity
other than that corresponding to the amount of strychnine present.
The corresponding salts of hydroxylamine did not give a positive result
either.

§ 3. In this paper only racemic salts of the type:
{Fe (C,H, 0,),{ Me, + nH,O,

are described in which Me is replaced successively by A, (VA), -

Na, Rb, Cs, and Tl. The Ba-salt could also be prepared, from
which other salts could be obtained by interchange with soluble
sulphates. The sodium-salt erystallises badly, and its description is
therefore omitted here.

From warm solutions often pale green salts are obtained, possessing
rhombic symmetry, and containing 1 H,0, not 2H,O as Scuorz’)
believed. The corresponding A-salt erystallises badly, and the results
obtained, although pointing in any case to a distinct isomorphism
with the other salts, have therefore not been separately given here.

The most common K-salt, however, is a triclinic salt, erystallising
with 4 1,0. Of the Rh-salt we obtained, besides the rarely occurring
rhombic erystals (+1 H,0), also darkly coloured triclinic erystals,
containing only 1 H,O too, but not well measurable. At least there
must, therefore, be three series of hydrates possible here: rhombic
and triclinie with 1 H,O, and triclinic ones with 4 1,0. But in no
case we met with the crystals indicated by Scnorz, containing
2H,O. A systematical investigation of the eventually possible hydrates,
is very desirable. ;

The different salts can be prepared from concentrated solutions
of the alkali-malonates by adding the calculated amount of free
malonic acid, heating on the water-bath, and by finally adding
freshly precipitated and well-washed ferrt-hydrovide prepared from

1) A. Senorz, loco cit. p. 443. 445.

ee Oe

285

the calculated quantity of ferri-sulfute. The beautifully green solutions,
after evaporation on the water-bath, deposited crystals of the salts

here described.

$ 4. PorassIUM-FERRI-MALONATE.

Splendid, pale emerald-green, ordinarily flattened, very big and
perfectly transparent erystals. They are well developed, but in the
zone of the prism and that of the clino-doma they often show
oscillations of the angular values to an amount of 3°. Analysis

proved the composition to be
(13,1 °/, H,0; 10,3 °/, Fe) : K,} Fe (C,H,0,), } + 4,0.

The salt possesses

apts Cane
Potassium-F err1-
Malonate
(+4 H,0).
Angles:

te at Gr on rl So ret aaa,

therefore 2 H,O more than mentioned by ScrHo1z ').
Triclinie-pinacoidal.
a:b6:c=0,4924 :1:0,4897.

A= 967 aa! ea aS, oo
ie == 10343: SOES:
C= 84°51" y= 82°52)!

Forms observed: b = {010}, large and lustrous;
m = $110}, and a = 100}, almost equally large;
p = {120}, somewhat smaller than m; ¢= {110},
narrow, ‘but well reflecting; o == {111}, and
g ={011}, well developed, and like r= {101},
giving perfect images; c= {001}, small; s=}\021},
well-developed; w = 14 1}, as a narrow truncation
of the edge 7:5. The external habit is elongated
parallel to the c-axis, and in most cases some-
what flattened parallel to {O10}.

Observed: Calculated:

:b = (100) :(010) =* 95°19’ —
:¢ =(010):(001) =* 83 27 —

a —= (001) (100) =* 76 17 ==
:m —(100):(110) =* 26 39 —

q =(010):(011) =* 59 13 —
:m =(010):(110) = 68 40 68°40’
:gq =(001):(011) = 24 14 24 14
:p =(010):(120) = 48 16 48 613
mn = (120):(110) = 20 34 20 332/3
:t =(100):(110) = 25 0 ek

: 6’ =(110):(010) = 59 43 59 34

1) A. Serorz, Monatshefte f, Chemie, 29. 445. (1908).

e465. — {00}:
s:b’ = (021):
6b: = (010)
sr =(111):
r: 6! =(101):
mer, = (ADE
r:q =(101):
q:m= (011):
"40:0 = (010):
ag (100):
q:o {Dl
oa =D)
a’:» = (100):
dg):
c: 0. (001):
onm (11):
Hse (VIO):
P:g={120)"
a:r—{100):
ree = 10h);

286
Observed: Calculaled:

(021) = 47 30 47 361), j
(010) = 48 59 48 562/3
(111) = 60 36 60 36
(101) = 20 5 20 10
(010) = 99 20 99 14
(101) = 59 59 60 0
(011) = 52 38 52 45
(110) = 67 14 67 151/,
(111) = 72°26 12:17
(011) = 80 26 80 24
(111) = 36 32 36 31
(100) = 43 44 43 53
(111) = 54 42 54 58/3
(011) = 44 42 44 3715
(111) = 38.32 38 20!/,
(110) = 35 34 35 481),
(001) = 74 16 14 9
(OND), 608 60 22
(101) = 51 59 52 31
(001) = 5148 _ - 51 391

No distinct eleavability could be found.

Distinetly dichroitie : on

{010} for vibrations parallel to the c-axis

green, for those perpendicular to the former: yellow. On p and m

the dichroism is only unap

preciable.

On 6 and m is the angle of extinction about 27°, on p 44° with

respect to the direction of
axes intersects the edge 5:

the vertical axis. The plane of the optical
gq on {010} under an angle of about 21°.

AMMONIUM-FERRI-MALONATE.

Fig. 2.
Ammonium-F erri-Malonate.

{| Fe (C,H,O,),} (NH), +1 H,0.

Pale green, flattened, very lustrous,

small crystals.
Rhombic-bipyramidal.
a:b6:c=0,9407 :1 : 0.6860.

Forms observed: a = }100}, predomi-
nant, giving sharp reflexes ; m= {110},
well-developed and highly lustrous; 0 =
{111}, well-developed, and like r=}{102}
yielding good reflexes; s={101}, and
e={001}, very narrow and badly
reflecting.

287

Angles : Observed: Calculated :
a:m = (100):(110) =* 43°15’ ek
a:o —=(100):(111) =* 58 59 ne

o:o =(lil):(lil)= 62 5 62° 3’
m:m=(110):(110) = 93 31 93 30
a:s =(100):(101)= 54 0 53 54
sir. = (101): (102)—. 15-53 16 4
a:r =(100):(102)= 69 52 69 58
rect (102) (OON 22008 20:72
r:o =(102):(111)= 32 33 32 48
m:o =(110):(111)= 44 59 44 58
c:o =(001):(111)= 45 2 45 2
o:o =(111):(111)= 57 52 58 . 0
os. = (111): (101) == - -28 56 29 0
m:o =(110):(111)= 92 30 92 28!/5

No distinct cleavability was observed.

The pale green crystals are distinctly dichroitie : on {100}; pale
yellowish-green for vibrations parallel to the c-axis, for those
perpendicular to them, pale green. The plane of the optical axes is
{001}, with the a-axis as first bisectrix, probably of positive character.
The birefringence is weak ; the apparent axial angle is very small.

RUBIDIUM-FERRI-MALONATE.

Fe (C, a; 0), kb, +1 Hap 0.1) ek
Pale green, rhombical limited erystals. |
Rhombic-bipyramidal.
a:6:c— 0,9442 : 1 : 0,6985.

Fig. 3. Rubidium-Ferri-Malonate. + 1 H0).

1) This content of water of crystallisation is adopted because of the isomorphism
with the other salts. The numbers of the analysis were unsatisfactory, the quantity
of material at hand being too small.

288

Forms observed: a={100}, strongly predominant and highly
lustrous; 0 = {111}, and r={102}, well-developed and yielding
good reflexes ; m = {110}, well reflecting ; 6 = {010}, and w = {122},
narrow, built exactly measurable; «— {112}, small, often absent.

Angles: Observed: Calculated:

a:o =(100):(111) =* 58°46’ ==
0:0 = (111):(111) =* 58 38 =
o:o =(111):(122) = 14 14 14°22’
a: =(122):(102) = 33 22 33 141%
a:m=(100):(110) = 43 20 43 211,
m:b =(110):(010) = 46 40 46 401/,
a:r =(100):(102) = 69 48 69 42
rir —(100):(192) = 40 24 40 36
o:x% =(111):(112) = 18 46 18 32
o:0 =(122):(122) = 34 0 33 44

No distinct cleavage.

Noticeably dichroitic: on {100} yellowish-white for vibrations
parallel to the c-axis, for those perpendicular to them: pale green.
The plane of the optical axes is {001}; the a-axis is first bisectrix.
The apparent axial angle is very small.

No exact measurements could hitherto be made of the triclinic
Rb-salt with 1 H,O, because of the bad erystals only at hand.

CABSIUM-FERRI-MALONATE.
We (CH, O).} Cs, + 1 H,O.

Pale green, kite-shaped
crystals.

gee rr

Rhombic-bipyramidal.
a:b6:c=0,9548 : 1 :0,7089.

Forms observed:

a = {100}, predominant and
giving good reflexes; 0 =
{122}, well reflecting, and,
like r = {102}, rather largely
developed; m—={110}, narrow and smaller than 7; 6= {010}, small _
but well measurable.

Fig. 4.
Caesium-Ferri-Malonate.

Angles: Observed: Calculated:

a:o =(100) : (122)=* 73° 9 —
a:r. =(100),; (102)==*.69 38 =
oro = (122). (122) = 233.42 33°42’
rir =(102) : (122)= 40 46 40 44
a:m=(100):(110)= 43 48 43 401),
mb =(110).2 (010) — 4642 46 191),
o:r =(122):(102)= 33 28 33 36
0:0 = (122) : (122)= 66 56 67 0
0:6 =(122): (010) = 56.30 56 24

289

No distinct cleavability could be found.

The crystals are dichroitic: on {100} greenish-yellow for vibrations
parallel to the c-axis, for those perpendicular to them: pale green.
The plane of the optical axis is {001}; the apparent axial angle is

small, with the a-axis as first bisectrix. The birefringence is weak.

THALLO-FERRI-MALONATE.
{He (C,H,0,)s} TI, + 1 HO.

Beautiful, pale green, flattened crystals with rectangular borders.

Rhombic-bipyramidal.
a:b:e=0,9615: 1 :.0,7050.

Fig. 5.
Thallo-Ferri-Malonate.

Forms observed: a = {100}, predominant and very lustrous;
r = {102}, large and lustrous; c= {001}, commonly hardly visible, |
sometimes however well developed and striated parallel to c : a;
m == {110}, commonly well developed, very lustrous, occasionally
narrow, while p= {120}, which form is ordinarily absent, is much
broader in that case than m, but much duller; 0,= {111}, well
developed; ¢== 121}, also well-developed and lustrous; w — {221},
much smaller than ov, but giving good reflexes; w= {122}, very
narrow, but exactly measurable; 6 = {010}, very narrow.

290

Angles : Observed: Calculated :
asf. ==1(100) 3102)", 6952! —
a:o =(100):(111)=* 59 4 =
r:r =(102):(102)= 40 16 40°16’
O:m:==(1bt):(122)— 14 6 14 15
oro == (122):(122)= 33 37 33 22
a:w =(100):(221)— 49 40 49 411),
w:t = (221):(121)= 17 5 17 1915
a:t =(100):(121)= 66 58 67 1
a:p =(100):(120)= 62 40 62 311/»
a:m=(100):(110) = 43 50 43 521/,
m:m —=(100):(110) = 92 20 92 15
m:p =(110):(120)= 18 50 18 39
m:b° =(110):(010) = 46 10 46 TU
o:r =(111):(102)= 33 4 33 6

No distinct cleavability.

The crystals are noticeably dichroitie: on {100} yellow-green for
vibrations parallel to the c-axis, and pale green for those perpen-
dicular to the former. The plane of the optical axes is {O01}, with
the a-axis as first bisectrix of positive character. The apparent
axial angle is very small.

THALLO-MALONATE.
GONS.

Crystallised from water, the salt is deposited in very big, trans-
parent crystals; they are
anhydrous (79,93°/, Tl; cale:
80°/,). The compound is very
soluble, and crystallisation
starts only in highly super-
saturated solutions.

Monoclinic-prismatic.
ú:bac= 051011 1 Wsan.
B= 81°304’

Forms observed: c= {001},
very ‘lustrous; 6 = {010},
gives good reflexes; in the
same way: m — $110}, and fig. 6.

6 = {141}; s = 101}, narrow Anhydrous Thallo-Malonate.
and often badly measurable; = {102}, not measurable, because the
faces are either concave, or strongly curved. Besides the forms

291

reproduced in fig. 6, also plates parallel to {010} as a predominant
form, are occasionally observed.

Angles : Observed: Calculated :
c:m-= (001) : (110) =* 82°361/,’ —
b:m=(010) : (110) =* 60 33), —

c:o =(001) : (111)=* 59 213), —
m:m=(110):(110)= 58 53 58°53’

z o:0 =(111): (111)= 50 28 50 29
b:o =(010): (111)= 64 46 54 451),
o:m=(l11) :.(110)= 23 141), 23 141/,
m:s =(110):(101)= 40 49 40 42
c:s =(001):(101)= 69 1 69 11/5

Very perfectly cleavable parallel to {001}.

According to Hausnorer*) the corresponding potassium- lt Crys-
tallises with 1 H,O, and is also monoclinic, but without distinct form-
analogy with the thallo-salt here described. (a: b:c = 1,4945:1:0,9174;
B = 61°15’).

Laboratory for Inorganic and Physical Chemistry
of the University Groningen, Holland.

1) K. Havsuorer, Zeits. f. Kryst. u. Miner 6. 120 (1881).

Chemistry — “On some isomeric, complex cis- and trans- Diethy-
lenediamine-Salts of Cobaltum, and on Tri-ethylenediamine-
Zinc-Chloride”” By Prof. Dr. F. M. JarGer and Dr. Jur. Kann.

(Communicated in the meeting of June 30, 1917)

$ 1. According to Werner’s theory concerning the stereometrical
configuration of inorganic salts derived from the complex radical:
‘MeX’,', there must exist two isomerides of derivatives containing

7!

)
ions of the special type: (ave y"

and trans-isomerides. If the six co-ordination-loci round the central
atom be considered as situated in space like the six corners of
a regular octahedron, the substitutes Y' are located in the cis-
derivatives as near as possible to each other, while, on the contrary,
in the ¢érans-derivatives they are elongated as far as possible from
each other, being placed at the two ends of an axis of the octabedron.

If in the complex salts of this kind, the four co-ordination-loci
X', be occupied by two bivalent radicals X",, it is obvious that
the configuration of the molecule in the czs-derivatives possesses the
axial symmetry of C,; the heteropolar binary symmetry-axis of these
complex ions joins of course the middle of the octahedron-edge Y’ Y’
with that of the opposite and parallel edge. The symmetry of these
ions is therefore exactly that of the monoclinic-sphenoidical class of
erystallonomy and to every configuration of this kind corresponds
therefore a »non-superposable mirror-image, because the complex of
atoms possesses only axial symmetry. The cis-compounds of the type

), which are distinguished as cis-

Bs
(ave a) must, for that reason, be considered as racemic compounds
rk

eventually resolvable into two optically active and oppositely rotating
antipodes. The possibility of such a fission is demonstrated by WERNER

in an experimental way for several salts of this kind.
all

The trans-derivatives of the same wpe ( Me ee however,

Nes
possess the symmetry of the group Dr Their configuration is there-
fore identical with its mirror-image, so that they are not resolvable
into such antipodes. *)

1) See: F. M. Jarcer, Lectures on The Principle of Symmetry and Its Appli-
cations in all Natural Sciences, Elsevier-Gompany, Amsterdam, (1917), p. 228—256.

293

In the following paragraphs some of these resolvable and un-
resolvable salts will be described more in detail.

§ 2. Racemic CIS-DrAMINO-DIETHYLENEDIAMINE-COBALTI-CHLORIDE.

NH),
(Eine),

| Co Cl, +1H,O.

Red-brown, well-developed,
and very lustrous small crystals,
which obviously are isomorphous
with the corresponding bromide
and iodide.

Monoclinic-prismatie.
a:b:e=1,1172:1:0,8325;
B = 87° 561.

Forms observed: r', = [101],
predominant; the external shape
of the erystals appears ordinarily

Fig. 1. flattened parallel to this form.
Furthermore: m=={110), well-developed and very lustrous; r, =
[101], small, but giving sharp reflexes; 0= [121] and o= [121],
both very narrow, and almost equally developed; a=!100], small,

but very lustrous; g={011], mostly narrow, but yielding splendid
reflexes.

ae : Observed: Calculated:
a:m =(100):(110) =* 48° 9’ <
a En =" 54 38 =
a eae ce =, 5159 ae
m :r’,=(110):(101) = 67 14 67°171/,
m:o saat = ae | 32 23
m:m=(110):(110) = 83 42 83 42
ro:q =(101):(011) = 52 17 52 23
r:r/,= (101) :(101) = 713 21 13 23
m:r, =(110):(101) = 65 24 65 45
org (121) (011) — 29,13 29 12
ri:q =(101):(O11) = 54 22 51 31
m:o =(110):(121) = 34 59 35 15

Perhaps cleavable parallel to m.
The crystals are only slightly dichroitic, in a way analogous to
that of the jodide. They are evidently identical with crystals described

294

previously '), if only the following symbols be adopted there:
a = [101], 0 = [110], r= [100], s = [101], and w = [121].

In contradiction to the data given in literature, all three halo-
genides must have the same content of erystallisation-water, and,
according to the analytical investigation, 1 H,0.

§ 3. Racemic cis-DIAMINO-DIETHYLENEDIAMINIC-COBALTI-BROMIDE.
A7

| co (A belie

(Hine),

| Br, +1 H,0.

fig 2.
cis-Diamino-diethylenediamine-cobalti-bromide.
The substance erystallises in flat and long brownish-red needles
prismatic in the direction of the c-axis, or in short, thick and small
crystals, having a slight elongation in the direction of the a-axis.

Monoclinic-prismatic.
a:b:¢c=1,1177 : 1: 0,8322.
B=88 >
Forms observed: m = [110], and g=(011], large and lustrous.

Occasionally m is predominant, and eventually g. Furthermore:
r, = [101] and r', = [101], almost equally large and giving good
reflexes; w—={121], commonly small, but also, if ¢g={011] be
only slightly developed, occasionally almost equaliy large as 7,,
a == [100], very narrow, and mostly absent. The substance is com-
pletely isomorphous with the corresponding iodide.

1) F. M. Janeen, Zeits. f. Kryst. 89. 545. (1904).

295

Angles: Observed Calculated:
a:m= (100): (110) =* 48°10’ —

r’y:m =(101):(110) =* 67 15 —
r',:q =(101):(011) =* 52 14 =

m:m=(110):(110)= 83 40 83°40’
9:9 =(O11):O1l)= 79 13 19 13
r'o:q =(101):(011)= 52 14 52 14
Tis Gj = (101) HOLD =< 51723'/. 151 2315
r,):m=(101):(110)= 65 48 65 51
m:e =(110):(121)= 32 19 32 23
@:q =(121):(011)= 27 52 28

The crystals are distinctly cleavable parallel to [110]. They are
slightly dichroitic, analogously to the zodide. The angle of extinction
also, about 20° with respect to the c-axis on m, has a size also
comparable with that found in the case of the todide.

§ 4. Racemic cis-DrAMINO-
DIETHY LENEDIAMINE-COBALTI-IODIDE.
NT
NH).
(Eine),

Big, splendidly developed, brown-red and
highly lustrous crystals with very constant

Vz

on

Y

PM HO:

8

eee ees oe & = oe

angular values.
Monoclinic-prismatic.
as pac =O : 0,8178 ;
B
Forms observed: m= [110], predominant and
very lustrous; a— [100], smaller than m;
g =(011], large and sharply reflecting; r, =[101 |

and 7’, = [101], almost equally well-developed

and yielding excellent reflexes; 0 = [121], and
w=[121], almost equally large and well fig. 3

. Dh eee te . ; Racemic cis-Diamino-di-
reflecting. The habit is prismatic parallel to Etenen ee

the c-axis. iodide.

rm tr mm wn me om ~ we wwe eee nn eee

\

Angles: Observed: Calculated:
a:m=(100) : (110)=* 47°38’ Le
9:q —=(Olt) : (Oli) =* 78 304, =
Pete — (1017 2.(101)=* 73:22 a

m:m=(110) : (110) — 84 44 84°44’
a’ :ra/=(100) : (101)= 54 59 54 5613
r‚:a =(101) :(100)= 51 44 51 40

m:o =(110) : (121) — 35 20 35 31.

296

Observed: Calculated:

org 1121) : (OLD a3 29 12
m:w =(110) : (121) = 32 14 32 20
m:r;—=(110) : (101) = 65 20 65 17
m:r{=(110) : (101) = 67 12 67 134
fig SAIN, 6011) =~ 2 1.16 51 6
ro:9 =(101) : @11N)=- 52° 3 52 9
m:q —(110) : (011)= 64 38 64 43
m’:q =(110): (Ol1)= 60 31 60 38
w:qg =(121): (Oli)= 2757 28 18

Perfectly cleavable parallel to m.
The crystals are feebly dichroitic: on a yellow-orange for vibra-
tions in the direction of the c-axisy and for those perpendicular to
them: red-orange. On m is the angle of extinction about 28° with
respect to the vertical axis.

§ 5. TRANS-DIAMINO-DIETHYLENEDIAMINE-
COBALTI-IODIDE.

NH),

(Hine),

Small, often badly developed crystals ”,
with a dark red-brown colour; their
aspect is that of hexagonal plates (fig. 4).

Co

Rhombic-bipyramidal.
a:b:ce=1,2449:1 : 1,2842.

Forms observed: a —= [100], distinctly
predominant and very lustrous; 0= [111],
and m= [120], well-developed and giving
sharp mirror-images.

Fig. 4.
trans-Diamino diethylene-
diamino-cobalti iodide.

Angles : Observed: Calculated :
a:o =(100):(111)=* 57°58’ —
a:m =(100):(120)=* 68 7 —
0:0 =(111):(111)= 64 12 64°14’
m :m = (120): (120)= 43 46 43 46
m-o — (120): (111) == 35°46 Shale
0:0 =(111):(111)= 83 58 83 57!/2

The crystals are slightly dichroitic: on a orange-red for vibrations
parallel to the c-axis, dark orange-red for such as are perpendicular
to them. The plane of the optical axes is [100]; the a-axis is first
bisectrix. The apparent axial angle is large, the dispersion is rather
strong, with @<v round the a-axis.

297

§ 6. Racemic cris-DIAMINO-DIETHYLENEDIAMINE-COBALTI-NITRATE.

NH),

Co, Eine),

CAO)

Beautiful, well-developed,
yellow-brown or red-brown py-
ramids, with lustrous faces. They
are nicely built, and have very
constant angular values.

Rhombic-bipyramidal.

fy ee en Ae ae ; Fig. 5.
BP aR ees aa eg ey cis-Diamino-diethylenediamine-Cobalti-
Forms observed: o ={111], Nitrate.

with highly lustrous faces. The external form of the crystals is that
of flattened octahedra.
Angles : Observed: Calculated:

0:0 =(111):(111) =* 91° 0’ =
Oo O° (11) (ti) —* . 61 11 ae
ero (AN 57 48 57°38’
brode (tij 89! 2 89 0

No distinct cleavage was observed.

§ 7. Racrmic CIs-DINITRO-DIETHYLENEDIAMINE -COBALTI-NITRATE.
(Flavo-salts).
„0,

(Kine),

Small, flattened, yellow-red prismatic crystals. They are identical
with those previously measured’) by us, which we had received
from Werner, but they have other combination-forms, and somewhat
deviating angular values. The aspect of the crystals is that of fig. 20
in the paper mentioned.

Monoclinic-prismatic.
ao 0 @— 15089 1 04072:
a 68°30°/'.

All forms previously observed were met with again; their relative
development was also almost the same, only g = [011] was appre-
ciably larger here. New forms are: 7 = [401], well developed and
yielding good reflexes; and s=—={[410|, very narrow, but lustrous.

wo)

Angles : Observed: Calculated:
m:m= (110): (110) =* 110°50’ =
c:r =(001):(401)=* 57 36 =

1) F. M. Jarcer, Zeits. f. Kryst. u. Miner. 39. 564. (1904).
20
Proceedings Royal Acad. Amsterdam. Vol. XX

298

Observed: Calculated:

m:c =(110):(001)=* 78 0 —
mb =(110):(010)= 34 35 34°35!
a:s =(100):(410)= 19 44 19 56
Sot (410) (LO 35/31 35 29
a7 =(100) 2 (N10) 55 25 5525
m:q =(110):(011)= 60 48 60 541,
bg = (OLO (OL Tj =" “6020 69 141,
e:q =(001):(011)= 20 40 20 451,

No distinet cleavability was observed.

On [010] feebly dichroitic: for vibrations parallel to the c-axis,
yellow-orange, for such perpendicular to them red-orange. The angle
of extinction on [010] is about 60° with respect to the c-axis, in
the acute angle of the axes a and c. The plane of the optical
axes is probably [010].

§ 8. ‘TRANS-DINITRO-DIETHYLENEDIAMINE-CoBALTI- NITRATE. (Croceo-salt).
Se NON pe
Co (Hine). (N O,).

This compound was obtained from the mother-liquid of the cor-
responding eis-derivative, in the form of splendidly developed, red-
dish brown, very lustrous and transparent, little crystals. They are
evidently identical with those prepared by WeRNER, and previously
described’); also in this case the angular values somewhat differ
from those formerly measured. As a new form, 6 = [010] was
found. The external aspect of the crystals is that of hexagonally or

Fig. 6.
trans-Dinitro-diethylenediamine-Cobalti-Nitrate.

octogonally bordered, rather thick erystal-plates.

Monoelinie-prismatic.
a:6:c=1,3040: 1 :1,0085:
lem Bote BD
1) F. M. Jagger, Zeits. f. Kryst. u. Miner. 39. 565. (1904).

299

Forms observed: a= [100], predominant and splendidly reflecting ;

c= [001], smaller,

often completely absent, but very lustrous;

6= [010], small, well reflecting; o= [111], somewhat broader

than w =(111], perfectly reflecting. The external habit is flattened
parallel to [100], with» a slight elongation parallel to the 0-, or

the c-axis.
Angles:

Observed: Calculated:

(100) : (111)=* 70°13’ as

van (hdl: Gi)" 77 30 Bt
b:o =(010) : (111)=* 55 54 =
ao (lll (HD «8353 83°52’
b= (Ol) (PI 48°) A 48 4
os (Adie (1 =~ Bi 26 57 35
dsor (100): (111) 52740 52 12
a:c —=(100) : (001)= 73 58 18 311,
eso. — (OO (ERI 45712 44 59
Eros (O01) (NLD 57 22 57 22
ov or (ALTE ATD 62 69 12

No distinct cleavability was observed.

The erystals are distinctly dichroitic: on a for vibrations parallel
to the c-axis: orange-yellow, for those perpendicular to them:
lemon-yellow. The optical axial plane is [010]; one optical axis is
almost perpendicular to a. Extremely strong dispersion, with @ < v.

§ 9. Racemic. cis - DINITRO-DIETHYLENEDIAMINE - COBALT - NiTRITE.

Tee eee wee ee ee

=

1
‘
‘
‘
‘
’
.
‘
’
‘
'
’
'
ij
’

Fig. 7.
Racemic cis-Dinitro-
diethylenediamine-Cobalti-
Nitrite.

(Flavo-salt).

a! (No,),
|oo (Eine), (NO).

Dark coloured erystals, which in their
external aspect, are highly analogous to
those of the corresponding trans-derivative.
They are very beautifully developed, and
possess constant angular values.

Monoclinic-prismatic.
a: b:¢e = 0,7382: 1 : 0,9094:
p= 82 07".

Forms observed: m = [110],and) = [010],
large and very lustrous; o=[111], large,
perfectly reflecting ; w = | 223], small but
well reflecting; c = [001], small, giving
good reflexes. The external habit is prismatic
parallel to the c-axis, and commonly some-
what flattened parallel to [010]. Occasionally
| 010 is absent, in other cases, however,
the same is true for [223].

20*

300
Angles: Observed: Calculated:

m :m = (110): (110) =* 72°27’ =
c:m — (001): (110) 84 19 see
0-0 (ME Gi) —" 59 35 =
b:m =(010):(110)= 53 49 53°46! 5
c:» =(001):(223)= 48 34 48 241/,
c:a = (001):(111)= 56 46 56 51
o:m=(111):(110)= 38 41 38 50
o :m —(223):(110)= 35 35 35 54!/,
b:o =(010):(111)= 60 10 60 11
b: = (010) :(223)= 63 37 63 38
w:0 == (223):(223)= 52 49 52 44

No distinct cleavability was found.

In contrast with the water-containing crystals of the trans-deri-
vative, these crystals do not manifest any trace of change. They
are distinctly dichroitic: on [010] orange-red for vibrations parallel
to the c-axis, for such as are perpendicular to them: blood-red. The
angle of extinction on b includes 33° with the direction of the
c-axis, in the quadrant of the acute angle a: c.

§ 10. Trans-DInITRO-DIETHYLENEDIAMINE-CoBALTI-NITRITE (Croceo-salt).

Co

i
1
t
|
1
1
‘
1
1
mM,
t
'
H
1
i
1

Fig. 8.
Trans-Dinitro-diethylenediamine-
Cobalti-Nitrite.

(No),
(Eine),

(NO) + 2H,0.

Big, splendidly developed, dark brown
erystals which, however, soon lose
their water of crystallisation, becoming
gradually orange-yellow, most rapidly
on the faces of the prism.

| Monoclinic-prismatie.
asbre= 1 0456-2790.
f—=68°33"/;-

Forms observed: m —= [110], predo-
minant and very lustrous; o = [111],
large and giving good reflexes, like
ec = [001], which is much smaller than
m; w = [225], very small, and often
absent. Notwithstanding the different
angular values there exists a striking
analogy in aspect and development
of these crystals, and those of the
anhydrous cis-derivative.

301

Angles: Observed : Calculated:

c:o (001): (111) =* 54°57’
oto: (ill): (ly. 72 31
o:m/(111):(110) =* 50 17
c:m(001):(110)= 74 41
m:m(110):(110)= 91 30
C:m (001):(225)= 50 59
om (225) :(110)= 23 42

A distinct cleavability was not found.

74°49’
91 32
ol. 5
23 44

The erystals are dichroitie in a way analogous to that found in
the case of the cis-derivative. On m is the angle of extinction oblique.

$11. From a solution containing both
the cis- and trans-derivatives, crystals of
the form shown in fig. 9 were deposited,
which imitate in a striking way the
aspect of the cis-derivative. In contradiction
with those of this substance, however,
the crystals of this kind do not decay,
nor do they lose any water of crystalli-
sation. Moreover, they have often rather
oscillating angular values in the zone of the
prism, and uneven faces. Their dichroism
is analogous to that observed with the
pure ¢zs-, or trans-compounds.

Monoclinic-prismatic.

a: ee — 10169; 1 -0;9030: B = 66° 16’.

Forms observed: m =(110] predominant
and lustrous; 0 =|111], large, giving
good reflexes; 4—{|010j, much smaller
than m;c = [001], small, but well measu-
rable. The external form is prismatic
towards the c-axis.

Angles : Observed :

c:m = (001): (110) =* 72°52’
b: m=(010):(110)=* 47 3
0:0 =(111):(111)=* 64 2914
m:m=(110):(110)= 85 58
c:o =(001):(111)= 48 50
onl) (110), == 5755
b:0 =(010):(111)= 57 47

Fig 9.
cis- + trans-Dinitro-diethylene-
diamine-Cobalti-Nitrite.

Calculated :

85°54
48 26/5
58 18
57 451/,

302

No distinct cleavability was observed.

Notwithstanding the striking analogy in form, a distinct and real
difference in the angular values and parameters is found to exist;
the ratio a:6 is analogous to that of the trans-compound, while
for 6:¢ exactly the same is true with respect to the cis-derivative.
It is difficult to make sure whether here a double-compound of
the isomeric salts is present, or a mixed-crystal between them. The
fact that the angular values are somewhat fluctuating, and that the faces
often reflect irregularly, might be looked upon as an argument of
the greater probability of the last mentioned supposition. In any
ease, the peculiar erystallonomical analogy, and the intermediate
shape of these crystals between those of the components, are highly
remarkable phenomena, perhaps explicable by the only feeble mor-
photropic influence of NO,-groups in general.

It is worth drawing attention, moreover, to the fact that all the
cis-Diethylenediamine-derivatives here studied, have monoclinic sym-
metry, however not that of the group C,. Probably there will be
salts of this particular symmetry amongst the optically-active com-
ponents of these racemic salts.

§ 12. Finally we give here a description of racemic Tri-
ETHYLENEDIAMINE-ZINC-CRLORIDK, the fission of which is also attempted
in this laboratory.

{Zn (Eine),} Cl, + 2 H,O.

Fig. 10.
Tri-ethylenediamine-Zinc-Chloride.

303

Beautiful, colourless, very lustrous and transparent crystals, which
allow very exact measurements. The substance was prepared from
ZnCl, and ethylenediamine in slight excess; it was recrystallised
from water. Analysis gave: 20°/, Cl, corresponding with a content
of 2 molecules of water.

Monoelinic-prismatic.
a:b:c=0,9238 : 1 : 0,6299;
B= 86°33".

Forms observed: 6 ={|010], commonly predominant and highly
lustrous, occasionally totally absent; a — [100], well-developed and
always present; 0 [111], yielding splendid reflexes; m = [350],
lustrous, and when present, rather large; c= [008], small, often
absent, giving somewhat dull reflexes; g—= [011] and ¢ = [031],
very dimly reflecting; w — [111], highly lustrous and well-developed.

The external form is that of hexagonally bordered, thick plates
parallel to [O10], or that of short prisms parallel to the c-axis.

Angles: Observed: Calculated:
:¢ =(100):(001)=* 86 33 —
‚5 =(010):(111)=* 63 11 —
ners (T00): (ELT) 62, 121/, —

B DJ 153 44 53°44’
aoe — (bd): (111) =~ 56°16 56 16
=(010):(111)= 61 52 61 52

om (ADEL = “59.50 —
:0 =(100):(111)= 51 58 57 49

ORT A OG Seo aS) Sis
e

dr 001)} (Ol) 32-11 32 92;
t —=(0!1}):(031) = 29 51 29 55
ss (031); (010) =~ 27 53 27 551/o
:m =(100):(350)= 55 0 55 47
m:m = (350):(350) = +67 58 68 16

No distinct cleavability was found.
On [010] is the angle of extinction 11° with respect to the c-axis;
the plane of the optical axes is probably [010].
Laboratory for Inorganic and Physical Chemistry of the
University, Groningen, Holland.

Mathematics. — “On Elementary Surfaces of the third order”.
(Second communication). By B. P. HAALMEYER. (Communicated
by Prof. Brouwer).

(Communicated in the meeting of June 30, 1917).

§ 4. If A is cusp in one and not more than one plane a, then a
is tangent plane.

Let a be cuspidal tangent and K, and K, the branches meeting
at A. The Jordan theorem for three dimensional space tells us that
within any vicinity of A the branches A, and K, are connected
by a set of points I and by another set H, both sets belonging to
F*. Again I and II have no points in common and are the con-
tinuous (1,1) representations of plane regions I, and II, which have
the character of Jordan regions in the vicinity of the point 4,
corresponding to A. Inside a finite neighbourhood of A all points of
F° belong to 1+ 11+ A, + K,. Let EF be a linesegment in «
crossing both branches K, and K,. Suppose I and II were situated
on: the same side of «. If a parallel linesegment converges from
that side towards LF, it would end up by having at least two
points in common with I and also two with II: a contradiction. Hence
I and II lie on different sides of a, for instance I above and II below a.

Let 8 be a plane through A not containing the cuspidal tangent
a. A cannot be cusp in 8 (@=|=e), and the results of § § 1 and 3
show that A cannot be isolated or double point in 8. Hence A is
in 8 ordinary point or point of inflexion. On the line of intersection
of « and 8, A counts double in @ hence according to the theorem
of p.117—118, A also counts double on that line in p and this excludes
the possibility that 4A is point of inflexion in 8. Thus only remains
the possibility that A is ordinary point in > and the line of inter-
section with « is tangent because A counts double on this line.

Thus has been proved that in any plane not containing the
cuspidal tangent in a the point A is ordinary point with tangent in «.

Remains to consider a section of #* in a plane 8 (=e) through
the cuspidal tangent a. Let 6 be a line through 4 in 8 (=|-a). We
consider a sequence of planes £,,8,,.... through 5 and converging
towards 8. The -lines of intersection of a and §,,8,.... @ are
respectively denoted by” a,,a,....q@ (all passing through A). In

305

every plane 8, the line a, is ordinary tangent at A. It is easy to
show that in the vicinity of A the curves in all these planes lie
on the same side of «a. However this result is not wanted: we
merely take a component sequence of planes in which the curves
depart from A on the same side of «, let us say above «.

The point A divides the cuspidal tangent a in two semilines,
let a’ be the one departing from A in the same direction as the
cuspidal branches and a" the other. The corresponding semilines on
the converging lines we denote by a,’,a,’... and a,",a,"....

In every plane 8, a branch departs from A above « in the
direction of a,". The reasoning used for the examination of a section
in a plane through a tangent at a double point shows here that in
8 the limiting branch departs from A above « in the direction of
a". The line a has only A in common with £* and considering A
cannot be double point or cusp in ~, the only remaining possibility
is that A is point of inflexion in ¢ with a for tangent.

This completes the proof that a is tangent plane.

§ 5. If A is cusp in two different planes, then A is exceptional
point.

In § 1 it was shown if A is isolated in a plane «, then @ is
tangent plane or A is exceptional point. In case « is tangent plane
it was found that A is ordinary point in every plane except a.
Hence when A is known to be cusp in some planes, and if we
want to show that A is exceptional point, then it suffices to prove the
existence of a plane in which A is isolated.

Point A is cusp in two different planes. We consider two assump-
tions: that the two cuspidal tangents coincide or do not coincide.

First assumption: A is cusp in the planes @ and 8 and the line
of intersection a of these planes is the common cuspidal tangent.
Let y be a plane through A not containing the line a.

In « the point A counts double on the line of intersection of
aand y, hence, according to the theorem of page 117 —118, Aalso counts
double on that line of intersection in y. The same holds for the
line of intersection of 8 and y. Hence in y two different lines exist
on which A counts double, and from this follows that A is in y
either cusp, double point or isolated point.

If A were cusp in y, then A would be cusp in two planes « and
and the cuspidal tangents would not coincide. This case shall be
dealt with later on, when the second possibility is assumed.

Hence to show that A is isolated in y it only remains to prove
that A cannot be double point in y.

306

Let c be an arbitrary line through A in y, not situated in « or
B. Let the plane y revolve round c. If in any position of y the
point A is isolated then our object is attained. The alternative is
that A is double point in all planes through c except the plane
through c and a. The foregoing results (p.113—114) show that the
only way to escape immediate contradiction is to assume A cusp in
the plane through c and a. But c was an arbitrary line through A in
y only subjected to the condition not to be situated in « or 8, hence
every plane through a would show a cusp in A and the reasoning
given on page 108 shows that then A would be isolated in every
plane not containing a.

Second assumption: The cuspidal tangents do not coincide. The
line of intersection a of the planes « and 3, in which A is cusp,
cannot be cuspidal tangent in either of these planes, because a has,
except A, another point in common with /’*. Hence the case indicated
in fig. 6 ineludes all possibilities. Let bHEF CD be a plane 1 a.

Fig. 6.

The semiplanes aH and aD contain no points of /* inside a certain
finite neighbourhood of A. On p. 104—105 it was shown that if A is
isolated in a plane «, then on one side of « there is a finite
neighbourhood of A containing no points of /’*. The demonstration
was entirely based on the analysis situs, hence it is of no consequence
whether the semiplanes in which « is divided by a line through A,
happen to make an angle of 180° with each other or any other
angle (=l= zero). Applying this to the case of fig. 6 it follows that
there exists a finite neighbourhood of A containing no points of
F* inside that part of space situated between the semiplanes a£
and aD and in which the semiplanes af” and aC are not situated
(in the semiplanes a/ and aC branches meet at A, so in this angle
between aH and aD the point A is certainly not isolated).

307

Now let the semiplane aH revolve round a towards af and aD
towards aC as indicated by the arrows. For aH there is either a
last position in which A is, or a first in which A 2s not isolated.
Let this be aH,. In the same way aD, for aD. If the angle between
aH, and aD,, in which af and aC are situated, is < 180°, then
at once planes can be found in which A is isolated. Thus it remains
to consider the cases in which the angle is 2 180°.

In every semiplane through a, in which A is not isolated, two
branches must meet at A, because if there was only one, the pro-
longation of this branch would be situated in the complimentary
semiplane, and these two branches would be connected inside any
vicinity of A, on both sides of this plane, hence through a there
would be no semiplanes at all in which A is isolated.

For this reason, if the angle between aH, and aD, were >180°,
then there would be a finite angle inside which every plane through
a has a double point in A. Let y be a plane inside this angle. The
semiplanes through a in which A is cusp are supposed to lie wnder-
neath y. If y be turned round a in either direction, A at first
remains double point. In y four branches depart from A, successi-
vely AP, AQ, AR and AS. Let a lie between AP and AS and by
consequence also between AQ and AR. Let 5 be a line through A in y
between AP and AQ, hence also between AR and AS. Lastly let
8 be an arbitrary plane through 5. In 8 two branches arrive at A
from above y, because above y, AP is connected with AQ and AR
with AS. The alternative that above y, AQ is connected with AR
and AS with AP, is excluded, because in the planes through a in
which A is cusp, the branches meet in A from below y.

Now the two branches in 2 meeting at A from above y cannot
form a cusp in A, because in that case, A could at the utmost be
isolated in only one semiplane through a. On the other hand the
branches in 8 meeting at A cannot form an ordinary point at A with
b for tangent, because then we could turn y round a to a position
" y’ in such a way that the line of intersection of y’ and @ would
have three different points in common with the curve in g. But in
y’ the point A would remain double point (provided the rotation is
small enough) hence this line of intersection of y’ and 8 would
have at least four points in common with the curve in y': a con-
tradiction.

In 6 two branches arrive in A from above y, but we found that
in 8 the point A can neither be cusp nor ordinary point with 6 for
tangent, hence A must be double point in 8. 6 however, was an
arbitrary plane through 6 hence every plane through 6 would have

308

a double point in A and this has been shown on page 114 to be
impossible.

It only remains to consider the case that the angle between the
semiplanes aH, and aD, is equal to 180°. Above it was shown
that in every semzplane through a in which A is not isolated, two
branches must meet at A hence in the plane formed by al, and
aD, there are four possibilities:

A is double point.
is ordinary point with a for tangent.

is cusp.

ag i
Ps DS he

is isolated.

To complete the demonstration of the existence of a plane in
which A is isolated, we shall show successively that 1, 2 and 3
lead to contradictions.

1. Let y be the plane of a £, and a D,. From the double point
A four branches depart in this plane, successively: AP, AQ, AR
and AS. The line a lies again between AS and AP, hence also
separates AQ from AR. The semiplanes « and p, in which A is
cusp are again supposed to lie below y. In the complementary semi-
planes A is isolated hence above y, AP is connected with AQ and
‘AR with AS, below y, AS with AP and AQ with AR. These last
two connections are via the branches meeting at the cusps in « and
8. Let d be a line in y through A, separated from a by the branches
RAP and QAS, and let d be an arbitrary plane through d.
In plane d two branches meet in A from above 7. Both these
branches have d for tangent, because A is isolated in every semi-
plane through a above y. Hence in d, A is ordinary point with d
for tangent. In the vicinity of A the curve in d lies above y, hence
below y the point A is isolated in d. This however holds for any
position of d (through d), but then it is impossible that the branches
meeting at A in « (org) are connected inside every neighbourhood
of A with the branches meeting at A in y.

2. Let y again be the plane of a £, and a D,. In this plane A
is ordinary point with a for tangent. The semiplanes with cusp in
A are again supposed to lie underneath y. Let d be a line in y
through A having three different points in common with #* and
let d be a plane through d (==). In every semiplane through a
above y, A is isolated and in every semiplane through a below y,
A is not isolated, hence if a semiplane turning round a converges
from below towards the semiplane of y in which A is isolated then
in these semiplanes ovals, passing through A and having a for

309

tangent will contract towards 4. This means that in the plane d the
point A counts at least double on d, but d has two other points
in common with /’*: a contradiction.

3. A is cusp y with 6(==a) for tangent. The original semiplanes
a and 3, in which A is cusp are again supposed to lie below y.
Semiplanes through a converging from below towards the semiplane
of y, in which A is isolated, again show ovals through A with a
for tangent and contracting towards 4. Let ce be a line in y through
A (=| a and =|= 6). The contracting ovals show that c is tangent
at A in every plane (=|=y). Besides in all these planes A is ordinary
point because c has still another point in common with /’*. ¢ is
an arbitrary line in y through A (=|=a or =|=6), hence in every
plane through A (except those through a or 6) the curve is, in the
vicinity of A, situated below y and the tangents at A all lie in y. Let
us now consider one of the original semiplanes with cusp in A, for
instance a. Applying the same reasoning given on p. 115—116 it is
found that every line through A in @ (except a and the cuspidal
tangent in «) must be tangent at A in every plane passing through
that line (except «). But this contradicts the result obtained above
that in every plane through A (except those through a and 5), A
is ordinary point with tangent in y.

§ 6. Through A passes at least one plane in which A is either
isolated, double point or cusp.

Let A be ordinary point in two different planes, such that the
tangents a and 6 in A do not coincide. In the preceding pages it has
been shown that when A counts double on a line in a plane, then
A also counts double on that line in any other plane. From this
follows that in the plane through a and 6, A counts double on both
these lines, hence in that plane A is either isolated, double point or
cusp. Remains to prove that through an arbitrary point A of f°
pass two planes in which A is ordinary point with non-coinciding |
tangents. Except isolated points, double points, cusps and ordinary
points there are only points of inflexion. We begin by showing that
not all planes through A can have a point of inflexion at A.

Let « be a line through A which has still another point in
common with F*, and thus can never be tangent at a point of
inflexion. Suppose every plane through A shows a point of inflexion
at A. In every semiplane through a departs from A a convex arch,
situated at first either above or below the tangent at A (at the
outset it was assumed that no line through A belongs entirely ot #*).

310

Now we obtain a contradiction if we can show that none of these
semiplanes in which the convex arch departs above (below) the
tangent in A can be limiting element of a sequence of semiplanes
in which the convex arch departs below (above) the tangent.

Let the semiplanes a@,,a@,,@,.... converge towards a. Let the
corresponding semitangents be 6,,0,,6,....6. In a,,@,,a@,.... the
convex arches are assumed to depart below 6,,6,,6,...., and in «
above 5. The lines 6,,6,,6,.... have at least one limiting line 6'
through A in «. There are three a priori possibilities: 6’ can be
situated above 6, below 6 or ean coincide with 6.

First case. Let 6" be a semiline through A between 6 and 0,
having a second point B in common with the convex arch departing
from A in « (fig. 7). Let 9" be the plane
through 6” La. Let the lines of intersection
of 8’ and a,, «,, a,.... be respectively
BD Dee Frou BD NO Clade
a component sequence :b,, bn, On; having
6' for sole limiting element. Corresponding
sequences are @n;,@ny,On3,---- and D'n,
b" 1) O'ng-- For n, large enough the semi-
lines bn, Ong... (converging towards 6’) are

Fig. 7. respectively situated above 6"n,, bns, Ôns -
(converging towards 6.

In a@,, a branch departs from A between 6,, and 6",,, in an, a
branch departs from A between 6,, and 6", ete. None of these branches
can cross bn, On, bug... (respectively) because these lines as tangents
at the point of inflexion A have no other points in common with
F*. Hence in order that in the limiting semiplane « no branch
departs from A between 6' and 6" it is unavoidable that in the
converging planes the branches cross 6",, 6,", 6",,... at points con-
verging towards A. But all the lines 6",, 6",,... are situated in
one plane 8", hence in this plane 6” is tangent in A. Then however,
it is impossible that A is point of inflexion in 8’, because 6" has,
except A, another point B in common with /'*.

Second case. Possibly the line « has, below A, one or two other
points in common with /*. Let C be the nearest. Again b,,, On,
represents a sequence of semilines having 0! for sole limiting element.
In an, @n,---. Curves depart from A below 5, ete. and to the right
hand side of a. These curves cannot recross 0,, etc. because these lines
have only A in common with F?, hence in the lower angle between
a and 5, ete. these curves connect A either with the line at infinity

311

or with C or with the third point of #* on a, situated beyond C.
Then however, for every q such that AC >q >>0 there must exist
a point P of F* in the limiting plane, such that AP==g and this
point P must be situated either on 6 or a or in the lower angle
between 6 and a. Thus, once more a contradiction is obtained.

Third case. This case is treated in entirely the same way as the
second.

It has been shown that a plane through A exists in which that
point is not a point of inflexion. Let « be this plane. If in « the
point A is not isolated, double point or cusp, the only remaining
possibility is that A is ordinary point in « Let a be the tangent
at A in «. a has, besides A, another point C in common with F'°.
When a revolves round a, the point A continues to count double
and (to count single on a. Assuming that in no plane through a
the point A is isolated, double point or cusp, it follows that in
every plane through a, A is ordinary point with a for tangent.
Now when « revolves round a and if we consider both semiplanes
in which a divides a, it is obvious that at least once a semiplane
in which A is isolated must be limiting element of a sequence of
semiplanes in which convex arches are situated, passing through A
and having a for tangent. Let a be this semiplane and a,’, «,',4,' ..
a converging sequence. In order that A be isolated in the semzplane
a’, it is unavoidable that the curves passing through 4 in a’,, a’, a',...
are ovals contracting towards A (it must be understood that the
converging sequence is started far enough). Let 6 be a line in a’
through A having two other points B and C in common with f°.
Let 8 be the plane through 1a’, and let 6,,6,,6,... be the lines
of intersection of Band a’,, a’,, a’, ... respectively. Every line },,6,...
interseets the oval in the corresponding plane at a second point,
different from A. When the ovals contract, these points converge
towards A. Hence in plane @ the line 6 would be tangent at A,
but this is impossible, because 6 has two other points B and C in
common with /*. This completes the required demonstration.

Second part. Let A be ordinary point in a plane « and a ordinary
line of intersection through A in «.

Theorem 1: If a sequence of lines in R, converges towards
line a, then points of F® on these lines converge towards A. *)

1) This theorem and its demonstration hold also when A is situated on a line
of F3, provided this line does not lie in «.

312 ‘

Let 4B and AC be the branches arriving at A in «a and B’C’ a
linesegment crossing both arches 4B and AC. The Jorpan theorem
for three dimensions shows that a double connection exists between
the branches AB and AC by means of two sets of points I and II,
having no points in common. If I and II were situated on the same
side of « then a parallel linesegment converging from that side
towards 5’C”’ would end up by having at least two points in
common with I and also two with II: an impossibility. Hence I
and II lie on different sides of « and inside any neighbourhood of
A points of #’* exist on both sides of a.

The vicinity of A on F*® is the (1,1) continuous representation
of the vicinity of a point in a plane. From this follows that
inside any finite neighbourhood of A the points of R, which are not
situated on F*, belong to either of two regions G, and G,, which
regions are not connected: within that finite neighbourbood of A. The
common boundary of these regions #* has Jorpan character. Inside
any vicinity of A we found points of F* on both sides of a.
Let A,, A,... be a sequence of these points converging from above
and <A,', A,’... a sequence converging from below towards A.
A, and A, can be joined by a single path, belonging entirely to G,
and by another belonging to G,. In the same way A, and A, etc.
By going far enough in the sequence it follows from the Unbewalltheit
that these paths can be kept inside an arbitrarily small neighbourhood
of A. Each path connects a point above « with a point below a,
hence in « points of both G, and G, exist inside any vicinity
of A. From this follows that in « the region on one side of the
convex arch BAC belongs to G, and that on the other side to G,.

Let D and & be points on line « on different sides of A. D |
belongs to G,, H to G,. Round D a finite sphere 6, exists, all
internal points of which belong to G, and round Z a sphere 6,
containing points of G, only.

Now let the linesequence a,,a,... converge towards a. Let on
these lines the points D,, D,... converge towards Dand ZE, E,...
towards #. PD, D,... end up by being internal to sphere 6, and
then certainly belong to G,. #, E£,... become internal to 6, and
then belong to G,. Hence for n larger than some finite number the
finite linesegment D, £, of a, carries at least one point of F* and
these points can only converge towards A, because this is the only
point of #* on the segment DE of a.

Theorem 2: If the planes a,, a,... converge towards a, then
the section of F* in a consists of the limiting set of the sections vn

313

a, or 18 composed of this limiting set together with an isolated point.

Because /”* is a closed set, the section in « contains the entire
limiting set of the sections in «a.

From theorem 1 follows that an ordinary point in « must be
limiting point of the sections in «,. For a point of inflecion in «
this is proved in a strictly analogous way. That a double point or
cusp in @ is limiting point of the sections in @, can easily be shown,
if we remember the connection existing between the branches
arriving at such a point. This connection has been treated in the
jirst part.

Remains only a possible ‘solated point in a. That such a point
need not be limiting point of the sections in «a, may be seen from
examples of cubic surfaces.

Theorem. 3: The tangent plane changes continuously with the
corresponding point of FE.

Let the points A, A,.... of #*® converge towards A. Ifa,a, ...a
are the corresponding tangent planes we have to show that a, a,..-.
converge towards a, and nothing but a@. Suppose a,a,.... had a

limiting plane 2, different from «, but of course passing through A.
Foregoing results show that A is in 8 either ordinary point or point
of inflexion, hence anyway a line a through A in 8 can be chosen
having three different points A, B, and C in common with F*. Let

a,a,... be a linesequence respectively passing through A, A,....
and situated a,a,.... and converging towards the line a in the
limiting plane 8 (these lines a,,a,.... can be fixed in different

ways by a simple condition). From theorem 1 follows that for n >
some finite number the lines a, carry points B, and C, of F*
converging towards B and C respectively. The point A, however
counts double on any line in the tangent plane @,, hence lines would
be constructed having four points in common with /*: a contra-
diction.

Theorem 4: An elliptical’) point of F* can only be limiting point
of elliptical points. |

Let the points A, A,.... of F* converge towards A. Corresponding
tangent planes a,a,....a. Suppose A, were for every 7 double
point or cusp in «a, Then in every @, a branch would connect A,

1) Points of F3 which are in the tangent planes isolated, double points or cusps
we call respectively elliptical hyperbolical and parabolical points. Except these
F can contain one exceptional point, the character of which has been dealt with
in the first part.

21

Proceedings Royal Acad. Amsterdam. Vol. XX.

314 3

with the line at infinity and in the limiting plane a the point A
could not be isolated, because a sequence of connected sets of points
each having breadth >> some finite value p cannot converge
towards a single point.

Theorem 5: A hyperbolical point of F* can only be limiting point
of hypertolical points.

A, A,.... converge towards A. Corresponding tangent planes
@,«,...a@. A is supposed to be hyperbolical. We shall show that
if A, is assumed to be elliptical or parabolical for every n, contra-
dictory results are obtained.

The points of space inside a sufficiently small but finite vicinity
of A which are not situated on /’* belong to either of two regions
G, and G, which are not connected inside that vicinity of A.

The results obtained when proving theorem 1 show that if we
move round a hyperbolical point in the tangent plane we alternately
pass through G, and G,, to be more exact: twice we pass through
G, and twice through G,. Moving round a parabolical point in the
tangent plane we pass once through G, and once through G,. In
the tangent plane of an elliptical point however, a finite surrounding
of that point belongs entirely to only one of the two regions, for
instance to G,. Hence an arbitrary line through an elliptical point in
the tangent plane departs on both sides in the same region. This
also holds for a parabolical point, provided we exclude the cuspidal
tangent.

From the converging planes a, @,.... we choose a component
sequence «a, a... such that in each of these every line through
An, An... departs on both sides in the same region, for instance

G7, (again cuspidal tangents excluded). In thé limiting plane « we
choose two points B and C of G, diametrically situated with regard
to A. Let 4, and 6, be spheres round B and C, all internal points
of which belong to G,. Let a be the line through C, A and Band
Gn, In, -.- a linesequence respectively passing A,, A,, ...and situated
in Gn, Cu, ---, COnverging towards a and containing no cuspidal
tangents (again a, dn,... can be fixed by a simple condition). In
the end the lines a,,a,,... will pass through the spheres 6, and 5,
but this means that a line through a point which counts double,
departs on both sides of that point in G, and further on on both
sides carries points of G,. Hence on both sides #* must be crossed
again and lines would have been constructed carrying four points

6h Ln

315

Theorem 6: If a hyperbolical point moves continuously, then
the tangents in the tangent plane change continuously also.

Let the hyperbolical points A, A,.... converge towards the hyper-
bolical point A. Corresponding tangent planes ea, a,...a. A sequence
of tangents through A, A,.... in @,@,.... cannot have for limit
a line through A in « which is not tangent, for such a line would
have, besides A, another point in common with the curve in a and
theorem 1 tells us that the converging lines would carry points of
F* having this second point of intersection as limiting point. This,
however, contradicts the assumption that the converging lines are
tangents at double points converging towards 4.

To prove theorem 6 it only remains to show that the tangents at

the double points in @, @,... cannot converge to only one of the
two tangents in «. Let a and 6 be the tangents in @ and let us
assume that the tangents in the converging planes a,a,.... have

only a as limiting element. For increasing 2 the tangents in a, form
a diminishing angle tending towards zero. The part of «„ inside
this decreasing angle converges to the Jine a only and considering a

Fig. 8.

does not belong entirely to #'*, it is unavoidable that for n > some
finite number the part of a» inside the decreasing angle contains
that part of the curve which is of the second order and besides
this loop of the curve has the point A for sole limiting point (this
“loop” can, of course be a projective oval, having one or two
points in common with the line at infinity). |
Besides the loop, two branches depart from A, in the planes «,,
belonging to the part of the curve which is of the third order. Let
a,a,.... contain a component sequence a, «,,-…. … Of planes in which
these branches depart in the direction of the semitangents converging
FN ij

316

towards the semiline AB. This implies no restriction, because with
respect to the curve in « both possibilities indicated in fig. 8 are
considered.

Let c be a line through A in a inside the angle LAB. This line
has, besides A, another point in common with the curve in a. Let
Cn, Cag een «be a sequence of lines through My.) Aye 5 respectively
situated in a, a, ....and converging towards c. The points Pr, Py, .-.
on these lines are supposed to have P for limiting point. Let a, dn, .-.
be a sequence of tangents in a, «@,... and lastly we assume that
the points bn, B,,.... on these lines converge towards B. (see fig. 8).

In a,, a branch departs from A,, between A,, Bn, and An, Pr,
in a,, a branch departs from A,, between A,, B,, and A, P», etc.
These branches cannot cross the tangents A,, B etc. again hence
in order that in « no branch leaves A between AB and AP it is
necessary that the branches in the converging planes cross An, Pn,

in points converging towards An: According to theorem 1, however,

1 > ze} D “OP > , VIN 0

the lines Cn, (of which An, Pn, forms part) end up by carrying

points of #* converging towards R and because A, counts, double
)

lines would exist having four points in common with #*.

The tangents at a double point divide their plane in two parts,
respectively containing the loop of the curve and the part of the
third order.

Theorem 7: If a hyperbolical point moves continuously, hence the
tangent plane with tangents also changes continuously, then the parts
of the tangent planes, containing the pieces of the third order, merge
in each other and it follows that the same holds for the parts con-
taining the loops. Besides the loop cannot switch round 180°.

A, A,.... converge towards A (all hyperbolical). Tangent planes
a, a,.... a. The tangents in a, form four angles round A,, succes-
sively I,, II, Il], and IV,. These converge respectively towards I, II,
Ill and IV in @. Suppose for every n the principal branch (that is
the part of the third order) in a, lies in I, + III. This means
that in @, branches depart from Ato both sides inside these angles,
which are connected via the line at infinity. But then it is unavoidable
that in the limiting plane « a branch departs in I and another in
III, hence this part-of @ again contains the principal branch.

Suppose for every » the loop in a„ departs in II,. In order that
A be isolated in « in the angle II it is necessary that these loops
contract towards A. Hence, in the end they cannot reach the line

317

at infinity and it follows that in « the point A must be isolated
also in the angle [V. It follows that the oval cannot switch round 180°.

Theorem 8: If a sequence of hyperbolical points converges
towards a parabolical point then both sets of tangents of the hiper-
bolical points converge exclusively towards the cuspidal tangent of the
parabolical point. Besides the direction in which the principal branches
(that is the parts of the third order) depart, cannot switch round 180°.

Let the hyperbolical points 4,,A,... converge towards the parabolical
point A. Tangent planes «a; a,... «. A line through A in « which
is not cuspidal tangent has, besides A, an ordinary point B in common
with the curve in a. Theorem 1 shows that such a line can never
be limit of tangents at double points converging towards 4. This
proves the first part of theorem 8. In the same way as when proving
theorem 6 we can show again that the loop in the converging planes
lies inside the diminishing angle of the tangents and converges
towards A.

In the case of theorem 7 it was shown that the loop cannot switch
round 180°, and it followed that the principal branch cannot change
its direction discontinuously either. Here, however, the loop has dis-
appeared in the limiting plane @ and a new proof is required that
the principal branch cannot change its direction in discontinuous
fashion.

Let AC be the cuspidal semztangent in « departing in the same
direction as the cuspidal branches and A45 the other half. Suppose
in the converging planes the principal branch departs from A, in
the direction: of the semitangents converging towards AB, but in « in
the direction AC. Let 6 be a line through A in «a (=l= BC) and
b,,6,... a sequence of lines converging towards 6 respect. situated
in a@,,a,... and passing through A,, A,...

The principal branches depart in @, from A, to both sides inside
the increasing angles of the tangents. These branches do not cross
the tangents again, but each goes in its own angle to the line at
infinity. Besides they depart in directions converging towards AB.
In order that in @ no branches depart from A to that side of 5 on
which AB is situated it is necessary that both branches in «@, (for
n large enough) cross the lines b„ (converging towards 6) on both
sides of A, at points converging towards A. However, on every
line 6, the point A, counts double, hence lines would have been
constructed having four points in common with F'*.

Theorem 9: If A is double point in plane « and if a pom

518

departs from A along the principal branch in a, then at first the
loops in the corresponding tangent planes will cross plane a.

Let us consider a sequence of points on arch CA (fig. 9) converging
towards A. Provided we start close enough to A all these points
are hyperbolical (theorem 5). We assume that the loops do not cross
plane « and we select a component sequence of points such that the
corresponding loops all arrive from the same side of @ for instance
from above. Let this sequence of points be A, A,.... Tangent
planes a, dj vet 2%

Fig. 9.

The tangent to AC in A, intersects arch AB in B. For increasing
n, B, converges towards A. Plane a, contains line A, B. Let A, Dn
be the semitangent in @, which converges towards AD. Let us con-
sider in a, that part of the plane inside angle D, A, B. For
increasing m no principal branch in «a, can continue to depart inside
this angle, because the principal branches depart from A, in direc-
tions converging towards AE and AF. B,, however, belongs to the
principal branch, as we assumed that the loop does not cross a.
This principal branch which crosses A, B, at B,, cannot connect B,
with A, inside angle B, A, D,, but on the sides of this angle 5, is
the only point of #'*, because A, counts double on A, 4, and triple
on A, D,, hence it is unavoidable that inside angle B, A, D, this
branch connects B, with the line at infinity. Angle B, An Du, how-
ever, converges towards the semiline 4D and B, towards A. It
follows that line AD would belong entirely to /°: a contradiction.
This completes the proof of theorem 9.

Remark: When a point departs from A along tbe principal branch
in a the tangent can be divided in a front half and a back half.
The preceding proof shows that at first the loop departs inside that
angle of the tangents, in which the back half of the tangent in «
is situated. We shall indicate this briefly by saying that at first the
loop trails behind.

319

Theorem 10: Round a parabolical point a finite surrounding exists
m the tangent plane containing hyperbolical points only (except of
course the original parabolical point itself).

Let A be parabolical point, « tangent plane and a cuspidal tangent.

The points A,, A,... on arch BA converge towards A. Corresponding
tangent planes @,,@,.... The lines of intersection of «and «a,
are respectively a,,a,.... These lines are tangents in « converging
towards a. The points of intersection of a,,a,... and arch AC we
indicate by C,, C,.... These points converge towards A. (Fig. 10).

Let 6,,0,.... be lines respectively situated in a,,a,... passing

Fig. 10.
through A,,A,... and perpendicular to a,,a,.... These lines
b,,6,... converge towards 6 La in a. Now let us assume all points
A,,A,... were elliptical or parabolical. In no case the lines 5, can

continue to be euspidal tangents at 4, in @ for even if A, con-
tinued parabolical the cuspidal tangents would converge towards a
(this follows again from theorem 1). Hence for n larger than some
finite number the line 6, has, except A,, another point in common
with #* which counts single.

b, divides the corresponding plane @, in two semiplanes. We
consider the one that does not contain C,. These semiplanes converge
towards the top one of those in which 6 divides «a (fig, 10). If A,
is parabolieal, then for » large enough the cuspidal branches depart in
the semiplane of «, which does „ot contain C',, because the branches
departing from A, cannot switch round 180° in the limiting case
(this is shown in a way analogous to that used for theorem 8 where
hyperbolical points converged towards a parabolical point).

Hence for n large enough no branch departs from A, in the
semiplane of a, containing C,. But C, lies on the curve in a,
and 6, carries besides A,, only one singly counting point of the curve.
But A, is elliptical or parabolical, hence the curve in @, (with possible
exception of A) is connected. From this follows that C, must be

320

connected with the line at infinity in the corresponding semiplane of @,.

Passing on to the limit the semiplane of @, containing C, con-
verges towards the top semiplane of «a (fig. 10) and C, converges
towards A. But then it is unavoidable that a branch departs from
A in the top semiplane of « (including line 6). Hence a contradic-
tion is obtained and it has been shown that the points 4,, A,...
end up by being hyperbolical.

Theorem 11: If a surface F* contains no straight line, it
cannot exist.

Juni has shown?) that a non-degenerated elementary curve of the
third order contains one and only one point of inflexion if that curve
has a double point or cusp and that otherwise the curve has three
and only three points of inflexion. We consider an arbitrary plane
section of F'*. If this section contains no straight line it has at
least one point of inflexion. This point can be hyperbolical or para-
bolical. In the latter case we can, according to theorem 10, find
hyperbolical points in the corresponding tangent plane. Hence in any
case a hyperbolical point of F* can be found. Let this be A with a
for tangent plane. We do not consider the loop of the curve in a,
but only the principal branch. This branch has, according to JUBL,
one and only one point of inflexion B. We consider two points
A'and A", departing from A in opposite directions along the principal
branch and meeting again at the point of inflexion B, after moving
continuously along the curve. Theorem 9 shows that the loops of
A' and A", at first cross « and besides we found that at first they
trail behind. Now in this state of affairs no change is possible before
A’ and A" meet again at B: The tangents at a double point
change continuously (theorem 6), hence only at a point of inflexion
can they pass through «. Besides the loop cannot switch round
180° or change into a principal branch (theorem 7). Lastly it is
impossible that on the way from A to B the point A’ (or A")
changes its character, for this can only happen via a first parabolical
point. Then, however, the angle between the tangents, inside which
the loop is situated, would tend towards zero, and considering the
loop crosses « all the time, the limiting position of the tangents
would be situated in « and this would mean a cusp or point of
inflexion in a, because the above mentioned limiting line coincides
with the cuspidal tangent (theorem 8). Hence A’ and A" arrive at
the point of inflexion B both hyperbolical with the loops crossing
a and trailing behind.

1) Proc, Danish Acad. loc. cit. § 5.

321

Theorem 4 shows that B cannot be elliptical, hence B is
parabolical or hyperbolical. in the former case the points A' and A”
would (theorem 8) prescribe opposite directions for the cuspidal
branches departing from B: a contradiction.

Remains the case that B is hyperbolical. Let a be the tangent at
B in a. Now A' and A” prescribe opposite directions for that branch
of the loop in the tangent plane of B, which has a for tangent,
(the loops trail behind and cannot switch round discontinuously).
Hence once more a contradiction is obtained.

CORRIGENDUM.

In the first communication on this subject page 103 line 1 and 2,
for: “twodimensional continuum” read “closed twodimensional
continuum”’. .

Chemistry. — “The boiling point line of the system: hexane-nitro-
benzene.” By Dr. B. H. Böcaner. (Communicated by Prof.
HoLrEMAN).

(Communicated in the meeting of June 30, 1917).

On studying the experimental data available, it occurred to me
that one may easily predict, whether a system of two partially
miscible liquids will show a maximum in the vapour pressure curve
or not. The following rule, indeed, may be enunciated: when
the difference of the boiling points of the two substances is less
than 100°, a maximum pressure is found; when the boiling points
differ more than 100°, no maximum occurs in the p, v-eurve. Of
course, this limit is not perfectly sharp; yet, the deviations are
remarkably few. 1 have been able to find only three systems '), which
may really” be considered exceptions to the rule, as one observes
a maximum, although the boiling points differ from 110° to 120°.

This made me sufticiently trust in the rule to predict with its
help the behaviour of systems not yet investigated, and to expect,
for instance, in the system n-hexane-nitrobenzene (boiling points resp.
69° and 210°) a p, #-curve, continually descending from the hexane
side. This is particularly interesting, because a research by KOHNsTAMM
and TimmeRMANs *), in connexion with a rule given by van Der W aars,
would lead to the conclusion that the p, e-curve must have a
maximum. ,

These authors, indeed, reckon the systems, which consist of nitro-
benzene and a hydrocarbon, among the group: ‘splitting up of the
plait”, whereas van per, Waars has shown that splitting up of a
plait is only possible, if a minimum critical temperature occurs in
the system (which is equivalent to a maximum in the vapour
pressure curve). KonnstaMM and T1MMERMANs, it is true, expressed
themselves with some reserve. While they were able to observe
experimentally the minimum temperature in the plaitpoint curve of
the system decane-nitrobenzene, they were prevented in doing the
same for hexane-nitrobenzene by the appearance of solid nitrobenzene.

1) Cf. BAKHurs RoozeBooM, Heterogene Gleichgewichte, zweites Heft, Il (in the
press).
*) These Proceedings, 15, November 1912.

323

dt
They only ascertained that the value of — is negative, continually
ap ;

decreasing, however, in absolute amount: it falls namely from — 0°,0164
to — 0°%0031. In analogy with decane-nitrobenzene and petroleum-
nitrobenzene *) they concluded the system hexane-nitrobenzene also
to belong to the type: splitting up of the plait. Since, however,
according to vAN DER WAALS, this goes necessarily together with the
existence of a maximum pressure, we get obviously in defiance of
the rule enunciated above, which requires no maximum. One of the
three statements, therefore, must be wrong ; I thought it, accordingly,
of some interest to investigate experimentally, if a maximum exists
or not.

Instead of the vapour pressure curve one may as well determine
the boiling point line, which is more convenient. The question then
becomes whether the boiling line exhibits a minimum or not.

The determinations were carried out in Professor Smits’ apparatus,
‘formerly described *); the substances used were carefully dried and
fractionated. The results of the measurements are joined in the table
below, and represented moreover in the figure. .

As appears most clearly from the figure, there is no question of a
minimum; the curve shows, on the contrary, the shape, often found in
systems with limited miscibility above the critical solution point *).
A comparatively quick rise of short duration is followed by a very
slow increase of the boiling temperatures over an extensive area ‘) ;
an inflexion point occurs and finally the curve rises very steeply
to the boiling point of pure nitrobenzene.

This result fully agrees with my expectation; the question now
arises how KoHNstamM and TrMMERMANS’ determinations may be
made to tally with it. It seems to me most likely that the system
belongs to the type “retrogression”, as the authors mentioned
previously believed themselves also. We have, then, only to remark
the particularity that the plaitpoint curve does not ascend regularly
from the critical endpoint, but rises in the beginning slowly, after-
wards more steeply. Thereagainst no objection, indeed, can be
raised; although in the single system that is fully investigated
(methylethyleetone-water), a uniform rise is observed, the known

1) For this system too, one gets in conflict with my rule; but it must not
be forgotten, that petroleum itself is already a mixture, and that we have, thus,
no binary system.

2) These Proceedings, April 1917.

3) In our system 19.2°: Timmermans, Zeitschr. phys. Ghem. 58, 186, 1908.

4) Nearly coinciding with the region of limited miscibility.

324 5

Boiling points hexane-nitrobenzene at a pressure
of 76 cm.

nenten

Composition in | zi B
: | Boiling point
mol.proc. nitrobenzene |

0 | 69.0
1.6 | 69.7
3.5 70.0
5.7 10.7
7.5 its
10.0 | ALT
14.6 | 72.3
18.5 | 72.8
24.0 73.3
24.8 13.55
34.4 14.1
41.7 14.9
47.6 5.3
52.7 15.1
53.0 15.9
51.6 16.3
62.5 | 19.3
10.2 85.1
83.5 114

85.4 | 121.5
88.5 | 134.5

100 | 210

data regarding the other systems are so scanty as not to allaw of
any positive assertion on the course of the curves. The shape, which,
in our system, I consider probable, may even happen to be the
most usual.

A further conclusion from this investigation is that systems, one
component of which is a member of a homologue series, need not
belong to tbe same type of demixtion phenomena. If we pass from
decane to hexane, the limited miscibility remains, it being connected
above all things with the chemical nature of the components, but

325

165

5 10 15 20 25 3035 40 45 5055 60 65 70 75 80 8590 95
Hexane Molproc. Nitrobenzene

the type of the vapour pressure (resp. boiling) curves does change;
with regard hereto the increasing difference in volatility of the
two substances is of dominating importance.

It need not be argued that a fortort no maximum pressure
occurs in systems with the lower members of the series, e.g. pentane
and butane‘); these too will belong to the type retrogression. This
is the more remarkable, because now the type: splitting up of-the

dt : :

plait with a negative value of oe which was already considered
ee

rare on theoretical grounds, loses the greater part of its few

representatives. 7) Kounstamm’s earlier opinion that this type would

hardly ever occur seems therefore to be right, in spite of his later

experiments which made bim withdraw his statement.

Inorganic Chemical Laboratory
University of Amsterdam.

1) As has been found, indeed, for isopentane by KonowaLow.
2) Cf. e.g. Nieuwkame’s review, dissertation Amsterdam 1915.

Physiology. — “Photography of the fundus of the human eye’. By

Prof. I. K. A. WERTHEIM SALOMONSON.

(Communicated in the meeting of April 27, 1917).

Since the discovery of the ophthalmoscope by HeLmnorrz, dis-
closing the interior of the living human eye, many different attempts
have been made to keep a permanent record of the aspect of the
retina on a photographic plate. This proved to be much more difficult
than viewing the background of the eye. The greatest difficulty was
caused by the reflexes given off on the surface of the cornea and
the anterior and posterior surface of the lens. Different ways- have
been tried to get rid of these reflexes and after more or less suc-
cessful attempts by BaGnéris, GUILL0z, GERLOFF and others, Dimmer
succeeded in obtaining satisfactory results. Shortly afterwards THORNER
and also Worrr, working on different lines, showed photographs of
the living human retina which were nearly as good as those of
Dimmer. His photographs are generally excellent.

Of the eyes of animals NicorarEw was also successfulin obtaining

- good negatives. But his method did not yield satisfactory results
with the human eye, the fundus of which is infinitely more difficult
to photograph than the animal fundus.

For practical purposes as yet only Dimmer’s and perhaps Worrr’s
method have to be considered. But Drvmer’s method necessitates a
costly instrumentarium, requiring much room and skilled assistance.
I do not know of its being used outside his own clinic, except by
a very few specialists (e. g. Hess.).

The different methods for obtaining a reflexless image of the
fundus have been ably discussed by GuLLsTRAND, who gave a clear
„and critical review of the general and special conditions necessary
for getting clearly defined ophthalmoscopic images, free from any
reflex. Finally his resalts were embodied in his large demonstration-
ophthalmoscope, constructed by Zeiss, which shows the ophthalmos-
copic appearance of the human eye with less diffienlty, more exten-
sively with a higher magnification and yet more clearly than any
other instrument of the same kind. As yet this instrument cannot
be used for photographic purposes. But it seemed to me that it

I. K. A. WERTHEIM SALOMONSON: “PHOTOGRAPHY OF THE HUMAN FUNDUS OCULI".

Fig. 1.

Normal fundus. Normal fundus.

Fig. 3. Fig. 4.

Normal fundus with choreoidal vessels. Secundary atrophy of the optic nerve.

Proceedings Royal Acad. Amsterdam, Vol. XX. DELTA VL EER OMS seas

327

-

might possibly be rendered suitable for such. After a few prelimi-
nary trials I had an instrument made for me, differing in many
respects from the original one.

The Nernstlamp was discarded and was replaced by a lamp of
greater intrinsic brilliancy. The arrangement of the illumination-tube
was slightly changed so as to allow a relatively greater part of the
light reaching the eye.

In the Zerss instrument the image of the Nernst filament is pro-
jected upon a slit and by means of a second condensor into the
pupil of the eye. The light, after leaving the second condensor is
deflected by a glass plate, making an angle of 45° with the axis of
the tube. The optical system for viewing the fundus looks through
this glass plate. With this construction about 8.5°/, of the light
leaving the second condensor is projected into the eye and 91.5 °/,
of the light leaving the eye reaches the objective of the viewing-
tube. L placed the glass-plate so as to make an angle of 65° with
both tubes, which allowed about 21 °/, of the light to enter the
eye and 79°/, to reach the observing eye. In this way the amount
of light falling on the photographic plate was about doubled.

The Nernstlamp has an intrinsic luminosity which I measured as
of 3.1 Hefnercandles per square millimeter. By the use of a special-
ly constructed halfwattlamp of low voltage I got an intrinsic bril-
liancy of nearly 29 units per square millimeter. A suitable small
camera having been adapted to the instrument I got, after a few
failures, usable negatives of a diameter of 26 to 30 millimeters,
showing about 27 degrees of the fundus and covering an area of
4'/, times the diameter of a normal optic disc.

The negatives were sometimes good, though very often blurred,
owing to the long exposure of 0.4 to 0.5 of a second. I have tried
to get better results with a small arclamp of about 5 amperes but
without much success. Though the intrinsic intensity was 3 times
greater, the area was notably smaller. With an entirely modified
construction and with an arclamp of 25—30 amperes better results
might be expected as the exposure might have been reduced to
1/, of a second. As the angle of view was also rather small and
could only be enlarged by a complete reconstruction of the appa-
ratus, I have kept the instrument as it was, and have tried to get
more satisfactory negatives in quite another way.

With the indirect ophthalmoscopy. we can entirely eliminate the
reflexes on the cornea and the lens by following GULLSTRAND's method.
But we always retain two reflexes on the ophthalmoscope lens.
These do not hinder visual observation as they are rather small and

328

by slight movements of the ophthalmoscope lens can always be
removed from any part of the real image of the fundus.

But their presence is an absolute hindrance to photography. They
cause the appearance on the negative of one or two large spots,
covering its central part and having a diameter of nearly */, of the
whole negative. The brightness of these reflexes is several hundred
times larger than that of the image of the fundus. I have tried in
many different ways to eliminate these reflexes and have found that
they could be reduced so as to be almost invisible by means of two
small screens.

In accordance with this principle I constructed a new photographic
ophthalmoscope. The diameter of the negative is 40 mm. The retina
is photographed with a magnification of 4.7 times, over an angle
of 33 degrees, giving an image with a diameter of about 5'/, times
as large as the normal optic disc. The small arclamp of 4 to 5
Amperes with which the instrument is fitted allows of exposures of
'/,, of a second, though this may be redneed to */,, of a second in
some cases. However as the reflex time for the orbicular muscle
reflex is much longer there is no advantage in further reduction of
the time of exposure. The exposures are short enough to give sharp
negatives even in a case of nystagmus.

The quality of the negatives is generally sufficient. They are
sharply defined. Generally the middle part is more strongly impressed
than the marginal parts, as was to be expected. Yet direct enlarge-
ments or prints can nearly always be made without any retouching.

The whole apparatus, which will be fully described elsewhere can
be used for both eyes without any alteration except the ordinary
focussing. The dimensions are only slightly larger than those of the
GULLSTRAND-ZeIss demonstration-instrument. Its use is not much more
difficult than the making of an ordinary photograph with a studio-
camera.

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS

VOLUME XX
N°5:

President: Prof. H. A. LORENTZ.
Secretary: Prof. P. ZEEMAN.

(Translated from: “Verslag van de gewone vergaderingen der Wis- en
Natuurkundige Afdeeling,” Vol. XXV and XXVI).

CONTENTS.

J. C. KLUYVER: “On hyperelliptic integrals of deficiency p= 2, reducible by a transformation of
order r=4”, p. 330.

J. P. TREUB: “On the Saponification of Fats”. II. (Communicated by Prof. S. HOOGEWERFF), p. 343.

ANNIE VAN VLEUTEN: “Do the forces causing the diurnal variation of terrestrial magnetism possess
a potential?” (Communicated by Dr. J. P. VAN DER STOK), p. 358.

ANNIE VAN VLEUTEN: “On the question whether the internal magnetic field, to which the diurnal
variation in terrestrial magnetism is partly ascribed, depends on induced currents”. (Communi-
cated by Dr. J. P. VAN DER STOK), p. 361.

J. E. VERSCHAFFELT: “The equation of state of an associating substance”. (Communicated by Prof.
H. KAMERLINGH ONNES), p. 365.

H. I. WATERMAN: “Influence of different compounds on the destruction of monosaccharids by sodium-
hydroxide and on the inversion of sucrose by hydrochloric acid.” IL. (Communicated by Prof.
J. BOESEKEN), p. 382.

A. SMITS, G. MEYER, and R. TH. BECK: “On Black Phosphorus”. I]. (Communicated by Prof. S.
HOOGEWERFF), 392. .

A. SMITS and C. A. LOBRY DE BRUYN: “On the Electro-Chemical Behaviour of Nickel”. (Communi-
nicated by Prof. S. HOOGEWERFF), p. 394.

S. DE BOER: “On the transmission of stimula through the ventricle of frogs’ hearts”. (Communi-
cated by Prof. G. VAN RIJNBERK), p. 404.

JAN DE VRIES: “Surfaces that may be represented in a plane by a linear congruence of rays”, p. 419.

J. BÖESEKEN and H. W. HOFSTEDE: “Observations about hydration under the influence of Colloidal
Catalysers and how to account for this process”, p. 424. (With one plate).

G. P. FRETS: “On Mendelian Segregation with the Heredity of Headform in Man”. (Communicated
by Prof. C. WINKLER), p. 435.

H. B A. BOCKWINKEL: “Some Considerations on Complete Transmutation”. (Third communication).
(Communicated by Prof. L. E. J. BROUWER), p. 449.

C. J. C. VAN HOOGENHUYZE: “Researches relating to the Etiology of Febris Exanthematicus.” (Com-
municated by Prof. C. EYKMAN), p. 465. (With one plate).

F. ROELS: “Intercomparison of some results obtained in the Investigation of Memory by the Natural
and the Experimental Learning Method”. (Communicated by Prof. C. WINKLER), p. 477.

J. J. VAN LAAR: “On the Fundamental Values of the Quantities 6 and p/a for Different Elements,
in Connection with the Periodic System. V. The Elements of the Carbon and Titanium Groups”,
(Communicated by Prof. H. A. LORENTZ), p. 492.

J. J. VAN LAAR: “Idem VI. The Alkali Metals” (Communicated by Prof. H. A. LORENTZ), p. 505.

A. SMITS and J. GILLIS: “On Milk-Sugat”. I. (Communicated by Prof. S. HOOGEWERFF), p. 520.

22

Proceedings Royal Acad. Amsterdam. Vol. XX.

Mathematics. — “On hyperelliptic integrals of deficiency p= 2,
reducible by a transformation of order r= 4.” By Prof. J.C.
KLUYVER.

(Communicated in the meeting of Sept. 29, 1917).

The conditions that an hyperelliptic integral of deticiency p= 2
is reducible to an elliptic integral by a transformation of order
y—4, have been assigned by Borza'), who used direct algebraic
methods and also by leer ?), who based his deductions on the trans-
formation of the double thetas. | will show that the geometry of a
linear system of conics affords the means to solve the problem, and
that geometric considerations enable us to add some results to those
previously obtained *).

Let the integral be of the form

Xda
Vwd,
where X is a linear quantie and w,,..,,w, are binary quadrics of
the variable w=—=,:,, then the integral is reducible under the
following three conditions:

1. There are three quadries &,,&,,,, each of which is a perfect
square, such that the quartics 88,8, W, are linearly connected.
Otherwise stated, these quarties are elements of an involution J
of order 4.

2. The involution J contains an element 7’, a quartic being a
perfect square.

83. The numerator X of the integral bas a determinate form
depending on ‚tw,

In fact, supposing the first and the second of these conditions to
be fultilled, we can take in J any two elements whatever A, and
k,, and k,,k,,k,,2 being certain constants, we have
EU = R,—k,R, Erb,=R B, EhB kB, TRA
then putting

1) Math. Ann., Bd. 28, p. 447.

2) Monatshefte für Math. u. Phys., Il, p. 157.

3) For a summary of the researches on reducible Abelian integrals see: W. C.
Post, Dissertation, Leyden, 1917.

331
we get

1
dt = (RR — R,'R,) de ,
Ry

V (t—k,)(t—k,)(t—h,)(¢—A) — se VEEE www,

and hence
dt oe dz (R,R,'—R,'R,)
VEN kt) Vy, TVEEE
Now the sextic (&,R,'—R,'R,) is evidently divisible by the quintic
TV =. EE, therefore the given integral is reducible, if we take
| _ (BEER)
TREE
Thus it is seen that the reducibility of the integral in the first
instance depends upon the existence of the involution J, and on the
possibility of determining the quadrics £,,&,,£,. The investigation of
the involution / and of its characteristic properties may be conducted

as follows.
With three binary quadrics

W,=4,07 + 2a,e-+a, , Ya, et + 2a,'x+a,’ , p,—a,"2*+ 2a,"e+a,"
six common invariants are associated. As such we get in the first
place the discriminants

A,,=2 (a,a,—4,”) ’ A,,=2 (aa, Sar ie, ’ A,,—=2 (a,'a,'—a,'”)
and further the harmonic invariants

: LEGE peat ie et " "

A,,—=(2, a, +a, a, — 2a, a, ) ’ Are (al a, ad, —2a,"a,),

A,,=(a,a,'++a,'a,—2a,a,').

a

The three quadrics themselves are connected by the identical relation

At 2 An w,
ats Ais ‘Al w, 0
A,, A, As w, is

a AS ED 0
which [| write in the form

K=ayrp,’ + by,’ + ep,’ + fd, + gb, + Www, =0.
Now this relation between binary quadries can also be conceived
as the equation in trilinear coordinates w,,y,,w, of a conic K,
and since each of the coordinates is a quadric in w, this variable
procures a parametric representation of the curve. From the same
point of view any homogeneous polynomial F'(w,w,w,) on the
22%

532

one side is a binary quantic in 2, on the other side it represents a
curve in the plane of the conic XK. An arbitrary quadric, for instance,
can always be thrown into ‘the form hw, +, py, +h, w,, and
therefore it represents a right line meeting the conic K in two
points, at which the parameter w is equal to either of the roots of
the quadrie A, w, +h, w, +,,. In particular, since the quadrics
Sher tet ardine iedee squares, the right lines §,, &,,5,, are tangents
of K with the points of contact, say, A,, A,, A,. In this way each
element of the involution / corresponds with a conic and the in-
volution J itself defines a linear system S of conics, such that the
system is determined by three of its elements. Evidently the system
S thus constructed must contain: the conic K, the three pairs of
right lines §,w,, §w,, &,y, and lastly the double line 7. Since S
contains a double line, it is not a wholly general system. Its Jacobian
breaks up into the right line 7’ and into a conic H. The Jacobian
passes through every point of contact of two conics belonging to &,
hence the conic H passes through the points A,, A,,A,, and meets
K in a fourth point A,. The tangents to A at the points A,, A,, A,,
i.e. the right lines &,,&,,§,, and also the tangent §, at the point A,
intersect H again respectively in the points 5,, B, B,, B, The latter
points, lying on the Jacobian, are the centres of degenerate conics
EWE, EW, and of a fourth degenerate conic §,,, and thus we
have proved that the involution / besides the three elements §,1p,,
EE, each having a double root, necessarily must contain a
fourth element &,w, that has the same peculiarity.

In a certain sense the tangent £, considered as a binary quadric
is directly connected with the reducible integral. Let us seek for
the points in which the conic K is touched by any conic of the
system S. If A, and A, are two arbitrary elements of S, the
equation of the system is

R,+ aR, +uk=0.

In order to find the values of the parameter z at the points of
contact, we must again conceive R, and R, as binary quartics and
the required parametric values are the roots of the sextic

(RL Te R,'R,) = 0.

Now, as was said before, the points of contact in question are
the points in which K meets the Jacobian. Hence the roots of the
sextic are the parametric values of « at the points A,, A, A, A,
and at the points of intersection of K with the right line 7. There-
fore the sextic is the product of the five quantics VEVE VEE,

333 .

and 7, and the numerator Y of the integral, which we have found
to be

(Rk, R,'— RR.)

is identical with the linear quantic V &,.

Obviously, we may now conclude that as soon as the given
integral is reducible, there are three other integrals of deficiency
i 2 anne with the involution //, i.e. the a seat

gean VES
Tant Bee nh rs ,
which can be reduced to elliptic integrals. Moreover, it will be evident

that the transformation of the four integrals will be effected by one
and the same transformation formula, and we may notice that

likewise the integral
F
= dx
Vwb,

of deficiency p — 3 becomes elliptic by that transformation.

In order to find how the involution / can be constructed from the
given quadrics y,,y,,~,, 1 will proceed with the analytical investiga-
tion of the system S. It is always possible by adjoining suitable
constant factors to the quadrics §,, §,, &, to ensure the identical
relation

Vee ME, Vel,
and hence the relation
5 +8 +8 — 26,6, —- 28,6, — 25,2 — 0,
an identity in the variable «w that denotes at the same time the
conic XK in the trilinear coordinates §,,§,, S,-

The point A, on AK, the coordinates of which in the system
&,,&Ȥ, are (0,1,1), has its conjugate with respect to the system S
at the point A’, where the tangent 5, of A meets the double line
T of the system.

Supposing 7’ to have the equation

| P= Te. A ME + NEO 3 een
this point A’, has the coordinates (0, —N, M), hence the coefficients
of the equation
15? + BS? + C5,° + 2FE,E, + 265,8, + 25,5, = U,
representing any conic of S, underly the condition
BN— FM + F(N—M)=0,

or

ie

MEN) MEN:
In like manner the points A, and A',, A, and A’, are conjugate
points of S, therefore we have also

ag ofa ony aby pail
TR Wet
M L
HS À

Ek pee 2g

L—M L—M

and the equation of the system itself may be written in the form
2M 2N

6 2N
as — set eh Bl ee Ent |+

2L : 2M
ee | yer ot aoe od

Since S contains the three degenerate conics £,y,, SW’, &,w,, it
is seen that the expressions between brackets in the above equation
denote the right lines hg es sy we may write

_aN
2L 2N
Er EA eat en eee
Paes 2L Spree: Nei Me
sw, =a | NLS ms _N Ala s |

where P,, P,, P, are determinate constants.

From these equations we deduce, always using the coordinates
5,,§,,§,, the coordinates of the points B,, B,, B,, the centres of the
conics AUS 52s, So,

Putting

L(M—N)=q, , M(N—L)=q, , N(L-M)=yq ..- (3) 5
so that the constants q,,q,, q, are related by the equation

„ntt,
we find for the coordinates of B, B,, B, respectively (O, q,, q;),
(q,, 0, 91), (Ya 91, 9). Incidentally we may remark that at the points
B,, B, B, the right lines &,,§,.§, are touched by the conic
K, = QS ar Q.'Ss ae Qs Ss an 2924s S253 7 29,9. 535, oP 29,9; 5:53 = 0,
and at the same time we conclude that the equation of the conic H,
that forms part of the Jacobian, must be
4 QS." a Hen al 935s an Qi53°s a 15: 5, 25 93515. = = 0.

For plainly this conic oie through the points A,, A,, A, and
also through B,, B,, B,.

The equations of the tangent &, and of the right line w, remain

335

to be found, and to this end I consider the pencil of quartic curves
MH? HEE EE = 0.

These curves have the right lines &,8,, 5,8, as bitangents and
"the eight points of contact are obviously the points A,, A,, A,, A,,
B,, B, B,, B. Now the product KK, is a quartic curve that passes
through 14 of the 16 fixed points that are common to the curves
of the pencil. For K touches the bitangents at A,, A,, Ay, A, and
A, has contact with §&,,&,,§, at the points B, B,, B,. Hence the’
curve AK, belongs to the pencil, A, touches &, at the point B,,
and there is a certain value 2, of 2 such that

4,1? ale En 5, E, Ly = lt KK,
It is readily seen that 4 — 1, that we must have u = 1, and from
5,556, KK, — 4°
we find

by = (9:—9;) (9, — 42) §, aa (9:—9:) (9. —s) En a (92—9s) (ge q,) 5, = 0, (4)
Putting
w= 1,5, Hu, Hu En
we write down that A,(0,1,1) and 4,'(O,—N,,M) are conjugate
points with respect to the conic &,w,. Thus we find the relation
u, Ms

EMH (MN + NL + LM) ZEN — (WN + NE + LM)

and a similar relation is obtained by means of the conjugate points
A, and A,„. In this way it appears that the right line w, will be
denoted by
wy, = — 3T + (f+ 545) (MN+NL+LM. . . (5)
; Ei MORSEN
In the preceding we took for granted that it was possible to
represent one and the same conic KX in two different systems of
coordinates by the two equations
K= ay,” + by,’ + ep, + WW, + 29y,p, + bw, = 9, . (6)
Ha be Et LE = 28.8 — Ob co ee ee Oye aan
the w-coordinates depending on the §-coordinates as is indicated by
the equations (2), and we have now to examine if these two equa-
tions are really consistent.
Here it is noteworthy that, after introducing certain constants
Fi Jo ho, the lefthand side of equation (6) becomes
w, fap, + (aA), + G49) + VLA +4), + op, + (AA) +

= Vlg), ri (f+ AY, JE cys],
and that the lefthand side of equation (7) can be written

336

Da (Pis, a PSW, A P&W).
The equations (6) and (7) therefore have the same meaning, as
soon as we have
PE, is Ps De
ay, kh) HIHI) (+h, + op, HA),
Pig;
= — (8)
(9—9.)¥i HAHA) + es
and it is only when these relations hold that either of the equations
(6) and (7) is a direct consequence, of the other.
To simplify somewhat the notation I put
M+N NIL LM
EET EI BENT idg (9)
the new constants a, 8, y being related by the equation
(Lt a)(l + BL Hp) HAB n= 0,

bape voi ee 0E ee . (10)
and consequently instead of (2) we may write
Py, = tn) + (14+ 8s,
Pw, =U + ys, —s. + (Ll — a) 5,
Py, =0-98+04+08—8. |
Comparing now the two sets of equations (8) and (11), we observe
that (11) defines a homographie transformation expressing the quan-
tities Py in the quantities §, and that the inverse transformation is
given by (8).
Writing down the determinant
=a, by A ae
l+y¥ — 1] l1—a

md eget le al —]

of the first. transformation and also the determinant
a Ath, gg, |

Pi BP, BP,

heh, b AHA

CPPI

Km Ene
P, PP, Pili Pit
of the inverse transformation, in which [ have interchanged
lines and columns, a known proposition says that the elements
of the latter determinant are proportional to the corresponding minors
of the former. Nine ratios therefore are equal to one and the same

or by

(11)

337
quantity 4, and so we have the equations

Fee Cn i al Ee ree ae Eb ee
PoP Pr P,P@+a—B—y + By) PP(aatBtrten)

from which we infer

SV be f+V bo
PP EP ER

or
Vo
beef sant
Similarly we obtain
gw
te +-ya = ova ’

and then by equation (10)
ft+Vbe gt+Vca h4+Vab
f- V be g— Vea h—Vab
Thus it appears that the reducibility of the given hyperelliptic
integral implies the relation (12) between the invariants common to
Ws Wo, Wo, and conversely as soon as these invariants, with a suitable
determination of the surds, satisfy the relation (12) the involution J
can be realised, and the given integral will degenerate.
Supposing that the condition (12) is fulfilled, we have

A= (12)

En By aad) b APEN 2e i Nt! WAG
TT Pte PB Psy P,P,2+6y) P,P,(24 Eye)
h fi I: hy

» (13)

— P,P,2+ap) ~ P,P(a-B-y) PP a-F Bey) BEATS —a-B+Y7)
and
áPV he = DBP Vie Sy PV ab en oe ap
From these equations the constants a, ?, 7, PP, PL, M, N55: Gur Vi
can be successively evaluated, we can find the quantics §,, &, §,,5,, Tae
and finally the transformation that reduces the integral.
To illustrate the method described, | will consider a numerical

example. Let the given integral be
X da

il, VOE — 120 + DO — 2e + 2) (Ta? — be +2)
that is, let us assume
Yy, = 5a? — 12e + 4, Wy, 5a? — 2a +2; Ww, = Ta? — 6a + 2.
Calculating the invariants, we have

An An Au Aus An An
—16 9 5 9 1 eee
AA 2 PGE,
AO Te en eevee ee
Now we may take
‚VE Va Va
—15 —10 6
and with this determination of the surds we get
AOR Gag hl a eg CI
f—V be g—V ca h—V ab

The sum of the three fractions is equal to 2, therefore the integral —
is reducible. At the same time we have found

py = 15, ya = — 10, ap = =10)
or
a 4, fi = 19, oe
so that we have either
(ie p= — 3, y= —5,
or
a= — 2, B= 3, ae

Two sets of values for the constants a, 8, y being admissible, we
infer that the given quadrics w,, w,, w, allow us to us build up two
entirely distinct involutions ./, and instead of a single reducible
integral of the given form, two such integrals are possible. This is
obviously in accordance with the known proposition that, as soon
as an integral belonging to an algebraic function of deficiency p — 2
is reducible, that function possesses a second degenerate integral.

I will take up the case

==, B= — 3, y= —5.
Then we have from (14) and from (13)
P, eN de er
ae
Ee nA
4 —10 0

hence from (8)

S, IRT ie Ge

or

939

Now equations (9) and (3) give
EM RM der ged ae SN
and then by equations (1), (4) and (5) we find
T=const.(52?—62+4), §,==const.(7* —2)’, Y,—=const.(350? — 64a — 16))
The elements £,y,, &,, SW EW, 7? of the involution J being
known, we may put for instance
Sys. x? (5a7— 12a -+ 4)
EW, (m1) (5472242) ©
and obtain consequently
t oy I! 1-¢t
POP 12244) (el (5a*—Ba +2) (Ta*—6x 42)
ay 375 t—-32 8—s3t
= (Tz—2)* (8507 —64a—16) (5a*— 62-4 4)*
The above transformation now will reduce four integrals of
deficiency p= 2, connected with the involution J, and we may
write down at once

—

lS (77 — 2) da sf og
Vor ns Gr kle Se ube Vill-D(8—35)

da dt
Ir nbr
V (ba?-12e+4)(52?-2042 (- 350°+64a+16) Vt(32-375t)(8- 30)

(a—1) da
V (5a? — 12e +4) (72? — 6" +2) Ea
= — va | 4 p
Vt (1—t) (82 —375t) (8 —3t)
ada

Vibe 20+ — Qe +2) (72? Ba 350° 1 640416)

on << ae (8—3t)
the constants me etc. at the right hand sides being easily found

by observing, that for small values of 2 we must have t= 2’.
As I have remarked before, the same transformation will also
reduce an integral of deficiency p = 3, connected with the involution -/.
In fact, we have

(Be —Or 44) da
en 2 12e 44) (be 2 + 2) (72° —Ôz +2) (35a? 1640716)
dt

a Vi(l- =f) (32-875 6) 5D

340

Again, if we had used the second set of admissible values for u, 8, y’
a=z— 2, == 3, =d
we should have found successively
Ee arp t lige st i AS age an 1

eae Pr LE LE TLD or ota eel
&, a Es En
4y,+8y,—8y, 9p,—27y, FOI Oe
ite c
Si AA Se
(G2)? (Gre EEN 4
AEN NN 6 f
a —6a+3 Dael
LS, MENNO En Le SOR ==

T = const. (25a@’—162+-4), §, == const §,, wW, == const. W‚
Now we may apply the transformation
Sp, — (@—2)? (5e7—122+4 4)
— Esp, (6e—3)* (5a*— 2a 42)’

whence we have

9 9
(e—2)? (5u7—122+4) (Or —3)? (522)
OTS ee:

~ (5a—1)? (Te? —6e 42) (25e? — 1644)
and we shall obtain
(w—2)da : dt

V(5a7—1 20+ 4)(52*—224 2) (7a? — 6e + a i Vt(1 —t(32—27t)
where the constant 4 is found by observing that #=— 2 + d implies
t= ris 0.

The involution ./, in this case, is somewhat special, because we
have now

y= , §=—6§,.

In the corresponding system S the points A, and A, coincide,
and the right line wp, passes through A,. Hence wedi tent reducible
integrals of deficiency p=—=2 in the general case connected with
the involution J, three degenerate here at once into ordinary
logarithmic integrals. The integral

{> Tde
Vpb.

in the general case of deficiency p==3, reduces here to an elliptic
integral of the third kind, but the transformation indicated above
effects still a further reduction, and we obtain another logarithmic
integral.

341

In fact, we shall have
Ae (252?—16a+ 4)du a tf dt
(5a?—12e+4)V (Tat 6a + 2)(5a*—2a+3) J Wit

As I remarked in the beginning, the principal condition for the
reducibility has been given by Borza and by Ierer. I will now show
that the invariant relations they deduced, may be derived without
difficulty from the results obtained here.

Borza and Ieren both introduce the anharmonic ratios 2, À,,, À
formed by the roots of each pair of the quadrics w,, w,, w‚.

The anharmonic ratio 2,,, formed by the roots of wp, and w‚, is
given by the equation

Al) a, (Ast 1)?
AEN os TS
and putting

Vi,,+1 HI
EE Vi ’
or | ul
the constant. u, is related to the invariants A, A,,,A,, by the
equation
See ee a
LAs As

Now we have
Ah 10 ne
—ca—g* abh" gh=af

23 38 23

and hence by equations (13)
AEN ae aS.
2P,/p6— A) 2P, yey) PBT aa),
where s stands for a+ 8 + y.
In this way we get

u, bat, w+ ae ae u
V By(s—py(s—y) —(2Byt+as) as ey
and we may take aas
BY
“= BE RS

(err
Similarly we obtain

Re En PA ya
aoa eee a B)

bah igh va ‘as SEL
a G+a(e+y’

342

and we infer
Baba se
PR soe
Hence we have between u, u,u, the set of relations
au, + Busy, + yeu, = 0,
au, + Bu, + yu, = 9,
au, + Buu, + yu, = 9,
and by eliminating a, 8, y, we obtain as the invariant relation between
the quadrics w,, w,, wp, Borza’s equation

as Us” + 2 4,40, = 0.
B, WM 1
When this relation is dauefied for any one of the possible deter-
minations of the constants u, U, t,, the quadris y,, w,, wy, are apt
to build up a degenerate integral.
As we have
Gp esa (ae
ul (u?—1)

we have also

ae ee

28° "51

(u, + 1)(u, +1)
NA
Now it follows from Borza’s equation that

(nd) Gx) == (u,u,—p#,)"

and since
at+3 3
at Gls Nn Pe En ’
we get |
var te Fet) 7 1
Pe : — =! in (YUaHs HVU (+ B), Us} y
Suu, S SU, UA {ls

Writing out similar expressions for ’2,,4,, and /2,,,,, we find
by adding them Ierr’s equation

ale ak a ODE

12°°31

IE

za 12 31°"3

Again, if for any one of the possible determinations of the surds
this relation is satisfied, a degenerate integral can be constructed
by means of the quadrics w,, w,. y,. Both the equations of condition
given by Borza and by Ierr involve rather intricate surds, and I
should say that they are less adapted for examining the reducibility
of a given integral than the equation (12) deduced in this paper.

Chemistry. — “On the Saponification of Fats.” IL. By Dr. J. P.
Treus. (Communicated by Prof. S. Hoocrwerer).

(Communicated in the meeting of June 30, 1917.)

§ 1. In a previous paper ') equations were derived for the case
that saponification takes place in emulsion, which equations indicate
the quantities of glycerine and fatty acid present in free state after
an arbitrary time.

These equations are:

2pq 3q óp
—)]~— e—3kt zt ape ee eee
: (2p-8) (7-8) EED Geena
Tl ee e—3kt 1 = (ap) —2pkt _ 3E —qkt , (2)

(2p-8) (9-3) 2p-3)(q-2p) 8) q-2p)-

Equation (1) represents: the fraction of the total number of
molecules of glycerine, present after the time tin free state. (Relative
concentration of the free glycerine after the time ). .

Equation (2) expresses: the relative concentration of the free fatty
acid after the time ¢.

In these equations p and q represent: the increase of concen-
tration of di- resp. monoglycerides at the surface of contact between
fat- and waterphase, in consequence of the adsorption.

‘When the velocity constant / is not constant in reality (as in the
saponification in emulsion, where the surface of contact between
fat- and water phase constantly varies in size), the equations (1)
and (2) have no physical meaning each in itself. When, however,
we eliminate Xt, which is possible for different numerical values
of p and q, we arrive at equations of the general form /(g, 7’) = 0,
which give the connection between the relative concentrations of
free glycerine and free fatty acid for the assumed values of the
adsorption constants p and q. As was proved’) these equations are
independent of the variability of the constant of velocity 4.

In the saponification in emulsion complications make, therefore,
their appearance, which in some cases entirely cover the stagewise

1) These Proc. 20, 35 (1916).
2) loc. cit. 41.

344

course of the saponification (saponification in alcalic surroundings),
and it is now of importance more closely to examine the ideal case,
in which such complications do not appear.

Let us consider the saponification of triacetine in solution, and
let us assume that the three ester-groups are perfectly equivalent *).
Then the saponification takes place according to the following scheme :

Pan ae =
WE En
hoe G
i ky
(r) (x) (y) (s)

In this A represents the triglyceride, B, C, and D the diglycerides,
EH, F, and G the monoglycerides and A glycerine. Let the number
of molecules of each of these substances, present after a time tf, be
represented by 7, #, y, and s, in which & and y indicate the number
of molecules of the three di- resp. monoglycerides each taken
separate, and let the constant of velocity at the splitting us of each

fatty acid group be 4,.
Then the equations of velocity are:

dr
— —=3k,
dt ,
da
on = k, r—2k, . #,
dy
dt. = 2k, ak, U,
ds é
=a ery
dt

Starting from a molecules of triglyceride, we may calculate from
this that the number of molecules of the different stages present
after a time ¢ amounts to the values from column 2 of table 1.

The relative concentrations (fraction of the total possible number
of molecules) of each of the glycerides is, therefore, represented by
the values from column 3 of table 1. The sum of these relative
concentrations is of course, always — 1.

When we now finally calculate the number of molecules of acetic -
acid (z) split off after the time ¢, we find from z= 3 w + 2y + 5):

. z = 3a(1—e—“t),

1) With regard to the validity of this assumption ef. § 3 and 4.

T ABilsB ok
ES SE SE SS a dS ES SE ee
1 2 | 3
. Number of roles: present B Relat. bondantations!
et after the time ¢ | after the time ¢.
Triglyceride | r=a, e shit vy =e Shit
Diglyceride | 8x = Sa.e— Me Ai deze AFF je Ab
Monoglyceride | 3y = 3a. Pages 6 rd m= ze ile Php

Bg gine Aa e= (ite tik

Glycerine

Where now 3a molecules of acetic acid are possible, the relative
concentration 7’ of the free acetic acid is:

T= 1 —e ht
6 aS eS EN aa

The two relative concentrations which are of most importance
here, are those of the free fatty acid and of the free glycerine, viz. :
G =O ee. 22 (8) and: oT Sea merk)

from which follows:
GG) hol DEN LEN ALLA DTE LEN

or in words:

In case of stagewise saponification of an ester of a tri-valent
alcohol (with eqauvalent OH groups) and a univalent acid, the relative
concentration of the free alcohol, independent of time, is always
equal to the third power of the relative concentration of the free acid,
provided that no complications (as in case of saponification in emul-
sion), take place.

When we saponify an ester of a trivalent acid and a univalent
alcohol, the reverse holds of course.

The equations (3) and (4) may be derived from (1) and (2) by
putting p=g=i.

We may now apply our considerations also to the opposite case,
viz. the esterification of glycerine with the equivalent quantity of
acetic acid.

In this case the reaction is bimolecular, as the reaction velocity
is not only dependent on the number of still unesterized OH groups,
but also on the number of molecules of free fatty acid. When we,
therefore, draw up equations of velocity as in the case of triglyce-
ride saponification, i e. when in our equations we only express the
number of free OH groups, which can become esterized, and not
the number of acetic acid molecules available to bring about this

23

Proceedings Royal Acad. Amsterdam, Vol. XX.

346

esterification, the velocity constant #, occurring in this equation *)
will in reality not be constant, but vary with the acetic acid con-
centration, hence with the time.

The formulae for g and 7, which. we obtain by integration from
these equations of velocity, will then have no physical meaning in
themselves. If, however, £, X ¢ is eliminated, which is always pos-
sible, the equation obtained gives the connection existing between
g and 7’ at any moment of the esterification, and that independent
of the variation of &,. .

The equations of velocity drawn up like this for the esterification run :

ds
— — = 3h, .s,
dt
d
- —k,s—2k, 4,
‘
da
di =, 2k, i) Ees Pd
dr
dt Dn 3k, ~a@y

from which after integration and introduction of the relative con-
centrations, follows:

ge ast, and T= e—t, and therefore g— TZ"

We see, therefore, that g= 7" holds both for the reaction in
saponifying sense and for that in esterizing sense. It: follows from
this that when the two reactions take place at the same time (i. e.
when an equilibrium sets in) g= 7" holds likewise.

This may be proved as follows:

For the reaction of equilibrium holds:

dg te Gy ok m dg,
dT dT,+mdT,

dg dg
when ia refers to the reaction in saponifying sense, and = to the reac-
1 12
tion in esterizing sense, in which m is a quantity varying with the time.
Because at any moment:

dg __ dg,

Se — 37
aT, = wae

also

') We shall denote here and later the velocity constant in saponifying sense by
k,, that in esterising sense by ky.

347

dg mnd hef
dT
independent of the value of m.

As further the equilibrium reaction begins at the same point as
one of the finishing reactions (viz. g= T=0, org = T=1), the
equation g = 7’* must hold also for the equilibrium reaction:

gaol

If the saponification did not take place stagewise, but if triacetine
directly split up into glycerine and acetic acid, then 1/n of the
acetic acid would be present in free state, when 1/n of the glycerine
was split off, so that in this case the relative concentrations of
glycerine and acetic acid would be equal to each other. For
the esterification of equivalent quantities of glycerine and acetic
acid the same thing holds of course.

The comparison of the relative concentrations of initial resp. final
products furnishes, therefore, a direct quantitative proof whether or
no a reaction takes place in stages.

§ 2. The case of the saponification of triacetine in aqueous solution
is not easy to study experimentally on account of the difficulty to
determine the free glycerine here quantitatively.

Esters, for which free alcohol and _ free.acid is easier to
determine, are the fats. Glycerine is soluble in water, the glycerides
of the higher fatty acids are hardly so, no more than these fatty
acids themselves, so that the split off glycerine will be easily
separated from it. |

The slight solubility in water of the higher glycerides, however,
involves that most of the saponification processes do not take place
in solution, but in emulsion.

A procedure, however, that takes place in solution at least partially,
is the so-called sulphuric acid saponification.

Van Erpik Tuirme'). has isolated the lower glycerides in this
method of saponification, which quantitatively proves the stagewise
course of the reaction.

The method comes to this: 5—10°/, strong sulphuric acid is added
to the fat, which is heated to 120° or higher. The action of the
acid is allowed to continue for some time, then the reaction product
is led into boiling water, and the formed emulsion is boiled, till the
required degree of decomposition is attained.

During the first phase of the process the reaction takes place in
solution. The action of the sulphuric acid is here twofold: 1 the

1) Thesis for the Doctorate, Delft 1911. J. f. pr. Chem. (2) 85 284 (1912).
23*

348

strong acid acts saponifying; and 2 it attacks the double binding
of the oleic acid present, in consequence of which this is partially
found back after the operation is over (saponification in emulsion
and distillation), as iso-oleic acid, which is solid at the ordinary
temperature. Accordingly this method of saponification is applied,
when a yield of solid fatty acid as high as possible is desired.

In order to examine the stagewise course of the saponification
with strong sulphuric acid quantitatively, it is necessary to start from
a saturate triglyceride in order to avoid this complication. A suitable
material is the trilaurine (to be prepared from Tangkallak-fat by
re-crystallisation from alcohol), the same triglyceride that was used
by van Epix THIEME. |

After repeated futile attempts to isolate the partially saponified fat
without further saponification in emulsion taking place, the following
method was applied :

5 grammes of trilaurine were carefully weighed in a glass beaker
of 250 ce. It was carefully melted, a certain quantity of strong
sulphuric acid was added, and everything was thoroughly mixed.
Then the glass beaker was covered by a watch glass, and left for
some time either in a drying oven, or at room temperature, during
which the stirring was several times repeated.

After the fixed time had elapsed, the beaker was cooled in ice,
and in order to separate glycerine and sulphuric acid from the
glycerides and the fatty acid, as much pounded ice was added as
was necessary to keep the temperature under 0° C. After mixing,
ether was added and it was all brought into a cock-funnel. After
shaking and separation of the layers*) the water layer was poured
off as sharply as possible, and collected for the determination of the
quantity of free glycerine.

The ether layer was rinsed with alcohol in a flask and titrated
with alcoholic KOH to fix the quantity of split off fatty acid. In this
titration a little sulphuric acid is always also titrated. This was
gravi-metrically determined after evaporation of ether and alcohol,
decomposition of the soap with hydrochloric. acid, and removal of the
fatty acids with ether, and then deducted.

The glycerine water was repeatedly shaken with ether to remove
traces of glycerides, then boiled till all the ether (and alcohol originating
from the ether) had again been removed. Then the glycerine was
determined by oxidation with potassium bichromate of known strength,

1) If a sufficient quantity of sulphuric acid is present a good separation is
obtained in a few minutes.

349

and retitration with sodium thiosulphate according to STRINFELS’
method’.)
The results obtained thus are recorded in table 2 (curve A of the

1009

figure). The relative concentrations have here been both given in
percentages. It appears that within a short time a state of equi-
librium is reached, the situation of which varies with the concen-
tration of the sulphuric acid.

TA BEEZ:
Saponification of Trilaurine.

1 2 3 4 Bari fake 22e We 8
| &§ | Ie ber of NS 1 men
Time | & | Strength ren Eee Mn RODE Zs 00 g
of action | ey | of the ea er. of ae — Remarks
(about) E HoSO, | tritaurine | found) | CCU | nd
WEEET lated
E
| |
take | 60°. 91.5% | 15 | 39.6 BEAR rd
dere GORDON | ag RGN NOA
BAE 05 ae. | Baul eis | (4.9) | carbonization!
4 > 15 Pit} ABS | 39.0 | 5.9 | 3.1
SS 950/o 0.25 | 15.35 aa:
1 > 60 > 40} 4de FEZ 8.9
he ae 30 | 50.4 12.8 | 10.0
15 min. | 20 — > 20 | 48.1 | 11,15 | 10.7

') Seifens. Ztg. 42 721 (1915).

350

Somewhat less split off glycerine was always found than corre-
sponds with g = 7". The deviation is slight, but on an average more
than the error of observation. In § 3 the importance of this will be
further set forth.

As, on account of the setting in of the equilibrium, only a small
region could be examined in the saponification, it was tried to
complete the found curve by starting from equivalent quantities of
glycerine and laurinic acid, and esterizing this in strong sulphuric
acid. The procedure was here quite identical with the preceding
case, instead of 5 grams of trilaurine 4.705 grams of laurinic acid
and 0.710 gram of glycerine were now weighed.

The results obtained in this way are collected in table 3 (curve
B of the figure).

It appears that in the esterification always more free glycerine is
present than corresponds with g = T° and the more as the esteri-
fication has proceeded less far.

It is now tbe question: how are the deviations found in the
saponification and the esterification, to be accounted for?

Ty A Bal ses:
Esterification of laurinic acid with glycerine.

i 2 Sher 4 5 6 7 8
i | © leo TI Number Arnen ey rad
Time of | 3 | saoy este voor 100 ¢
action | § | BED aurinic acid Beso sete:
(about) | = | 22E and 0.710 gr. found | found
| & [A lof glycerine.| |
5 hours 20° 950/ 20 43.8 |. 8.4 | 10.6 19.2
eae 4 A E 55.9 | 17.45 | 25.8 31.25
20 min. 4 : : 61.75 | 23.5 | 31.3 38.1
(woe AN : : 63.4 “| 25:14 Maga ey aes
| | | |
base piney ly ee bees ee) : | 67.9. | 31.3 aes 46.1

$ 3. Shortly after I had obtained the results recorded in tables
2 and 3 I imagined the question to be as follows: In virtue of the
fact that secondary esters are on the whole more slowly saponified
than primary ones it is natural to expect that the middle ester group
of the triglyceride will also be slower to react than the two others.
We may then expect in the saponification that g << 7°. If on the
other hand laurinie acid with glycerine is esterized, the secondary
alcohol group will probably also in this case react more slowly.

351

It seemed explicable to me therefore that in the esterification
reversely g > 7° was found. On second thoughts the latter, however,
appeared to be entirely erronéous.

Let us first study the saponification of the triglyceride, Pe let
the velocity constant of the saponification of the secondary ester-
group be —=p.k,, those of the two primary estergroups —4,. Then
we obtain the following scheme:

In the calculation of the relative concentrations of the different
mono- and di-glycerides we must now bear in mind that these con-
centrations are not mutually equal.

We now find that after the time t:

(Lh a th (lekke ee ek ae (6)

TST dekt kinine ad)
If in these equations we substitute p=1, we of course find back
the equations (3) and (4).

It is now the question: what is the situation of the curve
f(g, T) = 0, when pzi, with regard to the curve y= [2
- For pee we have according to (6):

1—e +t
g = (lekt (Leth = (Lent)? , —___,
: lekt
or
, eht — e—pkit
bi ee SSN igh) ASSO eee
g = (lekt) {i+ aes (8)
Further:
Pin Zeh! Es Jerk! == le htt 4 (eht — e-vhat),
or
: k ; t e-kat — e—phit

If we now put the fraction:
e—kit — e—pht
Aes bil) an
in which therefore A is positive or negative according as p is
greater or smaller than unity, the following equation follows from (8) :
g= (1 seh" aA A He LON
and from (9)

352

T=(1— eA (14+44)
or
T*—=(l — eb +iaAy. er .
From (10) and (11) follows:
Gist) 1+ ‘A at 1+A
THE TAG ee
This fraction is always <1, also when A is negative, because
the third power term can never preponderate. The latter is easily
proved as follows:
If A <0, it is required for the preponderance of the 3'd power
term that:

(12)

A Soa At hence: Art en
hence:
e—kit — e—pkit ae oe (hho k Geha t,
or
Bet + e—pPht > 9,
and this can never be fulfilled, because eht and e-#hit are always
smaller than 1. :

It follows therefore from what precedes that when p deviates
from the value 1, no matter whether p> 1, or <1, always less
glycerine will be liberated during the saponification than corresponds
with g= Shek ~

§ 4. Let us now consider the esterification which proceeds ac-
cording to the same scheme. Then we find:
ge ETD Shine ee een
and
T=terhtaekii vo ay
We now put the question: what is for values of p21 the
situation of the f(g, 7’)=0O determined by (13) and (14) with
regard’ to the curve g = 7”.
We find from (13) and (14) in the same way as before:

e Plat — e-kat
ge sl, F En

(15)

and

Tekst: | bi: (16)

Let us put the fraction:
e—Pkyt — e—kgt
TA

~

353

in which A is now positive or negative according as p is smaller
or greater than unity. Then follows again from (15) and (16):

9g 1 +4

CT ve eI cen

We see therefore that also in the esteritication g is always < 7”,
when p departs from the value 1, as here too A< — 9 is im possible.

When now equilibrium sets in, the reactions in saponifying and
esterizing sense take place simultaneously. When one estergroup
deviates from the two others, every change in either direction leads
to a relation g< 7”, in other words g < 7'* holds also here for
the reaction of equilibrium.

When we shortly sum up what precedes, it appears that:

Both in the saponification of triglycerides and in the esterification
of equivalent quantities of glycerine and fatty acid the relative
concentration of the free. glycerine is equal to or smaller than the
third power of the relative concentration of the free fatty acid,
according as the different groups are equivalent or not, provided no
further complications (as e. g. in the saponification in emulsion)
take place. .

The situation of the experimentally determined curve 4 (see the
figure) under the curve g= 7” can therefore have been caused
either by a saponification of the secondary estergroup that proceeds
more rapidly and by one that proceeds less rapidly compared with
the primary ester groups.

It is, however, natural to assume that the latter is the case.

§ 5. After the foregoing it is clear that the experimentally
determined curve B of the figure can never find its explanation in
inequivalence of the ester groups; accordingly there must be another
reason that causes g to be > 7’ in the esterification. This reason
is probably the bivalence of the sulphuric acid which serves as
medium.

To realise this let us first inquire a little more closely into the
process of saponification :

When trilaurine is put in contact with sulphuric acid, addition
compounds are formed. The sulphuric acid is very loosely bound in
these compounds, in water it is immediately split off again. The
binding between sulphuric acid and laurinic acid may perhaps be
most fitly compared with that between water and salt in a salt
hydrate. However this may be, the saponification may now be
imagined as follows:

354

The sulphurie acid bound to an estergroup, expels the laurinic
acid from it under formation of glycerine sulphuric acid, so that,
finally glycerine trisulphuric acid is formed. This may be expressed
as follows:

CH,OOC . R (H,SO,) — CH,OSO,OH (R. COOH)

| |
CH OOC. R(H,SO,) — CH OSO,OH (R. COOH)

|
(11,000 _R (H,SO,) — CH,OSO,OH (R . COOH)

What takes place now on the other hand in case of esterification
in sulphurie acid surroundings? When glycerine, laurinic acid, and
sulphuric acid are mixed the glycerine will immediately be bound
with the sulphuric acid, but certainly not only glycerine trisulphuric
acid will be formed then. When a molecule of sulphuric acid combines
with one of the alcohol groups of the glycerine, the chance is great
that the free acid group of the sulphuric acid still combines with a
second alcohol group of the same glycerine molecule, and that we
therefore get the compounds: .

OCH, Oan
0,86 we |
NOCH and 0,S¢ HC—0.S0,.OH
| |
H.C—0 SO, OH NOCH.

Le. sulphates of glycerine mono sulphuric acid.

We may imagine the esterification of these sulphates as follows:
ANY CH, OOC. R (H,SO,)
| (R. COOH) + R.COOH (H,SO,) |
CH — O CH O0C.R ,S0,)

ee . SO, . OH (R. COOH) eRe Cn, OOC. R(H,SO.),
\

Le. the two OH groups that are bound to one molecule of H,SO,

are esterized at the same time.

Let us now examine what relation there will be between the
relative concentrations of free glycerine and free laurinic acid. Let
us put the “constant of velocity” — pk, for reaction (1), and = &,
for reaction (2), we obtain the following scheme for the esterification :

Wea Fo
a

bet. tye (x) (r)

355

In this A represents the glycerine, which can be converted to
monoglyceride B with a “velocity constant” &,, and into diglyceride
C with a “velocity constant” pk,. Either of these can be converted
into triglyceride D with the given “constants of velocity”.

When we now determine the equations of velocity, then p and k,
occur in them, which are botb variable:

ds -

earn 0 1) A, .#,
dy
Jr) =k, s—pk, Ys
da
ap pk, e s—k, a,
dr
ee utpk,.y-

We may, however, put p and /, constant during a short period
from the beginning of the reaction, and ascertain the form of
f(g, T)=0 at the end of this short period.

As was already discussed before, £, does not occur in this last
function, from which follows that /(g, T)=0 is independent of
the value and of the variations of &,. It is now, however, the
question whether the factor p can likewise be eliminated, or whether
it remains in the equation. In the former case fy, T) =9 is also
independent of p, in the latter case it is not. It is, of course, directly
to be seen that the factor p will certainly have influence, and can,
therefore, not be eliminated.

This, however, causes the relation holding at the end of the short
period that p is considered constant, to change if the reaction progresses.
From this results that the found f(g, 7’) = 0 does not hold through-
out the course of the reaction, but only represents a curve which
may be serviceable as an orientation whether g > 7'* is possible at
the supposition made.

Moreover when equilibrium comes near, the reaction in opposite
sense begins to exert more and more influence. In view of the fact
that the sulphates considered are esterized to glycerides, but are not
restored to their original form by the opposite reaction, it is clear
that at last the same equilibrium is reached as in the saponification
of triglyceride.

It is, therefore, now the question: is there an initial value of p
to be found by means of which the curve B found experimentally
is to be explained.

356

Integration of the equations of velocity yields for the relative
concentrations of free glycerine and free fatty acid after the very
short time t:
giserke. MIN Ala Nee (17)
and:

Pe ES pat jee at) AME ELI KE
from which:

ry 1

is ss
=o ph + ght EE |
Equation (19) gives for:

(peal T= g's or eej ho
p= T= Ff 2 + g's} xs g=—,(V14+ 247-1?

When now the values found experimentally are compared with
the values of column 8 of table 3 and with the curve aaa |
the figure, it appears that the experiment can be explained by the
above reaction, if the initial value of pis p> 1.

The above should strictly speaking be completed with the deter-
mination of the quantity of “sulphate of glycerine mono sulphuric
acid” formed when glycerine and sulphuric acid are mixed, which,
however, will be exceedingly difficult to accomplish.

It is, therefore, only our intention to give a plausible explanation
of the experimentally found curve B of the figure. Whether this
explanation is perfectly correct may perhaps be decided by later
experiments.

We see, therefore, that in the esterification of glycerine with fatty
acid dissolved in strong sulphuric acid, complications are again met
with, which cause a deviation from the ideal case.

Probably others than glycerine esters will be better adapted to
realise this ideal case. It appears, however, from the above that it
is exactly the deviations that often furnish valuable data about the
mechanism of a reaction that takes place in stages.

In conclusion a few: words about the second phase of the sulphuric
acid saponification, viz. the boiling with water. When the reaction
product of the first phase is made to flow into boiling water, a very
fine emulsion is formed, the dispersity of which rapidly diminishes.
The temporary fineness is caused by the presence of monolaurine,
which greatly lowers the surface tension between water and fat.*)
This is attended with adsorption of monolaurine at the surface of
contact between fat and water (the place of reaction in saponification
in emulsion), which causes it to be quickly saponified, and the dis-

1) Trevs |. c.

357

persity of the emulsion to decrease.') A consequence of this is that
in the initial stage of the second phase a considerable separation of
glycerine takes place.

In the saponification in emulsion the relation between the relative
concentrations of free fatty acid and free glycerine is now deter-
mined by the adsorption of the lower glycerides at the surface of
contact between fat and water. In this case the three estergroups of
the triglyceride may be taken as equivalent with very close approxi-
mation. The equation 7’='/,(g + gs + g's), which is founded on
this supposition, and which holds for the TwrrcneLr-process, holds
also at the end of the second phase of the sulphuric acid saponification,
when the water contains but little sulphuric acid (from 1 to 2°/,). If
the acid content in the water is greater, then 7’ > '/, (g + gh + gh).

SUMMARY.

The relation between the relative concentrations of free glycerine
and free fatty acid was derived for the ideal case that tri-glyceride
saponification takes place in solution, the ester groups are equiva-
lent, and no complications present themselves.

It holds that the relative concentration of the free glycerine is at
any moment equal to the third power of the relative concentration
of the free fatty acid in case of stagewise saponification.

The same thing holds for the ideal case of esterification.

In the saponification of trilaurine with strong sulphuric acid only
a very slight deviation was found, caused by the not perfect equi-
valence of the three ester groups. This is a direct and quantitative:
proof of the stagewise course of the reaction.

In the esterification of equivalent quantities of laurinie acid and
glycerine dissolved in strong sulphuric acid deviations were found,
which are probably caused by the action of the two-basie sulphuric acid.

Gouda, May 1917. Laboratory of the Royal Stearine

Candle Works “Gouda”.

1) In treating fats, containing unsaturated acids, with strong sulphuric acid at
high temperature, compounds are formed that keep up the dispersity of the emul-
sion during a longer time.

~

Terrestrial magnetism. — “Do the forces causing the diurnal
variation of terrestrial magnetism possess a potential?” By
Miss ANNIE VAN VLEUTEN. (Communicated by Dr. J. P. van
DER STOK).

(Communicated in the meeting of June 30, 1917).

1. In their analysis of the diurnal field of variation, Scnuster *)
and FritscHE*) assumed that this field possesses a potential, the
necessity of this hypothesis not being proved a priori. For a test by
‘means of integration along a closed curve on the surface sufficient
data are still lacking, but another method enables us to inquire how
far the horizontal forces can be deduced from one and the same
function.

According to the Gaussian theory the potential of the terrestrial
magnetic field is determined, when we know:

either the Northcomponent over the surface, or the Easteomponent
over the surface and the values of the force, directed North along
a curve joining the two poles.

As in this case the forces are those causing the diurnal variation,
hence completely periodical, the Eastecomponent on the surface alone
is sufficient to determine the potential.

When for a number of points on the surface AX and AY are
given, we can deduce from AX a function a representing the
potential, if there exists such a function ; in the same manner we can _
deduce from AY a function a, and compare these two expressions.

2. From the diurnal variation at Pavlovsk (59°41 N. 30°29 E.);
Sitka (57°3’ N. 135°20’ W.); Irkutsk (52°16’ -N. 104°19’ E.); De
Bilt (52°6’ N. 5°11’ E.); Cheltenham (88°44’ N. 76°51’ W.); Zi-ka
Wet (31°19 N. 121°2’ E.); Honolulu (21°19’ N. 158°4’ W.); Bombay
(18°54’ N. 72°49’ B); Buitenzorg (6°35’ S. 106°47’ E.) and Samoa
(13°48’ S. 171°46’ W.), all for the period 1906—1908, for each of
the components a function has been deduced in this manner.)

1) The Diurnal Variation of Terr. Magn. Phil. Trans. Vol. 180 (1889) A.
*) Die Tägliche Periode der Erdm. Elemente St. Petersburg 1902.
5) This research is treated more at length in a thesis for the doctorate.

TABLE 1.
SE SS SSE SS SE SE ee ea nnen
SUMMER
| | aks VZ eel ) | Kie
EE

Hz i— 1.10 2.96 —11.45 1.81 — 8.33/— 1.61) 27.71)— 5.16\ 1.60/—4.36

Ay — 6.42— 1.89 — 3.02, 18.58 —34.41 29.77 46.67 29.88 —0.31/—3.36
| | | | |

Ut ede

Az 6.36) — 0.61|—13.91 “nm 1.78 — 0.871 22.99 — 1.30, 0.50/—0.51

Ty I. 34/— he 19 — 2.17) 23.88) 26: 26 —37. i ee 97 37.94|—3. a 150
| |

SUMMER

| 83 | hi | ee | he Les" hs gS | Ag | gt | hit

| ary | A
Nx 9.716/— 1.91) 2.62) 0.94) 0.26} 2.31/— 4.01 2.03 | 0.03/—0.43
| | | |

i | | | | | | }
Ty 11.63) — 4.02 1.22 — 0.86, os 1.59|— 4.44 3.11|—0.29) —0.36
| | | | |

WINTER

/
Î |

mz | 10. nf le sl 3.05) 1.96 — 0. 53 0.73|— 4.27} 1.73! 0.26 0.52

|
oy 5.74 — 8.17 3.78 0.45 — 0.88 — 5.23) 3.82) 0.31|—0.22

|

gn and hm are the coefficients of the function which Gauss used

in his “Allgemeine Theorie des Erdmagnetismus’, where:

| (9, cos m À + h sin m 2) B
4 represents the geographical longitude, V the potential, A the radius
of the earth, the functions P, differing from the spherical harmonics
only by a numerical factor.

It is worth notice that most of the coefficients in zr, are much
smaller than those in z,, whereas the sign is not always the same.

3. If we want to know what part of these differences may be
ascribed to the choice of the sets of observations, we have first to
find out how far these coefficients represent the results of observation.

In order to avoid disturbing influences as much as possible, we

360

used the diurnal variation on “quiet” days; in consequence of the
use of the “international” quiet days the material for all the stations
refers to the same days and therefore is as homogeneous as possible.

By means of the calculated values of g and fh the Fourirr-coeffi-
cients for the diurnal variation were obtained for the places used
in the calculation, and for the 10 stations an average deviation
between calculated and observed amplitude (positive and negative
values taken with the same sign) was found, amounting to

34°/, in the diurnal, 35°/, in the semi-diurnal period tor AX

DO a 38 3 AY Fee aha: +} 3 se ee

4. For a determination of the harmonic coefficients at other places
a graphic interpolation is the most suitable. The calculated points in
the diagram are joined by a smooth curve, by means of which tbe
coefficients for each intermediate latitude can be read off. Moreover
we can decide how far observations made at stations situated outside
the interval 60° N.—14° S. follow the curve of the calculated points.

A test in this manner gave satisfactory results and led to the
following conclusions :

a. In the main the curves would show the same character if
more stations had been used for the calculation.

b. The difference between 7, and 2, cannot be completely at-
tributed to insufficient observational material, i.o.w: The forces causing
the diurnal variation, taken as a whole, do not possess a potential,
although it remains always possible to deduce part of these forces
from a potential.

Terrestrial Magnetism. — “On the question whether the internal
magnetic field, to which the diurnal variation in terrestrial
magnetism is partly ascribed, depends on induced currents.”
By Miss Annie Van VLEUTEN. (Communicated by Dr. J. P.
VAN DER STOK).

(Communicated in“ the meeting of June 30, 1917.)

1. In 1889 Scuusrer ') suggested that the diurnal magnetic variation
might be explained by a system of electric currents exterior to the
earth, and currents induced within the earth by the former system.

Assuming that the conducting power of the earth increases with
the depth, he found the results of his analysis of the horizontal
and vertical forces in harmony with that hypothesis.

2. In 1902 Fritscnz’) starting from other data expanded the
potential of the diurnal field of variation in a series of spherical
harmonics, and in 1912 Srriner ®) examined whether these numbers
contained a corroboration of ScuustER’s idea.

It appeared that the mutual relations of the two parts of the
potential corresponding to the external and the internal field did
not correspond to the theory.

By taking into account also the magnetic permeability he could
obtain a reasonable agreement for some of the terms but in general
the above mentioned supposition was not confirmed.

3. In 1913 Frirscuy*) published new values for the potential
coefficients of the diurnal variation, using the same material as in
1902, except that of the polar stations, which seemed less reliable.

If we repeat Srriner’s investigation by means of these values,
these numbers are no more than the coefficients of 1902 in accordance
with the supposition, that the internal field is caused by induced currents.

1) The Diurnal Variation of Terr. Magn. Phil. Trans. Vol. 180 (1889) A.
2) Die Tagliche Periode der Erdmagnetischen Elemente. St. Petersburg 1902,
8) Die Bestimmung der Elemente des Erdmagnetismus und ihrer zeitlichen Aende-
rungen. Riga 1913.
4) Ueber die tagliche Variation der erdinagnetischen Kraft. Met. Z.schr. 29 (1912).
24
Proceedings Royal Acad. Amsterdam. Vol. XX

362

TABLE II.
ER SUMMER.
bj ‘a Leer nn eee ale Sea 4
; | | | FRITSCHE (1913) 3 | | | FRritscHe (1913)
Bors ne Oe = | = Pw ves Sal | e+al

aise: | |A Leta
1 g ! | 1
1] l 1

3.96 171°35'| 1.73 — 1.13) 2.02) 3309 | 4.24

1
|

P,!|— 5.79) 1.66) 6.02 196°0 ’ 245219’

P‚\\__0.03\ 2.97, 2.98 270 30 (12.70 212 15 |—7.26 7.22/10.24/224 50 5.19 187 52
Pi) \—21.25|—4.10 21.76 167 32 | 1.85 | 206 1 |—0.12 10.99 10.99) 90 37 | 5.41 | 18 24

| || | }
29.53, 4.89 29.93 350 36 || 7.63 les 19} 7.66 1.18 10.5 316 51 | 6.55 (298 8

|
|

|

|

P2\ 0.50'—2.56) 2.61 702 | 1.53 6213 0.15 — 1.30, 1.31) 83 23 || 1.67 89 19
| |

8.11/—1.82|| 8.31] 12 40 || 6.89 | 3131 | 2.59/— 1.15] 2.83 23 56 | 2.65 | 13 46
P| 0.14l 0.64

0.98 319 16 3.41 | 348 0 1.18|— 0.60]! 1.32] 26 57 || 1.71 | 78 34

P33| 0.07| 1.34!| 1.34'273 2 | 0.76 283 40 | 0.28 0.61) 0.66 294 41 | 0.76 |291 32

| | | | | | | }

pal 3.26 1.64, 3.64 206 42 | 3.39 | 214 4 0.97, 0.94) 1.38 223 50 | 0.89 216 21
| | | | |

PA |— 0.07/—0.32)| 0.33/101 13 || 0.27 | 145 43 —0.06/— 0.07. 0.10 130 34 | 0.20 182 52

|| | | |

WEN TEER:

| |

| |
2.49 109°45’|| 5.77 | 20°28] 4.60 — 1.56) 4.94) 18-23’| 1.56 |284929’

1.99 6.75, 7.04 286 27 12.92 | 204 36 |—9.74 6.07/11.48/211 58 || 4.78 \200 27

| | |
Pi |= 0.84l—2.34

Ps! |

P;! 20.88) 9.51 22.95 155 30 5.17 | 3262 | 4.14 9.48 10.34) 66 25 | 5.15 | 47 31
Pj! 33.04, 10.71 34.74 3422 | 5.88 | 3510 | 6.44 7.60| 9.97310 16 | 4.96 273 35

| ! |
|

Pa? — 0.94, 0.32) 0.99 198 52 | 1.64 | 221 47 |—0.31 0.20) 0.37/213 1 || 0.61 |216 18

8.80 —1.77, 8.97 11 23 | 6.00 | 1657] 4.16 — 1.87, 4.56 24 12 | 3.98 | 25 48
| : | | | | :
P,?2 |— 3.59! 1.47 3.88/202 18 || 0.75 | 259 17 |—2.02) 1.40) 2.46 214 46 ee 209 22
1]

| | | | | |
P33 | — 0.05 —0.08 0.09/121 10 | 0.72 | 48 58 [—0.00/ O.01| uncertain | uncertain

| | | | |
P38 |— 3.35) 1.58|) 3.71/205 16 || 3.45 | 212 27 |—1.40} 1.19 1.84/220 19 || 1.60 (229 19

| | | | | | | | | |
PA) _0.24—0.23| 0.33) 43 24 || 0.18 | 20 21 0.05 — 0.14 0.15) 70 50 | 0.08 330 15
| | | | | | | |
4. With a view to different objections, that can be raised
against SCHUsTER’s calculation as well as against that of FRrrTSCHr,
we thought it desirable once more to analyze the diurnal field of |

363

variation, and to separate it into an external and an internal part. *)
The notation is the same as that of Steiner. Instead of the
notation used by FritscHe in accordance with (Gauss) where

Ve n=

rn Ha, FS

R n= 1 ,
mn

P eae (9 cosma +h” sinm2?) P",
mei n n

2 representing the georgraphical longitude, V the potential, and &
the radius of the earth, the functions P, differing from the spherical
harmonics only by a numerical factor, we put: |

am (cos m 2 + EM) for the external field.

gr cosm ) + Am sinm A=
n n A” (cosmA + (e+ a)") ,, „ internal field.

A
5. In Tab. III e= — and a for each term represent the ratio
a

of amplitudes and the phase-difference between those parts that
correspond to the internal and external fields.

In the first column are given the values calculated for two
limiting values of the specific resistance, in the second our results,
in the third column the results of FrirscHe’s first calculation, in
the fourth those deduced from his last publication. (The first and
third column are taken from Steiner for the sake of comparison).

A comparison of the values shows that the newly calculated
values are more regular than those of Frrirscnw, a fact especially
demonstrated by the agreement between summer and winter.

With the exception of Pt, P', and P?, the ratio c is situated
between the limits calculated from theory, somewhat closer however
to that corresponding to the smallest g.

The phasedifference «a is negative for the first four terms, positive
for the other ones; all differences are smaller than those resulting
from theory, but approaching to the: limit corresponding to the
smallest resistance.

The regularities in the terms of higher order are in favour of
Scnusrer’s idea, but the fact that the principal terms P', and P!,
do not correspond to the theory indicates, that the cause of the
diurnal variation certainly cannot be ascribed to nothing else but
a system of currents exterior to the earth and currents within the
earth induced by the former system.

1) Some details of this research, which will be treated more at length in a
thesis for the doctorate, are given in the foregoing communication. For the potential
from which part of the horizontal forces is to be deduced we took Hz + 7),

24*

TABLE III.

364

EEE

P,

0 =3.1X 1014 ||
Q=3.1X 1012 ||

1

0 =3.7>< 10'4
1

2
0 = 3.7 x 102

@ =3.71>X 104
1

3
Q =3.7> 1012

4

Pe

0 =3.1X104
1
Q =3.7X 1012

Q =3.7 104
2
Q =3.7X 1012

0 =3.7X 10!

P22

3

0 =3.7>< 10”

8 =3.7X 104

-

3

+

0 =3.7> 1012

0 =3.7X 104
3

0 =3.7X 10!2 |

0 =3.7X 104 |

| Calculated

0 =3.7> 1022 ||

0 =3.7 104 |
} |

|

FRITSCHE FRITSCHE |
stome
B (1902) (1913)
I I Il] IV
| C a C a Cc a c a
0.03 /85°) 0.34 |—163°/ 1.90 | 779], 1.07 | 74°] Summer
0.40 |13 | 1.99 |— 91 | 0.44 |— 89 | 0.27 |— 96 || Winter
0.02 |87 || 3.44 |— 46 || 0.36 |— 45 || 0.41 24 || Summer
0.47 |22 || 1.63 |— 74 || 0.40 |— 26 || 0.37 |— 4 || Winter
0.01 89 0.50 — 77) 1.51 —119 | 2.96 |—128 | Summer
0.45 |30 || 0.45 |— 89] 1.13 | 25 || 1.00 | 81 || Winter
| |
0.01 (89 | 0.35 |— 34 || 0.91 183 || 0.86 79 || Summer
0.41 |38 || 0.29 |— 32 || 1.39 | 182 || 0.84 |— 77 || Winter
IN | | |
| | | |
0.04 85 0.50; 4 || 1.09 | 28 || 1.10 | 27 (Summer
ee | |
0.52 [15 || 0.38 | 14/| 0.37 |— 71 || 0.31 |— 5 || Winter
0.02 87 0.34 | 11 | 0.30 |— 28 || 0.38 |— 18 | Summer
0.53 |21 || 0.51 | 10 |] 0.64 2| 0.66 9) Winter
0.02 '88 | 1.35 67] 1.39 | 162 || 0.50 | 91 || Summer
0.51 |27 || 0.63 | 12 || 2.61 | 50 | 4.01 |— 50 || Winter
| | | |
0.04 86 0.50, 12) 1.02 | 8 |] 1.00 8 | Summer
0.56 [17 | || 0.36 | 39 || | Winter
0.02 187 0.37). ATA Orsi eee ge ones 2 Summer
0.55 220.50 15) 0.47, 19 0.46 | 17 | Winter
Kk | | | f
0.03 |86 0.30 | 29) 0.72) 41 0.75 | 37 || Summer
0.58 19 0.45 27 0.42 \— 41 0.44 |— 59 || Winter

4
0 =3.7 10!2 |

Physics. — “The equation of state of an associating substance”.
By Prof. J. E. VerscHarreLT. Supplement N°. 42a to the Commu-
nications from the Physical Laboratory of, Leiden. (Commu-
nicated by Prof. H. KAMERLINGH ONNrES).

(Communicated in the meeting of June 30, 1917).

1. A few years ago it was pointed out by Konnstamm and
OrnstTeIN') that vaN DER Waats’ equation of state is not consistent
with Nernst’s thermodynamical theorem. Taking this theorem in
the form which PranckK ’) has given it, viz. the entropy of asystem
in a condensed state at the absolute zero under its own vapour-
pressure, i. e. under zero pressure, is zero,*) which means that the
entropy-difference between this condition and any other (excepting
the ideal gas-condition) is finite, the contradiction consists in the
equation of state of vaN DER Waals giving for v= 6 (the limiting
volume) a value of the entropy which is lower than that for any
other volume by an infinite amount *). _

This objection only applies, however, to the equation ip its original
form, i.e. with a, 6, and R constant; for-it is evident, that it must
be possible by making these quantities change in a suitable manner
to bring about not only a qualitative, but even a complete quanti-
tative agreement between the equation of state and observation, and
thus also with Nernst’s theorem, if the latter is really in accordance
with the experimental system of isothermals. In particular it will
be clear that agreement with Nrrnst’s theorem’) can be obtained

1) These Proceedings. XIII (2), p. 700. Comp. W. Nernst These Proceedings
XIV (1) p 201. /

2) Thermodynamik, 4e Aufl., 1913, p. 266.

3) According to Nernst and PranckK the entropy at the absolute zero should even
be zero under any other pressure, i. e. should be independent of the pressure. Comp.

however Max B. Weinstein Ann. d. Phys., (4), 52 (1917), p. 218; (4), 53 1917, p. 47.
Vid. also Paut S. Epstein Ann. d. Phys., (4), 53 (1917), p. 76.)

v

A ;
4) Indeed, according to v. D. WaaLs Sv — Sp =[ (5) dv =|R log (v- |.
b

Ov
The two further important inferences to be drawn from the theorem: (5e) =0
p

and Cp =O for the condensed state at 7'—=0, which seem to be confirmed by
observations at low temperature, are not satisfied by this equation of state either.
5) That is: Sy — Sp can be made finite,

366

by making R approach zero at the same time as 7’ in an appro-
priate manner. A variation of A of that nature for a definite
amount *) of substance means that the molecules combine to aggre-
gates (association) 7) or undergo a different process, which resembles
the former very closely (quasi-association) and was repeatedly assumed
by van DER Waats*) and his pupils *) for the purpose of explaining
the incomplete numerical correspondence between the original equation
of state and the actual course of the isothermal. It was, however,
always assumed, that only one kind of molecular aggregate was
formed ®), whereas for Rk to fall to zero a much more complex _
association (or quasi-association) is required, viz. a process in which
the molecular groups which are formed become more and more
complex, according as the temperature falls, until finally at 7’—0
the substance behaves as if consisting of only one molecule, at least
at small volumes ‘). |

On this ground | have made an attempt to extend the theory of
association by introducing the hypothesis, that instead of one special
aggregate all possible molecular complexes are formed: double
molecules, triple, quadruple, ete. 7) In this attempt I think I have
succeeded, proceeding logically on the lines followed by van DER
Waars in his theory of ‘‘quasi-association or molecular aggregation”

1) R is only then a universal constant (the gas constant) when it refers to a
gram mol of the substance. When the molecules form groups, the number of
gram-mols in a given mass diminishes and R for that mass changes proportto-
nally to that number.

*) Previously F. Richarz (ZS. f. anorg. Chem., 58 (1908) p. 356; (1908) p.
146) has explained the fall of the specific heat by association. Comp. also J.
Ductavx, C.R. 155 (1912) p. 1015. The same hypothesis was shown by C. Benepicks
(Ann. d. Phys, (4), 42 (1913) p. 133) to be capable of giving an explanation of
Piancxk’s law of energy-distribution.

3) Continuitat, Il, p. 27. These Proceedings (1), p. 107 and 494.

4) J. J. van Laar, ZS. f. physik. Chem., 31, (1899) p. 1; Arch. Teyler, (2),
Xl, 1908.

G. van Ru, Proefschrift, Amsterdam, 1908.

5) This assumption was made for the sake of simplicity, although the probabi-
lity of different molecular complexes being formed was recognised. (See for in-
stance VAN DER WAALS, These Proceedings, XIII (1), p. 494).

6) Comp. H. KAMERLINGH Onnes and W. H. Kersom. Die Zustandsgleichung.
Encyklopadie der Mathematischen Wissenschaften, V, 10, p. 886. Suppl. 23 of the
Comm. from the Phys. Lab. of Leiden, p. 272

We shall not dwell on the difficulty which may be involved in imagining the
state in which during the transition to the molecular condition of one solid lump
the substance must have consisted of a small number of molecules.

7) This thought I developed many years ago for an associating substance
such as water in the dissolved condition (Acad. Roy. de Bruxelles, Mém. cour., 1896).

367

(loc. cit). The equation of state at which I have arrived in this
manner is no longer incompatible with the requirements of NerNst’s
theorem °). .

2. Let.1 gr. of the associating substance contain «, gr. of single
molecules, x, gr. of double molecules, etc. in general x, gr. of n-fold

= .. En
molecules *), so that Sw, = 1. The «; grammes contain „ Sram:
ni

mols (M being the ee ah of the simple substance), 80 that

there will be altogether ae “gram-mols per gramme. If «M be the

mean molar weight, « being the mean degree of association, the

1
number of grammols per gramme will be ae ne that
a

iy Me A ROSA ae

a Nn

We now assume, that for M grms. of an arbitrary unchanging
mixture of molecules of mean degree of association a, the equation
of state has the form used by vaN per Waats:

Tk
Pia a eee ae ta

where #, does not represent the gas-constant A corresponding to a

R 2 :
gram-mol, but A, == —®. Moreover az and bz are functions of the
a

quantities z,, 6; being also a function of v, whereas both a, and 6,
might in general be functions of the temperature; but on various
grounds, which were fully discussed by vaN per Waa ts‘), we shall
not introduce the last-mentioned supposition.

We shall also for the sake of simplicity following vaN DER W AALs *)
leave out a dependence of 6 on the quantities z,, i.e. we assume

1) According to E. Artis (Paris GC. R. 164 (1917) p. 593) this theorem
would be implied by the ordinary theorems of thermodynamics. This statement is
incorrect, however, and is obviously due to the author unconsciously introducing
suppositions which involve the theorem. As an instance he adheres to the identity
of adiabatics and isentropics down to the absolute zero, although at that tempe-
rature dQ=TdS=O does not necessarilly imply dS=0. He also introduces in
some wellknown relations such as HELMHOLTz’s theorem transformations which
are not in general mathematically allowable and therefore presuppose a special
course of the thermodynamical functions at 7=(.

2) n is supposed to assume all possible integral values from 1 to oo.

3) See page 366, footnote 1.

4) These Proceedings, XIII (1), p. 109; XIII a? p. 1213,
5) These Proceedings, XIII (1), p. 121,

368

that the volume of the molecules is not modified by the grouping
of the molecules, which agrees with vAN DER Waats’s manner of
considering associated (or quasi-associated) molecules as “mere com-
plexes of the simple molecules, which can be formed:witbout further
radical modifications in the structure of the molecules themselves’’).')

3. As to the quantity a, vAN DER Waals had originally 7?) made
some simplifying assumptions which resulted in this factor also
becoming independent of the degree of association; later on*) he
made his theory independent of these special suppositions on the
ground, that when molecular complexes are formed only a part of
the cohesion can make itself felt as internal pressure. In order to
conform to this view it is natural in our case to replace the form
dz = a|(1—a) + kr]? chosen by van per Waats in the case of only
one kind of complex molecules by the more general form “)

Ar =a (= kenen)” 50 eas tie Ke
where a is independent of the degree of association. In this manner
we obtain an equation of the following form:

BT a
= ESE hie de

b ORE

P
which is a generalisation of the one assumed by vAN DER WAALS
Ges. pe IAN

The coefficients 4, may no doubt be assumed to have definite
numerical values, only depending on the ordinal number n (4, = 1).

1) These Proceedings, V (1902), p. 304. Van LAAR on the other hand (Arch.
Teyler, (2), XI, 1908) assumes, that the molecular volume does change
(increases or diminishes) by association. This assumption is an essential one in
his theory, as he derives from it the possibility of a coexistence of the vapour
and liquid with a third (solid) phase (These Proceedings 1909—1911).

2) Continuität Il, p. 28. See also van Ris and vAN LAAR (loc. cit.).

3) These Proceedings, XIII (1), p. 119.

4) This form is arrived at by assuming in general a = San Zu? +2 D dy» Tu a,
and making the additional simplifying assumption a?» = au ay (c.f. v. D. WAALS,
loc. cit, p. 120).

5) We might have retained the equation used by vAN DER WAALS, assuming
that in it 2 represents that fraction of the substance which is present as complex,
n-fold molecules, ” being the mean complexity of all not-single molecules; the
possibility of such a generalisation was pointed out by vAN DER WAALS himself,
loc. cit., p. 494; would then have to be considered as a function of v and T,
connected with @ by the relation:

1 . n—l

L
Sag Mee a
«a n Nn

369

If all & were equal to 1, a, would be equal to a and the equation
would be

1 AL a .
p= er € e 5 8 . e é (4)
av—b v
If k, were equal to — *), we should have
RT a
ere me CS at Mie AD Au
wd a v—b av? tee
A more general assumption would be that
n-—l og!
ee, een EEE Ds that hey rl as 21)
n a
in that case*) :
RT oe te
zen 2 call REE p=(t- ) (5)
av—b v* a

4. The manner in which 6 changes with », independently of all
association, has also been fully discussed by VAN DER Waais. The
law of dependence sketched out by him®) may be very well
represented by the following expression :

| (lk): B
(bic — b) (v — hbe) = 4 (1 — 4283, Or b= be pa gout Ass BN (.))
v—k be
where 6, is the value of 6 for an infinite volume, whereas at the
smallest possible volume:
Viim == bolim — 4 (1+4) bs where 0 < bim U SOLE Lash el +1“). (6')

On this assumption as regards the relation between 6 and v the

equation of state (5’) becomes:

9

u\~ 5
') The special form proposed by VAN DER WAALS Ar =a € ao is arrived

at by putting kn @ =|=1) constant and equal to 1/,; even at the highest degree
of association (c=1) ar would still be 1a. It seems more natural, however, to assume
that as ” increases, ky becomes smaller ; if the diminution of An went so far, that
it approached zero, dr would ultimately become zero.

2) Although a given mean degree of association may be obtained in an infinite
number of ways, still according to the assumption (5) the cohesion (and thereby
the equation of state for a given «) is no longer dependent on the special way in

which the molecules are grouped.

2

LY 8
If ¢ = 4, so that p (a) =+} (1 ed ‚ the limiting value of ax will be equal
at

to 1 a, as according to VAN DER Waars’'s assumption.

3) These Proceedings, XV @), pr 113k
4) If k=1, b= constant =b,, as in the original equation of VAN DER WAALS.
With k= — 1 we should have vlim = blim = 0.

370

2k 5
jn RT Dm 1 A ke lim a p (a) (6"
es a (v—biim )? v*

This would therefore be the equation of state of an associated
substance of unchanging mean degree of association «. But we may
also look upon this equation as referring to a substance undergoing
a molecular transformation, if.we consider « as a function of v and
T expressing how the mean degree of association depends on volume
and temperature. This is the function which we propose to determine.

5. But before proceeding to do this we shall first consider (6")
once more as the equation of state of a substance with constant «.
The critical constants are found to be by the usual method

(a= Ky b, (pie = Ky

2 2
where the numerical coefficients A,, A, and K,, the so-called critical
coefficients *), have values which change with 4 *). Hence:

e)a = (ve), (Pela = (Pe), pla) — (Tk)z = (TH, @ y (a) *) © (7)

According to the above assumptions, therefore, the critical volume
of all polymers would be the same; i.e. the same as that of the
pure substance with the simplest molecules; the critical pressure
would also remain constant, if e were zero (equation 4), and Pz
would then be proportional to @, whereas with e=1 (equation 4’)
Pak would be inversely proportional to «?. and Tp inversely to a.
With the more general form (5) it becomes somewhat less simple:
Pak is then found to diminish continually with increasing a, and 7’,
first increases and afterwards also diminishes *).

") Vide H.KAMERLINGH ONNEs and W. H. Keesom, Die Zustandsgleichung, p. 703 (89).
*) The following results are obtained:

js | bb Ks K,=zr=—=0,0370 K‚= 8% = 0,296

27

pa) RUT = Ky ape) O

(Jae 3b. 2,80 0,0408 0,308
==0) Jb, 2,32 0,0540 0.342
k=—'/, A0 1,59 0,0952 0,422
k= — 0 0 oc 1
5) Although these relations are derived from a special form of the equation, it

is quite possible that independently of it they may be at least approximately valid,
in the same way as the law of corresponding states, although it was found by
means of the original equation of state of vAN DER WAALS, is not bound to this
particular one. But the necessary experimental data to test the equations (7’) by
experiment are not available.

*) In the well known case of polymerisation: acetaldehyde (C,;H,O:; th = 188°C.)-
paraldehyde (C3H,.03; tx =290° C) the relation (7’) gives a correct result for
the critical temperatures with €= $ about; unfortunately the critical pressures
and volumes of these substances are not known, so that a further test is impos-
sible. However, it is doubtful, whether the above theory would be applicable to
a chemical transformation of that kind.

371

From the equations (7) it also follows that

1 R (Tr, /
COMEN itis sae al ae
« (phx (vx)

6. We shall now consider the equation (4) as representing the

tj This constant, the critical virial-coefficient Ay (vid. KAMERLINGH ONNES and
Kerrsom, loc. cit., p. 752 (138)), again has a value which changes with k. For:
k= 1 "Ke 0 AR hy al
Ki == £23 2,67 2,69 2,73 2,80 3
The change of Ky, with k is seen to be comparatively small, and the relation
which we have assumed between b and v does not raise the coefficient K4 to the
experimental value of 3.6 about, holding for the so-called normal substances, or
the still higher value required by the associating substances; it is even doubtful,
whether this could be attained with any other relation b = y(v), unless a very spe-
cial form were purposely chosen (comp. on this point vAN DER WAALS, These
Proceedings, XIII (2), p. 1211; see also KAmertincH Onnes and W. H. Keesom,
loc. cit., p. 752). It is true that v. d. Waats (Boltzmann-Festschrift, 1904, p. 305)

b
found the correct value of K,, by putting )=6, (1 ey and k = 3, but it
v

is easily seen, that this expression for b cannot hold down to small volumes: in
fact for Viin —blim it gives an imaginary value.

In this connection it may be of interest to remark, that the author (vid. these
Proceedings, II, p. 558 and Arch. Néerl. d. sc. ex. et nat., (2), 650 1901) esta-
blished a purely empirical form of the equation of state

v,—b v,—) n
P=P+e Pr Aree peat
v—b v—b
according to which
RT b
Ws os =ng| 1 — —};
Pk Uk Uk

where it appeared that approximately n»= 4 and b=0,1 vk, which gives a value
for K, agreeing well with the experimental value.

Batscuinskt (ZS. physik. Chem., 40 (1902) p. 629; Bull. soc. imp. nat.
Moscou, 1903, p. 188) has made the relation (7”) with K,=4 the basis of a
determination of the degree of association for associating substances, in particular
for acetic acid at different temperatures, assuming that the degree of association
varies with the temperature, but not with the volume (an assumption which is,
however, in contradiction with the well known laws of dissociation and also with
determinations of the vapour-density). By comparing the separate isothermals of
the associating substance with the set of isothermals of a normal standard substance,
Batscuinsk1 determines the critical constants appertaining to each isothermal and
then calculates « by means of (7”). In this manner he finds, that the value of
(vk) is very much the same for all isothermals, whereas (7'%)z diminishes a little
towards the higher temperatures, whereby « also gels smaller and (pk)« increases
in the same direction; this agrees in the main with what we have just derived
about (vk)z, (pk)z and (T%)z from the equation of state.

372

set of compressibility-curves for various mixtures at a given temperature
and ask the question: in what manner does the isothermal of the
associating substance run through this set of curves, i. e. how does
the degree of association vary with the volume at a given tempe-
rature? For the solution of this problem we shall follow the way
given by vaN peR Waats (loc. eit p. 121): we first establish the
expression for the free-energy of a given quantity (M grms) of a
mixture with constant « and then write down the conditions that
for every elementary change of condition (change of the molecular
constitution) at constant temperature and volume the free-energy
remains unchanged (the free-energy having a smallest value in the
condition of equilibrium).
In our case the expression for the free-energy is *)

P= =RIOSS Eh) + RTE log 2, + EE or TEHe 18)
n (2) n

where #, is the internal energy of M grms n-fold molecules in the
ideal gas-state, Hn its entropy in volume 1%; /(v) stands for

dv
{SS so that according to § 4:

lk Diim
oo (8)
1+k v—Dbiim

The condition dF =O, connected with Ede, =0, by the usual
method gives an infinite series of equations of the form

Zie 2a
~ =) -
n

J ©) = log (v — dim)

4 RT RT
kn Eken + — loge, + == ETH 0 8
n n

v
The constant u?®) can be determined by the condition 24, = 1.
In order to make this summation possible it is necessary to make

1) It is only necessary to generalize the expression given by vAN DER WAALS
(loe. cit., p. 121).
*) En and Hn are functions of the temperature, connected by the relation
dE, dH, __, .
aT Lp (x, the specific heat of M gr. n-fold molecules in the ideal state
at constant volume.
%) Putting Z= F+ pv (thermodynamic potential) the equation may also be put

4 OZ
in the form Ds
Oz,

present the so called molecular thermodynamic potential of the n-fold molecules.
Equations (9) thus express the law that in the condition of equilibrium the molar
thermodynamic potential is the same for all the different kinds of molecules ; this
is a well-known theorem, which we might have used straight away to establish the
equations (vid. for instance van Laar, loc. cit.).

) __, Which shows the indeterminate constant gy to re-
P

373

a suitable assumption about the manner in which £,, H, and &
depend on n. The simplest supposition no doubt which we can
make regarding Z„ is, that in a union of M germs of simple
molecules in the ideal gaseous state with nM germs of n fold
molecules to form (n-+1) fold molecules a quantity of potential
energy & is lost which is independent of n*); on this assumption

we have:

n—1
EE Ea hat Dinge it ee, vac EMO
n
dE, dH,
d tl | ing that —- = 1 ;
and thus also, seeing that —, ZT )
n—l
Hi; — Hi, aa HH, . . e . 5 e (10)

where Z, and H, now refer to simple molecules *). These equations
are of the same form as that which was assumed above between
k and n and which we shall again introduce here.

Equation (8) now leads to:

FD erent” xt et Ne ar Pa en EE
where
a E la 2a 1 a—l
Rr OR I) — RTF ae
YES OaRReC = Nea? oe On 2a 1
Visi bata en el Sen
RT BSA o( a )
We thus find:
X Y ted
1 = Sa, = e-X. Se" ¥ zer ;°
x e e e ERE )
so that
Ge :
lte x
and hence

1) This assumption which is tantamount to assuming an equal binding of all
molecules in the complex, was made by me before (vid. Acad. Roy. de Belgique,
Mém. cour., 1896, p. 54).

H dE dH ans:
2) == = a = (, and also i ly IT (it will appear further on that E and

H must be assumed to be also functions of the temperature). It follows that

on —1\dE
sk ater aut ie
ae a

dT dT
3) Provided e-Y < 1, which is confirmed by the result, at least unless
e-X=0, or X =o, which is only the case at 7=0 or at v = bim:

374

xX
iy eee ee (11'
(ly 0-2
Finally we obtain:
1 Ly i 1
Ee Te Ed Ene EN fe 1 gx
hen TLN log E pox |=aeel + } = (12)

This is, therefore, the equation determining a as a function
of v and 7; it actually gives for a values enclosed between 1
(X = — oo) and infinity (X= -+ o)'). It is a fairly complicated

Er . 1
relation, as X itself contains a; but as — only changes between nar-
Qa

row limits, these limits themselves are not modified thereby and the
law of change of X is moreover mainly determined by that of v
and 7.

The manner in which a changes with v and 7 is most easily
understood by putting £=1 and e—0, that is 6 independent of
v and a, independent of «. Expression (12) then reduces to

i MS 1 e7 (12!)
EP og == AP en ds
where
E H
EE et 1 SAM. dhe ae, eee eee
q RE R at ) ( )

is now only a function of the temperature. In this case it will be
seen, that at a constant value of e7, i.e. at constant temperature,

OM ANG
— diminishes continually from 1 to 0, when v decreases froma to
a

6, so that along an isothermal « steadily increases from 1 (v =)
to a (v= 6). When the temperature falls towards 7’ —O at constant
volume the degree of association increases regularly up to a=,
at least if / and H depend on the temperature in such a manner
that at T'=0 g=o; similarly with rising temperature a falls
towards unity, if q approaches — oo ; it will appear in § 10 that
this must be so.

According to the law of dependence assumed between hb and v
the course of f (v) (equation 8’) remains in the main unchanged, so
that the change of @ with v and 7’ also remains very much the

1
1) The following table shows the manner in which — changes with X:
a AN

Kin Ond B el er 2 a) 4 5 Orr As
1
ite 1 0,99 0,98 0,93 0,85 0,69 0,48 0,28 0,16 0,08 0,03 0,01 0...

375

same. The introduction in X (eq. 11) of the term depending on the
cohesion (¢ >0O and even —1) will not produce any fundamental
change in the relation.

8. We shall now consider equation (5) as the equation of state
of a substance with molecular transformation. According to this
equation the isothermals representing the compressibility obtain a
shape which does not essentially differ from that of a substance
without molecular transformation (« = const). This is best recognized
in the simplest case where k—1 and e= 0, that is for equation
(4’) connected with (12’), when the equation of state would be:

eg a (13)
v—b eit TIAA ads

eee eh ef Nees Add 0
the function i log (1 =F =): which takes the place of ved

RT
p= bay (1 ~
ed

the original equation of state of van DER Waars has in the main
a similar course to the latter function, with which it coincides to a
first approximation (large volume or small value of 7, i. e. weak
association):.as v decreases from oo to b, the function increases

if
steadily from 0 to oo, more slowly, however, than nag It follows

that the isothermals of the associating substance intersect the normal
set (e/==0), towards lower pressures (consequently towards lower
temperatures), as v becomes smaller.

Again the change of 5 with v does not bring about an essential
difference in this result. As regards a dependence of a on a, this
again cannot modify the shape of the isothermals fundamentally ;
but it can have an important influence on the whole set of isothermals
in the sense, that the possibility of neighbouring isothermals intersecting
each other is not excluded, which might give rise to special
phenomena. But it is not our intention to inquire into this further
on this occasion.

9. Before going on let us for a moment longer consider equation
(13), in which a and 6 are constants. The critical point, as determined

0 0?
by the conditions (5) =d and ( 4) = 0, corresponds to a tempe-
Ov) 7 Ov? Jy

rature which is given by
_ a (l—u+ Yi—u+wv)l4+V1—at+ u*)
OR (2—u+V 1—u-+u?)?

Tr si ces. ei CA)

where

oy 4% Er HR ra
Te Be
and to a volume
ve b(2—ut+V1—u+u) , ahs CO

whereas equations (12’) and (13’) determine the critical degree of
association ap and the critical pressure pz. Moreover
pre __ 2—-utV1—w +u? oe (2 4 u ) ae
RT; u l—u+V1—u+u?
9—utV1—u+u?
lut Wu ul Vu + u”)
As wu increases from O to o (that is a, from 1 to @), the
expression (16) diminishes slowly but steadily from 3 to 0. Now
it is well-known, that for most (so-called normal) substances the
Pk Vk
k
substances it is even smaller. According to (16) this might be
explained by assuming, that even for the so-called normal substances
association (or quasi-association) exists at the critical point, to such
a degree that w= 7 about, giving vx — 1.56 and a, = 5. ')

(16)

value of is about .28, whereas for the so-called abnormal

1) If a and b are functions of the volume this value will probably be considerably
lowered (vaN DER Waats, these Proceedings, XIII (2), 1257 etc).

Similarly Scuames (Ann. d. Phys. (4), 39, (1912) p. 887; Verh. d. D. Physik.
Ges. 15, (1913) p. 1017.) tries to explain the deviations of experiment from
VAN DER WaAats’s original equation by association of simple to double molecules.
He starts from equation (4’), in which, however he considers & and a as functions
Tr a—1
Terry eS be

a—l ak Tr | EE
v aen De va — ’
ee v° a

in which « changes from 1 to 2 according to a law which is not specified, as »
decreases from oo to its smallest value. He further assumes the relations (7) (with

he thus establishes

ty a
of @, in such a manner that b= bk “and a= ak
a

the equation

ee ro weds 8
a=ak and by = bk and p (a) = 1), which involves the relation Ë = get Owing
PkUk

RT
to these assumptions Scuames finds for 5 = the experimental value 3.6 by taking
kk

aj, = 1,4 and for the smallest volume the value 1 vz instead of 4 vz. The fact, that

oT
not due to his association-hypothesis, but is the consequence of the further fact,
that he makes a change inversely as 7’ (for this well-known rule vid. e.g. J. P.
Kuenen, Die Zustandsgleichung etc., 1907, p. 194).

SARS)
for the critical coefficient al he also obtains the experimental value 7 is

377

10. We now return to the more general equation of state (6"),
where @ has the value given by (12).

As at very low temperatures eX becomes very large (infinite of
infinitely high degree), it is easily seen that, provided v itself is

not infinite,
dv ek 0 d any es 0 1

For the free energy by introducing (11) into (8) we find the
expression :
a Bea a &?
F= — RT log (1 +¢—-X)— —-—__| (1—s)*— — |+ £,—E—T (HH), (1 7)
4 as

a (2)

and therefore, (for the sake of simplicity assuming a and 6 inde-

pendent of 7’) *),
OF E E
= Reel log (1 +e-X)- me
sm (ar) =A 481+ gp Jetta ge
2a E rom a 1
Bae ede

Bat Peay ahs
At very low temperatures and moderately large volumes we
thus have:

SER and eee TSE, B-—(1—s) . (19)

It will, therefore, be seen that at low temperature the entropy
on the side of the condensed condition of matter is no longer a
function of the volume (or of the pressure) *), so that there can only
be question of one specific heat
dS „IHM dE)
al Nn dT dT 4

(20)

1) This agrees with what may be derived from the theory of quanta at low
temperatures (vid. e.g. P. Lancevin et M. pe Brome: La théorie du rayonnement
et les quanta, Paris, 1912, p 284).

2) Comp. VAN DER WAALS, these Proceedings, XIII (2), p. 1213.

os Ov
3) This follows also from the relation | — = — | 5, |, and therefore = 0;
Op Jr 07 p

OT Ov
4) In our equations ZE) and £ are indeterminate functions of the temperature
and we cannot, therefore, conclude from them that C approaches zero with 7.
It is clear why this is so. The dependence of C on the temperature is not solely
determined by the equation of state of the system of separate molecules, but
also by the internal mechanism of the molecule itself, that is at low temperature
and small volume the internal mechanism of the amorphous-solid body consisting

25

S Wee
similarly (55) = C ) = 0. This is again in accordance with modern views.
v T

Proceedings Royal Acad. Amsterdam. Vol. XX.

378

We may choose the arbitrary constant occurring in the function |
-
E, = eat such that at 7'=0 and v= 5 the energy U=0.
7 ~
Further utilising the fact that at 77—0O C=O, we can also choose

C.
7 dT and

Tr
the arbitrary constants of the functions //, el
0

a
dE :
vF | aa pee so as to make S,=—0O. The functions Z, EH, H,
a
0

and H are then completely determined by the values of C, and C,
except that H, and / contain another constant A which, however,
occurs in the constant of the vapour-pressure formula, the chemical
constant so-called (vid § 12), and is therefore determined by it.
Putting also:
1 ’ dH, ’ ry
Cy ee Eee Co TAO
and similarly, in agreement with the preceding discussion (Cr—o
must be 0),

dE dH

— = Ts (TL) « EN
Tar 0 th D (22)

it follows that

N
K=E+S+O7+ [Amar |
0

i
E=E,+0,7+ [Mar
; Br,

x
1
H,=h +, log r+ (har
0

Ji
1 ry)
ne CT + (7 jae ed |
0

11. What are now the quantities determining the coexisting

of one single molecule. Our equation cannot teach us anything about that
mechanism; how the properties of the amorphous-solid condition arise by
association is here left out of consideration.

379

phases at low temperatures, i.e. for infinitely low values of 7’?
They are determined by the relations p, = p, = Peoer. and Z, = Z,
(the index 1 referring to the condensed condition, the index 2 to
the dilute gaseous state) which for small values of 7’ are capable
of great simplifications. In fact at low temperatures v, differs infinitely
little from the limiting volume bj, that is: v, — bis infinitely small;
a, is infinitely large; wv, is infinite and peer. infinitely small;
assuming .further that «a, differs infinitely little from 1, as confirmed
by the result, and neglecting infinitely small terms, the conditions
of coexistence become:

saa bim 1 a (1—s)?
1+k (v,—btim)? |

Peoër =

ne =
EN a, b lim

and

RT log (46%) + Cl e= RT log (le).

Bim
In order that @, may be infinite and a, may differ infinitely little
from 1, according to (12) e& must beintnits and e*: infinitely small;
the ae condition of coexistence becomes as follows

log NE E 4- 1 4+- —_— (1—e)? + anr Se 9 |
? Pe EEN Jai i a
hence
log pcoer — log RT — RI asthe ie eens (i—e)? +. . . (25)
and
_ 9
dee ITO lim ie ee EN

12. The heat of evaporation at low temperature is found to be
(vid. equation 19)
4= T (8,—S,) = RT logv, + TH, — T (H,—H) =
i
(Le) FE FC, HDT + |A (DAT +...) (27)

0

=i

Putting d= A+ By, i. e. the heat of evaporation at
lim 3

1) This expression is also arrived at, if one starts from the relation

dlog p a
SRT" which holds for low pressures, i. e. for low temperatures, or from

aT ae

dh
the relation aT Cp — C repeatedly used by NERNST.

25*

380

h C
T = 0, and i = log RH — = — 1, i.e. the chemical constant 5,

equation (25) becomes,

E R

Arin tl odor aes AL apa) gga | ee; .

log p=——— + — log 1 tig f AOT art SDAL (25)
0 0

Tse) R

We thus find, as might be expected, a vapour-pressure formula
of the form assumed by Nernst. *)

13. The equation contains two indeterminate functions /,(7’) and
FT), the former being governed by the internal mechanism of the
single molecule and determining the change of the specific heat C,
with the temperature, while the latter depends upon the internal
mechanism of the associated molecule and in connection with the
former determines the change with 7’ of the specific heat C of the
condensed state. The fact that we are free to choose these functions
as we like might give the impression as if the equation of state
which we have deduced and which is independent of that internal
mechanism, would be compatible with the properties of the amorphous-
solid condition without any modification. Sull this is clearly impossible :
for it is evident that the pressure must also depend upon that
internal mechanism ®, and from this point of view something
must be lacking in our equation of state.

It is, however, not impossible that it will be sufficient to make
a small modification in the law of dependence of 6 *) in order to
obtain the devised correspondence with the equation of state of the
solid condition, and that by establishing the ‘‘equation of state of
the molecule” ’), not only of the single molecule, but also of the
complex one, the object might be attained. Still, so long as the
equation of state of the solid condition itself is not better founded
than it is, it would be premature to go into this question any further.

1) |. W. CeperBerG (Die thermodynamische Berechnung chemischer Affinitaten,
Berlin, 1906, p. 25) puts t=Jlogpk; in our result this constant remains in-
determinate.

2) This is known not to be the case with the original equation of state (cf. e.g.
PLANCK, Thermodynamik, p. 277).

5) Comp. H. KAMERLINGH OnnES and W. H. Kersom, loc. cit, p. 887 (273)
etc. Vid. also Max B. WeINsTEIN, Ann. d. Phys, (4) 51 (1916) p. 465; (4) 52
(1917) p. 208; (4) 52 (1917) p. 506.

4) As an instance: whereas according to the previous assumptions the condensed
state would be completely incompressible at 7'—=0, the hypothesis of a change
with pressure of the limiting volume blim would give the substance in that condition
a certain degree of compressibility.

5) See e.g. VAN DER WAALS, these Proceedings, III, p. 515, 571, 643.

381

14. In conclusion a remark may be added about the dilute
gaseous condition. It follows from equation (11) that:
kod} — — eX(n—1),
av,”
Representing by c, the molecular concentration of the n-fold
molecules, namely the number of gram-mols for unit of volume,

Xn

we have c, = — , and the above equation may be written as
Vv

nM
follows:
n 1
oP (Mag hye i een) Be OOD
: EN
For very large volumes this becomes
n 1
= -(Meynt=K,.... . . (28
Gi n

in other words, to a first approximation this ratio is independent of
the volume; this is the well-known law of GULDBERG-W AAGE, as
applied to the transformation of n-fold molecules into single ones.
It further follows that

dlog K _ i dX i E 29
pr tre: . - - 8)
the expression —(n—1) EF giving the heat effect of the transfor-

mation (vAN ’T Horr’s law).
Finally representing the total molecular concentration by cy, so

1 Ln jn |
A 1 . :
that c,, i Sate ap we find from (12)

1 2
Mem—=—slog(l + eX) . « . « ... (80)
veX
For v very large and also e* (strong association), we may write
by approximation
1
Men = log (Ll re) Ay) ne ee hn OP
Tis

where 7 represents the density and 9 =e, a coefficient depending
upon the temperature, that is: decreasing with rising temperature *).

1) This relation (but in that case for the dissolved condition, 0 being the
ordinary concentration by weight of the dissolved substance) was several years
ago derived by me empirically for water from determinations of the molecular
weight (cf. Acad. Roy. de Bruxelles, loc. cit., p. 20, 23 and 37).

2) This is also confirmed by the experimental data (cf. Acad. Roy. de Bruxelles,
loc. cit. p. 47). From the change of this coefficient with the temperature:
dlogr dX E

derden EE
2 M grms ol double molecules of water,

[ calculated (loc. cit., p. 61) the heat of dissociation FE of

Chemistry. — “Jnjluence of different compounds on the destruction
vf monosaccharids by sodiumhydroaide and on the inversion
of sucrose by hydrochloric acid IT”. By Dr. H. 1. WATERMAN.
(Communicated by Prof. J. BÖESFKEN).

(Communicated in the meeting of June 30, 1917).

In a previous communication’) it has been proved that in alkalic |
solutions amino acetic acid and « amino propionic acid behave about
as one-basic acids, because they retard the destruction of glucose by
alkali almost as much as an equivalent quantity of hydrochloric acid.

In acidic solution the said amino acids act about as monacidic
alkali since they retard the inversion of sucrose by hydrochloric
acid almost in an equal degree as the equivalent quantity of strong
alkali.

It could be expected that other amino-acids with a greater number
of atoms of carbon should in the same way show in alkalic solution
strong acidic, in acidic solution strong basic properties as well.
Experiment has confirmed this expectation.

The following «-aminoacids were examined :

CH,. CH,. CH(NH,). COOH a-aminobutyrie acid
Molecular weight: 103.

IN Bana ; «-aminoisovaleric acid
re „CH.CH (NH). COOH haalen:

Molecular weight: 117.

CH,

CH.CH, CH(NH,).COOH a-aminoisocaproic acid

CH,. (leucine).
Molecular weight: 131.

Behaviour in alkalic solution.

The just mentioned aminoacids prevent the destruction of glucose
by sodiumhydroxide as is proved by the following. A solution of
50 Gr. glucose in distilled water after being boiled was diluted
to 1 Liter.

1) Chemisch Weekblad, 14, 119 (1917). These Proceedings, Vol. XX, p. 88,
April Th, TOT.

YS an

383

From this solution 40 cm’. was taken with a pipette. I let it flow
into a 50 cm’. flask, added a fixed quantity of the aminoacid in
question to some of the flasks and added at the same time different
volumes of a solution of sodium hydroxide of known strength.

Finally the liquids were diluted with H,O to 50 cm’. and
shaken thoroughly. The thus obtained solutions were placed in a
thermostat with waterjacket (temperature 34°), the temperature of
the liquid in the flasks therefore rising gradually. At the beginning
of the experiment and later from time to time the polarisation and
the intensity of colour of the solutions were determined under
comparable circumstances. The results of these observations are found
in table I. (See table on the following page).

From the results mentioned in this table it follows in the first
place that the aminoacids in question practically do not influence
the polarisation of glucose (polarisation of Nrs. 6, 7 and 8 at
beginning).

After 3'/, hours the polarisation has diminished most in the flasks
that contained the largest quantity of NaOH (Nrs. 5 and 9: + 4°,6 V.).

Although the added number of em’. of the NaOH-solution with
the Nrs. 6, 7 and 8 is equally great as that with the Nrs. 5 and 9,
it follows from the table that the polarisation with 6, 7, and 8 has
diminished only to respectively 6,5, 6,6, and 6,5.

This number is less than that of N°. 3 to which 3 em*. of the
NaQOH-solution had been added. We may therefore conclude that
two milligram molecules of each of the aminoacids compensate the
action of about 2 em’. of 1,06 normal NaOH-solution. The intensity
of colour too of Nrs. 6, 7, and 8 (after + 21 and 6 X 24 hours)
that lies between that of Nrs. 3 and 4 was herewith in agreement.

The «a-aminocompounds of butyric acid, isovaleric acid and
isocaproic acid therefore behave in alkalic solution as about mono-
basic acid.

Behaviour in acidic solution.

The inversion of sucrose by hydrochloric acid was likewise
prevented by the said three a-aminoacids.

From the resuits united in table Ila and 115 especially from the
polarisation at the beginning of the experiments it follows that these
three aminoacids neither influence to an important degree the
polarisation of sucrose nor change polarisation of the solution by
their own optical activity.

Whilst the polarisation after addition of 5 cm’ 1,01 normal hydro-
chloric acid after 16'/, hours (Table Ile) has lowered to respectively

384

TABLE. x1,
Influence of aminobutyric acid, z-aminoisovaleric acid and z-aminoisocaproic acid
on the destruction of glucose by alkali.

Polarisation in

Sess grades VENTZKE eager ae the
SoG 2 dm. tube) ees
0 Added |S) 45 SCE es EN
NO. | Ene At begin- After Aft Ys | After +
sBs) |aingolthe) + 310 |i hours| 0X24
| | if 3” ent) | hours (co | hours
40 cm3. of a sol. | = | |
' containing _ colour- | colour-
! ‚glucose (50Gr. | | ? | 5 | +15) ILA! tess | less
hele SEE ONE U
| Vv |
: pale pale
2 id. 2 | 2 *) +81 yellow | yellow
= |
ot OPE eee s : = gut MAR n z
ES
3 | id. 3 | = 4) +68 yellow | yellow
S| | |
Ka r | 5 | er A TTE erie. 7
i 2 | yellow brown-
vibe id - |= | 5) 198 ‚brown yellow
E Maa a nn
| dee
E |
: | deep
5 id. 5 5 | +100 +46 | yellow-
| | zl brown | brown
REEN [206 milligram{ | 5 | ie ei 5 :
; z-aminobutyric lene | a. in
6 id. sad line on ty OP ae =
ca __grammolecule) _ = |} Ze if 3
ENT 234 milligram 7 es a
. z-aminoisova- oo | | Zi =
ae = leric acid (= 2 BY Wee, at 10,8) 528s eee és
: __imgr.molecule 5 _| SE 3
| 262 milligram | | 2 | 5 | aa
: Ze soca- | | | |
8 id. Kaate id 5 | g | +106 | +65 SrA 3
BS: (=2 mgr. mol.) Ag ics ee ed
lo | deeper |
: | © | deeper
9 id. 5 |Z | +103 | +46 | yellow-
: |i | ; brown: (Urows

+1,2 and +2,5 (Nrs. 4and 7) the addition of 2 milligrammolecules
of «-aminobutyrie acid and qa-aminoisovaleric acid causes that in
presence of the same quantity of hydrochloric acid the polarisation
has diminished only to respectively + 12,2 and + 11,9 (Nrs. 5 and
6). From this results that the added quantities of these amino acids

1) Between the dilution to 50 cm’ and the determination of polarisation of
course some time passes. .

2) The polarisation was not determined; the results obtained would lie be-
tween that of N°. 1 (+ 11,5) and those of Nrs. 5 and 9 + (10,0 and + 10,3) and
would diminish gradually from N°, 2 to N°. 4 (Compare the results obtained before).

385

compensate the action of somewhat less than 2 cm* of normal
hydrochloric acid, which is demonstrated too by the observations
made after 24, 40—41 and 72 hours.

Therefore they act about as monacidic alkali. From the observation
after + 89 hours it follows that finally the same end-situation is
reached so that the stated influence of the amino acids cannot be ascribed
to an accidental influence on the polarisation of fructose or glucose.

Oss
v ee eis Te} ae pi ead
=| aas e) Ee e) o fe)
(a: Wo IE gee ee a a a | a
fay |) onc Ve — — ~~ ~~ | he
5 En jes a ‘ i | ,
mms << Dn | mn. NL = | = | =
E | + | = Ne} 15 | it) pp hae
_ _ _— —_ ~_ —_—
be: z fon AA | Riches |
al AEN ae RE 7 Putas
SR ole foes al em al Gein q
A s Jes 35l 9 o 36 3 o
oO SEE tn = x en
sh EES (Sora a |A NS af
to) Zat | td es In Ik |
a A So = | = oct = ———— (Gee Pe.
‘= > | «a ere | En
g | Ss > Sa Si ze i=)
ABS | RU ea Feed eye an IES
SIS RM tes Ne eA eI es ende
= Je Gea 1 | fi | Em SI
= de | a
> a 2 es 60 nex | bel | — | nen
a 5 3 [855 o o OP if), 0
=| „9 lees = = Ne) =
w EI DO, CN a N Nn
wn | aS ts, ot ~~" ~—
je) BS. Bia [FS Dee ON et Se So Er Ane &
ENA id Soke Ge TA En
= n a) = eN al 5 al
a | 4 =f st se) + a) >
apy) | BS ia ee SN A el es ma tin co Se a
& wy © OL * oS Tey = NL
eis lee ens =?) oy a le) le) le) fe)
Ss < b= + + + a St bal +
—
SN, | ies Nn) el ile Ae Nee eee eee eye ia
: > : e ;
mos Diluted to 100 cm’ and placed in
SN thermostat (temp. 34°)
athe Number of |
je) cm? 1,01 | |
FE — |l Norm. HCI- > on RE nz eed i! oF
3 sol. added |
Ede rh eee a
>|
1S) ous)
4 m5 2
5 ERE
kn
Ss | = SO i
ae || T SO
a
3 ke} a
= < AGE.
le} Eo hd
iS a oN
— to .2
= I Eed Shar En Be
5 5
— | 215
© 59
Se
3 3o2
= Od
se nn
= | £0
= | Gea &
=e | Side = — = 3 xg 3
ofu
©
2 OS
=
oes
| pl
| REA
2

No
1
2
3
4
5
6
7

386

EAB Ex iia.
\ Influence of x-aminobutyric acid and zx-aminoisovaleric acid on the inversion of sucrose by hydrochloric acid.
| | Eas | Polarisation in grades VENTZKE (2 dm tube)
| | ES ern - =: =
oS5so | | | |
N°. Added Beag | Ib ee After | After | After | After After
| | ESS | ane + 161% hours | + 24 hours | 40—41 hours -+- 72 hours + 89 hours
| | | 4 | | | | | |
: | | | | Temp. of! ‘Temp. of! Temp. of! ‘Temp. of| ‘Temp. of
1 | roe eb 0 | 4 49,6 the pola- the pola-! the pola- the pola- the pola-
Se sor BON EMS. | : risation | _risation | risation | risation | risation
meee ; | LE liquid | ‚liquid _ liquid liquid ‚liquid
| 2 | | | | | |
2 id. | a | 9 + 49,3 H13,7 (23°) |+6,0 | (25°,5) |—5,9 | (24°,5) is 13,4, (232,5) - 14,5) (23°,5)
oA > é eee Aig Pome eter | : j | Rd,
sole | | | | | |
3 id. 4 | Se + 48,9|+ 6,7 | (23°) 7—0,5 | (259,5) re 10,3) (25°) — 14,7 (23°,5) |— 14,8 (249)
en | | | } |
| | 5e | en Ne. Ee enb |
40 id. | bee | oF I+ 48,4\-+-1,2 | (23°) (— 5,0 | (25°,5) |— 12,6) (259 |— 15,1; (239,5) |— 15,1} (249)
| | KS | |
(206 mgr. z-amino- ens RE reel | | | |
5 id. ‘butyric acid(=2 5 © 8 [+ 48,7/+ 12,2 +45 | Eel ld 48)
Ee es __\milligr.molecule),_ ROER | | | =<
234 mgr. z-amino-| |p| | | | | | | | | |
6 id. isovaleric acid | 5 SS |+ 48,8/+ 11,9] (23°) |+4,3 | (25°) j—7,2 | (24°,5) |— 14,0; (23°) |— 14,8) (23°,5)
| (= 2 milligr.mol.) eee | | | | | a
ae ES a : | |
7 id. 5 | 2 + 48,6 + 2,5 | 4,9") — 12,5 — 15,2) — 15,4 (23°)
es | | | ETR | | | 7
8 id. | RE | ee 49,0) + 14,8} (229,5) |-+7,0 | (24°) |—55 | (24°) |—13,5) (23°) |— 14,4)
| | | | |

387

TAB LE. lila,
Influence of asparagine, glutamic acid and tyrosine on the destruction of glucose

by alkali.
ry (os le Eg | Polarisation in_
| ee, ek: grades VENTZKE Colour of
N°. Added B45 (2 dm tube) __|the liquid
3 EL E At | After | after +
3 = 50 begin- _ + 3!/5 | 43 hours
| A ning hours
40 cm3 of a | | | | Bte
1 | sol. containing 0 | + 11,3 | + 11,2 eS
(50Gr. gluc. p.L.')) EE hy Her 4 le iS, Mie
: | | pale
2 id. | 2 + 10,5 | +8,0 yellow
3 | id. 3 + 10,3 | + 6,3 yellow
en a5 Oe =: rh sea : ded shane) DD
| . brown-
4 | id. 4 + 10,2 | +52 | Veiiow
| | +96 | + 4,4 brown

108 IE 5,0 brown

(temp. 34°)

| [141 milligram aspa-|

6 | id. ragine = + 1,06| 5
I ah _ milligrammolecule
157 milligram glu- |

hal id. [patie acid. ==. de 5 |

1,06 milligrammol.
| 193 milligram tyro-|
8, id. | sine = + 1,06 5
milligrammolecule |

5 | | + 9,7 + 4,4 brown

+103 + 6,3 2)) yellow

"498 | +59 | yellow

Diluted to 50 cm3 and placed in thermostat

|
|
|

TA.B-L,E We

Influence of tyrosine on the destruction of glucose by alkali.

| de Ee Polarisation in
| ZOL grades Vee Colour
i Beg dm tube) |
N | | Added 2% Z 5 KE] After after —+
| iS 50 begin- +5 ‘ 24 hours
2 5°” ning | hours |
40 cm3 of Bel 6.
1 {a sol. containing 3 5 =| 10,7 | +54 yellow
50 Gr. gluc. p. L.') PAs a=,
; ia 835 oe
; WO brown-
g id. 5 28E +108 | +30) ow
5 |S zin en tlg
192 milligram ZE =| rot yellow,
3 id. tyrosine (= 1,06 Ber Es ater +50 | ENE
milligrammolecule)|_ A = mined, No. 1

1) This solution was boiled for a moment and afterwards cooled till the tempe-
rature of the room was reached.
2) All the glutamic acid is dissolved.

388

From Table II’ it follows in an analogous way that 2 milligram-
molecules of leucin («-aminoiso-caproie acid) neutralizes the action of
+ 1'/, em? of normal hydrochloric acid. In acidic solution leucine
behaves as + °/, acidic alkali.

Then I set myself to the examination of three more complicated
compounds viz.

COOH CH (NH). CH. CO(NH) Asparagine
Molecular weight = 132 (mono-amide of amino-succinic acid)
COOH ..CH(NH;) .CH,.CH, . COOH ~Glutamie acid
Molecular weight —= 147 (a-aminoglutarie acid)
HO .C,H,. CH, . CH(NH,) . COOH Tyrosine
Molecular weight = 181 (p. hydroxyphenylalanine)

I observed that with my experiments acetamide CH, . CO (NH)
and urea CO(NH,), behaved in alkalie and in acidic solution as
practically neutral.

From this appears again a contrast between the acid amides and
the amino acids; I observed before this contrast in another direction. *)

Furthermore it has been proved in a previous communication °) that in
alkalie solution phenol acts about as a one-basic acid, whilst this
compound practically has no influence on the inversion of sucrose
by hydrochloric acid.

In agreement with these results it could be expected that in
alkalic solution asparagine possessing the carboxyl group should
behave as a one basic acid. For the presence of the amino group
asparagine should behave as monacidie alkali in acidic solutions.
Glutamic acid, which compound possesses two carboxyl-groups in
presence of sodium hydroxide should act as a two-basie acid; in
acidic solution it should behave as monacidic alkali (NH,-group).
Finally tyrosine for the presence of the phenolic hydroxyl-group and
the carboxyl-group in alkalic solution should be two basic acid; in
acidic solution the (NH,) group sbould render it monacidic alkali.
These predictions were confirmed by the experiments (Table IIIe,
III? and IV).

From the experiments mentioned in Table III* it follows that
under the circumstances described asparagine acts as monobasic acid.

1,06 milligrammolecule of asparagine neutralizes the action of about
1 em? 1,06 normal NaOH-solution, as results from the polarisation
after 3'/, hours.

ee, WATERMAN, Die Stickstoffnahrung der Presshefe, Folia microbiologica.
(Holländische Beiträge zur gesamten Mikrobiologie) 2, 173 (1913).

3) These Proceedings, Vol. XX, 88 (April 27, 1917).

TABLE IV.

Influence of asparagine, tyrosine and glutamic acid on the inversion of sucrose by hydrochloric acid.

1) All the glutamic acid is dissolved.

| = = E
| os Polarisation in grades VENTZKE (2 dM. tube)
Shit
N°, Added 22 Es ate = —— aie a
Eide begin- (After + 16'/, hours, After + 24 hours | After 40—41 hours | After + 72 hours | After + 89 hours
eef ning
50 cM3 of a solution | | Temp. of | | Temp. of Temp. of | Temp. of | | Temp. of
| which contains i the polari- ‘the polari- the polari- the polari- the polari-
1 . 0 + 49,6 ; 7 ; 5 :
| 130 G. sucrose p. sation | ~ sation sation sation sation
Ie (0 cM3 liquid | liquid liquid | liquid | | liquid
2 id. 3 | | + 49,3 | +13,7 (23°) + 6,0 | (25°,5) — 5,9 (242,5) — 13,4 | (239,5) — 14,5 | (239,5)
| | | |
=| | ae ae 5 st =
3 id. 4 + 48,9 | + 6,1 | (23°) — 05 | (259.5) | —103 | (25°) — 147 | (2395) | —148 | (24°)
= | | c ef Coe ee | aes kee:
|
4 id. 5 | + 48,4 | + 1,2 | (23°) — 5,0 | (25°,0) — 12,6 | (25°) — 15,1 (23°,5) — 15,1 (24°)
; a | ek
5 id. eae Se 5 5 | +49,2 | +10,5 | (23°) + 2,1 | (2595) | — 84 | (249,5) | —13,9 | (2395) | —145 | (24°)
a _—— — = — =
| 181 mgr. tyrosine a 5 |
6 id. = 1 milligrammol. 5 | = | + 48,2 | + 6,0 — 13 — 10,6 — 14.9
| Se | = = =h = nn = = = =: =
=a Sl! nD |
7 id 5 | B | +486 |+ 25 — 49 — 12,5 — 15,2 — 15,4 | (23°)
Ie '
—| 2 ——____ — pa ee ---
8 id, 3 E +490 | +148 | (222,5) + 7,0 | (24°) °: — 55 | (24°) — 13,5 | (23°) — 14,4
= = hel mt — LI EE
eten jrAt |
| = | begin- | After + 34 hours | After + 20 hours | After + 137 hours
| | ning EE | a le x zal he I Pa)
| | = > | | Temp. of Temp. of Temp. of
BS the polari-| the polari- ithe polari-
l= sation | sation | sation
| 3 | liquid liquid liquid |
EE sle | 7 = =
LS |
9 id. 0 | 5 | +496
lea
£ z= ns e = aie ES =a! 4 es
10 id 3 | 5 | +49,2| +434 | (2495) | +106 | (2325) |—149 | (24°)
| o
— as ee Es = = = : => = = “e
il id. 4 + 49,4 | 441,7 | (2495) | + 4,0] (249,5) | —14,9 | (24°,5)
|
L alle = L | —— ae
12 id 5 +49,0 | +39,4 | (25°) — 1,0} (24°) | —15,2| (24°)
. | | | | pe
= 7 io | |
147 mgr. glutamic | A
13 id acid = 1 milligr.mol. 5 not yet dissolved + 3,0') — 148
14 ~ id 5 +491 | + 39,4 = 0:90} oil
15 id 3 + 49,2 | + 42,7 + 10,6 | (23°75) | —15,1 | (249)

a. TABLE VI. Influence of aniline and pyridine on the inversion of sucrose by hydrochloric acid.
— ne ne mmm
; - —
| | Number of | | Polarisation in grades VENTZKE (2 d.M. tube)
M*. 1,01
N° Added Se == : zi ee
| \Normal HCI- AE be- ; 7
| \solut. added | ginning After 5%/, hours After 815 hours After 24 hours |After 4 >< 24 hours
1 1
F temp. of | temp. of | temp. of | temp. of
|50c.M3. of a solution Á att .
EE ‘ | +498) +494 edion | +495 (Aeon | con neen
r. sucrose p, L. i = | | liquid | liquid liquid liquid
| | Sei were
2 id. 2 +496 | 444,9 | | 41,2 | +217 — 12,2 | (16°,5)
3 id. } 4 | | +495| +407 +343 | (189,5) | + 58| (1825) | —163'| (170,5)
| ER Se salt je x = L = i a - Ss
| | | : ]
4 id. 6 | +495 | +363 | (199°,5) + 27,7 | (189,5) | — 4,2 | (189,5) — 16,8 | (17°,5)
| = | = = a ed = | =
5 id. 10 +495 | +212 | +155 182 | —16,7| (17°5)
6 | id. 1,72 Gram aniline 10 +495) +495 | (199,5) | +495 | (189) + 49,3
je = = a en IE r == 3
ah id. | 10 a +492 | +255} (1995) | +147 | (18°) | —12,7| (1895)
b. | = |\nedieeine | After 18 hours After 24 hours After 48 hours
L = _ v Ee at he | —
= „of i temp. -
1| id. 0 =| 440,7 | 4494 [polarisation iste ee ene
3 » 3 5 as ad liquid liquid En liquid = =<
©
2 | id. 2 E +498 | +271) +21,5 e
| B EN ov | " al: u en
3 | id. 4 = | +494) 4 126 | (16°) | + 4,9 — 8,7 (19°)
u = — — = he] — —— — 2
4) id. | 6 3 +492 | + 14) (179) — 52} (16°) | —136 | (199,5)
| a Am = a = We Ee as = a Dn
5 id. 10 = +489 | — 98) (179.5) | —136 | (16°) | —158 | (18°,5)
6 id, 0,50 Gram aniline 10 5 +494 | + 81 | (179) + 12 | (16°) — 11,5 | (182,5)
= = j = — im len Fer Fe = Sl =
1 | id. 0,64 Gram aniline 10 5 +492 | +178 | (169,5) | +11,2 | (16°) — 48 | (19°)
8 | id 10 S| +488| — 84| (16°5) | —12,7] (169) | —159 | (19°)
| Ln
Eeen Ee = =
c. im bepituing After 6 hours After 21/2 hours | After a long time
; "|temp. of temp. - 1p. of the .
1 id. 0 4.498 | +49,7 |poisrisation| +. 49,6 |poistisation| -+ 49,9. [polarisation
n # | liquid ¥ liquid : liquid
2 id, / 2,1 + 49,5 | + 42,0 + 14,6 | (20°,5) | — 82 | (21°)
3 id. | 4 +494 | + 35,3 | — 1,7 | (2095) | —14,6 | (20°,5) |
4 “id. 6 449,4 | 428,0 | — 95 | (219) | —147 | (20°5) |
= — | z / 220° ï
5 id. 10 | + 49,5 | + 18,0 | — 14,2 | (20°,5) | — 15,2} (20°)
: |
| |
6 id. 0,494 Gr. pyridine 10 +495 | + 35,3 | | — 08 | (21°) | —145| (199,5) | |
: |
hi
ij id. 10 | 449,2 | +168 — 149 | (20°,5) | —15,7 | (19°) |

a

389

Much stronger is the defending influence of 1,06 milligram-
molecule glutamic acid and of 1,06 milligrammolecule tyrosine,
which compensate the action of about 2 cm° 1,06 normal NaOH-
solution (polarisation after 3'/, hours).

Glutamie acid acts just like tyrosine as almost two basic acid,
which was once more confirmed for the latter compound by the
experiment described in table 1115.

It must be remarked that the glutamic acid (N°. 7) did not quite
dissolve even after being shaken repeatedly.

Nevertheless + 14 ecm’ of the clear solution were used at the
beginning for the determinion of polarisation.

In the remaining alkalic liquid all the glutamic acid was dissolved
after some time (within 3'/, hours). The tyrosine (N°. 8, table Illa
and N°. 3, table 1115) was dissolved but little. At the addition of
the NaOH-solution after being shaken it was quite dissolved. *)

The addition of hydrochloric acid too causes the rapid solution
of tyrosine.

This compound in this regard resembles substances such as zinc-
hydroxide and aluminiumhydroxide.

In acidic solution the glutamic acid dissolved but gradually. 1
did not determine polarisation before all had dissolved. Although
the glutamic acid therefore could not be active to a certain extent
at the beginning of the experiments, from the results obtained it can
be concluded with rather great certainty that glutamic acid in
acidic solution behaves as monacidic alkali.

Tyrosine too (table IV, N°. 6) behaves in acidic solution as
monacidie alkali; 1 milligrammolecule compensates the action of
about 1 em? of normal hydrochloric acid.

Asparagine acts as °*/,-acidic alkali; 2 milligrammolecules com-
pensate the invertive action of about 1,5 cm* of normal hydrochloric
acid (Table IV, N°. 5).

Afterwards aniline and pyridine were subjected to research, in
how far these compounds influence the destruction of glucose by
alkali and the inversion of sucrose by hydrochloric acid.

GH NH. Aniline.
Molecular weight = 93
C,H,N. Pyridine.

Molecular weight = 79

1) In heating the solution to the boiling point but without the addition of NaOH
the tyrosine dissolved almost quite, but in cooling till the ordinary temperature
was reached an important quantity crystallised, which was dissolved rapidly at
the addition of NaOH.

The aniline present in
drying distilled in the ordinary way.

and after
was 180°.

390

the laboratory

was distilled with steam

The boiling point.

The pyridine of the laboratory was distilled in fractions. The
fraction which boiled between 115° and 117° was used for the research.

It could be expected that in alkalie solution
neutral,

should behave
monacidic base.

in

acidic

both compounds

solution they should act as

The referential experiments which are mentioned in table V and
VI have contirmed this expectation.

Ti AcB LES Ve
Influence of aniline and pyridine on the destruction of glucose by alkali.
a. Aniline.
er ay ke mt.) | Polarisation in Fa
tek grades VENTZKE
| 53 (2 dm tube) | Colour after
NO, Added Ee ‘3 Aba fu Riter
3 E EE begin- | #3 | + 24 hours
| A ning hours |
80 cms of a solution,
1 | which contains 0 -+ 11,0 | + 11,1 colourless
a 50 Gr.gluc. p. Liter ')| ae 3 ¥ MERA
3 u
2 id. 10 E ee 99 | 4 An
| £ di brownyellow
3 id. 158 Gra | 40 de sene ann
| Ws deeper)
| ES
4 id. 10 | EE + 96 | + 7,2 PRE.
mn In = = ie == wiles ee EE
SE | At | After
b. Pyridine. | 2 & | begin- | + 54 |
5 | ming | hours |
oS
le) id. 10 = 102 Le 3,4 \ brownyellow
en | | 2 Che Ce ea
| : 1,66 Gram ss
2 id. pyridine 10 | £ +102 + 3,4 | brownyellow
4 = > emeente i} _ rn pave a | — —
3 | id. 10 +108 ig 3,5 | brownyellow

From table Vla it follows that the retarding. power of aniline on
the inversion is very great.
proves that practically no sucrose has been

The polarisation

inverted.

') The solution was boiled for a short time and afterwards cooled down to the

ordinary temperature.

3914

Herewith the fact was in agreement that after 24 hours the liquid
of N°. 6 (Vla) did not possess any reducing power on FrHLING’s
solution. ie

A iodometrical determination of invert-sugar proved that less than
60 milligrams of invert-sugar was present per 100 ecm’.

If the liquid of N°. 6 (Vla) after 24 hours is boiled for some
time the reducing power on Feurine’s solution becomes greater.
Boiling with an extra quantity of strong hydrochloric acid gives a
liquid that after being neutralized with alkali possesses a strong
reducing action on FeHriNG’s solution.

From table V1é it follows that 500 milligrams and 640 milligrams
of aniline compensate the action of respectively + 5 cm’. and 7 cm’.
of normal hydrochloric acid, which proves that in acidic solution
aniline behaves as monoacidic alkali.

From table Vic it can further be concluded tbat 0.494 gr. of
pyridine compensates about 6 ecm° of normal hydrochloric acid,
pyridine acts therefore about as monacidic alkali in acidic solution.

The basie and acidic character of the compounds described in the
above is in accordance with the constitution-formula, which nowadays
are assumed for these compounds.

The method of research described can help to find a better con-
stitution-formula in cases where the said accordance does not exist *).

For the rest one may be astonished a little by the strong neutra- '
lizing action against hydrochloric acid on the one hand, sodium-
hydroxide on the other hand of compounds being generally known
as. feeble acids or basic substances.

Frequently we can make good use of this neutralizing action of
substances with a but feebly acidic or basic character in watery solution
in order to compensate the influence of strong alkali or strong acid.

In many experiments in the laboratory as well as in technical
processes we have often to struggle with the formation of strong
basic substances or strong acids. In such cases we can compensate
the action of the strong alkali or acid by the addition of efficient
amphoter or weak electrolytes.

Dordrecht, June 1917.

1) A first example of this was given in the preceding communication with the
betain.

Chemistry. — “On Black Phosphorus’. II. By Prof. A. Smits,
G. Meyer, and R. Tu. Beek. (Communicated by Prof.
S. HoOGEWERFF).

(Communicated in the meeting of June 30, 1917).

As was already communicated before, our researches carried out
with Bripeman’s black phosphorus have corroborated the supposition
that this new form of phosphorus is always metastable under the
vapour pressure.

That this is the case at the triple point temperature of the black
phosphorus, is beyond doubt, for it appeared that the black phos-
phorus melts + 2° lower than the violet phosphorus, hence at
+ 587.5°. The vapour tension determinations, however, gave results
which, though this did not seem probable, pointed to the possibility
that below + 560° the black phosphorus, and above that tempera-
ture the violet phosphorus would possess the smallest vapour tension,
or in other words that there exists a transition point between black
and violet P at 560°. This conclusion seemed, however, by no
means necessary, as the results could also be explained by a too
slow establishment of the internal equilibrium at temperatures under
df 008:

To ascertain whether the black phosphorus under 568° is really
stable, the following experiments were made. Equal quantities of
violet and black P were heated with 1 °/, Iodine in a tube of glass
that melts with difficulty in vacuum 13 days in succession in a bath
of KNO,—NaNO, at + 480°.

Then the tube was quickly taken out of the bath, the contents
extracted with CS, etc. and then the specifie weight is determined
according to the suspension method. It then appeared that almost
everything had been changed into a substance with a specific weight
2.3, some particles still possessing the spec. weight 2.7. Accordingly
it already follows from this that 560° is no transition temperature,
for at 480° the black P was still converted into the violet phos-
phorus and was therefore metastable *).

In the following experiment the proportion between black and

iy Without contact with violet P we have not been able to convert the black
P into violet P. Even after 4 hours’ heating of black P with 1°/) IT at 580° not
the slightest change had occurred.

393

violet phosphorus was chosen differently, viz. 0.9 black P and 0.1
violet P, and 1°/, I was again added to this mixture. The result
was that after 16 days’ heating at 450° practically everything lad
obtained a specific weight of 2.4, from which therefore in agreement
with the result of the preceding experiment, it follows that the
violet P is metastable at 450° under the vapour pressure.

It was now the question whether this could also still be demon-
strated at lower temperature.

Two tubes were taken of glass that does not melt easily ; one
filled with */, violet P + '/, black P, and the other with */, violet
+ ‘*/, black P. Again 1 °/, I was added to both mixtures. After
being pumped empty and fused off, the tubes were heated for 3'/,
months at 380°. When then the contents of the two tubes was
examined, it appeared that the mass, which at first consisted for
*/, of violet P and for '/, of black P, had been quite converted to
the violet modification, whereas the mixture that at first consisted
for //, of violet, and for */, of black P, had not appreciably changed.

To what it is owing that when the black phosphorus was greatly
in the minority, it was entirely converted to violet phosphorus,
whereas in the other case nothing could be perceived of a conversion,
cannot yet be stated with certainty; the one positive result, however
proves already that also at 380° the black modification is the meta-
stable one under the vapour pressure, and this makes the view, set
forth in the preceding communication '), greatly gain in probability.

Amsterdam, April 10, 1917. Anorg. Chem. Laboratory

of the University.

1) Proc. 18, 992 (1915).

26
Proceedings Royal Acad. Amsterdam. Vol. XX.

Chemistry. — “On the Electro-Chemical Behaviour of Nickel.” By
Prof. A. Smits and C. A. LoBry pe BRUYN. (Communicated
by Prof. S. HOOGEWERFF).

(Communicated in the meeting of June 30, 1917).

1. Nickel is a metal that assumes internal equilibrium exceedingly
slowly, and can therefore very easily be disturbed. When nickel,
immersed in a solution of NiSO, or Ni(NO,), is made anode, resp.
cathode, a very strong polarisation is found already at very small
current densities. Also when attacked by chemical reagents as HNO,,
H,SO, etc. nickel is very easily disturbed in noble direction.

In a solution of NiSO,, which is in contact with the atmosphere,
nickel does not assume the equilibrium potential, simply because
the attack to which Nickel is subjected under these circumstances
by the air oxygen dissolved in the electrolyte, is sufficient to give
rise to a pretty great disturbance of the internal equilibrium in the
metal surface. That these small quantities of oxygen exert so great
an influence is owing to this that the oxygen is at the same time
a negative catalyst for the setting in of the internal equilibrium.
What is remarkable is this that not only oxygen but also hydrogen
has appeared to be a negative catalyst for this process, so that
nickel does not assume the equilibrium potential in a solution under
a hydrogen atmosphere either.

2. These circumstances being unknown, the nickel potential has
always been measured in a hydrogen atmosphere or in air and it
was thought that in this way the equilibrium potential of this metal
was measured. Only ScHoca has measured the nickel potential also
in vacuum, and found that it differs from that which is found in
a hydrogen-atmosphere.

Led by the new considerations about the internal state and the
chemical and electromotive behaviour of metals, we have made some
experiments with the result that the behaviour of nickel, which is
still much more remarkable than we thought, can be explained in
an exceedingly simple way.

To show this it is necessary to discuss the condition for the
hydrogen generation from a solvent containing hydrogen ions by
means of a metal (here nickel).

395

As was already stated before, in this case we have to do with
the two following equilibria (in the liquid)
Mz Nin + 20

hd

and
AH, 2 2H + 26.

When the electron concentration. of the nickel equilibrium is
greater than that of the hydrogen equilibrium corresponding to a
hydrogen pressure of an atmosphere, hydrogen generation will have
to be found.

We have:

L yi = (Ni) (0)? and = Ly,= (A)? (0)
from which follows:

LM: Ly
Gyr See and ONE ai :
( tne (Ni) ( ) Ly, (A)
so that the condition for the H,-generation is:
Lyi Lp,
Mi) TY
or
4 (Ni)
LM > L 2
ay eee
Pati (Ne) tj then.
ie
Lyi :
N > ry
or
Lu,
A: 2 2
( ) Pa.
Now

Mi 102% — 48 and: +. Zing LON 48
hence (H:) must be > 10-°. |
When therefore the hydrogen-ion concentration is more than 10’
times the concentration in pure water, the metal nickel must give
hydrogen generation, which, however, is not the case.
How to account for this will appear from what follows.
For this purpose we consider here the equilibrium that we have
to do with in the system Ni — electrolyte, viz. :
Nis2 Nig + 208
Me Me + ob
Now we know that hydrogen generation occurs when the electrons
combine with the hydrogen-ions to hydrogen molecules. These
26%

396

electrons would, therefore, have to be withdrawn from the metal
equilibrium, here therefore from the nickel equilibrium, which would
cause this equilibrium to be disturbed. If nickel behaved normally,
this disturbance would be immediately negatived, and it is easy to
see in what way this disturbance would be annulled.

This cannot take place through the reaction:

Nip— Nir + 207
for, the concentration (.V2;) being very small, this is a reaction
that can produce but exceedingly few Nz-ions and electrons per
second. What would have to happen is this: the electrons and
nickel ions would have to go into solution from the metal and in
the metal: the reaction

Ng — Nig + 20s
would have to take place.

In this case the potential difference during the hydrogen generation
would be exclusively determined by the Mi-ion concentration
prevailing at the moment.

The metal nickel however, behaves quite differently.

As was already observed, nickel is exceedingly inert, Le. its
internal equilibrium is very slowly established, and to this comes
that the hydrogen, which dissolves in the nickel to a small degree,
greatly retards the setting in of internal equilibrium.

Now we may imagine the phenomenon to be like this: when
electrons have been withdrawn from the solution by the reaction:

oH + 26> H,
and in consequence of this ions and electrons have gone into
solution from the metal, the disturbance of the internal equilibrium
is not abolished any longer, and the potential difference z-electro-
lyte has consequently become less negative.

For nickel, which is so exceedingly inert under the influence of
the dissolved negative catalyst, hydrogen, the supposition suggests
itself that this process can continue till the electron. concentration
of the nickel equilibrium in the electrolyte has become equal to
that of the hydrogen equilibrium.

It is self-evident that it is supposed here that the electrolyte is
perfectly free from oxygen and that the result in question is to be
expected when we work eg. in vacuum or in a hydrogen
atmoshere. We shall now examine. what the potential difference
nickel-electrolyte has become in this case.

For this purpose we return to the derivation of the equation of
electrons for the potential difference. The condition of equilibrium

397

for the equilibrium between the electrons in the metal and in the
coexisting electrolyte is:

in which (ta,), 9 and(ug,),_ indicate the molecular thermodynamic

potentials of the electrons in the metal and in the coexisting elec-
trolyte for the case that the potential difference = 0, Vs and Vy,
being the electric potentials of the metal and of the electrolyte, so
that FV gs and FV; denote the molecular electrical potentials of
the electrons in these two phases.

It now follows from this equation that when we omit the index

== 0

RN Lee (2)
F

As it was our purpose to derive an equation for the potential
difference in which not only the concentration of the electrons in
the electrolyte occurs, but also that in the metal, the splitting up
of the molecular thermodynamic potential into a concentration-free
term and into a concentration member, viz. :

es a RT WC AANEEN ee aT
has been applied both to the electron in the electrolyte and to the
electron in the metal. We then get:
Wa. a Mo, + RT In (6s) — RT In (Or)

A= | 5 eh (EM)

If we now put:
6. — wo, = RT In Ko sienna eeen)
we get:

nij adik Sas)
jd (07)
the electron equation, derived by Smits and AreN, tor the potential
difference.
When we now again return to equation (5), and add & 7'/nôs to

the two members, we get:
Wog + RT ln (3) = RT In K'4 (03) + lon» 2

(6)

or
RT In Ko (Os) = wg — Wo, Man AA B re ate EAN)

We get for the potential difference of 2 different metals:

DR l : yl Ore ae eS
‘a (Or) ©).
and
5 ns = AAS 5 . e . . . .
vs (Or)

From which follows that the electromotive force of a circuit con-
sisting of these two metals immersed in the corresponding salt
solutions is:

! ~

rr. doe Os) ARES (B)

; A in — — In te eae ee
yi Ks, (Os) I (Or,

Let us now apply the just found relation (8), then we get:

Mes, au B MS Er el 38 eh RT x (07)

F F F (Ors)

Now Wo, is the same for different solutions with the same sol-

A,—A,= (12)

vent, so that we get:

Mo — U6, RT 7]
ant [NS = Si pa ln ( Li) 5 ze he (13)
F PE)

The term:

represents the volta-effect, viz.: the potential difference that appears
when the two metals are brought in contact in dry condition. Now
we know that this potential difference is very small, so that this
equation (18) tells us that when the electron concentrations (47,)
and (@7,) have become equal, A, — A, will be very small or zero,
so that then
Ar

will hold in first approximation.

This result tells us, therefore, that finally the potential difference
nickel-electrolyte will have become equal to the potential difference
hydrogen-electrolyte. The experiment was, in fact, entirely in accord-
ance with this.

3. An NiSO,-solution, in which a slight cloudiness had been
brought about by the addition of a little NaOH, to make the solu-
tion as little acid as possible, was brought in a vessel with 5 tubes.
These were closed by means of rubber stoppers, through which were
led two nickel electrodes [very pure KAHLBAUM Ni-wire 3 mm. thick,

399

fastened in a glass tube by means of sealing-wax], a platinized
Platinum electrode, which could be immersed in the liquid just
before the measurement, a bevel, which was in connection with
the calomel electrode, and a supply and exit tube for the hydrogen.
In the middle tube there was placed a stirrer with mercury closure
to make stirring possible if necessary. Very pure hydrogen which
was obtained by leading the bydrogen from a cylinder or from an
electrolytic hydrogen apparatus (with nickel electrodes) through a
glass tube with Pt-asbestos, which was heated up to + 500° in
a furnace, and then through two blown washing bottles with a
suspension of Fe(OH),, was first led through and later over the
NiSO, solution, while from time to time the potential differences of
the different electrodes were measured.

The result was:

Mi— 0.640 V | electrolyte 2,5 NiSO,; potential
Pt — 0640 V \ with respect to 17 calomel-electrode.

As follows from Wismore’s') calculation and Scuocn’s determi-
nations*), as also from determinations made by us, the equilibrium
potential of Ni lies at + —0,48 V with respect to 1n calomel
electrode. Hence the above found nickel potential is not the equilibrium
potential, but the potential of a state disturbed superficially in the
base direction. We see here that the nickel has assumed the same
potential as the H,-electrode, which is in agreement with the above
given theoretical derivation.

When we consider the solutions of NiSO, with increasing H-ion
concentration by continually adding more H,SO,, we see that the
equality continues to exist also for other values of the H-ion con-
centration :

Ni Pi
— 0.640 V — 0.640 V
ENDE, — 0,517 „
=) 0.350, ee

On measurement in hydrogen atmosphere the found Ni-potential is
accordingly quite dependent on the H-ion concentration, and always
equal to the H, potential.

Before going on we will anticipate and mention already here that
according to the here given theoretical considerations the potential
of the unary nickel, or the nickel in internal equilibrium,
can only be measured in a solution in which the electron concen-

1) Z. phys. Chem. 35 [1900] 291.
») Amer. Chem. J. 41 [1909] 208,

400

tration of the hydrogen equilibrium is smaller than that of the
nickel equilibrium. We shall, therefore, have to make the H-ion
concentration as small as possible by the addition of a base. In this
case a few nickel ions and electrons will go into solution, and, no
hydrogen separating on the metal, the internal equilibrium can set
in. In the determination of the equilibrium potential, however, we
may have no hydrogen atmosphere, for hydrogen of a pressure of
1 atmosphere would render the electron concentration of the hydrogen
equilibrium too large. Nor may we have an oxygen atmosphere,
hecause when there is oxygen present in the solution, the equilibrium
0,+2H,0+4024 0H’
exists, in which case the electron concentration is much smaller than
would correspond with the unary nickel equilibrium. The conse-
quence is then that electrons are withdrawn from the nickel equi-
librium, and the nickel sends ions and electrons into solution. The.
nickel is therefore attacked, and it results from the inertia of the
nickel, together with the negative catalytical action of the dissolved
oxygen that this disturbance is not negatived, and the nickel is super-
ficially in such an ennobled condition that the corresponding electron
concentration is in agreement with the electron concentration of
the oxygen equilibrium. Accordingly the potential of Ni with respect
to a nickel salt solution measured in contact with the air, hes at a
much less negative value than would correspond with the equilibrium
potential.

It is therefore clear that to sind the unary equilibrium potential
of nickel, we shall have to work in u solution, in which for
(Ni°)=1 the H-ion concentration is <10-5, and in an atmos-
phere free from H, and O, or in vacuum.

Thus we really easily find the unary equilibrium potential.

ScHocH') and in imitation of him later on ScHILpBAcH?) are the
only investigators who as far as we know, have also worked in
vacuum. They have, however, not succeeded in finding the true
explanation of the disturbing influence of H, and Q,.

In a determination of the equilibrium potential in a 1 N . NiSO,-
solution boiled out in vacuum, the following values were found:

Potential _—0453V —0477V.—0480V —0.480V
of the nickel: after4hours after 47 h. after 52 h. after 70 h.

Then hydrogen was led through the same solution, and the H,-
potential was measured. After 48 hours we found :

t) loc. cit.
4) Z. f. Electr. Chem. 22 [1910] 977,

401

Nita 03680. 7
Pt (H,) — 0,640 V

and the nickel electrode had therefore again assumed the H,-potential.
That in this measurement the H-ion concentration must be smaller
than 10-3 follows from an analogous determination with an NiSO,-
solution boiled out in vacuum and acidified with H,SO, as electrolyte.
The Ni-potential was now constant at — 0,317 V immediately
after the boiling.
On conduction of H, through the same solution — 0,315 V was
found later for the H,-potential.

4. When we consider the different determinations of the equili-
brium potential of Ni, the following facts are worth noticing :

Neumann ') finds — 0,588 V for electrolytic Ni, hence Ni charged
with H, and disturbed towards the base side. He does not speak
about the atmosphere in which the determination has been made.

PFANHAUSER*) and Siemens’), who worked in the same way as
Neumann, found values deviating comparatively little from the
real value.

MutumMan and FraunmpBereer *) find — 0,880 V as most negative
value for nickel charged with H,, which value they took for the
equilibrium potential. The values found by PraNHAUsER and SIEMENS.
can probably be explained by this, that two counteracting influences,
the H, charge and the air atmosphere, compensated each other.
PrANHAUSER could only obtain the most negative value — 0,466 V
found by him when the electrode was measured in the vessel in
which it was formed by electrolysis. Transmission through the air
always yielded Jess negative values.

Kister *) found — 0,800 V as minimum value for the tension of
separation of nickel.

SCHWEITZER °) carries out the measurement in the H,-atmosphere,
because he wanted to exclude the influence of O,. He finds — 0,616 V,
a value, which as was set forth above, is not the equilibrium potential,
but a value which is determined by the H-ion concentration in the
electrolyte used.

1) Z. phys. Chem. 14 [1894] 215.

2) Z. f. El. Chem. 7 [1901] 698.

8) Z f. Anorg. Chem. 41 [1904] 249,

4) Sitz.ber. Bayr. Akad. Wiss. 34 [1904] 201.
5) Z. f. El. Chem. 7 [1900] 257.

6) Z. f. El. Chem. 15 [1909] 602.

402

Scuocu ') found — 0,48 V by the method described above, and
also when measuring in an N,-atmosphere.

ScniLpBAcH ®), working according to the method Scrocn, likewise
finds a value which is in very good agreement with that of ScHocn.

It was of importance to examine whether in a solution of NiCl,
the same influences of H, and QO, would be found. The Cl-ions
exerting so strong a catalytic influence on the setting in of the
internal equilibrium of iron, it was possible that here too the influence
was noticeable, and would prove to be stronger than the disturbing
influence of O, and H,. This has not appeared to be the case.

In 2.5 n. NiCl, in H,-atmosphere we found :

Nil —0.604 V, Nill —0.600 V, H, — 0607 V
after addition of HCI:
Nil — 0.379 V, Nill —0.372 V, H, —0.376 V.
The behaviour of nickel in NiCl, is therefore quite analogous to
that in NiSO,.

SUMMARY.

The researches on the electromotive behaviour of some metals,
made in the Amsterdam laboratory of late years show that one of
the most characteristic properties of a metal is the velocity with
which it assumes equilibrium at a definite temperature and pressure.
At the ordinary temperature and pressure this velocity is on the
whole exceedingly small for a metal in dry condition. In contact
with an electrolyte this is, however, quite different, but at the same
time we meet then with complications through the appearance of
catalytic actions, both positive and negative ones, which render a
comparison of the behaviour of different metals difficult. It occurs
namely that in the same metal the equilibrium sets in quickly in
one electrolyte and slowly in another; thus iron assumes internal
equilibrium e.g. with great velocity, when it is immersed in a solution
of FeCl,, and this takes place much more slowly in a solution
of FeSO

There are, however, metals that also in contact with an electrolyte
assume internal equilibrium slowly, and nickel, which has been
discussed in this communication, is one of these metals.

Now it has appeared of late, that not only oxygen, but also
hydrogen can be negative catalyst for the establishment of the internal

1) Le.
2) Z.f. Electr. Chem. 22 [1910] 977.

403

equilibrium, and the remarkable feature in the behaviour of nickel
is this that the influence of these negative catalysts is exceedingly
great.

This ‘is accompanied with something very remarkable, viz. this
that when hydrogen of e.g. 1 atm. is conducted through an NiSO,-
solution, in which Ni-electrodes stand, the metal through the deposition
of electrons + nickel-ions resp. through the going into solution of
these compenents, is disturbed so far in base resp. noble direction
till the electron-concentration of the metal equilibrium in the liquid
has become equal to that of the hydrogen equilibrium in the
electrolyte.

It can be demonstrated that in this case the potential difference
of the nickel with respect to the electrolyte, with exception of the
Volta-effect, must become equal to the potential difference of the
hydrogen electrode.

Experimentally it was found that under the circumstances given
here, practically equality of the two potential differences is found.

The theoretical considerations led us further to expect that the
unary equilibrium potential for nickel was only to be expected in
the experiment in an atmosphere free from O,- and H, or in vacuum,
and when an Ni-salt solution was used in which the H-ion
concentration is smaller than 10~% for an Ni-ion concentration of 1.

Experiments made in this direction gave as result —0,480 V.
with respect to the 1n. calomel-electrode, a value that agrees entirely
with the value calculated by Wimsmore and also with that found
by ScHocH in an experiment in vacuum.

In the many potential measurements of the metal nickel which
were carried out in the aw or in a hydrogen atmosphere, the equili-
brium potential of the unary nickel has not been measured, but the
potential of a state of this metal that was disturbed in base or in
noble direction.

Amsterdam, June 29 1917.
Laboratory for Anorg. and General
Chemistry of the University.

Physiology. — “On the transmission of stimula through the ventricle
of frogs’ hearts”. By Dr. S. pr Borm. (Communicated by Prof.
VAN RIJNBERK),

(Communicated in the meeting of June 30, 1917).

The following facts were among others stated by me in the phar-
macophysiological investigations I made into frogs’ hearts, after
| had poisoned them with veratrine or digitalis.

1. The duration of the refractory stage of the ventricle-muscle
increases after the administration of each of the two poisons, and
so does likewise the a-v-interval; at last the contractility of the
ventricle-muscle decreases.

2. As soon as the relative duration of the refractory-stage
(aren of the total refract. stage

RTT surpasses the value 1, suddenly
duration of a sinusperiod ) P )

or gradually the normal ventricle-rhythm changes into the halved one.

a. The sudden halving of the ventricle-rhythm comes about in
the following manner: |

The duration of the refractory-stage of the ventricle has increased
during the normal rhythm of the ventricle for this reason that the
ventricle-muscle was not yet entirely restored at the beginning of
every ventricle-systole. What was still wanting to this restoration,
was called by me the residue refractory stage.

The periodical refractory-stage was added to it by every systole,
so that the total refractory-stage consists of two components. If now
the relative duration of the refractory-stage has become longer than
1, the next following ventricle-systole falls away, and a protracted
ventricle-pause is the consequence. This protracted pause influences
the two components in an opposite sense.

The ventricle-muscle’ restores itself better, so that the residue-
refractory-stage decreases. But after a protracted pause the next
following systole of the ventricle is considerably enlarged, consequently
the duration of the periodical refractory-stage of the ventricle increases.
If now this increase of the duration of the periodical refractory-
stage surpasses the decrease of the residue-refractory stage, then
suddenly halving of the ventricle-rhythm sets in.

b. The gradual transition to the halved ventricle-rhythm however

ae

_—

405

takes place, when the decrease of the residue-refractionary-stage
surpasses the increase of the periodical refractionary-stage. For, if
this takes place, the normal ventricle-rhythm continues after a
protracted pause, till by accumulation the duration of the residue
refractory-stage causes again the falling away of a ventricle-systole,
and the normal ventricle-rhythm is resumed again. So groups of
ventricle-systoles come into existence, which become gradually
smaller and smaller, till in the end the halved ventricle-rhythm is
reached in this way.

3. Spontaneous alternations between the halved ventricle-rhythm
and the normal one occur frequently. The cause of these alternations
lies in the fact that during the halved ventricle-rhythm the katabolic

duration of the refract.stage of the ventricle

duration of a ventricle-period — )
decreases again by restoration, till it has become less than */,. Then
the normal ventricle-rhythm sets in again. In this twice as rapid
ventricle-rhythm *), the katabolie index of the ventricle increases
again, and consequently the halved rhythm of the ventricle sets in
again. So these alternations can repeat themselves again several times.

4. By extra stimulation of the ventricle the halved veniricle-
rhythm can artificially be converted into the normal twice as rapid
rhythm by the addition of one little ventricle-systole. This proves,
that during the halved rhythm of the ventricle the sinus-impulses
that are not answered by the ventricle, did really reach this part
of the heart, but rebounded on the not yet irritable ventricle-muscle.

The normal ventricle-rhythm can likewise be converted into the
halved one by extra-stimulation. The enlarged post-compensatory-
systole fixed then the ventricle in the halved rhythm.

I attributed these and many other results, not mentioned here, to
the fact, that an important factor of the action of the heart, viz
the refractory stage had been modified under the influence of the
employed poisons. lts duration increased by veratrine and by
digitalis. These poisons had no further possible mysterious actions
for the results, mentioned above.

The following observations made with regard to not poisoned
frogs’ hearts afforded an unmistakable affirmation of this fact. The
before mentioned sudden and gradual transition into the halved
ventricle-rhythm occurs likewise in the not poisoned frog's heart, the
spontaneous alternations between the halved rhythm of the ventricle
and the normal one can also be stated.

index of the ventricle (

') During the normal ventricle-rhythm the katabolic index of the ventricle is
equal to the relative duration of the refractory stage.

406

In fig. 1°) we give a reproduction of the suspension-curves and
the electrograms of a frog’s heart (rana esculenta). More than an
hour after the suspension this heart shows constantly repeated

1) Constantly 1 electrode was placed on the auricle — and 1 on the ventricle-
point in the following reproductions.

407

alternations between the normal ventricle-rhythm and the halved
one. I succeeded in photographing such a spontaneous alternation
under simultaneous registration of the action-currents.

This reproduction shows a great number of important details,
and affords a formal confirmation, likewise for not-poisoned frogs’
hearts, of the theoretical explanations communicated by me in former
essays. In the figure we see suddenly appear the halved ventricle-
rhythm after 4 normal ventricle-systoles. Three of these are still
registered.

l intend more explicitly to explain here the following details, which,
in my opinion, are of interest for my subject.

1. As [ indicated in my former investigations the a-v-interval
increases during the normal ventricle-rhythm till the halving of the
ventricle-rhythm sets in. Afterwards the duration of the a-v-interval
decreases. The suspension-curves of this figure show a much shorter
a-v-interval after the halving than before it. But the electrograms
indicate these differences much sharper. The P-R-interval increases
still during the last 4 systoles. The first curve of the halved ventricle-
rhythm shows a much shorter P-R-interval of the normal ventricle-
rhythm. The restoration of the ventricle-muscle in the halved rhythm
is even distinctly to be seen in these 3 first curves of the halved ventricle-
rhythm. The P-R-interval of the 2rd systole is shorter than that of
the first, and that of the 3rd still shorter than that of the 2rd.

We must attribute the shortening of the P-R-interval after the
halving fo a shortening of the electric latent stage, as all sinus-
impulses reach the ventricle along the connection-systems (BUNDLE of
His), and consequently the time of conducting along these has not in
the least changed. It appears that this shortening still proceeds
from the moment of the first ventricle-systole of the halved rhythm.

2. The duration of the R-oscillation is after the halving shorter
than before it. This duration is now also again shorter during the
2nd systole than during the first, and at the 3'¢ systole shorter than
at the 24.

In the halved ventricle-rhythm the conductivity through the
ventricle is consequently better than in the normal twice as rapid
rhythm of the ventricle. From the first systole of the halved ventricle-
rhythm the conductivity still improves from systole to systole.

The. P-R-interval and the duration of the R-oscillation conse-
quently sustain alterations in exactly the same sense. We must attri-
bute both these alterations to the changed metabolic condition of
the ventricle-muscle (katabolic index). This metabolic condition deterio-
rates in the normal yentricle-rhythm. If now -the rhythm of the

408

ventricle suddenly halves, the metabolic condition of the -ventricle-
muscle suddenly improves much, but also in the halved ventricle-
rhythm this improvement increases from systole to systole.

This reproduction which for the present moment will remain most
likely exceptional among my material, afforded me an irrefutable
confirmation of the theories I explained before. For the present |

of

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409

shall most likely be compelled in my further investigations to restrict
myself to artificial transitions of poisoned frogs’ hearts, and, when
doing so, I shall at the same time register the action-currents.

[ am likewise in possession of beautiful examples of the slow
transition to the halved rhythm of unpoisoned frogs’ hearts. One
example of these is reproduced in the figures 2, 3, 4, and 5.

=
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=
=
=
2
ne
2

=

Proceedings Royal Acad. Amsterdam. Vol. XX.

410

The heart of a rana temporaria was suspended and soon showed
group-formation, because constantly 1 systole of the ventricle fell
away. The groups grow gradually smaller, till groups of 2 and 3
systoles (fig. 4) form the last transition to the halved ventricle-
rhythm. (fig. 5). We see during the groups the duration of the a—v
interval increasing splendidly; again and again the ventricle-systole
sets in later in the auricle-diastole, till one ventricle-systole falls
away. After this the interval is shortened again, to be protracted
again in the same way during the following group. The ventricle-
systole of each first curve of the group commences in the figures
2, 3 and 4 close to the top of the auricle-curve. The ventricle
systole of each last curve begins at about the middle of the diastolic
line of the auricle-curves. This is the case with the large groups,
but also with the little ones (bigeminus groups). Consequently in the
beginning more systoles of the ventricle are required than later to
protract the a—v-interval as much. The deterioration of the meta-
bolic condition of the ventricle-muscle is announced here by the
formation of smaller groups. It is likewise clear, that during the
groups the metabolic condition of the ventricle-muscle deteriorates,
and improves again after a protracted pause. In my opinion we
must here also attribute the protraction of the a—v-interval again
to a protraction of the latent stage of the ventricle-muscle.

It is the active contracting terminal organ, the ventricle-muscle,
the refractory stage of which increases during the groups and so
does at the same time likewise the latent stage. The increase of
the refractory-stage is here likewise caused by the increase of the
duration of the residue-refractory-stage by accumulation. During the
protracted pause after a group the decrease of the residue-refractory-
stage surpasses the increase of the periodical refractory-stage. In
this way the constantly decreasing groups come into existence, which
ends in the halved ventricle-rhythm.

The conductivity through the ventricle was examined by me
still in another way. In a former communication it was already
stated, that the T-oscillation had altered in a negative sense
after extra-stimulation of the ventricle-basis or of the auricle. A
positive T of the normal ventricle-systole decreased during the extra-
systole which was excited in this way, a negative T increased. In
some cases a positive T became negative. The T oscillation had
changed in a positive sense after extra-stimulation of the ventricle-
point. A negative T decreased, a positive one increased ').

1) Zeitschrift für Biologie. Bd 65, Seite 428.

411

These modifications of the T-oscillations were now examined by
me more carefully, and it appeared to me, that the conductivity of
the ventricle plays here an important part. J stated in this evamt-
nation that the T-oscillation varies the more in a negative sense, after

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412

extra-stimulation of the ventricle-muscle or of the auricle, in proportion
as the conductivity of the ventricle is worse at the moment, when the
extra stimulation or the “Erregung” conducted after the provoked auricle-
extra-systole reaches the ventricle at an earlier epoch of the ventricle-period.

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413

The positive T-oscillation of the normal systoles during an extra-
systole can even produce a negative T, if at an early epoch of the
ventricle-period the ventricle is excited to an extra-contraction, and a
decreased positive T, if at a later epoch of the ventricle period the
ventricle is excited to an extra-contraction.

A few examples may explain the above more accurately.') In the
figures 6 and 7 extra-stimula were applied to the ventricle-basis at
the upward oscillations of the signal. In Fig. 6 at 1 an extra-
stimulus is applied to the ventricle-basis immediately after the com-
pletion of the preceding T-oscillation. The T-oscillation of the extra-
- systole which is positive at the normal ventricle-systole, becomes
now decidedly negative. ‘At 2 a following extra-stimulus hits the
ventricle-basis at a much later epoch; consequently the negative
T-oscillation is now much smaller. Here I already fix the attention
to the fact that the enlargement of the positive T-oscillation is so
much the more important during the postcompensatory-systole, in
proportion as the preceding extra-systole has been brought about at
an earlier epoch of the ventricle-period.

In fig. 7 an extra-stimulus hits the ventricle-basis at 1 soon after
the completion of the preceding T-oscillation. This causes a great
negative T in the electrogram of the ventricle-extra-systole. At 3 the
extra-stimulus hits the ventricle at a later epoch, this makes the
T-oscillation smaller, at 2 the extra-stimulus comes still later, an
extremely little negative T-oscillation is the consequence.

[t is beautifully brought out here, that the enlargement of the
positive T-oscillations of the postcompensatory systoles is so much
the more important, in proportion as the extra-stimulus has hit the
ventricle-basis at an earlier epoch of the ventricle-period.

The extra-stimulus was applied to the auricle in the figures 8 and
9, which proceed from the same frog’s heart. At about the same
epoch of the ventricle-period the ventricle-systoles set in that are
brought about after auricle-extra-systoles by the conducted “Erregung”’
after the extra-stimula applied at 1 and 4. The T is negative and
of the same dimension at both the ventricle-systoles brought about
in this way. The ventricle-systole sets in at a much later epoch of
the ventricle-periods after the extra-stimulus applied to the auricle
at 2. The T-oscillation remains now positive, though distinetly reduced.
_At 3 an extra-systole of the ventricle is brought about through
current-loops directly by the stimulus.

1) In the following figures extra-stimula were only applied at the upward oscil-
lation of the signal,

414

Fig. 8.

At 5 and 7 the ventricle-systoles caused in the same way set in
at a much earlier epoch of the ventricle-period than at 1 and 4.
In conformity herewith both the negative T-oscillations are now
also much greater. The ventricle-systoles that are brought about

415

after 5 and 7 teach us how exactly the dimension of the negative
T-oscillations is determined by the epoch of the ventricle-period at
which the ventricle systoles are brought about. After the extra-
stimulation of the auricle at 5 the string is a little longer in the

416

position of rest, after the completion of the T-oscillation of the
ventricle, before the R-oscillation of the anticipated ventricle-systole
begins, than after the extra-stimulation at 7. The difference is little
but it can distinctly be observed. Entirely in conformity herewith
the negative T is after the extra-stimulation at 5 smaller than after
the extra-stimulation at 7. At 6 the anticipated ventricle-systole
begins, after the extra-stimulation of the auricle at a much later
moment of the ventricle-period. The T-oscillation remains now
positive, but is somewhat reduced. The extra-stimulation at 8 has
evidently hit the auricle at the same moment, as it was reached
by the sinus-impulsion.

In these representations the enlargement of the positive T-oscil-
lation of the post-compensatory systoles is also the greater, in pro-
portion as the extra-systole of the ventricle sets in more anticipated.

If now we try to answer the question, why. the T-oscillations
of the ventricle-electrograms change the more in a negative sense
after extra-stimulation of the ventricle-basis and the auricle, in
proportion as the extra-ventricle-systole begins earlier in the ventricle-
period, then we must look for the cause of this phenomenon in the
conductivity of the ventricle. At an earlier epoch of the ventricle-
period this conduction is slower than at a later period.

This is the cause that the negativity of the point at an extra-
systole, which has been brought about at an earlier epoch of the
ventricle-period, begins later than at an extra-systole that has been
brought about at a later epoch. Consequently the point-negativity
domineers the more in the latter part of the ventricle-electrograms,
in proportion as the extra-systole has been brought about at an
earlier epoch of the ventricle-period. The earlier the extra-systole is
brought about in the ventricle-period, the more the T-oscillation
changes in a negative sense. It is likewise moreover of importance
in this respect that the contractility of the basis at an early epoch
of the ventricle-period is still trifling; when from there the contraction-
wave reaches the point in a slow tempo, its contractility has become
more intensive. But this factor can be reduced again to a slackened
conductivity at an early epoch of the ventricle-period. This theoretical
explanation corresponds so perfectly with the experimental results,
that I ean diseretionally produce extra-systoles with reduced positive
T-oscillations and with negative T-oscillations.

I wish to fix here the attention to one point that is distinctly
demonstrated in the figures 8 and 9, I pointed out in my former
communication already, that after extra-stimulation of the auricle
and of the basis ventriculi the T-oscillations of the ventricle-

417

systoles were modified in an equal sense. This appears most
distinctly from the figures 8 and 9, in which at about the same
epoch of the ventricle-period, at 3 by extra-stimulation of the
basis ventriculi (by current-loops), and at 5 and 7 after extra-

418

stimulation of the auriele, an extra-ventricle-systole is brought about.
We see now the T-oscillation after extra-stimulation of the basis
ventriculi about as large as after extra-stimulation of the auricle.
The anticipated ventricle-systole is brought about in the latter case
by the “Erregung” that reaches the ventricle along the usual con-
nection-systems. The epoch of the ventricle at which the anticipated
ventricle-systole begins, and not the place where the stimulus attacks,
determines the extent of the T-oscillations. It is obvious that this
does not hold, when an extra-stimulation hits the point. But here
the T-oscillation of the ventricle-systole varies the more intensively
in a positive sense, in proportion as the extra-stimulus reaches the
ventricle at an earlier epoch of the ventricle-period. This is distinetly
to be seen in Fig. 10. At 1 and 3 the extra-stimulation reaches the
ventricle-point at an early epoch of the ventricle-period. Now very
large positive T-oscillations set in. At 2 the extra-stimulation hits
the ventricle-point at a later epoch. Consequently the positive T is
now smaller. If at 4 the ventricle-point is hit by the extra-stimulation
at a still later epoch, the positive T is again still considerably
smaller. At 4 the basis-negativity had already begun, when the
extra-stimulation set in, and brought the string back to and beyond
the O-position. It appears consequently that the ventricle-point is
irritable in the ascending line of the R-oscillation.

The basis-negativity is consequently so much the more intensive
in the ventricle-electrograms of the extra-systoles after extra-stimul-
ation at the end of these electrograms, in proportion as the extra-
stimulation hits the ventricle-point at an earlier epoch of the
ventricle-period.

Mathematics. — “Surfaces that may be represented in a plane by
a linear congruence of rays’. By Prof. JAN DE Vartks.

(Communicated in the meeting of April 27, 1917).

1. In order to obtain a representation (1,1) of a cubic surface
in a plane we may make use of the bilinear congruence which has
two straight lines lying on that surface as directrices. To this pur-
pose the linear congruence (1,3) may also be used, which is formed
by the bisecants of a twisted cubic lying on the surface *). It would
also be possible to make use of the congruence (1,3) of the rays
that intersect a twisted cubic and one of its bisecants, provided
that those two directrices are lying on the surface.

We shall now consider, more in general, the surfaces that can
be represented by means of a linear congruence of rays (1,2).

The representation of a surface ®+r+! with an n-fold and a
p-fold straight line by means of a bilinear congruence has been
amply treated by Guccia and Minxo ’).

2. Let now be given a surface P+! with an n-fold twisted
cubic a’. Through a point P of ® passes a bisecant of @, which
intersects the plane of representation rt in P'; in general a point
P has one image P', and, inversely, a point P' of t is the image
of one point P.

Let us now consider the ruled surface & formed by the bisecants
t projecting the points of a plane section y?”+1. In the plane of that
curve lie three straight lines ¢, © is consequently of order (2n-+-4).

That cone, which projects a° out of one of its points, determines
on y?"+! evidently (n-+2) points P, consequently « is an (n+ 2)-fold
curve on @2+4.

The image of y+ is therefore a curve of order (2n+4) with
three (n+2)-fold points A; this curve will be indicated by the symbol
c2n+4(3 A"+?) }

The images of two plane sections have the (2n+-1) points P’ in
common, which are the images of the points P lying on the inter-
section of the two planes.

1) See e.g. R. Sturm, Geometrische Verwandtschaften, IV, 288

2) Guccia, Sur une classe de surfaces, représentables point par point sur un
plan (Ass. francaise pour l’avancement des sciences, 1880). Mineo, Sopra una
classe di superficie unicursali-(Le matematiche pure ed applicate, volume I, p. 220).

420

Besides the 3(m-++-2)? intersections lying in the principal points A
they have consequently moreover m= (n°+2n43) points G in
common; they are evidently the intersections of as many bisecanís
g of a° lying on #?+1,.

On the surface lie therefore at least (n?4-2n-++3) straight lines. |

From this it ensues in particular that any twisted cubic lying on
a cubic surface has six straight lines of ®* as bisecants.")

The complex of plane intersections of "+1 is consequently
represented by a complex (0°) of curves c++, which has three
(n+2)-fold and (n?+2n-+3) simple base-points.

3. The curve a’ is represented by the ruled surface U of the
bisecants ¢, which touch at 2+! in one of its two points of support.
The straight lines y are double generatrices of dU; for they may be
considered to touch in two points of @’.

Let now « be the order of 4, y the multiplicity of a* on that
ruled surface. The intersection on t is then a curve a%(3A’,mG?’).
As «° has evidently 3x points in common with a plane section
y2+l, the consideration of their images produces the relation

Onde = 8(n+2)y + 2(n?+2n-+3) + 3n.

As two biseeants of «* can only intersect on that curve 4 has
2y points in common with an arbitrary bisecant; so we have c=2y.

We now find y= 2n+ 3, e=4n-+4 6.

The image of the curve a° is therefore a curve a*t6(3.A2"+4, mz”).

4. Each of the n planes that touch +! in a point A of a’, con-
tains a generator ¢ of the ruled surface 2%, which moreover intersects a?
in a point S. The remaining (7 + 3) straight lines f meeting in / touch in
Pnt in another point of «°. The pairs of points R, S belong to a
correspondence with characteristic numbers (n-+3) and. The points
S belonging to the same point R form pairs of an involutory corre-
spondence with characteristic number (24-3) (n—1L); the coincidences
originate from points AR, where two of the tangent planes coincide.
On «@ lie therefore 2 (n4-3) (n—1) cuspidal points.

To each point A correspond n points of the image a, so the
points of a are arranged in an involution /,.

5. Let in the plane rt a curve f be given of order p, which passes
a times through the principal point Az, and gz times through the
principal point Gj. With the image CRIA, G;) it has apart from
the principal points a number of points in common, indicated by

p* = (n+2) (2p — Zar) gz.

1) For »=2 we find the surface 5 with nodal curve 2? amply discussed by
R. Sturm, (Geom. Verw. IV, 311).

421

This number is evidently the order of the twisted curve ®, which
has f as image.

As f has apart from the principal points, a number of points in
common with det (Ar ES, G2), represented by

n* = (2n + 3) (2p — Zar) — 291,

the curve ® rests in n* points on the curve a’. A straight line /
is therefore the image of a A*”+4, which intersects « in (4n-+-6) points.

6. For the simplification of the representation we submit the
figures in tT to a quadratic transformation, which has A; as principal
points. By this the curve c?"++ is transformed into a curve c’t?,
which does not pass through A;, but does pass through the (n° +2n-3)
points ©, in which the principal points G are transformed.

To the curves y?"+1, in which the surface ®?"+' is intersected by
the planes y of a pencil, correspond now the curves of a pencil
(ent?) Among them there are 3(n-++1)’, which possess a nodal point,
which is then at the same time the case with the corresponding
curves y2"+1,

The surface Dt! is consequently of class 3 (n-+1)?.

The straight line ©, 6, is transformed by the quadratic trans-
formation into the conic /?(A; G,G@,) and the latter is the image
of a twisted curve ®”, which rests on a° in (2n—1) points. For
through f* and @* passes a hyperboloid, which has the curve /? and
n times the curve «*® in common with &2"+1; the residual section
is the ® in question.

7. We shall now consider a surjace p" tr! that passes n times
through a twisted curve a? of order q, and p times through a straight
line B, which is (q—-1) times intersected by a’.

The straight lines ¢, which intersect a and 8, form a linear
congruence (1,9), by which @ is represented in a plane rt; for t
intersects @, except on @ and 8, only in one point P more.

The ruled surface €, which represents the plane section y"t?+! is
of order (n+-p+q-+1); for in the. plane y lie g generators.

Out of a point of 8 the curve a? is projected by a cone of order
q with (g+1) fold edge 8, which intersects pret! in (p+q)
points P. Consequently € passes (y-+-q) times through 8.

Out of a point of a the line 8 is projected by a plane that deter-
mines (n+1) points P on y. Consequently «a? is an (n +1) fold directrix.

The image of y"t?+" is consequently a curve ertetatt (q Ant, Beta),

Two curves c have apart from the points A and B and the images
of the (n+p-++1) points in the section of their planes a number of
points G in common represented by

422

mln p+ q+ Ie HIG (pDA I=
—=n(n +1) +(2n + 1)p—(n? —1)q.

From this it ensues that the surface &»+r+' contains at least m
straight lines. |

If we take here g—1, it appears that a surface Dr+Pt! with an
n-fold and a p-fold straight line contains 2np +n -+ p-+-1 straight
lines resting on the multiple straight lines *). For n=1, the number
m appears to be independent of q; we find that a surface det?
with p-fold straight line contains m=3p--2 straight lines g, resting
on «a and 8. A plane passing through 3 and one of those straight
lines contains one more straight line A, wich does not rest on
a; on r+? lie therefore at least 6p-+-4 straight lines.

8. Let us now determine the image of the straight line 2. A plane
y passing through 8 contains one more curve y"+!, which has on a but
not on 3, an n-fold point Q. As it intersects 8 in (+1) points, the
plane contains (n+-1) straight lines ¢, which touch the surface in
points of 8, consequently are generators of the ruled surface ® by
which 2 is represented; «a is therefore an (n-+1) fold directrix. As
Q comes to lie on @ for (q—t) different positions of the plane y, ¢
will as many times coincide with >, consequently 8 is a (g—1) fold
torsal line. But each of the p planes that touch at ® in a point of
8, contains one straight line ¢; consequently the multiplicity of 9
on % is equal to (p + g—1) and the order of 8 equal to (n+p+4).

The image of the straight line 8 is therefore a curve brea (g A+!
Bral, mG).

This result may be controlled by determining the number of points
that 6 has in common with a curve c apart from the principal
points A, B,G; we promptly find then the number p, being the
number of common points of @ and prte+!,

9. Let us now determine the image of ay. Each of the n planes
that touch at ® in a point Q of a, contains one straight t of the
ruled surface AU that represents «, consequently « is an n-fold directrix.

If 8 is a y-fold directrix a plane passing through 8 contains a
section of order (n +4), and the image of a has as symbol a™ty
(qA", Br, mG).

By combining with chtptatt (qA"t!, Beta, mG) we find for the
determination of y, the relation (n-+p-+-q+1) (n+y) = n(n+1)q +
+(p+q)ytm-+ng, in which it has been taken into account that a
plane section has nq points in common with ug.

1) See Mineo ibid. page 221 or J. pe Vries, Surfaces algébriques renfermant
un nombre fini de droites (Archives Teyler, serie Il, tome VIII, p. 262).

423

From this relation it ensues that y =p + gq.

Consequently the image of a? is a curve an+e+4 (qA", Beta, mG).

The combination with the image of 8 produces a control; from
which it appears that the curves a and 6 have promptly 2 (q—1)
points in common, apart from the principal points.

10. If we write in: the results arrived at, n=1, g=3, p=1,
we obtain the representation of the cubic surface to which we
referred in § 1. The directrices of the linear congruence (1,3) are
then a twisted curve a° of ®* and one of the bisecants of a’ lying
on ®’.

The image of a plane y° is then a c° (3A’, 5%). If the six bisecants
mentioned above are indicated by 6;, and if 6, is the directrix of
the (1,3), the five straight lines cj, are represented by points.

The image of a* is a curve a’ (3A, B“, 5C’) the image of b, a
curve 0° (3A*, B*,5C); these curves have, as they ought to have,
two more points in common, which are the images of the points of
support of the bisecant 5.

It is easy to determine from these data the images of the remaining
21 straight lines of #°.

Chemistry. — “Observations about hydration under the influence of
Colloidal Catalysers and how to account for this process”.
By Prof. J. Bénsexen and Mr. H. W. Horsrepe.

(Communicated in the meeting of June 30, 1917).

Some time ago') we performed and described a number of hydrations
with the aid of colloidal catalysers. At the same time we gave an
outline of what might be expected during the development of this process,
and what we shall therefore observe in measuring the absorption of
the hydrogen, viz:

The hydrogen and the matter to be reduced will pass from the
gas-space into the liquid medium, then they will pass one or more
layers covering the atoms of the catalyser and finally they will
coalesce. Hence we first observe a succession of diffusion-processes,
before the catalytic-chemical reaction enters. It was assumed that
the rapidity of the latter process is always considerably greater than
that of the former, so that the observed rapidity of absorption,
would seem to refer to a diffusion-process.

The assumption of the all-surpassing velocity of the catalytic-
chemical process is arbitrary; the rapidity will of course depend
on the nature of the catalyser. But we have been guided in the first
place by the consideration that the action of platina-metals on some
of the processes brought about by them is exceedingly great indeed,
so that few thousandths millimol are sufficient to bring molecular
quantities to a fairly rapid chemical change.

It appears then that we want far greater quantities of catalyser,
say from 50—100 millimol, to reach the result, obtained in the
observations made. So it is obvious that we must rather look for
the cause of that very moderate velocity in the possibility, that of
these 50—100 millimol only a very small part can be reached at
the same time by the reacting substances, than to suppose that the
process of these particular reactions is such an extraordinarily
slow one.

In my opinion the very potent catalytic action of the metal-atoms
results from the nature of these atoms themselves. Owing to their

1) Recueil 35 260 (1916).

425

capacity of rapidly transmitting electricity from atom to atom they
must be pre-eminently adapted to decrease chemical resistances, the
latter being probably occasioned by electrical tension.

Secondly we have been guided by the consideration, that the
“activation” of a colloidal catalyser is frequently due to’ a finer
division i.e. to an increase of the number of atoms, coming into
contact with the reacting substances at one time so that inversely
the paralysis must be caused by a decrease of this number. We have
assumed that this is the result of the presence of the above-
mentioned layers about the atoms of the catalyser, which may include
both the layer of the protecting colloid (gum-arabic, sodium-salt of
protalbinic acid, ete.) and the layer of all other molecules existing in
the medium — also that of the catalyser itself *). (I. ¢. p. 262—263).

We shall call these layers the ‘“paralysis-layers”’ *).

Following up this assumption we have argued that with a not very
active catalyser the process is bound to proceed almost till the very
end with constant velocity, if only care is taken to bring the H,
with sufficient velocity into the liquid space. If the concentration
of the substance to be reduced is not too slight, then the outer side
of the paralysis-layer will remain saturated with the mixture of
this substance + hydrogen, whereas on the side of the catalyser
the concentration of this mixture (or rather of one of the components)
is kept at zero as a result of the great velocity of reaction. Thus
we measure a diffusion-process with a constant difference of level.

Seeing that the concentration of the substance to be reduced
decreases towards the end, so that the paralysis-layers on the outer
side no longer remain saturated, the velocity of absorption is bound
to decrease towards the end.

The saturation of the paralysis-layers with gaseous hydrogen was
brought about by me by forcibly shaking the colloidal solution with
H, and by raising the number of shocks in one minute, until the
velocity of absorption when using a very active catalyser no longer
increased. In this way we could make ourselves independent of the
first phase of diffusion (le. p. 262): the dissolving-velocity of the
gaseous hydrogen.

But it is especially by the following observations we believe we have

1) This may be observed as agglutination, flocking out or even crystallization.

*) We have imagined these layers to be close and more or less permeable; it
may be, though, that with very slight concentrations of the interacting molecules
they must be represented as impermeable and as having smaller or larger inter-
stices. It stands to reason that in this case a quite different explanation than
the provisional one we have given, would be required.

28
Proceedings Royal Acad. Amsterdam Vol. XX.

426

demonstrated that the analytic-chemical reactions proper have
certainly not been measured by us.

1. The velocity of reduction of «-crotonic acid, isoerotonie acid
and tetrolic acid under the influence of a very active palladium-
catalyser (PaaL')) was perfectly equal under otherwise equal
conditions.

2. The reduction and substitution of trichloroacrylic acid to pro-
pionie acid and the substitution of pentachloropropionic acid took
place without any sudden changes of velocity and even for these
two acids with almost equal velocity.

3. The two double bonds of the sorbinie acid were hydrated
without any sudden change of velocity, and the velocity was equal
to the substitution-velocity of the two chlorated acids mentioned
under 2.

4. It was ascertained (with a less active catalyser) that cinnamic
acid, muconie acid, malonic acid and vinylglycolic acid were hydrated
almost quite as rapidly, if only the decreasing activity of this catalyser
was taken into account.

5. The result obtained with a relatively large quantity of palla-

diumsol (Skrra ?)) that under otherwise equal conditions, equimolecular -

quantities of cinnamic acid, glutaconic acid, muconic acid and
vinylglycolie acid were hydrated with approximately equal velocities
whereas itaconic acid, mesaconic acid and citraconic acid were
hydrated, with somewhat lesser velocities, though mutually equal
ones, than the former acids.

6. It is true that during the reduction of the cinnamic acid under
the influence of finely divided platinum, a decrease of velocity
was noticed after the absorption of about one molecule of hydrogen ;
it could be determined however, that this divergence decreased in
proportion as a greater quantity of the catalyser was taken. This
cannot be accounted for if we assume a chemical cause of the
decrease in velocity, which is besides very slight and not very per-
ceptible. .

7. It was found that the velocity of the hydration was largely
dependent on the condition of the catalyser p. 275—279 and p. 286.

8. The temperature-coefficient was but a small one (We do not set
great store by this argument, because an increase of temperature
may cause a diminution of the surface of the catalyser, the result
of which will be a decrease of velocity).

1) B. 38, 1401 (1905).
2) B. 3%, 24 (1904), 40. 2209 (1907), 41. 805 (1908).

427

§ 2. Now the very rapid bydration of undecylenie acid-sodium,
with palladium-sol, during which process the contents of the reaction-
vessel were changed into a froth (p. 270), had revealed to us that
the method of investigating pursued up till now, did not sufficiently
guarantee the demand that we should keep the paralysis-layers
about the catalyser saturated with H,. S

Hence it was necessary :

1. to improve the hydration-vessel ;

2. to take a quite definite catalyser, if possible the activity of the
same must be constant, and at any rate easy to control;

3. to work in a constant temperature.

As regards the first requisite we selected the apparatus employed
by Mr. Conen and one of us in our experiments on light, with which
the H, is spouted in small particles with great velocity through the
liquid '), the contact-surface of the gas with the colloidal catalyser
being thus very considerably enlarged. This is brought about by rotating
a hollow stirrer with great rapidity through the liquid; in the gas-
space above the surface of the liquid is an opening. During the
rotation the gas is sucked through the stirrer and dashed into small
bubbles against a wave-breaker. This wave-breaker keeps the fluid-
level constant.

A revolution-counter indicated that the number of revolutions was
at least 2200 a minute. It is probable that we succeeded in keeping
the paralysis-layers saturated with hydrogen, as with a much slighter
velocity of rotation the same velocity of hydration was obtained in
otherwise similar conditions and when applying but moderate quan-
tities of the catalyser. .

As catalyser we used the palladiumsol prepared according to SKITA
and Meyer”), which proved to be very active, so that we needed
but few milligrams of metal to obtain a velocity of hydration that
could be measured. The hydration-apparatus was placed in a
thermostat, in which was also an electric lamp, so that the vessel
could be observed without taking it out of the thermostat. We were
enabled to do so, because the hydration-apparatus had been fixed
in an iron frame-work that could be moved up and down with the
apparatus.

The hydrogen was purified by conducting it through alkaline and
acid permanganate of potassium, through silver nitrate, through alka-
line pyrogallol solution and finally through concentrated sulphuric acid.
1) Proc. Kon. Ak. Wet. 25 March 1916.

2) B. 45, 3579 (1912); one c.c.m. contained one m.g. of palladium.
28*

428

Before use the palladiumsol had been saturated with hydrogen, |
the gas was also conducted for some time through the hy dration-
vessel, while the spray-stirrer had been brought into action, and the
substance to be reduced was already in the vessel.

By means of a three-way tap the hydration-vessel was connected
with a graduated Lunge-burette, which the H, filled. By means
of a simple lever-apparatus gas-volumes could always be read during
a constant mercury-level. The accompanying figures show the complete
installation in elevation and in plan. By way of elucidation it may
be remarked that in the drawing we find between the purification-
flasks of the hydrogen and the dryingflasks a copper tube with
iron mantle, filled with copper-shavings. This tube is heated in
a chamotte-oven and serves to free the hydrogen from oxygen.
The ends of this metal tube are provided with refrigeration-jackets.
(See below).

The first experiments were made with cinnamic acid in aqueous
solution at high temperature and without thermostat; afterwards in
96 °/, alcohol at the usual temperature. Herein the catalyser frequently
proved to flock out, so that we finally worked in 80 °/, alcohol,
always about 25°. Besides cinnamic acid, we also examined other
substances.

With the readings of the hydrogen that had been absorbed we
noticed the barometer and the temperature in the neighbourhood of
the gas-burette; as a rule we took 3 mg. palladiumsol against
0.5 to 1.5 gr. of the hydrated substances.

We subjoin a tabulated summary of a series of experiments.

The summary gives two series of experiments; in the first series
of 15 the H, had not been conducted along a red-hot copper-spiral.

Though the character of the results obtained is in accordance
with what was found before, yet it appears even in a superficial
investigation that great irregularities occurred.

The velocities of hydration diverged in a rather considerable
degree under comparable conditions; this is especially striking in
the reduction of the cinnamic acid-aethylesters where the initial-
velocity of absorption was found to be within the limits 5.5 and
80 ¢.m, per 10’ (N°. 8—11).

The cinnamic acid-methylester too revealed strange leaps, the
commencing velocity being between 10 and 40 cem. per 10.

Then we were struck with another phenomenon, viz. that not
unfrequently the number of absorbed ec.m. H, surpassed the number
calculated to a considerable extent; we had observed this before
and this fact has been repeatedly observed by other investigators

on OO

24

25

26 |

Substance

Cinnamic acid |

”

Undecylenic acid-

aethylester

”

vy

Cinnamic acid-
aethylester

Cinnamic acid-
methylester

‚ Cinnamic acid-

aethylester

Quant.
in gr.

0.7265
0.4792
1.3010
1.6514
1.1263

1.2387
0.8909

1.0971

0.9533
1.3059

0.6840
0.8447
3.0390

2.1397
8.0519

2.0832

not meas

2.0210

2.3121

2,2081

Solvent ,

H,0

96 9 alc.

”

75

”

429

EEEN NOEDELS IEEE

Pd-sol. |

in ccm.

2
after 30’
still 3

1
| after 30’
| still 1
etc.
| (c.f. text)
1

after 30’
still 1
etc.
dito

ecm

ccm

EAA | |
Velocity |
gote Ansen, in ies 7. | Observations
lated bed Ber |

17 | dts 25 94° | Irregular

62 18 iy 5 Idem.

146 144 13 18 Very regular.

147 | 149 | 80 18 After the first 10’a
considerable retard-
ation; flocking- out.

189)" 192i eae 16 Idem.

132 130 | 92 ~=17.7 | Retardation; no

| flocking-out.

144 143 105 18.2 | Idem.

113 5.526 The course of the
hydration was a
very slow one,cause:
\very pure H,(c.f.text).
ebr 13S: ia „ | Hydrogen contain-
ed O,; course very

‚regular.

155 |7 O33 80 „ | Idem.

28 » | Course auto-cataly-
| | tic, was not con-
| tinued upto the end.

158 «_ 103 40 ; Hydrogen contain-

ed Op.
rt eran mae Great retardation,
‚cause unknown.
‚ 493 447 35 Zi Hydrogen contain-
| ed Op.
1351 | Sho 40 ‘ Idem.
| 1276 | 1161 | + 50 pe Idem; course auto-
| | | catalytic.
ae aL te 1 ‘ H, very carefully
| | | purified, from
| oxygen.
pada ss HSH kn H, not conducted
| over Cu-spirals.
Wee te 56 ee Cu-spiral was used;
| but Hg still contain-
| ed Os; very regular
| reduction. ~

314 308 B27 on Strongly inclined

curve.

303 | 299 45 jk Idem.

295 | 294 : jd Idem; towards the

end almost com-
pletely paralysed.

193 | 189 8 a Partially flocked-

after 30’: | out, after the addi-

100 tion of 3 ccm Pd.

sol unexpectedly
strong acceleration,
then paralysis again.

283 | 270 | (c.f. text) | ,

* No flocking-out; but

decrease of velocity.

313 | 288 Idem.

430

as well. In our first experiments we had used the hydrogen out of
a Kipp apparatus and had conducted it through a flask with alkaline
pyrogallol-solution; the quantities of hydrogen that had been caleu-
lated and measured did not diverge greatly.

Afterwards we employed a hydrogen-bomb and obtained far too —
great absorption-numbers (nos. 9, 10, 12, 14 and 15). At first not
being prepared for the presence of oxygen in the purified gas we
excluded as well as we could all other sources of the coming in
of oxygen in the apparatus by repeated washing, preliminary
treatment of the palladium, and the avoiding of rubber-junctions.

When however the surplus continued, the presence of oxygen
was thought probable and determined in the following way.

1). By hydration of larger quantities of substance in the same
quantity of solvent and with the same amount of catalyser.

And indeed it was found that on applying:

+ 1 gram cinnamic acid-methylester the surplus = 55 cc.m. N°, 49:
3 Li} >) > 99 2’ —= 46 ” be) 14.
8 bi 3) 9 99 +B = 115 39 x” 16.

In case of the presence of oxygen in the hydrogen the surplus
used must be approximately proportional to the quantity of methylester
reduced, if at least the amount of the oxygen in the H, is constant.
Whereas we conducted the gas through an alkaline pyrogallol ~
solution, this will surely not have been the case, but an increase
of the surplus when applying more substance is indubitable.

2). By some blank experiments.

For this we took 100 eem. 80°/, alcohol and 3 eem. palladiumsol
saturated with H,, they were put into the hydration-vessel, after it
had been filled with hydrogen in the usual manner.

We actually observed an absorption of gas in four control-
experiments, it amounted to between 20 to 30 eem.; it is curious
that though far more gas was in the burette, the action came to
an end with this.

The cause of it may be, that on account of diffusion all oxygen
had come from the burette into the hydration-vessel, or that the
catalyser is paralysed; the former cause seems the more probable.
Now the hydrogen was freed from oxygen by means of a tube
with red-hot copper shavings; hereupon a decrease of only one
cem. took place in 60’. |

But when we were now going to hydrate the cnnamic acid-methylester
with this very purified hydrogen, which had also been carefully freed
from oxygen, the process was an extremely slow one. (N°. 17).

Of course this might be due to an accidental paralysis of the

431

catalyser; in order to investigate this the Cu-spiral was switched
off without substituting a pyrogallol-solution, the reduction was once
more very reguiar and even took place with greater velocity than
when the pyrogallol-solution was used. (N°. 19).

The Cu-spiral was switched on again, but this time it had
evidently not completely freed the hydrogen from oxygen, as the
course of the reaction was fairly rapid and regular. (N°. 19).

Therefore .some quantitative experiments were started with the
Cu-spiral as above; the reaction took place; but no paralysis set
in, it appeared finally that 6 eem. too much had been absorbed.
As the contents of the hydration-vessel are + 600 cem., and 300 eem.
have been received from the burette, the total amount of gas that
has been in contact with the catalyser amounts to + 900 eem. The
surplus 6-cem. corresponds to 2 eem. O,, hence its amount is
+ 0.2°/,. So this quantity proves to be amply sufficient, to keep up
the reaction. (N°. 20).

With a subsequent experiment (N°. 21) we had reduced the
O,-quantity (calculated from the surplus gas that had been used)
to 0,1 °/,; otherwise everything being similar; the velocity of reduc-
tion was appreciably less and a stronger retardation set in.

This was still more obvious, when we had decreased the O,-quantity
to 0,003 °/,; towards the end the velocity had decreased to zero,
so that a small amount of catalyser had to be added to bring the
reduction to an end.

To demonstrate the activating effect of the oxygen still more
clearly, the hydrogen was mixed with a still far greater quantity,
viz. with 4,8°/, and compared with hydrogen that had been care-
fully purified (but evidently still containing oxygen). :

2,0336 Gr. cinnamic acid-methylester hydrated with 4 eem. sol,
absorbed 391 ccm, calculated 306 ccm.

The blank experiment also with 4 eem. Pd sol showed volume-
contraction of 65 cem. It is remarkable that this reaction i.e. the
water-formation, proceeded rather slowly, the sol being completely
paralysed, even after the addition of another emm. Pd-sol.

The control-experiment with purified H, with a very slight oxygen-
quantity was performed with 1,9958 substance, 4 eem. Pd-sol and
like the two preceding ones at 25° C. and with 2250 rotations of
the spray-stirrer.

Absorbed 291 ccm, calculated 294. (This deficiency is probably
to be attributed to a not inconsiderable change of temperature in
the neighbourhood of the gas-burette, during the experiment, on
account of -which the volume-calculation has been less accurate).

432

The reduction took place altogether regularly without any consider-
able paralysis, as a sign that the H, contained some O, after all.

Yet the difference in velocity with the H, containing + 5 °/, O,
is very striking as may be seen from the subjoined table.

» Gas-absorption of the 5 10 15 20
cinnamic acid methylester min. min. min. min.

Duration NO. of the
of graphic re-
‚reduction presentation

66°} 120.4 1155 112230 60° 27

] In Hp» with 4.8°/, O,
Mal Es is... “aotrace’ Os 25 40 52 64 180’ 28
Ill | Combinationof H,andO,| 20 35 44 49 | 29
Difference I—III 46 | 85 | 131 | 181

EN

When we also observe, that the oxydation of the H, proceeds
rather slowly (ID) we draw the conclusion, that the two reactions:
the oxydation of the H, and the reduction of the ester, influence
each other very favourably. °

This migbt lead us to suppose that owing to the presence of the
oxygen the palladium is continually freed from a paralysis-layer,
so that the number of atoms becoming available for the reduction-
reaction is greater than when no O, is present. But if this were so,
the paralysing of the catalyser, when no reducible substance is
present would be unintelligible. It may be, that the solvent, the
80 °/, alcohol, which, no doubt will be acted upon by the oxygen
and converted into aldehyde, plays an important part.

A closer investigation will have to elucidate this question, but it
appears at any rate that this catalytic reduction is far more com-
plicated, than would appear at first sight.

It is certain that the velocity of the hydration proper:

H, + substance to be reduced = reduction-product,

can be considerably modified by a by-reaction, so that we feel still
more positive in concluding, that we have not measured the velocity
of this reaction.

The question, whether we shall ever be able to measure this
velocity of reaction when using colloidal catalysers cannot be
solved yet.

It would be indispensable for us to be able to introduce these
catalysers into the reaction-mixture in such a state and to bring
about the course of the reaction in such a manner, that the forma-

J. BOESEKEN and H. W. HOFSTEDE: “Observations about hydration under the influence of Colloidal Catalysers and how to account for this process”.

Nd “0 FS,
Ne oo

1. He bomb. 9. Mercury-flask.
2. Rinsing-flasks 10. Escape-tube for the Ho.
3. Cu-tube. 1], Cup-shaped glass with water in which the
4. He SO, rinsing-flasks escape-tube empties
5. Glass spiral to saturate Pdsol with He 12. Motor.
6. Hydrogenation-apparatus 13. Thermometer in the thermostat
7. Circulation-tube for the Ha 1d. Electric lamp „
NS. Gas-burette 15, Thermo-regulator
Fig. 2
| a
=p 6
ee
Pe

Proceedings Royal Acad. Amsterdam. Vol. XX.

Jon

250

ft
~_
y
©
&

200

150

100

50

10 20 30 60 120

250

200

100

60

eee: 30
Ens

120

433

tion of paralysis-layers is altogether excluded; but the part the
oxygen has played in our experiments, gives but little hope that
we shall be enabled to fulfil this condition in the near future.

§ 3. On examining the graphic representations we notice especi-
ally in those hydrations with gas-mixtures deficient in oxygen, a
strong curved line, pointing to paralysis. This paralysis evidently
enters sooner with the cinnamic acid aethylester than with the
methylester, moreover part of the catalyser repeatedly flocked out.

To make the phenomenon stand out the more, we at first added
somewhat less Pd-sol.

With N°. 23 we first used 2 cem.; after 30' another 3 ccm.
„ were added. Though the velocity after the second addition was very
considerable, yet a strong retardation soon set in.

When starting from 1 ce.m. sol, the velocity had soon decreased
to zero, but when a second ce.m. sol was added, the initial velocity
was of a similar order of magnitude and occasionally even some-
what greater than if we had immediately started from 2 eem. sol.

With a view to the ever varying quantity of oxygen we must
be. very cautious in drawing a conclusion. But it may be in indubitably
concluded that the paralysis of the first quantity of sol does not
happen to the same extent, as otherwise hardly any hydration would
have set in. The first quantity of sol will probably fix an impurity
of unknown nature; perhaps the second quantity can partially take
on the paralysing substance from the first quantity. A closer in-
vestigation is desirable.

§ 4. With the undecylenic aethylester too, a great retardation in
the second part of the hydration had been observed, during the first
experiments. (N°. 4—7, without thermostat).

The same took place with the undecylenic acid when using H,
freed from oxygen, 2.0237 gr. dissolved in 100 ccm. 80 °/,
alcohol + 4 eem. Pd-sol, employed 252,5 ccm. calculated 253-cem.
(N°. 30).

The first 130 ccm. were absorbed in 5 minutes, then a very
pronounced retardation set in, in the following 5 min. 24 cem.
were absorbed and in the then succeeding 5 min. 16 etc. ~

Simultaneously a flocking-out had set in, which proved to be com-
plete when the reduction was over.

As a solution of undeeylenie acid + 4 cem. Pd-sol remained
unchanged for days, it may consequently be ascribed either to the
undecylenic acid formed, to the reduction itself, or to both.

434

From an experiment with undecanic acid it appeared that this
substance indeed, flocked out the sol in 80°/, alcohol.

Now a floeking-out need not necessarily cause a paralysis; but
as the first phenomenon is a visible indication of the diminution of
the surface of the catalyser it is most probable that in this instance
the decrease of the velocity of reaction is related to the flocking-out.
It» stands to reason that in determining the velocities of hydration,
the possibility of the paralysis must always be reckoned with, and
that any cases in which flocking-out sets in are of but little value
for the comparison of these velocities.

We consequently give a short summary of the substances, which
at a normal temperature immediately or after a short time completely
flock-out the Pd-sol : ;

in aqueous solution: diluted HCI, diluted KOH; in 80°/, alcohol
undecanic acid, cinnamic acid, phenol, acrylic acid methylester, iso-
erotonie acid, oleic acid, glacial acetic acid, propionic acid, butyric
acid, valeric acid, capric acid, caproic acid, laurie acid, palmitic
acid.

On the other hand the sol remained unchanged for a considerable
time: in aqueous solution on the addition of diluted sulphuric acid,
acetic acid to 80°/,, diluted soda, undecy lenic acid, undeeylenie acid
aethylester, cinnamic acid methyl- and aethylester. Diluted nitric
acid dissolved the metal.

On summarising the phenomena described in these pages, we
undoubtedly get an idea, that even in this apparently so simple
catalytic. reduction, viz. an irreversible reaction with an elementary
catalyser, the events are far more complicated and far more sensitive
to by-cireumstances than could be expected.

Not until these events have been sufficiently studied, so that they
can be entirely brought under the control of the experiment, can
a mathematical treatment produce good results.

Anatomy. — “On Mendelian Segregation with the Heredity of
Headform in Man”. By G. P. Frets. (Communicated by
Prof. C. WiNkLER.)

(Communicated in the meeting of June 30, 1917).

The significance of the shape of the head as an anthropological
characteristic was brought to light by the investigations of ANDERS
Rerrzius (1842—1860) '). Rerzius introduced craniometry and based
the classification of races on the dimensions of the cranium. He
discriminates the brachycephalic or short and round, occasionally
square, and the dolichocephalic or long and oval skull type.

With this method of investigation began a new period in anthro-
pology. The task was, to examine with various peoples and races
the index of the cranium i.e. the shape of the skull, expressed in

100 idth
got It was further inquired into — fixed values
the length

having been accepted for the dolichocephalic and the brachycephalic
skull — how dolichocephaly and brachycephaly are spread over
the various countries.

For the experimental doctrine of heredity the significance of the
shape of the head is a different one, viz. its behaviour in heredity.
This question is not entirely strange in anthropology, but was eli-
minated there. A. Rerzius e.g. points out that — in order to be sure
of having to do with the pure type, — one has to choose material
from the country, and when he disposes of a considerable collection
of skulls, e.g. Swedish ones, he selects by a first comparative ex-
amination, a few of those that do not show accidental or strange
properties. |

The experimental doctrine of heredity examines the heredity with
regard to its significance in connection with the problem of the
origin of species. Besides with regard to its practical significance
for man, as to the breeding of animals and the cultivation of plants,
and also with regard to diseases of man and the improvement of
race. This latter part of the science of heredity is distinguished as
eugenics.

Only few investigations into the heredity of race-characteristics

the proportion

1!) A. Rerzwus. Ethnologische Schriften. Stockholm 1864.

436

have been made as yet. The inheritance of the headform has not
yet been investigated methodically. Hurst’) mentions in a list
of properties, which segregate according to MENDEL’s rules, that round-
headedness is dominant over longheadedness and E. FiscHer *) con-
cludes, from his bastard-material, that the headform is most probably
hereditary according to the rules of Menper. FiscHem gives moreover
quotations from the literature of non-mendelistic investigations, which
are in favour of the theory that brachycephaly is dominant over
dolichocephaly.

The above-mentioned twofold signification of analytic investigations
into heredity in man has induced me to investigate the heredity
of the headform. The present first communication regards the results
of a thousand measurements. My material consists of the visitors
calling on the patients of the lunatic asylum Maasoord of Rotterdam.
By numerous journeys to Rotterdam and other places, consequently
somewhat in the manner of the fieldworkers, I have measured as
completely as possible all the members of those families for whom
this was of importance. Also a few other families have been inserted
into the tables. The extensive tables, on which the communication
rests, will be published later. All the measurements and calculations
have been executed by myself.

The anthropological knowledge of the headform may likewise
serve as a guide for the choice of the shape of the head for
a Mendelean investigation. A. Rerzius has not given any fixed values
for the dolichocephalic and the brachycephalic skull. In a letter to
Duvernoy (1852) he says*) that with the dolichocephalie skull the
length as a rule surpasses the width by 4, whereas with the brachy-
cephalic skull this difference varies between + and 4. These figures
mean for the dolichocephalic skull an index value < 75 and for
the brachycephalic one an oscillation of the index between 83 and
88. G. Rerzius adds to this information the interpretation that his
father left a space between the typical measures, fixed for the two
groups; he fixed centres, round which the various sizes of the skull
can be arranged.

If this should be so, one might expect that, in case a population
contained besides dolichocepalic also brachycephalic race-elements,
the indices calculated from many observations, being united in a

1) G. C. Hurst. Mendelian Characters in Plants, Animals and Man. S. 192.
Mendel Festschrift in Verh. d. nat. forsch. Ver. in Briinn. 49. Bd. 1910.

*) E. FiscHer, Die Rehobother Bastarde und das Bastardierungsproblem beim
Menschen. Jena 1913.

3) lc. p. S. 18.

437

curve, this difference would appear in the shape of the curve, viz.
a twotoppedness. One top would be found at about 75 and another
at 85. As far as in literature such curves have been communicated,
no distinct twotoppedness is to be seen (Rerzius and First‘), BorK®),
E. Fiscuer*); comp. also Werckrer*) S. 42.) Also in the curves of
my material they are failing (Fig. 1—5). This is possible, when the
dolichocephalic type, whilst crossing with the brachycephalic, origin-
ates equally all possible intermediate shapes, but still the more so, if
there are not two types, but several. The solution of these problems
is being searched for by the experimental science of heredity.
Generally biological the question is: are the dolichocephalic and
the brachycephalic headforms characteristic marks of distinction
between two races, consequently of elementary species, and expressed
mendelianly: is the headform determined by units of heredity
and by how many? Thus the investigation of the headform is a
hereditary-analytical problem, to be studied by family investigations.

The material, of which the composition is represented by the
curves (Fig. 1—5), consists of families from one generation (brothers
and sisters), from two generations (parents and children), and from
three generations (one to four grandparents, parents and children).

At a first glance at the material the impression is made that of
a family of brothers and sisters the brothers often have lower indices
than the sisters. In the curves 6¢ and 6? 169 brothers and 169 sisters
have been brought together, who are chosen in such a way that
constantly from each family as many brothers as sisters have been taken.
If consequently two brothers and two sisters of one family have been
measured, then two brothers and two sisters-(i.e. the two elder)
have been inserted into the curves. From these curves it appears
still more distinctly than from the curves of the total number of
men and women (Fig. 2 and 3) that as a rule the men have a
somewhat lower index than the women. Of the men (Fig. 2 and 3)
one top lies at 77,5, and another at 82, of the women the top lies
at 80; of the brothers (Fig. 6%) the top lies at 79, of the sisters at
81,5. 33 brothers and 12 sisters have indices under 77. The
variability of width of the sisters (74—89) is likewise somewhat
smaller than that of the brothers (72—90).

1) Rerzius und Furst. Anthropologia Suedica. 1902.

*) L. Bork. De bevolking van Nederland in haar anthropologische samenstelling.
(Uit Gallee. Het Boerenhuis. Utrecht 1908).

3) E. FiscHer. |. c.
» 4) H. Wetcker. Untersuchungen über Wachsthum und Bau des menschlichen
Schädels. 1, Theil. Leipzig. 1862.

= ww we = wy oa F5 @
— (=)
AG 5 B 2g o = iA & “= en 2 cS > c co o i) =
ER
e
sy A “3
=d
E al
_
DN
&
—
“st
> ES
/
a Ss
Ca
© >
hd
= >
Se
a ed
Co
G ord
ao
= =
oS.
i S
Go
= 2
be =]
-H co
—H
co
==) ao
@
2 =
per
wo
© vo
©
e
Le
\o
‘i =
Fig. 3. Fig. 2. Fig. 1.

AN

d Rn AE dl

Fig. 1—5. Curves giving a general survey of the material. Fig. 1. The material containing 1014.
measurements. Fig. 2. The measurements of women 661. Fig. 3. The measurements of men 353. Fig. 4.
120 natives of the islands of South-Holland; of the parents and likewise usually of the grand-parents
it is known that they were born there. Fig. 5. 100 persons, natives of Rotterdam, as likewise parents
and grand-parents.

PSU

439

If now we pass to the examination of the behaviour of heredity
of the headform of the material at our disposal, then we follow,
when doing so, the way taken by other investigators (Rüpin,
DAVENPORT, LUNDBORG): we examine if the data admit of a Mendelian
explanation. We assume in this respect that the index of length-

23 745 YOY FES 801 B23 B 5 867 RE 9~90

OLS aes Hijs dos Oy 5 8 7 88 3 go
Fig. 6.
169 brothers and 169 sisters. Of each family as
many brothers as sisters have been inserted.

width types the headform. A. Rerzius has answered to the objection ')
that the index represents only two dimensions, that experience
teaches the index may in reality be taken as the expression of the
headform. Another question is, whether the Mendelian analysis of
the factors of heredity of the shape of the head can restrict itself to
trace the heredity of the index. In this case we should have to do
here with one pair of units of heredity (for the large and small
index, or for several different large indices). If the length and the
width mendel separately, we have to do either with two pairs of
units of heredity or with two progressions of it (respectively for the
large and the small length and idem width or for several different
large lengths and different large widths). Both possibilities will be
examined. If the length and the width of the head mendel sepa-
rately phenomena of segregation will come to expression in the index.
We begin by tracing these. This gives us the advantage, that the
complete material, likewise that of not fullgrown persons, can be
considered. In the first place we examine if there is segregation.
In favour of segregation plead these cases where, with little differences
in the indices of the parents, the children show a great divergence
of values, or if a single child has a strongly deviating index. In
the tables I and II some of these cases are collected.

1) A. Reraws, lc. S. 57.

440

TA Ball:
Little difference of the indices of the parents, great difference of those of the children.

Mylo e ae EN e
vie 2 | 2 Î Sons | Dn | Be
number (| = | — a8 ee ES

1/2) 3] 4)-5 46]. | 2) 8) a agli

XLVIII 79.7 78.7 79.9 83.6 86 83.1,79.1/80.6 82.4) I
LVI 80.1/79.9 76.3 84.8 | I
LXIVb |77.6|76.2 73.2 80.9 I
CXXIVa 80.1 79.8 78.9 82.1 86.5 85.1 85.979.883.179.7 | I
CXXXVII 80 82 1 76.9 82.9181 .4|82.5 i Ment
CLXXIII 80.181.3 87 81.8 83.8 83.2 82.8 I
CLXXVII 81.4 81.4 81.4 84.5 'g1.3/79.4|83.6|79.7, || 1
XL 81.8 83.6 80.5 11.8 83.6 81.7 82.1 | Il

CXIV 82.5 81.8 76.7 78 lu
XXXIVa_79.7/79.1 78.7 81.9 83.6 80.3 84.8 80.1 81.7 In
LXXXVlle 82.2/81.8 81.8 84.7,79.1/80.3) | Hi
LXXXVIId 82.7 82.2 75.9 80.9 84.3 79.8) Lof int
LXL 80. 80.4 81.3 79.7/73.7 80.8 81.7 79.2 80.1 88.8 Lm
XCIIla 81.8 82.6 83.7 78.4 81.7 86.1 eae
CLXXIXa (80.8 80.9 80.9 84.7 88.9 86.7 86.682 81.6) III
XXIld 81.4 81.7 80.4 82.9 93.5/79.6, | | (IN
XXe 80.2 80 is3.8/78.6/ | | an
LXXXIV 84.785 85.4 /80.8 i ect ea
CLXXVIld 8181.5 75.2 77.9 Bees
CLXXXIId 83.7 81.3 82.6 81.7 79.4 77.5 84.7/85.3/80.8| || v

The examples of table I and Il may be regarded as the expression
of the formula DR x DR= DD+2DR+4+ RR, those of table II
of a more complicated segregation (vide page. 441). When perusing
the lists made from the material, we are struck both by the great
variability and also by the irregularity of the indices. The great
variability renders it unlikely that heredity should simply be deter-
mined by a pair of units of heredity brachycephaly-dolichocephaly.

1) Number of the large tables which will be published later.

441

If this were the case we should more frequently meet with homo-
zygotic pairs of parents, to whom would consequently belong a
group of children with very slight differences of the indices. These
however do not occur in the lists. The Table which contains the
families with four grandparents of which consequently three com-
plete generations are known furnishes moreover the indubitable proof
that the headform is not inheritable on account of a single pair
of allomorphie factors.

Consequently we reject the indication of the motion of heredity
according to the monohybridie scheme.

It is very well possible that inheritance of the headform is deter-
mined by a progression of units. Heredity goes then according to a
type that Ninsson-En1.e') (1909—1911) has systematically worked
out for cereals and that according to different investigators (LANG,
T. Tammes a.o.) may be applied to their results. If the head form
in the. conception of NirssoN-Enre is determined by a few pairs of
units, consequently A,A,A,... and a,a,a,... then of a pair of
parents, the father can possess a factor of beredity that the mother
misses, and the reverse, or in heterozygotic form. Among the children
may then, by combination of factors, occur headforms with indices
larger, resp. smaller, than those of the parents. Generally a great
variability is to be observed with mutually slight differences, which
in a restricted number of individuals may give the impression of
intermediary heredity. By special combinations which, as can easily
be calculated, are very rare, greater deviations may occur. By a
separate breeding of the third bastard-generation (/,) it can be
proved that a cross-breeding follows this scheme. Then different
proportions of number must occur, according to the number of
Mendelian factors which has been adopted for the explanation of
the second bastard-generation. This took indeed place in Nitsson-
Enre’s experiments.

The data of different families plead for the fact, that, if the
heredity of the headform follows the rules of segregation of MENDEL,
factors working in the same direction in the sense of NILsson-
EnLE must be admitted. So e. g. those, that have been gathered in
table VI, where one index deviates considerably from the others.
Likewise those of table VII, where the indices of the children
surpass those of the parents on both sides or on one side.

We find consequently in the collected material indications for the

1) H. Nitsson-Exte. Kreuzungsuntersuchungen an Hafer und Weizen. I und IL
1909 u. 1911. Acta Universitatis Lundensis Lund.
29

Proceedings Royal Acad. Amsterdam. Vol. XX.

442

TABLE I.

Series of indices a single one of which strongly deviates

Children bs
prey 2 E | Sons é Daughters - BS
Paty | a | | 52

| it RB aca ba | 4 | ae
1 | 76.3 | 80.3 | 80.3 | 80.2 81.1 | 80.6 84.3 | | I
Il 76.5 81.5 | 77.3 79.2 84.1 | 82.7 80.9 | 78.6 | 81.3 I
XVII | 77.8 | 81.7 || 78.7 | 79.2 | 79.5 || 77.7 | 88.4 | 84.8 | I
CLXIV | 79.2 | 77.8 || 76.3 | | 79.7 76.8 | 75.8 | 77.8 | 83.5 || 1
CXCVI 74.7. 79,5 74.1 82 | 18.3 | | | I
CLIV_ | | | 77.8 | 75.1,| 79.6 | 82.5 | 79.2 || Il
XXXVII | 77.3 | 76.8 || 77.9 82.3 | 79.6 I]
CCXV | 78.1} 81 |] 71.5 85.6 | HI
XVI | 80.5 || 83.8 | 73.5 | 84 | 85.6 | 82.3 82 II
CXVb | 16.3 | 16.8 | 71.7 83 75.4 19.6 “19 IV

view, that segregation occurs with the heredity of the shape of the
head, and that some pairs of factors of heredity are concerned in it.

A second question is, if, with this alternative heredity, the first
bastard-generation (/’,) is intermediary or that there exists dominance.
If, as Hurst indicates from literature, brachycephaly is dominant
over dolichocephaly, only dolichocephalic children will be born from
the marriage of two dolichocephalic persons. For the brachycephalic
headform is then defined by the factor D (dominant) the dolicho-
cephalic one by the factor A (recessive), the former can be represented
by the formula DPD or DR, the latter only by RR and the pairing
of two RR’s gives only AR descendants. The great variability, the
non-occurrence of families in whom a definite headform is nearly
constantly inherited, the .complication in the cases, in which three
generations are completely known brought us to the conclusion,
that not one single pair of factors determines the inheritance of the
headform, but we think it possible that some more factors working
in the same direction are active in this respect. Moreover dominance
may occur, and the occurrence of families (table III) among whom
only comparatively low head-numbers are found, seems to speak
in favour of it. Families (table IV) of whom one of the parents is
brachycepbalic, and all or most of the children are brachycephalic,

443

occur likewise. There are however also families (table V) for whom
this is not the ease, likewise families are met with (table VI), for
whom the indices differ very little, but are somewhat higher than
those that hold for dolichocephaly. Consequently we cannot admit
beside segregation simple dominance of brachycephaly.

The cases mentioned already (table VII), where the indices of
children surpass those of the parents, either on one or on both sides
are especially in favour of the possibility, that the inheritancy is
determined by a series of factors working in the same sense, which
each individually give an intermediary first bastard-generation (/’).
This method of explanation can moreover very well be applied
to other phenomena of our material. The great variability of the
indices of the children and the slight differences of those of the
parents (table I) can e.g. be occasioned by the fact that the parents
contain the factors in a heterozygotic form, so that exteriorly they
are intermediary. Consequently different homozygotic, thus greatly
divergent values of indices, will occur among the children. The
separate occurrence of a strongly deviating index-value is caused,
according to this view, by the rare combination of two or more
factors working in the same direction in a homozygotic form (one
of the 16 cases in the dihybridic scheme, one of the 64 cases in
the trihybridic’ one). The slight mutual differences of the tables UI
and VI can be explained by admitting, that the forms of issue contain
nearly the same factors, so that consequently no new divergent
combinations occur in the children. The cases of table VIII contain-
ing examples of extreme values of indices of one of the parents
may be explained in the same way. A very low index-value is
caused by the deficiency of, a very high one by the existence in
homozygotie form of some factors working in the same direction.
It is consequently clear that among the children, as a rule, the
extreme value will not occur.

What gives likewise significance to this manner of explanation,
is the possibility of explaining the limitedness of the selection: by
selection no more can be reached than the form that contains all
the factors of heredity working in the same direction in a homo-
zygotic form. Examples in our material of “selection-working”’ (ac-
cording to NirssoN-Eare's explanation, consequently of combination
of factors) are the fam. III and XXXIV (vide the genealogical trees,
p. 448). The parents and eleven children of fam. XXXIV have
_rather low indices viz. the parents 79 and the children 72—79.
If now we admit that the parents possess each a different factor
for brachycephaly, with which consequently absence of a similar

; 29%

TAB LTE

444

Ill.

Low indices of the parents and the children.

| dE =
Number | 5 | 5 | R 58
of the | 3 r= | Sons _ Daughters Bw
ama ES in crepe ARTI TT Ze
| MRE Beaken” =
XXI 15.8 71.9 79.171. 1 76.9|74.2 81.2| | | I
XXVII |76.8/78.9|'76.3; | | ||79.8/79.4/78.8 I
XXXIV 19.1/79.7/73.6 75.3 73.1 72 | 119.1 15.6 15.8 77.9 71.8 71.4 78.6 I
LXXXVI /79.2)78.7/78.7/79.5/75.5, |75 77.8 78.5 I
CXXXVI_78.5/80 | /78.3| 177.6 I
CCV 78.1/78.9|'78.2 Irie I
Da RR | (r.al78.2178.2) | I
XXXlla '76.5|74.6/73.7| |75.9 eae
XXXVIla 76.7) _ |/19.2/76.5/75.1 11.2/78.2 76. 877.1/76.8 79. 8 Il
Lx1| [79.5 | 14.2/76.9 71.6 Il
LXXXIII | = |79.2| 16.5|74.9|76.6 II
GEN ee ABS 78.8 78.5 I
CXV | __ [73 eae (74.9 76.3) | I]
CLII 78.8 19.97.11) | ll
CLXL | |79.3|/77.3|78.8| \77.2/78.1|78.1]79.2; {| I
CLXXXII |77.6,79.8 78.3,78.3,78 |74.5|80.7/77.4 79.5, EI V
TAB LE IV:
Families of which one of the parents and most of the children have a high index.
] BE 2
| || Children =
Number | 5 5 || 5 Is
Dd ES PEES Sons | Daughters 55
family th = Ki aS =
| Keeler gall deld Sal nn 6 5
IIIb | 82.4 | 78.3) |83 | 87.2|83.9|84.7/83 |86.1]] 1
LXXXIII | 77.3 | 85.3 | 84.7 | 84.1 | 87.3 | I
LXXXV |75 | 84.7//83.6| 78.9 | 79.4|/81.3/ 79.2179 | 80 l
LXXXVIII | 78.3 | 83.2|/83.3) | — |/76.4/ 84 lll
CLXVII | 76.4 89.2) | 85.8 86.1 | 83 79. Bien I
CCXXXIV | 82.2 | 87.3) (86.3 | 84.5 | 86.7 87.9 I
CLXXVIIL | =| 75.5|| 80.4 81.6 83.5. II
CLXXIX | (74. 3, 84.2 | | || 82.5 | 80.7 | | | I
XVle | 77.8 | 85.6|/85.9/ 84.3) || 81.6 | 84.2 82.2 80.2 87 UI
LXXXVlle | 75.4 (87.8 || 78.7. 86.2 81. 9 84.6 .6 | 83.1 | I]
CLXXIXb 77.2 | 85. 2 80.6, He ‚8 83.2 80.8 83 (rn
CCXII_ ine | 82.5 | 83.7 | 83.4 | 84.9 I
Ila | |)84.5 | 82.4 | 83, 6 82. 9 8: 5 II
CXCVII | 186.5|85.3| || 85.7 | | II

445

TAB LEN.
Family of which one of the parents has a higher index, whilst only one of the children has
a high index.

Children | 5
Number; 5 | 5 Ik:
of the 5 = Sons | Daughters | =
Family | ™ | = | 5 Td SE ES Ee S

EN eh pe De aa pal

Illelg3 |g6 ||79.1/81 |86.1|78.5/85.2'78.2| 80") | 81.8 | 80.8 84.5.86.5 84.2; || II
CXLVII 85 (79 ||79.3 80.2/84.4 80.8 85.278.5)| 78") 86.8") /81.2*) 80.2 81.8 | I
CLXXIX¢ 79.7 82.2 80.8 80.7 ES ANBO PAS RITE be | foe |e
CXCV 79.2 83.3 78.8 | || 81.5 | 82.6 | 81.8 Hi
VII (84.2//74.5/710.7 | 16.7 oes | Rak
XCVI| [84.679 ara steerer vaags PERS Ra
XVIb 78.7 84 (|71.9 80.7,79.7 83.4 80.7 80 | 79.7 | 79.5 |79.7 81.8 81.6 83.2) Ill
X1 83.415 (76 81.8 | 80 {80.6/79.4/ | | I
XIX|78.5|84.2!'77.6,79.4| | 49.8 '-|.80.5: 9755 (IB: Blus Us I
XVlla|76.7/84.7 16.6/16 |76.2! 75.5 | 18.8 |. | | I
XXXII f 79.2 84.1) 78.7 Bis Tie Baes Agen ce hehe ew oe
LXVIII 79.8 83.6 78.7 77.8 79.8 | 71 Bt recl I
LXXVI 78.7 85.9 77.4 fg BP See Re Tee | I
CXXX 84.9 78.4//75.5 | 265 | I
XXXIVe 75.3 83 75.5 85.2 Rae 16.8 | 76.3 ea | I
KXIle|75.6|83.8/81.1/81.1; | | | 79.7 | 80.2 Fae: Dink dl rige CR
XII 79.6 82.1) 80.8) ee BLA LIONS [ILA a a ce

*) Sons, consequently 7 sons and 5 daughters and 9 sons and 2 daughters.
**) 2 daughters of the first marriage of the mother.

TABLE VI.

Somewhat higher values of the indices of the parents and of the children.
Children 2
Number 5 04 LN te ken Ee
of the | 5 = Sons Daughters =
Family | & & | ee ae
Vien SE A Paya is ae pea eA SE
| | | | | | Z
XII | 79.6 | 82.1 || 80.8 | 81.4 | 19.2! 78.1| 78.4 I
Xv | 81.2 | 84.4 || 83 | 82.2 81.3|78 | 78.5) 83.5 82.2/82.4 || I
CXXXla | 81.8 | 80.4 || 78.7 | 78.8 | 81 ||79.7| | | | i
CLX | 81.8 | 77.8 71 | 77.5| 77.6, 79 | 78.1|78.7 || 1
LI| .- | 80.2 |/80 | 80.4 EEL | oat | im)
LX | 81.6 | 81.6 | 81.1 ices et
LXXXVIIb | 80.3 || 80.7) 81.3 79.9|80 |79.8)/82.1) | HIV
CLXXla | 79 | 80.5 79.8 81.6) ll
CLXXVIIb | 82.1 82.3 || 18.8 || 82.6 | 81.3] 82 | UI

TABLE VL

Transgression of the variation-width to both sides towards the dolichocephalic and the
brachycephalic side.

Number
of the
Family

XXII
XXXVla

CXCIII
XVa
XXXIVa
XXXIVe
XXXIV
Ille

Illf
XXIIb
XXXVIB
LXXXIVa
LXXXIX
CIX
CXIII
CXXIV
CXLVIII
CXVe
XXII
CXVa
CLXV
CXVIII
LXXXVII
CXXII
XXXlle

Children | 5
= 8 | Sons 4 | outers baal =
Ee NRE a tek
: | 1] | | 12
77.9 81 [76.3 15.6/19.7/76.5 81.7 79.4 82 82.2 | I
19.8 81.5 85.8 84.1 76.4 78.1. see

CLXXI 78.3 79.8 | 77.4 81.7 82.7 81.5 81.2 81.8 | 77.3 82.8 82.6 79.8 80.3. | I
81.8 16.7, 85.9 85.2 | | il
83 80.2) 17.7.87.7/88 Lm
73.176 | 74.1,70 vv
81.3 78.3) 73.2 (16.776 80.3 | 4
79.1 79.7 | 73.6|75.3/73.1 72 78.6%) | 79.1 75.6 75.8.71.9 71.8 a I
84.5 80.2) 83.3 80 83.7 86 83.5 86.7 Im,
82.2 83.5 | 87.591 — | I
82.478 | 81.584 84.6 (79.7 AE
78.884 ||86.7/88.2/19.7 83.1 83.5 81.2 87.4 80.2'86.7) 1
80.4 75.5 | 84.7 19 81.3 81.5 83.3 82.2 | 81.1 ed | I
[81.3/78.9|/83.7/@5.1| | | | e141 ein
15.8 80 | 83.2 | 83.8.80.3 | 1
83.1 83.7 | 89.5 84.1 | Shes
82.2 86.9 | 80.8 84.9 86.3 | 87.4 88.1 86.6 ALR
13.8 77.4 | 81.5 | 83 | | I
77.8 76.3 | 80.781 83.7/80.5 | 80 Il
16.3/16.2||78.2/76.7 | | 82.2 11
74.3118.2 || 719.8/76.8 79.6 81.5 80 | I
85.1 83.8 | 88.2 89 | ae
| 81.1 ||87.1 |83.71,85.1 | ll

81.2 85.6/82.2 82.2|87.8 II
80.9 |89 (83.2) 86.8 84.1 80.5 82.6 | Il
76.7 81.7) 85 ‘84.7 87.5 | Vv

“) Seventh daughter,

447

factor’ (by which dolichocephaly comes into existence) corresponds,
and that they possess e. g. still a third factor in heterozygotic form,
so that the formulas of the parents are A,A,a,a,A,a, and a,a,4,A,A,qa,
then the low values 72 and 73 of the children are explained e. g. by
the absence of the two factors for brachycephaly (a,a,q@,a,A,q,).
The daughter and the two sons who marry into families where
several higher indices occur, will consequently among their seven,
five and four children see higher values show themselves, the son
73.1 however who marries a daughter 76 of a family whose members
have also rather low indices, has a great chance, that there are
among his children some who represent a combination of factors,
that answers to the absence of e. g. three factors of heredity
(a,a,b,6,c,c,); suchlike children surpass then the low indices of the
parents towards the dolichocephalic side. This is indeed the case in
this family: The two sons have an index of 74 and of 70. The
same reasoning, but now for the combination of the factors whose

TABLE VIIL
Extreme value of one of the parents.
EPT NE
Children 5
Number 5 Re eeN ess $$ || +
of the 5 = Sons Daughters =
Family | & | = ||-——-———— ~~ ir 5
1 2 3 1 Mlk eh Re BENE
a
LVII | 72.6*)| 78.9 || 81.6 81.3 I
| |
LVIla | 72.6") 15.3 | 80.5 | 79.8 | II
XLIII Gee er Ur NI gi. | 7a" et.3 II
LXXXVlle | 75.4 | 87.8 || 78.7 | 86.2 | 81.9 || 84.6 | 83.1 | Il]
CVIII 87.6 | | 82.2 | 83.1 | CNR I
CXII 85.9 || 81.8 | 82 | | Il
CXV | Tse all THB | 74,3: 14.9 | 76.3 | | 1
| |
XVla| 73.5 | 82.3 || 80.4 | 82.2 | 75.5 || 82.5 | 19.7 | 80 | 80.5 I
XXXIVb | 73.6 | 80.2 || 75.5 | 79.2 | 79.7 || 82 | 80.6 u
LXVI | 88.5 | 81.1 | 91.4 | 89.5 90 | 86.9 | 86 |92 |85.8 || III
CLXXilla| 87 | 78.5 | 75.6 II
LVlla | 72.6 75.3 | 80.5 19.8 | IV
CLXXVIla 85.7 || 80 | 79.8 | 81.3 | IV

*) The same person; first and second marriage,
Pp } 8

448

gives for Fam. III the explanation
of the “occurrence of very high indices 90—95 in the third genera-

existence cause brachycephaly,

which our results still contain for the mendelian

explanation, will be dealt with in a following communication.

tion.

Difficulties,

Jee. LT
TCE Le,

He Ie HO CHHOCCS © fF I

DODO DDMD Ld Leer L

men HIG Ma D/A
eenn |
NEI CPE) OO DD © I
| UT d Ile mf Weg
Fig. 7a.

Lam. XXX XXXIV e
BO CHC AQ Be 1

OF O2 CBA FHeeeee© BOOe 2
en

"H@OOREAM FCC CHCA MF x

XXXIV a XXXIV £ XXXIV C XXXIV d
Fig. 70.

Mathematics. — “Some Considerations on Complete Transmutation.”
(Third Communication). By Dr. H. B. A. BocKWINKEL. (Com-
municated by Prof. L. E. J. BROUWER.)

(Communicated in the meeting of November 25, 1916.)

14. In connection with. the preceding considerations the con-
tinuity of a transmutation we shall refer to an inaccuracy that occurs
in the proof of theorem X of Bourrer, which will give us an op-
portunity to observe that the theorem itself stands in need of a
clearer statement. j;

Theorem X is as follows Toute transmutation additive, uniforme,
continue et régulière est donnée par la fomule

Tu au trut ut ff eee rs ret LN

où a, a,,... désignent des fonctions régulières et u’,u’’,... les deri-
vées successives de la fonction régulière u.

If this theorem, and the proof Bourter gives of it, is considered
more accurately, its meaning appears to be not over-clear, and this
is to be attributed to the fact that BourLer has omitted to fix a
functional and a numerical field in which the transmutation is to
be thought as defined. Are we to suppose that the transmutation,
for any function that is regular in the neighbourhood of a certain
point, produces a transmuted that is regular in the neighbourhood
of the same point? Are there points that form an exception to
this? Or functions for which this is the case? Are only analytical
functions to be thought of as objects of the transmutation? Or also
such as coincide in a certain N.F. with the analytical function f,(w),
and in another N.F., lying outside it, with the analytical function
Fy (a)? Is the transmuted an analytical function?

The matter does not become quite clear, for the definitions given
by BovrLer, are all highly incomplete. By a regular transmutation
he understands, according to his own words, a transmutation that
makes a regular function pass into an equally regular function. But
what is a regular function? BourLeT answers this in a footnote:
“Je prends pour définition de la fonction regulière celle de M. M.
Méray, Weierstrass, Fucus, etc.: ‘Une fonction de la variable a
est dite régulière, dans le domaine de rayon g, autour du point

450

„=d, si elle est développable en une série ordonnée suivant les
puissances croissantes de « — «,, pour toute valeur de « telle que
Von ait | e—vx, | <<.” All right, it is clear now, what has to be
understood, when is said: a function is regular in a domain with
radius @. But we do not know yet when a function is to be called
regular, if the emphasized words are omitted. It is not to be sup-
posed that by this should be meant a function that is regular in
the neighbourhood of any point, for in that case, a simple function

could never be object of the transmutation. Heceptional

as U =
Stn

points must therefore be admitted. But how many, and how situated?
A function which within a certain circle is equal to 1, outside it
equal to 2, and on its circumference indefinite, is that a regular
function? Ete. ;

All these and similar questions BourLer has avoided by adding
at once, in the definition he is going to give of a regular function,
the words dans un domaine de rayon 9. lt may be assumed however,
that BourtuT, when he spoke of a regular function, always bad in
view, consciously or unconsciously, a definite domaim, in which
the function in question is to be regular, no matter how it behaves
in a domain outside it. Indeed, if matters are put in that way,
everything tallies. Moreover, the short parenthetic clause, thrown in
at random, as it were, speaks for the correctness of this view.

We consequently assume as Bourter’s intention that the functions
to which the transmutation is to be applied, are regular in a certain
circular domain round a point 2,, no matter how such a function
has been defined elsewhere; further that the same holds for the
transmuted of these functions, and that this marks the transmutation
to a regular one, independent of the question whether or not it
produces functions with the same property for a// functions that
are regular in a neighbourhood of #,. When, therefore, BoURLET
speaks of a regular transmutation, we shall assume that is meant:
regular with respect to a certain functional field.

With regard to this F. F., Bourrer tacitly makes two important
hypotheses. He begins his proof with the construction of the series
of functions

Mar Mian Gis as.) NE

with which the series (1) is going to be built up, and he deduces
it from the functions into which the series of integral powers of z

Bs Wann ween ice Bs os) onl cS CNE
is transformed by T. Consequently it has been tacztly supposed that

/

451

the functions (21) form part of the F.F., of T, in other words,
that any function of the series

Cassie Ace he CH al Hee eT Ed

into which (21) is transformed by 7, is regular in a neighbour-
hood of ,.

This tacit supposition, however, is essential. To see this we
observe that, if we. start from the supposition that a series of the
form (1) represents the given transmutation, the existence of the &’s

follows from this. For, that series is finite for any function of the

series (21), consequently its convergence is beyond doubt; we find,

if m,, m,,... are the binomial coefficients of m,

Em ca, + maria, +... + am=(eda)” .-. « (23)
in which the last member is a symbolic form, indicating that in
the expansion of the binomium a* must be replaced by «zp, and in
the term without a, that is the one with zw”, the factor a° must
be added.

Thus, if we should suppose that a” has not a transmuted for
every integer m, the existence of the series (1) would at once be
excluded. |

On the other hand, however, the a’s are completely determined
by the &’s; for from (23), by respectively writing m == 0,1, 2...
we can solve the functions a,,a,, d,,.. „ giving

Om == (80) eae), warm Ae ed

where the second member is again a symbolic binomium, in the
expansion of which §* must be replaced by &£ whereas in the
term without §, that is in (—.)”, the factor § must be added.

The second supposition we have in view, not less essential for
the theorem, is since follows: There is a circle (6) to which all &’s
belong. If this were not the case, the lower limit of the radii of
convergence of the functions (22) would be equal to zero (without
the limit being reached, since this would not agree with the supposition
already made, that all &’s are regular in w,). The existence of the
series (1), in a neighbourhood of «,, for other than rational integral
functions, is then apparently excluded as well, however small this
neighbourhood may be taken. Only in the point w, itself a result
would not necessarily be impossible, but in every case x, could
not be considered as a centre of ‘a domain. : .

We may therefore assume that it must have been Bourrert’s
intention to make this second supposition as well.

That there is a circle (0) to which all &’s belong, also agrees

452

with our conception of the nature of a functional field, as we
set forth in N°. 9: we stated there that the F. F. will of course be
of such a nature that all its functions are regular in a common
domain, which we have called the numerical field of the functions
(N.F.F.); this is here the circle (0). It further follows from (24)
that all a’s belong to that same circle. On the other hand, it can
be inferred from (23) that, if all a’s belong to a circle (6), this is
also the case with the §’s, but PINCHERLE proves that the domain
of validity of the series (1) extends moreover over an infinite aggre-
gate of other, transcendent functions, of which all transmuted belong
to the circle in question.

The functions a, having thus been constructed, BcurLET says:
“Considérous alors la série (1) .

2!

pour toute fonction régulière w qui rend cette série convergente,
cette egalité définit une transmutation additive, uniforme, continue *)
et régulière”. These statements are not further explained by him.

Here, in the first place, the mode of expression does not seem
exact to me; we can say: “pour toute fonction régulière cette égalité
définit une transmutation wniforme et régulière, but not : “pour toute
fonction régulière cette égalité définit une transmutation additive” or
“continue”; for an operation cannot be additive for one function,
nor can it be continuous since for the idea “additive” at least two,
for the idea continuous, in any case an infinite number of functions
are wanted. The meaning, however, cannot be other than the one
which would be expressed if the words “pour toute fonction régulière”
were replaced by “pour l'ensemble des fonctions régulieres considéré.”
But in that case it follows at once from the investigation in N°. 13
(2"¢ communication) that the statement, at least so far as regards
continuity, cannot be true in its generality ; for, we found there
that the series Pu is convergent for all functions belonging to a circle
(3) in all the points of the corresponding domain («), but that it
does not represent an operation that is continuous in the functional
field of these functions with regard to the N.F.O., the continuity
being disturbed in the point «=a.

Since, however, the further reasoning, from which it must appear
that Pu is identical with Tu, as based on the pretended continuity
of Pu, the proof must be considered as incorrect. It might be still

a, | Us "
ihe CR +—yw4t.,..;

‘) The italics are my own. The statement that the transmutation is regular
cannot be objected to if the convergence of the series is uniform in the N.F.

453

meant that the continuity of Pu could be deduced from the supposed
continuity of Zw, and that after this the identity of Pu and Tu
might be proved. But this idea would not do either, and it may
be convenient to set forth this somewhat more clearly, as it may
contribute to enable us to understand the rather restricted tendency of
the theorem X of BourLer. This theorem is of great importance as
it states, broadly speaking, that an arbitrary transmutation may be
developed into a series of integral positive powers of the special
transmutation 0; in this respect it is to be compared with the
theorems of TayLor or Mac-LAURIN in the theory of functions.

15. Before starting with the statement in question we shall for
the sake of brevity introduce a name for the transmutation we have
gradually begun to realize ; we shall call such a transmutation normal.

A normal, additive transmutation is therefore a transmutation for
which the following holds:

1. There is a functional field #'(7), all the functions of which
belong to a circle (6), for which functions the transmutation produces
as result functions belonging all to a circle (a).

2. All rational integral functions belong to the field.

3. The transmutation is continuous in the pair of associated fields
F(T) and (@)

A normal transmutation therefore is always regular, without the
reverse being necessary. It further follows from the definition in
connection with the property of continuity of a complete transmutation
that any such transmutation is also normal. On the other hand,
however, so far I can see, it cannot be proved that the series,
which answers to a normal transmutation, has also necessarily the
property of completeness.

The circle (6) may be equal to and smaller or greater iba (a);
the first is for example the case with the transmutation D-—, ie
second and third with the operation of substitution S,, if w is
respectively equal to $a and 2a (a, = 0).

We now develop an arbitrary function u of F(7) into a series
of powers

uc, He, (a—wx,) + ¢, (e—«,)? +
If we write
Pm (x) =, + ¢, (w@—wx,) +... Homer)” . . . (25)
the fundamental series
DG), Pla), « «Pen B) sas), LE ER (AO)
converges in the domain (0) uniformly towards the function u,
because w belongs to (60).

\

454

The functions a, have now been constructed in such a way that
for all integral values of &

; A i mt (x*)

On account of the additive property of 7’ and P, in connection
with the fact that the quantities px) are rational integral functions
of x, it-follows from this that

BADEN PE (pin) NEP DE iat One > (27)

Since, now, the series of functions (26) in the domain (6) con-
verges uniformly to u, it follows from the supposed continuity of
T (cf. the second form of the definition of series continuity in N°. 9,
nd communication) that the series of functions

T (ps (2), Tp, @) Tom @)…
in the domain (a) converges uniformly towards Tu. The same there-
fore takes place according to (27) with the series
P(g, (@)), P(P, (#)) 5+.» P(Pm(#)) s+ +s
which we express in the usual notation
E(u) = dam. Boal gn Se

m == 00
with the restriction that this equation is valid in the circle (a).

It appears therefore that from the premises follows that lam (Pp, («))
exists in the whole N.F. But the existence of P lim (p (a)) = P(u) *)
does not follow from this. Neither had this been asserted by
BourLeT, who had expressly assumed that Pu exists. But even
if this is done it has not yet been proved that 7u —= Pu, since we
are not allowed to conclude without more that

hin P (pia(a)) =P (lon pn ()) EE

m = oo m ==

This conclusion might be drawn if it were already known that
the transmutation P is continuous in the F. F. and the N. F.
considered, but from the example quoted already in N°. 13, we
may infer that the existence of P in a pair of associated fields
does not imply its continuity in the same fields. Yet, some one
might ask: Is it perhaps possible to prove that P is continuous for
the special fundamental series (26), of which u is the limit, and
which merely consists of rational integral functions? In that case
too (29) would have been proved.

Let us suppose therefore that the series Pu exists in the domain
(a) for a certain function u of (7), of which the circle of
convergence (7), on account of the form of the series P, is of
course greater than (@). We substitute, in the terms of P, for the
function wv and its derivatives their developments in series, which

1) Cf. the examples in Nos. 16 and 17.

455

now also hold in the domain («). For shortness we write wv — 7, = y,
and indicate the coefficients in the various derivatives by means of
accents. We then obtain the following schemes
P (Gm (@)) = a, (Co + +--+ Cm”) +
SN ag ee eg a a an
zl

Lan ie debet Bel | ae
+ ‘
+ Am Bs
P(u)= ale, +. + eny™) + @, (mpi y™t! +...) +
Ha, (do +. + Cmr PTI) + a, (my +...) +

a
+ Gn De Bee teen gean (cn it] Ten ee Ne
+
m) m
ai amc + Am (c! dy aoa
+

Now, it is first possible to find, corresponding to a given arbitrary
small number ¢, an integer Q, such that we have in a point 2 of
the domain (a) at the same time

|P(u) — Pr(u)| <e ‘
| P (Pm (©)) mn (pm (z)) | Se | forn>Q;

in this P,(2) represents the sum of the first 2 +1 terms of the
series Pu. If this has been done the only thing left is to compare
Pu and Pe,(x) for values of n and m>Q. It is sufficient to think
men, for if we choose n>m, the contribution which the terms in
Pu, for which n>>m, produce for the difference between P,u and
Ppa) would be smaller than 2e, because we have already m>Q.

If we therefore choose mn, we have for the difference between
Pu) and Pep,(x) the following scheme

P,(u)—P(gm(#)) = a, (Cm ymtt +...) -f
+ a, (Cny™ + +++) + |
za un 2 alan CA See)

(30)

Wi |
Fan Opt) +

456

1 (n+-1
= An+1 (Cane ae any seert: Greg ym—n—l) +
+.

+ an en

Now, it is far from certain that this amount may be rendered
arbitrarily small. When „» has been fixed (n —n,> Q), an integer
M >> n, may be chosen such, that the part indicated by the bracket
is smaller than ¢, for m> M, but since M for this purpose has
perhaps to be chosen much larger than n,, the remaining part of
the scheme may contain an important number of terms, so that
nothing is to be said about the total amount of them. For, though
the complete, that is infinite series produce an amount smaller than
2e in the horizontal rows from the ordinal number ” + 1 down to
m, — as it follows from the first of the inequalities (30), — this need
not at all be the case with the finite parts of those series, if terms
of different argument occur in them.

So it appears impossible to prove, even when both members evist,
the equality (29), and with this Bourrer’s theorem X in its generality
becomes illusory. Also PincHERLE, who gives the same development
and calls it “série fonctionelle de Tayror”’, points out that the latter -
does not hold without more, and that it has to be investigated in
each special case. He does not mention, however, a category of
cases, in which its validity may be stated, and from the preceding
investigation it is clear that this is not at all easy : we see for instance
that it is not sufficient that the expression (31) for the natural majorant

u of the function w, that is

ulo + aly + lel y? +... (Ye — a)

may be made arbitrarily small in all the points of the domain (a),
in order to deduce from this that this holds as well for w itself;
this would only be possible if all a’s were also majorant functions.

Though the preceding investigation leads to a negative result, yet
it is not without interest, as from it clearly appears, what is wanted
for a statement of the theorem: without a special supposition con-
cerning the a@’s we cannot arrive at a result. As to this point, we
have seen that the series P is, shortly speaking, continuous, if it
represents a complete transmutation *).

1) It ought not to be supposed that if Pu exists in the domain (g) for an
arbitrary function of F(T), the completeness of P would follow from it. This
would only be the case if the functional field F(7') contained all the functions
belonging to a certain circle (9), but it is not at all unimaginable that T'is defined
in such a way that this is not the case.

457

We therefore suppose, more exactly, that the transmutation P is
complete in the circular domain (a), with centre z,, which forms
the numerical field of operation of 7’; further that there are among
the functions belonging to the domain (8) corresponding to (a), such
as are contained in the F.F. #(7’) of 7; the aggregate of these
functions we represent by #'(7’). The equation (28) now holds in
the N. F.Q. (a) of 7’ for all functions contained in the F.F. of 7.
A fortiori the equation holds in the domain (a) for a function u of
F(T). But we further have

EP (pi, (OP BAER VAA Ie 4E Tat (29a)

me

For 1° there is, because u belongs to (8), a circle (o) > (8), in
which the fundamental series (26) converges uniformly towards u,
and 2° the series P, on account of its completeness in (a) with
regard to (8), is continuous in any associated pair of fields, of which
(a) is the N.F. and the F.F. consists of the functions belonging to
(o) (cf. N°. 12, 24 communication).

From (28) and (29) it now follows that

T (u) = P(ú)
this equation being valid in the domain (a) and for the functions of
PECL).

We now have proved a limited validity of Mac-Laurin’s*) theorem
for the functional calculus, which is expressed in the following
proposition :

If the series P, answering to a normal additive transmutation T,
is complete in the circular domain (a), which forms the N.F.O.
of T, we have in this domain

Fu) = Pa)

for such functions of the functional field F(T) of T as belong to
the circle (B) corresponding to (a).

With this for the first time, in so far as I know, an exactly
defined category of cases has been given, in which Tay1or’s theorem
for the functional calculus may be stated.

It follows from it that, before we can proceed to the treatment
of this theorem, we have first to consider those series of the form
(1), which represent a complete transmutation. The treatment of
theorem XI of BourLET must therefore precede the one of theorem X.

Observations 1st. We have not violated the generality by suppos-

!) We call the proposition especially theorem of Mac-LAURIN when we oppose
it as a special case to the general theorem of Taytor, which we are going to
treat in NO. 19.

30

Proceedings Royal Acad. Amsterdam. Vol. XX.

458

ing that P is complete in the whole N.F.O. (a) of 7. For if P were
only complete in a certain part (a@,) of this N.F., we would simply,
in order to be able to apply the theorem, have to replace the first
pair of fields by another, of which the numerical field is the part
(a,) mentioned.

gnd, We have spoken of the part /’,(7’) that the functional field
F(T) of T has in common with that of P, formed by the functions
belonging to (8). In this room has been left for the possibility that
not all functions belonging to (8) form part of #(7’). Should this
ease, however, be realised, then the series P produces of course
at once the ‘analytic continuation” of 7’ over the functional field
formed by the functions in question. Any definition of 7 for these
functions that should lead to another resulf, would make of 7 a
transmutation, which would be no longer continuous for the extended
F.F. The reason for it is that a function of the F.F. added to it,
is the limit, in the domain (3), of the fundamental series (26), all
functions in which form already part of the uncompleted FF.
Consequently we have again for such a function in the domain («)
the equation (27)

L (Gna) = PG), met ee
from which we deduce here, by passing to limits
lim TP) SR (U), Pee soe * (26a)

the counterpart of (28). If, however, 7’ should keep its property
of continuity in the extended F.F., the left-hand member ought
to be made equal to 7u. For, if we have p> o, the function u,
which in the domain (9) is the limit of the fundamental series (26),
is this a fortiori in (6). And if we have 8<{o, and we want to in-
clude all functions belonging to (8) in the new F.F. of 7, the
numerical field of the functions is no longer (6) but (3), so that now
in the latter we have u= lim gp» (2).

16. We now apply ‘“Mac-Lavrin’s theorem” to a few examples.
We consider a neighbourhood of the origin O, and the operation
T= D~'!, which transforms a function u into the integral of that
function, the origin «=O being taken as the lower limit of the
integration. This is a normal additive transmutation. For in the
first place it produces for all functions belonging to a certain circle
(0), functions belonging to the very same circle, with which the conditions
under 1 and 2 for a normal additive transmutation are satisfied
(with #6). It is further continuous in tbe pair of fields connected
with this; for, corresponding to any arbitrarily small number r,

459

there is a number o, such that in the whole field (a)
[DE (i) <a eit, <0,
where M, is the maximum modulus of wu in that same domain (a),
which here represents both the N.F.F. and the N.F.O. of 7.
D-— is a fortiori continuous with regard to the same N. F. (a) in
any F.F. of functions belonging to a greater circle than (a).
For the construction of the series P we have °

If this result is substituted in the formula (24), we find, if mj
indicates the binomial coefficient of m for the ordinal number #,

m—k
Am = » i mr — y= =

=

m

— gat!
toed ) NE (mt De HH.
oi

ml

We now have
m m1
Ni (m+-1)p44 (I= Ni (m+ 1);(—1) = (1—1)"+1 = 0.
= 7
so that

m

yi (m+ Vea (IH = — 1.

0
Consequently
jn 1 ym amtt

ee Sa
D-!u therefore answers to the “series of Mac-LAUurIN”’

im (—-1)" gm yn)
es ml Lm!

GC

Am =

The upper limit
1

ia faethe NRE ae ee ne)
m—=o

is here equal to |x|; the upper limit a (a) of a, for the domain (a),
is therefore equal to «a, and the transmutation is complete in the
domain (@), with a corresponding domain (8), the radius of which,
according to the formula (7) (ft communication), is determined by
p= 2 e= 2a.
30*

460

According to the theorem proved, we have now in the domain (a)
Blauv== DF Ww;
for all functions belonging to (2u), since these functions a fortiori
belong to («), and therefore form part of the functional field of D!
The series Pu does not converge, in the circle («), for functions
that belong to («), without belonging to (2e). On account, however of
what we stated at the beginning of this number about the continuity
of D—', the relation (28) must hold, for the functions mentioned, in
the domain («); this relation here takes the form
DAG) hen P (ga (E)) aen ae (32)

We shall illustrate this by an example. Let be given 4} <Ca<1,
we have then

1
bt 2 ern tm Aje en
(1— 2)’

in which

Pm (2) = 14+ 244+...4 (m+ lem,
is a function that does belong to («) but not to (2a) The trans-
formation D-—'! gives for this in the whole domain (a)

Dt = ) fan me > ~ D = + (33)

The series P becomes

1 1 4 ve \m+1
Ged ier a

This series diverges in a certain part of the circle (a), for example
in the point on the real axis rs =a, since a >> 5. We shall show
now in a direct way that the limit (82) in the whole domain (a)
exists and is equal to (33).

We therefore write, somewhat extensively

gmt + 2a? + 327° + Amt +... (m+ 1), omt

1/
Ee É
ey, f a ht of eee (m + 1): am+i
a?
eg tes a det +... 4+ (m+ 1), oH

4 ant (m)
(—1)m EE Pm == (— ])m (m + er ml,

461

If these equations are added according to columns, the sum of the
coefficients for the A column is equal to 1 + (1—1)*, that is 1 for
every column, and we see how in this way the binomial coefficients,
though increasing towards the right, balance each other. By this it
is explained how it comes that the series artsen after the summation
in question
w (1—-a"T"! )

1—#x

P(gm(e)) =a@+ a? +...4+ e711 —

’

has still a limit for al/ values of « in the domain («) for m =o,
whereas by summation according to horizontal rows when they are
infinite, a limit does not exist for a// such like values. We finitely
arrive at

lim P (gn (2))= ee,

m =o Iz
agreeing with the right-hand member of (33). Thus the general
formula (28), according to which lim Pp” (x) in the whole F. F. of
T, therefore eventually also outside the F. F. of P, must exist and
be equal to Zw, has been proved directly in this special case.

17. As a second example we consider the substitutton. Let w (x)
or shortly w be the function that is to be substituted for w, and
let us take again as N.F, of 7'=$S, a neighbourhood of O;
we have then to suppose that O is an ordinary point of w (x) and
that the N. F. of S, is a domain within the circle of convergence
(A) of w, say a circle with radius @ < A. In order to see which
group of functions have a transmuted in («), we consider the cor-
responding domain 2 of (@), which is obtained by conformal repre-
‚ sentation of the relation vz == w (x), and is a domain round the point
z—=w(0). Let the maximum modulus of this domain be o, then
S,, produces for all functions belonging to the circle (o) a uniquely
determined transmuted v(v) in the domain (a). These transmuted
belong to the circle (a); for it follows from the equation

va) = Sy (u (2)) = u [© (z)],

on account of the well-known theory about functions of functions
that vv) is regular in the domain («) including the circumference.
The conditions under 1 and 2 for a normal additive transmutation
are therefore satisfied. S, is further continuous in the pair of fields
in view: if w(x) tends towards zero in the N.F.F., that is the
domain (0), this is a fortiori the case in the domain @, which lies
within (0); thus v(e) = u [w(e)] tends towards zero in the N,F.O. (a).

The transmutation S, therefore is normal, For the construction

462

of the corresponding series P we have §, =”, which, substituted
into (24) gives
dn = (w — x)"
S,, (uw) answers therefore to the “series of Mac-Laurin”’

(oa
Pe) = Vi APE ulm) ,

The functions a,,(«) are regular in (a), since it had been assumed
that « is smaller than the radius of convergence A of w (x); the

number
1

m ==
is limited in («) and its upper limit is
a (a) =| (am) — am |,

if «, is the point on the circumference of (a), where a, attains its
maximum value. The series Pu represents therefore a transmutation
that is complete in (a), with a corresponding domain (9) the radius
of which is determined by

B=a + | w (en) — em |-

The F. F. of P is in most cases a part of the one of S,, because
as a rule 8 > o. For we have in general, if x, is the point on the
circumference of (a) where w(x) assumes its maximum modulus 6,

B lem | == Ko (tm) — 2m | De 2, | + |W (7) — ds|
> [es 4+ W (ee) — #.| = |e (2.)| =O. ‘

If, however, w(z,) has the same argument as z,, 3 is equal to

(olan); the result 6 >o or

| (am)| > |W (e2)|
would then be inconsistent with the meaning of w (x,); in this case
the points z, and x, must coincide, as well as the circles (0) and (9),
so that now the F.F. of P is exactly equal to the one of S,.

In any case all functions belonging to (8) form part of the F. F.
of S; therefore according to the theorem proved we have for those
functions, in the domain (a),

Sole (m)t

We already observed in the discussion of an example of this
series P, in N°. 7 (2"4 communication) that it might be deduced at
once from the ordinary theorem of TayLor of the theory of functions.
Indeed, if we write

S (u) = u [wm (#)] = ula + (w (2) — 2)],
it appears that S,w in the point z, of the domain («) may be
developed in the series of TAYLoR

463

= W (vw — #@ m
Su (u(#,)) = Yn VAE re
ECA He

um) (7)

if the circle with w, as centre and
4 |w (z,) — 2, |
as radius lies within the domain of convergence of u (x). But this
is evidently exactly the case when w belongs to the circle with O
as centre, and
8 Se a | (tm) DEE ea

as radius, owing to the signification of 2,. We arrive thus at the
same result as higher up, so that the theorem of Tay.or for the
functional calculus, in case the operation is a substitution, may at
once be deduced from that same theorem of the theory of functions.

Now that this has become evident, the fact that the theorem of
TayLor is exactly valid for the functional calculus, if the series P
in question represents a complete transmutation, appears in a clearer
light. For with regard to the convergence of a series, the behaviour
of the terms at infinity only is decisive, and we can say now that
any series of the nature in question behaves, as regards its terms
at infinity, in a certain sense as a substitution. For, since

1
az = lim | am |"
m=

we have for large m approwimately (at least for an infinite number
of terms)

m
| Am | — az

so that, in a definite point w, the modulus of a, behaves for great
values of m as the mit power of a positive number a,. This, now,
is also the case for the substitution, where this number is az = |w(e)—el;
only the behaviour in question has already been present here from
the first values of m.

The domain in which the series P corresponding to a transmuta-
tion converges for a certain function w is naturally smaller than the
circle of convergence (7) of u, because the series P consists of powers
of Du. If therefore the numerical field of operation («) of 7’ is
greater than the numerical field of the functions (6), Pu will not
exist everywhere in (a) for such functions of F(T) as do belong
to (6), but not to (a). But the limit-relation (28),

Ri MEIR (PANEEL Nee ve les AS)
must necessarily be fulfilled everywhere in («) for the last mentioned
functions.

464

The operation of substitution furnishes simple examples of this.
If we have w(e)—= ta, a N.F.O. (2) = 20 answers to the N.F.F. (0).
Let us suppose }< 6 <1, then we have 1<a@< 2. The function

b= forms part of the F.F. of S;,, but does not belong to the

—&£
circle (a); the transmuted of it is

1 2
v—'Sir =S REEN CDE (34)
1—2 Din

and does belong to («).
For the series P we have here

Dll ned oy ne a JE
le) 1e NS

which form becomes illusory in the point a =1. We shall now
show in a direct way, that the limit (28) does exist everywhere in
(a), and is equal to (34).
We have here
Or la tae a5 a + hen En am
EP f

— 4 4—2 (4) #?—3 (4) a? —....—m, (4) a”

+ 5 es (4)?27 3) a? HH, (4)tam

z\em
(—1)" @ ; — = (—1) (4)” am
mM.

If these equations are added according to columns, the sum of
the coefficients for the (4 + 1) column is equal to
| (1 — IE = (4),
so that we arrive at

m

P (gm (&)) = Si (4 «)*
0

This expression has indeed, since « < 2, in all points of the circle

(a) a limit for m= o, the latter being equal to the right-hand
member of (34).

Bacteriology. — ‘Researches relating to the Etiology of Febris
Exanthematicus. By Dr. C. J. C. vaN HooGENHvyze. (Commu-
nicated by Prof. C. Eykman).

(Communicated in the meeting of June 30, 1917).

Many researchers have been engaged in studying the causation of
Febris exanthematicus, and several organisms have already been
looked upon as the causative agent of this disease, such as protozoa,
spirilla, diploeoeei and diplobacilli. Considering, that even in such
cases in which the researcher succeeded in growing pure cultures
from the blood or from an organ, the several findings did not agree,
or did not seem to agree in every respect, so that we do not know
what to think of them, all these inquiries are looked at askance
and it is generally acknowledged that the real etiological factor has
not yet been found. Every one pursuing this line of research should,
therefore, record his results for the purpose of comparison with
those thus far obtained. Fortunately cases of spotted typbus are
few and far between in Holland. When, therefore, some cases oc-
curred at Amsterdam a few months ago, I seized the opportunity to
study them, and determined to publish my observations for the
reasons stated above, although I for one feel convinced that one
investigation is not of great value. The patients were treated in the
Wilhelmina-Hospital. Through the kindness of Dr. Kvreer, the Director,
and of Dr. v. ZADELHOFF, the physician of the ward, I was in a
position to examine the blood of one of the patients who was still
very ill.

This patient F. W. R., 59 years of age, was admitted to the
Hospital on Feb. 24. On the 20 he had had chills. On the 24st he
was laid up with head-ache, gripes, and back-ache.

Status praesens: Patient does not look ill, does not complain;
throat: somewhat red; tongue: coated and tremulous when put out;
full pulse, not dicrotic; Lungs, heart: no anomalies; Bowels: present
nothing particular. Patellar reflexes: high; plantar reflexes: normal.
Many roseolae on chest, belly, back, arms and legs. WipaL negative.
In the urine much albumin, much urobilin. Sediment: some red,
and white blood corpuscles, kidney-epithelium, some cylinders. Blood :
number of leucocytes 7000. Temperature: See list I.

466

Temperature [. Pat. F. R.

Febr. March.
245 26°21 28" 1 Zas 4D ABD ADA

In
ge nâ, a

390 | | | / \ AVA

EEP | \ | Ak
38° bia th A |
37° eea | i

On March 2 I drew by means of a syringe + 20 ec. of blood
from a brachial vein (of course with the necessary aseptic precau-
tions). Immediately I transferred '/, to 1 c.c. of the blood to tubes
filled with 10 ecc. respectively of tapwater, 0.25°/, NaCl, 0.5°/,
NaCl, 0.75°/, NaCl., peptonewater, ascites-glycerin-bouillon, (1 cc.
ascites fluid, 9 ee. of bouillon, 0.4 ¢.c. of glycerin) and 2 °/, ammo-
nium oxalate, two tubes of each sort. The fluids were heated to
body-temperature and in going from the laboratory to the hospital
and vice versa I took preventive measures to insure security from
cooling down. I made use of water and several amounts of common
salt, because, if NicorLe, Conor and Conse. are right in asserting
that the virus is contained chiefly in the leucocytes, I could thereby
liberate the organisms by destroying the leucocytes. According to
GOLDBERGER and ANDERSON, however, who made similar experiments
to those of the researchers just mentioned, and who pointed to their
faulty conclusions, it may be more readily assumed that the virus
occurs isolated in the plasma. However this may be, I thought fit
to take note of both investigations. 1 grew at 37° C. organisms both.
in aerobic and anaerobic cultures. To obtain the latter I put some
pyrogallol and potassium-hydrate upon the cotton plug and shut up
the tubes with rubber stoppers. Of course | also made the necessary
control-experiments.

After intervals of 24 hours preparations were made of all aerobic
cultures; the anaerobic cultures were examined only macroscopically,
to see if there was any growth. After 3 X 24 hours preparations

467

were made also of the anaerobic cultures. The aerobic cultures
yielded the following results: |

Blood in water: a few diplobacilli. Gram_ positive.

Blood in 0.25 °/, NaCl: a few diplobacilli. Gram positive.

Blood in 0.5 °/, NaCl: a few diplobacilli. Gram positive.

Blood in 0.75 °/, NaCl: no micro-organisms.

Blood in peptone-water: no micro-organisms.

Blood in ascitesglycerin-bouillon: a few diplobacilli. Gram positive.

Blood in ammonium-oxalate: a few diplobacilli. Gram positive.

From the tubes, in which diplobacilli had been found cultures
were made upon sloped agar. 24 hours later everything was examined
again. The agar-tubes had all remained sterile. In the other tubes
the following was observed:

Blood in water: a few diplobacilli, to all appearance not many
more than the first time. GRAM positive.

Blood in 0.25 °/, NaCl: many diplobacilli. Gram positive.

Blood in 0.5 °/, NaCl: many diplobacilli. Gram positive.

Blood in 0.75 °/, NaCl: no micro-organisms.

Blood in peptone-water: no micro-organisms.

Blood in ascitesglycerin-bouillon: rather many diplobacilli. Gram
might be called positive as well as negative.

Blood in 2°/, ammonium-oxalate: some diplobacilli, not many
more than the first time. Gram positive.

From the tubes in which diplobacilli were found again cultures
were made upon sloped agar.

After 24 hours it appeared that besides the remains of red blood-
corpuscles, also diplobacilli, very much like those found previously,
were present in the liquid expressed from the agar of the agar-tubes,
inoculated with the blood + 0.25 NaCl and the blood + 0,5 NaCl,
likewise in the liquid of the agar-tnbes that had been inoculated
the day before. All the other agar-tubes had remained sterile. The
preparations of the liquid-tubes were similar to those of the
second day.

The anaerobic tubes appeared yet to be sterile.

The question now arose if what had been found in the fluid
squeezed from the agar, meant growth or whether it had been
evolved by inoculation. I collected the fluid from the several tubes
and inoculated it into some sloped-agar tubes, made a stabeulture
and added broth to the rest to the ratio of 1 expression of water
to 10 of broth. After 24 hours, preparations were made of the
broth and many diplobacilli were found.

When watching the agar-tubes, especially those that had been

468

inoculated with 0.25°/, and 0.5°/, NaCl + blood, the surface did
not any more appear to me quite clear and transparent, though I
did not see any colonies either. Next day little more was noticeable.
However one preparation made of it revealed diplobacilli. 1 now
took a piece of this somewhat turbid-looking agar and ground it
in broth.

In the depths of the stabeulture something similar was to be
seen; preparations of it contained the same diplobacilli. Of this agar
I also put some pieces in broth.

Preparations made from these broth-tubes showed numerous
diplobacilli. I made a subculture of some of this broth upon sloped
agar. Only after 24 hours did I see a few colonies appear. Likewise
a few colonies were visible upon the agar inoculated with expression-
water, very much like the preceding. Repeated subcultures upon
agar resulted in more colonies; though the growth was rather poor,
it gradually seemed to progress.

The colonies produced looked white with incident light, were
clumped and could be easily shifted bodily with a needle all over
the agar-surface. It seemed as if the colonies floated upon the agar.
With transmitted light they looked bluish and translucent.

The bacilli are not all exactly alike as to shape. We found short
rods and longer ones. The short ones generally occur in pairs
joined end to end, so that they might be called diplobacilli. Most
often the centre of the longer ones is of a lighter colour, which
renders the bacillus dumbbell- or biscuit-shaped. Maybe we have to
do here also with two bacilli joined end to end, or all of them are
single bacilli depressed in the middle? However this may be, it is
evident that the two parts are linked together. In some cases we
might even term them diplococci, the cocci being slightly elongated
(coccobacilli). See Fig. 1 and 2 of the plate.

The bacilli (1 will stick to that term) have no spontaneous
motility. They readily stain with the ordinary dyes, most distinctly
with carboi-gentian-violet, which, therefore, I made ample use of,
also in my investigations of the bacilli in the digestive and intestinal
tract of lice. With Lörrrer’s methyleneblue we can see at the poles
more darkly stained granules, occasionally distributed all over the
body. They are Gram-positive and non-acidproof. In ascites-glycerin-
bouillon, and in ordinary broth cultures, especially tbe older ones,
other bacilli are found among the Gram-positive bacilli, that look
exactly like them, but are decolorized.

The bacilli are clumped, which renders suspension difficult.

The best temperature for the growth seems to be 37°. At room-

469

temperature they do not grow at all. At first a culture was made
every day; after some experiments the interval could be lengthened,
at most to about 3 weeks, this being the longest interval after which
growth seems to be possible.

Upon gelatin at 22° the growth is exceedingly tardy. No lique-
faction of gelatin takes place.

Upon LörrLer’s serum: rather poor growth; especially the above-
mentioned granules are made out here.

Upon Endo the growth is also poor. The colonies are white and
change to light red after a few days.

In bouillon a sediment is formed. The broth does not get
evenly turbid, only in the lower layers.

Indol is not produced. Milk does not coagulate. Saccharose-,
lactose-, mannite-, glycose-, maltose-, raffinose-bouillon: no acid-
formation, no fermentation.

In caviae, inoculated with a suspension of this broth, a rise of
temperature was observed after 4 or 5 days, that persisted for a
shorter or a longer space of time as is shown in List II.

Temperature Il. Cavia.
Celsius

39°

38°

37°

epee Ne eee

/

Subcutaneous injection of a suspension of the bacilli.

They then sat quietly in a corner of the hutch and appeared to
be ill. To a second injection a short time after the temperature had
become normal, there was no response and the temperature remained
constant. To all appearance they had become immune as is shown
in List III.

In order to make sure that this was not brought about by a less
pathogenic condition of the cultures, also healthy caviae were
inoculated at the same time. With them, a reaction really took place.

We managed to cultivate the same bacilli with the same properties
from the blood drawn from the heart of one of the caviae by
means of a puncture and that on a day on which the temperature

470

Temperature III. Cavia.
Celsius

39°, i
3g0|_| |

370 NA

36° BEE:

* Subcutaneous injection of a suspension of the bacilli.
*" Inoculated for the second time.

was very high. This experiment was conducted in the same way
as described above.

Lastly I have also been able to study the agglutination. On the
9th of March, ie. 7 days after the blood had been drawn for the
culture, I again drew some blood from the patient. The serum
agglutinated the bacilli in a 1:100 dilution; the serum of two
sound subjects and a sick one (with a high temperature) only in a
1:25 dilution. In a second patient, who had also suffered from
typhus exanthematicus I performed an intravenous puncture 11 days
after the crisis (See List IV). I did not succeed in cultivating bacilli
from the blood. The serum agglutinated the bacilli described above
in a 1:100 dilution. The history of the case is briefly as follows:

17 Febr. pat. (M. J. F. V. 21 years) was down with a bad
head-ache, gripes, fever, pains in the arms and legs, chills. For
about 3 weeks previously he had been languid. Status Praesens:
pulse: quite full, not dicrotic; Coniunctiva slightly red. Diffuse
bronchitis in both lungs; patellar reflexes very high, spleen did not
seem to be enlarged, as appeared from percussion, not palpable.
Scattered roseolae on chest, back, arms, legs, belly, also some in
the face. Hemorrhage from the mucous membrane of the mouth;
here and there spontaneous bleeding. The urine presents nothing
peculiar. Number of leucocytes 8000. Wipar negative. Diagnosis:
spotted typhus.

It appeared, then, that the serum of both patients, one of whom
was recuperating, agglutinated the bacilli in a 1:100 dilution; the
serum of a patient. suffering from another disease with high fever
in a 1:25 dilution; the serum of sound subjects also 1 : 25.

In connection with the communications of pA RocHa Lima and
Téprer I have examined some clothes-lice, taken from the body-
linen of the first patient (R). I prepared as well as possible the
digestive and intestinal tract from the body and ground it-on an

471

Temperature IV. Pat. M. V.
Febr. March.
Dre 28 25526 27) 25 ABD WET ae} BP ATO Te fB 00, OURO DL
| | | | |
Celsius |

|

ape ALA de Rn

BAASIN

38° |: LL
31e BEREK | | ‘A |

| edad

objeet-glass. I was fortunate enough to demonstrate organisms,
exactly similar to the bacilli described above and to those described
by pa Roca Lima and Toéprer. On the photographs 3, 4 and 5
some bacilli are distinctly visible. I could not demonstrate these
organisms in clothes-lice or head-lice taken from sound persons nor
in house-bugs. |

The only difference was the staining, after Gram, as these
organisms were easily decolorised (as pa Rocua Lima also tells us)
whereas the cultivated bacilli were not. For want of material I was
not enabled to grow cultures from the organisms found in lice.

Finally I examined several coverslips of the patients blood,
stained after Giemsa, to detect bacilli and the bodies that various
researchers discovered in the white bloodcorpuscles; here and there
I saw in some leucocytes in the protoplasm beside the nucleus now
punctiform bodies, now again rods with rounded ends, sometimes
indented in the centre. This is exemplified in the figures 6, 7, 8,
9. It is especially the latter that remind one forcibly of the
organisms described above. Staining with methyleneblue made
everything clearer still.

Whether we really have to do here with bacilli which, on account
of their being disposed differently in the microscopic field, present
different shapes, I will not venture to decide. On slightly turning

SS

472

the screw of the microscope it seems as if the point is shifted a
little, so that it may be taken for a rodlet slightly inclined from
the vertical. However this may be, it is at any rate remarkable to
observe the resemblance in the shape of the bacilli found in the
first fluids, the bacilli cultivated, the bacilli in the intestinal tract of
the clothes-lice and the bodies (such I think they may be called)
in the white blood corpuscles of the patiënt. Besides these shapes
in the leucocytes I also, though rarely, detected a diplobacillus in
the coverslips, after a prolonged research.

It must be observed that not in all the fluid tubes bacilli were
found, although they all contained blood from the same syringe
(with a control for an aseptic operation). They were found only in
those tubes in which the fluid was hypotonic relating to the blood;
it should also be observed that after some subcultures, the cultivation
was considerably easier and the intervals between two successive
cultures could be lengthened; again, that the bacilli then seemed to
be less closely packed together and less pathogenic.

It is worth while to compare this finding with those of other
researchers with numerous material.

In my judgment there is a great resemblance between the organisms
found by pa Rocna Lia and Toprer in the body of lice, without
their being able to cultivate them, and those demonstrated by me.
Owing to lack of material I was not in a position to perform
sections in order to observe the different epithelial cells, as described
by pa Rocna Lima and was limited to coverslips only. This may
account for my organisms lying free in the field of view. Entire
epithelial cells I did not see, only fragments of them.

I am personaly inclined to think after consulting the literature,
accessible to me, that the bacilli detected by us when compared
with cultures grown by various experimenters resemble most those
of RagiNowirscH. For the sake of brevity I shall not passin review
all the researches and only refer to the extensive publications of
RaBiNowirscH, who comes to this final conclusion, that in reality all
investigators have found one and the same organism, viz. short
shapes growing in pairs. The slight deviations recorded, are owing
to the different methods of staining. The properties also agree rather
well. This is also the case with the bacilli cultivated by me. Only
the description of the agar-colonies do not apply to my cultures.
According to him they first resemble colonies of streptococci (dew
drop) and afterwards assume a pale yellow colour. My colonies are
compact, white and remain so. With incident light they remind one
of colonies of staphylococci. For the rest they are not at all like them,

473

as set forth in my description. The properties and animal experiments
correspond to those of RaBINoOwirscH and others. RaBiNowirscu believes
that the reason why so many investigators could not produce a cul-
ture, is that the bacilli occur in the blood principally in a certain
period, viz. at the conclusion of the fever-period, immediately before
the crisis, so that, as he puts it, it is not the day of the beginning
of the disease, but the day of the crisis that is of prime importance.

I happened to collect blood from the patient on the very day
when his disease had reached its acme and his life was feared for.
The following day, however, he suddenly took a turn for the better,
but in consequence of several complications occurring with the disease,
the temperature was not indicative of the time of the critical period.

Those who have grown these bacilli agree that it is exceedingly
difficult to produce cultures. Frequently the rods were demonstrated
in the microscopic preparations (e. g. in broth or expression-water),
but they could not be developed any farther. Rasrnowitscu holds that
the blood-serum of the patients inhibits the growth also when diluted.
Very weak dilutions would, therefore, further the growth. Even
the fluid expressed from the agar he considers to be a dilution.
Without touching the agar surface‘he first puts the blood in the expres-
sion-water and only then, after shaking it well he pours it out over
the agar. My success in producing cultures may be owing partly to
similar dilutions in my procedure, and partly to the use of hypotonic
fluids to put the blood in, and to the lucky circumstance of drawing
the blood at the proper moment.

Prorz and later on Poporr are the only investigators who detected
an organism considered by them as the etiological factor of typhus
exanthematicus, differing essentially from all other organisms detected,
this bacillus typhi-exanthematici being obligate anaerobe.

Several investigators succeeded in cultivating organisms not only
from the blood, but also from the urine and the feces of sufferers |
from typhus exanthematicus. They were agglutinated by the blood
of the patients. Hortucu! e.g. produced from the feces and the urine
a paratyphus-like bacillus; Witson obtained from urine and feces
bacilli differing from the Bacillus coli only in that they did not
convert lactose; PREDTJETSCHENSKY managed to grow from urine,
sputum, and bronchial mucus the same diplobacilli as from the
blood; Kropnirzky grew from house-bugs, obtained in a typhus
ward, a culture of small motile very virulent bacilli, called by him
bacillus violentus. However he refrains from considering them as
the agent exciting typhus exanthematicus. Prrruscaky collected from
sputum rodlets which he took to be the cause of the disease; Wer

31
| Proceedings Royal Acad. Amsterdam. Vol. XX.

474

and Ferix grew proteus-like bacilli from urine; Drees managed to
grow from two clothes-lice taken from a patient proteus-like bacilli,
resembling those of Wetn-FEnix.

I will not attempt to criticize, as I am fully aware that one case
cannot yield conclusive evidence. Still, it favours the supposition
that in this patient the bacillus detected was indeed the cause of
the disease, which bad been clinically diagnosed as ‘‘typhus ex-
anthematicus”’. Also in my investigations questions are still left
open, owing to lack of material.

We now summarize our results:

From the blood cultures were grown that may be called diplo-
bacilli on account of the form and the location. The colonies upon
agar look white with incident light and faint-bluish transparent with
transmitted light. They are detached from the agar surface and can
easily be shifted bodily with a needle.

The bacilli possess no motility, take up all dyes and their appear-
ance varies with the dye selected, (finer, coarser, granular, dumbbell-
or biscuitshaped). They are Gram-positive (in older cultures occasionally
Gram-negative); they are non-acidproof; they form no spores.

No growth at room-temperature. Tardy growth at 22°. They grow
best at 37°.

Gelatin is not liquefied; upon LOrrLEr’s serum there is only a
poor growth; milk does not coagulate; the bacilli do not form acid
nor indol. No fermentation in saccharose-, lactose-, mannite-, glycose-,
and raffinose-bouillon.

After repeated inoculations the properties change.

They are pathogenic for caviae (i.e. they cause a rise of tempe-
rature) [could again be cultivated from the blood of these caviae]
and render them immune. Through the serum of the patient bimself
and that of another who was convalescent they were clumped in a
1:100 dilution; through the serum of a sufferer from another disease
and two sound persons only in a 1: 25 dilution.

In the digestive and intestinal tract of clothes-lice, taken from the
patient, organisms were found resembling the above-mentioned bacilli
and the Rickettsia Prowazeki of Da Rocna Lima. With them no
culture- nor animal-experiments could be made for want of material.
The Gram-staining was negative here.

In the coverslips of the blood of the patient very rarely a single
diplobacillus was found. True, some forms were detected outside
the nucleus in some leucocytes that reminded us forcibly of the
above-mentioned organisms (also those in lice).

The Amsterdam Public Health Service Bacteriological Laboratory.

475

REFERENCES.

1. J. F. Anperson and J. GorpBeReER. Collected Studies on Typhus. Treasury
Department United States Public Health Service, Hygienic Laboratory. Bulletin
N°. 86. October 1912.

2. L. Drees. (Deutsche Med. Wochenschr. 1917, N°. 15, p. 461). Das Weil-
Felixsche Bakterium.

3. W. Dreyer. (Arch. f. Schiffs- und Tropenhygiene Bd. XV, 1911, p. 319).
Untersuchungen über den Typhus exanthematicus in Aegypten.

4, B. Ductaux. (Extrait des Annales de l'Institut Pasteur, 1912, April NO. 4,
May N°. 5). Recherches Expérimentales sur le Typhus exanthématique, entreprises
a linstitut Pasteur de Tunis pendant l'année 1911.

5. Fuertu. (Zeitschr. f. Hyg. und Infektionskr. Bd. 70, 1912 p. 333). Die Fleck-
fiebererkrankungen des Frühjares 1911 in Tsingtau und Untersuchungen über den
Erreger des Fleckfiebers.

6. GOLDENSTEIN. (Centr. bl. f. Bakt., | Abt., Orig. Bd. 78, Hft. 12, p. 82, 1916)
Zur Bakteriologie des Flecktyphus. *

7. GorscHLicH. (Deutsche Med. Wochenschr. 1903, N’. 19, p.329) Ueber Proto-
zoén Befunde (apiosoma) im Blute von Flecktyphuskranken.

8. HeyLeR und v. PROWAZEK. (Berl. Klin. Woch. NO. 44, 1913, p. 2035) Unter-
suchungen tiber Fleckfieber.

9. Horrucut. (Centr. bl. f. Bakt., [ Abt. Orig. Bd. XLVI, Hft. 7, p. 586, 1908)
Ueber einen neuen Bacillus als Erreger eines exanthematischen Fiebers in der
Mandscharei während des japanisch-russischen Krieges. “Bacillus febris exanthematici
Mandschurici”’.

10. Krreerr. (Centr.bl. f. Bakt., | Abt. Orig. Bd. XXXVIII, Hft. 5, p. 518 1905)
Bakteriologische Untersuchungen des Blutes bei Flecktyphus.

11. Kirstein. (Veröffentlichungen aus dem Gebiete der Medizinalverwaltung,
Bd. IV, Hft. 9, 1915) Das Fleckfieber und seine Bekämpfung.

12. Kuopnirzky. (Centr.bl. f. Bakt, | Abt. Orig. Bd. 67, Hft. 5, p. 338, 1913)
Beobachtungen über Flecktyphus in Astrachan in den Jahren 1907—’O09.

13. KROMPECHER, GOLDZIEHER und AvuGyaNn. (Centr.bl. f. Bakt., | Abt. Orig.
Bd. 50, Hft. 6, p. 612, 1909) Protozoonbefunde bei Typhus exanthematicus.

14. Lewin. (Centr.bl. f. Bakt., 1 Abt., Orig. Bd. 60, Hft. 6, p. 498, 1911) Zur
Aetiologie-frage des Flecktyphus. .

15. Mark. (Arch. f. Schiffs- und Tropenhygiene. Bd. 17 N°. 23, 1913 p. 805)
Flecktyphus auf Schiffen.

16. MULLER. (Münch. Med. Woch. N°. 25, p. 1364, 1913) Vorläufige Mitteilung
liber bacteriologische Befunde bei Flecktyphus.

17. G. Nrcorre. (Extrait des Annales de l'Institut Pasteur Janvier 1911) Recherches
expérimentales sur le typhus exanthématique entreprises 4 l'institut Pasteur de
Tunis pendant l'année 1910.

18. CG. Nrcorre. Recherches expérimentales sur le Typhus exanthématique. 1911.

19. PETRUSCHKY. (Centr.bl. f. Bakt., | Abt , Orig. Bd. 75, Hft. 5 and 6, p. 497, 1915).

20. Prorz, Orirsky and Barur. (The Journal of infectious Diseases Vol. 17,
1915. p. 1) The Etiology of Typhus exanthematicus. .

21. Poporr. (Deutsche Med. Woch. N°. 16, 1916, p. 471) Ueber den Bacillus
typhi exanthematici Plotz.

fa

476

22. PREDTJETSCHENSKI (Centr.bl. f. Bakt., I Abt. Orig. Bd. 55, Hft. 3, p. 213, 1910)
Zur Frage über den Flecktyphuserreger.

23. —— Abstr. from Arch. f. Schiffs- und Tropenhygiene. Bd. 15, 1911, p. 807.
Weitere Untersuchungen zur Frage des Erregers des Flecktyphus.

24. H. pa Rocua Lia (Münch. Med. Wochenschr. N°. 39, 1916). Untersu-
chen über Fleckfieber.

25. —— (Archiv. für Schiffs- und Tropenhygiène Bd. 20, 1916) Beobachtungen
bei Flecktyphusläusen.
26. —— (Beiheft zu Bd. XXVIII Centr.bl. f. allgemeine Pathologie und Patho-

logische Anatomie 1916. Kriegspathologische Tagung in Berlin am 26 und 27
April 1916). Zur Aetiologie des Fleckfiebers.

27. RaBiNowirscH (Centr.bl. f. Bakt., 1 Abt. Orig. Bd. 52, Hft, 2, p. 173, 1909)
Ueber die Flecktyphusepidemie in Kiew.

28. —— (Arch. f. Hygiene, Bd. 71, 1909, p. 331) Zur Aetiologie des Flecktyphus.

29. —— Deutsche Med. Woch. 1912 N°. 43, p. 2018) Ueber die Komplement-
bindung bei Flecktyphus durch den wässrigen Extrakt aus dem Flecktyphuserreger.

30. —— (Münch. Med. Wochenschr. 1913, N°. 44, p. 2451) Ueber den Fleck-
typhuserreger.

31. —— (Berl. Klin. Wochenschr. 1914, NO. 31, p. 1458) Ueber den Flecktyphus-
erreger.

32. SERGENT, Forey, VIALATTE, Sur des formes microbiennes abondantes dans
les corps de poux infectés par le typhus exanthématique et toujours absentes dans
les poux témoins nontyphiques. Soc. de Biol. 13 Juin 1914. Abstract from Archiv.
für Schiffs- und Tropenhygiene, Bd. XIX, 1915.

33. W. StempeLL (Deutsche Med. Woch. NO. 15, 1916, p. 439) Ueber einen
als Erreger des Fleckfiebers verdächtigen Parasiten der Kleiderlaus

34. —— (Deutsche Med. Woch. 1916, NY. 17, p. 509) Ueber Leukozytenein-
schlüsse bei Fleckfieber.

35. C. S. Stoxvis. Ned. Tijdschr. v. Geneeskunde 1915, 2e Helft, N°. 8, bijblad
(sectieverslagen). Wijze van verspreiding van typhus exanthematicus.

36. H. Töprer und H. ScuiissLer (Deutsche Med. Woch. 1916, N°. 38, p. 1157).
Zur Aetiologie des Fleckfiebers.

37. H. Téprer (Deutsche Med. Woch. 1916, NO. 41, p. 1251) Der Fleckfieber-
erreger in der Laus.

38. E. Wein und A, Ferrx (Wien. Klin. Woch. 1916 N°. 2 p. 33). Zur Sero-
logischen Diagnose des Fleckfiebers.

39. Witson. (The Journal of Hygiene, Vol. X, 1910, p. 155) The Etiology of
Typhus Fever.

Other works on this subject were not accessible to me.

Experimental Psychology. — ‘“J/ntercomparison of some results
obtained in the Investigation of Memory by the Natural and
the Experimental Learning Method.” By Dr. F. Roms. (Com-
municated by Prof. C. WINKLER).

(Communicated in the meeting of June 30, 1917.)
Il.

This paper is to be considered as a sequel to a similar commu-
nication published in the Proceedings of the Meeting of March 31,
1917 Vol. XXV p. 1309, and treats more particularly of the way
in which the experimental method benefits the spontaneous tendency
to use rhythm in learning by heart.

For the method and the technique of our investigation we refer to
our first communication (Ibid p. 1310).

It has long been known that, under the formal conditions of
imprinting the material, rhythm tends largely to save repetitions and
learning time. In daily life people are fully convinced of this; when
they have to iearn something by heart they use rhythm. In our
experiments this tendency was strong with the natural method (I) ;
with the experimental method (JJ) it is still noticeable, though less
distinct. The following tables illustrate the rhythm of the first
repetition of the series and also how the rhythm varies when
the observer gets more familiarised with the material. The
tables were constructed as follows: The duration of the
repetition was split into groups, corresponding with the grouping
of the material according to the rhythm used by the observer. Then
I determined the time required for every syllable of the several
groups by dividing the time required for every group by the number
of the syllables it contained. From these timevalues the means were
calculated. The intervals between two successive groups were deter-
mined from the time elapsing from the moment the last syllable of
a group was pronounced to that on which the first of the following
group was read. In short, we determined the mean time falling to
every syllable of the group in a learning (l.) and a repetition (r)
experiment; when compared with the time of the syllables consti-

478

tuting a group, the time required by the last syllable of a group
indicates the rhythm used by the observer.

As to the tables illustrating the changes which the rhythm undergoes
when the familiarity with the material increases, we have divided
the number of repetitions required in learning and repeating a series
into three groups of successive repetitions in the manner just described
(Vide Proceedings of the meeting of 31 March 1917, Vol. XXV
p. 1316). We then determined the mean time of every syllable of
the various groups and intervals. R. and D. learned the series in
three groups of four syllables. M. in two groups of six, each sub-
divided into three groups of two. The stress fell invariably on the
last of the two syllables. For the rest the tables are self-evident. All
timevalues are expressed in seconds. Tables I, HU, and III refer to
the rhythm of the first repetition, while Tables IV to [X show the
changes in the rhythm resulting from the observer’s greater famili-
arity with the material.

When studying the data of Table 1, Il and III we see directly
that our observers used rhythm already at the first repetition, with
I as well as with II. M. alone, seems to make an exception with
Il; anyhow with him the time falling to the last syllable of the
group is shorter than the mean duration of the remaining syllables
constituting the group. With all our observers the rhythm with I
undergoes a slight acceleration from the very beginning, which is
rather peculiar. (M 0.82, 0.80; R 1.07, 1.04, 1.01 ; D 1.02, 0.99, 0.86).

While the mean time of the syllables, constituting the various
groups, is gradually diminishing from the beginning to the end, the
intervals decrease for R also (1.51, 1.40); for D however they remain
approximately constant (1.43, 1.45)..Since M learned the series in two
groups of six syllables, no data referring to him could be procured.

In the repetition experiments the duration of every syllable is, as
a rule, shorter than the corresponding duration in the learning-
experiments. The same may be observed regarding the intervals,
viz. a decrease with all observers. The acceleration of learning in
the repetition-experiments benefits the group as well as the inter-
vals. Just as the rhythm of the learning-experiments that of the
repetition-experiments is also slightly quickened from the beginning
towards the end, as the acceleration, observed in the learning-and
in the repetition-experiments, follows, almost without exception, the
order of the groups and the intervals, i.e. the farther group and
interval are advanced in the series, the greater is the decrease.

With II there is in the learning-experiments a tendency to
constantly increase the velocity while retaining the rbythm, as the

479

TABLE I. Observer M.

| Arithm. Mean :
Groups | mean deviation Median
| | 0.82 0.06 0.79
[st
r 0.82 0.05 0.82
I | 1631 0.16 1.22
interval |
(20) r 1.29 | 0.10 1.30
l 0:80 hs 0706 0.80
2nd
ee: 0.83 0.06 0.84
| 1 6:96 “=| 0.05 0.94
[st
r 0.94 0.06 0.92
(hit | 0.84 0.09 0.80
interval |
(20) ie 0.88 0.12 0.80
| | 0.80 0.09 0.79
2nd
r 0.78 0.06 0.76

mean duration of the syllables decreases in the order of the groups.
(M 0.96; 0180: Rs 945 0:73: OMD De 093079 045). Le
duration of the interval decreases a little with R; with D, however,
it rather increases. With a single exception only (insignificant for
R, more significant for D) the mean duration of the syllables is in
the repetition-experiments smaller than the corresponding time-values
of the repetition-experiments. The relation of the lengths of the
intervals in the learning-, and in the repetition-experiments presents,
in the main, no uniformity.

For M it decreases a little; for R the intervals increase, while
for D the first interval increases, the second decreases. Irrespective
of the exceptions just alluded to, it may here also be observed that
the acceleration, which the learning undergoes in the repetition
experiments, increases in the order of the groups.

A comparison of the values for the mean deviation is of some
importance as it shows us in how far the rhythm, irrespective of
the peculiarities alluded to above, (acceleration in the order of the
groups etc.), remains constant. Now it is manifest that with I, as
well as with II, in the learning-as well as in the repetition-exper!-
ments, the mean deviation is considerably greater for the intervals
than for the groups. This goes to show that the observer sticks
better to his rhythm in the groups than in the intervals, lt also

480

TABLE II. Observer R.

| Groups | Arithm. mean Mean deviation Median
| | |
| | | 1.07 0.10 1.06
| 1st
| | r 1.02 0.09 1202
Cage | L351 0.40 1.67
‚interval { |
Wes 1.31 0.35 1.45
I eel 1.04 0.12 | 1
¢ 2nd |
ey | |r 0.95 0.10 0.95
| eet 1.40 0.40 1.55
‚interval { | .
r 1.17 0.38 127
| | 1.01 0.12 1.03
3rd
| r 0.88 0.11 0.90
l 0.94 | 0.11 0.90
lst
| r 0,91 0.04 | 0.90
l 1.04 0.12 1
interval
| r 1.06 0.18 1.10
del | 0:13. # Lof 0208 0.73
2nd |
(19) r 0.74 0.09 0.69
| 1.01 0.17 1
interval { |
r 1.05 0.18 1
1 0.72 0.11 0.70
3rd |
r 0.67 0.06 | 0.67

appears, that the mean deviation with I as well as with II, for
groups and intervals is most times greater in the learning than in
the repetition-experiment; in the latter the rhythm is consequently
more constant. It also appears that the experimental method is more
favourable to a constant rhythm than the natural method, the mean
deviation being generally smaller with II than with I. With II it
is especially the intervals that are more equalized.

Concerning the influence exerted on the rhythm by an increased
familiarity with the material, we see in tables IV, V, VI, VII, VIII,
IX in the learning-experiments with I, an increase of the time
falling to every syllable of the group (1, 2 and 3) from A to C,
ie, in the case of greater familiarity with the material, For M and

_

C. J. C. VAN

Fig. 7.

Fig. 8.

Fig. 1. Fir
Fig.i2; Cul
Fig. 3 Pri
thé
Th
Th
(At
(At
(At
(Au

Fig.
Fig.
Fig.
Fig.
Fig.
Fig.

OE

The photo

Fiend:

|

Proceedings Rc

|
|

C. J. C. VAN HOOGENHUYZE. ‘Researches relating to the Etiology of Febris Exanthematicus”.

- x >
,
N »
pe E der
, > gm
: ’ e
ne .
: = ,
t =
)
7 a
Fig. 1. Fig, 2.

Fig. 3. Fig. 4.

EXPLANATION OF THE PLATES.
Fig. 1, First obtained culture from blood of the patient, on agar. (Augm. 1560).
Fig. 2. Culture after inoculation on agar. (Augm. 844).
Fig 3 Preparation of the digestive and intestinal tract from a clothes-louse of
the patient, ground on an object-glass. (Augm. 1560)
Fig. 4. The same of a second clothes-louse of the patient. (Augm. 1560).
Fig. 5. The same of a third clothes-louse of the patient. (Augm. 1560).
Fig. 6. (Augm. 1540),
Fig. 7. (Augm. 1560), ) Leucocytes of the blood of the patient with several
Fig. 8. (Augm. 1500), \ forms of corpuscles outside the nucleus.
Fig. 9, (Augm. 750).

The photos have been made by Dr. L. TH. REICHER, whom I render my sincere thanks.

Proceedings Royal Acad. Amsterdam. Vol. XX.

Fig. 5:

hen

Fig. 7.

Fig. 8.

Fig. 9.

481

TABLE Ill. Observer D.

|

Groups _ Arithm. mean /Mean deviation Median
| | |
|
a! Bay 1.02 |, ts 0.97
Bae Mpa Ci er Se IF. 0.87
ne) 1.43 bo: 18 1.42
interval | |
r | 1.31 0.20 1.32
ial 1 0.99 |. 0.15 0.97
and } |
(18) | | Lat 0.83 AGT se | fen OUT
bree aby «|. i, 20 1.32
interval
r 1.24 40.21 1.20
| |
| | 1 0.86 etl 0:16 0.71
3rd | Í |
A 0.75 | 0.18 0.72
| OAN BO 0.98 0.09 1.02
| | r | 1.07 A ae a ie: 1,07
| | |
| ie | 1.04 0.19 1
| interval
| all 1.06 EAMES 1.10
me || le vee oven nen 0: 05 0.80
II
Eten
8) r 0.72 04 0.72
(| vt 1.13 0.19 1.10
: | interval |
rive 1.09 0.14 1.02
| | 1 0.75 0.07 0.73
| 3rd
| r 0.65 0.08 0.61

R there is always a progressive increase, viz. it is larger from B
to C than from A to B, whereas for D there is in two cases a
decrease, in one case an increase from A to B. The tendency of the
rhythm to accelerate in the order of the groups is maintained also
with a greater familiarity with the material. For M and D the time
invariably decreases from the first to the last group; for R the
time-values of the third group are always smallest, whereas those
of the first are sometimes smaller, then again greater than those of
the second.

In comparing the time-values of the intervals from A to C and
from 1 to 3, again a certain uniformity is to be observed. With
R there is without exception an increase from A to C; with M the

482

TABLE IV. Observer M.

Ist group | interval 2nd group
| | |

erie? Ores tle, Ve 0.83
A || MD. 0,05 | 0.15 | 0.05
| ei ORE EE
| A.M. 1 1.41 | 0.92
LAB | MD er EE 0.07
M. 1 eee eames)
AM: | tate ete 107
Vic MD, || “dap wals (0.18
ee.) eae 1.400% ad Od
a Dm AM age 1 Salvors Des
| ure M.D. 0.05 | 0.41031) 20206
| M. 0.84 | 135 | 0.86
sh. 1 WOMEN) 5494 eee
ee en te MD. ly) ones eon noun
| eas 0.99 | 1.37 0.96
A.M. Ee) Ries (wie (ee Jara epi fog
ly Pha | | MD. 0:37,1 10-89 0.35
| | | | M. | dds | 1.30 | 0.93

interval increases from A to B; from B to C it remains constant.
With D the interval from B to C always increases; from A to B,
however, it always decreases.

Leaving a few deviations out of consideration we can state that
the first interval is either larger than the second or equal to it.
The tendency of the second interval to decrease in relation to the
first, which we found vaguely indicated in studying the tables relating
to the rhythm of the first repetition, is thus seen to maintain itself
all through the learning-process.

With I the data yielded by the repetition-experiments present the
same uniformities, though less distinctly. A greater familiarity with
the material induces a progressive increase of the time falling to
every syllable of the groups 1, 2 and 3 for M, to those of groups
1 and 2 for KR. The time-values for D that formed an exception

483
TABLE V. Observer M.
| | Istgroup interval | 2nd group
A.M. 0.94 0.84 0.79
A MD. | 0.06 0.07 0.04
M. 0.93 0.83 0.78
A.M. 0.93 — 0.85 0.79
l B M.D. | 0.05 0.08 0.04
M. 0.93 0.81 0.79
A.M. 0.94 0.82 0.83
iG | M.D. 0.06 0.05 0.05
Mii og 0.81 0.82
de | AM. | 0.94 0.88 0.80
A | _ M.D. 0.06 0.11 0.06
M. 0.92 0.86 0.77
A.M. 0.91 0.83 0.78
r B | M.D. 0.05 0.08 0.05
M. 0.91 | 0.80 0.77
AM. 0.95 | 0.5 0.80
C | M.D. 0.06 | 0.09 0.06
| M. 0.94 0.81 0.77

even to the general rule in the learning-experiments do not present.
anything uniform here either, except perhaps the average duration
of the syllables of the last group, which is always smallest.

The accelerations of the rhythm in the order of the groups appears
very distinctly in the repetition-experiments with R. and in a
smaller degree with M. With R the decrease of the mean time for
one syllable of the several groups, proceeds regularly from 1
to 3; with M a decrease reveals itself from A to B, while the time-
value in A is abont equal for the three groups. With D decrease
in A is regular; in B and C we observe an increase from 2 to 3.

[f we eliminate D, it appears that the second interval is apt
to increase, in relation to the first as well from A to C as from
1 to 3. This is most distinct for R, who presents only a single
exception. With M we observe an increase from A to B, a decrease
from B to C, whereas the time-values for D are not uniform at all.

484

TABLE VI. Observer R.

| | Ist group | interval 2nd group| interval | 3rd group
A.M. 1.07 1.55 | 1.04 1.40 | 1.06
A | MDI enz Ost | 0.12 0.39 0.21
M. 1.07 1.62 | 1.02 1.55 1.08
AM. | 1.15 1.60 | 1.18 1.50 | 1.05
IASB | M.D 0.18 04 a so 0.53 | 0.21
M 1.10 1.75 1.13 1.50 1.07
A.M 1.29 Be Dd A ales 1.82 (712
| C | M.D 0.52 0.78 | 0.39 0.99 0.50
| M 1.05 1.40 1.14 1-45 IT 084
SAM 0:02 1.29 | 0.95 Bik 0.87
A | | MD. | 0.12 Oar BS 0.40 | 0.13
| M. 0.97 1.50 0.93 1.25 0.90
| AM. 1.04 1.44 0.98 1.75 0.68
r (|B | MD: ||, < 0.22 0.48 0.20 0.79 0.15
|M. 1 1.20 0.92 1.65 0.65
| AM. | 1.16 1560 808° > -1.57 0.79
é | M.D. | 0.42 0.74 Dee 1.44 0.22
MENE 1,07 1.60 ata 1 |" 0.98

Whereas in the learning-expiriments with I the time falling to
every syllable of the group generally increases from A to C, hardly
any regular increase is discernible with II. For M e.g. the time values
remain nearly constant; for R they regularly increase in the first
group from A to C, while in the second and in the third groups
B is greatest. For D we always observe an increase from B to C,
whereas from A to B the time alternately decreases and increases.

The acceleration of the rhythm in the order of the groups is
very distinct with M and R. For both observers the decrease from
1 to 3 in A, B and C proceeds regularly. With D we observe the
same in A and B; in C, however, an increase is observed from
1 to 2, a decrease from 2 to 3.

Whereas the intervals for M in A, B and C are about of the
same length, with R and D we observe a regular increase from A

485

TABLE VII. Observer R.

| ist group | interval | 2nd eon interval | 3rd group
| | AM | 0.93 te | on | 12 0.70
hen de by 0.14) Of +O-083 17 Ogi “| 40.08
Men 2056 | (soap qisr0,41 | 1 0.72
ols AM op 20.8%. || 1 Bhi) 90.90 kelder je 0:08
1 (|B {| MD. | 0.13 | 0.57 ‚| 0.00, | 0.08 | 0.00
| Bae wie es bes lt vovom Ap $408. ED ed
AM. | 0.86 | 1.59 | 0.84 | 1.10 | 0.64
Ey EMD, ol 0:88 nj ORs ok 0520, OAT POM
| M.. | 0.85 Haak dl 0.74y | bb 0e
: VAM of: G.8b op ot 0.7 | 1.05 | 0.70
a )| Mp. | 0.04 Gedy Gh 40.08y1 4009: (| one
sa Nie Nl 0.9 | 1.10 | 0.69 1.05 0.68
A.M. | 0.86 blFe gl 068, 4 | 1.14 0.61
r B Ms (0,10. 1040) “008 0.06 | 0.18 | 0.10
Min lb 65 1.20 G64.) Scag a, abe
MAGNE. ols abn (laeten 0.69, | 1.32 0.61
Vs leo | M.D. | 0.15 0.66 Ole boas | 0.48
| He Mr niee ROG glo NL OO) NN in Gd | 1.15 | 0.67

to C. The second interval evinces, here also, a tendency to decrease.
Anyhow, with R it is, as a rule, smaller than the first.

Whereas in the learning-experiments with II we could hardly
distinguish a regular increase of the mean time falling to one
syllable of the several groups, in the repetition-experiments there is
even a tendency to decrease from A to C.

Whereas with M the time-values for A, B and C are still about
equal, with R we see already a tendency to decrease, and with D
this tendency comes to the front unmistakably (only one exception).
The decrease from 1 to 3 continues regularly; only D presents an
exception.

The intervals in the repetition experiments with II increase
regularly for R and D from A to C, for M they remain about
constant. With R the second interval is always smaller than the
first, with D it is always greater.

486

TABLE VIII. Observer D.

| ist group | interval |2ndgroup interval | 3rd group
| AME d 1.45 0.91 1.45 0.81
A M.D. Deen octal Alo Boe 0.20 0.07
A. 0.94 1.46 0.84 1.36 0.75
AM. 0.90 1.40 0.87 1.36 0.83
I B | M.D. 0.17 0.14 0.15 0.19 0.12
| M. 0.89 1 40 0.80 1.32 0.81
ads poser? {le gash A nadoen
€ || M.D. 0.18 0.33 0.27 | 0.38 | 0.48
| M. 0.84 | 1.35 | 0.91 1,31 0.79
A.M. 0.89 1.29 0.81 B 0.73
A M.D. 0.16 0.19 | 0-169) GONG" LONT
M. 0.90 1,825 | 0.800 | 0.70
A.M. 0.81 1.26 | 0.96 1.33 0.71
r (|B MD. 0.16 0718" | 0.28" 0.800 KOS
| M. 0.83 13e Dol 1.281 | area
A.M. 0.82 1.40 | 0.83 1.22 | 0.79
C | M.D 0.14 0.48 0:10 702287. at
| M. | 0.83 1.20 | 0.82 | 1.20 | 0.73

Considering the values of the mean deviation it appears that,
just as we observed about the mean deviation in the first repetition
it is considerably greater for the intervals than for ‘the group, which
proves that, just as with the first repetition, with I as well as
with II, the rhythm is more constant for the groups than for the
intervals all through the learning- and the repetition-process.

The rhythm adopted by our observers, in the learning- as well as
in the repetition-experiments, with I and II affects the recitation in
that here also the mean time falling to a syllable of a group, is
generally shorter than that of the last syllable of a group, in other
words the recitation also proceeds in groups and intervals (See
Tables X, XI and XII, which are constructed in the same way as
Tables I, Il and III). We shall not enter into a discussion about
the question whether this is owing to the direct tendency of the

487

TABLE IX. Observer D.

Ist group | interval | 2nd Spotl interval | 3rd group
A.M. 1.04 1.05 0.76 Lele 8.78
A MD. 0.06 0.17 0.05 0.15 0.06
M. 1.07 1.01 0.75 1.13 0.73
A.M. 1.01 PAZ 0.71 1.26 0.64
1 a M.D. 0.06 ° 0.18 0.03 | 0.21 0.08
| M. 1 1.15 OM: O28 0.66
ME | 1.28 0.72 | 1.28 0.68
| M.D. | 0.09 0.31 0.08 DOT 10:09
M. 0.99 1.33 Ons a 1520 0.65
is A.M. 1.09 1.06 0.69 1.12 0.65
‚A | M.D. 0.04 0.13 0.04 | 0.14 0.07
M. 1.07 1.07 0.69 dn 0.63
A.M. 1:02 1.15 0268. vj 4515 0.71
v | M.D. 0.06 0.16 ODA ONS 0.16
M. 1.01 Le 0.67 1.16 0.71
A.M, 0.99 1.20 . 0.66 1.24 0.70
fe M.D. 0.10 0.18 0.04 0.18 0.12
M. 1.01 1.19 0.64 1.29 0.72

observers to recite with rhythm, or whether it results from the
circumstance that the rhythm used in learning and repeating the
series, the groups, being apperceptive wholes are easier to reproduce
than the first syllable that forms the transition from the one group
to the other. It is a fact that the rhythm of recitation bears a great -
resemblance to that of the learning and repetition-experiments. In
the same way we notice in the recitation that, broadly speaking,
the mean time falling to a syllable of the groups decreases in the
order of the groups.

This is tbe case in the recitations of I and II and those of the
learning- and repetition-experiments. The time required in the recita-
tion of the last group of the series is remarkably short, a fact that
we also noticed in considering the changes undergone by the rhythm
in the learning process. In this respect the intervals are less uniform.
This is not surprising if we consider that, as stated above, the first

488

TABLE X. Observer M.

EEE EE NN EE

Groups Arithm. mean | Mean deviation Median
|
l 1.36 0.44 1.15
{st
r de BY 0.29 1.06
I | | 2 1.26 Los
interval}
(20) r 2:15 1.05 1.78
l 0.97 0.31 0.87
2nd
r 0.82 0.13 0.77
| 1.20 0.29 1
{st
r 1.04 0.23 1.06
II l 1.26 0.54 0.92
interval
(19) r 1.59 0.98 0.90
| l 0.24 0.92
2nd
r 0.97 0.23 0.86

syllable of a group is much more difficult to reproduce than the
others, in consequence of which the tendency to use rhythm inter-
feres with reproduction-tendencies of different degree.

The process of the recitation agrees in another respect with that
of the mean duration of the time falling to a syllable of one of the
groups, viz. in this that with | as well as with II, the mean duration
is almost without exception, longer in the learning-experiments than
in the repetition-experiments. In a much smaller degree this also
applies to the intervals, their duration being in most cases also
shorter in the repetition- than in the learning-experiments.

The mean deviation is for all observers, with I and with II, in
the learning- and in the repetition-experiments greater for the in- ,
terval than for the groups, from which follows that here also the
rhythm is kept up better for the groups than for the intervals.

The mean deviation being in the majority of cases smaller with
I than with [1 it would seem that, broadly speaking, the natural
method is more favourable than the experimental to an orderly
recitation of the learned series.

CONCLUSIONS.

1. With only a few exceptions all our observers used rhythm

489

TABLE XI. Observer R.
EE IT LEE EL SS SS EE EE es omne en mn

| Groups | - _ Arithm. mean Mean deviation | Median
ea 1.03 0.30 | 0.96
[st | |
| r 1.10 0.39 | 0.90
poo 1.91 1.16 | 1.45
interval | |
| hae eRe 1.68 0.95 | 1.20
I {| asa 1.21 0.56 | 0.88
2nd |
(20) | Herd 0.97 0.28 | 0.83
| 1.42 0.73 | 1.20
| interval |
End Af 1.02 0.38 | 0.90
| AE: 0.82 0.41 0.58
NRE 4
thor 0.73 0.25 0.65
| acd 1.07 0.51 0.77
| [st ‘
| E 0.96 0.40 0.87
| ine 2.01 | 1.56 1.20
| interval
r 1.34 0.64 1
ane A | | 1.31 0.96 0.43
| Ind |
(19) | bate 1.07 | 0.61 0.73
Bayt 1.83 | 1.04 1.40
interval |
r 1.80 | 1.09 1.07
1 1.03 1.03 0.65
3rd
r 0.72 | 0.31 0.58

in the first repetition: i.e. the imprinting occurred already with the
first repetition in groups and intervals.

2. The rhythm of the first repetition shows almost invariably the
tendency to slightly accelerate in the order of the groups. Witb I
(natural method) this quickening confers a benefit on the groups as
well as on the intervals, both in the learning- and in the repetition-
experiments, though in the latter less distinctly than in the former.
With II (experimental method) this quickening is clearly demonstrable
only for the groups. In the repetition-experiments the time-values
of the groups and intervals are always smaller than the correspond-
ing values in the learning-experiments. In the first repetition the
experimental method is generally more favourable than the natural
as to consistency in the rhythm; especially the duration of the in-

32

Proceedings Royal Acad. Amsterdam. XX.

490

TABLE XII. Observer D.

|

Groups | | Arithm. mean | Mean deviation © Median
| |
—
| 0.90 0.16 | 0.83
| [st | |
| r_ 0.79 0.13 | 0.80
| lien 1.28 0.37 | 1.05
interval | *
r 1.11 | 0.20 | 1.10
a Ue ae) 1.05 0.37 | 1.02
zn
(18) | Ne had 0.81 | 0.23 0.75
| Ts 1.60 | 0.65 | 1.30
‚interval { | | |
lr 1.99 | 1.06 | 1.40
alan 0.96 0.21 0.95
3rd |
Re 0.72 0.16 | 0.72
oe 0.99 0.37 | 0.82
Ist { |
| r 0.80 HOI OHS 0.77
ihe 1.43 0.55 | 1.25
| interval { | |
| } or 2.36 1.24 | 1.80
LM Pl En Doane 0.91 0.42 0.66
| 2nd {| |
(8) | r IMT Rag 0.17 0.60
Reape al 1.66 | 0.98 1.07
interval | |
| Ree | 1.50 | 0.74 1.35
| ei | 0.79 | 0.39 0.65
3rd | | |
IGE 0.90 | 0.69 0.47

tervals. is generally more uniform with II, though here also the
rhythm is as a rule preserved better for the groups than for the
intervals.

3. With respect to the changes in the rhythm with greater fam ili-
arity with the material, we observe that most often the time, falling
to ‘every syllable of the groups into which the series is divided,
increases progressively as the observer gets more familiar with the
material. The tendency of the rhythm to quicken in the order of
the groups is kept up when the familiarity with the material
increases. Barring an occasional deviation, the first interval is either
greater than tne second or equal to it. The tendency of the second
to decrease, in relation to the first, as vaguely indicated already in
the rhythm of the first repetition, is kept up all through the learn-

491

ing process. In the repetition-experiments the same similarities may
be discerned. They are however hardly noticeable in employing the
experimental method, though here also, with a greater familiarity
with the material the rhythm is maintained better for the groups
than for the intervals.

4. The recitation occurred in groups and intervals; the mean
time falling to one syllable of the groups is here also in most cases
shorter than that of the last syllable of a group.

With I and with II, both in the learning- and in the repetition-
experiments the mean time falling to one syllable of the groups
decreases in the order of the groups. The intervals are less uniform,
which is not surprising, if we consider that the first syllable of a
group is much more difficult to pronounce than the others, so that
the tendency to use rhythm interferes with reproduction-tendencies
in different degree.

In the recitation also the mean time, falling to one syllable of
one of the groups is longer for the learning-experiments than for
the repetition-experiments with I as well as with II. In a much
smaller degree this holds good for the intervals; nevertheless their
duration is, in the majority of cases, shorter in the repetition- than
in the learning-experiments.

The rhythm of the recitation is also kept up better for the groups
than for the intervals. It seems moreover that the natural method
is more favourable than the experimental to a rhythmic recitation
of the learned material.

32*

Physics. — “On the Fundamental Values of the Quantities b and
Va for Different Elements, in Connection with the Periodic
System. V. The Elements of the Carbon and Titanium Groups.”
By Dr. J. J. van LAAR. (Communicated by Prof. H. A. LorEntz).

(Communicated in the meeting of May 26, 1917).

A. The Carbon Group.

1. For the calculation of the critical data and the values of a
and 6 following from them the elements of this group offer very few
data indeed. Even the situation of the melting point is still insuffi-
ciently known for carbon; of other elements the accurate knowledge
of the boiling point is wanting. Yet it is exactly this group — though
we have often to be satisfied with a rough estimation — that gives
a fine confirmation of the fact set forth by me in the preceding
papers that the values of Wa for elements, which (at least for high
temperatures) occur as separate atoms and not as molecules (either
partially dissociated or not), will suddenly become very much higher
than the normal (rest) values for compounds.

We already found a first example for Antimonzwm and Bismuth,
where War (per gramme atom) present instead of the normal rest
values 9, resp. 11.10 the greatly increased values 32,5 and
36.10 2. The value of Wa, was only partially increased for other
elements, as Arsenic, Selenium and Tellurium, owing to the fact
that these elements are only partially dissociated to separate atoms
at the critical temperature.

For the carbon group we find values for a, which lie between
32 and 40.10-?, so that al/ these elements appear in the form of
separate atoms, as the rest values for Waz, which would hold for
compounds, lie much lower, viz. between 3 and 11 .10-?.

2. Carbon. The melting point lies very high, viz. at 4200°
absolute. Probably this is only the sublimation point at 1 atm. pres-
sure. According to LumMeR’s interésting experiments, described in a
paper published by Virwee: ‘“ Verfliissigung der Kohle” *), the carbon

1) Sammlung Vieweec, Heft 9/10, 1914.

493

melts namely at the ordinary pressure (see inter alia p. 64—65)
exactly at 4200° C. abs. (According to accurate spectrometrical deter-
minations). Though the contents of the said treatise is pretty confused
and incoherent from a physico-chemical point of view, and very little
indeed can be concluded from what is stated for other pressures, yet
this one fact — viz. 7, == 4200 — seems pretty firmly established.
But then the critical temperature lies certainly not below 6000 a
7000° abs., and Vaz must be at least —=32.10-2, when bz =
100.10, as we found before (in I). For the formula

8
Wie Agee
27 by
in which (see earlier papers) the factor 4 at a temperature of 6470°
(see below), where y=—2,11, has the value 0,781, so that with
R=1: 273,09 we get:
‘ ay, a,
Peg h 91 tt OBI 2 22963, 19
bz bz
then gives:
1024. 104

PR keen ATO aha
k By okie Ae

We remind of the fact that the factor A s=s4(4). in
ee a ee
which y represents the reduced coefficient of direction of the imaginary
straight diameter between Dj, and D, in a D,T-diagram *).

It is now the question whether the value of y is confirmed to
some extent by another way. We derived before that 6; : 6, = 2y
(see “New Relations” I). As 6, is expressed in so-called “normal”
units, the value of 5, must still be multiplied by 22412, and this
product divided by the atomic weight A, to obtain 6; in cm’. per
gramme. And as 6, =v,—1: D,, we get:
be X 22412 X D,

2
Y A

?

so that D, can be calculated from
ay kA
ie Beek 22412
in which (cf. also “New Relations” III) 2y can be calculated in
approximation from the formula
F | y= 0,04 77.
We, therefore, calculate the value 1 + 0,04 « 80,4 = 4,22 for
') See for all this my series of papers: “A New Relation, etc.” in These Proc.

of March 26, April 23, May 29 and Sept. 26 1914; resp. p. 808, 924, 1047 and
451; especially 1,

494

2y, so that y has the high value 2,11. [For “ordinary” substances
with’ critical temperature of about 400° to 625° absolute (125° to
350° ©.) we find for this the value 0,9 a 1}. For all other elements
of the carbon group we shall find values for 27, lying between 4
and 3. |

Hence we find:

- 4,22 « 12 _ 50,6
100.105 x 22412 22,4

For graphite (at lrigh temperatures all carbon forms are converted
to graphite, and this is therefore the only stable form at 7%) D = 2,10
to 2,25 at ordinary temperatures, according to different statements.
Thus among others Morssan gives from 2,10 to 2,25, Meyer from
2,144 to 2,25; LE CHATELIER has found 2255 for artificial Acheson-
graphite. The value for tbe limiting density D,, calculated. by us is
therefore in excellent harmony with the experimental value at the
ordinary temperature, which will be only very little lower.

The value of pr is now found from

J

= 2,26. (calculated)

et 1 ag
EEE
in which 2==0,781 (see above). Hence with Wapr—=32.10-?,
Di OO

ore ee oen
Pk == Vole xX 100 10-8 == atm.
From the formula
fk
log? PE — f° (> a 1)
Ps Ts

follows at 7, = 4200° (7; is properly speaking the boiling point,
but probably represents the sublimation point at 1 atm. here) and

Ps = 1;
6470
A168 7," |e ree
4200

from which follows /s'° = 3,473:0,540 = 6,43, ie. f, = 6,43 X
X 2,303 = 14,8. This value is very well possible, as according to
one of our formulae (see “New Relations’ I) f, = 8y, when a
and 6 at 7, are independent of 7’, so that fp would be = 16,9.

And f, is always somewhat smaller than /,.
If 7} is really = 6470° abs, 7}: 7; would be = 1,54 for carbon.

3. Silicium. If we assume here a, = 84.10-2, we get with
A= 0,816 (2y = 3,81, see below) and 6; = 155 . 105 = 15,5 „10%:

495

66,03 X 1156

Te = 4920° :

. 15,5 aes

The melting point, lying at 1426° C. = 1699° abs. (98,9°/, Si)
according to DopriNcKeL (1906), 7%: 7; would be = 2,90.

If we repeat the above check-calculation (§ 2), it follows (2y
becoming — 1+ 0,04 X 70,2 = 381) that
8.81 5< 288.0 oe 1078
Oo = 735, 10-8 x 22412 34,74
This value seems somewhat too high, as at the ordinary tempe-
rature for Si values have been found in the neighbourhood of 2,50
(Wönrer gives 249; in Ruporr’s book on. the periodic system we
find 2,48; etc), but these earlier values may be too low.
For pe we find with 20,816 (see above): -
_ 0,0302 x 1156.10~4
Pk ~~ 940,25.10-8 |
If we assume the value 1,6 for the ratio T,: 7, 7 (the boiling
point) would be about 3080° abs. V. WarreNBere’s value 1902),
viz. >>1205° C. (the melting point n.b. lying as high as 1426° Co,
is therefore rather euphemistic.
The … yalué py, =—.0,)61 20,6 == DA, tenes A= 19,47 would
correspond with 7’, = 3100°. For fe may be expected fi, — 8y = 15,2,

= 3,10. (calculated)

= 1450 atm.

4, Germanium. With Va,= 386. 10~? we find (2 = 0,835, as
2y is = 3,58), br being = 210 00
ge BODE Fae end
21,0 — '
According to Brurz (1911) the melting point lies at 958° U. —
= 1231° abs., hence 7%: 7, would here be — 3,39.
For 2y is found 2y—=1+0,04< 64,6— 3.58, so that we
calculate : . eae

3,58 X 72,5 259.6
— 210.10 22412 = 47.07 OO (calculated)
Winkter found 5,47 for 20° C., so that the agreement is again

striking.

We further calculate for pj:
Pa _ 9,0809 XxX 1296. NO es otal ata

441.10-8 en

With 7: 7,=1,6 we should find about 2600° abs. for 7,
Ruporr gives 7’, > 1300° C. Remark as above for Silicium.

For Jen we calculate 2,959: 0,6 = 4,93, hence /,= 11,4, while
Jk would be = 14,3. iia

496

5. Tin. When we put Va, again 2 units higher, viz.
Vazr— 38.10-*, we calculate with bk = 265 . 10, 2y = 3,44 (see
below), 40,847:

1 ee == aid aks,
26,5 en

At the time (1887) GurpaerG calculated the too low value 3000°
abs. from various data.

For 2y we find | + 0,04 « 61,1 = 3,44, and thus we have:

3,44 « 118,7 Ss 408,3
°—~ 265.105 X 22412 50,4

At —163°,4 Conen and Orie (Zeitschr. f. ph. Ch. 71, 400; 1909)
found for white tin, 7,35 and for grey tin 5,77. The value calculated
by us les near that of white (tetragonal) tin, which is stable at
higher temperature. TRECHMANN (1880) found the value 6,5 to 6,6
for the density of the rhombic tin, which is stable above 161°. Our
value 6,9 lies, therefore, between that of tetragonal (7,3) and rhombic
tin (6,6) 5.

And now the vapour pressure determinations, made by GREENWOOD
in 1911 (Z.f. ph. Ch. 76, 484). Let us, however, first calculate the
probable value of the critical pressure.

For this we find:

0,0314 x 1444104

= 6,87. (calculated)

ES = 650 ;
a 702,25.10-8 AA li
As Greenwoop found 2270° C. = 2543° abs. for the boiling point,

the ratio would be 7: 7, = 1,47, which is too low in my opinion,
so that either the temperature is still higher than 3700°, or — what
in connection with what will appear for lead, is by no means
improbable — the boiling temperature determined by GREENWOOD
has been given too high (or the vapour pressure at that temperature
too low). If 2543° were correct, a value about equal to 2543 >» 1,6 =
= 4100° abs. might be expected for 7, in consequence of which
also Va; would have to be raised to about 40.10-2 instead of
ao 10-7

If the value 3730°, calculated by us, is correct, the ratio 74: 7,

') When we take the density of liquid tin at the melting point, viz. 6,99
(Vicrentin1 and Omoper, 1888), as standard, we might certainly expect Dy to be
>.7. But then it is overlooked that the solid rhombic modification, which bas the
density 7,2 at the melting point (232°), has the so much smaller density 6,5 a
6,6 at the ordinary temperature. Abnormal changes of density are, therefore,
also to be expected in the liquid phase at decreasing temperature, if this phase
could be realized below the melting point.

497

becomes — 3730: 505 — 7,4, which is very high; with 7% — 4100°
or still higher this ratio would even become 8 or 9.

GREENWOOD’s vapour pressure determinations gave the following
result.
2543 abs.

T= 2243 | 2373
1 atm.

p= 0,133 0,345

; i;
From the well-known formula log En =/( 7 =. 1) Gr lag n=
P ;

= (f+ log pp —2 we find:

T, log p,—T’, log p,
(Eee Gia
every time from two successive observations (between which / is
supposed constant), after which 77% can further be calculated from
ST. = T(ft+log pk) — Tlogp. Thus we find from
log p= 0,12385(—1) 0,53782(—1) karo
Tlog© p= 2118-2243 1276—2373 6
resp. f** + log*® pe = 6,681 and 6,453; f'° Tr, == 16950 and 16410.
(GREENWOOD calculates for this the too low values 29: 4,571 — 6,344
and 73900 : 4,571 == 16167).

Hence f'° becomes resp. = 3,868 and 3,640 with p, = 650 atm.,
log’ py = 2,813, from which 4380, resp. 4510, mean 4450° would
follow for Fr. And a modification, even a considerable one, in the
assumed value of px has little influence on this.

The value of /.’° lies here, therefore, in the neighbourhood of 3,75,
i.e. f; in that of 8,6. This value seems too low to us, as 2y—= 3,44
already corresponds with the critical temperature 7), = 3730°, so
that fj, then would be = 13,8. And 2y would be = 3,7 with 4500°,

i.e. 7; = 14,8. Everything points therefore to the fact that the boiling
point determined by GREENWooD is too high, or rather that the
vapour pressures determined by him, have been given too low.

If we retain the value 7);,—3730°, calculated by us, the real
boiling point temperature 7’, would be = 2330° abs. with: 77: 7; ==1 6,
instead of 2543° abs. as GREENWoOOD gives, and the value /,’° would
then be = 4,69, i.e. f, = 10,8.

I + log pe =

?

6. Lead. If we assume here)’ a; = 40.10”, and bj = 320. 10-5
again 55 units higher than tin, though this cannot be ascertained
in default of compounds, the critical temperature and pressure of
which are known), we get with 2y = 3,35, 4 = 0,855:

498

‚16 « 1600
== pote ay = 3469° abs.
32,0

The value of 2y is then =1 + 0 04 x 58,8 = 3,35, and we find
for Di:

we dn AEN = dn = 9,68. (calculated)
320.10-5 X 22412 7172 ——
D has been found Sd at the ordinary temperature, so that 9,7
would be about 15 °/, too low —- unless lead, like tin, is converted
to a less dense modification at lower temperature. Conen and Her-
DERMAN’s researches (cf. among other things Z. f. ph. Chem. 74, 202
(1910) and 89, 733 (1915)) suggest already the existence of even
more than two allotropic forms.
We find for the critical pressure:
ok geh le > 1600.10 ghd NA hfe
1024 .10—8 Sa
For the boiling point GrrENwoop (1911) found 1525° C. = 1798°
abs.; v. WARTENBERG (1908) found 1580° C. = 1853° abs. The two
values do not differ much. If we assume the mean value 7’, = 1825°

abs. as correct, Jr: 7, becomes = 1,90, which is very high, and
fs becomes = 2,69: 0,9 = 2,99, i.e. 7; =: 6.9, whereas f; = 8y = 13,4
is expected. The value of f, would, therefore, be much too low.
And as T,, — 327°,3 C: = 600°,4 abs., 7: 7, would become 5,76.
Possibly the critical temperature has been assumed somewhat too
high. From Grexnwoop’s vapour pressure determinations a tempe-
rature would follow for 7} somewhat below 3000°. But then 4;
must not be assumed to be — 320. 105, but e.g. again 55 units
higher, hence 4/k= 375.10 5. In virtue of the Cerium-Tantalium

period, inserted after the tin, this may not be improbable. With
2y = 3,19, 2 = 0,868 we should then have obtained for 7% the value

70,23 X 1600: 37,5 = 3000° abs., in perfect agreement with GREEN-

woop’s experiments. But a still lower value would then be calcula-
ted for D,, viz. 660,9: 84,05 = 7,86. The value of pz would be

= 0,0321 Xx 1600.10-4: 1406,25.10—-8 = 370 atm., and the two ratios
Tr: Ts and 7;: Ti, would be found resp. = 1,64 and 5.

GREENWOOD’s determinations now gave what follows.

T = 1588 | 1683 | 1798 | 2143 2313 abs.
| | |

p = 0,138 0,350 Kei | 6,3 11,7 atm.
| | |

log p = 0,13988(—1) 0,54401(—1) | 0 | 0,79934 | 1,06819

T log'0 p = 222,1—1588 91571683, ved aise 2535

499

A similar calculation as that for tin yields between each succes-
sive couple of observations:

f'%+ log'p p= 6,301 6,672 4,965 3,573

fT, 11312 _ 11996 | 8927 504

From the two last calculated values for f*° + log’® pz, and f° Ts
appears the inaccuracy of GREENWooD’s vapour pressure determinations,
especially at temperatures higher than 2000° abs. (the same
therefore as for tin). For as log'® pj = 2,69 to 2,57, hence on an
average about 2,63, it would follow from the last determinations
that f'’is = 2,34 to 0,94, which is quite impossible. And if reversely
we take the probable value 4 for f'°®, the much too low values
2200° to 1500° abs. would follow for 7; from the said determina-
tions. ‘ Accordingly, both for tin and for lead, we must reject the
vapour pressure determinations above 2000’ as quite erroneous. It
we only take into account those below 2000°, we find with
log** pr = 2,63 the values f'° = 3,67, resp. 4,04, yielding 7; — 3100
to 29705 mean 3035° abs.; in excellent agreement with the value
of 7; computed by us, when by = 375.105 is assumed (viz. 3000°).
Very little in agreement, however, with the calculated value of 7%,
on the assumption of 6; = 320.10, viz. 3460° abs. It is really
difficult to make a choice here; we do not know in how far
GREENWOOD’s observations below 2000° are reliable. That, however,
v. WARTENBERG found an only slightly divergent value for the
boiling point; is a reason to assume the middle value 1825° for the
boiling point as pretty accurate. But then the critical temperature
can probably not be higher than 2900° or 3100°. (According as
7T;: 7, is taken 1,6 or 1,7). If we, therefore, retain the value
3000° abs.*), which was obtained with 6, —=375.10-5, 7}: 7s
becomes = 1,64. As log'* 370 = 2,57, f,'° becomes = 2,57: 0,64 —=4,0,
ie Js = 9.4, while the value 12,8 is expected for f,=4 X 2y.
(With dog’? p, = 2,57 the value 3,73, resp. 4,10 follows for dak
from GREENWooD’s observations, which is in good harmony with the
above mentioned value 4,0).

7. Recapitulation. Combining the values of Wax, bx, Tx, px, ete.,
found above for the elements of the carbon group, in a comprehen-
sive table, we get the following survey. For lead we have given

1) Gurpsere (1887) gives also for lead the much too low value 7% = 2000" abs.
For mercury the same: 1000° abs., which is also too low.

500

320.10—5 and those corre-

215.10.
Let us in conclusion briefly discuss the minor-

IN | |
|| / (‘4p wo.) |
|_é 98'L || 898'0| GLE || OLE 000E || SLE |
009-| zs || FTI 2 89'6 || cc8‘0| CES | 067 2 O9PE | (oze)| OF |
(419) PRAS (419 wou) | |
& S92 | | é OShP | |
goe |(o¢ez)|| s'L voo | 189 || Lego] vre || oso | cre || goz| ge |
aM l els Jule ge IE
O€ZI (0092) Lv‘G zes || ces‘o| see || 016 | OLIp || O12] WE |
| OOLI \ogoe) | ocz | gore || 918'0! 18'e |) OSFI lohan || (Sel ave |
| — | ooze || eee | oz || ieLol ze lots oto || oor| ze
| | | |
| | Ee
wT s (punoy) | (‘area ee CHE) CSQB) Te alae zl
LL | og og ee Beed ee 7:01 D/ |
rn: || | : |

B. The Minor-group Titanium, Zircon, Cerium, Thorium.
Titanium.

both the values corresponding with bj,
8.

sponding with 6;

501

group of Titanium. Here data are still more greatly wanting, as we have
not even an idea of the probable values of 6; through compounds of
which the critical temperature has been determined. We can, there-
fore, only assume them in approximation between those of the
elements of the principal group. Thus we might put:

Ti Zr Ce Th
Das 109 == 28525 237,5 292,5 402,5
VYa,.10?= 35 37 39 41
With respect to Titanium we find with 2y = 3,68, A = 0,827:
66,92 « 1225
De a ?
k 18,25 4490" abs

In consequence of this 2y—1- 0,04 Xx 67,0 becomes — 3,68.
We then calculate for D,:

Ni eee ese Bias, EE ean 4,33. (calculated)
17 182,5.10-5 & 22412 40,90 ==
At ordinary temperatures Weisz and Kaiser (1910) found 3,99
(amorph, 85,65 °/,, ie. 3,2°/, O, + 11,15 °/, iron) and 5,17 (molten
97,4°/,). Morssan (1895) and Muter (1899) found 4,87 (molten 2°/,C).
At last Hunter found the value 4,50 (100°/,) in 1910. The real value
will, therefore, no doubt lie in every case between 4 and 5, so
that the value calculated by us is again in fairly good agreement.
For the melting point Bureess and WartenNBere (1914) found 1795°,
Hunter (1910) from 1800° to 1850°. If we take the round value
1800° C. = 2073° abs., Tr: 7, becomes = 2,17. We have finally

for px:

D= LE AEN EL = 1130 atm.
333,06 .10—8 Dennen
9. Zircon. Here we have with 2y = 3,51, 1= 0,841:
LA BE LU? OM = 3920 abs.
23,75 —

From this follows 2y = 1 + 0,04 x 62,7 = 3,51, so that for D,
we calculate: rae
op 8,51 X 90,6! «817,6
Ap 28955 100 SLAAT rn
This is again in fairly good harmony with the experimentally
found value, viz. 5,95 to 6,39 (WepexKinp and Lewis, 1910), and
6,40 (Weisz and Neumann, 1910). The former two authors worked
resp. with powdery (96,5 °/,) and molten Zircon (91,3 to 96,5 °/,);

= 6,02. (calculated)

502

the two latter with almost pure molten material (99,7 to 100 °/,).
The value calculated by us seems to be slightly too low. Possibly
br may be assumed too high.

For the boiling point the exceedingly high temperature 2350°
(97 °/,), i.e. 2623° abs., has been found by v. Botton (1910), and
also by Weprkinp and Lewis (1910), so that then 7%: 7, would
amount to = 1,50. Also from this too low amount it would follow

that the critical temperature has been calculated too low in conse-
quence of the value of b,, which has been assumed too high.

If for 6,.10° instead of 237,5 we assume the value 210, which
also holds for Germanium, 7% becomes about 4400°, and the ratio
T,.: T,;, rises from 1,5 to 1,7, the calculated limiting density becoming
slightly greater than 7, somewhat too great therefore.

We calculate for pz:

ea 0,0312 « 1369, 104

— =: 760 atm.
EE 564,06 . 10-8 A as

10. Cerium. As 2y = 3.39, ) = 0,851, we calculate:
68,86 « 1521
Ce 2 8G 55
For 2y follows from this 2y =1 + 0,04 « 59,8 = 3,39, so that
we get: Ss

= 3580° abs.

_ 3,39 X 140,25
“292,5 x 22412

6,92 (98 °/,) was found by Hirscn (1912), and 7,04 by MurHMaANN
and Weisz (1904). The calculated value may possibly be somewhat
too high.

As for the melting point 635° (98 °/,) was found by Hirscn, and
623° C. by Mutamann and Weisz, we may assume the middle value
629° C. —902° abs. to be pretty accurate, so that 7%: 7, becomes
me

For pr we find:

00315 1521 : 104
FES (955.6, 10-8

= 7,25 . (calculated)

= 560 atm.

11. Thorium. As 2y appears to be = 3,17, and therefore 2 is
= 0,870, we get:

70,49 Xx 1681
NAE = S,
ta Aue 940° abs

This makes 2y really =1-+ 0,04 x 54,.2—3,17, and we find
for Dr

503

3,17 x 232,15 135,9
er AOS Dr Orr Lr 90,2

This value agrees pretty well with the earlier density determina-
tions (1863), which gave 7,7 or 7,8 — but badly with the later
ones by Nirson (1882), who found 11,0, and those by v. Botton
(1908), who even found 12,2. The value of bj has, therefore, possibly
been assumed too high.

This is also evident from the value of 7%: 7;,. For the melting
“point Warrenserc (1910) found, namely, 1700 to 1755° C. (The
Thorium contained only 0,15 °/, C.). If we assume 1727° C., i.e.
2000° abs., 7: 7; would become — 1,47, which seems again too

= 8,16. (calculated)

low. The same thing, therefore, as for Zircon. If instead of 402,5.10—6
we assume for 6; the value 375.10—5, which also holds for lead,
Tj, becomes somewhat higher, and then a value in the neighbour-
hood of 9,5 follows for J, hence still too low.
Finally we find for pz:
0,0322 X 1681 . 10-4
SAE 1620 . 10-8

= 330 atm.

12. Recapitulation. When we recapitulate what was found
here, in a table, we get the following survey.

Ty , | P | | Do Dy | 7,

TEEN ec orn OEE ED En nit Wants
H i

{|

Va, .102! b 105! |R 27 i nih
| k | k | (abs.) (atm) | | | (calc.) ‚ (found) br

Ti 35 | 1825? || 4490 | 1130 || 368 \0,827 | 433 | 4a5 2,2

|
Zr 37 | 237,57 || 3920 | 760 || 3,51 | 0,841 || 602 | 6a6,4 || 1,5?
Ce 39 | 29252 || 3580 | 560 | 3,39 0,851 || 7,25 | 69a7,0 || 4,0
| | | | | |

Th | 41 | 40252 | 2040 | 330 3,17 0,870 || 8,162 | 8a 12 1,5?

]

12. Conclusion. It follows most convincingly from all that
precedes that — in order to determine the elements of the carbon
group (and of the Titanium group) — the values of the molecular
attractions War.10-? must be taken very high, ranging from 82 for
carbon to 40 for lead. These values are very much higher than the
residual attractions for the compounds of these elements, which range
from 3 to 11, as we saw before. (See I to IV). This means, there-
fore, simply that we have to do here with free atoms which ex-
hibit a so much greater attraction than the bound atoms in the
molecule.

504

Thus Va=0O e.g. for carbon, which is surrounded symmetrically
on all sides by atoms or atom groups, as in CH,, CCL, C,H, ete.
For doubly bound C, as for C,H,, C,H,, C,H,S, we-find Va=1,55.10~%;
whereas for triple bound C, as for C,H, (likewise for CO, CO,, CS,, ete.),
the full residual value VWa=3,1.10—5 is found (see 1). But
only for the free atoms in the element carbon (C,) we find the so
much higher, ten times higher value Va = 32.102.

And the small deviations between theory and experiment which
still remain cannot detract from this fact — not for the other elements’
either. Whether the value 32 will perhaps have to be replaced by
33 in the end, or the value 40 by 41 or 42 — this does not affect
the above in the least. And it is noteworthy that also the elements
of the minor-group Ti, Zr, Ce, Th., of which so little is known, yet
confirm this important fact in the clearest way. Besides we found
this already indubiously expressed for Antimoniwm and Bismuth (see
IV) with a, = 32,5, resp. 36. 10-2.

As far as the values of bj are concerned, they appear to be the
same as those which are also calculated from the compounds (if
present) — which might have been expected beforehand.

In my next paper I hope to treat the exceedingly important
elements of the group of the alkali-metals, besides those of the
minor-group Cu, Ag, Au.

Clarens, May 1917.

Physics. — “On the Fundamental Values of the Quantities b

and Va for Different Elements, in Connection with the
Periodic System. VI. The Alkah Metals”. By Dr. J. J

VAN LAAR. (Communicated by Prof. H. A. Lorentz).
(Communicated in the meeting of June 30, 1917)

1. After the group of the noble gases, the halogens, the elements
of the oxygen and nitrogen groups, and those of the carbon group,
we will, for practical reasons, first treat the group of the alkali
metals, and not until after this can we treat the intermediate
Beryllium and Borium groups, and the remaining minor groups
with some certainty. | |

The task undertaken by us to compute with some certainty the
values of the critical quantities, and those of bj and Waz, with: the
required accuracy, gets more and more difficult. For the alkali
metals e.g. compounds of which the boiling point or the critical
temperature are known, are entirely wanting, and thus we are deprived
of a valuable test. Nothing is known beforehand that could be
used in any way as a foundation ; everything must be calculated
anew, estimated, weighed, and considered. For an element as
recalcitrant as e.g. Carbon or Silicium, the critical temperature of
which is quite inaccessible, we know at least the value of. bj from

compounds, from which — in connection with other data — the
values of az, Tj and pj can be calculated with almost mathematical
certainty.

This is not the case for the alkali metals. Here nothing is known
beforehand concerning bz, and in most cases we shall therefore
have to be satisfied with defining limits between which the required
values of bz and az must lie. Fortunately these limits are com-
paratively narrow, particularly when the course of the vapour
tension curve is sufficiently known, so that the values calculated
by us can yet lay claim to a satisfactory degree of accuracy.

It will appear that for the alkali metals we are very near the
truth with respect to the critical temperature, when we multiply the
absolute temperature of the melting point by 5'/,, and that of the
boiling point by 1,7. Thus we have the following survey.
Proceedings Royal Acad. Amsterdam, Vol. XX.

| Liane Ne K Rb NE
nn
Ti er 342A | 310,6 | 335,6 | 311,6 301,3
Hei en PE 56,0 “1035,3 971,1 | — 949,1
| |
TX Sa 2411 | toa 1790 1652 1607
RO | = 1965 1150. 47 Aton 1603
= = : Ep — u t nn
TX 4 = il Way ATZD ah BG 1452 1406
ets ka [01734 1554 [edere oo A
And now we shall see in what follows that — at least for Na,
K, Rb and Cs — the real values of 7} lie between those obtained

with the factors 5'/, and 1,7, and those obtained with the factors
4?/, and 1,5. Generally nearer to the first group. Besides we know
already from former considerations that the factor by which the
absolute boiling point temperature must be multiplied to obtain the
absolute critical temperature, lies in the neighbourhood of 1,7. This
factor can also be smaller, but it seldom becomes smaller than 1,5.
For Lithium there is reason to suppose that 7% lies probably higher
than would follow from 7%, XX 5'/,. The factor is there with pretty
great certainty = 5,6.

It is certainly rather remarkable that the ratio 7: 7), is so
constant for the alkali metals, viz. about 37/,.

2. Lithium. The melting point lies at 1790° C. = 452,°1 abs.
According to the above we may expect the critical temperature
between 452,1 x 5'/, = 2411° and 4521 Xx 5 = 2713° abs. We
shall perform the subjoined calculations for both values of 7,
rounded off to 2416° and 2700°.

The value of y follows from our formula 2y =1 + 0,038 / 7%.
It gives with V 7; = 49,09 to 51,96 the values 2,865 and 2,975,
ie. y=1,43 to 1,49. This is accordingly the (reduced) coefficient
of direction of the ‘straight diameter between Dj, and D,.

ENE, i Y
From formula A= —— | —— |, with —— = 0,58 r 0,598,
(4 wi ait Or Ror

hence (5) — 0,347 or 0,358, and with 8y—1 — 10,46 or 10,90

we further find the values 2 —= 0,895 or 0,886 for the factor A in
re erkent AG oel pj: d : i
Mae Pe eat as is renders Ari à: R with

507

8
R=1:273,1 resp. = 72,44 to 71,67 (the value nk R is = 80,915).
(

Let us now calculate the value of 6;. From 6;:6, = 2y follows
wilh seo as De
2 2 A
by == i Ok u x aie |
D, X 22412
when bp is calculated in ‘normal’ units, and per Gr. atomic

weight. We must, therefore, know D,. From the relation for the
ideal straight diameter (while namely the vapour density can be

7

0

2(1+ D

bs
neglected) D= D, — 2D, De follows with D=
k

WE
BSD (: +=) :
1+y 7%

Unfortunately, however, the liquid density for Lithium is unknown
But D = 0,5935 holds for solid Lithium at 15° C. For liquid Lithium
D is therefore slightly smaller than this value, perhaps 2°/, smaller.
Thus we have:

5 D 0,589 x =) = 0,930 D, |
< 05935 = (1- 5 X a) =o f

’

288
or = D, | 1—0,598 « —— | = 0,936 D,
2700

so that D, becomes < 0,638 or < 0,634 (according as 7% — 2410° or
2700° abs.) Hence
2,865 X 6,94 19,88
kP 0638 x B2412 — 14310
: « 22412 14310
2,975 6,94 20,64
or Re ah EE 45
0,634 X 22412 14210
The value of 6, can therefore be at most 2°/, greater, i.e. from
142 to 148105. Now 6, —=:55 for FE, = 70 for 0, = 85 for N,
= 100.105 for C.; we might, therefore, expect for B the value
115, for Be 130, and for Zi the value 145.10—5. If this last value
is correct, 7% would have to lie between 2400° and 2700° for Lithium,
e. g. it would be about 2550°.
Now the value az follows from
Tr X br

af —— ’

p

9 108

St
in which g= ay (see above). This gives:

33*

508

10 X 13,9.10-4 _ 38500 Ba

är kee. 104 = 462,5. 104
72,44 pe

2700 x 14,5. 104 39150 |

ES — 10-4 = 546,3. 104

ee 71,67 a |

from which we find: |
Vaz > 21,5 or > 23,4, 10-2(e.g. Warp = 21,7 a 23,6. 10-2).
For pj, immediately follows from (as 8: R= 8 Xx 273,1 = 2185)

RET se,
the value 2410 2410 a
Pk == = ——— = 777 atm:
2185 x 142.10-5 3,103
2700 2700 Ù
OR Pr =— — 835°" .,
2185 x 148.10— B 3,234

from “which log’ pz = 2,890 to 2,922 follows, which we should want,
when a series of vapour pressures above the melting point were
known of Lithium. But of this exceptional element literally nothing —
more is known than the melting point and the density at 15° C.
Not even the boiling point or the density at the melting point.
Still less the coefficient of expansion in solid and liquid state.

3. Sodium. Here we tread on firmer ground. The melting point
lies at 97°,5C. = 370°,6 abs. The boiling point lies at 882°,9 C.=
= 1156° abs. according to Herycock and LamPLoven (1912). (Rurr
and JOHANNSEN gave 877°,5 C in 1905). Hence the critical temperature
lies in the neighbourhood of 370,6 x 51/, = 1977, resp. 1156 « 1,7 =
= 1965, mean 1970° abs. ; or, as lowest limit, in that of 370,6 X 4?/, =
= 1729, resp. 1156 x 1,5= 1734, mean 1730° abs. We carry out
the calculations again at these two temperatures.

With 7; = from 44,38 to 41,59, we find from our formula (see
for Lithium) 2 y= from 2,686 to 2,580, yielding y—1,34 to 1,29.
For y: (A + y) we find further 0,573 to 0,563, hence for y?: (1 + y)?
the values from 0,329 to 0,317, so that the factor 2 becomes from
0,910° to 0,919, and p from 73,67 to 74,38.

VicenTini and Omoper (1888) found 0,9287 (liquid) for the density
at the melting point 97°,6, so that D, can be calculated from

370,7
0;9287 =D, (1-0. 573 AS ) == 0,892).
970
370,7
or = Dl 10 70 es) == 0,879 DS

yielding D, = 1,041 to 1,056.

509

Then bx becomes:

2,686 < 23,0 61,78 ,
a == == 260%. 10
1,041 X 22412 23330
2,580 X 23,0 _ 59,34 |
of br — 351; 105
1,056 X 22412 23670
We find for ak:
1970 x 26,5.10-—4 52210
AA ende gs ter ear ia ha)
73,67 73,67
1730 Xx 25,1.10-4 48420 5
pat eel Ts DBA 10
anes 74,38 74,31

so that we have Vaz = 26,6 to 24,2 .10-2.

We calculate for the critical pressure:

41970 EE Ee
PRATEN BBR LOSS ABO neen
1730 __1730

OE Pia ED 4: |

2185 x 251.10-5 5,484
which renders log'® pp = from 2,532 to 2,499.

The found values of y, viz. 1,34 to 1,29, may be tested, however
little, by the experimentally found value of the coefficient of expan-
sion a in liquid state. In order to reduce « to y we can derive the
following relation. From

1 vn,
a ee
v, tet,
follows immediately :
1 it
az=D. Le 5 Lan tas Ds:
tt a ae ny

1 3 2 1

so that the quantity y' in D,—D, =’ (t,—-t,) is found from

ya D,, or a from ee
D,
Now (reduced) d,—d, = 2 y (m‚—m,), when the vapour densities

can be neglected, hence as d=D: Dx, and m=T: Tj, also D,- D, =

D
= riba Wy yes T), so that we have:
Ti
Dy
pre SG
aa

and

9 Meel
Y : 0
But Jk EE D, ( =e) and Dr 2 (1 y so that
NY ue RES
Wee pan LEY
OR 7
OLR NPD

EET TE

T,, is therefore always the higher of the temperatures, between

which the experimentally determined expansibility holds. When we

now apply this to Sodium, where 278.10~° has been found for «

between 101° and 168 C. (Hagen), we calculate (7, = 168+4273=441
0,573 _ 0,573

a 0e
1970 —0,573 441 1717

0,563 0,563 ;

OF: a= - _ eo TZ if =s80.10-+|
1730—0,563441 1482

As the coefficient of expansion near the melting point will prob-
ably be still somewhat too small (we need only think of water,
mercury, ete.), the found value 278 is probably to be raised to 334.
If we have to make a choice, the higher of the two assumed critical
temperatures, viz. 1970°, seems in any case to be nearest the truth.

If we assume that the determined coefficient of expansion really
holds for the mean temperature (101 + 168): 2 — 1384°5 C. =
— 408° abs, we should have calculated the values from 330 to
375.106, which are only slightly lower.

Also from VANsTONE's density determinations we can determine
the value of 7’, hence also of a. Vanstone found namely at 110°,
184° and 237° C. resp. the values D = 0,9265, 0,9058 and 0,8891,
yielding y' = 280.10 between the two first, and y = 315 .10-*
between the two last. Or 295.10 between the first and the
third. If we assume this last value to hold at the mean tempera-
ture 173°,5, at which D — about 0,909, then
ged 0 ss
Od
hence very near the above calculated value 330.

If we now assume the newer value 325 .10~® to be more accurate
than the much older value 278, determined at somewhat lower tem-
peratures, then the value calculated above from y with 7% = 1970°
appears to be much nearer 325 than the too high value 380, cal-
culated with 77,—=1730°. 7% lies therefore near 1970°, and (according
to the density determinations) sooner somewhat higher than somewhat

= 325 .10-§ follows from « =y’: D (see above),

511

lower than this temperature; e.g. 7% — 2000° (extrapolated 1997°).

A second test is furnished by the vapour pressure determinations.
We owe the following data to Hacksrirr. (1912).

t= 350 355 365 390 307 | 883° C.
T= 623 . 628 638 663 610 | 1 156° abs.
p= 0,08 0,12 0,15 0,21 0,26 | 160 mm.

log’ p= 0,903(—2) 0,079(—1) 0,176(—1) 0,322(—1) 0,415 (—I) | 2,881
According as 7}=—=1970° is assumed or 1730° abs, we find
log’? pe = 2,532 or 2,499 (see above), and we have therefore, this
being in mm. — 5,413 or 5,380:
Pk

logos = 6,509 6,333 6236 6,090 . 5,998 | 2,532
or = 6477 6301 6,204 6058 5,065 | 2,499
a le EPO OST! | POR ti POT 1,940 0,1042
nld bee. 1710 or 1,609". 71,582 |), 04065
EN bE Ss = 3 2s Bete 8 os
Hence 10 = 3,02 2,96 2,99 3,09 3,09 | 3,60 —> 4,67
or = 3,64 3,59 3,62 3,76 3,77 | 5,03—>»448

We see, therefore, from this that the vapour pressure factor f is
pretty well constant at the lower temperatures 350 to 400° C., but
at the boiling point (the values of the last column on the righthand
side of the vertical line refer namely to the boiling point) it has
considerably increased in both cases: from 3,1 to 3,6, and from 3,8
to 5,0. The latter increment is much too great, the more so as the
limiting value at Tr, viz. fy = 8y = 10,74 or 10,32, so f,'° = 4,67
to 4,48, would be smaller in the latter case than the value at 7,
viz. 5,03, which in view of the great increase of f at higher tem-
peratures is quate unpossible.

We see from the above how exceedingly sensitive the method of
the vapour pressures is, especially at higher temperatures. This is
owing to this, that then (7%: 7) —1 is exceedingly variable on only
a slight variation in the value of 7. In our example from 0,70 to
0,50 for a decrease from 1970 to 1730. And in consequence of this
also the value of f is changed in the same degree (from 3,6 to 5,0).

We can therefore conclude from the vapour pressure determina-
tions for Na with great certainty that 7% —=1730° will be out of
the question, and that 7%—=1970° will be near the truth.

dt
Heiscock still gave Fp 0158 (p. in mm.) at the boiling point
Pp '

temperature 882°,9 C. Now follows from

512

; Ty.
og Es = = 1),
p

hen #’ ts S
WwW n represents ——:
ene eee dv
ee: i nT
re de md (z- 1);
p dT | 1

hence at the boiling point:

Te TA dp bab) Gites 2
i pad aes
ve Tr 5 Be ae 5 )|
This gives with 7% = 1970°, dif kel

_ (1156)

1 n
S= X 9,4843 | —_——__ 0,704 f' |.
a 1970 dn: Peewee iu / |

We can put in approximation 0,51 :486 — 0,00105 for /’, so
that f,'® would become

fs'° = 678 & 0,4348 (0,00860 + 0,00074) = 295 & 0,00934 = 2,76.

According to the above table, this value is too small, as it would
be still smaller than the value of f'° at 397°, viz. 3,09. We expect,
indeed, a lower value than 3,60, as the latter represents the chord
in the curve y= f(z), and 2,76 the tangent — but not a value so

ah
much smaller. The value —- 0,153 given by Hetscock is therefore
° wp
probably too high *) — or else the value assumed for f” is too low.
Also the value 7/=—= 1970 can have been assumed too low.

4. Potassium. After the above explanations we can be briefer,
and simply repeat the same calculations as above.

The melting point lies at 62°,5 C. = 331°,6 abs. The boiling point
at 762°,2 CU. = 1035°,3 abs. (Herscock and L.). Rurr and Jon. give
757°,5 C. The critical temperature lies, therefore, at 335,6 x 5!/, =
== 4790; 1085,3°% 1,7 = 1750, mean 1770°*ortat335 0 > 47
= 1566, 1035.3 « 1,5 = 1554, mean 1560° abs. These will again
appear to be the limiting values.

Thus 7; becomes = 42,08 or 39,50, hence 2y = 2,599 to
2,501, y=1,30 to 1,25. This gives the value from 0,918 to 0,926
for the factor 4, and the value 74,29 to 74,90 for g. For a
we have used y: fy + 1) = 0,565 to 0,556.

1) Or would 0,153 be a printer’s error for 0,135 ? See Tables Annuelles of 1912,
which are full of misprints. Then 860 would become 975, and 0,00934 would
become 0,0105, because of which fs'° would become from 2,76 to 3,09. And this
value is very well possible and would — like the expansibility — point to a value
of Tj; which would be slightly higher than 1970, e.g. 2000° abs,

513

Vicentin1 and Omopri found D = 0,8298 (liquid) at 62°,1. From
this follows therefore:

0,8298 = D,| 1 0 565 See — 0,893 D,

RE en 1770 |
335,2

or =D, —0,556 zer Koi

1560 _ |

from which D, = 0,929 to 0,942.
From this follows for bp:
2,599 X39,1 101,6
TT 0,929 X22412 20825

2,501X39,1 _ 97,79
ee Ee 63 POD
0,942 22412 21120

And further for az:

1770X48,8. 10 t 8638
wes bile es 1168 104

— 488. 105 j

of U ==

Ak =

74,29 T 74,29
3.10-4 _ 72230 :
or er ne esa ave 04s 9084
74,90 — 74.90

vielding Vay= 34,1 a 31,1. 10}.
For pr we find:

1770 ABO
Pe ages des 10—5 10,660.21 ul
1560 1560 ie |
OFS: <= —= =. ”
Pk 978552463. 10—5 10,12

in consequence of which (og’® px = 2,220 to 2,188.
From the formula for the calculation of the coefficient of expansion
« from y, derived above for sodium, we then find:

0,565 peg 565 :
els — 361. 10-6
rea ays 0,565 x 363 1565 |
0,556 0,556 he
Dru = == 4091 rte

1560—0,556X363 ” 1358

„And as « —=299.10-® has been experimentally found by HAGEN
between 70° and 110° C. (mean temperature 90° C = 363° abs),
it follows from this that the value, calculated from y with 7’ —= 1770°,
is nearer the truth than that value calculated with 1560°. (Just as
for sodium the expansibility at 90°C. will probably be lower than
the normal value at higher temperatures, so that 299 will be too
small. Indeed all HAGEN’s values seem to be too small. For Na 278
had to be raised to 325 through the later determinations of VANSTONE).

514

And now the vapour tensions. HackspiLL has found :

t = 264 316 331 340 350 360 365 GRS: ae

T = 537 589 604 613 623 633 638 1035° abs.

P= 0,1 0/15 1,15 1,35 1,75 2,13 2,3 760 mm.
log0p = —1 0,875(—1) 0,061 0,130 0,243 0,328 0,362 2,881

From the above found values of log'°p; we find in mm. log’ p, =
— 5,101 and 5,069, so that we have:

logioP® — 6,101 5,226 5,040 4,971 4,858 4,773 4,739 2,220
ane 5,194 5,008 4,938 4,826 4,740 4,707 | 2,188
1 = 2,296 2005 1982 1,887 184 1,196 1,774 | 0,097
of=1,905 1,648 1,584 1,544 1,504 1,466 1,445 0,5067

yielding f10=266 261 261 263 2,64 266 267 3,13->4,52
of=3,19 3,15 316 3,20 321 323 3,26 | 43244

From the same considerations as for Na it follows also here again
very clearly that the upper series of values is better than the lower
one, and that therefore 7;—=1770° is preferable to 1560°. The
limiting values of f for 7; are viz. fp = 8y = 10,40 to 10,00, or
fie’ = 4,52 to 4,34. Probably the accurate value of 7% lies some-
what below 1770° abs.

Herycock gave 0,135 for 7 When we assume /' = 0,46: 397 =
—0,0016, the following formula follows for f,'° with 7, = 1770
T, = 1035 from the formula derived in § 3:

2 (1085) |
A ee 0,135 « 760

= 606 X 0,4343 (0,00975 + 0,00082) — 263 « 0,0106 = 2,78.

This value may be correct. It is larger than 2,67 and at the

same time smaller than 3,13 (tangent and chord, see for Na).
Perhaps 7; lies again somewhat below 1770° abs.

zb KO)
s

< 04543) 0710 Se 0,00116 |

5. Rubidium. 38°,5 + 273,1 —311°,6 was found for , the
absolute melting point temperature. This multiplied by 5'/, gives
T;, = 1662°. Rurr and Jon. (1905) found for the boiling-point
691° C. = 971°,1 abs. This 1,7 gives 1651. Let us take the round
value 1660. On the other hand 311,6 Xx 4°/, = 1452, and
971,1 X 1,5 = 1457, averaged and rounded off 1450.

In consequence of this 7% becomes = 40,74 to 38,08, hence

515

2y — 2,548 to 2,447, y= 1,27 to 1,22. We find therefore 0,560 to
0,550 for y:( + y), hence according to our formula 2 = 0,922 to
0,930, and p= 74,60 to 75,25. Hence
Hy 2,548 « 85,45 _ 217,7
~ 1,648 x 22412 36930
2,446 « 85,45 < 209,0
1,673 x 22412 37500
D, being =1,648 to 1,673. For, according to Hacxspit, D = 1,475
(liquid) for 38°,5, hence according to the formula derived by us:

: — 590.10-5

’

or by = —= 557.105

311,6
= 1— 0,560 ——_ | = 0,895 D
1,475 », ( eo) :
311,6
== a ——— | = 0,882 D
or D, (1 0,550 vet ) i

We find further for az:
1660 X 59,0.104 97940

apen RE a Qeek ot (818. 104
74,60 74,60
1450 X 55,7.10-4 80770 |
ari ape me hand EO SENT ih
75,25 . 75,25

giving War =36,2 a 32,8. 102,
We find for px:

1660 ane en
TEE m.
Pk =~ 3185 X 590.105 12,89 |
1450 1450
Of Pk = ——— >> SE DD „

2185 X 557.10—-5 12,17

which causes /og’® py, = 2,110 to 2,076.

From the above values of y we find for the coefficient of expansion
a, resp. with 7% —= 1660 and 1450 abs.:

0,560 0,560
Oe ee a ee a Ome
1660 — 0,560 x 363 1457
0,550 0,550
or = = 440.10—6

“=~ 7450 — 0,550 X 363 1250

- while HackspirL gives 339.10-® between 40° and 140° C. (mean

temperature 90°C. = 363 abs.). Here too the first value 38.105

is nearer the experimental value 34 (which will have to be raised

somewhat, see for potassium and sodinm) than the second 44.105.
HacksPILL gives for the vapour tensions of Rubidium:

516

fi) 292 305 … 930. 333-340.” 346.350... 353, “256! 13091030 698° C.
15 565 578 603 606 613 619 623 626 629 638 640 971°abs
p =0,06 0,98 1,46 2,66 2,95 3,29 3,67 40 4,25 451 5,51 614 | 760mm
log!0 p =0,778(—2) 0,991(—-1) 0,164 0,425 0,470 0,517 0,565 0,602 0,628 0,660 0,741 0,788 | 2,881
As log'*p, in atm. = 2,110 to 2,076, this is in mm. 4,991 to
4,957, and we find then:
login” = 6,213 5,000 4,826 4,566 4,521 4,474 4,426 4,389 4,362 4,331 4,250 4,203 | 2,110

Tk

conse- f10= (2,86)

quently

of = 6,179 4,966 4,792 4,532 4,487 4,440 4,392 4,355 4,328 4,297 4,216 4,169 2,076
A 1= 2,174 1,938 1,872 1,153 1,739 1,708 1,682 1,664 1,652 1,639 1,602 1,594 | 0,7096
of = 1,772 1,566 1,509 1,405 1,392 1,365 1,343 1,327 1,317 1,305 1,273 1,266 | 0,4933

2,58 2,58 2,60 260 2,62 2,63 264 2,64 2,64 2,65 264 2,07—»4,43

of = (3,49) 3,17 3,18 3,22 3,22 3,25 © 3,27 3,28 3,29 3,29 3,31 3,29 | 4,21—>4,25

Here too the first row of values appears to be nearer the truth
than the second. As / is still greatly increasing at 7’, the values
past 4,21 cannot possibly remain below the limiting value 4,25 (for
T;). This limiting value is namely pz = 8y = 10,19 to 9,79, i.e.

k’’ = 4,48 to 4,25. Perhaps 7% lies slightly below 1660° abs.

6. Caesium. At last the last member of the group. Here the
triple point is at 28°,25 C. —= 301°,3 abs.; the boiling point is at
670° C. (Rurr and Jou.) = 943°,1 abs. So that 7% will lie between

301,3 < 5'/, = 1607, 943,1 X 1,7 = 1603, mean 1605° abs, and

301,3 X 4°/, = 1406, 943,1 « 1,5 = 1415, averaged and rounded off
1410° abs.

We find therefore 40,06 to 37,55 for 7%, yielding 2y = 2,522
to 2,427, y=1,26 to 1,21. The value of y:(1 + y) becomes
0,558 to 0,548, so that 2 becomes = 0,924 to 0,931, and p = 74,77
to 75,36. And we find for bj:

2,522 x 182,81 334,9

eN in a
+ 9.061 Sc 29412 46190 q |

)

== 666% ol

’

2,427 Xx 182,81 322,3

OM ys — SS PRs
k 3,090 X 22412 46840

because D, = 2,061 to 2,090. For the density at the melting-point
28°,25 HacksPILL gives namely D= 1,845, so that

517

301,3
1,845 = D, | 1—0,558 —— ] = 0,895 D,
1605

301,3
or = D,( 1--0,548 ——— | = 0,883 D,
1410
And we find for a; :

1605 X72,5.10-* 116360
Oh ee a 10-4 5000
74,77 eg |

1410x68,8 . 104 97010 |
Dn ED Oa eae gg
75,36 — 75 36

giving Va, — 39,4 a 35,9. 10-2.
And for pj, we find:

1605 EREN om
dead —— 5 = SS ESS a m.
Pk = 9185125. 10-5 15,84
1410 _ 1410 :

Pk oie5sces8 10-5 15,08

giving log’’p, = 2,01 to 1,97.
The coefficient of expansion may be calculated from

0,558 10; 558 Cee ta
— 1605—0,558 8387 1417
0,548 0,548 |
Of U ZZ Er => 447 . LO

1410—0.548 337 1225

The value 39.10-5 was found between 17° and 100° C. (mean
63°,5 C = 337° abs.) by HckHarp and GrarkFe (1900). As these
experimental values had to be raised a little nearly everywhere in
order to get into agreement with the normal expansibility at higher
temperatures, given by y (the older values of Hacen for Caesium
lie still lower, viz. mean 345 .10-®), it is possible that the critical
temperature of Caesium will lie between 1605° and 1410° abs.

Let us consult the vapour tensions. Hackspi1 found:

= 230 244 212 308. 315... 330°. 333 , 350 . 365, 397 670° C.

T = 503 517 545 581 588 603 606 623 638 670 | 943° abs.
p=0,2 0,29 0,99 2,58 3,18 4,27 445 6,72 9,01 15,88 | 760 mm.

logip =0,301(—1) 0,462(—1) 0,996(—1) 0,412 0,502 0,630 0,648 0,827 0,955 1,201 2,881

We find 4,886 to 4,853 for /og'*p,, and further:

518

logi0 B 5,585 5,424 4,891 4,475 4,384 4,256 4,238 4,059 3,932 3,686 | 2 006

of = 5,552 5,391 4,857 4,441 4,351 4,223 4,205 4,026 3,898 3,652 1,972

ee —1=2,191 2,104 1,945 1,762 1,730 1,662 1,649 1,576 1,516 1,396 | 0,702

of = 1,803 1,727 1,587 1,427 1,398 1,338 1,327 1,263 1,210 1,104 | 0,495

hence f!0—2,55 2,58 251 254 253 2,56 2,57 2,58 2,59 2,64 | 2,86—>4,38
of = 3,08 3,12 3,06 3,11 3,11 3,16 3,17 3,19 3,22 3,31 | 3,08— 4,22

It also appears clearly from these values of f, that the real critical
temperature will lie between 1605 and 1410. The limiting values
of fat Tj, are f, = 8y = 10,09 to 9,71, hence f7'° = 4,38 to 4,22.
The value f,° lies slightly too far from 4,38; the value 3,98 lies
too near it.

7. Recapitulation of this group. In accordance with the
course of the vapour tension factors f we shall assume the critical
temperature of sodium to be 2000° abs., i.e. */, of the difference
between 1970° and 1730° higher than the first of these values.
Further that of Potassium to be 1710°, i.e. */, of the difference
between 1770° and 1560° abs. lower than 1770°; that of Rubidium
to be = 1590°, i.e. '/, of the difference between 1660° and 1450°
lower than 1660°; that of Caesium — 1510°, 1. e. */, of the difference
between 1605° and 1410° lower than the first value. At last that
of Lithium, in virtue of the value of 6; (which we assume = 145 . 10—5)
halfway between 2410° and 2700°, i.e. = 2550°, so that we get
the following survey.

| | [ i] 1] ;
| | || Tp) 7 | bp | V |
loen | ik Tp || k k pitt RS Ak | Pk fi

| | rr ll | £,10
| Tel Zelle p< rospie| (atm) Tr.—kp. Sk
Lithium | ‚452 14507 2550 | [5,6 | 1,16? 1,46 | 145 | 22,6 (806) — [5,1

Sodium 371 1156 | 2000 54 1,73 | 1,35 | 266 | 26,9 | 343 2,9 35 4,7

Potassium 336 1035 | 1710 | 5, | 1,65 || 1,29 | 481 | 33,2 || 163 | 28-34 | 4,5

| | | | | || |
Rubidium 312 o71 \1590 51 1,64 125 510 354 |
Caesium | 301} 943 | 1510 15,0 | 1,60 |) 1,24 | 207 | 37,7 ||

126 | 28-34 4,4
98 28 3,4 | 4,3

These are the most probable values following with pretty great
certainty from the available data, The inaccuracy will at most amount
to 1 or. 2°/,-

With regard to the values of bj, we will only observe that they

519

are about in the following ratio 1:2:4:5:6. For 145:1—145,
266% 22=—too, 481 34190) 579: 5 == MORTON 6=1 18:

From the value of 7; and pz for Hydrogen, determined just now
by K. ONNEs c.s., would follow the value 59.10-5 (per Gr. atom)
for bz, i.e. exactly half the middle value 118 for K:4, Rb:5 and
Cs:6. So that the ratio of the b-values for H to Cs would become
ele ee ed 6.

Possibly the ratio values for Li and Na will later have to be
rounded off to | and 2 on more accurate knowledge of some data.

And it appears again from the values of Waz, for which rounded
off we may write 28, 27, 33, 35, 37,5, that all these metals occur
atomically with the very much increased valency attractions. If they
were bound to Li,, Na,, ete, only the ’’rest-attractions”’ 3, 5, 7, 9,11
would have manifested themselves (per G. atom). For Li it is possible
that undissociated molecules of L, are still present at 7 (the
abnormally low value 23 would point to this), but it is also possible
that this is not the case. All these questions must be left open for
the present till the whole periodic system sball have been examined.

In a following paper the minor group Cu-Ag-Au will be treated,
besides Manganese and the Iron-Platinum group.

Clarens, June 1917.

Chemistry. — “On Milk-Sugar’. 1. By Prof. A. Smits and J.
Gius. (Communicated by Prof. S. HOOGEWERFF).

(Communicated in the meeting of Jnne 30, 1917).

1. Introduction.

When in 1880 SCHMOEGER *) and ERDMANN *) began their investigations
about milk sugar, besides the hydrate, an anhydride was known, which
was obtained by heating the hydrate in a drying stove at 125°.
Later on this anhydride, in contradistinction to another anhydrous
modification, was called the «-form. This other anhydrous modifi-
cation, the p-form, was obtained by SCHMOEGER and ERDMANN by
evaporation of a saturate solution of milk sugar at the boiling
temperature (+ 108°).

That this was another anhydrous modification than the «-form
followed from this that while the «-modification is very hygroscopic,
gives a clear yeneration of heat when brought into water, and
yields a solution, the optic rotatory power of which decreases with
the time, the #-modification on the contrary is not hygroscopic,
dissolves under heat-absorption and yields a solution, the rotation of
which decreases with the time.

Hupson *) was the first to consider the problem offered by the
milksugar, from a physico-chemical point of view. He demonstrated
that whatever form of the milk-sugar is dissolved in water, the
final condition is always the same, and represents an equilibrium.

Hupson showed further that the muta-rotation has the same course
as a mono-molecular reaction; he determined on one side 4, + k,,
on the other side &,, and thus found 4, indirectly; in this way he

k
got K’ == 1,6 for the constant of equilibrium at 11°,2.

2
Hupson started from the supposition that the hydrate possessed a
high rotation, and the g-anhydride a low rotation, and he was of

opinion therefore that the said equilibrium was to be represented by
the equation:

1) Berichte 13 1915 (1880).
2) Berichte 18 2180.
*) Zeitschr. Phys.-Chem. 44 487 (1903).

521

or
C,H, OF — Cy. 05, a H,O
K (Anhydride) (Water)
C= —
(Hydrate)
or |
el sel (Anhydride)
(Hydrate)

Through the slow setting in of the internal equilibrium in solution
milk-sugar hydrate presents the remarkable phenomenon that the
initial solubility is much smaller than the final solubility.

As, when the total concentration is not too great, it may be
assumed as certain that the concentration of the hydrate in the
solution in equilibrium with the solid hydrate, remains permanently
the same, the constant of equilibrium will follow from the determina-
tion of initial and final solubilities, for then:

final solubility — initial solubility __ (Anbydride)
> initial solubility (Hydrate)

In this way was found A’ — 1.44 at 15° *), hence a value which
is considerably smaller than that found from the reaction velocities,
from which it therefore follows that the total concentration for this
determination of A’ was already too great.

From initial and final rotation of hydrate and g-anhydride also
the constant A’ can be determined. |

Hupson found [a] a 86°,0 as initial rotation for the hydrate,
and [a]2—35°,4 for the g-anhydride. On the other hand the
value 55°,3 had already been fixed for the final rotation by SCHMORGER.

From this follows for 20° *) :
186,02205.8
558354
a value which lies between the two others.

In consequence of the slow setting in of the internal equilibrium
in the solution it was possible to determine the initial solution heats
of the different modifications, and likewise the heats of the trans- |
formation of the hydrate and of the «-anhydride in the g-anhydride
in dissolved state.

In this Hupson and F. C. Brown*) succeeded according to a
method of H. T. Brown and Pickerine ‘).

ree:

Ki ad

1) J. Amer. Chem. Soc. 26 1074 (1904).
% J. Amer. Chem. Soc. 30 1781 (1908).
3) J. Amer. Chem. Soc. 30 960 (1908).
4) Chem. Soc. 71, 782 (1897).
34
Proceedings Royal Acad. Amsterdam. Vol. XX.

522

The following results refer to 1 gram of anhydrous milk sugar
and to the temperature of 20°.

Cag (solid) — a, (dissolved) — 12,6 cal. (initial heat of solution). (1)
BL A BC Ne eet otk 5 ae)
EN) > ct, ( … )+ 7,3 „ (hydration heat + inital
heat of solution of «‚,). (9)
Ban (dissolved) > Brie ae) + 1,05 ,, (heat of transformation
in solution '). (4)

From these data the following calculation can be made:

from (1), (2), (4): aag (solid) — B (solid) + H,O — 9,25 cal. . (5)
from (1), (3): a (solid) + H,O — ata, (solid) + 19,9 cal. . (6)
from (5), (6): a (solid) — 9 (solid) + 10,75eal. . . . . (7)

In these determinations HupsonN found also that the difference
between initial and final heat of solution, both starting from
a-anhydride and from hydrate, had exactly the same value, from
which, therefore, follows that «-anhydride in contact with water
hydrates immediately, and that then the following slow conversion
takes place: .

Hydrate — g-Anhydride + H‚O.

We also come to the same conclusion when we consider that
a-anhydride and hydrate show the same initial rotation.

Hupson has also determined the temperature where the trans-
formation of hydrate to g-anhydrate takes place, i.e. he determined
the temperature at which the coexistence Sy + 5e + L occurs.

From the final solubility of the hydrate between 0° and 89° and
from two final solubilities (0° and 100°) of the g-anhydride followed
92° as transformation temperature. *)

He further determined this point from the intersection of the
three-phase lines of Sy +2+G and Sq +8S;-+G, and then
the temperature of the quadruple point Sg + Ss + L + G was
found at 94°.

Attempts to determine the transformation point thermically and
dilatometrically proved unavailing, because of the exceedingly small
velocity of conversion.

On the ground of these data Hupson in 1910 drew up a general
theory of miuta-rotation*), which for all sugars can be given
schematically as follows:

!) In this the situation of the equilibrium at 20° has been taken into account.
2) J. Amer. Chem. Soc. 30 1775 (1908).
5) J. Amer. Chem. Soc. 33 893 (1910).

523

a-sugar + H,O 2 hydrate = g-sugar + H,O
great rotation small rotation.

The equilibrium 1 sets in with great velocity, whereas equilibrium
2 does so slowly, so that the mutarotation lies in the second
equilibrium process.

2. Hxperimental part.

a. Inquiry into the stability of the occurring solid phases; deter-
mination of the transformation point Hydrate—B-anhydride + solution.

This was the state of affairs when we began our examination of
the milksugar, which examination was very attractive, because here
a very slow establishment of the internal equilibrium was found,
which opened tbe possibility to get to know something about the
relation between the pseudo-system and the setting in of the internal
equilibrium.

In the first place we had to examine what was the stability of
the solid phases, hydrate, p-anhydride, and a-anhydride in the
system water-milksugar. It appears clearly from the method of
preparation and the investigation of Hupson that the hydrate is
stable below 93°; accordingly both the g-ahydride and the a-anhydride
always yielded the hydrate below 93° in contact with water.

It further appeared again both from the method of preparation
and from Hupson’s investigation that above 93°, g-anhydride is the
stable solid phase; accordingly the hydrate gave the g-anhydride
above this temperature in contact with the saturate solution, and
the a-anhydride always gave the 8-modification under these circum-
stances.

It followed, therefore, from this that the a-form is metastable not only
below 93°, but also above it. -

The question is now whether there exists a transition point
between the « and $-modification at higher temperatures. For this
purpose the final solubility was first determined from 93° to 200°,
both starting from the 3-and from the a-modification.

The result was that the «-form was always first visibly converted
into the 8-form, and that the found points lay without exception on
the solubility curve of the s-modification.

These determinations could now be made to fit in with Hupson’s
results about the hydrate, as the 7x-figures 1 and 2 express.

From the graphical representation in fig. 3, in which log ~ is

]
represented as function of ple’, follows the temperature 93°,5 for

200

150

100

50

0
WATER

200

150

50

be

02522

d
102258
5 b1°%6
EUR
Cc
A
4
1
4
Vi
1
‘
'
ri
'
1
i
ee eas re F MILKSUGAR

50% 60 70 80 95%
HYDRATE (WEIGHT IN o/°)
Fig. 1.

MILKSUGAR

50%

Gata, H,0 (moree
Fig. 2.

525

the tempeparature of the three-phase equilibrium hydrate + (-
anhydride + solution.

a 0,25 0,5 0,7 1,0 1,25

Fig. 3.

The «-moditication appeared, therefore, to be metastable below
200°, and .now it was the question how it is at the higher tempe-
ratures. The solubility lines could not be pursued towards higher
temperatures, because the milk sugar decomposes during the time
necessary for a determination.

The only thing that offered a chance of success was the
determination of the melting point in exceedingly thin capillary
tubes, according to the method of Socu.

This investigation gave a perfectly convincing result, for the
a-modification melted in 2 seconds in a bath of 222°8, while the
3-modifieation melted in the same time in a bath of 252°,2. Hence
the 3-modification melted 29.4° higher than the «-form, which shows
that the a-modification is metastable up to its melting-point.

On this occasion also the melting-point of the hydrate was. deter-
mined according to Socu’s method, for which was found 201°,6. It
may serve as a proof for a very small velocity of transformation
in the solid substance, when as here, the melting-point of a hydrate
is to be realised more than 100° above the already discussed dehy-
dration point. |

b. The hydrate is a hydrate of the a-modification.

All this did not solve the problem offered by the milk sugar,
however, by any means, for another highly remarkable peculiarity

526

presented itself for the hydrate, a peculiarity which as far as we
know, has never yet been observed for another hydrate, though it
is probable that it will occur in more cases.

We found, namely, that the hydrate when heated at 125° in dry
condition, always yields the a-modification, whereas in presence of
the saturate solution the hydrate at the same rale always
passes into the 2-modification.

To study this interesting phenomenon more closely the hydrate
was brought in contact with a vessel of strong sulphuric acid of
ordinary temperature at temperatures between 0.5° and 200° in
vacuum. It appeared in all these experiments that the hydrate was
exclusively converted to the a-anhydride, and this happened both
below and above 93°.5.

This result enables us, as we shall see presently, to consider the
milk sugar problem from another point of view, which will make
it possible to account for all the phenomena observed up to now
in the system water-milk sugar in a simple way.

Hupson already assumed the transformation :

hydrate = 6 + H,O.

In this the expectation that the hydrate is a hydrate of the B-
modification is of course implied.

Now Huvpson assumes further that the water in the hydrate is
bound in this way

ZOE

mea ee)

NH
from which would follow that the hydrate might just as well be
called a hydrate of the «-modification as a hydrate of the 8-modi-
fication, because the stereochemical difference in the final carbon
atom, which he assumed for the 3- and g-modifieations'), viz.

H—C—OH and HO—C—H
Za as

has perfectly disappeared in the structure formula for the hydrate.

If this were so the hydrate would be neither «- nor 8-hydrate
and then it could absolutely not be understood why the dry hydrate
always gives a-anhydride. also above 93°, whereas the damp hydrate
passes into B-anhydride above 93°.

We must, therefore, certainly reject this supposition ; the hydrate
must be a hydrate of « and g-anhydride, and now all the experi-

1) J, Amer. Chem. Soc. 31 66 (1909).

527

ments made by us, in which it was found that the hydrate in dry
condition gives the a-anhydride at all the temperatures examined
point to this that the milk sugar hydrate is the hydrate of the
a-modification.

c. The hydrate presents a transition-dehydration: point.

This conclusion throws another light on the problem: the trans-
formation point of 93°.5 is no ordinary transformation point, for in
this point takes place the conversion

a-hydrate — B-anhydride, + solution.

If we had to do here with an ordinary transformation point, the

conversion would be the following :
a-hydrate — a-anhydride + solution.

For milksugar, however, the dehydration is accompanied with a
transition of the a-modification into the B-form, so that here a pheno-
menon is met with, which as far as we know, has never been
observed as yet. To express this particular behaviour also in the
denomination of the transformation point we shall call this point
henceforth the transformation-dehydration point, resp. transition-
hydration point.

d. The system water-milksugar must be considered as pseudo-
ternary. Derivation of the isotherm-diagram.

This remarkable result must be expressed in our way of repre-
sentation of the system considered here.

As the occurrence of the said transition-dehydration point shows
with the greatest clearness that we must consider the system water-
milksugar as pseudo-ternary, we have begun collecting data for the
representation of the solubility-isotherms of the system H,O + a-milk-
sugar + @-milk-sugar at a temperature below 93°.5.

In fig. + the points a and 6 indicate the initial-solubility of the
hydrate a, and of the g-anhydride determined by Hvupson *) at the
temperature of 0°. The three-phase-equilibrium «@,, + 6 + h was not
determined, and was found by us by starting from these three phases,
and by squirting off the liquid through a filter after 1 hour’s vigorous
stirring. The total concentration of this liquid was determined
by evaporation and weighing, i.e. the total quantity of milksugar.
In this way we find, therefore, on what line drawn parallel to the
line ag, the point D lies. In order to be able to indicate the place
of the point D on this line, a second quantity was pressed through
a filter, and then through a bent tube provided with a refrigerator,
through which water flowed of the same temperature as that of the

1) J Amer. Chem. Soc. 80 1767 (1908).

WITH MIXED CRISTALS.

Fig. 4.

WITHOUT MIXED GRYSTALS.

batn. Directly after it had left this tube, the solution was received
In ice, to fix the equilibrium in the homogeneous liquid. By deter-

529

a
mination of the initial and the final rotation the Bate was found.’)

In the determination of the final rotation use was made of the
positive catalytic influence of a drop of a solution of ammoniac.

. . a . . . .
Is is clear that when the ratio 2 is given, it is not stated that the

molecules of the two modifications « and 8 really occur in perfectly
unhydrated condition in the solution ; it only expresses what the ratio
is between the concentration of « and 8, leaving it quite an open
question in how far these molecules are hydrated.

Thus also the point D was determined, which is the point of
intersection of the isotherms of the «,, and of the $-modification.

It has already been ascertained by Hupson whether the situation
of the equilibrium between « and 2 in solutions of different total
concentration shifts with the concentration. The result was that the
equilibrium af does not change on dilution of the solution, as
was indeed to be expected in dilute solutions, as we have there to
do with isomers. We can, therefore, represent these equilibria in our
triangle by a straight line starting from the point H,O.

As it had appeared that aaj below 93°,5 is the stable solid phase
in tbe system water-milksugar, it was certain that the said line for
the homogeneous equilibrium would have to intersect the isotherms
Of aag: 2

This point of intersection is now determined by shaking a, with
water for 2 or 3 days at O° with the aqueous solution. On analysis
of this solution in the same way as this had been done with the
liquid D the point Z was found lying on the isotherm of a,,. Hence
the phases «a, and L and the homogeneous equilibrium line H,O—E
denote the binary equilibrium system water-milksugar at the existing
temperature.

Now it is clear that tbe observations must show that at 93°,5
the equilibrium line H,O—E must pass through the three-phase
point D as Fig. 5 expresses, so that then a,, + 8 + L can coexist
in the binary equilibrium system.

Tod The ratio between initial rotation 7) and final rotation 7, was determined
at O° for hydrate and g-anhydride mixtures of different concentration. The gra-
phical representation of this gave a straight line, which enabled us, not only to

7 Yr
determine the ratio 8 from ae, but also to find the accurate value of the equi
a
iy
se rer Yo
librium-constant K’ at 0, because K’ = —, when iks ile
a 00

Kk’ = 1,65 was found.

530

Above this temperature the equilibrium line H,O—E must inter-
sect the isotherm for g-anhydride, which is represented in fig. 6.

B &

Se)
an
=

531

a. Experimental confirmation of the validity of the pseudo
ternary views.

To show that this is actually the case, it was examined how the
situation of the points D and ZL changes with increase of temperature.
The shifting of L with the temperature was easy to examine, and
we have therefore different observations at our disposal, which, as
the figures 7 and 8 show, prove that the equilibrium az shifts
a little to the a-side on increase of temperature.

IN WEIGHT °

p

100

or
HOG 0 20 30 40 50 95 %

Fig. 7.

It follows already from this that also the point D will have to shift
towards the a-side, and to a much greater extent too.

The determination of the shifting of the point D with the tempe-
rature does not present any difficulties at temperatures below 50°
as the velocity of transformation of B-anhydride into aq, takes place
slowly, so that we can take care, e. g. by continually adding g-anhy-
dride, that there is always g-anhydride present by the side of the aay,

At higher temperatures this velocity of transformation increases,

532

p

IN MOLEC %

H,0 1 2 3 4 5 10
Fig. 8.

however, greatly, and on account of this 50° was the highest tempe-
rature at which a reliable observation could still be made.

As, however, appears from the figures 7 and 8, the points D
shift exactly as had been predicted, and the line traced through the
points D points with perfect certainty to an intersection at + 93°,
so that a complete confirmation of the supposition made has been
obtained in this way.

We are therefore justified in saying that above 93°,5 the diagram
of isotherms with the binary system of equilibrium lying in it will
be as was represented in fig. 6. (To be continued).

Amsterdam, July 1917. Laboratory for General and Anorg.
Chemistry of the University.

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS

VOLUME XX
N°. 4.

President: Prof. H. A. LORENTZ.

Secretary: Prof. P. ZEEMAN.

(Translated from: ‘Verslag van de gewone vergaderingen der Wis- en

Natuurkundige Afdeeling,” Vol. XXV and XXVI).

CONTENTS.

P. ZEEMAN: “Some experiments on gravitation. The ratio of mass to weight for crystals and radio-
active substances”, p. 542.

S. DE BOER: “Are contractility and conductivity two separate properties of the skeleton-muscles
and the heart ?” (Communicated by Prof. G. VAN RIJNBERK). p. 554.

J. TEMMINCK GROLL: “The Influence of Neutral Salts on the Action of Urease”. (Communicated
by Prof. G. VAN RIJNBERK), p. 559.

N. H. SIEWERTSZ VAN REESEMA: “The Use of the Thermopile of Dr. W. M. MOLL for Absolute
Measurements”. (Communicated by Prof. J. BOESEKEN), p. 566.

A. SMITS and J. GILLIS: “On Milksugar”. II. (Communicated by Prof. S. HOOGEWERFF), 573.
H. I. WATERMAN: “Influence of different compounds on the destruction of monosaccharids by

sodiumhydroxide and on the inversion of sucrose by hydrochloric acid. III. Constitution-formula

p. 590.

of the hydroxybenzoic acids and of sulfanilic acid”. (Communicated by Prof. J. BOESEKEN), p. 581.
A. W. K. DE JONG: “The Structure of Truxillic Acids”. (Communicated by Prof. P. VAN ROMBURGH),

L. RUTTEN: “Old Andesites” and “Brecciated Miocene” to the east of Buitenzorg (Java),” (Com-
municated by Prof. C. E. A. WICHMANN), p. 597.

H. B. A. BOCKWINKEL: “Some Considerations on Complete Transmutation”. (Fourth communication).
(Communicated by Prof. L. E. J. BROUWER), p. 609.

A. DE KLEYN and W. STORM VAN LEEUWEN: “Concerning Vestibular Eye-reflexes. I. On the Origin
of Caloric Nystagmus.” (Communicated by Prof. H. ZWAARDEMAKER), p. 622.

40
Proceedings Royal Acad. Amsterdam. Vol. XX.

Physics. — “Some experiments on gravitation. The ratio of mass
to weight for crystals and radioactive substances.” By Prof.
P. ZEEMAN.

(Communicated in the meeting of Sept. 29, 1917).

Jy vOurr ideas concerning gravitation have been so radically
changed by EINsreiN’s theory of gravitation that questions of the ut-
most interest in older theories are now simply discarded or at least
appear in a changed perspective. We cannot try anymore to form
an image of the mechanism of the gravitational action between two
bodies, and we must return to the older theories in order to justify
the suspicion, that the structure of substances might influence their
mutual attraction. In most erystalline substances the velocity of
propagation of light, the conduction for heat and the dielectric con-
stant are different in different directions, and we might then suspect
that the lines of gravitative force spread out from a crystal un-
equally in different directions.

A. S. MACKENZIE!) in America, and Poynting and Gray’) sought
for evidence of a directive gravitational attraction.

Mackenzie proved with an apparatus like that used by Boys in
his beautiful researches on the gravitation constant, that when the
axes of calcspar spheres were set in various positions the maximum
difference of attraction amounted to less than >!" part of the
total attraction.

PoyxrinG and Gray proved that the attraction between two quartz
spheres with parallel axes, differs less than 1 in 16000 from the
attraction between these spheres with crossed axes.

KREICHGAURR®) sought for a change of weight of sodium acetate
when this substance crystallized from the fluid (supersaturated)
state. It appeared that the change amounted to less than 4.10 of
the total weight.

2. The weight of quartz spheres in different positions.
Determinations of the weight of crystals in different orientations
have, I believe, never been published. Some years ago I decided upon

1) Physical Review 2. 321. 1895.
2?) Philosophical Transactions. A. 192. 245. 1899.
3) KREICHGAUER. Verhandl. Berliner Physik. Ges. 10. 13. 1892.

543

carrying out such experiments, but only now, in connection with
connected material I intend to give an account of the results. Weighings
were made with different crystals, but the greatest accuracy was
obtained in a comparison of the weights of two quartz spheres,
42 m.m. in diameter and weighing about 127 gms. Each sphere
was mounted in a ring of argentan, and could be turned about a
horizontal axis, the ends of which were supported by the suspending
wires of the pans of the balance. The spheres were of nearly equal
weight, so that it was only necessary to nearly cancel the difference
of the weight of the spheres. The optical axes of the spheres hanging
from the left and right arms of the balance were placed alternatively in
vertical and horizontal positions by means of a simple mechanism,
allowing the necessary operations without opening of the balance case.

From 15 different series of weighings it resulted that the double
effect sought for is less than 0.01 mg. in 127 ems. or less than 1 in
13000000.

I have much pleasure in thanking Miss C. M. PrrreBoom, phil.
nat. cand., who has taken part in the investigation, and made many
weighings.

Experiments with the torsion balance.

3. By means of the common balance we are able to ascertain the
equality of two weights. In how far equal weights correspond to
equal masses, in the meaning introduced in the science of mechanics,
can be found out only by experiments. A rough experiment to prove
the proportionality of weight and mass consists in observing the
equality of the time of fall of various bodies. Much more accurate
results were obtained by Nuwron’s pendulum experiments (descent
along a circular arc). Pendulums of equal form and hanging by
equal threads, but of various composition, have the same time of
oscillation. In his fundamental experiments Newton ') was able to
ascertain the equality of the times with an accuracy of one part in
1000 and this therefore is also, at the same place, the accuracy of
the proportionality of weight and mass. BrsseL *) refined NEwron’s
method, and came to the conclusion that a difference of attraction,
experienced by various bodies of equal mass, must be less ‘than
1/60.000 of the total attraction. .

A much more accurate result was obtained by von Eérvés with
CavVENDISH’s torsion balance, which he brought to a high degree of

1) Newton. Principia.
2) Bresser. Abh. Berliner Akademie. 1830.
40*

544

perfection, after many years of continuous, very refined, studies on
the local variations of gravity, and which he applied also to the problem
now under consideration.*) The force acting on a body at the
surface of the earth is the resultant of two forces: the attraction
of the earth and the centrifugal force. The direction of the resultant
is dependent upon direction and magnitude of these components.
At a given place of the earth the centrifugal force is directed per-
pendicularly to the earth’s axis and dependent upon the mass. If
for various substances of equal masses the attractions were different,
then the resulting force for these substances would have different
direction, and a couple would act on a torsion balance, the rod of
which is placed perpendicularly to the meridian and carries at
its ends different substances.

Von Eörvös used a torsion balance with a rod, 25—50 em. long;
the torsion wire was of platinum, 0,04 mm. thick, and charged
with various substances all of 30 gms. weight. The rod is placed
perpendicularly to the meridian and its position relative to the case
of the intrument determined accurately by means of mirror and
scale. The whole intrument, rod with case, is then rotated through
180 degrees, the substance that first hung at the east side, now hanging
at the west side. The position of the rod relatively to the instrument
is now read again. The kind of effect considered must produce a
torsion of the suspension wire. With a brass ball at one end, with
glass, cork or stibnite crystals at the other end of the rod no effect
was to be observed. A difference of weight of various substances
of equal mass, must be for brass, stibnite and cork less than one
twentymillionth, for air and brass less than one hundred thousandth.

4. The astonishing fact of the equivalence of mass and weight,
the expression of the narrow tie between the phenomena of inertia
and gravitation is of fundamental importance for EiNsTriN’s theory
of gravitation. This theory, only possible, if there exists a field of
force giving the same acceleration to all bodies, even enables us to
“create” a gravitation field by a transformation of coordinates. ®)

The fact mentioned therefore merits to be tested in all possible
directions. It has been my aim to extend the work of von Eörvös
in two directions, viz.: by the investigation of orientated crystals and
of radioactive substances. I also hoped, that I might be able to

1) v. Eörvós. Ann. d. Phys. 59..354. 1897, especially p. 372—373, and
Mathem. u. Naturw. Berichten aus Ungarn. 8. 64. 1891.

2) Einstein. Ann. d. Phys. 49. 769. 1916.

545

introduce some changes, securing, at any rate, the independence of
my results.

An investigation of crystals seemed important to me, because
von Eörvos in his investigation of stibnite does not mention, whether
attention was paid to a definite orientation. The orientation relative
to the vertical however might be of supreme importance. An invest-
igation with radioactive substances is of interest, because it enables
us to verify the proposition that energy possesses mass. We know
that if AZ denotes a change of energy of a body, c the velocity
of light, then to AZ corresponds a change Am of the mass given

ea .
by the formula Am —= —. Because c’ is extremely great, we can
c

by ordinary methods only obtain inappreciable changes of mass.
We cannot hope ever to be able to measure the changes of mass
caused by the effect of temperature or by chemical transformations.
In. the case of radioactive bodies the processes of disintegration
entail losses of energy of another order of magnitude than in the
case of chemical transformations. During the transformation of uranium
into lead and helium an enormous amount of energy must be relea-
sed. This is already the case during part of the necessary transfor-
mations, for in the course of its life one gram of radium with its
transformation products including radium F' emits about 3,7 10°
calories *).
3,7X10°X4,18107 _
Oar Sone

‘This corresponds to a change of mass equal to

= 0.6 X 10- += per: sram.

If this energy possesses mass, but no weight, then pendulums
with lead, helium, uranium must give values for the acceleration,
differing by more than 1 part in 10,000.

Already several years ago these considerations were given by
J. J. THomson, who also made experiments with a pendulum the
bob of which was made of radium. It was, however, impossible to
obtain a high degree of accuracy as the quantity of radium available
was very small. Afterwards SourHerns’) made experiments in
TuHomson’s laboratory with pendulums with uranium oxide and red
lead. He came to the conclusion that the ratio of mass to weight
for uranium oxide, does not differ from that for lead oxide by more
than one part in 200,000.

1) Rutnerrorp. Radioactive Substances. p. 582.

2) SourHerns. Determination of the ratio of mass to weight for a radioactive
substance, Proc. Royal. Soc. London, A. 84, 325, 1910,

546

Hence we must conclude, within the limits of experimental errors,
that if energy possesses mass it also possesses weight. Now we can
considerably restrict these limits by the use of the torsion balance
and this justifies us, I think, in applying it to the investigation of
radioactive substances.

5. New experiments. My own experiments were made with an
apparatus, principally after the design of that of v. Eörvös, but of
much smaller dimensions. The weights at the end of the torsion
rod were each 30 grams in v. Körvös’ experiments; in my apparatus
the weights were each of 1 gram. The weight of the torsion rod
with mirror was only about 1,5 grams. The distance between the
centres of the cylindrical weights at the ends of the torsion rod is
about 10 ems. The smallness of these weights enabled me to take
advantage from the properties of fine quartz wires, not yet discovered,
indeed, at the time v. Körvös began his researches.

The torsion wire in my apparatus was 22ems.long, and about0,0l mm.
thick. The time of oscillation ranged from about 350 to 400 seconds.

In order to protect the apparatus from thermal and electrical
perturbations we used, as also did v. Edétvés, double and even triple
walls, of brass, about 3 mms. thick.

Manipulations with an apparatus of so great a sensibility as this
torsion balance, requires exceptional stability of the surroundings.
The mounting on the brick piers of the Amsterdam laboratory proved
to be ‘quite insufficient. I, therefore, resolved to construct an arrange-
ment, probably securing the apparatus against vibrations and per-
mitting its rotation, with scale and telescope, about a vertical axis
through 180°.

The principle of this arrangement is the one used by MicHELson
and by EINTHOVEN for similar purposes.

I have much pleasure to thank here Mr. W. M. Kok, phil. nat.
cand. for his assistence in the construction of this arrangement and
during the continuation of the present investigation.

The principal part of the arrangement is an iron basin floating
in a tank with thick oil. Tank and basin are of annular form; the
central part being open, it becomes possible to suspend an apparatus
from a vertical bar through the centre of the annular basin and
to fix it at different heights.

A more detailed description is reserved for another occasion.

It was found that this arrangement gave excellent protection against
vibrations of short period and permitted also to give slow and
smooth rotations,

547

On the contrary the protection against vibrations of long period
was very bad. Experiments with the suspended torsion balance
soon taught that, even during the most quiet hours of the night,
the torsion rod was never at rest. Sometimes the amplitude of the
oscillations gradually diminished to zero, but then the amplitude
increased again to 5 m.m., not to mention the extremely annoying
nutations of the mirror, which, indeed, never ceased. Apparently
vibrations of a period of 300 or 400 seconds (the period of the
torsion balance) are never failing in the marshy land of Amsterdam,
at least in the neighbourhood of the physical Laboratory.

[t was therefore hopeless to work with the torsion balance in
Amsterdam, and I resolved to continue my experiments in the cellar
of a country house near Huis ter Heide (prov. Utrecht).

To my surprise I found that the stability of the balance, at the
new station, was most excellent. The motion of the mirror, about a
horizontal axis, was entirely absent and the amplitude of the oscil-
lations always decreased with time. After about one hour the image
seen in the telescope was at rest. The apparatus was placed upon
a wooden table, resting on the cellar floor. Even hard stampings
upon the floor in the neighbourhood of the apparatus had not the
slightest effect.

Of course the temperature of the cellar was very constant. One
disturbance had an effect on the observations, viz. the magnetic
action due to the iron beams of the cellar vault. The constant
displacement of 0,3 m.m., noticed in the experiments with quartz _
and recorded later on, is probably due to this cause.

In view of the accuracy aimed at in the experiments, this amount
could not be neglected, though in some experiments its influence is
eliminated. I therefore transferred the apparatus, first to a second
place in the cellar, where presumably the perturbations would
be less.

Afterwards the apparatus was placed in the vestibule. Also here
the stability was excellent, but of course the constancy of tempera-
ture, though satisfactory, somewhat less. Several excellent series of
observations were obtained. As they extended, however, over the
whole day and the principal entrance of the house was then put
out of use, I restricted these observations to a rather limited
number of days.

6. For Amsterdam the latitude p = 52,4 and g = 981,3 cm/sec.
The angle « between the attraction of the earth 6, and the resultant
of attraction and centrifugal force a, becomes 5'42" = 342",

548

: , : a sin p
As is easily seen from figure 1, a= NIER and hence da =

en db = B if the attraction 6 changes with the amount dod.
a By the change of direction da, the force acting
H in tbe horizontal direction H becomes Rda, R
6 being the resultant of a and 5.

If there is a change of mass a with da, then

da sin p da

de = Ee

Ee b a
BE Let the difference of attraction for two sub-
stances be 1/1000000, then da == 0,000342" or in radians 1/600000000.
In order to give an idea of the sensibility of the apparatus and
to calculate the effect to be expected with a given value de, I now
give some details of the arrangement, a sketch of which is given

in figure 2.

Experiment with quartz cylinders, mean weight of each 0,888 g.

Time of oscillation (complete) of torsion rod with quartz cylinders
350 sec.

Time of oscillation (without cylinders) 186 sec.

Distance of the centres of quartz cylinders 7,6 cm.

Moment of inertia of quartz cylinders K, = 2 x 3,8? X 0,888 =
05,68 em:

Moment of inertia rod + cylinders K = 25,6 + 10,1 = 35,6 g.cm’.

549

5. ee35, 1
For the torsion couple S we find from = 35: == 0,0195

em sec. 2.

Hence force per radian twist 0,0115/3,8 = 3,03 x 10 3 dynes

=a,00 2 10-8 gp:

Distance scale to mirror = 540 mm.

A displacement of 1 mm. observed in the telescope corresponds
to a rotation of —— 2540:

A difference of the positions of the balance equal to 1 mm.,
when pointing first West than Hast, corresponds for the single effect

to

"| and therefore equal to
4540 8

3,09 x 10-6
ae =a eae ‚
2160 be 5 |
The vertical force acting on the cylinder is 0,888 g.
1
If the effect is ————, then we have (see above) Ada =
1000000
seb, 888

10-8 = 144 < 10-9 g.

ca therefore see that with the sensibility used, 1 mm. of the
scale corresponds to an effect of 1 in 1000,000. In many cases the
result is certainly smaller-than 0,1 mm.

7. Results. We will now summarize the results obtained. In
-the first experiments the position of rest of the rod was deter-
mined from three succeeding extremities of vibration. The presence
of the observer, however, brings about a marked disturbance by
convection currents. Preference was given in the subsequent obser-
vations to the noting of the final position of rest, actually attained
after about one hour.

Experiments were made with quartz, calc-spar, lead oxide, uranium
oxide, uranyl] nitrate.

Quartz. The 2 cylindrical quartz rods were 25 mm. long and
of 4,5 mm. diameter. The axis of the cylinders lay in the vertical
plane through the rod of the balance. The crystallographic axis was
perpendicular to the axis of the cylinder. I determined its position
before the beginning of the experiments, by means of observations
with the polariscope, and noted it by means of a small cross, cut
by a diamond, in the surface of the cylinder.

Experiments in the cellar. The annexed table gives an example
of results for cylinders with crossed and with parallel axes.

550

Quartz cylinders: axes 1. Date: August. 2.

Telescope West Telescope Fast.
Reading scale 66.8 67.2
in m.m. 66.8 67.2
67.0 67.3
67.0 67.4
67.0
66.92 67.28, hence E > W 0.36.

The results are exhibited as a curve in fig. 3. The abscissae give
the hours of the observations. The temperature of the instrument
diminished in the course of the day a few tenths of a degree.

Quartz cylinders. Axes //. Date: August. 5.

Telescope West Telescope East.
Reading scale 39.7 40.0
in m.m. 39.7 40.2
39.8 40.1
39.8 40.1
39.75 40.10, hence E > W 0,35

These results are represented in fig. 4.

Quartz-axes 1 [Aug. 2 cellar]

ag) mM 2
pate
iL 66
i at = 1

9 10 11 12 1 2 3 4 5 6 7 8 9 10 hours
: Cr eS
Quartz-axes || [Aug. 5 cellar]
+ mM |
4 vs o o 0

bh Ei ; ‘
. 4 il ! L [ L

9 10 11 {2 1 2 3 4 5 6 7 8 9 10 hours

These observations exhibit a difference of the readings for the
position of the torsion rod after rotation of the apparatus from the
E-W to the W-E position. Moreover it appears that it makes no
difference whether the crystallographic axes of the quartz cylinders are
parallel or crossed. At least this difference is only 0,36 —0,35=0.01 mm.

The constant difference of 0,35 mm. was traced to be probably
due to the asymmetrical, magnetic action of the heavy iron beams
of the cellar vault on the weakly magnetic torsion rod of hardened
copper of the balance. A magnet placed in the neighbourhood of
the apparatus caused a small deviation.

The balance was then transferred to the vestibule, a place of rather
constant magnetic potential. The difference between the E-W and

dol

W-E observations had disappeared. As an example I give again a
table of results, also plotted in fig. 5 and fig. 6.

a :
= 545 ——, —.—_
7% a

Quartz. [vestibule
: : axes | Aug. 27]

3 4 hours 7 8 9 10

"hours
Fig. 5. Fig. 6.
Quartz cylinders: axes //. Date: August 27.
WV, OP
74.9 75.2
Todi 75.0
74.9 74.9

74.97 75.03, hence E > W 0,06.
Quartz cylinders: axes |. Date: August 27.

NN 0.
54.1 54.0
53.8 53.9

53.8

53.95 53.90, hence E << W 0.05

The above mentioned and further observations justify us in con-
cluding that. an influence of the orientations of a quartz crystal on
the ratio of mass to weight is less than 1 in thirty milhons.

Subsequent observations were made with two cale spar cylinders
with the same results as obtained for quartz.

These observations were also made in the vestibule.

The results obtained are in agreement with the conclusion in § 2.

8. The torsion rod was next charged with a small glass tube
with yellow lead oxide at one end, and one of the quartz cylinders
at the other. A difference of the ratio of mass to weight for these
substances was certainly less than 1 in twenty millions.

9. Radioactive substances. Observations were begun with uranium
oxide, included in a thin cylindrical glass tube. The results were
rather puzzling. A first series of observations in the cellar gave, in
the above used nomenclature, E >> W 1,2 mm. In this series lead
oxide was compared with uranium oxide.

On August 24 I began observations with a second glass tube
charged with uranium oxide. I resolved to compare this second tube

552

with the first in the torsion balance, expecting to find a devia-
tion of, at the utmost, a few tenths of a mm. when observing in
the cellar, and practically zero in the vestibule. An observation of
August 24 in the cellar however gave, using the torsion rod, charged
with the two uranium oxide tubes, a difference of W > 42,1 mm;
in the vestibule an observation of August 25 gave W > 22,2 mm.

The first tube was incidentally broken and part of the contents
lost. The further experiments were made with the second tube
(balanced against quartz) and gave on August 28, 29, 31 and
September 2 the results W > KE 1, 1, 2.5, 3 mm.

The deviations found widely exceed the errors of observation.
They prove (observations of August 24 and 25) that the two samples
of uranium oxide are not identical. Probably both or at least one
of the uranium oxides are contaminated by iron.') By this hypothesis
we may understand that the magnitude of the deviation changes
with time (observations of August 28 and following days), and even
exhibits contrary signs (first observations in the cellar compared
with the later observations in the vestibule).

I had not yet an opportunity to test quantitatively the suggestion
as to the influence of a contamination by traces of iron. This must
be reserved to another occasion. Meanwhile, I have now to record
observations with wranyl nitrate, which go far to prove that radio-
active substances also follow the law of proportionality between
mass and weight with great accuracy. The uranyl nitrate was in-
cluded in a thin cylindrical tube and balanced against quartz. The
results of observations of September 10 and 11, made in the vestibule,
are plotted in figs. 7 and 8.

The curves are not quite parallel to the axis of abscissae, ‘a case

Fig 7.

Uranylnitrate [Sept. 11, vestibule]

w

SATO San 12 1 2 3 4 5 6 7 8 9 1C hours

Fig. 8.

1) Owen (Ann. d. Phys. 37, 686, 1912) finds that the three preparations of
uranium, which were used in his magnetochemical experiments, contained much
iron. See also there (p. 672) some remarks on the omnipresence of iron.

553

realized in the cellar observations Figs. 3 and 4, but exhibit a
small slope due to the rise of temperature in the vestibule. This
slope is however the same for the E. and W.-curves. The lines are
even practically coincident for the greater part of the observations
of September 11, and the small residual differences exhibit contrary
signs in the observations of September 10 and 11. The observations
justify us in concluding that for wranyl nitrate a deviation of the
law of the constancy of the ratio of mass and weight is less than
1 in twenty million. |

It seems extremely improbable that the behaviour of uranium
should be otherwise, as far as a so fundamental property as mass
is concerned, in an oxide than in a nitrate. The deviation found in
the case of an oxide, is therefore most probably due to a magnetic
contamination. If the deviation found in the case of the oxide were
really due to a change of mass, than the nitrate should exhibit
about half the effect of the oxide or about 1,5 mm., allowance being
made for the quantity of uranium in the two combinations. But the
effect is most certainly less than 0.2 mm.

Perhaps I may be allowed to add that electric disturbances during
the observations were excluded sometimes by a short exposure of
the inner case, opened to this purpose, of the balance to radium
rays, sometimes by placing a few scrapings taken from the leaden
box containing the radium preparation.

I have projected recently some improvements of the apparatus
and the method of observation, by which I hope to be able to
increase the accuracy, according to an estimation at the safe side,
ten times.

I hope to return to this subject on another occasion.

Physiology. — “Are contractility and conductivity two separate
properties of the skeleton-muscles and the heart?’ By. Dr. S.
DE Boer. (Communicated by Prof. G. VAN RiJNBERK).

(Communicated in the meeting of September 29, 1917).

In 1888 BiepERMANN made experiments on skeleton-muscles, from

which he concluded that under special Gircumstances these muscles:

are still irritable and have still retained their conductivity, whilst
the contractility has ceased to exist. He placed the sartorius of frogs
over a certain length in water. When it had lain in it for some time
he stimulated the end of the sartorius that had been in contact with
the water. The result was, that the stimulated part of the muscle
that had been in the water, did not contract, but the other part did.
ENGELMANN repeated the experiment, and obtained the same result.
ENGELMANN applied this experiment likewise to the heart. He plunged
the auricles of frogs’ hearts for some time into water, thereupon he
stimulated the auricles, and saw after doing so that the ventricle
contracted, whilst the auricles did not show any contraction at all.
ENGELMANN communicates his results in the following phrases : “Von
der Richtigkeit der Thatsache hatte ich mich durch eigene Versuche
am Sartorius curarisirter Frösche überzeugt. Die Bestätigung ist
so leicht, wie das Resultat überraschend. Der Muskel wird in der
ganzen Ausdehnung, in welcher das Wasser ihn seiner Contractilität
beraubt, gleichsam zum Nerv. So nun auch die Muskelbündel der
Vorkammern: sie verlieren im Wasser ihren Charakter als Muskeln,
und behalten ihre Function als motorische Nerven der Kammer”.
Further: “dass die Muskelfasern der Vorkammer auch nach voll-
ständiger Auf hebung ihrer Contractilität doch den Bewegungsreiz für
den Ventrikel noch fortzupflanzen im Stande sind, und zwar mit
einer Geschwindigkeit durchaus derpelben Ordnung, wie wenn das
Verkürzungsvermögen erhalten ware”

It has now appeared to me, that (ie conclusions made from their
experiments by BIRDERMANN and ENGELMANN, are entirely incorrect.
This may appear from the following experiments which I made with
regard to this problem. In the first place about the skeleton-muscles.
I attached a m. Gastrocnemius of a frog to a lever and plunged
the muscle into a solution of Rincer. Then I induced the muscle

et ee ee eee DP ee a

555

to contraction by direct stimulation, and registered it by means of
a stationary drum covered by smoked paper. Then I substituted
water for the fluid of Rincer. After 5 minutes the point of the
lever had already distinctly risen, then the drum is turned round a
little way, and by direct stimulation another contraction is brought
about and registered. In this way the drum is turned a little way by
the hand every five minutes, and afterwards a contraction is registered on
the stationary drum. The result is reproduced
in Fig. 1. We see that after every five minutes
the point of the lever has risen, and that
after 25 minutes the point of the lever has
mounted over the top of the first registered
curve. Then on a stimulation the muscle
no longer reacts with an abbreviation. This
experiment can be explained as follows:

Fig. 1. Because the muscle has been so long sub-
merged in water, it becomes saturated with water, swells and assumes
rather a globular form. This causes the ends of the muscle to come
nearer to each other, and an inflation-abbreviation is the consequence.
As soon as this inflation-abbreviation surpasses the height of the
“Zucking”, the mechanical proportions have assumed such a character
that an abbreviation can no more become manifest after a stimulation.
The irritability and conductivity are intact, and the abbreviation
exists already on account of the inflation. The active abbreviation
of the fibrils no more approaches the ends of the muscle nearer
to each other, because the inflation has already approached them at
the smallest possible distance. What is decisive in this respect is
the fact, that such a swollen muscle can no more dilate.

I arranged my experiments on the frog’s heart in the following
manner. I fastened a canula of Kronecker through the sinus venosus
in the auricles, after I had destroyed the septum atriorum and the
atrioventriculary-valves. Then I imbibed the heart with the solu-
tion of Ringer under the pressure of a column of water 9 mm.,
and registered the heart-curves by means of suspension on a drum
covered by smoked paper. Then I substituted by water the fluid of
Ringer. Within a short time the heart stood still and indeed at a
level with the tops of the formerly registered curves (vide Fig. 2).
If thereupon the heart is again imbibed with the fluid of Rineur,
we can make the heart pulsate again, after the inflation-abbreviation
of the heart-muscle has first diminished. The heart has, like the
skeleton-muscles, sustained a rather important inflation by the im-
bibition with water, and the distance from the basis to the point of the

556

ventricle has decreased. Here also the mechanical proportions have
consequently assumed such a cha-
racter that an active abbreviation
of the heart-muscle cannot express
itself. After the substitution of the
water by the fluid of Ringer the
restoration of the systoles sets in,
but first an allongation of the heart-
muscle has occurred again on ac-

Fig. 2. count of a decrease of the inflation.
Fig. 2 shows these proportions distinctly. The restoration of the
systoles is not complete here, because the stagnation has lasted
rather long. A complete restoration of the systoles of the ventricle
can however easily be obtained by imbibing the heart for a short
time with water.

Consequently the disappearance of the contractility, whilst the
conductivity and irritability continue to exist, as BIEDERMANN and
ENGELMANN supposed to be the case for the skeleton-muscles and
the heart, are only an apparent phenomenon. In order to show,
that the processes that are the causes of the contraction really take
place in the heart that is swollen by water, I had recourse to the
string-galvanometer. After having placed 1 unpolarisable electrode
on the point of the heart and 1 on the atrioventricular limit I
registered the action-currents from the imbibed (with the solution
of Rinegr) suspended frogs’ hearts. When now I imbibed the heart
with water, it stood soon still in the maximal abbreviation-state on
account of the inflation. The action-currents continued for some time
in the beginning of the stagnation. (vide fig. 3). In this way it was
ascertained, that during the stagnation in the maximal abbreviation-
state the automation and the conductivity of the heart had remained
intact.

I may be allowed to devote a single word to the criticism that
Kaiser thought necessary to pronounce with regard to BIEDERMANN's
and ENGELMANN’s experiments. This physiologist attributed the results
of BIEDERMANN and ENGELMANN to currentloops that should have
induced to contraction from the stimulator the part of the muscle
(of the heart) that had been in contact with the water. They do not
deserve this criticism. My experiments, in which the stagnating
imbibed frogs’ hearts produced still electograms, teach us, that in
reality the processes that cause the contraction and. the conductivity
can continue to exist. The experiments of BIEDERMAN and ENGELMANN
remain consequently unimpeachable.

Time 1 min.

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aaa totclodoheboteletdetetedsh-tefofetotetetetatakee de iet-fepeta
Jo hoe i EEREERE | PTT
GENK EARKERSAAKAENEEREER KERN rep pepe epen
Fo ol vod ot rere ee rer
erder reper ered) sperer separ pepe) rerererer CET
Pol LL hdl Ll odes Shep
PP PPP ttt pl fel -folet hb dott lol be dale bels d-bdehed
teboelotebested tek eb LPN EH lob bokerbe beedehede helde ade hadde bedeeld
DEUIMEERELERKD Ree wae Heeceuens KEEL Lieh
ab Elke behold behe eds balade lok: KEN ERD
‘Gur EPEEKDEEN EHS
Ld PEEL J | | J
“ll gah aa PELLET TLN
sg! ace ETS ee
IT HEEE

Kiemen

ae Li

BERNER Bannan (od nà
rd peter pete PPR
fel ae Oh ae ee ee a a et ae ok
PE ey eee wake eh

+ ll Han

Proceedings Royal Acad. Amsterdam. Vol. XX

41

558

Their far-going conclusions, however, which have connection with
important generally physiological problems are not justified. Their
experiments do not furnish the proof that conductivity and contract-
ility are 2 separate properties, because the muscles (heart) swollen
by water are already maximally abbreviated, and consequently an
active abbreviation cannot become manifest.

In this short preliminary communication I wish to desist from
far-going conclusions and considerations, but 1 observe only that
no more than the contractility and the irritability, the contractility
and the actioncurrent may be separated.

Last year E1nraoven') has moreover pleaded in one of his essays
for the connection between contractility and impulse-currents.

1) Pfligers Archiv Bd. 166, Seite 109, 1916.

Physiology. “The Influence of Neutral Salts on the Action of
Urease’. By J. TEMMINCK Grou. (Communicated by Prof.
G. VAN RIJNBERK).

(Communicated in the meeting of September 29, 1917).

Ferments being colloids it is not impossible that the influence
that electrolytes exercise on the action of enzymes must be attributed
to modifications of the dispersity of the ferment.

If this were really the case, then the character of one of the
two ions would have a domineering influence over against that of
the other, as was likewise found by Harpy for the flaking of colloids:
it appeared namely that with colloids moving cathodically, which
are consequently charged positively, the anion of an electrolyte has
the greater influence, whilst with colloids moving anodically the
influence of the kation was domineering.

A ferment will be either a positive colloid or a negative one,
and a ferment that has its action especially in alkaline surroundings,
will usually be negatively charged, whilst the ferments acting in
acid surroundings will be positive. We can consequently expect a
great influence either of the character of the anions or of that of
the kations, according to the ferment being either positively or nega-
tively charged. Moreover with colloids the phenomenon presents
itself, that both the kations and the anions can be placed in a special
series according to their being able to cause a modification of dis-
persity either in a greater or in a more restricted degree.

With regard to the kations this series is not always entirely the
same, but in.the main there is no difference with various colloid-
phenomena.

Some metals occasionally change place, or as FREUNDLICH remarked :
“Die Reihe der Kationen ist wieder etwas verschränkt”.

Usually the series of the kations is: NH,, K:, Na‘, Ca’, Srv,
Ba::, Mg.

The series of the anions is CNS’, J’, Br’, NO’, Cl’, SO,’’.

In order to examine in how far the influence of salts on a fer-
ment-action corresponds with that on colloid-phenomena, it is advisable
to make use of an enzyme that decomposes a crystalline substrate
to crystalline products of decomposition, for if one takes a colloidal

41*

560

substrate, the salts might modify the substrate and make it con-
sequently more or less susceptible to be influenced by the ferment.

The phenomenon to be studied would problaby become more implicate.

The urease, the ureum decomposing ferment, which is found in
soybeans is very fit for such like experiments.

The experiments were executed as follows: 3 ce of a solution of
ureum (1°/, percent) a definite number of ce of the solution of salt
and water to constantly the same volumes (100 cc.) were put in a
number of flasks, asarule 7. The flasks were placed in a flat-bottomed
basin in water of the temperature of the room. After the flasks
had been so long in the water till the temperature in each of them
had become the same, 3 cc. of the urease-preparation, according to
JANSEN, was added to them. One minute elapsed between the addition
of the ferment to each flask. After 50 to 60 minutes the ammonia
that had been formed, was titrated with methyl-orange as indicator,
every time with an interval of one minute. Consequently the action
was of equal duration in each flask. It appeared at a preliminary
experiment that equal results were obtained indeed in flasks of the
same composition.

1. Kations.

The kations used were K:, Na, NH:,, Mg, Ba‘, Sr, Ca’. Of all
metals the chlorides were used.

The following results were obtained: (Vide Tab. on the next page).

If now we express the above figures graphically in such a way
that the salt-concentration is indicated on the abscis, and the quan-
tity of ammonia that has been formed on the ordinate, then irregu-
larities strike us with two metals viz. calcium and magnesium.

Fig. 1.

561

Concentration salt Nanitee Concentration salt niente
(grammol. C.c; 1/19 HEL Reduced (grammol. ec. 1/49 HCL. Reduced
per liter) per liter)

| pear) 7.9 100 Ba“ 0 8.5 100
1/29 6.8 86.4 1/90 5.4 63.7
3/29 6.0 16.2 3/30 4.2 49.6
5/20 5.2 66.0 5/50 3.4 38.9
19/99 4.3 54.6 10/5 1.7 19.5
15/op 3.75 47.6 5/29 1.0 11.8
79/20 3.3 41.9 20/oy 0.5 5.9

Na: 0 7.6 100 Spin 0 8.1 100

‘og - 6.2 81.6 '/o0 5.5 67.7
3/50 5.0 65.8 3/50 4.9 60.3
5/20 4.4 57.9 5/20 4.6 56.6
10/99 3.6 47.4 10/99 4.1 50.4
Er) 2.9 38.2 15/og ga 38.1
20/20 25 32.9 20/59 1.9 23.4

NH, 0. 8.2 100 Mg” 0 10.1 100
fap 8.1 98.8 "59 6.5 64
3/20 / 16 93.0 3/20 6.1 60
5/og | 7.0 85.4 6/s0 5.9 58
10/50 6.0 13.6 10/og 5.5 55
15/ag 5.4 65.9 15/0 4.6 46
20/,0 4.6 56.1 20/59 3.5 35

Ca” 0 7.9 100
0.9 twentieth Bie) 69.8
1.8 i 5.5 69.8 |}

5.4 = 5.8 73.7
9.0 5 See 66.0
13.5 ‘ 4.5 al
18.0 i 3.5. 44.5

562

These curves indeed do not, like the others, descend regularly
but show first, to about '/,, grammolecule, a considerable descent.
and afterwards suddenly descend much less, so that they cut a
number of the other curves.

I constantly found this sudden modification of direction at '/,,
grammolecule in repeated experiments.

With regard to the other salts, they appear to retard the urease
stronger in the following order :

NE Kreeg Nar Sr Bas
Up to '/,, grammolecule per Liter the curve for Ca” corresponds
about with that of Sr”, and that of Mg” with that of Ba” . With
the experiments with colloids we do not find as a rule the whole
series mentioned, the series is most complete in the experiments of
Pauli about the increase of the temperature of congealment of
solutions of albumen viz.:

Bor < ON ea ig

With the exception that here Mg” comes after Ba’ , whilst these
two metals on urease to */,, grammol. per Liter show an equal action,
there is perfect agreement between the two series.

The order of succession NH,’ , K° , Na’ , Mg” occurs a.o. in
experiments about the coagulation of albumen by an earthalkali salt.

The coagulation is namely favoured by ‘salts of these four metals
in the indicated order. —

The series K° < Na’ < Sr’ < Ba” < Mg” is found at lowering
the melting-point of a gelatine-gel by neutral salts.

One might draw the following normal series, from the different
series that occur in literature for colloid-phenomena:

NH,” < K° <Na’ < Sr <(Ba\< Me; which correspon
consequently with that which is found for the influence of the
action of urease. Occasionally Ca” occurs in another place in one
series than in the other, now between K° and Na’ ‚now after Ba”.

This depends presumably on the concentration that has been used;
it occurs likewise with urease, as the calciumcurve cuts a number
of other curves above the concentration */,, mol. per Liter.

Il. Anzons.

When we studied the influence of anions on the action of urease
we used potassium salts. Only those acids were used the potassium
salts of which are neutral, the action of urease, like other ferment-
processes being very sensitive to modification of the hydrion-

563

concentration, so that a retardation or an acceleration might be caused
both by a modification of the hydrions and by the anions.

The hydrion-concentration can indeed be kept constant by means of
buffer-mixtures, but as these are likewise composed of salts, borates,

phosphates, citrates, we should make the results more complicated.
by adding at the same time other anions.

The following potassium-salts have been made use of: KCl, KI,
KBr, KNO,, KCNS, K,SO,.

The experiments were executed in entirely the same way as those
with different kations. The obtained results were the following:

Concentration salt | Number | Concentration salt} Number
(grammolecule Biet, Reduced (grammolecule of GC. Reduced
per liter) l/o N* HCI. per liter) Lio N° HCI.

Cl’ 0 7.9 100 So,” 0 | 100
1/50 6.8 86.4 0.7 twentieth 1.5 82.5
3/20 6.0 76.2 | si 5.9 64.9 ©
5/20 Di 66.0 3.4 pn 5.1 56.1
10/59 4.3 54.6 6.9 - 4.0 44.0
15/20 3.75 41.6 K0,3 - > y 3.5 38.5
20/a9 3.3 41.9 13.38.15 5 3:1 34.1

CNS’ 0 9.9 | 100 id 14.3 | 100
'/20 8.7 88.9 "/20 12.6 88.1
3/20 7.6 76.7 3/50 10.9 76.2
5/20 ey fa Aaa 5/20 9.95 69.6
l/o 5.9 59 6 10/o9 8.7 60.8
15/20 5.3 53.5 18/29 7.8 54.5
20/50 4.7 41.5 2/20 12 50.3

NO,’ 0 8.9 100 Br’ 0 9.1 100
1/20 7.8 87.4 1/20 | 7.9 86.9
3/20 6.5 72.8 3/50 6.7 73.7
5/20 5.9 66.1 5/o9 5.9 64.9
'°/20 4.8 53.8 10/50 4.9 53.9
"5/20 4.2 41.0 15/20 43 | 47.3
20/59 3.8 42.6 20/59 3.8 | 41.8

564

If we represent these figures likewise graphically in the same
way as with the kations, then we obtain a number of closely
cramped curves from which appears, that with the same kation the
nature of the anion has only an unimportant influence (fig. 2).

KCNS en KJ
KCL, KBR en KNOs
Kr SO,
2

Fig. 2.

In so far as we can still speak of a regular succession it appears
that CNS’ and I’ retard the action of urease least, and SO", most.
Cl’, Br’, NO', lying between these, are so close to each other, that
the differences do not surpass the errors of observation. As has been
remarked before, we find at different colloid-chemical phenomena
the series CNS’, I', Br’, NO',, Cl’, SO",

The faculty of precipitating colloids rises e.g. in this succession.

CNS' and SO", are here likewise the two extremes of the series,
whilst I', Br’, NO', and CI lie between them. But as the differences
between CNS' and I' and between Br’, NO’,, CI are greater than at
the urease-retardation, it is possible to place them in a definite
succession.

The comparatively slight influence of the anions over against that
of the kations is likewise the reason, why in fig. 1 the concentra-
tions are indicated in grammolecules per Liter, whilst in fig. 2 at
K,SO, they are indicated in half grammolecules per Liter, consequently
as the normalities. . N

[In the first case we have e.g. for NaCl and BaCl, it is true, the’
prejudice that the action of 1 Na +1 CI is compared with that of
1 Ba” + 2Cl', but after the unimportant influence of the anions was
proved: by the 2nd series of experiments, this seems to me to be
preferable to a comparison between 1 Na’-++ 1 Cl’ and */, Ba” + 1CTl’,
which would be obtained from the comparison of normal solutions
instead of molar-solutions. |

In Fig. 2 on the contrary e.g. KCl and K,SO must be compared.

565 ~

As now the influence of the kations is greater than that of the
anions, we had better compare

1K°+10Cl' and 1 K+ '/, 50,"
than 1K-+1Cl' and 2K: + 1 SO,

It appears from a comparison of MgSO, and MeCl,, which exercise
about the same influence at an equal concentration of magnesia,
that really for the same kation the action of 1 SO," corresponds
with that of 2Cl', consequently that of '/,SO," with that of 1 CI.

Mg sulphate Magnesium chloride Concentration
100 100 O grammol. p. L.
64 64 Jeg onda ae?
60 60 So» „on
55 58 5/20 se rene
47 go? 10/59 5 he
45 46 lom » oo»
35 35 1 ‘ wo»

We may deduce from the results that we have obtained, that for
the action of neutral salts on the decomposition of ureum by urease
the kations have a domineering influence, and that the succession
in which those kations can be placed, in accordance with their
faculty of retardation, is about the same as that which we find at
the flaking and at other colloidal phenomena.

The nature of the anion has comparatively little influence; in so
far as we can still observe at anions an increasing faculty of retard-
ation, the series into which they can be placed corresponds like-
wise with the series occurring in colloidal chemistry.

Consequently it is possible that the influence that neutral salts
exercise on a ferment-action consists in the fact that the dispersity
of the ferment is modified by the ions in the same way as with
other colloids. .

Physiol. Lab. of the Univ. of Amsterdam.

Chemistry. — “The Use of the Thermopile of Dr. W. M. Morr
for Absolute Measurements.” By N. H. Strwertsz vAN REESEMA.
(Communicated by Prof. J. BöESEKEN).

(Communicated in the meeting of Sept. 29, 1917).

For a number of Photochemical researches, carried out in the
Phys. Chem. Lab. of the Technical University of Delft, use was
made of the Thermopile and the Galvanometer of Dr. Mot.

In order to be able to express the measured light absorptions in
an absolute measure, it proved necessary to gauge the thermopile.
What follows here is a preliminary communication about the
measurements referring to this, the particulars of which will be
published in my Thesis for the Doctorate.

The idea to gauge the thermopile by means of a Hefner-lamp or
another normal lamp or also by an incandescent lamp tested e.g.
by the ‘“Physikalisch Technische Reichsanstalt’” was relinquished.
A direct method was preferred, (without use being made of auxiliary
light sources, Hefner-lamp or other normal lamp), which could be
carried out by the investigator himself in a simple way, independent
of the measurements of others made in other laboratories.

Besides it would be possible, as will appear, to avoid the
measurement of illuminated surfaces (here therefore a thin line of
light on the thermopile) and the measurement of the distance from
illuminated surface to light source, which becomes necessary in the
use of a normal lamp in the indirect method.

At first it was my intention to make use of the compensation
Pyrheliometer of ANnestrém or of a ‘simplified application of the
principle on which it is based.

The course of procedure would have been as follows. A quantity
of light in the form of a thin streak of light is made to fall on the
platinum plate of the Pyrheliometer, and the electrical equivalent by
compensation is measured. An electric current is namely conducted
through another plate of the same shape. Behind the plates there
are found thermo-elements, which have been adjusted in such a way
that their electric forces work in opposite directions. Thermo-elements
and plates are of the same shape. If the quantity of absorbed light
in one plate is equal to the quantity of heat generated by the

567

electric current in the other plate, the two E.M.F. must neutralize
each other, and then we may write: [== r, in which / represents
the quantity of light absorbed per time unit, 4 the electric current,
r the electric resistance in the plate. When / has been measured
in: this way, the pyrheliometer is removed and the thermopile is
put in its place. |

If U; is the deviation that the galvanometer gets in consequence
of this, then:

Ur
kU; =f =r, hence ks;
1
k is a constant. In this way the pile and the galvanometer are
gauged, & indicates the quantity of energy falling on the thermopile,
which causes the unity of deviation.

In a conversation on this subject Dr. Morr suggested a great
simplification in December 1915. Instead of the pyrheliometer a
blackened platinum plate might be placed immediately before the
pile. Then no special apparatus is required for the gauging, but a
plate is simply slid before the pile, which plate is subsequently
illuminated, then an electric current is passed through it, while the corre-
sponding deviations in the galvanometer are measured.

Dr. Motu was so kind as to give me a quantity of Wollaston
plate, for which I express my indebtedness to him here, and through
which he enabled me to work out and apply his excellent idea.

The mode of procedure was now as follows:

A plate was slid before the pile, an electric current was conducted
through it.

Let us then call the deviation in the galvanometer belonging to
the thermopile, U.

The strength of the current led through the plate, 2.

Its resistance r.

We then write:

~
>

Uik ae ee ae a ts BNG

Then a beam of light originating from some constant source of
light or other, for which in my case a Nernst-lamp served, was
thrown on the plate.

Let us call the deviation of the galvanometer U’; and put the
quantity of light = /, then:

OMEN en OE ENEN RN |)

Now the quantity of light has been gauged, £' is eliminated from
(1) and (2) and we get:

568

pap oe 3
ee . . : . . . . .
Uy (3)
Then the plate is removed, and the same J is thrown directly on
the thermopile. We then get:

Da Up hh ot eae oie io ee

if U, denotes the deviation which the galvanometer now gets.
From (3) and (4) now follows:
vr U';
1 Ui
in which & again represents the quantity of energy required to
impart the unity of deviation to the thermopile.

Now the thermopile and the galvanometer have been gauged.

As now the sensitiveness of Dr. Morr’s thermopile is variable over
the width of the pile, it must be defined at the gauging of the pile
which spot has been gauged. For the same reason it is not indiffer-
ent with what width of beam of light we work. To obtain an idea
on both points, the sensitivity of the pile was determined as function
of the width of the pile, and at the same time of the length of
the pile. ,

For this purpose a very narrow beam of light, 0.2 mm. wide and
5 mm. long emitted by a Nernst-lamp, was thrown on the pile.

The rays fell at right angles to the latitudinal direction of the pile.
The latter stood on a heavy block of wood, which could be displaced
in a direction normal to the direction of the rays. A simple arran-
gement was applied to measure the displacements.

At every position of the piece of wood a number of readings were
made of the deviation. given to the galvanometer, when the beam
of light fell on the pile. Then the pile was shifted vertically, so that
another spot in the longitudinal direction of the pile could be treated.
Also for the new longitudinal position the function of the latitude
was determined.

At last the lines were obtained which are adjoined here.

The thermopile was made in the year 1914 by the firm Kire
and Son at Delft (May, year 1914).

The thermoelements were made of copper-constantan. The pile was
+= 20 mm. long and 8 mm. wide. The pile had to be repaired once,
but showed still a sufficient uniformity.

It follows from the steep course of the curves how greatly the
sensitiveness of the thermopile depends on the latitudinal direction
of the pile.

S&serceesks

“~nwPUaANoewds

569

Lower at of the ORL Middle part of the pile.

|

Ear in mm. | | | | | | |
normal to the direction Nae | | | | á |
of the rays of ght. 13) 14/15 161718 17 .5)19/20/21/13 14 (15/16 17/1 vetje 0) 1920
Deviations of the gal- | | | | | Pel |
vanometer in mm. Op 4 18/27/28) 31 16 3 O| |+1) 4 16 28 28, 31 16 3

Upper part of the pile. or ae
Displacements in mm. | | | | | | | | | a |
normal to the direction, ad Ie AM eN! | | a
of the rays of light. =| | | | jiqjisii7.5jio) | | || Al 1817.5,
iy TTT ENOR rei id im | ; | ;
Deviations of the gal-| | | | | | Nae | | he
vanometer in mm. | | | Pope 29 no || 28 28 30 is

lower part of the thermopile
———-— middle part of the thermopile
oe 1pper. part
-------- part between toppart and middle.

Horizontally the shiftings are indicated in
mm.

Vertically the deviations of the galvanometer
through the illumination of the pile, likewise

570

The necessity of sharp adjustment at a definite spot is very evident
here. If e.g. we have gauged the thermopile at adjustment 16, and
if we use the pile later at adjustment 17, it follows from the table
that errors can be made of the order of 50°/,, at least when again
beams of 0.2 mm. are used.

Hence a displacement of only 1 mm. can already have a great
influence. It is also clear that the deviation is greatly dependent on
the width of the used beam of light.

When we consider e.g. 2 beams that send the same quantity of
energy to the thermopile per second, the centres of which coincide
in the maximum of sensitivity of the pile, but the widths of which
differ, and may be successively 1 and 3 mm., it follows from the
table that in the case of the narrow beam a deviation is obtained
about 20°/, greater. In this calculation the mean values of the sensi-
tivities have been used.

It follows further from the graphical representation that the ther-
mopile has been made very uniform. The maximum sensitivities all
lie at 17,5 mm. This uniform and sharp appearance of the sensitivity
maximum enables us now to use this as eriterion of adjustment.

The procedure in this is as follows: The function of width is
determined with the beam to be used, as has been done above;
from this it is possible to determine accurately with what adjustment
the beam can show the greatest deviation. Then we adjust the thermo-
pile in the required position, after which we carry out the desired
measurements.

This operation, which can take place quickly, is carried out
before the gauging as well as before the use of the thermopile.’)

In this way the difficulties with the adjustment have been solved.

With regard to the width of the beams it may be observed that
in these experiments use was made of a blackened platinum plate
1 mm. wide, which was at a distance of 1 mm. from the thermopile.

This plate can now be used in a simple way to make also the
beams of light about 1 mm. wide.

In the gauging the used beam of light was obtained by cutting
out by means of a screen from a larger parallel beam of light origin-
ating from a Nernstlamp. The sereen is provided with screen doors,
so that the beam to be used can be made broader and smaller.

1) With regard to this adjustment Dr Morr informed me lately, that for relative
measurements he also works with the greatest sensitiveness, by rotating the thermo-
pile to and fro. As a very accurate adjustment is necessary for the gauging, I
prefer to determine for this the function of width in order to determine the maxi-
mum adjustment graphically from the different points that have been found.

571

When first we make the beam fall over the edges of the plate
that is placed immediately before the thermopile, we see on the
thermopile, besides the illuminated plate, 2 lines of light, with the
shadow of the plate between them.

We now slide the doors towards each other, till the two lines of
light have just disappeared from the thermopile, which can be very
sharply observed. Accordingly the beam of light now falls exclusively
on the plate before the pile.

When the thermopile is used the width is made equal to about
one mm. in a corresponding way, only instead of sereen doors use
is made here of a cylinder lens.

With regard to the accuracy we observe that it follows from the
table that 2 beams of equal intensity, but of the widths 1 and 0,5
mm., both adjusted at the maximum, will show about a difference
of deviation of 2.5°/,. It follows from this that if we perhaps make
the widths equal to about 0.1 mm. in this way, no great errors will
henceforth be made with this either.

The question whether the above given formulae may really be
applied here, is fully entered into in my Thesis for the Doctorate.
Nor will the constructions be discussed here, which were executed to
my great satisfaction by the chief instrumentmaker of the laboratory,
Mr. Jon. DE Zwaan.

As a further elucidation of the investigation we shall proceed to
give a numerical example.

Voltage of U U U
source of light I I IU; heat Vale
110 Volt 74.1 m.M. 112 m.M. 0.436 0.434
KOBUS, 61,0 STD 0.451
102); 47.3, 109.0: „ 0.434
08) A SOu2 6 86.8 , 0.417
Generated heat PrX0.24. cal.
i U; in cal. per sec. TR In am, | mean value
r><0.24 l î
0.015 amp. |-179.9 m.M. | 0.0152 2.47 x 0.24 742107 7.40 X 107
0.020 , 3244 , 0.0202 2.47 Xx 0.24 1:80 X¢ 10-7

0.025 , | 500.4 , | 0.0252%2.47><0.24 1.49>< 10-7
0.030 , | 720 , | 0030X24TX04 | 714110-7 | °

572

By making the Nernstlamp burn at different tensions, the gauging
could always be carried out for different intensities of light. Likewise
different intensities of current were always used. These were measured
in a milliampere meter of “Koeler”. The resistance of the pee
plate was measured with the ket bridge.

The values 2,43 2, 2,50 2, and 2,48 £2, hence on an average
2,47 £2 were found for 7, the electric resistance.

pe gt eee Eke .
= ae ie ee oe

== 7.40 « 10—' & 0.434 = 3.2 >¢ 10—" eal. per secant:
ie. that a deviation of 1 mm. is caused by 3.2 X 10 cal. per
second. It should be stated here that the distance from the galvano-
meter mirror to the lath amounted to 174.3 mm. in these measurements.
I owe a few words of cordial thanks to Prof. Dr. W. REINDERS,
who enabled me to carry out this investigation in the Phys. Chem.
Lab., and for the encouragement he gave me during the work.

Delft, July 15, 1917.

Chemistry. — “On Milksugar’. Ul. By Prof. A. Smits and J. Guus.
(Communicated by Prof. S. Hoogewrrer.)

(Communicated in the meeting of Sept. 29, 1917).

The Ternary Pseudo T, x-figure.

When we draw up the pseudo-ternary 7’, a-figure, we come to
the following representation. (Fig. 9).

In the side plane for a-anhydride— H,O
we know the initial solubility from O°
to 25° (determined by Hupson loc. cit);
further the situation of the eutectic point
at —-0.3° and 0.27 mol. °/, « (point d).

We found the melting point of «a at
222°,.8 and that of Qa, at 201°,6. With
regard to the real transformation point
of ag, 1. e. the point where under the
pressure of 1 atm. the following con-
version takes place:

Gag >edL

we have been able to ascertain that it
lies above 100°; for when a-anhydride
was brought at 100° in a saturate milk-
sugar solution, a marked generation of
heat still took place as a proof that at
this temperature hydration of the solid
a-anhydride still occurs.

If instead of a saturate milksugar

Fig. 9. solution pure water of 100° is taken,
this generation of heat is not observed, because the velocity of
solution is very great at this temperature, and the negative heat
of solution then hides the smaller positive heat of hydration from us.

In the side plane of the system p-anhydride-water we know the
initial solubility at 0°, the eutectic point d' lying at —2°,3 and 2,2
mol. °/, 8, and further the melting point of H,O (point a), and the
melting point of B-anhydride 0’, lying at 252°, 2.

42
Proceedings Royal Acad. Amsterdam. XX.

574

In the side plane for a-anhydride + g-anhydride we only know
the meltings point 5 and 5’ of a and g-anbydride, it being assumed
here that there occurs a eutectic point (point d").

The melting surfaces in the space are clearly visible in the spacial
figure. At the place where the melting surfaces or solubility surfaces
intersect, there arises a three-phase line, and at the place where
these three-phase lines meet — and this meeting always takes place
by three three-phase lines at the same time —' there arises a
quadruple point.

Two quadruple points may be pointed out here in our spacial
figure, first the point c' lying above 93°,5, where coexist:

Saag + Sy --- Se + \
and secondly the ternary eutectic point e‚ where coexist:

Saag + Sed So d L

lying according to the calculation at — 2,6°.
The T, x-spacial figure of the binary system.

In the ternary pseudo figure described here lies the binary system.
We get this system, when we draw the surface of equilibrium
for the equilibrium «Sg in aqueous solution at different total con-
centrations and different temperatures through the axis for the
component H,O.

This surface of equilibrium intersects the melting point surface
of g-anhydride along the line /m, from which it follows that in
the binary system water-miiksugar the saturate solutions in stable
state coexist with g-anhydride from m to /. A change sets in in this
in the point m, for below this point the equilibrium surface does
not intersect the melting point surface of B-anhydride, but the
melting point surface of Gag, SO that it is clear from this that the
point m represents the found remarkable transition dehydration
point lying at 93°,5, and for which on supply of heat the trans-
formation

Saag > StL
takes place.

Further the said surface of equilibrium of course intersects also.
the ice-plane, and the line of intersection here indicates the melting
point line of the ice in the binary system. This melting point line
of the ice and the melting point line of the hydrate intersect in n
(at 0,65° and + 0,6 mol. °/,), where the surface of equilibrium meets
the eutectic line de of the pseudo-ternary system.

575

It should be pointed out here that the shape of the surface of
equilibrium could be derived from the final solubility of Sag and ‚Sz

from 0° to 170°, and further from the projection of this line on
the ground plane over the temperature range O° to 100°.

Theory of the mutarotation.

On the ground of the new views to which we were led by our
experimental investigation, we now arrive at a theory on muta-
rotation which is essentially different from that drawn up by
Hupson.

As follows from the here given explanation of the behaviour of
the system water-milksugar, this system must be considered to be
pseudo ternary, in which water is then a component and a and 8
milksugar are the pseudo components. From this it follows imme-
diately that it must be assumed that these different sorts of molecules
occur side by side in the liquid phase, in which the phenomenon
of the mutarotation takes place. We know further that for ‘milk-
sugar a hydrate ag, is known in solid state. This alone would
already point: to the presence of this hydrate also in the solution,
for which, as we have seen, other phenomena plead too.

Now it is evident that when the e-anbydride can combine with
water to a hydrate, this will also be the case with the 8-modifica-
tion. That we do not know this hydrate in solid state, does not
plead at all against this assumption, for this tells us only in this
connection, that the solubility of the 3,, must be greater than that
of Bg-anhydride, which is just the reverse for the other pseudo-
component.

In favour of the assumption of 8,, in the solution pleads further
that for maltose, a substance which also presents mutarotation, the
Bag is known in solid state’), but the aq, is not, and also that here,
therefore, we find exactly what is still wanting for the milk sugar.

We assume, therefore, that in the aqueous solution the following
equilibrium sets in:

B a H,O — aq:
In this symbol there are two conversions of which we can say

that they take place with great velocity.
First the reaction

1) J. Amer. Soc. 31 76 (1909) en 32 p. 894 (1910).
42*

576

a + H,O pas Gag:
This follows from what was stated in the first communication.
With regard to the conversion
B+ H,O 2 Bay
we must refer to the system H,O-maltose, in which it was also
found that this reaction proceeds very rapidly. Hence it might be
said that for the sugars the hydration equilibrium seems to set in
very rapidly, and on this ground we assume that the hydration
equilibrium of B-anhydride sets in very quickly.
In this way we come, therefore, to the conclusion that the setting
in of the equilibria

a Gag
Nl and {f
B Bag

must be held responsible for the mutarotation, in other words that
the establishment of these equilibria takes place slowly.

In this view the difficulty that lies in the mutarotation theory of
Hupson, and is also acknowledged by himself, is entirely obviated.
Hupson namely assumed that we should have to do with the
following equilibria :

1 2
dann. + HO, @ Hydrate @ Bann. + H,O

in which the equilibrium 1 sets in with great velocity and 2 very
slowly.

On account of this view HupsoN is forced to assume that for
maltose exactly the reverse takes place, and that it is there exactly
the equilibrium 2 that sets in very rapidly, and that 1 sets in very
slowly.

Hupson says about this: ‘“Why the monohydrate should change
instantly to the a-form for some sugars, but to the 9-form for others,
is entirely unknown, and is a most interesting problem.”

Hupson’s view was somewhat forced, as he was obliged to assume
for two perfectly analogous reactions, viz. hydrations of sterioisomers,
that one proceeds very rapidly, the other very slowly. According to
our view it is assumed that the reaction velocity of the said per-
fectly analogous conversions differs but little, whereas a great differ-
ence is assumed to exist between reaction velocities, one of which
is a hydration and the other an intra-molecular conversion.

Mutarotation would, therefore, not be due to a slow dehydration process
in the sugar series, as was assumed by Hupson, but to the slow

577

establishment of the internal equilibrium between two stereo-isomeric
forms.

Constitution of the mono-hydrates in the sugar series.

TorreNs was the first who ascribed a lacton structure to the a-
and J-modifications of the sugar series.
For glucose we should e.g. have the two following configurations :

CH,OH . CHOH . CH . CHOH . CHOH . O< pj and
et ied gl

CH,OH . CHOH . CH. CHOH . CHOH . C<
Kg a ee |

OH
H

which, therefore, only differ by the asymmetry of the final carbon
atom. These compounds contain, therefore, no aldehyde group, and
it is in accordance with this that they do not colour fuchsine sulphuric
acid, form no bi-sulphite compounds, and do not exhibit any tendency
to polymerisation.

Simon, and especially ArMsTRONG have succeeded in showing the
probability of this structure, the latter by demonstrating that the
two methylglucosides «a and 8, which certainly do not contain an
aldehyde group, are converted to «a and p glucose by hydrolysis
with enzymes.

Hupson *) advanced facts in 1909, which gave a very great probability
to this formula.

He says namely this: if we assume that the a- and 2-modifieations
of the aldoses possess the structure proposed by Torrens, the molecular
rotation of one may be represented by the sum + A+ B (A=
rotation of the asymmetric final carbon atom, B = rotation of the
rest of the molecule) and of the other by — A + & (rotation of the
group B diminished by that of the levo-rotatory final carbon atom).

The difference between the molecular rotations of the «- and g-
modifications is therefore 2A and the sum 24. From this it follows
that all the aldoses not substituted at the final carbon atom will
bave to exhibit a constant difference of 2A, whereas the same aldoses
with different groups at the final carbon atom must all yield the
same values of 25.

Hupson found this prediction actually confirmed, not only for the

1) The signification of certain numerical relations in the sugar group. J. Amer.
Chem. Soc. 81 66.

578

unsubstituted sugars, but also for the glucosides, the acetates, the
hydrazons, and the compound sugars, so that this gives a very
weighty support to the validity of the assumed structure for the
a- and $-modifications.

Hupson, however, has overlooked a very important point as far
as the structure of the sugar mono-hydrates is concerned.

In this he has not been very consistent, as he assumes e.g. for
the sugars lactose, glucose, arabinose, and galactose as rotation for
the a-modification (a rotation which is not directly to be determined
because the «-modification of those sugars passes into hydrate on
contact with water), that calculated from the rotation of the hydrate,
for which he gives the structure as follows e.g. for glucose:

CH,OH. CHOH.CHOH.CHOH. CHOH. CH(OH),.

This formula cannot be correct, as in this way the asymmetric
final carbon atom of ToLreNs’'s formula disappears, which means
that the rotation of this hydrate will no longer be A + B, but
only B’, which quantity is equal to the algebraic sum of the rotations
of the asymmetrie carbon atoms indicated by X:

x x x
CH,OH . CHOH . CHOH . CHOH . CHOH . CH(OH,)
J Y B a
and this sum is no longer equal to B, because the asymmetric
carbon atom has changed now too.
The observed regularities for the aldoses are, therefore, no longer
to be explained in this way. We should e.g. have for glucose:

a-hydrate: B; sum
B-anhydride: — A+ B | —A+ B+ B’
On the other hand we have e.g. for the ethylglucosides a and 8
(both anhydrides)

a-ethylglucoside: + A’ + B ) sum
B-ethylglucoside: — A’ + B | 2B

Hence the sum of the molecular rotations of the a and 92 glucose
cannot be equal to that of the « and the 8 ethyl glucoside, 23200
and 25230 being found, hence a pretty good agreement. The same
thing holds for galactose and ethyl galactoside, for which is found
34700 and 36400.

For other glucosides e.g. methyl-d-glucoside, methyl-d-galactoside,
and methyl-d-xyloside, of which the @-modifications of the two first
contain resp. 4 H,O and 1 H,O, and of the third the a and 2
modifications are anhydrous, Hupson’s law holds good in spite of

579

this, though the rotations of these substances are taken as if they
were really anhydrous.

It is therefore evident that the Bonen superposition takes place
here whether these substances contain water or not, and that for
all the’ asymmetric final carbon atom with the lacton ring occurs.

Accordingly we arrive at the result in this way that for the
unsubstituted aldoses e.g. glucose, galactose, milksugar etc. the presence
of H,O does not change amything at all in the structure of the
asymmetric final carbon atom.

No more, therefore, than tor methyl glucoside will the water be
bound to the final carbon atom in glucose itself. We may now
question how it is that Hupson notwithstanding this inconsistency,
has arrived at satisfactory results. The answer to this question is
very simple. In his calculations for the rotation of the anhydride
Hepson has namely taken the rotation of the hydrate, which is
only allowed when the water is bound in such a way that it
cannot influence the rotation of the final carbon atom.

If e.g. milk sugar hydrate is a molecular compound of C,, H,, O,,
with one mol. of water, and quite to be compared with CuSO,H,O,
then it is clear that the water bound to the sugar molecule does
not affect its rotation or only very slightly. If we, therefore, know
the rotation of «-milksugar hydrate, this rotation is the same as
that of milksugar anhydride, as nothing has been changed in the
grouping of the active carbon atoms, and thus it may be understood
that though Hvpson has executed his calculations with the rotations
of hydrates, they have yet led to good results; but in this way it
has been proved very convincingly that the hydrates do not contain

the group Eo au

SUMMARY.

The results of this research may be summarized as follows:

1. The final solubility curve was determined between 89° and
200°, starting both from g-anhydride and from a-anhydride, and
this curve, in connection with Hupson’s determinations, proved the
existence of a break at 93°.5.

2. From the fact that a-anhydride is always converted into
B-anhydride above 93°.5, and also from the melting-points of e-anhy-
dride (222°.8) and of g-anhydride (252°.2) determined for the first
time follows with certainty that above 93°.5 the a-modification is
metastable.

580

3 Tt could be established by dehydration experiments of the
solid hydrate at different temperatures that this hydrate is a hydrate
of the «-moditication.

4. It was demonstrated by experiment that 93°.5 in the system
water-milksugar is no ordinary transformation point, but a point
that we propose to call a transition-dehydration point, as at this
temperature the hydrate of the a-modification under the pressure of
1 atm. dehydrates and is also converted to the B-modification.

5. The observed phenomena have suggested that for milksugar
two kinds of molecules « and 8 have to be assumed, on account of
which the system water-milksugar must be considered to be pseudo-
ternary. Of this system the isotherms have been determined under
the pressure of 1 atm. at different temperatures. By also indicating
the curve which represents the situation of the equilibrium between
« and 3 milksugar at different total concentrations the situation of
the binary equilibrium diagram at different temperatures could be
pointed out in the pseudo ternary system, which enabled us to
explain in a simple way all the phenomena observed.

6. Then the pseudo ternary 7’-a-figure of the system H,O + «
milksugar + 8 milksugar with the surface of equilibrium lying in it
was constructed.

7. A new theory was drawn up for the mutarotation, and in
conclusion it was still proved that the monohydrates in the sugar
=(OH),

H .

series cannot contain the group —C_

Amsterdam, June 29, 1917. Anorg. Chem. Laboratory
of the University.

Chemistry. — “Influence of different compounds on the destruction
of monosaccharids by sodiumhydroxide and on the inversion
of sucrose by hydrochloric acid. 1. Constitution-formula of
the hydroxybenzoic acids and of sulfanilic acid’. By Dr.
H. I. WATERMAN. (Communicated by Prof. J. BÖESEKEN).

(Communicated in the meeting of September 29, 1917).

Benzoie acid, salicylic acid and meta- and para-hydroxybenzore acid.
In a previous communication ') | have proved that in alkalic solution
phenol behaves as monobasic acid. It can be expected that in alkalie
solution benzoic acid will act as monobasic acid. This expectation
has been confirmed by the experiments.

According to the said properties of phenol on the one hand,
benzoic acid on the other hand it might be supposed that in alkalic
solution the three hydroxybenzoic acids would behave as two-basic
acids. It has been proved that this is only the case with meta- and
para-hydroxybenzoic acid. Salicylic acid on the contrary in alkalic
solution acts as mono-basic acid. The results of the referential experi-
ments are united in table Ie and 1?

From the polarisation at the beginning of the experiments is proved
that neither benzoic acid nor the three hydroxybenzoic acids have
considerable influence upon the polarisation of glucose (table [4 and
16). Whilst the presence of 5 em’ of 1,06 normal sodinm-hydroxide-
solution after 3} hours has lowered polarisation from + 11,1 to
respectively +-5,8 and +5,9 (N°. 5 and 8, 1%, the addition of 1 milli-
gram-molecule of salicylic acid and benzoic acid has caused that the
polarisation has only been lowered to + 6,4 and + 6,6 (N°. 6 and 7,[*®).

Analogous results were obtained with the same experiments after
+ 64 hours; under the influence of 5 em? of 1,06 normal NaOH-
solution the polarisation has here been lowered to respectively + 3,3
(N°. 5) and + 3,1 (N°. 8), at N°. 6 and 7 only to + 3,9 and + 4,0.

1 milligrammolecule of the said acids compensates the action of
something less than 1 em? of 1,06 normal NaOH-solution; salicylic-
acid and benzoie acid behave therefore as about monobasic acids.
If salicylic acid would act as two-basie acid, the polarisation of
N°. 6 (I*) after 34 and 64 hours should not be respectively — 6,4

1) These Proceedings, April 27, 1917. Vol. XX p. 98.

582

and + 3,9 but about + 7,6 and + 5,4. The intensity of yellow-

colouring too was herewith in agreement.

qual

v

,

(I?) after 24 hours was nearly e

+

The colour of N°. 6 and 7

to that of N°. 4 (I©).

Benzoic acid ') and salicylic acid 1) in alkalic solution.

TABLE la.
En Numb.of cms. Polarisation (2 d.m.) in grades VENTZKE Colour of the
E Added 1.06 N. NaOH, ee anaes tat oe ee SOLUTION after
5 added. | at beginning jafter +3'/, hours after +6!/, hours 24 hours
40 cm3 of a solu-! |
1 Lee An 0 5 + 11.2 + 11.1 ‚not determined} colourless
ucose 2
zor 97/0 B'UCOSS: En = = See ee — ; Z
: = | rather pale
2 id. 2 By + 10.7 + 8.9 + 6.9 yellow
es Ee ie a . “— - a Been En
‘ Rila, |
3 id. | 3 ae + 10.2 + 7.6 + 5.4 | pale yellow
| | a. |
oe il eee —
. | A 3 deep
4 id. 4 Ei + 10.3 + 6.9 | + 4.1 | pale yellow
5 id. | 5 | SE Pee 51S. (ae oo ee svelfow
| É : E | wk 3 or Se eee AC
| Ref ; ; Os |
‘ 138 milligr. salicyl. acid > | | deep
6 | id. = 1 milligr.molecule | 9 es) 5 pn ea Bane | Pale yellow
es | |
— —-|—— nds el Te | En
| : 122milligr.benzoic acid s 5 deep
1 id. = 1 milligr.molecule | 5 = | + 10.1 + 6.6 | + 4.0 | pale yellow
2 bat nge ae | dE: s Le :
8 id. | Biga poes Sne el yellow
| |

') Preparations from the laboratory-collection.

583

5

table 1% it follows that with N°.

and 9, where 5 em* of 1,06 normal NaOH has been added after
three hours the polarisation has been lowered from + 11,4 to

+ 5,3 (N°. 5 and 9, 14).

in

results united

the

From

06 milligrammolecule of meta- and para-
has caused that the polarisation has been

lowered only to +7,5 and + 7,7 (N°. 7 and 8, IP) about equal

The presence of 1

acid

hydroxy benzoic

=~

TABLE Ib, o.-,') m.-,') and p.-') hydroxybenzoic acid in alkalic solution.

a Numb. of cm3. delen de, lut grades ‚ Colour of the
= Added 1.06 N. Na OH ee od es solution after
Z| added at beginning | after 3 hours | + 2! hours
| 40cm3ofasolu- |
1 | tion containing | 0 | + 11.4 | not determined colourless
__| +59 glucose 4 tos ie Pas EEE = an te
2 id. 2 = + 10.9 4 8.5 pale yellow
| So 5 2 sie
| | | 65
3 | id. | 3 E | + 10.5 1-45 yellow
= = ae | 38 De kn
4 | id. 4 SEE Ot oen enen
| Te | : |
| ‘ES Pepi 5 Er ae
5 id. 5 is EEE) EE brown
‘ o | |
est |- : SE aca i
| 146 mgr. salicyl. acid = Sy | :
6 | id. 1.06 milligrammolecule 5 Ze + 10.0 + 6.4 yellow-brown
en = (3) = = 3 =
eee ellow some-
: 146 mgr.m. hydroxyben- EE Ye
Zj id. zoic acid=1 06 mgr.mol. 5 38 + 10.2 eas ns deep
EE = ee en Be
8 id. pp eae tae sie 5 2 10.4 ae yellow
7 | e= Ee Roel.
9 id. 5 + 9.8 + 5.3 brown

') Preparations of “KAHLBAUM”.
2) CsH4 (OH) COOH. 1 Aq.

584

to the polarisation of N’. 3 (14), where 3 em’ of 1,06 normal NaOH-
solution had been added. 1 milligrammolecule of m.- and p.-hydroxy-
benzoic acid compensates therefore the action of 2 cm° of normal
NaQOH-solution.

To avoid every accidental deviation in the comparison, experiment
N°. 6 (12) served, from which could be concluded once more that
in alkalic solution salicylic acid behaves as monobasic acid.

The intensity of colour of the solutions, which with these experi-
ments (16) was observed after 21 hours was in rather good agree-
ment with these results. *) |

In the above we have met a difference in properties of salicylic
acid on the one hand and the meta- and para-hydroxybenzoie acids
on the other hand, which is not expressed by the usual constitution-
formula: |

OH OH OH
4 \ Coon 4 ps ( 5
he \/COOH hes
COOH
ortho meta para
hydroxybenzoie acid.

Indeed this difference had been stated in other directions, viz. :

1st. The slight solubility of salicylic acid in water and the strong
solubility in oil. *) .

Ind, The slight liability to attack of salicylic acid by organisms,
compared with the strong liability to attack of the meta- and para-
hydroxybenzoic acids. ?)

3°. The antiseptical action of salicylic acid, which is strongly
connected with the properties mentioned under 1 and 2.)

Furthermore in such compounds the presence of hydroxylgroups
increases the solubility in water and decreases the solubility in oil.

Finally in many cases the augmentation of the number of
hydroxylgroups causes that the compounds can be attacked easier by
organisms. ”)

These considerations necessitate to reject the usual constitution-

1) It must be remarked that the observations of intensity of colour have no
absolute but only a relative value; the results of a series of experiments therefore
can only be compared mutually and not with experiments belonging to another
series. To a certain degree this is also the case with the polarisation because the
temperatures at the beginning of the experiments are not always the same but
depend on the temperature of the laboratory.

2) J. BOrSEKEN and H. I. Waterman, These Proc. November 25, 1911 p. 608.

a

585

formula of salicylic acid, because this formula contains a hydroxyl-
group in the nucleus next to the carboxylgroup.

H A formula as proposed by Brunner *) is
G in accordance with the just mentioned pro-
ZENE porties of the salicylic acid.
HC Ge)
| |
HC C— oh H
NN
C H
H
Lactic acid, Male acid,
CH, . CHOH . COOH COOH . CHOH . CH, . COOH
Tartaric acid, Citric acid.

COOH. CHOH. CHOH. COOH COOH.CH,. C(OH).(COOH)CH,COOH.

The research of these four acids was quite in agreement with
their constitution-formula, in connection with my experiments described
before.

In alkalie solution the alifatic alcohols, such as methylic alcohol
and aethylie alcohol, have no acidic properties. *)

It could be expected that lactic, malic, tartaric, and citric acid
would behave as respectively one-, two-, two- and three-basic acid
in accordance with the presence of an equal number of carboxyl-
groups.

The referential experiments are united in table II.

Hippuric acid and _ sulfanilic acid *)
C,H,CO.NH.CH,.COOH p.C,H,(NH,)SO,H.

Although about these compounds with certainty nothing could be
foretold in accordance with the neutral character of acetamide
(CH,.CONH,) and urea (CO(NH),) in acidic and alkalie solution *)
it could be expected that the aminogroup of the benzoxylamino-
acetic acid would not enable this compound to fix hydrochloric acid
under the circumstances of my experiments. Therefore it might be
supposed that hippuric acid in acidic solution behaves neutral, in

1) Compare A. F. HorLEMAN, Die direkte Einführung von Substituenten in den
Benzolkern; Leipzig, 1910, p. 180.

2) Chemisch Weekblad 14, 119 (1917).

3) The used preparations were from the laboratory-collection.

4) These Proceedings, June 30, 1917.

586

TA

BEEN;

Ila. Malic acid and tartaric acid. ')

Lactic acid, malic acid, tartaric acid and citric acid in alkalic solution.

g Numb. of cm3. ban (2 dm.) in grades | Cojour of the
E Added 1.06 N. NaOH | ENIZKE solution after
= Sap A aig tee eR Fe
a | ao At beginning After + 63 hours, + 24 hours.
40 cm3. of a sol. | Dn
l cont. +59/, gluc. 3) ed sg + 10.3 + 5.7 pale yellow
zs | Bel § |
2! id. | 5 aes hoe eee +3.6 | — yellow
| RA os FEM | =
A 150 mgr. tartaric aci BE ps eae
3 id. — | m.grammolecule?) 5 ee 5 = 5 + 11.0 + 5.7 pale vellow
a 134 milligr. malic acid —| Bee ST 8 EN ae
‘ | : = = GE
4 id. ‚1 milligrammolecule 2) 2 B £ 5: ata re pales velo
IIb. Lactic acid and citric acid. ')
FS | | RE 5
3 | ‚Numb. of cm3, | | polgueatigh (2dm.) in grades \ Colour of the -
| Added 1.06N.NaOH | | ENIZKE solution after
= “= — = — = mom,
Z a | added | At beginning ‘After + 6 hours + 24 hours.
40 cm3, of a sol. | | u | |
1 ‘cont, + 50/, gluc. | 0 5 [ie te Te er Ne | colourless
ze = Be ESS wo | | ed =
2 id | | 2 Es aj 10.7 GA | rather pale
id. Î sE ne ere | yellow
ass pt ASH ze of a ie a |
| kaf | |
| id 8 a not determined + 4.8 pale yellow
er - En | =
4 id. | | 4 B = st not determined + 88 | yellow
= wea : nd ee |
| | oY |
5 | id. | 5 288 | + 96 | + 2.6 | yellow-brown
—— —— —_—_—___- == Or | oo — -
| : 210 milligr. citric acid | el | | rather pale
BY ia '(+1Aq.) =1 mgr molec. 3 i 3 ziee LOST + Ol yellow
| — = - = 33 | |
| : 90 milligr. lactic acid = =H |
id, 1 milligrammolecule 5 a = | sy t0ee Tose TENE
<a av ms E E
8 id. | 5 Ss | + 9.9 + 2.8 | yellow: brown

1) The preparations employed were from the laboratory collection.

*) The polarisation caused by tartaric acid and malic acid is slight.

587

alkalic solution as monobasic acid because this compound possesses
one carboxylgroup.
The referential experiments were in agreement with this expectation.
When we suppose that sulfanilic acid possesses the open SO,H
constitution-formula we might expect that this compound
should behave in alkalic solution as monobasic acid and | |

in acidic solution as monacidic alkali. Se Oi
However it was proved that in acidic solution sulfa- NH,
oo 1 | le) on + oo nw — | Number
| tee
as Sg
— —e a — — — — =8 Ped
a a. a. (Sh a. a. a me
a | ian Oo
| = 2.
| 5 50
Bon F
Sk a Do
o| 32 3 =]
3 oO =: >
= 3 |) ahs
Gee ele =
32.08 | © 9g > fc
=. JR a m
s | A3: = |
Sl Eu a =
Shes = S
5 ES nova =a :
=
EERE
ay ee 23 =
ke)
Es eZ e a 2
— Z =
| p= ie
| eos fe
Bee 18
or or oO or > ie) Do Co oO Ze a:
ane
CEN EE
Tw a
; Ex : a)
Filled up to 50 cm. and placed in thermostat =
with watermantle. (Temp. 33°—34°.) 2
‘ 3 je) | en as
a = = o i»
ep © IG.
o | 5 ana | Geen Pt, Ke
© _ Le) a 5 5 1 o =. o |i
=] =] S|; <5 f=
| © ® CEM ee ED
p| Ros
| ee
+} + ) + | + + IE
B sg aS
cm) vo o2 = > id eee a es
~ m oo EN m 1S 5
5 jen
E a| 8
oo iss)
= = See ne © 8 8 5
© bne | Bao 5 te oe ogee ae)
< BES net = ®, a (ee ° Ho
an : = eg ws
Si) eee bee Soe | oe | Bee |e | ues
© ® a) le) oO ES) 4 arts)
s oS 3) #05 = s > wn e
a er) 5 =? 3 a

588

nilie acid behaves neutral and in alkalic media as monobasic acid.
The referential experimenis are united in table Ille and III’.

= MVR Peat adel AEE re! = Number
Ww ee = | = eS ae ee Mo En
w
eae ee 88
| | be 1 ge |
om Ww
Breese 2 ZT EEM NRN
Re de a a a a a a Avene
= > 6 2 SS =p
2 @ oe mo p
en DD NW
ae | | raa
208 —_____— - Sos = =:
BB rca rte any | Seg ioe D3 |
hele) © Ss | ta
Bo be ee bape eae | is
= LS = = | = = | | | ee)
Bog GEB 1 te oe | | > =
eee Sas | ga: | = m
Vv AT \ 3 | Hee? || @ “
5 io) 5 | 3 | a. —)
Es Ot Cie as >
A ony || eon Meare | :
ae Eee E |
ws zin IA En {oo
Sip ED | eed
= 3 TI : ! ae
ES) | o 5
= iz Is
ues —5
az: | SS 5
a ar ur On as Se) bo o acs fc.
ae | | NA Tee
ONE | To |P
ze | 13
oo | | | | | | Q a.
Ee ee en —— |
ES Diluted to 100 cm.3 and placed in =
aS thermostat with watermantle. (Temp. 33°—34°).
ae ae - =
— J 5, | a _
B rl Seeka ten oe ee 5
EE St ae | ae De EN 5 5
as Pia am a ROR I ene Le m | DN © — | oo fe.
Bee Sins gli) SA Er Ss
(eo) > | | | | Ox a 5 me
en fo) hia ras oe = ET > | 5 =
ae | . 5 Valen,
Ww 5 .
aoe temperature of the polarisation ele IS
See liquid was 26°.5 alo le
ET Ne ese IAR ae ; ; Te ae
Bu Bers Se + | Bl ee
| | | PEER a | —
a> on On on ol a | a © id 5 ="
hs =) CE KE le) (de) Do … nl IS
Si gn REE
< © ; me te ZEK
@ pw temperature of the polarisation Satay
7 2. liquid was 28° ah
EN ren 2 ak’ a # NL » EDE ae ie 2
| 5
5 EN
EEND | | | | han Beate p 8
7) — — — _ _ ae oO <a
a Ae Te Se SIEN ere = a eae
= al nl EN See fe Ox a Le Aes
» Pe @ al
_ z fo)
Q temp. of the polaris. = = | a
- > fe) 3
= liquid was 28°.5 a w |

From the experiments 6 and 7 (III*) we see that after + 6 hours
polarisation has been lowered to respectively + 3,6 and + 3,7.
This number is almost equal to the polarisation of Nr. 4 after the
same time. From which it may be concluded that 1 milligram-

589

molecule of sulfanilic acid and hippuric acid each compensate the
action of something less than 1 cm? of 1,06 normal NaOH-solution.
In alkalic solution they act practically as monobasic acid.

The mentioned acids practically have no influence on the inversion
of sucrose by hydrochloric acid (III).
‚ This gives rise to the supposition for sulfanilic acid in alkalic
solution of the “open” constitution-formula, in acidic solution of the
“closed” formula.

NH, NH,

ES Were cade Bs ae
awe, Sulfanilie acid ee ep ELD,
we ed

SO,.OH SO,
in NaOH solution in acidie solution

Dordrecht, August 1917.

43
Proceedings Royal Acad. Amsterdam Vol XX.

Chemistry. — “The Structure of Truaillic Acids.” *) By A. W.K.
DE Jone. (Communicated by Prof. P. van ROMBURGH).

(Communicated in the meeting of Scpt. 29, 1917).

Up to now the following truxillic acids are known; a, 8, y, J, «
truxillie acid and 3 coecaic acid, which belong to two series, because
the members of one series cannot be converted to those of the
other without previous depolymerisation to cinnamic acid.

The y acid belongs to the series of the a truxillie acid; it is
formed by heating of the a truxillic acid with acetic acid anhydride *®);
the 2 cocaic acid, which is formed by melting from a truxillie
acid with KOH, *) belongs to the same series.

The second series is derived from 3 truxillie acid, which through
melting with KOH passes into d truxillie acid *). The last acid yields
e truxillie acid through heating with acetic acid anhydride; the «
truxillie acid possesses the same melting point as y truxillie acid;
a mixture of the two acids melts, however, about 20° lower, from
which the difference of these acids can already appear. Also in the
solubility of their salts there are found great differences.

By the formation of « truxillie acid from @ normal cinnamic acid ®)
and of 3 truxillic acid from 8 normal cinnamie acid ®) it is known
that one of the 4 following structural formulae, corresponding with
the 4 different ways in which 2 molecules of cinnamic acid can
combine under formation of a tetramethylene ring, belongs to these
truxillie acids.

These 4 structural formulae belong to 2 series viz. the two first
to one, and the two last to the other series. The members of these
series cannot be transformed into each other without previous
depolymerisation to cinnamic acid.

We know from LIRBERMANN's researches that 8 truxillic acid
yields benzil?) on oxidation with potassium permanganate, from

1) Ber. 22, 2255; Ber. 28, 2516; Ber. 26, 834 ; Ber. 27, 1410.
3) LIEBERMANN, Ber. 22, 2240.

3) Hesse, Ann. 271, 202.

4) LIEBERMANN, Ber. 22, 2240.

5) RUBER, Ber. 35, 2908.

6) These Proc. 1915, Vol. XVIII p. 181.

1) Ber. 22, 2253.

43*

592

Cuo H COOH
lll IV

which it appears that for this acid the C,H,-groups must be found
at 2 adjacent C-atoms, so that in connection with the formation
from normal cinnamic acid one of the formulae II] or IV must
be assigned to this acid. |

The 3 truxillie acid forms an internal anhydride‘), and is not
changed into another truxillic acid by heating with acetic acid
anhydride *). It appears from this that the COOH groups are placed
on the same side of the closed-chain of four carbon atoms; hence
we must give formula [II to the acid.

Through melting with KOH go truxillie acid is formed from 8
truxillie acid ?); this acid cannot have arisen from the former by
displacement of one COOH or one C,H, from one side of the ring
to the other side, because then in the former case J truxillic acid
with acetic acid anhydride would have to yield 3 truxillie acid,
and in the other case d truxillie acid would not be changed by
heating with acetic acid anhydride, neither of which is conformable
to the facts. It must, therefore, be assumed, that 2 groups exchange
places at the same time, viz. a C,H, group and a COOH group,
because, as is easy to see, the formation of e truxillic acid from d

1) Ber. 22, 834.
2) Id. 2240.
SA:

593

truxillic acid through heating with acetic acid anhydride could not
be explained in another way.

These exchanges of place of a C,H,- and a COOH-group from
one side of the closed-chain to the other side can also take place
for the groups of 2 adjacent C-atoms of the four ring, and also
of 2 C-atoms placed opposite each other. In one case formula IV
is obtained, and in the other case the following formula is valid.

%

CHs

COOH
Vv

This formula is built up from 2 molecules of allo-cinnamic acid.

The d truxillie acid now is formed from normal cinnamic acid,
viz. through the illumination of the stable lead salt of this acid. *)

0,466 gr. of cinnamic acid, as lead salt, gave 0,075 gr. of truxillic
acid and 0,095 gr. of d truxillie acid after 27 hours’ illumination.

The formula [V must, therefore, be assigned to the d truxillie
acid, whereas the « truxillie acid possesses the following formula
(See form. VI following page).

By heating with hydrochlorie acid at 180° and also by melting
with KOH it is changed into d truxillic acid. These two acids are
in the same relation to each other as fumaric and maleic acid, ¢ is
the cis-acid, d the trans-acid.

1) Shortly an extensive paper will be published about the action of solar light on
cinnamic acid salts,

594

[It is clear that one of the 2 structural formulae I or II must be
assigned to the « truxillic acid.

Coo
VI

This acid does not give an internal anhydride, *) and is converted
by heating with acetic acid anhydride into y truxillie acid,*) which
acid is again converted into @ truxillic acid by heating with hydro-
chlorie acid.*) These two acids possess therefore a similar isomerism
as the cis- and trans-acids, in which the « truxillic acid possesses
the trans-form. In the y truxillie acid the COOH-groups are on one
side of the ring, this not being the case for the a truxillie acid.
Accordingly this latter acid must possess formula I, and the y
truxillie acid is formed from the a truxillie acid by displacement
of one COOH-group, which causes its structural formula to assume
the following form. (See formula VII).

The 8 cocaic acid arises from « truxillie acid by heating with
KOH. *) Through heating with acetic acid anhydride it is not changed
into another truxillie acid. In this acid the COOH-groups must
therefore be situated on one side of the ring, just as in y truxillic
acid. It can however, not have arisen from a truxillic acid by

1) Ber. 26, 834.
9) Ber. 22, 124.
8) Ber. 22, 2245,
4) Lc,

595

displacement of a COOH-group alone, because then it would have
to be converted into « truxillic acid by heating with hydrochloric
acid, whereas it is not changed by this operation. There must,

C4,

OOC Coo H

GA,

Vil

therefore, two groups viz. a C,H,- and a COOH-group have been
displaced from one side of the ring to the other in the formation
of B cocaic acid from a truxillic acid. This can take place in two
ways, in which in one case formula II arises and in the other case
all C,H,- and COOH-groups will lie on one side of the ring.

Formula IL is built up of 2 molecules of normal cinnamic acid,
and the other formula is formed from 2 molecules of allo cinnamic
acid. The 8 cocaic acid was found on illumination of the stable
barium salt of normal cinnamic acid together with > truxillie acid.

0,593 gr. of cinnamic acid, as barium salt, yielded 0,168 gr. of
B truxillie acid and 0,092 gr. of 8 cocaic acid after 27 hours’
illumination. ;

It appears therefore from this, that formula II must be assigned
to B cocaic acid. Besides it is very improbable that a substance for
which the heavy groups are all found on one side of the ring would
arise by melting with KOH.

After what has been said about the conversions of the 6 truxillic
acids, the following rules may be given:

596

1. The truxillic acids for which 3 large atom groups are situated
on one side of the ring are converted through heating with KOH
or HCI to truxillie acids with 2 heavy atom groups on one side.

2. By melting with KOH those forms arise for which the heavy
atom groups are situated alternately on one side or on the other
of the ring. ;

3. The truxiliic acids for which 2 heavy atom groups are placed
on one side of the ring are not changed into another truxillie acid
through heating with hydrochloric acid.

In the foregoing exposition it has been assumed that the truxillic
acids possess a tetramethylene ring. The proof for this has not yet
been furnished; their properties found up to now can very well be
reconciled with this conception.

I hope shortly to be able to communicate the results of an in-
vestigation in this direction.

Geology. — “Old Andesites”’ and ‘‘Brecciated Miocene’ to the east
of Buitenzorg (Java). By Dr. L. Rurren. (Communicated by
Prof. Dr. C. E. A. WicHMANN).

(Communicated in the meeting of September 29, 1917).

In their “Geology of Java and Madoera” *) VERBEEK and FENNEMA
deal with the development of the Neogene formations about as follows:

During the oldest period of the Miocene numerous and very
extensive eruptions took place all over Java; the eruptive rocks
were for the greater part andesites. Where they emerge they are
in many cases difficult to distinguish from the andesites of the
present volcanoes; in other cases, however they differ from the
recent andesites in being weathered to a greater depth.

In a subsequent period the old voleanoes were denuded again
considerably, their detritus forming round the volcanic nuclei a
system of stratified breccia, conglomerates and andesitic sandstones
(m 1), which may locally be found to be interstratified with clay and
sometimes with limestone or marl.

This period was succeeded by one in which fine-grained layers,
especially marls, were deposited upon the old miocene breccia and
conglomerates (m 2).

Subsequently in the most recent part of the Miopliocene especially
limestones were formed (m 3).

Now after the Tertiary had been folded, the present volcanoes
arose in Java and spread their discharge over vast areas of the
tertiary strata, which had meanwhile been considerably denuded.

In the period m 1 and m 2 only few voleanic eruptions occurred ;
in the period m 3 there was no volcanic activity at all.

It is obvious, that the authors, who. were thrown upon their
own resources when called upon to construct, in comparatively few
years, a geological map of an island, four times larger than Holland
and difficult to traverse over long distances, had to make a number
of working hypotheses on the stratigraphy, the tectonic and the
geological history of the island, if they would make anything of
their injunction. It will be well, therefore, to look upon their maps
and profiles of Java — in which their views of the neogene, as

1) R. D. M. VerBeEK and R. FENNEMA, Geology of Java and Madoera. 1896.

598

described above, have been laid down graphically — first and
foremost as the embodiment of their working hypotheses, which in
some details are reliable, but whose general trustworthiness must be
called in question.

The maps and profiles cannot be unreservedly accepted. Only
more prolonged and more minute investigations than V. and F. were
able to perform, will have to demonstrate for every part of the
island whether the writers are right in their generalising assertions
about the geology of Java, or whether their mapping has to undergo
a thorough revision.

On the geological map of Java old andesites, brecciated and marly
miocene and recent eruptive rocks have been indicated to the east
of Buitenzorg '). A number of excursions made in this district enable
me to form an estimate of this mapping.

Up to about 4} km. before the Poentjakpass the great highroad
from Tjiandjoer to Buitenzorg, runs over the young, volcanic mantle
of the Gedeh-Pangrango-massif. According to the geological map it
then crosses a mountain ground of old andesites and basalts, which
in the South leans against the Northern declivities of the Pangrango,
and extends towards the North, broadening rapidly, as far as nearly
15 km. north of the Poentjakpass. The southern part of these old
eruptive rocks (G. Gedogan, Djoglok, Soemboel and Gegerbentang)
is, according to the geological map, as it were, pinched off from
the more northern tracts by the young voleanic massif of the G.
Limo. In the landscape these “old andesites” do not seem to be
distinguishable from the “young volcanoes”. Nor do the large num-
ber of andesite blocks on the Batavia-side of the Poentjak, visible
from the tea-gardens, look less fresh than those found on the slopes
of the Gedeh. It appears, then, that after traversing the district once
no evidence whatever can be adduced, to show the existence of two
volcanic massifs of entirely different age.

The other arguments brought forward by VERBEEK and FeNNEMA
in their “Geology of Java and Madoera” to support their bypothesis
that in this region the old miocene eruptive rocks are detached
from the young volcanoes, are not quite satisfactory. In the French
edition (p. 506) we read: “Autant qu’on ait pu juger par les afleure-
ments insuffisants, les sédiments tertiaires semblent reposer au Nord
et a [Ouest sur l’andésite et contre celle-ci”. It seems, therefore,
that, as regards the normal superposition of the tertiary sediments
on the old andesites — which would indeed prove the age of the

1) Sheets All, Alll, BH, BILL.

599

latter — the writers themselves are not quite satisfied. Further on
they assert that several eruptions must have taken place in the
great old-andesite-massif, and try to substantiate this view by pointing
to the facts that, petrographically, the rocks differ rather much,
and that remains of different crater-rims are still extant. One

of the peaks, however, of these “old” crater-rims — the Goenoeng
Limo — is according to the writers a young volcano, while with
respect to a second peak — the G. Kentiana — they also suggest

the possibility of a younger date, “mais les autres points d’éruptions
appartiendront sans doute a l’ancien massif andésitique’. (p. 506).

No conclusive arguments for the high antiquity of the “old”
andesites, east of Buitenzorg, are to be found in the text; on the
other hand indications are found in the work to show that the
writers themselves were wavering in their opinion, and there is,
moreover, one important fact that seems to point to a later date.
Indeed VerBEEK and Fennema also allude to traces of rather distinct
crater-rims inside the massif. If, in addition, we consider that, as
they suppose, the “old voleanoes” had to furnish the material
for all the strata of the miocene breccia and conglomerates to a
thickness of sometimes thousands of meters, and if we also keep in
view that, after the miocene, the Tertiary of Java, has undergone
at least one intense plication, we can hardly realise that volcanoes
should have outlived a very lengthy period of denudation and an
intense plication so long as to contain even now distinguishable
crater-rims.

These considerations, which force themselves upon the geologist
who travels from Tjiandjoer to Buitenzorg, induced me to make
excursions with a view to farther inquiries into the relative age of
the Tertiary and the eruptive rocks round about Buitenzorg. From
Sept. 1914 to May 1917 I spent 21 days in fieldwork.

It stands to reason that a minute study of the Sedimentary Tertiary
round Buitenzorg can a priori be expected to yield sufficient evidence
to decide whether the “Old Andesites” are really older than the
sedimentary Tertiary and, consequently much older than the volcanic
mantles of Gedeh and Salak, or whether all eruptive rocks about
Buitenzorg are coeval. If the “Old Andesites”’ belong to pre- or old
miocene, the miocene strata round these massives must be very
coarse-grained; they must include many fragmentary materials from
the old voleanoes near them, they must be ‘“brecciated’’, as indeed
also V. and F. have suggested. If, however, the oldest part of the
sedimentary Tertiary in the neighbourhood of the “Old Andesites”’
appears to be fine-grained and free from or poor in volcanic discharge,

600

we can only assume that the andesites were not formed before
the deposit of the Tertiary. Again, if the large masses of the old
eruptive rocks existed before the folding of the Tertiary, we can
expect the tectonic of the Tertiary in the neighbourhood of these
massifs to adapt itself in some measure to the large eruptive masses.
If, however, the folding of the Tertiary is older than the “Old :
Andesites”’, the general tectonic appearance if the Tertiary continues
undisturbed up to the eruptive masses. Besides, if the sedimentary
Tertiary is younger than the “Old” Andesites we may look for the
“Qld Andesites’” normally disposed, underlying the sedimentary
Tertiary in its deep-seated plications. If the Andesites are subsequent
in age, we can hardly expect to find their points of emergence in
a country so difficult to traverse, but in their neighbourhood the
sedimentary Tertiary might show traces of contact-metamorphism,
whereas enclosures of sedimentary Tertiary might be found in the
Andesites. Facies and tectonic of the sedimentary Tertiary may
therefore yield valuable material. We can a priori expect little from
a direct comparison of the “Old Andesites” with the recent eruptive
rocks in their vicinity. V and F. have already alluded to it that
the “Old Andesites” of Java differ from the recent Andesites only
in being more intensely weathered. If, however, we reflect that the
weathering of some recent volcanoes has advanced so far (as instanced _
by the slopes of Salak and Gedeh) that from the volcanic tuffs

colloid matter has been derived, still including recognisable andesite
fragments, of a waxy softness, we can realise, that generally the
“Old Andesites” can hardly be weathered to a still greater depth.

Now let us discuss the results of our local inquiry in the direction
alluded to:

When starting from the country-seat Tjiloewar on the road from
Buitenzorg te Batavia fowards the South-east,') one first moves
along brownish-red grounds in which occasionally andesitic blocks
are revealed at the surface; they are the typical weathering products
of the recent Salak-Gedeh tuff breccia and agglomerates. Prior to
reaching the river Tji Keas one descries on some hills yellow
grounds, which at the Tji Keas prove to be the weathering product
of a bluish-grey, bulbous shaly hard clay (shale), containing in some
places little Foraminifera (Rotalidae, Globigerinidae). On the Tji
Keas a Globigerina-containing marl-bank occurs in this badly stratified
clay (shale), so that the strike and the slope can be measured
(N 60°.0, 10°). In some. places the clay — generally fine-grained —

) See the accompanying map and profiles.

601

is slightly sandy: the residue appears to contain many pyrite-
granules and quartz-splinters, while plagioclase, amphibole, pyroxene
and andesitic ground mass are decidedly wanting. Between Tji-
Keas and Tji Teureup we find again at the detached hillock Pr.
Bondol perfectly analogous clays sloping towards the North at an
angle of 14°. A road runs from the Pr. Bondol towards the south,
crossing the ridge of the Pr. Maoeng (See map). When I visited
this district, the road was under repair, it was even partly torn up,
which had required a great deal of digging, so that the disclosures
were very interesting. Up to a short distance from the ridge of the
Pr. Maoeng the presence could be ascertained of the hard, greyish
blue, bulbous shaly clay, getting plastic in a moist condition,
yellowish-white when weathering, and constantly declining towards .
the North. The strike is invariably about N. 70°0, the slope increases
from N. to S. from about 15° to beyond 55°. At first we are
surprised to see in all brooklets and on the hills immense fragments
of andesite, but on closer investigation we discover that they have
nothing to do with the rocks of the interior of the hills, as they
are the relics of a recent tuff-breccia formation. In one regress of
the road we could beautifully observe how a red volcanic area,
interspersed with andesitic fragments, overlay discordantly vertically
erected, denuded tertiary clay that had been weathered white.

A geological survey is largely impeded by the circumstance that
all over the environs of Buitenzorg suchlike young volcanic agglo-
merates overlay the Tertiary, because in many cases the solid stone
is found only in deep indentations of the river. Perhaps this is why
V and F have indicated so many “miocene breccia” in this
district.

At the watershed of the Pr. Maoeng the regress of the road is
10 m. deep; here also are we confronted with the typical blue
clay, now, however, declining 28° towards the South. Here we are
in the heart of an anticline, in whose northern arm we have
encountered a very uniform clay-formation of 1500—2000 m.
thickness. The clay in the kernel of the anticline also appeared to
be next to quartzless and completely devoid of andesitic matter.

On the Southern slope of the Pr. Maoeng again the Tertiary clay
may be noted repeatedly, now sloping down southward. Whereas,
however, in the northern arm of the anticline the strike was rather
constantly N. 70°0. it becomes in the Southern arm N. 70° W. to
N. 50° W.; it seems, then, as if the anticline again dips towards
the South. South of the confluence of the Tji Djajanti and the
Tji Keas disclosures of Tertiary rocks are over some distance

602

wanting in the latter river. In the upper course of the Tji Djajanti,
however, the bluish-grey concretionary shales occur again, which
are also noticeable on the South slope of the Pr. Karet at an angle
of 55° towards the South. Also on the Western slope of the Pr.
Karet we encounter clay-formation, sometimes including Globigerinidae.
At one place the clay inèluded a sandy thin layer, very much
weathered and about 5 cm thick (N. 69° W. 24° 5). After washing
a small specimen, the sandy intermediate layer appeared to contain

)

A. Andesite (solid
rock)

Cc : 5
T. Andesite (tuff
breccia) ss
G. Andesitic Con- fr S Kelapa Noenggat

glomerates S
Z. Sandstone
K. Limestone
t. Shale andi
¥marl

G. M. Globi-
gering marl
Q Hot
spring
z Field of
Andesitic
blocks

Tt Leungst

G Karang
t30k kk

603

many fresh plagioclase-splinters and magnificent idiomorphic biotite
laminae. |

Where the valley of the Tji Keas breaks Southward into the
uplands about 13 km. to the south of the embouchure of the Tji
Djajanti, the disclosures reappear; they are the familiar, somewhat
marly hard clays, which, however, here include bands of marl,
sandstone, and conglomerates, which incline 30°—45° towards the
South. The strike is N 65° W. On this level, then, which lies
stratigraphically at least 1700 m. above the core layers of the Pr.
Maoeng anticline, we find for the first time numerous coarse
clastic bands intercalated into the tertiary formation. The marl is
not fossiliferous, the sandstone and the conglomerates are calcareous,
and all stones are very rich in andesitie constituents. In a small
affluent of the Tji Keas, running Southward, it is admirably revealed
that in still more elevated levels, true andesitic tuff-breccia succeed,
which, in their turn, alternate with sandstones and clays. In the
most Southern affluent of the Tji Keas, the Tji Bedoeg, we again
encounter hard shales, including besides bands of andesitic sandstones,
conglomerates and breccia, also a layer of Globigerina marl and
a thin bank of coralligenic limestone. In the latter Lithothamnia
and Amphistegina occur. The direction is N. 55° W., the slope
30—-31° S. The southmost disclosure of the Tertiary I found in a
rivulet north of Gadok, where the andesitic sands and tuffs dip
20° Southward.

In the profile described above we have encountered a tertiary

Profile I.

S Andesite and allied rocks.
EE Coarse clastic rocks with
andesitic material.

EA Limestone.

EA Shale and mari.

formation whose lowermost portion consists of homoplastic clays
that bear no or hardly any volcanic matter. Their marine origin

604

is proved by the sporadic Foraminifera. Higher up the clays are
gradually pushed back by coarse-clastic layers, built up of volcanic
material, while the most recent layers of the plicated Tertiary
consist exclusively of voleanic sands and tuff-breccia, dipping beneath
the Gedsch-massive at a slight angle.

/

The clay-formations can be traced towards the East up to the
spot where VerBEEK has roughly indicated the western limit of
the “Old Andesites” — i. e. along the upper course of the
Tji-Teureup. Between the rivers Tji-Djanjanti, Tji Teureup, and
Tji Keas, the bulbous shale, free from volcanic material, can
again be seen wherever the solid rocks emerge. Between the upper-
course of Tjikeas and Tji-Teureup, over a zone of little breadth but
of 1 km. length (direction about N. 20° W.) many huge ‘wool-
packs” (some of them 100 m® of andesite are lying in-or on the
clay-formation, forming an immense field of blocks. It cannot be
made out, whether these blocks — like the numerous smaller ande-
site fragments occurring all over the area (cf. page 601) — are the
remains of a discordant young covering of tuff-breccia, or whether:
they are the line of outcrop of an andesite dyke breaking through
the clay-formation.

In the upper-course of the Tji-Teureup we find the familiar clay-
rocks, here including Globigerina and moreover slightly silici-
fied, so that it can be ground. It slopes down towards the South-
west amidst disclosures of andesite-rocks which, downstream, pass
into a granular-crystalline, diorite-like rock. It is interesting to note
that the hollows of the Globigerina are often silicified. A most
peculiar brecciated limestone, including Lepidocyclina and Amphi-
stegina, occurs in concordant arrangement with this clayrock. It is
made up of irregular limestone fragments, some of them highly
crystalline, cleft by thin intermediate layers of the clayrocks just
described. It seems as if the limestone has been broken to pieces
during the plication of the mountain ground and particles of the
plastic clay have been intercalated by pressure. So we see that close
to the contact with the old andesites, nay, surrounded by them on
nearly all sides, we find a tertiary formation, absolutely devoid of
voleanic material.

The claystones occurring here, differ from the shales, found more
to the West, in being more consolidated on account of only a slight
silicification.

When going down the Tji Teureup, encountering all the way
normal, imperfectly stratified shales, we approach the magnificent,

>

605 7

cone-shaped Goenoeng Pantjar, 870 m. in height, recognized from
afar as a typical volcano, and, indeed, indicated on V and F’s map
as “Old Andesite”. This is highly surprising as, with the exception
of a few huge Andesite blocks — perhaps the outcrop of adyke —
we can only descry, either round or on the mountain, bluish white,
more a less silicified shales, again without any trace of volcanic
material. Moreover we find on the western base another calcareous
sandstone, built up mainly of quartz-splinters. In this rock voleanic
material is also lacking. The pebbly shale is so hard in some places,
that the natives use it as flint. In Profile Il we assumed, that the
nucleus of the coniform G. Pantjar is constituted by an andesite
mass, occasionally also a granular crystalline rock (see notes of
interrogation) and that the silicified shale, a silicification perhaps due
to contact-metamorphism, covers the voleanic nucleus, as a mantle.
Two geysers on the northern slope of the G. Pantjar lend support
to this view.

In the Tji Teureup as well as in a rivulet east of the G. Pan-
tjar loose fragments of Andesite were found, which unmistakably
include tertiary shale. Though it seems reasonable to look for the
origin of these andesite blocks in the southern massif of the ‘Old
Andesites”’, still it may be possible that they arise farther away in
the Gedeh massif. On this account their discovery is not conclusive
for the relatively recent date of the “Old Andesite’.

In the profile across the G. Pantjar (N°. 11) the axis of the
Pr. Maoeng anticline is found rather to the North of this mountain,
as shales sloping southward were still noted to the N.E. of the
G. Pantjar. In the northern arm of the anticline lies in this profile
the G. Hambalang which, like the more eastern G. Karang, looks
from afar — e.g. from the road Batavia-Buitenzorg — like a plane
distinctly sloping: northward. On the G. Hambalang the disclosures
of the Tertiary are few and far between, as it is largely overlaid
by volcanic material of a recent date. Due east from the G. Ham-
balang the Tji Leungsi exhibits an almost uninterrupted disclosure
of the Tertiary. On its left bank, at one place in the youngest parts
of the clay-formation, a Cycloclypei-coral limestone has been inter-
calated, which towards the east rapidly increases in thickness, emerges
there through the steep south slope of the G. Karang as a white
outcrop, and spreads horizontally to the North of G. Karang all over
the area as far as Kalapa Noengal (Profil III). These lime-stones
contain besides Cycloclypeus also Lepidocyclina and Amphistegina.')

1) At one place in the youngest part of the shale-formation the Cycloclypeus
lime stones are covered by peculiar felspathic sands.
44
Proceedings Royal Acad. Amsterdam. Vol. XX.

606

Also to the south of the G. Karang, does the Tji Leungsi, which
flows here through a narrow cleft, present an almost continuous
disclosure of Tertiary. Again we observe every where the bluish-grey,
occasionally marly shales, containing locally many calcite dykes and
septarian nodules, composing the whole formation ; volcanic material
is absolutely wanting down to the core of the anticline, so beauti-
fully disclosed here, the eastern elongation of the Pr. Maceng anti-
cline. It is true though, that discordant, thick river-sediments of
volcanic material are here seen to overlie the denuded shales. This
material comprises some huge blocks, most likely remains of a young
tuff-breccia, which formerly overlay the whole area of shales. In the
profile of the Tji Leungsi the core of the anticline described, is
arched nearly horizontally ; farther on the South arm. however,
many disturbances are to be seen, while still farther, quite near
the shales — some kilometers west of the Tji Leungsi — the solid
andesite rock can be detected. Generally the shales dip in the direction
of the andesite.

A last series of researches was made east of the Tji Leungsi —
in the basins of the Tji Djanggal and Tji Pamingkis. In the Tji
Pamingkis we find below the embouchure of the Tji Handjawar
only the typical clay-formations of a peculiar position ; the strata
are sloping mainly to the east. Here the vast anticline, which we
saw emerging from the West, east of Buitenzorg, may be supposed
to dip again towards the east.

In the upper course of the Tji Djanggel shale was found sloping
towards the south at only about 10 m. from the steep blocks of
horpblendie andesite and its breccia. This again bears witness to the
fact that the shale dips in the direction of the andesite, but it is
still a matter of conjecture whether the andesite overlies the shale
or whether it is separated from it by shifting. Perfectly analogous
phenomena were noted between the Tji Danggel and the Tji Handjawar
in a small river, the Tji Soerian. Here we see moreover distinctly
in some places that the hornblendic andesite breccia rest on shale.
It may also be, though, that such brecciated masses have glided
down from the neighbouring mountains and are now, as a secondary
formation, superposed on the clay-formation, so that also this obser- ~
vation does not afford conclusive evidence for the relatively recent
date of the “Old andesites”, although it lends plausibility to the view.

Finally a very interesting observation was made in the Tji
Handjawar, west of the steep, coneshaped Goenoeng Handjawong,
970 m, in height. On its left bank, at the base, we find consolidated
andesite rock, superposed on this a fine-grained, greyish-green breccia

607

of little thickness, this again being surmounted by very thick, coarse
blocks of breccia. All the breccia are sloping towards the West and
the whole arrangement seems to be the normal one, resulting from
their being deposited as effusiva of the G. Handjawong. In the fine-
grained breccia — an integral and transported component of the
“Old andesites” — numerous, small enclosures of tertiary shale
were found.

In the foregoing we saw then that by far the greater part of the
sedimentary Tertiary, east of Buitenzorg, is built up of a very fine-
‘grained clay-formation, in which volcanic material is all but wanting
or completely so. Higher up, however, course clastic banks of vol-
canic material found their way to the surface. It appears, then, that
the state of matters is just the reverse of VERBEEK and FENNEMA’s
hypothesis. They supposed the deep-seated portions of the Tertiary
to be “brecciated’’, the upper portions to be “marly”. This paralyses
their main argument for the high antiquity of the ‘old andesites”.

Let us now summarise our arguments against a high antiquity
of these rocks :

1. In no place in the deepest plications of the Tertiary did we
encounter disclosures of andesite.

2. The fact that in many places, close to the andesites, a sedi-
mentary formation, free from voleanic material, was found, goes
very much against a bigher antiquity of andesites than of sediments.
For this contact the only explanation could be furnished by an ex-

tremely complicated system of shifts -— of which there are no
indications — if the andesites were older than the sedimentary
Tertiary.

3. The fact that in several places, near the contact, the tertiary
clays dip in tbe direction of the andesite goes very much against
a higher antiquity of andesite than of shales.

4. The silicification of shales at the G. PANrJar can be explained
satisfactorily only when assuming that the andesites are younger
than the shales. |

5. The occurrence of clay-enclosures in loose andesite fragments
west and east of the G. Panrsar also points to a recent date of
andesite.

6. The fact that between Tji Djanggel and Tji Handjawar
andesite breccia are in some places superposed on shales is best
explained by assuming that the andesites are younger than the
sediments.

7. Lastly the discovery of enclosures of shale in breccia of the.
“Old Andesites” at the G. Handjawong proves unequivocally that

44%

608

the “Old Andesites” originated only after the deposit of the Tertiary
clay-formation. ;

CONCLUSION.

The old andesites east of Buitenzorg are younger than the en-
circling sedimentary Tertiary. The bulk of the sedimentary Tertiary
east of Buitenzorg is developed as a clay-formation: only in the
recent levels breccia and allied rocks occur. Now it is still a moot
point whether the “Old Andesites” are the places of eruption, from
which the conglomerates and the breccia of the young Tertiary, in-
cluded in the folding, were extruded, and consequently are a little
older than Gedeh-Salak, or whether they are contemporaneous with
these volcanoes. That they are less high and that their volcanic
shape is less perfect, would seem to favour the first view. This |
question, however, can be positively solved only through minute
researches in this volcanic region, so difficult to traverse, and through
petrographic investigations in connection with them.

In the above we have pointed out, for one district, that the
geological map of Java, requires a thorough revisal. It would
not be difficult, even by dint of comparatively little fieldwork,
to enlarge the number of instances. We do not mean to censure
the makers of the “Geological map”. Those who hold that respect
is due to the men who performed a comprehensive task with only
little assistance, may still feel called upon to point to the short-
comings resulting from a superficial examination. The only fault of
the authors is, that in publishing an illusive map, indicating in
every place “continued” formation outlines, and accompanied by a
great many profiles, they have made us believe that we really
possess a rather detailed geological survey of this island. This is
why after the investigations by V. and F. the, necessity of further
geological work in Java has been given hardly a moment’s
consideration.

May this paper tend to produce the conviction that a minute
revision of the geological survey of Java, is of prime importance for
geological science. In practice a similar wish has frequently been
expressed, but in vain.

Buitenzorg, 1 Jan. 1917.

Mathematics. — ‘Some Considerations on Complete Transmutation.”
(Fourth Communication). By Dr. H. B. A. BocKwINKEL. (Com-
municated by Prof. L. EB. J. BROUWER).

(Communicated in the meeting of December 21, 1916.)

18. In this paper we shall discuss the general theorem of Taylor
for the functional calculus, a particular case of which, that we, as
such, called Mac LauriN’s theorem, was discussed in the preceding
paper. Before proceeding to this, we shall, however, treat an impor-
tant proposition of the normal additive transmutation of which we
shall avail ourselves in the future.

The proposition in question is a special case of another that holds
in general for a continuous transmutation and which, generally
expressed, states, that such a transmutation keeps its additive property
with regard to an infinite sum. The proposition is as follows

Lf a series the terms of which are functions that form part of
the F.E. of a continuous additive transmutation T, converges
uniformly in the N.F.F. towards a function u, which also
forms part of the FF ., the series the terms of which are equal to
the transmuted of the first mentioned terms, converges uniformly
in the N.F.O. towards Tw).

For if we have

uu, HU Fe HF Ume
such that the series converges uniformly in the N.F.F. of 7, this
“means: corresponding to any given, arbitrarily small amount d there
is an integer N such that in the whole N.F.F. of 7

Heet for vn

Further, on account of the continuity of 7, there corresponds
to any arbitrarily small amount t an amount o such that, if v be
a for the rest arbitrary function of the F.F.

ik OS Bit in the: NEO or 57;
bie rt 0 in the N.F.F. of 7.

Thus there corresponds to any arbitrarily small amount + an

integer N, such that we have in the N.F.O. of 7,

610

x
(> in) | Et AN RAD ANN
| |
Now we have, on account of the additive property of 7, in the
numerical field of operation

@
Tu=tfu, + Tu, +... Tura + “(ys ms
n
from which it follows in connection with the result just arrived at,
that we also have in that field ,

oo
TU m i ba Uns, «

0
the series converging uniformly there. Thus the proposition has
been proved.

If 7 is a normal transmutation both the N.F.F. and the N.F.O.
are a circle with centre 2,. The radii of these two we call respect-
ively (0) and (a). The characteristic expansion of u is in that case
the one in a power series

U Gho ern a Cea eG WE Ee

Then, if we call 8, the function in which (v—z}j” is transformed
by 7 (the latter exists and is regular in («) since the rational integral
functions form part of the F.F. of a normal transmutation), we
have in («)

li Oe ee (oN A Sage

and the series converges there uniformly. This is the proposition
we referred to and which we may express as follows: |

If T bea normal additive transmutation, of which the circle
(a) forms the N.F.O., and the circle (6) the N.F.F. the transmuted Tu
of a function u of the FF. may be found by applying the trans-
mutation term by term to the power series, in which u may be deve-
loped in the domain (6), and the result is a series that converges
uniformly within (a).

19. We shall now discuss the generalisation of Tarror’s theorem
for the functional calculus.

This is concerned with the development of the transmutation 7,
applied to a product w—=vu, according to powers of Du, if the
transmuted 7 of v and an enumerable infinite number of other
quantities, which may be called the derivatives of the transmutation

611

T applied to v (the name has been introduced by Pincurrre) are
known; a development of Tw in ‘the point w=v, or in the
vicinity of w=v, as we might say, while the special development,
according to powers of Dw, might be called one in the point w=1.
(Cf. the general development of a function f(z) in the point «—a
and the particular one in the point «= 0).

We again make the following suppositions:

1. Tis normal such that the N.F. is a circle («) with centre z,,
while the functions of the F.F. belong to a circle (0), concentric
with (a).

2. The series P belonging to 7’ is complete in (a) with. corre-
sponding domain (9).

It will further be easy in the future to suppose that as a pair
of associated fields of 7’ each such pair of fields of P comes into
consideration, which involves that for (6) may be taken any circle
greater than (5) (see N°. 12, 2rd communication). Possibly (@) itself
cannot be taken as N.F.F. together with (a) as N.F.O. (N°. 13); but
in any case we have now according to the functional theorem of
Mac-Lavrin, Ju Pu, for all functions belonging to (8), because,
if w belongs to (8), it also belongs to a somewhat larger circle.
According to the final observation in N°. 15 (8"' communication),
if 7 were not yet defined for all these functions, we might extend
the F.F. of 7 over them by simply writing for these functions
Tu = Pu. And if 7 should be defined a priori for a certain part
of the functions belonging to (8), such that we had for them
Tu=— Pu, T would not be continuous in the total field, and in
that case we should not retain thes 7’ in the considerations we are
concerned with here, but put another operation in its place, which
for the functions mentioned is identical with P.

If therefore w be a function belonging to (8), we shall have in
the domain (a)

le}
<< Ay Ww”)
M= Sn n :
a n!

Now we have, since w—vu, according to the formula of Lerpniz
win) = yur) + n‚v'ulr-l) +... + vu
consequently

les}
LI
Tw Nn — (vur) + n‚vuri) 4+... + vl)u),
oon ni

In each member of this last series, from and after the one
for which „==m, occurs a term with ve); taking these terms

612

together, we can, at least formally, represent the transmuted 7'w
as follows

um) pk)

Tw= Su Nb antr: - 5 . . Fy (35)
0 0

We shall now assume that w and v both belong to (8), and then
prove that the doubly infinite series in the right-hand member con-
verges absolutely; then, as is known, the validity of the change in
the grouping of the terms, has been proved, and consequently the
correctness of the last formula.

If w and v belong to (9), they belong also to a somewhat greater
circle (9) with a radius ge =8 + d, (d> 0). If further the maximum
modulus of uw as well as of v on the circumference of (og) be smaller
than VW, we have in the domain (a)

| ol) | oM u(n) oM
AI (Bap * mf “(Bard

(36)

Since, further, the transmutation is complete in («) with correspond-
ing domain (3), there corresponds to any given arbitrarily small
number «, an integer 2. valid’) for the whole domain (@), such that

| an (Bate, for nS ae be D> Ree

Consequently we have for £2 n., corresponding to any integer m
(not negative) and in all points of (a)

| vik) oM (B —a+e) B—a+e-k
| Om+k ! | et N et 12'S
| kl | B—a+o B—a+to
If we now assume s chosen smaller than d, the left-hand member
of this inequality is in the domain (@) comparable with the general
term of a decreasing geometrical series of constant positive terms.
The series

(38)

0,
vk)

in ae Ank Ty Se a a
. !
converges therefore absolutely (and uniformly) in the domain (a).
Since the first series in (85) arises from it by multiplying the terms
by the factor uw) :m/, which is independent of k and limited in
(a), the latter converges as well absolutely in (a) for all integral
(not negative) values of m.

There still remains to be proved the convergence of the series

1) See the so-called uniformity supposition in the 2nd paragraph of NO. 4, Ist
communication.

613

ce (m)
ces (40)
J m!
0
in which
bm = YE | dupe or (39a)
0

The inequality (38) holds also, according to (37) and the first part of
(36), for all integral (not negative) &, and in all points of (@), if
man: We therefore have in («)

oM Ae
bm Eh aan (8—a- e)” ON MES Wes EMI ea 1e (41)

from which it follows in connection with the second part of (36)
that, from and after some fixed term, the terms of the series (40)
are in the whole domain (a) comparable with those of a decreasing
geometrical series of positive terms not depending on z. The. series
(40) therefore converges uniformly in («), and thus the absolute and
uniform convergence of the system of two successive series in (35),
and consequently the validity of this formula has been proved.
According to (89) we can write that formula as follows:

0

EA Reds yr)
Tw = T (vu) = N m Am raga et (42)
ence m!
0

This form is the development in question of 7 (vw) in a “series
of Taytor’. In order to accentuate the analogy we introduce the
following symbolism. The development

Go

~ ap vik)
ie % "i E

0

suggests to represent the transmutation 7’ by the symbol

oo bea:
DS Viz
wm li
0

If this symbol is differentiated with respect to D as if it were a
power series in that letter, and if we represent the arising symbol
by 7', we have ;

Ba ar DE
he
0

614

Going on in this way we find, by indicating the result of m-fold
repeated differentiation by 7”

Do

Te —~ Am k Dk
TH) = Yi ase REN ce
0
According to (39) we can consequently write
dn TOORN oy a fae ne de NN

and tor formula (42)
Tm)
(OT ie Nn LTM) Di enn en
sv!
0

Since the symbol indicating the operation 7 bas arisen by m-fold
repeated differentiation from the symbol for 7, we may call the
operation 7” the omt! derivative of 7’; PincnerLe introduces this name,
though in another way. We further observe that the transmutation
D answers in the functional calculus to the function y= in the
theory of functions, because the derivative of D is equal to 1, that
of Dm to m D»—1,. and the (m-+ 1)" derivative of Dm equal to
zero. In consequence of this we may call the transmutation indicated
by a rational integral function of the symbol D a rational integral
operation. Infinite power series in D represent then what we may
call a transcendental operation.

The formula (42’) is now clearly recognizable as a development
in the series of Taytor, that is a development of Tw in the point v
according to powers of the simplest operation D, calculated for the
increase u; we have to take the term increase with u in the meaning
of geometrical increase, that is multiplication by u. We may also
speak then of the development of Tw in a geometrical vicinity of
functional point w= v. For v= 1 the series passes again into the
particular one found before; it appears in that case that the coeffi-
cient a, is equal to 7) (1), that is the mt" derivative of T applied
to the functional origin w(e)=1. For this particular value of
we) the derivative in question may therefore also be found by
means of the formula (24); for w= v, as is to be found in PINCHERLE’S
paper, and as we are going to verify presently, the following
generalisation of (24) holds

dn == Tv) = T(e™v)—m, aT (av) +... (lar (ev) (45)

From the proof we have given of the general development it
appears that it holds under the same conditions as the particular,
provided that the “beginning point” v(r) and “‘increase” u(x) belong

615

both to the F.F. of the series P corresponding to 7. Hence the
general theorem of Tarror for the functional calculus has been
proved, in the following form:

If the series P answering toa normal BAG ve transmutation
T be complete in the circular domain (a) that forms the N.F.O.
of T, Tw) may be developed in this domain in the functional series
of Tarror, in the point w=v, if both point of origin v and increase
u (w= vu) belong to the circle (8) corresponding to (a). ')

. 20. We may consider both members of (42) as an operation on
the function u, and thus read this formula as a development of the
transmutation 7, — Tv, applied to the function u in a “series of
Mac-LauriN”. Since the result we arrived at in the preceding number
states that this series (which we shall indicate by P,, since it
answers to the transmutation 7) converges in the domain («) for
any function belonging to (8), provided this is also the case with
the fixed function », the series P, is likewise complete in
the domain (a), such that the corresponding domain is at most
equal to (3). This may also be derived directly from the inequality
(41), which a fortiori holds tor | a’. For it is evident that, instead
of the statement of which that inequality is the expression, we may
as well say: Corresponding to any arbitrarily small chosen number
e there is an integer n, such that

Kl (Base) ODE MS neit 8 le)
and from it follows, if the second part of the proof in N°. 4
1st communication) is consulted, the completeness of the series P, in
(a), with a corresponding domain that at most is equal to (9).

We may therefore interpret the result, arrived at in N°. 18 in
this way: /f the series P answering to a normal additive trans-
mutation IT’ be complete in the circular domain (a) forming the
N.F.O. of T. with corresponding domain (B), we have 1° that the
series P, answering to the transmutation T= Tw), in which v is
a gwen, fired function belonging to the circle (B), is also complete
in («) with a corresponding domain that at most is equal to (B), 2°
we have in the domain (a)

| T, (4) = P, (w)
for such funtions of the F.F. of T, as belong to (8).
We further add the following remark: Since the last equality

1) We have not expressly stated again that w,v,w, form part of. the F.F. of
T, since this is a matter of course now that these functions belong to (8).
according to our agreement in the beginning of this number.

616

|

according to “the theorem of Mac-Lavrin” would c.p. just hold if
T,==Tv were a normal transmutation, the question arises whether
this is the case. This question must be answered in the affirmative.
For, if a function w of the functional field MT) of T belongs to (9),
this is also the case with the product vu, so that 7(vw) and conse-
quently also Zu == 7(vu) exists, according to our agreement in the
beginning of N°. 19, in the domain (a). From this it follows that
there is for the transmutation 7, a pair of fields to be indicated,
the numerical field of which is the circle («), and the functional field
consists of the functions that belong to the circle (9). Since thé
rational integral functions are also among these, 7, satisfies already
the conditions sub 1° and 2°, which we indicated in N°. 15 (Srd
communication) for a normal additive transmutation. But 7’, is also
continuous in the pair of fields mentioned; for, if w in the domain
(3) tends towards zero, this is also the case with the product vw
of w and the fixed function wv belonging to (3). But in that case
Twu) tends to zero in the domain («) according to the property of
continuity of 7, and since 7(vu) is identical with 7, the property
of continuity of 7, has been proved.

If the expression Zvu is considered as the result of the transmuta-
tion 7, — (Tw), applied to the function u, the following establishment
of the expression (45) for the coefficients a’, == 70 (v) of the series
P corresponding to 7, is at once obvious. If the function in which
vm is transformed by 7, be denoted by &',, we have

Em (Toa Drs),
and the application of the symbolic formula (24) according to which
we have
Qn = (§ —2)*;

leads at once to (45). From this the other representation (39) of
a, may be found then, by first deriving from (45) that we have
form == 4

; PVT) Se oY le ae
and further the recurrent relation

Tm) (ov) = Te Deo) — 2 TM) (ce) Ys ne eee

According to the “theorem of Mac Latrin” we have further in

the domain (@)

eo

/ yin) x
To) = Se anr DL RNA NN

0

1) The relations (46) and (47) occur in Pincuerte’s paper as definition of the
respective derived transmutations of 7.

617

it yn—1) ;
T (zv) = za,v + > ae era 1) EE:
n—

The latter sum may be divided into

A y(n) fn—1)
» n Lay + DE (een | zen . . 9 3 (50)

For the first of these two series, with the restriction that the
term with m=O does not occur, is equal to the series (48) multi-
plied by z, and therefore it converges, as well as the latter, in the
domain («) and produces there, together with the term za,v, the
result «7v. The second series is now equally convergent in (a),
since the undivided series in (49) converged there; if further £ + 1
be substituted for in the second series, the result will be

x

vik
T (av) = 2 T(©) + DE ar EH
0
so that, in connection with (46),
hts vk)
T(t) = Yk anys, eT ew

0

This series differs only in so tar from (48) that all a’s have moved
up one place to the left; from this remark and the fact expressed
by (47), that any following derived operation is obtained in the
same way from the preceding one’), the correctness of the repre-
sentation (39) for 7, (v) in the domain («) follows at once.

Let now v(z) have the radius of convergence r, and let r, be the
a-value to which, for the series P, r corresponds as a 3-value. The
series P, answering to the transmutation 7, = (Tv) is then at all
events complete in any domain (e) < (r,), with corresponding domain
(B) < (vr). But the series P, will often be still complete in a domain
greater than (r,), as may be derived from formula (45). For according
to the latter the quantities a’,, are regular in a domain, if this is
the case with the transmuted of xv, for arbitrary integral positive
values of m, and we amply explained in N°. 7 and 8 (second com-
munication) that the transmuted Zw of a function w with radius of
convergence 7 may very well be regular in a circle greater than

1) We have also to take into account that the formula (51) holds if r is replaced
by zv, which is caused by the fact that it holds for an arbitrary function belonging
to (6). ‘

618

the cirele (r,), to which (7) corresponds for the series P belonging to 7.

As a rule 7(av) will further, together with Jv, be regular in a
domain. For let 7 be normal, such that the N.F.O. is a circle (a) >(7,),
and the N.F.F. a circle (6) < (r). If we know that all functions
belonging to (6) form part of the F.F., nothing is to be demon-
strated as in it is included that both v and 4” have a transmuted
in («). But if it is provisionally only given that v forms part of the
F.F. we may explain in the following way that this as a rule
includes that wmv as well forms part of it’). If

ease, fey +e. + ey" + ..- (y = # — 4),

be the power series in which v may be developed in the vicinity
of a= .wz,, we have according to the proposition of N°. 18, in the
domain (a)

Tone, Seher Sir piece GaGa weg
if &, be the transmuted of (a—.,)". If now T is applied term by
term to the development of the series of yv, we arrive at
Co En as En + ...-4- En eye = Pane

From the convergence of the first series that of the second follows
if the quotient of §,41, by &, remains within jinite limits for all
integral » values. This now is often the case, (for instance with S,
and D~-!) and if so, it is clear that the series, emanating when
yrv is substituted for v, converges as well. If, for a moment, we
write the result as 7, (y”"v), this need not be equal to 7’ (y'"v) for other
values than m=O, but in any case a normal transmutation does
exist, namely 7, which, with preservation of the N.F. (a) >(r,),
contains both v and y”v, and consequently «vv in its F.F., and it
is evident that 7 in common cases will be defined in such a way
as to be exactly that transmutation. | |

If now Tv and T (av) be regular in a domain (a) greater than

(r,), this is necessarily the case for d'n = 7\™(v), and there is every
1

reason to expect that a’, = lim ALS will be MZmited in such a
greater circle, consequently that the series P is complete in it. Hence
we arrive definitely at the statement:

The series P, belonging to the transmutation T, (Tv) is not

1) It is not inconsistent with our agreement made in the beginning of N°. 19
that we speak of the case in which 7(xv™"v) is not at the same time regular with
To in domains greater than (rj) It only follows from that agreement that we only
take notice of a 7' that for al/ functions with the radius of convergence 7 produces
a transmuted regular in a domain smaller than (7), because to the latter a domain
smaller than (r) corresponds for the series P.

619

/

only complete in any domain smaller than (r,) but in most cases in
any domain smaller than (1) if r is the radius of convergence of Tv.

If now the radius of convergence of the function w be greater
than 7, the series will converge in a domain greater than (r, We
have in this case been able to find for the transmuted Tw of w,
by developing in the point wv according to powers of D ee
an expression representing this transmuted in a greater numerical
field than would have been the case if we had developed in the
point w=1. Or we may also consider the thing in tis manner '
that, with the preservation of the numerical tield @ > (r,), the func-
tional field that at first consisted of functions belonging to a circle
(3) > (vr), has now been extended to some functions not belonging to (9),
namely such as show the same kind of singularity in the domain
(B) as a fixed function v (wz), their quotient by v(e) being regular in
(3); functions of which we might say that they lie a a geometrical
vicinity of the functional point v(x). Taken thus we observe an
analogy in the present phenomenon with the one called analytical
continuation in the theory of functions (Cf. N°. 8).

21. We shall now elucidate the more general theorem of TAYLOR
by two examples. As transmutation we take in the first place, in
a vicinity of the origin, the substitution S,, in which w is a function
of w, which has «=O as ordinary point, the radius of convergence
for that point being denoted by A. We fully explained in N°. 17,
(3rd communication) that S is a normal transmutation, the numerical
field being a circle with radius « < A, and the functional field con-
sisting of the functions belonging to the circle (6) concentric with
(a), the radius of which o is equal to the greatest modulus of w in
the domain (a). We further saw that to this transmutation belongs
a series P, which is complete in (a), with a corresponding domain
(3), which is determined by the equation

B=a+ | wlan) — Um |,
where 2, is the point on the circumference of (a), where \w
attains its maximum; we pointed out that 8 is at least equal to o.

We can therefore apply the functional theorem of TAYLOR to a
function w==vu, provided we suppose the radius of convergence 7
of the fixed initial function v to be greater than |w (0). The circle
(7,) to which, for the series P, (7) corresponds is determined here
by the condition that in it the maximam modulus of w—w is equal
to r—r,, while the circle of convergence (7) of S,(v) satisfies the
condition that in it the maximum modulus of w is equal to 7. The

|

620

development in the series P,w will in the domain (a), for a function
u belonging to (@), certainly hold, if a<r,, but probably also, if
r,<(a<r’. We shall see how this is and calculate to that purpose
the coefficients a',, by means of (45). We arrive at

d'n = (w—a)" v [w (v)] == (w—a)” S,, (w).

According to (42) the series P,w becomes therefore

a (m)
S (vu) = Sn Sj (v) (w— a)" mi
™m!

0

If in it the factor S,(v) is brought outside the symbol of sum-
mation the remaining series is precisely the series Pu, which
answers to the transmutation JS, (w). Apart from the factor mentioned
the series therefore converges in any domain (@) < (A), provided u
belongs to the domain (@). With the factor S,(v) added to it the
series converges consequently for such a function in any case, if «
is smaller than the radius of convergence 7’ of S,(v). Thus our
presumption has been affirmed.

If for instance w(2)= te, we have a, — (4 x)", consequently
O(a) == +o: and. P= 4 a; so that

ie
But the circle of convergence (1) of Sy, (v) has a radius

r — 2r

so that the series P, is complete here in a greater domain than (7).

22. As another example we take the transmutation D-!, also in a
vicinity of «=O. In N°. 16 we saw that D-! is a normal trans-
mutation such that the numerical field is an arbitrary circle (a),
and the functional one consists of the functions belonging to the
same circle. The series P belonging to D ! was further complete
in («) with corresponding domain (2a). Thus if we keep with respect
to the fixed function v the same notations as before, we have in
this case, ;

Td
while the circle of convergence (7’) of D-!») is the same as the
one of (v), so that

'
== Te

We expect consequently that the series P,, which belongs to the
transmutation D-!(»), will not only be complete in («) fora < $7,
which is necessary according to the theorem, but also if $r <a <1.
The result confirms this. Formula (45) gives

621

Gn [tv (2) deme fet v(t) dt + ...+(—l1)"™am fr (¢) dt
0 0

0
x
= fer v(t) dt,
0

which result, to begin with, is regular within the circle of con-

vergence (7) of D-‘v. In order to find a majorant value of |’ we
reduce as follows
HH

jan, |<} |ti—a|™.[v@®
0

=

T
„lat <erM of | dt |
0-
or
| al | Fai om M (a), |
if a—= |rt) and M(e) be the maximum modulus of v on the circum-
ference of the circle (a).

From this it follows for values of «<r
1

Gs = hin ! rie | mea
m0

consequently —

a’ (aye oe |
according to the general result (41a) in N°. 20, but here extended
to domains greater than (47), provided they are smaller than (7).
Thus, in fact, the series P, is complete in any domain (a) < (r)
the corresponding domain (8) being at most equal to (2e). The
series Pu, determined by (42), converges therefore within the circle
(4s), if the radius of convergence s of u is smaller than 27, and
otherwise within the circle (7).

45
Proceedings Royal Acad. Amsterdam, Vol. XX.

Physiology. — “Concerning Vestibular Eye-refleres” By Mr.
A. pe Kueyn and Mr. W. Storm van Leeuwen. I. “On the
Origin of Caloric Nystagmus.” (Communicated by Prof. H.
ZW AARDEMAKER).

(Communicated in the meeting of June 30, 1917.)

Although it has long been a matter of cominon knowledge that
a nystagmus is induced by allowing cold or warm water to flow
into one ear, and this has been the subject of a considerable amount
of experiments with men as well as with animals, the origin of
this calorie nystagmus is still a moot point.

Among the various views taken of the subject those of BARANY
and BarTEeLs stand prominent.

Barany holds that streams in the endolymph evolved by a change
of temperature excite the coloric nystagmus, whereas BARTELS inclines
to believe that by applying cold water the function of the labyrinth
is eliminated entirely or in part and that by applying warm water
the labyrinth is stimulated.

It being our purpose to endeavour to clarify our ideas concerning
the cause of the caloric nystagmus, we first of all made a series of
experiments to ascertain whether the caloric nystagmus may perhaps
be provoked by arresting the function of the labyrinth through cold
water. The results of this inquiry will be discussed in this paper.

In our investigation we started from the following considerations:

If the caloric nystagmus is brought about by eliminating the
labyrinth on the irrigated side, the caloric nystagmus must present
the phenomena resulting from an extirpation of the labyrinth.

These phenomena are:

a. The calorie nystagmus would correspond in direction, frequency
and nature, exactly with the spontaneous nystagmus occurring after
extirpation of the labyrinth.

6. It would be possible to arrest the nystagmus, occurring directly
after extirpation of a labyrinth by ejecting cold water into the other
labyrinth.

c. When in an animal one labyrinth has been extirpated, a
spontaneous nystagmus will ensue, which will disappear again after
some time; the nystagmus is then what is termed ‘“compensated”’.

623

In this phase of compensation extirpation, or a squirt of cold water
into the other ear may excite nystagmus. If a squirt of cold water
into the ear is equivalent to elimination of the labyrinth on the
same side, then both labyrinths will have been eliminated in case,
in the .compensatory stage, the unimpaired ear is irrigated. It
follows then that the nature, the direction or the frequency of the
forthcoming nystagmus can no more be modified by altering the
animal’s position in space. 5

d. A concomitant flow of water into the two ears, generally brings
forth a condition in which no nystagmus manifests itself. If therefore,
squirting cold water into the labyrinth is identical with elimination,
"a concomitant flow into the two ears cannot provoke a rotatory
movement or a deviation of the eyes in case the animal's position
in space is altered.

These phenomena have been examined in the following way:
Ad. a. In order to ascertain whether the caloric nystagmus (which
term here always indicates the nystagmus occurring after squirting
water, of from 11°—12°, into the animals ear from a height
of 1,5 m.) corresponds completely to the spontaneous nystagmus
occurring after unilateral extirpation of the labyrinth, a number of
experiments were undertaken. In these experiments the animals (in
this series only cats) were tied to an operation board and subse-
quently the calorie nystagmus was examined in 6 different positions
of the animal:

1 ventral position.

2 dorsal position.

3 vertical position, head up.

+ vertical position, head down.

5 lateral position, the irrigated ear up.

6. lateral position, the irrigated ear down.

After this the labyrinth on the side of the irrigated ear was
extirpated. The technique of this operation has been described by
one of us in another paper. ') After the animal recovered from the
narcosis, the nystagmus was again examined in the six positions
just mentioned. With some animals the examination was repeated
the next day. We noted the frequency, the direction and the nature
of the nystagmus. The frequency of the nystagmus cannot be
accurately demonstrated in these experiments, it being apt to change
every moment as well in the spontaneous nystagmus as in the
caloric nystagmus. We are entirely ignorant of the cause of it.

') Pflügers Archiv. Bd. 145 p. 549,

624

Spontaneous eye-movements may come into play here. Sometimes we
fancied to note an increased frequency when the animal got restless.
We feel assured, however, that a change of the animal’s position
can exert an influence upon the nystagmus, so that e.g. the fre-
quency is always less with ventral position than with a dorsal
position, and this holds good for the caloric as well as for the
spontaneous nystagmus showing itself after a unilateral extirpation
of the labyrinth. .
Similarly it is often very difficult to accurately determine the
nature of the nystagmus. It happens again and again that the caloric
nystagmus in a certain position is perfectly horizontal for some time
and suddenly turns rotatory, without any assignable cause. After a
change of the animal’s position we invariably delayed noting down
the nature of the nystagmus for some moments, in order to preclude
the chance that the nystagmus brought forth by the change of the
animal’s position should interfere with the caloric phenomenon. The
terms: nystagmus to the left or to the right always mean a nys-
tagmus with the fast component to the left or to the right. From
this-inquiry it became evident that the direction of the spontaneous
nystagmus is not in all positions identical with that of the calorie
nystagmus, as if appeared that the spontaneous nystagmus occurring
after unilateral extirpation of the labyrinth, has in all positions the
fast component turned towards the unimpaired labyrinth, whereas
the caloric nystagmus is, with ventral position and with vertical
position (head up), turned towards the non-irrigated ear; with dorsal
position and with vertical position (head down) it is turned towards
the irrigated ear; with lateral position (5 and 6) the caloric nys-
tagmus is mostly towards the same side as with ventral position.
Ad 6. Directly after extirpation of a labyrinth a nystagmus reveals
itself spontaneously towards the side of the unimpaired labyrinth.
As appeared sub a, this nystagmus does not vary its direction with
a change of the animal’s position in space. When both labyrinths
are removed during the operation with only a very short interval,
no nystagmus will occur. If, therefore, caloric nystagmus should be
identical with elimination of the labyrinth, it must be possible to
arrest in all positions the nystagmus occurring after unilateral extir-
pation of the labyrinth, by allowing water to flow into the unim-
paired ear. This, in fact, is not the case, as was evidenced by four
experiments upon eats, which all yielded similar results. This may
be instanced by the following abridged protocol:
31-5-1917. Cat. A flow of cold water 11—12° into the right
meatus auditorius from 1,5 m.

625

Animal in ventral position; horizontal nystagmus to the left side.
Animal: vertical, head down, horizontal and rotatory nystagmus
to the right side.
Extirpation of left labyrinth under narcosis of ether ; operation
finished at 11.15.
11.45. Spontaneous nystagmus to the right side in all positions.
11
Ventral position, frequency per 10 seconds i
12

1
Head down, frequency per 10 seconds {15.
1

12.10. A flow into the right meatus.
In ventral position, nystagmus to the right side lessens considerably;
we did not succeed, however, in removing it altogether.
Eud. head down: nystagmus to the right side (horizontal + rota-
tory) grows much stronger and rapid. Now the frequency per
24
_ \23
A0 See" 18" Jan.
23
24
5.15. Spontaneous nystagmus to the right side in all positions.

. . . 1
Ventral position: horizontal nystagmus, frequency per 10 sec. {11.
1

1
Head down: horizontal + rotatory nyst.; frequency per 10 sec. |16.

A flow into right meatus.

Ventral position: eyes perfectly quiet. O.S. distinet downwards
deviation.

Head down: strong horizontal rotatory nystagmus to the right side,

23
frequency per 10 sec. ee
22

Again in ventral position eyes perfectly quiet.

5.30 extirpation of labyrinth on the right with ether-spray.

8.30 not the slightest indication of nystagmus.

It then appeared from this experiment (and the other experiments
were conducted in the same way) that after bilateral extirpation of
the labyrinth there was no nystagmus, after a unilateral extirpation
there was a spontaneous nystagmus which could not in all directions
be arrested by a flow into the meatus of the untouched ear. From

626

this we may safely conclude that the phenomena, associated with
the irrigation, cannot be caused by an elimination of the labyrinth.
The objection that in these cases the unsatisfactory outcome of the
irrigation depended on mere chance, is refuted by the fact that

1. irrigation was very effective before the extirpation of the
labyrinth

2. irrigation after unilateral extirpation of the labyrinth in ventral
position actually arrested the spontaneous nystagmus and

3. with head down the spontaneous nystagmus was not unaltered,
but was considerably stronger.

Ad c. When, in one experiment, both labyrinths are removed,
violent movements are observable, but not nystagmus. If, however,
we first remove one labyrinth, the spontaneous nystagmus will be
seen to vanish after a short interval. When subsequently the second
labyrinth is removed, nystagmus returns and that to the side of the
labyrinth that was first extirpated. We shall not enter into a
discussion here about the question how this nystagmus, first described
by BeCHTEREW '), is originated, and which parts of the central nervous
system come into play here for a nystagmus to manifest itself. This
is being further investigated at our institute.

The time that has to elapse after the first extirpation of a labyrinth
before a nystagmus can reveal itself after the second extirpation,
is varying, but after a few days it is sure to appear.

If directly after the unilateral extirpation of the labyrinth water
is sent through the meatus on the unimpaired side, the spontaneous
nystagmus resulting from the unilateral extirpation of the labyrinth
can be arrested in the ventral position by the irrigation as described
above. A repetition of this procedure at different intervals after the
extirpation of the labyrinth will prove that the spontaneous nystagmus
cannot only be arrested in a ventral position, but also that after
some time a nystagmus toward the opposite side is evolved by a
flow of water ie. towards the side of the first extirpation of the
labyrinth. In this phase we may look for the spontaneous nystagmus
detected by BrcHTEREW, as quoted above, when the second labyrinth
has been removed. If, therefore, irrigation is identical with elimina-
tion of the labyrinth, then the latter compensation-nystagmus
must correspond entirely to the nystagmus obtained when some
days after the unilateral extirpation of the labyrinth, the meatus
of the unimpaired ear is irrigated. It appears, however, that

1) Pflüger's Archiv Bd. 30 p. 312.

627

there is not such marked concordance, as can be evidenced in the
following way :

Of a cat the right labyrinth is extirpated; a nystagmus occurs to
the left side. This nystagmus disappears after a few days. A flow
through the meatus of the unimpaired left side yields nystagmus to
the right side in ventral position; but on changing the position of
the animal a change of the direction of the nystagmus is produced;
in a dorsal position e.g. it is always to the left. If on the same
day the left labyrinth is extirpated, the BeECHTEREW-nystagmus to
the right side appears, whose direction cannot be changed by
changing the position of the animal (nor indeed can its nature or
frequency).

This, then, also proves that a flow of cold water cannot completely
arrest the function of the labyrinth.

Ad d. If the function of the labyrinth were arrested by ejecting
cold water into the meatus, there could not be any vestibular reflexes
after ejecting it into the two meatus simultaneously. Rabbits are
highly serviceable for an inquiry into this matter, these animals
making hardly any spontaneous eye-movements, besides having very
strong eye-reflexes. We, therefore, experimented upon rabbits to see
whether after ejecting water into both ears rotatory movement or
deviation could be provoked through changing the animal’s posi-
tion in space. From an investigation made by v. p. Horvr and
DE KrryN it appeared that in normal rabbits the greatest difference
in rotatory movement of the eyes may be observed, when first the
animal is examined with its head down and then with its head up.

We subjected five rabbits to the following experimentation.

First of all we ascertained whether no nystagmus at all was
discernible after water had been ejected into both ears. This was
necessary, as it was evident that with some animals the effect of
the irrigation is not quite the same on both sides, so that a slight
nystagmus still remains. Such animals were of course not suitable
for an investigation. When this inquiry had been made, a photo-
graph was taken of the head of the animal in two positions, with-
out irrigation, viz. .

animal vertical, head down
animal vertical, head up.

In order to determine the intensity of the rotatory movement a
cross was burned into the cornea after cocainization. This, according
to VAN DER Horvr and DE KLzEYN’s experience, somewhat slackened
the rotatory movement, so that it is somewhat less marked in our
experiments than is usually the case with normal rabbits. Consid-

628

ering, however, that we only endeavoured to ascertain whether
or no with bilateral irrigation a rotatory movement occurred, this
did not matter at all. So when the two photographs had been taken
prior to the irrigation, a simultaneous flow of cold water into both
meatus of the animal was performed, and when no nystagmus was
distinguishable any more, again two phothographs were taken of
the head, in two positions. A measurement of the rotatory move-
ment, caused by the change of position could now be made on the
photographs. The difference between the movements in the two
positions (head down and head up) amounted to

Exp. Without irrigation after bilateral irrigation
d 66,5° oh
2 70° 65°
3 59? ot:
+ 88° Je
5 63° 78°

From this table it is obvious, then, that after a flow of cold water
into the two meatus of rabbits a strong rotatory movement is still
effected by a change of the animal’s position in space, perhaps as
strong as with normal animals.

For the present we refrain from theoretical speculations on the
basis of these experiments. Our object in publishing our experience
was only to prove that the caloric nystagmus cannot possibly be
provoked by a complete elimination of the labyrinth by cold water.

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS

VOLUME XX
N25:

President: Prof. H. A. LORENTZ.
Secretary: Prof. P. ZEEMAN.

(Translated from: “Verslag van de gewone vergaderingen der Wis- en

Natuurkundige Afdeeling,” Vol. XXV and XXVI).

CONTENTS.

W. STORM VAN LEEUWEN and C. DE LIND VAN WIJNGAARDEN: “On the Effect of Lobelin on the
Rise of Blood pressure by Nicotin”. (Communicated by- Prof. H. ZWAARDEMAKER), p. 630.

H. R. KRUYT and Miss J. E. M. VAN DER MADE: “Contributions to the theory of dyeing”. (First
communication). (Communicated by Prof. ERNST COHEN), p. 636.

H. C. BURGER: “On the Theory of the Brownian Movement and the Experiments of BRILLOUIN.”.
(Communicated by Prof. W. H JULIUS), p. 642.

F. A. H. SCHREINEMAKERS: “In-, mono- and divariant equilibria”. XVIII, p. 659.

H. J. HAMBURGER and R. BRINKMAN: “Experimental researches on the permeability of the kidneys
to glucose. III. The NaHCO3 percentage in the transmission-fluid”, p. 668.

P. EHRENFEST: “A paradox in the theory of the Brownian movement”. (Communicated by Prof.
H. A. LORENTZ), p. 680.

Miss H. J. FOLMER: “A new Electrometer, specially arranged for radio-active Investigations.” Part II.
(Communicated by Prof. H. HAGA), p. 684. / — D tah.

S. DE BOER: “On the ventricle-electrogram of the Frogs’ heart”. (Communicated by Prof. G. VAN
RIJNBERK), p. 696.

Miss H. J. FOLMER and A. H. BLAAUW: “Researches into the Radio-Activity of the Lake of
Rockanje”. (Communicated by Prof. H. HAGA), p. 714.

B. P. HAALMEYER: “On Elementary Surfaces of the third order”. (Third communication). (Com-

4 municated by Prof. L. E. J. BROUWER), p. 736.

Proceedings Royal Acad. Amsterdam. Vol. XX.

Physiology. — “On the Effect of Lobelin*) on the Rise of Blood
pressure by Nicotin”. By W. Storm van Leeuwen and
C. pe Linp van WIJNGAARDEN. (Communicated by Prof. H. -

ZWAARDEMAKER).

(Communicated in the meeting of September 29, 1917).

When one of us was carrying out an inquiry into the effect of
various lobelia preparations, the results of which will be published
in the “Nederlandsch Tijdschrift voor Geneeskunde”, we found that
the rise of bloodpressure, evoked in a cat by nicotin, was greatly
influenced by a previous injection of lobelin. We decided to look
more closely at this phenomenon, as we presumed this to be an
instance of Bürers “potentiation” (Potenzierung) ie. the effect of a
mixture of two poisons is stronger than is expressed by the algebraic
sum of the action of the component parts. The facts revealed in
this inquiry are chiefly the following:

When a cat‚ which — to eliminate central influences — had been
previously decapitated after SHERRINGTON, was given intravenously a
small dosis of nicotin, e.g. 0.1 mgr, a fall of the bloodpressure
generally ensued, which was succeeded by an increase of bloodpressure.
The first phenomenon probably results from an inhibiting influence
of nicotin upon the heart, the second from a stimulation of
sympathetic ganglia, causing a constriction of the bloodvessels. The
course of such a nicotin-curve is irregular. When, however, from 1 _
to 2 mgr. of atropin has been previously injected, the fall of the
nicotin-curve disappears completely or partly and only the rise
remains. The current opinion is that the first fallin the nicotin-curve
is effected by a stimulation of the vagus by nicotin, which is again
inhibited by atropin. Why, in some cases, the atropin does not
prevent the primary fall, has not yet been explained.

1) When our supply of lobelin was exhausted and none could be procured in
Holland, Prof. Scuoort had a fresh supply prepared in his laboratory, for which
we acknowledge our indebtedness.

631

When, after an injection of atropin, a decapitated cat is subjected
to a series of injections every time of the same amount of nicotin,
the subsequent increases of the bloodpressure are relatively equal,
as one of the present writers has pointed out. *) The amount of
nicotin, injected every time, must-be small: from O1 to 0.2 mgr. ;
larger doses bring about before long a paralysis of the sympathetic
ganglia, while no rise of the blood pressure is evolved. The same
occurs when more than 8 or 10 small injections have been
administered. When, after the 2rd or 3rd nicotin injection a similar
quantum of lobelin is injected, a rise of bloodpressure reveals itself,
as high as the rise after the nicotin, as shown in Table I.

TABLE I.

The effect of 0.1 mgr. of nicotin compared with that of 0.1 mgr.
of lobelin, after the injection of atropin.

Te

Exper. rise of the blood — rise of the blood —
pressure after 0.1 nicotin pressure after 0.1 lobelin
18 66 mM Hg. | 54 mMHg
30 46 | 56
44 | 13 19
57 30 31

average 38,75 average 37,5

The concordance between nicotin and lobelin need not surprise
us as Dreser’) and Epmunps*) have shown before that lobelin and
nicotin act. similarly on many organs The quantitative concordance
between the two substances could not be surmised, especially also
because the structural formula of lobelin is not known. The empiric
formula was given as C,, H,, ON (®).

If, however, nicotin is injected before lobelin, and after this another
similar quantity of nicotin, the rise of bloodpressure after the latter
amount of nicotin is much higher than that oceasioned by the former
(Cf. Table II).

1) W. Storm van LEEUWEN, Geneeskundige bladen 19e reeks N°. 5.

2) Dreser. Arch. fiir exp. Path. u. Pharm. Bd, 26, page 237, 1890.

3) Cr. Epmunps. Americ. Journ. of physiol. vol. XI, page 79, 1904.

Cr. Epmunps. Journ. of pharmacol. and exp. therap. Vol. I, page 27, 1909.

46*

652

TABLE Il.
Cat; decapitated; subsequent injection of atropin;
Rise of blood pressure by 0,1 mgr. of nic. prior to and
after the injection of lobelin.

a. injection of 0.05 mgr. of lobelin

Rise of blood Rise of blood Rise of blood
Exp. pressure by pressure by 005 pressure by 0.1
0.1 mgr. nicotin mgr. of lobelin mgr. of nicotin

35 45 mmHg | 12 mm Hg 69 mm Hg
38 DA Tan, DV be 31 »
49 Lite Ps OA Wass
54 ZEN Pk Enne ay RT

b. injection of 0.1 mgr. of lobelin

Blood pressure blood pressure by Blood pressure by

Exp.
by 0.1 of nicotin 0.1 mgr.of lobelin 0.1 mgr. of nicotin
18 66 mm Hg 54 mm Hg 76 mm Hg
30 46 5 56 3 59 is
44 13 ” 19 ” ) 31 ”
aM 30 nd 31 4 70 je
c. injection of 0.5 mgr. of lobelin
|
Ex Blood pressure by | Blood pressure by Blood pressure by
re 0.1 mgr. of nicotin | 0.5 mgr. of lobelin 0.1 mgr. of nicotin
46 5 mm Hg 30 mm Hg 18 mm Hg
47 24 E | 90 5 90 .
48 fe oS | 28 d 42 i
53 Soa tig Cher 88, a
d. injection of 1.— mgr. of lobelin
Ex Blood pressure by Blood pressure by Blood pressure by
ke 0.1 mgr. of nicotin 1... mgr.oflobelin 0.1 mgr. of nicotir
50 20 mm Hg 122 mm Hg 19 mm Hg
51 6 5 82 ; LOS at at!

635

It appeared, moreover, that 0.06 mgr. of nicotin + 0.04 mgr. of
lobelin acted more forcibly upon the rise than 0.1 mgr. of nicotin
or lobelin alone. This, then, is a case of true “Bürer-potentiation”’.
This is illustrated by Fig. 1.

Stijging = Rise. — Lobeline = Lobelin. — Nicotine = Nicotin.
Fig. 1.

Bloodpressure, decapitated cat after atropin injection. a. 0,1 mgr. of nicotin
yields a rise of bloodpressure of 30 mm. Hg.; 5. 0,04 mgr. of lobelin + 0,06 mgr.
of nicotin yields a rise 59 mm. Hg. ; rm

When we endeavour to find an explanation for this phenomenon,
it would seem as if, after the lobelin injection still some portion
of this substance is left behind in the cats body, and that this, of
itself incapable of any action on the blood pressure, adds its effect
to that of the nicotin subsequently infused, so that this would seem
to be a case of “addition” and not of ‘potentiation’. However, the
matter is not so simple, for if this were so, we might look for a
higher rise of the blood pressure after a second lobelin injection.
This is not the case. On the contrary the action of successive lobelin
injections upon the blood pressure is most often progressively dimi-
nishing. Besides, the hypothesis alluded to is disqualified by the
fact that a simultaneous injection of lobelin and nicotin is also
attended by a “potentiation”, as stated above. Lastly we have
ascertained whether other substances affording a rise of blood pressure
are ‘potentiated’? by lobelin. This appeared not to be the case;
adrenalin e.g. yields perfectly similar effects before and after lobelin.

Ultimately we have tried to make out whether, conversely, the
nicotin could potentiate the lobelin. The result negatived the suppo-
sition. With regard to the rise of blood pressure the effect of lobelin
after nicotin is not increased, even diminished.

So far as we are able to judge the potentiating effect of lobelin
for nicotin is specific. It is analogous to the re-inforcing effect exerted

634

by thyroidin upon the action of adrenalin on the rise of blood pressure.

We wished to know whether larger doses of lobelin had a stronger
potentiating effect than smaller ones. With a view to this several —
injections of 0.1 mgr. of nicotin were administered, subsequent to
an atropin injection, to a number of decapitated cats, and then again
varying amounts of lobelin and finally again 0.1 mgr. of nicotin.
The results are given in Table II, in which we bave expressed the
rise of the blood pressure in m.m. Hg after the several injections.

In Table III we have tabulated for different doses of lobelin the
difference in blood pressure afforded by 0.1 mgr. of nicotin before
and after the lobelin.

TABLE III.

Mean values of the rise of blood pressure by 0.1 mgr. of nicotin
before and after lobelin.

a b Cc d
mean rise of blood pr. mean rise of blood pr.
dosis of lobelin | by 0.1 mgr. of nicotin by 0.1 mgr. of nicotin difference
before lobelin after lobelin

0.05 mm Hg 25 mm Hg 38 mm Hg 13 mm Hg
0.1 " 25 . 59 ; 34 J
0.5 d 25 a! 4 34.5 a
1.— ‘ 25 . 117 E 92 a

Lastly the curve in Fig. 2 represents graphically the ratio between
the amount of lobelin and the change in the nicotin action, brought
about by those amounts, so this curve shows the relation between
the values in columns « and d of Table III.

From the table and still more so from the curve it is evident
that the effect of the amount of lobelin is very considerable. In our
opinion the shape of the curve in Fig. 2 cannot be a basis for
further speculation, for which we need the mean values of a larger
number of experiments.

Since the publication in 1909 of Bérai’s investigations other
researchers and himself have detected a number of combinations of
potentiating drugs. As expatiated upon by one of us in another
paper, only in a very few cases has the presence of such a “poten-
tiation’ been proved.

Since ‘‘potentiation” is doubtless one of the most remarkable
phenomena in modern pharmacology, we thought fit to communicate
this new, striking instance of true potentiation, Further research

635

will have to furnish an explanation for the phenomenon in this and
in other cases. It is remarkable though, that in the case under

et 0.1 65 5
Fig. 2.

Relation of the amount of lobelin to the influence
exerted by lobelin upon the rise of blood pressure after
0,1 mgr. of nicotin;

abscissa: amount of lobelin in milligrammes.

ordinate: difference in nicotin action before and after
the lobelin injection, expressed in mgr. Hg.

discussion a potentiation has been found in a mixture of two sub-
stances, no doubt pharmacologically closely allied, and on that
account should, according to Birel’s preconceived idea, not display
the phenomenon.

In conclusion we wish to say that we do not ignore that the
tinctura lobelin is seldom prescribed in Holland, but it is used rather
largely in other countries, so that this potentiating effect upon nicotin
(and maybe also upon other drugs, not investigated’ by us) has
also its practical side.

Chemistry. — “Contributions to the theory of dyeing” (First commu-
nication) by Prof. H. R. Kruyr and Miss J. B. M. van Der Mabe.
(Communicated by Prof. Ernst COHEN).

(Communicated in the meeting of June 30, 1917).

1. Three different explanations have been suggested for the way
in which dyes are taken up by fibres and other solids. First a
merely chemical combination of dye and fibre was suggested; but
a chemical theory gave rise to so many contradictions, that, as soon
as the theory of solid solutions was developed, it was tried to explain
dyeing in agreement with the new theory. But in this way it was
just as impossible to bring experiments and theory into harmony,
and the new knowledge in the chemistry of colloids, especiaily that
of the phenomena of adsorption and of the electricity of contact,
gave a better starting-point for the explanation of the way in which
wool, silk, cotton etc. take dyes from their solutions in water,
both when electrolytes are added and when this is not the case.
FREUNDLICH, in collaboration with several pupils (Losev *), NEUMANN ’)),
stated the fact, that amorph. carbon takes the dye from its solution
in quite the same way as textile-fibres do; Perer®) has made
numerous investigations with his pupils on this subject‘). It may
be stated that nowadays the interpretation of the process of dyeing
as a phenomenon of electro-adsorption is generally accepted.

This point of view may be summarized as follows: the faculty
of the fibre to take up the dye depends on the electric potential
at the confines of fibre and solution, in connection with the sign
of the electric charge of the dye-ions. The fibres and charcoal are
charged negatively with regard to water, therefore they generally
adsorb the basic dyes better than acid dyes. Every influence increasing
the negative charge of the adsorbentia, will increase the adsorption

1) Zeitschr. f. physik. Chem. 59, 284 (1907).
*) Zeitschr. f. physik. Chem. 67, 538 (1909).
5) The results of these investigations are brought together in the interesting
monograph Perer-Jouver, Die Theorie des Färbeprozesses (Dresden, 1910).

*) Bancrort and his pupils too have paid much attention to the process of
dyeing of late years. A summary of the theory of dyeing can be found: W. D.
Bancrort, Journ. of physical chemistry 18, 1, 118 and 385 (1914).

An

637

of basic dyes. All inorganic compounds and their ions are adsorbed
in nearly equal amounts from solutions of equal molecular concen-
tration, excepted the ions of -H, OH and of the heavy metals, which
show an abnormally high adsorption. So the hydroxides of the aleali-
metals will add to the negative charge, for the OH ion is adsorbed
more fully than the metal-ion. The same will be caused by alcali
salts of polybasie acids, as the negative ion has a greater electric
charge than the kation; and again the same will be the case by
adding salts of organic acids, because organic ions are more fully
adsorbed than anorganic ones. When dyeing with acid dyes, we
shall meet with an exactly opposite effect of electrolytes, dyeing being
increased by acids, by salts of bi- and. trivalent kations and of
organic ones. -Influences, diminishing the process of dyeing, may
easily be inferred to the same line of thought.

PELET’sS investigations are in perfect accord with the above theory,
both for the adsorption of the dyes itself and for the influence of
dyes and electrolytes on capillary electric phenomena.

2. W. Reinpers') lately published a paper in which he drew
the conclusion that there remained much doubt about the connection
between dyeing and the phenomena of adsorption. He had made
investigations on the ratio in which a solution in water gave the
dye to another phase, which was not a solid with great develop-
ment of surface, but which was a liquid; the dyesolution was shaken
with isobutylic alcohol. First the distribution of the dye in the
two layers showed to be not in harmony with the distribution-
law. In the second place he remarked in those cases, that whether
acid or alcali increased the process of dyeing, the transition of the
dye into the alcohol layer was also favoured. He came to the
conclusion that these facts pointed clearly to a solution theory, and
that it was of no use to refer to the theory of adsorption.

The interesting question, that was brought forward again by that
paper, induced us to make the following investigation. First of all
we are of opinion that for a good solution of this intricate problem
it is of the greatest importance to avoid all unnecessary complication ;
therefore we shrink from drawing conclusions from experiments,
when alcali or acid is present together with the dye. For these
bodies often cause a chemical change in the molecule of the dye,
in which case we have to deal with a chemical phenomenon and
the process of dyeing simultaneously. So we have studied the

1) Koll. Zeitschr. 18, 96 (1913).

638

distribution of the dyes between water and isobutylie alcohol when
only the sodium salts‘) of different acids were present.

1 eem. of the solution of the electrolyte (12 millimol p. L.) was
added to 2 eem. of a 0.5°/, solution of the dye (the first experiment
was made, of course, with 1 eem. of pure water). Then 2 eem. of
isobutylie aleohol were added and the whole was shaken in a tube
with a glass stopper.

As the result we give the series in which the salts favour the
transition from the waterlayer to the alcohol. From the left to the
right -the ions are written in the same series as they cause the
alcohol to take a darker colour, resp. the waterlayer to take a
brighter tinge. When two ions are united with a brace, we wish
to express that the difference of colour was too small to decide
about their sequence.

rystal volet.
EGO SPO SO EN bere NG EEND

Fuchsine.
LEGO S02 "POS El Bron TiC Nes

Methylene blue. -
CEO SOR, “PO; Clap NOs Loe hs

The first thing that strikes us is the fact that the ions are placed
in the sequence of the so called /yotropic series of anions. The
sequence of the ions is the same as regards their influence on the
solubility of other bodies; the salting out of proteids; the influence
on surface tension; enlargement of swelling; the kinetics of many
reactions ete. These three dyes were basic, but when acid ones are
taken we can expect the anions to have either no influence at all
or an opposite one. Indeed we found such an influence though
very weak. ,

Naphtolyellow S.
!' 'Br [H,O] "SO, "PO,

Alcaliblue.
FEO} Cole eon, EO,

3. Perer’s theory of dyeing suggests an increasing adsorption of
basic dyes when NaCl, still more when Na,SO, and even much more
when Na,HPO, is added to the dyeing tub; we found however that
the adsorption by the alcohol was not increased according to the

1) We only used NH,CNS in staed of NaCNS.

639

basicity of the anions but according to their lyotropic strength.
Lyotropically SO, and PO, are nearly equai, but Cl has a sharply
different position, contrary to the sequence of basicity. Operating
with neutral salts we therefore seem to have missed the parallelism
Reinpers found with acids and alcalies.

One is inclined to consider the transition of the dye from the
water to the alcohol layer as the process of salting out, so common
to the emulsoids. Indeed the CNS ion, the extreme term of the series,
caused precipitation after 24 hours,') when brought to the dye
solution in water to an amount of 4 millimol p. L., (the same
concentration as the electrolytes had in the water-layer before the
alcohol was added). It is evident that dyes, showing in many
respects the properties of lyophylic colloids, are salted out by
electrolytes in tbe sequence of the lyotropic series; and it is therefore
easy to understand that lyotropic influences (it would be preferable
to say Aydrotropic here) cause a distribution of the dye in such a
way, that the most active ion causes the greatest transition of the
dye to the alcoholic layer.

4. Our knowledge about the influence of neutral salts on dyeing
is however small. The dissertation of Baccio Beccari’), a pupil of
PELET, is the only systematical investigation we know, and even
in this methylene blue is the only basic dye and Na,SO, and
Na,HPO, are the only neutral salts considered. The sequence in
which that dye is taken up is this: solution in water — solution
containing Na,SO, — solution with Na,HPO,, all according to
PELET’s theory *).

As we thought it interesting to know whether lyotropic phenomena
had any influence on the process of dyeing, we have made investi-
gations about the- quantity of dye adsorbed in the presence of the
sodium: salts. of PO,,.sO,, Cl, Br or NO;.

For dyes we took methylene blue, crystal violet and auramine;
bloodeoal was the adsorbens. In each series the initial concentration
and the amount of coal were exactly equal. The concentration at
the end of the process was estimated by Peter and Garutt’s volume-

1) When in contact with isobutylic alcohol, a precipitate could only be seen
after some weeks at the boundary of the layers.

2) Diss. Lausanne: Etude des rélations des phenoménes de teinture et d’adsorb-
tion. (Florence 1908).

3) Investigations of A. WILD, mentioned in PELEt-JoLivet’s book p. 98, show
the following sequence in the case of methyleneblue: HyO—CI—SO,—PQ,.

640

metric method *); methylene blue was measured by naphtolyellow 5,
crystalviolet by picrate of sodium and auramine by alcaliblue.
The adsorption showed to have been increased by the salts in
this sequence:
Methyleneblue B extra
[HGO | CTS), nea Pr SND
Crystalviolet
[HOBO SO. Cl Br iNO.
Auramine O

[H,O] 'Cl "SO, 'Br 'NO, PO,

In these series the ion which effects the adsorption is mostly placed
at the right.

5. These results of our first series of measurements indicate that
the interpretation must be more complicated than one should think
from Peret’s theory. When only the ions of 'Cl, "SO, and “PO, are
taken into consideration our results are in harmony with this theory
with regard to methyleneblue and auramine, the increasing influence
is added to with the basicity of the anions.

But the ‘Br and 'NO, ion already show the importance of lyotropic
influence. So we may draw the conclusion that dyeing is affected
by the two sorts of phenomena; — when two ions are of nearly
equal lyotropic strength, it will be possible for their electro-adsorptive
character to preponderate and therefore to fix the sequence; when,
however, there is great lyotropic difference between two ions, this
fact will be decisive. With crystal violet the lyotropy of the salts is
so predominant that no other influence can be noticed in the sequence.

An exact interpretation of lyotropic phenomena is always difficult
because lyotropy can have so many different effects. Even the
supposition that the cause of lyotropy is to be found in the power
to combine with watermolecules, the so called ‘“solvatation’” —
though acceptable in many respects — leads to so many consequences,
that the explanation remains totally arbitrary, as long as we
cannot verify our theoretical conclusions with many series of expe-
riments. We intend therefore to extend our researches to several
dyes, electrolytes and adsorbentia. But we may call attention now
already to the fact that the importance of lyotropy is not in contra-
diction with a theory regarding the process of dyeing as a- pheno-
menon of adsorption, though this complication has remained un-

1) Bull. de la Soc. Vaudoise des Sc. nat. 43, 1 (1907).

641

considered till now’). For without sufficient reason it is only electro-
adsorbative-phenomena that have been compared with the process
of dyeing; the investigation of FrrUNDLICH and Poser?) showed
however that such a way of interpretation is quite insufficient.
Adsorption is not determined by electrical causes only, besides
there is certainly a non electrical adsorption, determined by the surface
tension according to Gipss’ law. As the lyotropic series shows the
sequence in which the ions influence surface tension, it will be
clear that the principle of adsorption explains both the importance
of valency .and lyotropic strength.

Attention must all the same be directed to the possibility, that
the electrolytes do not leave the dyesolutions itself*) unaltered. The
peculiar condition of dyesolutions, often polydispers with the character
of a pseudo-binary system‘), make the problem all the more com-
plicated; only a wide investigation of each sort of influence by
itself can make it clear. Hence we already made a_ beginning
with such experiments.

Utrecht, June 1917. van ’T Horr-laboratory.

1) The only investigation showing the lyotropic influence on dyeing is to be
found Biochemical Journal 1 175 (1906). Baytiss made some researches on the
dyeing of filtering paper by congo-red in presence of salt; the influence of these
salts proved to be as according to the lyotropic series. BANCROFT (loc. cit.) does
not mention this paper, Perer-Jouiver does, but only in connection with other ~
questions. So Baytiss’ paper came to our knowledge only when this investigation
was closed.

3) Koll. Beih. 6, 297 (1914).

3) See I. TRAUBE, Koll. Beih. 3, 237 (1912).

4) The remarkable observation DROOGLEEVER FoRTUYN published [These Proceed-
ings 23 1380 (1915)] have convinced us again on this point. It is obvious that
in a solution of new fuchsine in water there are two kinds of molecules, a coloured
and a non coloured one, which are in equilibrium with one another. And in water
this equilibrium is reached only after a long time and as by absorption only the
coloured modification is taken away, some days are wanted to colour the water
again after it bas been decoloured by coal (acetic acid seems to catalyse). It is a
wellknown phenomenon that a solution of fuchsine is decoloured by alcali as the
dye is changed into the pseudo base; still such a solution will dye silk (JACQUEMIN,
C.R. 82, 261, (1876)| and wool. But the red colour returns too when the solution
is shaken with isoamylic- or isobutylic alcohol [Wirt, Farber’ Zeitung 1, 1 (1891).
The same thing occurs. when we shake a solution decoloured in the same way
as followed by DROOGLEEVER FortuyYN, with isobutylic alcohol; this suggests that
water is a solution of a pseudo base, which was present already in the primary
solution. Perhaps this is also the reason, why the adsorption of dyes seems to
take much more time than that of other bodies. The regeneration of dye, the
only component which has really been adsorbed, from the carbinol base, which
is even not in real solution (as an ultramicroscopic investigation shows), makes
it appear as if the process of adsorption should be slow in this case.

Physics. — “On the Theory of the Brownian Movement and the

Experiments of Briurouin”. By H. C. Bureer. (Communicated
by Prof. W. H. Jurrus.)

(Communicated in the meeting of April 27, 1917).

The purpose of this inquiry is of two kinds viz:

1st. to give a rigorous proof of some well known formulae of
the Brownian movement, with and without external force.

2nd, to trace the boundary-condition at a fixed wall and to inter-
pret some experimental results concerning this.

§ 1. When a material particle is suspended in a liquid, one may
inquire after the probability that, in consequence of the Brownian
movement, it will in the time ¢ in the direction of X cover a
distance between w and w + dr. This probability is determined by
v. SMOLCCHOWSKI *) by using the following image. Suppose that this

particle regularly after a time +t covers a distance d, with equal
probability to the right or to the left. Now, when one makes d and

d pe
t zero, but keeps — — D finite, one finds for the required probability :
T

9
a

x2
4Dt
F(a, t) de = —— e da. viii = er
2V_Dt
From this formula we find for the mean square of the displacement,
a? = 2Dt.

By entering closer into the mechanism of the phenomena, we
may find the last equation, in which D then proves to be:

ete Rivet a

= Hee

where B, 7’ and N have the usual meaning, u is the coefficient of

viscosity of the liquid and a the radius of the particle.

Now we may deduce (I*) however in a perfectly rigorous manner,

without making use of any image, whereby however the constant

D remains entirely undefined; and at the same time we may prove,

') Vorträge über die Kin. Theorie der Materie p. 101, 1914 and Bull. Acad.
Cracovie p. 419, 1913.

643

that an infinite number of images lead to the same final-equation.
We may viz., as has already been shown by v. SmoLucHowskt *)
come to an integral-equation for f(z, t) in the following manner:
When the particle in the time ¢ has covered a distance between
a and a + da, this may have taken place in such a way that:
in the time 7, a distance is covered between «, and «, + dr,
ed eles a ASSE Ke ~ te en BE » & + de,

Ne Wee hy Re ek 5 ps 3 Vn 5, In + dr,
in which 27,=t and Zw, lies between a and a + da. When we
suppose that the times ¢, are not too small so that the number of
shocks in ¢, is great enough to allow the use of probability-calculus,
the probability of this occurring is given by

Pe Batali CAA LEED Wate) Gade da,
in which / is the required function. Now this product, integrated
for all values of w,...%, for which wz, lies between a and a + da,
must give the probability, that the particle in the time ¢ has covered
a distance between a and a+ da, so f(a,t) da, in which this f
will be the same as the “elementary” / from the above product,
when only ¢ is not too small.

So we get:

ijn in
| ds free) eed Cpt) do, 4 ede ee (4,2) da,» (21e)
Here 4 is a factor of discontinuity that satisfies the conditions
A0 when 2%, < a or wara da
ad Wie a Lx,<a-+ da.
When n= 2 (Ile) becomes:

[Fey se ©, t,) dx AN t nie t.) . . . ° (ZP)

One easily sees that inversely (II*) follows from (II’) so that each
of these equations may be used as a basis for the following surveys.

We will now show in different ways how we may deduce the
formula (I*) from the equations (II).

I]. Take in (II*) the times ¢,...¢, small then m is great *).

1) Comp. M. v. Smotucnowsxt. Ueber Bown’sche Molecularbewegung ete. Ann.
der Physik. 48 p. 1103, 1915.

*) When we want, that in the first member of (Il¢) f(z, é) is the required
function, then ¢, must be larger than a certain very small time, which EINSTEIN
has estimated at 107 sec. for particles of a radius of | 4. (Ann. d. Phys. 19.371, 1906).

644

In connection with the physical meaning of f(z, ¢) we know that
the function f is even in ve, and for a small ¢ differs only from
zero for small values of z. From the proof of the law of errors as
given by BrsseL'), we know that under these conditions a product
that is built as the first member of (II), becomes for great values of n:

Fahy da Sper VE da he ee
in which p and w are functions of ¢ still unknown. To determine
these, we substitute (III) in (Il). This gives after integration

‘yew os Kn) te)
a i p( 4) td } Y(t HY te) —_ (t, As t,) eh +42)
f(t) + W(C)S

From this follows
DR NANG oe me EN Plt P(t) Vz

w(t,) + w(t.)

|

w(t, + t,) Vilt) +(e,

The solution of the first equation is: vO").

This, substituted in the equation for gy, gives:

g(t, )P(t,V att,
Y (t, + t) = ee
Welt, +t.)
This functional equation is solved by
any
p (t) = — et - *)
at

When we substitute this in (III) and consider moreover that:

| flat)dai=1,

then it is proved that c’ —0 and we find

C t

i (et) = =e e

1) Astr. Nachr. 15 (1838) p 369 = Ges. Abh. 2. p. 372,
2) Put:
Hi) Mty+t,) HE) HE)
w (t) ==, so —— sy p
t lysate t,k(t,) +t, k(t.)
When we put herein successively : f) = tj, 24), Sh, then appears: A(t) = k(2) =...
so k(t) is a constant

C .
5) Put viz. OVS — L(t).than L(t + ta) = L(t) L(t) or L(t) = ett in which

c’ = constant.

645

’ Lae :
When moreover we put herein c = tp ™ which D is a new

constant, we get exactly (I,)’).

Hl. From (ll”) we can deduce a differential equation for f(a,t).
When viz. we expand in the first member /(a—z,t,) in a series of
powers of ¢,, we find:

0°/(a,t,) iG f(a,t,)dze +... =f (at) + ¢, ue

ts)
2/da? , |

flat.) + Eelen He

’
2

1) We may give this proof in a slightly different way yet, viz. by using only
(lla). Take in (lla)t, =t, =...=dn= tr, and make m great. As is known from
the quoted proof of BessEL (I. c.):

oo
1 ae
7 (at): == | by (Apr cos aa dt ,
; 2%
a

in which

X (A) = fi (x,7) bbs vh da.

By expanding cos zò, we get:

ur
CA) St -5 f*s (&,x) de +...

When zr is small, the coefficients of the powers of A diminish rapidly and are
small, so:

gef
x¥(4)=1— | eren dz = 1 — p‚d'

and:
OP = (1 — ph eve,
while p is small and » great. Finally:

°

a?

Die We ie rappers
3 4pon
cos a A e—Ppon® — mae Ao
2a UW apn

ee GO

vi (a,t) =

Now nr=t. Further take Ps = D,then we find again the formula (Ia). Herein
ia

D is independent of +, while from the found form of f follows that py is pro-
portional to r.

This proof shows that always the same f(4,t) is found, independent of the
choice of the “elementary” f in the first member of (Ila). The value of D however
in this case depends on the form of this function.

47

Proceedings Royal Acad. Amsterdam, Vol. XX.

646

When we consider that

As (z,t) de = 1). and fom Alestidae =O

60

When in this we take ¢, small enough to allow the neglecting
of following terms and further

1 a
x | ef (ait) des Den Va oes NEN
2t,
we get the equation, .
òf Oe
DE ne er ot Se
Ot 0a? ( )
òf ak ay Te ; ;
As here ag and aa are functions of a and ¢,, D will not contain
a

t, and therefore will be a constant. *)

It is remarkable that we find here the diffusion-equation as a direct
consequence of the integral-equation (HP), while (LV*) immediately
connects the diffusion-coefficient with the mean displacement.

This equation (LVP) has to be solved without boundary-conditions
and with the beginningconditions :

1° | fda SS iat 14,0) = Onexcopt A0r da =.

This is a well-known problem of diffusion or conduction of heat,
the solution of which is given by (I*).

The exponential function of (I*) satisfies the integralequation (I1*)
for arbitrary » and £,. When we take however for the “elementary”
function f(z) in the first member an arbitrary function, that only
answers to the conditions ,

1 F(a.) 1s even,

2 f(«) differs perceptibly from zero only when a, has a small
value,
the integral in the first member of (II*) will, for great n always
give the same f(a,t.*) Therefore we understand that the simple
image of the Brownian movement, that is given by v. SMOLUCHOWSKI
yields a right result, in spite of the fact that it approximates the
true movement very badly.

1) When we introduce the solution of (IVb) i. e. (la) in (IVa), D becomes
actually a constant, as is supposed in the solution of (IVb).
2) See note 1, p. 645.

647

When one chooses however another image, i.e. another formr for
the “elementary” function /(z,), one gets the same formula (I*),
but then D depends on the choice of /(#,) in another way. *) Only
when the “elementary” function is the required f(z, t,), D will be
an indefinite constant, which is determined by the nature of the
Brownian movement.

§ 2. In the preceding paragraph we have always supposed that
the Brownian movement is not influenced by an external force. In
consequence of this, we might, for reasons of symmetry, accept
f(x, t) to be even in wv. When however a force acts upon the particles,
the probability of a certain displacement will be no longer an even
function of this displacement. Nor will this probability only depend
on the displacement, but also on the starting-point. First we will
consider displacements in the direction of X only. When we call
the abcissa of the particle at the time t— 0 «, we may represent
the probability that this abcissa at the time ¢ lies between p and
p+dp, by f(z, p,t) dp. One easily understands, that this function
f also satisties an integral-equation viz. :

fre Ps t,) fF (p, st) dp = plaipayt, ONT, (V)

From this integral-equation v. SmoLucnowskt*) for the case of a
quasi-elastic force, has determined f by repeated integration, but
hereby he remarks, that this way is only serviceable when we make
a simple supposition about the force y, and that, in general it is
preferable to determine f from a generalised equation of diffusion.
This equation is deduced by v. S. on a somewhat phenomenological
manner, without making use of (V). We will now show how this
equation may be deduced from (V), when we use suppositions that
are also put forward by v. S., determining f by repeated integration.

When namely in (V) we give ¢, a very small value, I (p, 4, t,)

1) Suppose e.g. that regularly after the time 7 the displacement 5 may occur
1—x
to the right or to the left, each with the probability ae ae that the chance

of remaining at rest is«. By this the form of f (ay) is defined and we find for D:

ere J? (1—x)
D Poe sh. EE wy? f(a.) da, ot ( TI a «
T 2r : Qt

2) See also M. Smotucnowsxr Ann. d. Phys. 48 p. 1103, 1915.

3) Ann. d. Phys. 48 p. 1104, 1915.
47*

648

is in first approximation an even function p of the displacement
p—a’) , because the influence of the external force for small times
is small with regard to that of the Brownian movement. To take
into consideration the influence of the force, we determine the
displacement d, that the particle, starting from p, would undergo,
when there was no Brownian movement. So, the displacement in
consequence of the Brownian movement in the time ¢, is p+ d—a,
when we suppose that the two displacements are simply superposed,
1, being small.

So for small ¢, we may substitute p(p + d—a,t,) for f(p, a, t,)
whereby p is a function that is even in its first argument. As the
influence of the force is entirely included by the quantity J, p,
which relates to the undisturbed Brownian movement, has the
following properties :

0

frena= and ferens=en oo: re

—_ 0
To compute J, we make use Sor the fact that small particles,
under the influence of a constant force, cover a distance that is
proportional to the time, when we leave the influence of the
Brownian movement out of account. As however the force y(«) is
a function of x, the distance d, which has been covered, is a function
of ¢, that will also contain the higher powers of f,, being expanded
into a series of ascending powers of this quantity. As ¢, is chosen
small, so that in every instance we keep only the first power, we

may take for d:
C= Bt WP) ea TLS. noses 1

So (V) becomes:

afr (x, p, t,) yp {p—a + Bt, w (p), ts dp =f (a, a,b, Ht).

By taking p —a + 5 we may De f and ¢ as follows:
0?
es
feat St) =f (@, 4 ty) + et MERE
owl) oy (Bis)
da 0g
1) Though we do not want the form of this function, we may remark that
this will be:

pip—a+Be wlatS§), Lip (Et) + Be

ppi) *

649

We also expand the second member of (V): iN

Òf (x, a,
feat HIJ=f@ at) +t, Ee) 5

After having multiplied and performed the integrations, we find,
making use of (VI), when we keep only the terms with ¢, and drop
those with higher powers:

Of (a, a, t, 0? 0
Dl wo Saen
or
Die dd,
Den Beep eds Vil
fe ee EN

This is the generalised equation of diffusion that v. SMOLUCHOWSKI
uses, however, deduced from the integral-equation (V). é

When one does not restrict oneself to the problem of one dimension,
but investigates the case of three dimensions, the probability that
a partiele with the coordinates x,y,z will have after the time ¢
coordinates between p, q, r and p + dp, q + dq, r + dr may be
represented by /f(«,y,2, p, q, r‚ t) dp dq dr.

We easily understand, that this function satisfies the following
integral-equation :

ff SF (GY 52 sQrt sty) f (21927145, €,64) dp dq dr =f (x,y,z,a,6,c,t, +#,). . (7X)
When we call again the force, that is now a vector, w, we find
for f in an analogous manner as in the preceding, the equation of
diffusion :
of
les agibme coed: EERE RE Po)
The operations A and div. here refer. to the dE three inde-
pendent variables in f, on which w also depends.
The truth of (X) is also easily proved in another way.

With the help of (VIII) we can solve the following problem:
The plane «=O is formed by a fixed wall, which, on a particle
in the liquid by positive x, exercises a repelling force, which becomes
perceptible in a point A at a small distance of the wall, increases
by approaching the wall, and becomes infinite when 2 = O, remaining
so for negative #*). What is, the probability that a particie for which
1

1) A wall with the described properties will reflect the particles colliding against
it, without ever one of them adhering.

650

vt, when t=O, has at the time ¢ an abscissa between x and
For this case (VIII) becomes:
òf 077 | Poe
pets gn
sp ir,

in which f(z,w,t) de is the required probability.
This equation, integrated with respect to # from a point 5 with
negative z to A, gives:

A
= Ay hoe 4 ud | — w | =F
Ot ae GE) (=), | BLG Wa — (fv) 8}.
B -

0 ”
As in A w=O and in 5 f and a will be very strongly zero
Xe

by the immense repelling force, we get:

A
òf. òf
5 |i de =D (55)
B

When we now make the thiekness of the layer in which per-
A
ceptible forces act, very small, f j dx disappears for every time,

B
because f is in this layer certainly not infinite. So we get at the

wall:
òf
id
(=): )

How f may be determined from this equation, needs not be
explained here.

$ 3. The boundary-condition at a reflecting wall may be easily
deduced from (VIII), and this is perhaps also possible for an ad-
sorbing wall in the latter case, bowever, it seems simpler to take
another way. We suppose that the adsorbing wall may detain a
partiele that collides against it, and that such a detained particle
never gets free again. This property may be observed under several
circumstances.

Suppose that a particle at the time ¢ has a distance x from the
wall that is situated at «—0O. After a time ¢ the particle may be
adsorbed by the wall or may still be free. So if naturally suggests
itself to introduce in this case the two following functions :

') Comp. M. v. Smorverowskr, Phys. Zeitschr. 17 p. 587, 1916,

651

1st. The probability that the particle mentioned has at the time
ta distance from the wall between p and d + dp: /(2, p,6 dp.

Jed, The probability that, at the time ¢, it has been adsorbed by
the wall: x (2, 2).

Since a particle, that is not adsorbed at the time ¢, + f,, is also
free at the time ¢,, f will satisfy the integral-equation :

[LG PALO 01) dp = Fea boty on ee)
0

We can: further obtain a simultaneous integral-equation for f
and xy: by remarking that the probability x (z, ¢, + ¢,) consists of
two parts viz. the probability that the particle was already adsorbed at
the time ¢, and the probability, that at the time 7, it was somewhere
in the liquid and is adsorbed by the wall during the time ¢,. This
leads to the equation :

geht) == (er) + fr pr be Grt) do ADE)
0

As, after the time ¢ the particle must be somewhere, we finally
have as third equation for f and x:
[rep i) dp = yf (ai, ele eee. ae, CAERE)
0
Just as in § 2 we are able to derive from (XI) a differential-
equation for 7, with this difference however, that now in the liquid
the external force w= 0. *)
So in the liquid f(w, p, t) satisfies the equation:
òf òf
An
BOL Op?
However to determine f we need a boundary-condition. This
may be found from (XII), (XIII) and (XIV).
When viz. we write for (XII):

(XIV)

Ale, t, = t)—X t,) ae
t

Mh

- | SF (&,p,t,) Po (pt) dp AT es NJ (Xi)
: 3 : 0
and make ¢, very small, then the second member of (XI1*) becomes :

1) The forces, that the wall exercises on the particle, are accounted for by the
function x.

652

x
eat). f (rss) dp,
2
0
because, in consequence of the meaning of 7(p,t), this function
for small ¢ only differs from zero when p is small too. Now, while
the limit in the first member exists and is finite, this will also be

the case with the limit of the second member and is therefore:

rf k p,t.) — 7 (p,0 ROD
Lim zo p= Lim | reer dp= Opr) dp=x,
t=0 dt, 0 1,=0 dt, tfo—=0
0 0
where w is a constant.
So it follows from (XII*¢):
Oy (a,t
cae == HIDDE) tt en NE
1

_Now, when we integrate (XIV) with respect to p from zero to
infinite, we find:

OO «
.

OE Te os Le a ee
tend Ak ee
0

In reference to (XIII) and (XV) this gives the boundary-condition:

o() = % fp=0 ner tr ema (ee
Op /—o

From (XIV), with the belp of this boundary-condition and the
beginning-condition, we may determine f. f being found, x follows
from (XIII).

While the functions f and y, used above, are probabilities, we
can test the obtained results only by making a very great number
of observations. This is easily done by bringing not one but many
particles in the liquid. Though we cannot say then how many
particles after the time ¢ will be adsorbed by the wall, we may
calculate the most probable number from which the true number
will differ the less, as there are more particles.

Imagine a cylindric volume, limited only at one side by the
wall that lies at «=O. In this volume are, at the time {5=0 a
great number of particles homogeneously dispersed, so that the
‘number of those particles being at a distance from the wall between
v and wv + dz, is n, dv. Now we ask after the most probable num-
ber of particles, that at the time ¢ will be adsorbed by the wall.

Of the n, de particles at a distance from the wall between # and
vt dr there will probably, after the time ¢, n, 7 («, ¢) dx adhere

—_ <i 4

653

to the wall, so after the time ¢ there will be adsorbed in total
oo
NI if ZX (a, t) du particles.
0
To compute this number it is not necessary to know f. As namely
the beginning-condition for f is less simple, we introduce a new
function :

F (pl) =n, di GF (eae ee en ETD)
0

Integrating (XIV) and (XVI) with respect to ez from zero to in-
finity, it is easily seen that /’(p, ¢) satisfies the same differential-
equation and boundary-condition as f(w,p,t), considered as a function
of p and ¢. Further it follows from (XIII) and (XVII) that the
beginning condition is #’(p, 0) = n,.

The solution of the differential-equation for #’, regarding the
beginning and boundary-conditions, is:

xt
‚Pp t Pe
1-0 nj) XVIII)
B Dn

78
at op
© u

5 P : IDV AID)
F( grog) + (&
5 ADEN
© ( ) 2 i > d
yy (u) — — Cm RUU
Va
0

The number of particles, adsorbed in the time f follows from
(XV):
ora t Xn t

Ni IE (4,1) de — nae a F (@,0;t) dae= «| F (0,t) dt ,
0

0 0 0

in which,

which, after integration, gives:

~

7 Zz

x24
DD t TT 0 BY Sata

n= — —- ze ol 5) Se re ate.” Ge 1) (X1X)
% D n x yi

With this the problem is solved that is mentioned by
V. SMOLUCHOWSKI *) with respect to the fact that this theory was not
in accordance with the experimental results of Broun *). The

» 2

1) F(p.t) is the concentration of the particles at the time ¢ at a distance p
from the wall.

*) H. Weger, Die Part. Diff. Gl. der Math. Phys. ll, p. 95.

3) M. v. Smorvenowski, Phys. Zeitschr. 17, p. 570, 1916.

4) L. Brirvovin, Ann. Chim. Phys. 27, p. 412, 1912.

654

latter has experimented with particles of gamboge in a mixture of
glycerine and water. The number of the particles that was adsorbed
by the wall could be determined by counting them on a microfotograph.

Now, while Brittouin has concluded, that his observations agree
with theory, v. SMOLUCHOWSKI has pointed out the incorrectness of
this method of reckoning and has substituted this by a better one.
Thereby he supposes however that every particle that collides
against the wall, sticks to it, and he further solves the problem by
using the very schematic image of the Brownian movement that is
mentioned in § 1. The result of this computation agrees very badly
with the observations and v. SmoiucHowski thinks that this is a
consequence of the fact, that a particle, colliding against the wall,
does not adhere at once, but on the average has to collide several
times before being adsorbed. So this would mean,that the boundary-
condition has to be altered.

As now is demonstrated by v. SmorucHowski '), we find the
solution of the problem as given by him, from the diffusion equation,
when we use F'(o,t)=0 as boundary condition, while (XIX) is
dedueed with the general limiting-condition :

D ie = KF

So we may expect that by a suitable Elites of x we get a result
that is more in accordance with the experiments of BRILLOUIN.

To be able to decide in how far this ‘is the case we compute from
the data furnished by BrrrLOvIN, 7, and D the latter with the help
of (16). When we now choose a certain value of x, we can represent
nm, graphically as a function of ¢. This is to be seen in fig. 1, in
which the abscissa represents the number m of the adsorbed particles,
and the ordinate, in accordance with BritLouin, the square root of
the time, expressed in hours. Further, for practical reasons, not: the

, x
values of x but those of «= 7 have been written at the curves.
vB

Por’ «== 0. (dS) Wessels a straight line, agreeing with the
theory of v. SMOLUCHOWSKL One sees that the observations of
Bri.LouIN, indicated by crosses, do not at all correspond with them.

Smaller values of x or « give curves that agree better with the
observations, though none of the curves gives an entirely satisfying
result. Probably e=0.003 («=1.5>10~8) comes nearest to the truth,
when we bear in mind, that the observations made after short times

1) M. v. Smotucnowski Phys. Zeitschr. 17, p. 585, 1916.

655

are least to be trusted. Then the influence of small differences of
temperature may viz. still be obserbed; this influence disappears
after a long time. Also, because we can only determine theoretically

12
11

10

0 100 200 300 400 500 600 700
Fig. 1.

the most probable number, the observations after a long time, which
deliver larger numbers, have more importance.

It is not to be denied bowever, that there seems to be a systematic
error. Perhaps this may be sought in the fact of the particles not
being exactly equal, or in a slow change of the electrical double-
layers at the particles and the wall, in consequence of the numerous
collisions. Further observations of the behaviour of colloidal particles
with respect to a fixed wall, will have to decide whether this
explanation is tenable.

A remarkable circumstance is, that the curves practically coincide

656

with the straight line for «=o (a=o), when x or ua are above
a certain small value. With regard to the precision of the measure-
ments all cases in which x >5 x 10-7 (a >0.01) may be treated
as if =o (a—o). If the observations agree with the theory for
x=, we may consequently only conclude from this that x is
larger than this small value.

The quantity x occurring above has the dimension of a velocity
and to know what the values of x mentioned before mean, we have
to find out with what velocity x has to be compared. So we have
to enter into a closer investigation of the physical meaning of x,
also to decide whether x may assume all values between zero and
infinity as has been tacidly supposed up till now.

From the way in which (XIX) is deduced, follows:

dn;

el —xF (0, t) a TAAL Oe ak te case (XX)

: z dn
Now bearing in mind, that en represents the number of the

particles that sticks to the wall in a unit of time, it naturally
suggests itself to compare this with the number of the particles that
collides against the wall. This number, as is known from the kinetic:
theory of gases and liquids, amounts to:

Fae FOV=% FOD), J ie ae

where v is the mean velocity of the particles that is given by the

known formula:
Ug
ye bp ae ne
Nm

Here R, 7’ and N have the known meaning and m is the mass
of the particle. With the help of the values given by BRILLOUIN we
find for the number of the collisions in a second:

“, F (o,f) = 0.1 F (0, t)

From this follows, that of the particles, colliding against the wall
only a very small fraction ¢ sticks. This ¢, that consequently represents
the ne that a colliding particle adheres, is to be found from :

» When we want to take the change of the concentration with the distance from
OF

the wall into account, we have to substitute F'y—=o + A = ) 0 = 0 for Fo = 90,

where A is a length of the order of magnitude of the mean free path of a particle.

As D (Ge) —) = *Fp=o, the ratio of the correction to the used value is

me Le. always very small.

657

BEEN ij (XXIII)
Siete CR VEN ONION RSE RER

So it follows from the observations of BrILLOUIN that ¢ = 1.5 « 107.
As the maximum value of the probability is 1, x can never be
larger than x, i.e. for the particles used by Brittouin x < 0.1. But
x, will never be so small that, when e= 1 or a not too small
fraction, we cannot take «=o, without appreciably changing the
result, so that for these cases the boundary condition F'(0,t) = 0
gives a serviceable result. Strictly speaking the boundary-condition
can however never be F'(0,f)=0, because, then x must be infinite,
which is impossible.

The probability « will be connected with the mechanism of the
collision of a particle against the wall and to get any insight into this,
it is desirable to know e as a function of different circumstances.
Except the experiments of BRILLOUIN, no other experimental data to
determine «, are known to me. .

$ 4. Though the boundary-condition in § 3 is deduced only for
a plane wall, we may undoubtedly also apply it to a curved wall
in the form:

Dy eee EDTA

where the horizontal line denotes values at the wall and » is the
normal directed towards the liquid.

This we may apply to the case of a fixed sphere with a radius
R, surrounded by a liquid in which at the time t == 0 many particles
are dispersed homogeneously (7, per unit of volume). The solution
of this problem is namely used by v. SmMoLucnowski') in his theory —
of the coagulation. Where he however uses as boundary-condition

f= 0, we will here take (XXIV) as limiting-condition. The number
of particles that sticks in a unit of time after some computations
proves to be:

dn; “xR ex,

0 — “lf ; ih as ’ (XX V)
RA) ex, RAD
when we, like v. SMoLucnowskl, only consider times that are large with
2

r
respect to —.
P -D

1) M. v. Smotucnowsk!, Phys. Zeitschr. 17, p. 594, 1916.

658

When e=1 or is at least not too small, we get very approximately
ex, R
———— = 1 and (XXV) becomes:
ex, Rk + D
dn;
dt
as is also found by v. SMoLucHowskKI. If eis however very small, then
the same formulae remain valid as is also remarked by v.S., when
we only multiply ¢ with a constant factor. This factor however is
ex

== 40 BD nier A ON

R

1

——. So, as long as « does not
ex, R + D . ;

become very small, the formula (XXV*) of v. Smor.ucHowski holds .
good. When however e becomes a very small fraction, then the
number of particles that sticks per unit of time also becomes smaller,
so that in this way the slow coagulation may be explained.
Finally I wish to express my sincere thanks to Prof. Dr. L. S. ORNSTEIN
for his kind help and advice at the composition of this communication.

not, as v. S. assumes «, but

Utrecht, April 1917. Institution for Theoretical Physics.

Chemistry. — ‘/n-, mono- and divariant equilibria’. XVIII.
By Prof. SCHREINEMAKERS.

(Communicated in the meeting of October 27, 1917).

Equilibria of n components in n phases at constant temperature.

In the considerations of the previous communication XVII we
have changed the temperature under constant pressure, now we
shall change the pressure, while the temperature remains constant.
Then we find similar properties as in the previous communication, e.g.

“In an equilibrium of components in n phases at constant 7
the pressure is maximum or minimum when a phase-reaction may

7

occur between the phases”.

“When at constant. 7’ the pressure on the turning-line is a
maximum (minimum), then from this turning-line two leaves of the
region go towards lower (higher) pressures and no leaf towards
higher (lower) pressures’.

We see that the figs. 2 [XVI], 4 [XVI] and 7 [XVI] are in
accordance with this.

We call the equilibrium ZE, which occurs under the maximum-
or minimumpressure Pr again Hr and the equilibria which occur
under Pe + AP {|AP>0O when Pr is a minimum, AP <0 when
Pr is a maximum] again E’ and #". The rules a, 6, and c,
which have been deduced in the previous communication now apply
again to the position of those three equilibria with respect to one
another.

In order to examine whether the pressure is a maximum or
minimum, we can use again formula 15 [XVII]; this now becomes

TOV). AB ae Cree ern ()
Herein is:
SAV) ==, +aVit... +4 Ve
consequently the change in volume, which occurs at the reaction:
FE, +4, FP, +... HAB = 0

Just as in the previous communication also now we may apply
this formula (1) to special cases.

660

Equilibria of n components in n phases, between which a phase-
reaction may occur, at variable T and P. The turning-line Ep.

For the equilibrium = F,+ F,...-+ Fn the equations (2) [XVII]
and (3) [XVII] are true. When the equilibrium £ passes into an
equilibrium Ep, then a, y,...2,¥,--- ete. have also to satisfy (13)
(XVII). From those latter equations which we once more mention
here sub (2).

PGES oo Mara: Ape the +1,=0
gg abe eo Sey (2)
a; +s Ys hoh nn =O |

follows one single relation between the variables 2,y,...a2,y,..;
we are able to find this relation by eliminating from (2) 4, ... an.
We shall call this relation, resulting from (2), which we may write
also in the form of a determinant, equation (2).

Now we have n?-+1 equations and „° + 2 variables; conse-
quently the equilibrium Zr is monovariant; it is represented,
therefore, in the P, 7-diagram by a curve, e.g. curve ef in fig. 2
(XVI) and fig. 4 (XVI).

From (2) [XVII] now follow the m equations (7) | XVII], they
are of the form:

— VAPH H,AT +2, [d@,+--J4+y[d@+.--.-J+--- | 3)
. $40Z,+4027,+...=—AK

From (3) [XVII] follow the n(n—1) equations (8) [XVII]; they

are of the form:

da), +4 (2), +...=da),+4@@,+-..=...=AK,

on 4)
dy), +4), +-.-= Ay), ++ dy), +...=...=AK, (

etc. When we differentiate the equation resulting from (2), then we
obtain another relation between Az, Ay,... Az, Ay,... For our

purpose we may find this in the following way. It follows viz. from (2)
Aa, + A2, H.H An =0 | |
a, Oa, +4, AA, t+... a, Ad, + 4, Aa, +... An Amu =O; - (9)
y, MA, Hy, AA, +... + yn Aln + 4, Ay, HH An Btn = 0)
etc. As relation (2) exists between «,y,..., we may eliminate
Ai,...44, from (5). For this we add the n equations (5) after
having multiplied the 1s* by u,, the 2°¢ by u,, ete. Now we may
put :
Wott tay +-.-=0
uut Hug +... = 0

ete. Then we have n relations between the n—1 ratio's uw, ... un;

(16)

661

it is apparent from (2) that we may also satisfy (6). Now follows
from (5) in connection with (6):

a, [us Aw, Hu, Ay, +} + 4, [u, Av, Huy, +...) +
eis | ft, Learnt RN yp

Now we have in (7) the equation sought for; for the ratios
between 4,...4, we have yet to substitute their values from (2)
and for the ratios between u,,..y, still their values from (6).

In order to calculate Sr, we add the ” equations (3) after having
multiplied the 1st by 4,, the’ 2™¢ by 4, ete. By using (2) and (4)
we find:
—TAV)AP+2I0H).AT142004+352(d0'Z)+.=0 (8)
or:

ap _ ECH 5
dln (4)
Herein is:
2(H)=14,4,+2,H4,+...+4, HH.
the increase of entropy, and
Sa Vy IV La, Vee ae
the increase of volume, which occur at the phase-reaction
FHA, FH... + An En = 0

The direction of the tangent to a turning-line Zp is, therefore,
defined in each point by (9) consequently by the same conditions
as a system of n components in » + 1 phases. It appears from (9)
that this curve has a point of maximum or minimum-temperature
when the phase-reaction proceed swithout changeof volume [2(À V)=0};
it has a point of maximum-pressure, when > (4 H) = 0, consequently
when no heat is taken up or given out with the phase-reaction.

Now. we shall examine whether a singular point may occur on
the turning-line; then AP and AT’ have to be of higher order.
For this purpose it is necessary that we are able to give the value
zero to AP and AT in (8) and (4) without all other increments

Ax, Ay, ... becoming zero also.
Consequently we must be able to solve from:
mdr, + y,dy), +...-=—AK
mdr), + y,d(y), +...=—AK Sos
ete. and from:
der =d), Seek AK,
das dj rn Sa Ay | En

48
Proceedings Royal Acad. Amsterdam. Vol. XX.

662

etc. and from (7) the ratios between the increments. In (10) and (11)
the sign d indicates now that we have to differentiate with respect
to all variables, except to P and 7. We now have n* + J equations
between n?—1 ratios of the n° increments; consequently relations
must exist between the coefficients. It follows again from (10) and
(11) that a,y,... have to satisfy (2), which is also the case here. (7),
however, must also be satisfied. When we compare (6) with (10),
then we see in connection with (11) that we may put
sul ts ade) SEN EN
is == AO) =a. =e.
When we substitute those values in (7) then we find:
1d Ze Zo Ae A ZS, 7 ee

(Prof. W. van per Woupr has drawn my attention to the fact
that we are able to express generally in a determinant the conditi-
ons in order that (7), (10), and (11) may be satisfied. We then obtain
the same determinant as that one to which attention is drawn in
the previous communication. We have, however, to add to this
determinant still a series, which follows from (7). The conditions
songht for are then, that all determinants which may be formed
from this, are zero].

The turning-line Er has, therefore, a singular point when (12)
is satisfied, then it follows from (8):
: —ZTAV).AP+ ZAM). AT+AZzOA+....=0

dP
Consequently for ae the same values as in (9). AP and AT itself

are values of the second order; when we express them in one of
the others, e.g. in Aw, then we may write
AP=aAar* +6 Az, +...
AT =a Aar? + 6,Aa,i +...

It is apparent from (9) that a: a, must be—= > (AH): = (AV).
We now give to Ar, the two opposite values + S and — 5; in
the one case we go along curve Er starting from the singular point
towards the one side, in the other case towards the other side of
the curve. It follows for Av, = + $ that:

AP=aS'+ 6S'+... and AT=a,S?+ ),S* +4... (43)
for Az, = 49 that:
AP = aS? BS 4, cand AT SST DSE ee

Consequently AP and A7’ have the same sign in (13) and (14),
curve Hp consists, therefore, in the vicinity of the singular point
S of two branches Sw and Sv with the common tangent Sw; the

663

direction of this tangent is defined by a:a, or by (9). Consequently
curve Hr has a form as u Sv in figs. 1-or 2. In fig. 1 it forms a
turning-point in S, in fig. 2 a cusp. It follows, however, from (13)

ww”
u

ww” u
raad

ie 5

Fig. 1. | Fig. 2.

and (14) that AP: AT is larger in the one case and smaller in the
other case than dP:d7’=a:a,; consequently curve Ep has a
turning-point as in fig. 1.

In our considerations on the region # in fig. 7 [XVI] we
have already used this result; we have viz. drawn there the
turning-line M Sm in the point S with a turningpoint.

In the previous communication we have deduced for an equili-
brium of m components in „» phases under constant pressure:

When (AH) and 2(4d?Z) have the same sign, then 7’is maximum;

When (AH) ,, (Ad?Z) have opposite sign, then 7’ is minimum;

When Z(id'Z)=0 then 7’ is neither maximum nor minimum.

Similar properties are valid for equilibria of n components in n
phases at constant temperature.

Let us assume now, for fixing the ideas, that (2H) is positive
on the turning-line MSm in fig. 7 | XVI]. Then it follows from the
rules mentioned above, that 2(4d?Z) must be positive in each point
of the branch mS and negative in each point of branch MS. In
accordance with (12), however, >(2d?Z) =O in the point S. When
we draw in the figure a horizontal or vertical line through the point
S, then we see that this line does not trace the three-leafed-region ;
under the pressure Ps, therefore, the temperature is neither maximum
nor minimum and at the temperature 7's the pressure is neither
maximum nor minimum.

Equilibria of n components in n phases in the concentration-
diagram.

Until now we have considered the equilibria Kin the P, 7-diagram ;
now we shall briefly discuss their representation in the concentration-

diagram. The composition of a phase, which contains » components,
48*

664

may be represented by n—1 variables; in order to represent it
graphically, we want, therefore, a space with n—1 dimensions.
We now take an equilibrium A= #,+...-+ ¥,,' under the

pressure P,; we call this the equilibrium 4(P= P,). Besides the

n(n —1) variables 2,y,...a,y,.., ete. we then have also the n +1
variables 7’ K K,..., consequently in total n?-+ 1 variables. They
are connected to one another by the „? equations (2) | XVII] and
(3) | XVII]. We imagine now all variables to be eliminated, except
those, which relate to a phase /; [consequently except ajy;.. .}.
Then we keep n—2 equations between the n—1 variables a;y;...
Consequently the phase #; follows in the concentration-diagram on
change of 7 an (n—1) dimensional curve; we call this “curve
F;(P= P.)’. Of course the position of this curve depends on the
assumed pressure P, and it changes with this pressure.
Consequently the pressure is P, in each point of this curve
F;(P = P,). T changes however from point to point; it is maximum
or minimum when a phase-reaction can occur, consequently when
the equilibrium # passes into an equilibrium ZR. |
As the equilibrium E(P = P,) contains n phases, it is represented,
therefore, by n curves /; (P= P,) in a space with n—1 dimensions.
Now we take an equilibrium # at constant temperature 7,; we
call this #(T'= T7,). A phase # of this equilibrium now follows
on change of P a curve F;(T = 7). Of course the position of this
curve depends on the assumed temperature 7, and changes with
this. Consequently the temperature is 7’, in each point of this curve;
the pressure changes, however from point to point and is maximum
or minimum when the equilibrium £ passes into an equilibrium Zp.
Finely we still take an equilibrium of 2 components in 7 phases,
between which a phase-reaction may occur, consequently the equili-
brium Hr. Each phase F; of this equilibrium follows a curve E (B)

in the concentration-diagram. The P and 7 change along this curve:

from point to point.

Consequently we have the following. Each of the equilibria
E(P= P,), E(T=T,) and Eg is represented in the concentration-
diagram by n curves F;; these are situated in a space with n—1
dimensions. When one or more phases have a constant composition,
then the corresponding curves disappear of course. AS we may
change P, and 7’, an infinite number of curves /;( P= P) and
E(T = T,) exist, therefore; one single curve /; (R) exists however.

We now take a point A on curve /i(/); through this point goes
a curve /;(P= Px) and F;(7' = Tx), which touch one another in
the point X. Curve #;(/?= Px) has viz. a maximum- or minimum

tn nn et

665

temperature in the point X; the increments Avr;Ay;... have,
therefore, to satisfy (9) [XVII] and (10) [XVII]. This is, however,
also true for the increments Aw;Ay;... in the point X of curve
Pi Tf x), as this curve has a maximum- or minimum pressure
in the point X.

In the same way the following appears yet also. When the curve
Fi (Rk) has a turning-point S, in-the P, T-diagram, then in the
concentration-diagram the three curves F; (Rk), F; (P=P,) and
F; (1 =T,) touch one another in the point S.

Corresponding points on the n—1 other curves belong to each
point X (or ‘S) of a curve F;(R); consequently the properties
deduced above are valid also for each of those curves.

We shall apply now those general considerations to the ternary
equilibrium H= B+ L+ G, which we have discussed in commu-
nication XVI in connection with the figs. 6 and 7. As this equilibrium
is a ternary one, it may be represented in a plane viz. in triangle
ABC of fig. 6. As B is a phase of invariable composition, each of
the equilibria is represented by two curves. viz.:

E(R) by the curve L(R) and G(R),
A ea PN if L(P= PiyandG (P =P)
lt SEA AED zn ay (Teen

Curve L(R) is indicated in fig. 6 by curve mSM (in fig. 7 this,
curve mSM has a turning-point in S); curve G(R) is not drawn
in fig. 6. Further in fig. 6 we find several curves L(T = 1);
a bed represents viz. a curve L (T= T,) (we have to bear in mind
that Lr ij Tdi; ef gh represents a;curve. (EST),
1Sk a curve LT T;) and ln a curve L(T=T)). The little
arrows indicate the direction in which the pressure increases along
these curves; this is a maximum on branch MS, a minimum on
branch mS of the turning-line. The reader may imagine the curves
L(P=P), G(T=T,) and G(P=P,) which have not been drawn
to be also indicated in fig. 6.

Let us now imagine in tig. 6 to be drawn through a point z
of curve mS M the curves L(T= T,) and L (P= P,); in accord-
ance with our general considerations, those curves must, therefore,
come in contact with one another in the point 2. When we take
the point b on mS M, then curve LZ (7’= 7), which is represented by
abcd and the curve L(P= P,), which is not drawn, touch one
another in 6, therefore, In the point c curve L(7=T,)=abcd

666

and the curve (P= P.) which is not drawn, touch one another,
ete. Consequently curve abcd touches two curves L (P= P,) viz.
in b curve. L(P —/;). and in.ec curve L( P= P‚).

In the point S three curves touch one another viz. curve
L(R)=mSM, curve L(T = Ts) =1Sk, and the curve L(P= Po),
which is not drawn.

Some special cases.

Before we have already drawn the attention to the fact, what
changes have to be made in the conditions (2) and (3) { XVII], when
one or more of the phases have a constant composition. When,
however, all phases have a constant composition, then the conside-
rations must be somewhat altered. Let us take a phases-complex

EEF +... A
in which all the phases have a constant composition. We now may
distinguish two cases, according to the fact whether in this complex

a phase-reaction may occur either never or always. In the latter
case K passes into an equilibrium He and the phase-reaction is

17, +...+4 EH... Han En =0
in which 4,... are independent of P and 7.
The condition for equilibrium becomes now:
444+4,Z,+...+4aZ = 0.
As Z,... depend only upon Pand 7’, the equilibrium is represented
therefore, in the P,T-diagram by a curve. The direction of this
curve is defined by (9).

We may also imagine the case that phases occur with limited
changeable composition viz. phases in which one or more components
have a constant composition and the other components a variable
composition. This is eg. the case when two hydrates A.n H,O
and 6.nH,O form mixed-erystals; then they have the composition
PA.(1—P) B.nH,O, in which P may change from 0to1. We
may also represent the composition of this mixed-erystal by:
aH,O+yA + (1—a— y)B, in which e=n:(n + 1) andy= P:(n+1).

When f, represents a similar phase, in which rz, and y,, are

z
constant, z,u,... variable, then we have to omit in (3) [XVII] aa
zy

0
and = and we must replace them in the first equation (2) [XVII]
Yi

by the corresponding quantities of another phase F,.

Ean

667

We have assumed in our previous considerations that at least
one of the variable phases contains all components; we may imagine
also, however, that this is not the case. Let us take e.g. the ternary system:
A + water + alcohol, in which A is a salt, which is not volatile
and which forms a hydrate A.n H,O. The equilibrium 4 = A + A
.nH,O + Gisthen ternary, but the variable phase, viz. the gas G,
contains only the two components water and alcohol.

For the contemplation of similar equilibria it is in general easier
when wedo not use the general formulas (2) [XVII] and (3) [XVII]
but the conditions of equilibrium, which are true for the special
case. In the communication ‘Equilibria in ternary systems” XVII
I have treated a similar case. I shall refer to this in a following
communication.

(To be continued)
Leiden, Inorg. Chem. Lab.

Physiology. — “Experimental researches on the permeability of the
kidneys to glucose.” By Prof. H. J. HaAmBurcer and R. BRINKMAN.

(Communicated in the meeting of Sept. 29, 1917).

Ill. Tar NaHCO, PERCENTAGE IN THE TRANSMISSION-FLUID }).
a

In a former paper’) on this subject we discussed the reason which
induced us to enter upon a systematical investigation of the effect
which the composition of the Rincer-fluid had upon the retention-
power of the frog’s kidney with respect to glucose. This investigation

TAB LsEy A

Effect of the Ca-concentration in the RiNGER-fluid on the retention of glucose.

] | | ij

Is &£ Reduction of
le = urine
| | LS gid : Retention of
Olo NaCl | Oo NaHCOs | °/, KCl Ke BE q glucose.
| | | H
| | ies | Left | Right |
| | | | |
| | | | | |
O16. 4) 02020. "1307010 0.000 0.098 0.095! 0.096 0
| | | | v. porta
0.6 | 0.020 | 0.010 0.001 0.10 0.095 0.094 0} renalis
| | | | ligatured
0.6 | 0.020 [0.010 | 0.002 |0.090/0.092/ 0.088; 0
076141107020 | 0.010 | 0.004 '0.090!0.090| 0.090 0
0.6 0.020 | 0.010 0.006 |0.098/0.10 | 0.096! 0
In like manner rising with 0.002 %) CaCl,6aq. to 0.012 0/, no retention of sugar.
0.6 |: 0.020 {0.010 0.012 |0.098| 0.080} 0.082 0.017
0.6 | 0.020 | 0.010 |. 0.014 0.098 0.076. 0.075 0.022
0.6 0.020 0.010 0.015 ‘0.09 | 0.060 0.061 0.030
| | | v. porta
0.6 0.020 0.010 | 0.016 ‘0.096/'0.066 0.068 0.030, renalis
| | | ligatured 3)
0.6 | 0.020 [0.010 | 0.018 |[0.102/0.102/ 0.06 | © 0

0.6 | 0.020 {0.010 | 0.020 (0.098|O.10 | 0.10 | 0

1) A more detailed account will be given in the Biochemische Zeitschrift.

2) HAMBURGER and BRINKMAN: Verslagen van de Koninklijke Akademie v.
Wetenschappen van Jan. 27, 1917, p. 944.

>) Obviously the results relate to the glomerulusproduct. Cf. the above-mentioned
paper p. 946,

ma a or

669

brought to light that, apart from other factors, the permeability of
the glomerulus membrane is, to a high degree, dependent on the
CaCl, percentage of the transmission-fluid and further that this
permeability is also affected by the KCl and NaCl concentrations.

We subjoin a series of experiments which demonstrates the effect
of calcium and which was not published in our tirst paper. (See
Table I, preceding page).

This series of experiments was carried out in February— March
tene)

Evidently glucose-retention took place only when the CaCl,-
concentration varied between 0.012°/, and 0.016°/,, that is to say
the Ca-concentration has its strict limits, and admits of but little
variation. Of the + 0.1°/, of glucose in the transmission fluid at
most 0.03°/, of glucose was retained.

To determine the effect of Potassium in the transmission-fluid the
KCl concentration was modified while the NaCl, NaHCO, and CaCl,
remained the same. Increasing quantities of KCl were therefore added
to the fluid composed of NaCl 0.6°/,, NaHCO, 0.02°/,, CaCl, . 6 aq.

TABLE: ft.
Effect of the KCl concentration in the RINGER fluid. Experiments of March 1917.

‘ | | | 0/0 | pecuceey of | Reduction urine | 8e
ls NaCl, Of, NaHCO 3} 0% KCl | CaCl, 6aq.| Pues! ! RE 2 ij
| | | | fluid. _ Left Right me
06 0.020 0.000 0.015 0.095 | 0.070 0.072 0.025 »)
0.6 0.020 0.000 0.015 _ 0.090 | 0.068 0.069 0.021
0.6 0.020 0.002 0.015 0.095 0.070 0.070 (0.025
0.6 0.020 0.004 0.015 0.115 0.092 0.088 0.025
0.6 | 0.020 0.006 | 0.015 0.10 0.098 0.10 0
0.6 0020 0.006 0.015 0.085 0.080 « 0.084 0
0.6 | 0.020 0.008 0.015 0.10 0.070 0.070 0.03
0.6 | 0.020 0.010 0.015 | _ 0.098 0.070 | 0.070 0.03
0.6 | 0.020 | 0.014 | 0.015 | 0.10 0.065 0.070 0.03
0.6 0.020 0.016 | 0.015 010 0.070 \ 0.075 0.028
0.6 | 0.020 | 0.018 | 0.015 | 0.092 0.094 | 0.092 0
0.6 | 0.020 | 0.020 0.015 | 0.080 0.070 | 0.010
0.022 0.015 | 0.098 0.095 | 0.098 | 0

0.6 0.020

1) It is desirable to know whether summer- or winter frogs are used for the
experiments. Cf. Verslagen van Jan. 27, 1917 p. 949.

670

0.15°/, and glucose 0.1°/, Some of the results are given in Table II.

It becomes evident that if from the Rinerr-fluid containing the
right CaCl,.6 ag. concentration viz. 0.015°/, (see Table I) all the
potassium is omitted, glucose is still retained viz. + 0.02°/,; if the
KCl is increased to 0.005 —0.006°/, all the glucose passes through;
at a further increase to KCl 0.008—0.017°/, the maximum amount
of glucose is retained (0.03°/,); at higher KCl concentrations the
retention decreases again.

Hence we see that the potassium ts not absolutely necessary *); the
chief function of the K in the transmission-fluid is probably to
balance an excess of Ca.

It appeared from out last paper that the NaCl concentration also
affects the results.

The composition of the transmission-fluid thus found could, however,
be hardly looked upon as the optimum one since from a transmission-
fluid with 0,1°/, of glucose at most only 0,03°/, was retained. And
this value decreased even when the glucose-concentration in the
transmission-fluid was lowered. The reason why also experiments
with glucose-concentrations below 0.1°/, were made was due to the
fact that the normal glucose-concentration of frog’s blood varies between
0,03 and 0,06°/,. If we used a glucose-concentration of 0,03—0,04°/,
not 0,03°/, was retained but at most 0,015°/,. The glucose retention
was, consequently, dependent on the glueose-concentration of the trans-
mission-fluid; a decrease in the glucose-concentration causes a
corresponding decrease in the retention. In spite of a great number
of experiments, we did not succeed in obtaining a glomerulus-filtrate
which contained no glucose. But even if the transmission-fluid
contained 0.1°/, of glucose and moreover the abovementioned
favourable Ca- and K-concentrations were used, it not unfrequently
occurred, more especially in summer when the frog’s have less
vitality, that little or no glucose was retained. Probably the most
effective composition of the Ringer-fluid had, therefore, not been
arrived at.

Increase of the usual NaHCO,-concentration.

We, therefore, attempted to improve upon our transmission-fluid.

1) The fact that it is necessary in the transmission-fluid for the heart need not
surprise us, because the heart uses potassium in its muscular labour; things are
different for the kidney, which is mainly a passive though complicated and sensitive
living filter. Besides, the same arterial blood must supply all organs and provide
every one with what it needs. Thus it may be understood that the most effective
artificial transmission-fluid need not have the same composition for every separate
organ.

671

The effect of CaCl,, KCl and NaCl had already been determined;
it only remained to examine the effect of the NaHCO, concentration.

Since RINGER it has been generally assumed that in artificial
transmission fluids NaHCO, is indispensable. The present researches
have likewise shown that it cannot be dispensed with in the
transmission-fluid. One of the functions of NaHCO, consists as we
know in maintaining a very slight actual alkalinity of the body-fluids
which would otherwise, owing to the continual formation of acids,
pass into an acid reaction. Like serum protein it acts as a buffer:
hence we also speak of a tampon or moderator. Besides a specific
HCO’,-action may have to be assumed (E. Laquwur).

It appears already theoretically that a concentration of NaHCO,
0.01 °/, is too low to act as a sufficient buffer. We shall revert to
this later on, in connection with other more theoretical considerations.

Ringer himself added 5 ec. of a 1°/, NaHCO,-sol. to 100 ce. of
fluid. Trropr even used 0.1 °/, of NaHCO,. But 0,02°/, of NaHCO,
is the rule in Rrnegr’s fluid. *) That the usual concentration of 0.02°/,
of NaHCO, is too slight for frogs’ kidneys could be determined ex-
perimentally in the following manner.

If namely to a transmission-fluid composed: NaCl 0.6°/,, NaHCO,
0.02 °/, CaCl, 6aq 0.015°/, some neutral red’) is added, the colour
-of the indicator is orange yellow (slightly alkaline), which corresponds
with [H-]=1.10~-%. It is necessary to use boiled out aq. dest. and
to prevent the absorption of CO,. Now we need only shake this
fluid for a moment with air or lead it through an india rubber
tube and the colour turns to pink, which points to an acid reaction
of [H]>>110 7. If, however, one is careful in preparing this fluid
then one succeeds in keeping it slightly alkaline. Now if this fluid
is transmitted through the kidney, the latter becomes evidently acid,
which is manifested by the red colour it assumes and also the meta-
bolism preducts which have passed into the urine, colour the indicator
red after some time. We have made the oxygenation in the experi-
ment as intense as we could to eliminate metabolism products of
greater acidity as much as possible, without succeeding, however,
in keeping the reaction of the urine neutral.

What is the reaction of the normal urine of the frog?

It is not difficult to obtain it by squeezing out the bladder of

') Cf. e.g. Bayuiss: Principles of General Physiology, 1916, p. 211.

ZWAARDEMAKER and his collaborators also use this concentration of 0.02 0/,.
(See eg. Proceedings 1916, April 28, May 27, Sept. 30).

2) The reader will be aware that the colour of this vital indicator is at
[H-] = 1.107 pink, at [H'] =1.10—8 orange-yellow and at [H']=1.10—9 yellow.

672

the animal. It appears then that the liquid is slightly alkaline.
The same can be demonstrated in the following way. If1 ec ofa
saturated watery neutralred solution is injected into the back
lymphsac, an investigation, half an bour after, brings to light
the following facts: skin, muscles, brain and spinal cord are
pink, intestine yellow and pink, depending on place and degree
of peristalsis, but the urine is yellow, and is, therefore, though only
slightly, alkaline. When we followed the practice hitherto adopted
and transmitted a Ruinegr fluid containing 0.02°/, of NaHCO,, the
quantity generally used for the heart, then the urine after some
time became permanently pink, that is to say, acid. Hence we see
that the protective value of NaHCO, 0.02°/, is not great enough.
At the same time it appeared that the acidity of the urine and the
diminution or loss of the kidney’s retention-power to glucose went
hand in hand. As an example we add the following experiment.

Transmission from the aorta with a sol. of NaCl 0,6 °/5, NaHCO, 0,02 °/,
KCI 0,01 °/,, CaCl,.6 ag. 0,016 °/,, and glucose 0,098 °/j, saturated with Os; no
india rubber tube was used; the colour of the solution is sorange owing to
neutralred. The first urine is yellow and has a reduction of 0,06 ®/,; the latter
red, its reduction being 0,090 °/,, in other words: now that the urine has become
acid, the kidney is found to have lost the power of retaining glucose.

The obvious course was now to increase gradually the NaHCO,-
cone. of the transmission-fluid. It was raised to 0,090 °/,. Now we
had therefore a transmission-fluid of the following composition :
NaCl 0.6 °/,, NaHCO, 0.90°/,, KCl 0.010°/,, glucose + 0.1 °/, and
had to discover the suitable CaCl, 6 aq. concentration, Table III
contains the results of these experiments.

In the first place it is observed that a much greater quantity of
glucose is retained than before. It amounts to no less than 0.079°/,.
But this requires a concentration of CaCl, 6 aq. of 0.024— 0.030 °/,.

Below this concentration and above it little is retained. The CaCl,
6 aq. cone. necessary for a maximum glucose retention has, there-
fore, risen from 0.015 °/, (Cf. tables I and II) to 0.024 °/,—-0.030°/,.
This need not surprise us for the concentration of ions of Ca is
repressed by NaHCO, and the ions of Ca are an important factor.
It may, therefore, be said that an increased NaHCO, conc. in a
transmission-fluid with +010°/, of glucose raises the maximum
retention from 0.03 °/, to an average of 0.06 °/,.

Further increase of the NaHCO, concentration.

We did not stop short, however, at this increase of the NaHCO,

673

concentration. We have namely made the titration-alkalinity of our
transmission fluid equal ‘to that of frog’s serum.

FA BEE UE
Effect of an increased NaHCO3-concentration.
Transmission from the aorta of NaCl 0,6 %, NaHCO; 0,090 °,,,
KCI 0,010, and CaCl, . 6 aq. 0,020 0/,—0,050 9/,; colour of transmission-
fluid orange-yellow caused by neutral red.
(Experiments of June—July 1917).

| Reduction |

CaCl».6aq. transmission Reduction oe | Colour urine !)
: urine. glucose |
, fluid. | | |
0.020 | 0.100 0.098 . 0.020%, | colourless
| |
0.020 0.105 ‘ 0.080 | 0.025 | colourless
NOSE: ie: VOMODE al o o0-0BO te Ar Ot ORE. «| cdlourless
|
0.024 0.115 0.062 0.053 _ light-yellow
0.025 | 0.100 0.040 0.060 © light-yellow
0.025 0.10 0.041 | 0.059 | yellow
| | |
0.026 = 0.115 0.058 | 0.057 | yellow
OL0282 2 OCS 0.064 0.051 | greenish yellow
0.028 0.111 0.052 | 0.059 | yellow
0.030 0.105 0.042 | 0.063 light yellow
0.030 0.105 0.026 0.079 | light yellow
0.030 0.105 0.031 4-7 v0078 light yellow
|
0.031 0.115 0.102. 7} 1 OLON colourless
| |
0.032 0.115 0.10 | 0.005 | very light yellow
page |
0.032 0.115 0.091 | 0.024 | colourless
0.035 0.10 0.089 0.011 | colourless
first light yellow
0.040 0.102 < | 0.090 0.022 | afterwards
colourless
0.045 0.098 | 0.075 0.023 | colourless
0.050 0.098 = 0.080 0.018 | colourless

For this purpose frog’s serum was titrated with '/,, normal tar-
taric acid with neutralred paper as an indicator, according to the
method of Snapper *). 1 cc. of defibrinated only slightly haemolytic

') For the meaning of this column see p. 677.
4) J. SNAPPER: Biochemische Zeitschrift 51, (1913), 88.

674

frog’s serum required 0.85 ec. of *,, normal tartaric acid. The
titration alkalinity of frog’s serum is, therefore equal to that of a
0.034-normal or a 0.285 °/, NaHCO,-sol. We have, therefore, given
a NaHCO, cone. of 0.285°/, to our transmission-fluid; to prevent
a resulting increase of osmotic pressure the NaCl cone. was lowered
to 0.5°/,. Now again it was obvious that the suitable CaCl, 6 ag.
cone. would have to be raised again, as the cone. of the free ions
of Ca would again be repressed.

The result will be found in Table IV.

The maximum quantity of glucose begins to be retained at CaCl,
6 aq. 0.030°/,; so this concentration is still somewhat higher than
if NaHCO, 0.9°/, is used (then the cone. of CaCl, 6 aq. was, as
appears from Table III 0.024 °/,).

Table III shows that if the CaCl, 6 aq. rose to above 0.030 °/,,
the retention of glucose began to decrease. In Table IV, however,
when a higher cone. of NaHCO, was used, this was not the case;
even if the CaCl, 6 aq. conc. rises to 0.080 °/,, the glucose-retention
remains pretty well invariably high viz. an average of 0.07 °/,. One
will be inclined to assume that this is due to the fact that in the
latter case the most favourable conc. of ions of Ca is brought about
automatically.

Indeed when through the RiNGer fluid containing 0.285 °/, of
NaHCO, and 0.080 °/, of CaCl, 6 aq., oxygen is led for some time,
a precipitate is formed of CaCO, *). The following physico-chemical
exposition will make matters clearer.

[Ca “J HOO, == K, or Ga : Pad eae
HCO; Bikes

The latter formula teaches that the concentration of the free ions
of Ca is only dependent on the cone. of the ions of H and those
of HCO, or also that the amount of Ca salt makes no difference,
when [H° | and [ HCO,” | are present in a certain suitable pro-
portion. Hence we see that there must be a buffer-system for ions
of Ca in this fluid.

To sum up: in order to maintain a proper concentration of ions
of Ca it appears that not only the cone. of ions of HCO',, but also
that of ions of H is of importance. A satisfactory regulation of the
cone. of ions of H is not so easy to arrive at in our circumstances,
where, if the kidney is to function well, the fluid must be saturated

1) We have invariably observed at the ultrafiltration of bloodserum that the
clear filtrate becomes troubled when shaken with air, owing to the formation of
CaCO;, which was kept in solution by CQ,.

675

with O,; this regulation will have to be further investigated.
An experimental confirmation of our view was obtained by deter-

EA BL EIS

Effect of a still greater increase of the NaHCO,-Concentration.

Transmission of NaCl 05 ©, KC1 0,010 0, NaHCO; 0,285 °,,
CaCl,.6 aq. 0,028— 0,080 "lo, glucose + 0,1 0/,.

All solutions have been made again in boiled out water and saturated
with oxygen.

(Experiments of July 1917.)

Se a neemen eneen
|

| Reduction | Reduction Dittevence
0„CaCl,.6 aq. transmission bis oa (Retention
| fluid. | of glucose).
ee ee EN
0.028 | 0.14 | 0.130 | 0.015
0.028 0.091 | 0.076 | 0.015
0.030 0.091 | 0.038 0.055
0.030 0.092 0.027 0.065
0.032 | 0.088 0.066 0.022
0.032 | 0.091 -| 0.056 0.035
0.034 | 0.098 0.042 0.056
0.034 0.091 0.040 0.051
0,036 “| “0.098 | “0,052 0.046
0.036 0.125 (0,040 0.085
0.038 | 0.125 |.» .0:035 0.090
0.040 = 0.106 = 0.031 0.075
0.042 | 0.105 | 0.029 0.076
0.044 | 0.105 | 0.045 0.060
0.048 0.105 | 0.035 0.070
0.050 0.105 | 0.031 0.074
0.052 0.105 | 0.053 0.052
0.056 0.105 | 0.050 0.055
0.060 | 0.115 0.062 | 0.053
0.064 0.115 0.058 0.057
0.080 0.115 | 0.041 0.074

minations of the electric conductivity of the system NaHCO, and
CaCl,. This will be discussed in a subsequent paper.

676

But the state of things in the Rrineer fluid is still more complicated
than in the system CaCl, and NaHCO,, especially because the fluid
contains a rather considerable quantity of NaCl. This renders the
determination of the conc. of free ions of Ca rather difficult.

It seems that the equilibrium of the system CaCl,, NaHCO, and
NaCl, so important to life, has hitherto not been studied. We intend
to revert to this subject later on. At any rate we have now obtained
a transmission fluid of which, of the 0.1°/, of glucose, on an
average 0.07 °/, is retained, and in which automatically that conc.
of ions of Ca sets in which causes + 0.07 °/, of the OA °/, of glu-
cose to be retained.

It is this transmission-fluid which has enabled us to obtain a wrine
Free from sugar, which had hitherto been found impossible. Accord-
ing to Bane') frogs’ blood gives a reduction which corresponds to
0.03 —0.05 °/, of glucose. We accordingly found in September a
reduction value of 0.04— 0.06 °/,. Now the question was: will the
kidney be able to keep back all the sugar from a Rineer-fluid of
the above composition and containing 0.05 °/, of glucose. The una-
nimous result of our experiments proved that this was indeed the
case,

All glucose was likewise retained, even when the Rincer-fluid con-
tained 0.06°/, of glucose.

Now it will be of importance to determine to what pitch hyper-
glycaemia can be raised before glycosuria sets in, in other words
how much sugar the kidney can bear. This question will be treated
in a subsequent paper.

CONCLUSIONS:

The fact that by modifying the composition of the Rrncxr-fluid the
colloid state of the glomerulus-epithelium can be regulated in such
a manner that it either admits or does not admit sugar, seems to
us of great importance, for now it has become superfluous to assume
as an explanation of physiological glucose-retention, that substances
are found in the serum which keep back the glucose in colloid
compounds and that the glucose cannot pass through the glomerulus-
membrane in that form. That this supposition is no longer necessary
will afford satisfaction, after Micrarris and Rona and also ABEL by
dialysis-experiments and we by ultra-filtration have found that
parchmentpaper and ultra-filters of celloidin allow all glucose in the

1) J. Bane, Der Blutzucker 1913. J. F. BERGMANN, Wiesbaden.

677

serum to pass, which as we observed before ') is not a strict proof
that sugar cannot be present in a composition with a serum-compound
which can pass through these two membranes, but not through the
glomerulus-epithelium. Now, however, it has been demonstrated that
the glomerulus epithelium can keep back the glucose as such.

We have evidently to deal here with a new form of permeability :
cells, here the glomerulus-epithelium, allow salts to pass, but not
the likewise crystalloid sugar, which under the circumstances is
highly useful; for thus a substance necessary for our nutrition is
kept in circulation. As far as we can see we find ourselves con-
fronted here by a phenomenon not observed before. The intestinal
epithelium and likewise the pleura and the peritoneum are perme-
able to salts as well as to glucose; the red blood-corpuscles of most
animals are impermeable to salts and to sugar both *).

Finally we wish to point out another fact. An examination of
table III makes it evident that although the transmission-fluid con-
tained neutral red, mostly a colourless urine was obtained in these
cases, therefore, the neutral red had been kept back by the glome-
rulus-epithelium. That the urine was free from neutral red appeared
from the fact that neither the addition of acid nor that of alkali
to the urine caused colouring.

Hence we may assume that if the NaHCO, concentration is high
enough, the glomerulus-membrane is impermeable to the colloid
neutral red. If the NaHCO, cone. amounts to only 0.02 °/, then the
glomerulus-filtrate becomes red, because the Rineer-fluid, on being trans-
mitted, grows too acid. That this is really only a glomerulus-produet
appears when for instance the porta renalis is ligatured, for then the
urine-secretion through the tubulus epithelium is prevented (cf. our
first publication) *}. According to table III, however, the urine in
some of these experiments was yellow, but this colouring originated
from neutral red, which is excreted by the tubulus epithelium ; this
is confirmed by the experiments of HöBer and KöÖNIGsBERG, to which
we shall have to refer again presently.

Our experiments also throw a light on the contradiction between
the results of the experiments of Gerzowirscn *) and those of Héprr ‘).

1) Cf. our first paper in Verslagen Jan. 1917.
2) Only some blood-corpuscles viz. those of man, of the monkey and of the dog
seem, to a certain extent, permeable to sugar. ;
3) Meeting of January 27, 1917.
< 4) GerzowiTscH: Zeitschr f. Biologie, 66, 391, (1916).
5) HöBer und KöniesBere: Pflügers Archiv 108, 324, (1905).
49

Proceedings Royal Acad. Amsterdam. Vol. XX.

678

GERZOWITSCH namely dissolved neutral red in ordinary Ringer fluid, the com-
position of which is not stated, and obtained at arterial transmission a “coloured”
glomerulus filtrate; he does not say whether the colour was red or yellow orange.
Hoper on the other hand injected neutral red into the back lymphsac, and on
examining the capsule microscopically he saw a “colourless’* glomerules filtrate.

Probably the contradiction may be thus explained: GeRZOWITSCH uses “eine für
den Frosch physiologische RinGER-lésung’’. This must have been one, containing
0,02 °/, NaHCO, (see above p. 672) and this gives an acid i.e. a pink urine.
HöBEr and KöNrosBERG, however, worked under physiological conditions, for normal
blood flowed through the frogs; only some vital colouring-matter had been
introduced into the back lymphsac. The glomerules fillrate was, just as with us,
colourless, but in its passage through the ducts it took up neutral red, which was
secreted by the epithelium of the tubuli. This would be in conformance with the
yellow colour of the urine, which we obtained when under practically physiolopica
conditions a suitable Rin@er-fluid was used.

SUMMARY.

1. If, the usual Rincur-solution containing 0.02 °/, NaHCO,, passes
through the kidney, then it is found that of the 0.1°/, glucose
at most 0.03 °/, is retained (table II) in however favourable a manner
we may vary the Ca and K percentage.

A considerable increase of the glucose retention may be attained
if the NaHCO, conc. of the transmission fluid is raised from 0.02 °/,
tv 0.090 °/,.

2. Experiments with neutralred teach that the cause of this phe-
nomenon is connected with the reaction of the transmission-flucd.

If the alkalinity of the latter (i.e. its protective value) is so slight
that on being transmitted it is easily acidified, then the urine formed
gives an acid reaction (neutralred becomes pink) and little or no
glucose is retained.

If, however, the NaHCO, cone. is raised to 0.090°/, then the
artificial urine remains alkaline (neutral red remains. yellow) and of
the + 0.1°/, of glucose about 9.06 °/, is retained.

In order to obtain this favourale result, however, the Ca-concen-
tration, the most effective cone. of which amounted hitherto to
CaCl, 6 aq. 0.015 °/, (see table 1) must be raised to 0.024— 0.030°/,
(table Lil), but not higher. That the CaCl, cone. should have to be
raised if the NaHCO, conc. is increased need not surprise us, since
an increase of the NaHCO, cone. impedes the dissociation of the
CaCl, and a sufficient concentration of ions of Ca in the transmission
fluid is of great importance.

3. The kidney can retain even more than 0.06 °/, of glucose if

679

the NaHCO, conc. is raised to 0.285 °/, i.e. the conc. which corre-
sponds to the titration-alkalinity of frogs’ serum. But then again
more CaCl, must be added, at least 0.030°/, (table IV).

4. It is remarkable that otherwise than in the experiments in
which NaHCO, 0.09 °/, was used (table III), now that the conc. is
0.285 °/,, an addition of more CaCl, 6 aq than 0.030 °/,, even of
much more, does not impair the retention (table IV). There are
reasons to assume that the most favourable conc. of ions of Ca brings
itself about automatically, when more CaCl, is added. The RiNGeER-
sol. in the latter case, when of + 0.1 °/, of glucose upon an average
0.07°/, was retained, was composed as follows: NaCl0.5 °/,, NaHCO,
0.285. °/,, ACI 0.01 °/,, CaCl, 6 ag. 0.040 °/,.

5. If the transmission-fluid contained 0.05°/, of glucose, the
average concentration found in frogs’ blood, then a sugarless urine
was obtained. This was even the case when the RINGER-sol. contained
0.06 °/, of glucose. |

6. This result seems important to us from a physiological-clinical
and from a general biological point of view; from a physiological-
clinical point of view, because the retention of sugar by the kidney
has now been reduced -to a question of permeability, so that the
supposition that glucose is bound by one of the serum substances
(sucre virtuel of Lépine) has become altogether unnecessary. Evidently
the chemical composition of the transmission-fluid determines the
state of the glomerulus epithelium, and consequently the permeability |
of the membrane to sugar. The results are important from a general
biological point of view, because we have to deal here with a new
form of permeability, one in which cells under physiological condi-
tions, though easily permeable to salts, are impermeable to the likewise
crystalloid glucose, a form of permeability hitherto unknown and
very useful under the circumstances.’

Groningen, September 1917. Physiological Laboratory.

49*

Physics. — “A paradox in the theory of the Brownian movement’.

By Prof. P. Exrenrest. (Communicated by Prof. H. A. LorEntz).

(Communicated in the meeting of September 29, 1917).

§ 1. Suppose a small sphere to be suspended in a fluid. Suppose
further that its Brownian movement is observed and consider a
moment when the sphere has a rather great velocity, e.g. upward,
May we expect the surrounding fluid to move with the sphere?

Prof. J. D. van per Waats Jr. and Miss A. SNETHLAGE have
recently shown'): 1st how the answer to this question is connected
with Ernstgin’s theory of the Brownian movement,

2d that the statistical theory of the molecular movement demands
that such a common motion does not exist. This theory implies namely
that for a given place and velocity of the suspended sphere and
for a given configuration of the surrounding particles of the fluid, equa/
and opposite velocities of these particles are always equally probable.

The authors cited remarked already that this result is somewhat
paradoxical; and therefore subjected it to. a detailed discussion.

In the following we shall make the paradox still more acute by
considering an analogous question for an extremely simplified model.
This will show us that two closely connected material points m,
and m, of which our model will consist, may on one hand possess
mutually independent velocities, while on the other hand they still
accompany each other (because of the close connexion).

§ 2. We consider two material particles with the masses m, and
m, and the following properties:

1st Both are constrained kinematically to move along the axis of X,

2nd By a field of force their distance can never become greater
than D?), where D may be small compared with the displacements
of the two points in the course of time along the X-axis:

eel SD ne Bale ee
3rd Let this pair of points be placed in an infinite extension

1) These Proc. p. 1322.
2) Let e.g. m, be a shell in which m remains enclosed.

681

filled with a gas which is in molecular-statistical equilibrium. Its
molecules collide both with m, and m,.
Considering a corresponding canonical ensemble, the expression

20 (ay > dm uy? mg uy?

const. e kT de,de,du,du, . . .'. . (2)

(where x,, #,, uv, and uw, are the coordinates and velocities of the
two particles, and ® (,, z,) is the potential energy of the force
holding them together) gives the number of individuals of the en-
semble for which w,, 2,, wu, and u, have their values between
specified infinitely close limits. For given values of 2,, 7, and especi-
ally also of wu, (2) gives the same number of individuals in the
ensemble for equal and opposite values of wu, i.e.: equal and
opposite values of u, are still equally probable, u, is “independent”
of u,. On the other hand the Brownian movement will in the
course of time cause great displacements along the X-axis. At the same
time. the points will remain close together in virtue of the inequality
(1). This is the paradox mentioned at the end of § 1.

§ 3. Let us first leave aside the molecular-statistic side of the
problem and put the following purely kinematical question. During
a long time @ the two points m, and m,, may be conducted along
the z-axis in an arbitrary way, only restricted to the conditions that

a. the inequality (1) shall remain valid;

6. the distance between the final and original positions of the pair
of points may be great compared with D. This implies that m,
accompanies’ m,. Now we ask: does this imply that always the
mean with respect to the time

ii *
ed duw 0. EEE OE Ee 2)

will be positive, or is it possible that the integral can be zero or
eventually even negative ?

The sign of the integral (3) indicates in a natural way in how
far the two points move more in the same or in opposite directions.
But we can point out that for the above described motion of the
pair of points m,, m, the inequality (3) need not be fulfilled. This
will become evident by an example of a case for which the integral
becomes negative. In fig. 1 the two zigzag lines represent a possible
x, t-diagram of the two points m,,m,. We see that the conditions

Fig. 1.

a and 5 are fulfilled and that still w, and uw, have continually
Opposite signs, so that the integral (8) becomes negative.

§ 4. Now we have:
4 uu, = (u; oh! dw) le a ae

The sign of wu, u, is determined by which term is the greater
of the two.

When the motions of the pair of points obey the equipartition
theorem, w, wu, is just equal to zero. (See the appendix).

From the above it is evident that a motion of the pair of points
is possible, in which they remain close together and at the same
time travel through great distances, while still at every moment
the velocity uw, is “independent” of w,. The paradox mentioned in
§§ 1 and 2 proves thus to be apparent only. Therefore there is no
objection against Eistemwn’s assumption that a suspended sphere during
its BROWNIAN movement imparts its motion to the surrounding fluid in
the same way as in the case of a systematic motion under the influ-
ence of a constant force.

$ 5. In the positive proof however that Einstein's assumption
follows from the fundaments of statistical mechanics we meet with
the following difficulty: Let us demand, (to stick to our example), that
the inequality u, <{ u, < u, te exists ;

1st at the instant 4, ;

2°¢ also during the interval from ¢,—7 till t,; and let us ask
what can be said of the occurrence of different values of w,. In

683

„

the first case we have to take from the canonical ensemble an
easily defined sub-ensemble M,, in which w, has equally often equal
and opposite values (and therefore is “independent” with respect to
u,). For the second demand a more closely limited sub-ensemble M,,
has to be selected from the mentioned ensemble M/,. It is however
„hardly possible to determine J7/,,. Still this should be done in order
to decide whether the distribution of the values of w, in it does or
does not agree with ErNsrriN’s assumption.

!

APPENDIX.
Let
me, m,r, 3 mt Met,
Per de == CT ALK sij m, Jm, =M,
de, dz,
nies Bt, ae oe
Minne « NE k
u, mone +4) > tian oe Rn ae)
Then:

Di

M? . en Sin ic
3 (m,u,° 35 mu”) FE nt Eq Ms Gs. M 2 19s (2, ih m,)§ =

8m,m,
Let p,,p, be the momentum corresponding to the coordinates
Qi qe, then we have

ON dann Mg,” aa: (m, as m,) 119s} Te en NE: OG (3)
yes

- M* en Ren

qaP2 — FE Mg, ag (m, Sr br Fs Gah a) lace a a he (y)
rig

and because of the equipartition theorem the mean values with respect
to time of (3) and (y) are both equal to #7, so that their difference
M*

oa ee Ua ys Scored. wake
ae 3 Ya § \ )
On the other hand («) gives:
J ype
Er 4 — (q,° oe q2 ) Se | SN Rae Oar eA: (e)
m,m,

From (d) and (¢) combined we find: u, u, = 0 (q.e.d.).

Physics.

— “A new Klectrometer, specially arranged for radio-active

Investigations”. Part Il. By Miss H. J. Former. (Communicated

by Prof. H. Haga).

' (Communicated in the meeting of September 29, 1917.)

In Part I, communicated by Prof. H. Haga in the meeting of
May 30 1914, the following brief description was given of the elec-

trometer which is represented in figure |

once more:

Fig. 1.

and reproduced here

The apparatus consists of two
separate spaces, viz: the measur-
ing space c; a brass cylinder of
small height, and the ionisation
space proper f: a brass cylinder
of volume 1 litre; the two cylin-
ders are insulated from each other
by ebonite.

In the measuring space c is
the metal needle 4, supported in
the middle by a second needle
d, insulated by amber; 6+d
together form the conductor,
which is charged by the ionisa-
tion current.

In c is also found the very
thin aluminium strip a, which a
few mm. above 6 is fastened to
a thin metal rod with mirror,
suspended on a Wollaston wire,

which is fastened to a torsion head insulated by means of ebonite.
Through a perforation in the amber and in the ebonite a rod /can
be brought in contact with the needle d.

In this way, a, 6-+d,c and f can therefore be separately brought
in a conductive connection with a storage battery or with the earth ;
c rests on a brass bottom plate to which legs are fastened which

685

support the apparatus. Here follows a more detailed description of
the arrangement *) *).

Description of the apparatus:

It is illustrated by the following reproductions :

Fig. 2 has been taken from a photo representing the apparatus
as if seen somewhat from the top.

Fig. 3 is a vertical section through the plane of one of the rods,
viz: the capacity rod /,, while Fig. 4 shows the ebonite disc in
horizontal section, the two last being at a third of their real size.

Starting from the central part of the apparatus we shall find that
the needle d is exactly in the middle; it is made of platinum and
consists of two parts, the lower part of which slides tightly
into the upper; in the middle this upper part is surrounded by a
very small cylinder, which contains two small cavities in order to
promote the good contact of the needle with the two rods which
touch it on two sides; viz. the charging rod /, and the capacity
rod /,. (Cf. Figures 3 and 4). The needle d is insulated by ambroid
(dotted in Fig. 3) which consists of two cylindriform pieces, to the
lower of which the needle is fastened, while the upper part, provided
with two wide perforations for the rods, enclose the needle loosely.
The ambroid is entirely surrounded by a brass tube serving as
guardring, in order to prevent loss of charge of the needle as well
as electrostatic disturbances upon it. This tube also consists of two
parts: of a small lower cylinder with a thick outer rim at the bottom
containing screw-perforations (in Fig. 4:1) by means of which the
guardring with the ambroid and the needle can be fastened to the
surrounding ebonite plate. The lower cylinder fits loosely in the
excavation of the ebonite and slides tightly in the upper part of
the guardring, which, besides two wide perforations for the rods
l, and /,, is yet provided with a screw-perforation for the third rod
L,, which brings the guardring to the potential value desired. (Cf.
Fig. 4). The ebonite plate itself is fastened to the lower side of the
brass bottom plate 72 of the measuring space c by means of three
brass screws and nuts (Fig. 4:2; Fig. 3: the screws to the left of

!) The electrometer was constructed in the workshop of the Physical Laboratory
at Groningen by Mr. H. J. Sips, who, with great devotion, surmounted in such a
masterly way the many difficulties which arose when he performed his task.

Mr. D. A. Vonx, as chief of the workshop, in many respects gave also valuable
indications.

2) The Instrument-Manufacture and Trade late of P. J. Kipp and Sons, (lim.)
Delft (Holland) is willing to construct the apparatus described here on sufficient
demand.

686

the middle); the nuts are pressed against the ebonite by screws.
Besides, the lid & of the ionisation cylinder f is attached to the
ebonite plate; for this purpose the ebonite plate is perforated in six
places, so that from the top countersunk screws can be driven into
the lid; they will penetrate the lid only half way (Fig. 4:3). Six
clamping screws cause the upper edge / of the ionisation cylinder
to press closely against the lid. At the top the brass ionisation
cylinder is partly shat off by the lid, partly by a small brass basin
consisting of two parts, with an opening in the centre for the
needle; bayonet closure unites these two parts, as well as the outer
part with the lid. The basin is to hold CaCl,, which has to
protect the ambroid from moisture; at the same time the needle
in the ionisation cylinder, except for the small space of air mentioned
above, is thus quite surrounded by metal, which is desirable for the
measurement of ionisation. The bottom of the cylinder consists of a
separate brass plate, pressed against the rim of the cylinder by
clamping screws. As to the upper part of the apparatus, the chief
part certainly is the brass measuring cylinder c,. the dimensions of
which are chosen in such a way as to cause the upper needle to
undergo a thorough damping during its motion. This cylinder fits
into a ring-shaped groove of the bottom plate and is provided with
a broad rim with two circular slits through which two serews with
notched heads fasten the measuring cylinder to the bottom plate.
This enables the measuring cylinder to move over a rather large
angle. (Cf. Fig. 2). In order to be able to check the state of things
inside the cylinder and to see whether the upper needle is in the
right position, there are in the walls of c, diametrically opposed,
two oval openings, covered with celluloid, which correspond with
two openings of the same size in a second brass outer cylinder,
which is revolvable, so that during the measurement the inner space
can be entirely shut off by metal.

The bottom plate 4 of the measuring space c is fastened by
screws to brass legs, which support the apparatus. These rest on a
triangular wooden base with levelling screws fitted with ebonite
insulating toes. .

To the lower needle 6, which slides tightly into an excavation
of d,a definite position can be given with the help of a scale made
on the bottom plate of c.

The upper needle consists of a small aluminium strip, 0.05 mm.
thick, and is fastened to a thin aluminium rod with mirror and
mirror-supporter ; the suspension consists of a thin platinum Wollaston
wire.

SS

687

The needle-system is arranged so as to let vibrations, which act
on the system from without bring about as little disturbance as
possible *). In order to obtain as much symmetry of inertia as possible
with respect to the suspension wire, a disc of aluminium of the
same size was affixed behind the mirror by way of counterweight.
Moreover particular care was bestowed on the shape of the thin
connecting hooks between rod and wire; finally the planes of mirror
and needle were placed perpendicularly with each other *. With
these precautions a fine, restful motion of the needle could be
successfully obtained. The brass tube with glass window that encloses
it, is fitted at the top with an ebonite torsion head and with an
arrangement which makes it possible to raise or lower the upper
needle without making it turn round.

The following still requires to be said about the arrangement
how and the way in which the three rods (cf. especially Fig. 4)
can touch the needle or guardring through the ebonite: rod /, is
the one that, touching the lower needle 6+ d, brings it to the
desired potential before measuring; at the beginning of the measure-
ment this contact is stopped and the charge of the ionisation enrrent
can be carried to the then insulated needle. The rod is made of
brass and fitted with a subtle platinum point, in order to make sure
of a good contact with the platinum needle. In order to bring about
the insulation of the needle from the observer’s place behind the
telescope the rod has been placed in a brass tube which can be
pushed tightly into the ebonite; this tube has been shut off at both
ends by small brass covers each with a round opening through
which the rod can pass freely without much friction. Round about
the rod between the outer small brass cover and a thicker part of
_the rod a steel spiral spring has been placed which is tightened
when the rod is drawn out. Now, the arrangement is chosen in
such a way as to make a weight which hangs by a cord over a
pulley draw the rod out and consequently break the contact with
the needle, whereas by raising the weight the spring reestablishes
this contact; this raising and lowering of the weight can be brought
about from a distance by means of a cord over a pulley.

Rod /,, which brings the guardring to potential, is fastened in
an ebonite tube, which has been fixed by means of an enclosing
small brass cylinder into the ebonite perforation. Rod /, does only
duty when capacity has to be measured (Harms-method); it is made
of brass with a platinum point; it is insulated from the enclosing

Zij Cf. a.o. H. E. J. G. ov Bors and H. Rusens: Wied. Ann. 48 p. 236, 1933.
2) In Fig. Ill these planes were put parallel to each other for clearness’ sake.

688

brass tube by two small pieces of ambroid; round about this we
have once more a brass cylinder as mentioned above.

At the ends of the rods terminals are affixed in order to fasten
the required connecting wires to the storage battery; also the screw
of the torsion head, the bottom plate z and that of the ionisation
cylinder f possess such screws, in order to bring the upper needle
a, the measuring space c, and the ionisation cylinder f to the
potential desired.

The arrangement of the electrometer having been explained in
this elaborate way, some particulars should now be added in relation
to some special purpose for which the apparatus has to be employed.
If, namely, one wishes to use it for measuring the radio-activity of
emanations, the ionisation cylinder must be exhausted, and therefore
it must be possible to close it hermetically. Without taking particular
precautions this cylinder would communicate by various ways with the
air outside; among others along the axle and the walls of the ambroid
cylinder; to prevent this, the needle in the lower ambroid cylinder
has been cemented air-tight, while between this and the upper rim
of the lower cylinder of the guardring a ring-shaped cavity has
been filled with piceine (cf. Fig. 3). Thereupon, in order to prevent
leakage along the lower rim of the guardring and then along the
screws or to the centre, a rubber ring was inserted (in Fig. 3 the
first ring mentioned from the centre) which fits closely in a ring-
shaped groove in the ebonite, cut a little outside screws 1, and
prevents the air to enter. In the same way a second rubber ring
on the inside of screws 2 (ef. Fig. 3) prevents leakage from the
cylinder along these screws or to the outer rim. At the bottom the
guardring g, besides having a wide outer rim, still possesses a narrow
rim turning inside, to prevent the ambroid, in consequence of
difference in pressure of air, from being pressed inward. Further
the closure of the cylinder at the top (by the lid XK) is brought
about in the same way as at the bottom (by the brass bottom plate)
viz: by means of rubber rings.

In the wall of the ionisation cylinder are two hermetically closing taps
of glass for the filling or exhaustion of air or emanation. With all these
precautions it appeared to be possible to bring the pressure inside
the cylinder down to 2 mm. with the pump (GAEDE's new single
barrel air-pump), while only after three days it was raised one mm.,
which is quite sufficient for the purpose we have in view.

If, however, the measurement must be done with regard to solid
substances (direct method), which one must be able to exchange
quickly and in which renewal of air should be avoided as much

Miss H. J. FOLMER. “A new Electrometer, specially arranged for radio-active
Investigations.” (Il).

Proceedings Royal Acad. Amsterdam. Vol. XX.

Miss H. J. FOLMER. “A new Electrometer, specially arranged for radio-active Investigations.” (Il).

Proceedings Royal Acad. Amsterdam. Vol. XX.

rw = 8?

689

as possible, then the bottom plate must be replaced by a ring (ef.
Fig. 5) which bears on the lower side in two places diametrically
opposed, two flat brass rails, along which one can slide a deepened
bottom plate with the sides dovetailed, which forms the bottom of
the ionisation cylinder. A second plate, fashioned in the same way,
can be slided along the same ways and replace the first. The ring
is pressed against the cylinder by two clamping screws.

Some particulars on insulation and arrangement.

In order to make sure that the ambroid really insulates the lower
needle, several experiments were still made; thus a tension of +10
Volt was given to the guardring, f brought to the same tension as
6: (O Volt) in order to avoid an ionisation current, then 5 insulated,
so that the charging of 5 could only be the consequence of a transition
of charge from the guardring via the amber to 5. With a sensitive
state of charge the needle displacement amounted to no more than
1 a 2 mm. per minute. If we take into consideration that in measur-
ing, the differertce of tension between 6 and the guardring is very
small — the latter is kept at V—O — and that the rise in potential
of 6 amounts during the measurement only to a small fraction of
a Volt, this will sufficiently prove how excellently the ambroid in-
sulates the needle, and that the leak it causes is of no account.
As to the arrangement of apparatus, storage-battery ete., it is such
as to make it possible to perform all the manipulations necessary for
the preparation of the measurements from the place at the telescope.

First of all we find here within the observer’s reach the storage-
battery from which our wires start, in order to bring a, 6, c, and
f to potential. The connection with a, 6, and f is direct, as these
conductors are always charged to a potential given by a whole
number of accumulators; c on the other hand receives exactly that
potential wanted to bring the needle back again to its untwisted
state after having charged a. Therefore the desired potential is
obtained by means of an adjustable laboratory rheostate working
as a simple type of potentiometer through which a small current
is carried of an accumulator, whose one pole is in connection with
a storage-battery. Looking through the telescope at the position of
the needle, one can at the same time regulate the tension at will
by adjusting the rheostate.

If, in this way, some state of charge has been given to the appa-
ratus, and / brought to potential, then the measurement can be
started simply by insulating 5 from a distance with the assistance
of the pulley-system described above.

690

Theory of the apparatus.

Of late years numberless new electrometers have been constructed
which, for the greater part, possess great sensibility and are to be
considered as modifications of two of the principles, known until
now; viz: that of W. THomson’s ‘‘quadrantelectrometer’ and the
principle realised in the “HANKEL-BORNENBERGER” electrometer. To the
first belong among others the measuring instruments of: DoOrpzALEK)),
Minty”), HorFMANN ®), Parson‘), to the second principle those of
Lutz and EDELMANN ®), Erster and Guiter*), Wor‘).

Besides the part of the measuring system which is charged to
the tension to be determined, there are also in all these electro-
meters two conductors, which are kept at constant potential during
the measurement. The electrometer described here possesses, it is
true, this latter quality, but yet cannot be reduced to any of the
principles mentioned; in shape it somewhat resembles the antique
measuring instrument of KonnrauscH-DeLLMAN *), which also has a
cylindrical measuring space with two metal needles. As these needles,
however, are charged together to the tension to be determined and
in consequence repel each other, so the similarity. spoken of here
is not mentioned with regard to the principle of. measuring, but
only with regard to the exterior of both instruments and the system |
KOHLRAUSCH-DELLMANN has to be looked upon more as a realisation
of the simple gold-leaf principle, while torsion has been made use
of at the same time. The electrometer which concerns us here,
however, strives after the combination of the following conditions:

1. Simplicity in the arrangement of the system (Cf. I, pp. 22 and 26).

2. Great sensibility by making use of the small torsion of thin wires.

3. Utilizing as much as possible the lines of force which arise
through addition of charge to the system for the motion on the
movable conductor. ;

As to the third condition, in communication I the motives were
already indicated why 1 thought better to abandon entirely the
principle of the quadrantelectrometer ’) (cf. 1 p. 26); at the same

1) F. DoLezaLEeK, Ann. d. Pbys. 26, p. 312, 1908.

3) C. Mürry, Phys. Z. 14, p. 237, 1913.

3) G. Horrmann, Ann. d. Phys. 52, afl. 7, p. 665, 1917.

4) A. L. Parson. Phys. Rev. N. S. Vol VI. p. 390, 1915.

5) CG. W. Lurz, Phys. Z. 9, p. 100, 1908.

6) J Exster and H. Gerven, Phys. Z. 10, p. 664, 1909.

7) THeop. Wurr, Phys. Z. 15, p. 250, 1914.

8) Pogg. Ann. Bd. 72.

%) The drawback of the horizontai wing-surface holds for the measurement
of a definite quantity of charge, of course not of fixed potentials.

691

time light was thrown upon the fact that the advantage of the
system with regard to this had been obtained by the fact that the
lines of force which undergo a change by addition of charge to the
system, act especially on the one vertical side of the upper needle,
i.e. will especially cause a moving couple. Let it be added that,
the arrangement once having become such as to show an asymme-
trical character, there had been introduced into the system at the
same time the principle called by Horrmann the “Labilisierungs-prinzip”,
which, considered by itself will yield “under certain conditions’ a
decisive advantage in relation to the sensibility of the apparatus, as
will become clearer yet from the following considerations.

For this it is necessary to account for the behaviour of the electro-
meter in the various states of charge, as these are realized before
the measurement takes place. Suppose that one of the states of
charge has been given to the apparatus, e.g. a + 12 Volt, 6 O Volt,
c — + Volt, wire untwisted, angle of needles 30° (cf. 1). The equilibrium
then arising is shortly due to the following: in consequence of
charging ato 12 Volt — if 6 and c are still supposed to be 0 Volt
yet — a greater density of lines of force arises between a and 6
than between a and ec, in consequence of a slighter distance between
a—b in relation to that of a—c; on account of this a resulting
electric couple will act on a, which can be compensated, however,
by a second electric couple in an opposite direction, which takes
place in consequence of charging c to negative potential (—4 Volt);
for 6 acts as a screen to the lines of force a—c (cf. also I pp. 24
and 25). If it is supposed that the needle has been suspended in
this condition without torsion, then, in theory at least, the equilibrium
will continue; however, this is an unstable equilibrium, for with
constant potentials at a slight turning of the upper needle into the
direction that will decrease the angle with 6, the density of the
lines of force between a and 5 will increase and a resulting electric
couple will arise according to the direction of the movement. The
equilibrium will also be unstable in the opposite direction, because
with an increase of the distance a——d, there will be a decrease in
the influence of 6, and the influence of the negative of c will be
preponderating. The torsion of the suspension wire, however, can
yield a couple, if sufficiently large, which brings about a stable
equilibrium; the torsion, however, can have a value too, so much
so that it does not counterbalance these above mentioned electric
couples, in which case the equilibrium remains unstable. Given a
definite height and angle of needles there will exist two conditions
by which these cases are determined: |

692

1. the value of the force of the torsion, consequently the thickness
and nature of the suspension wire;

2. the state of charge in which we can distinguish high and low
states of charge, meaning that the potential a—h can be large or
small thus e.g. the state (a -+ 30 Volt, 60 Volt, c—8 Volt) is a
higher state of charge than (a + 12 Volt, 60 Volt, c—4 Volt).

The meaning of condition 1 is sufficiently clear in itself; as
regards 2, if, with a definite wire one will always try to realise
higher states of charge, in the end the equilibrium from being
stable: will always become unstable. For with a higher state of
charge, the density of the lines of force between a and 6 and of
course also those between a and c (for there is a greater potential
difference between a and c at the same time) will always be greater;
then also the electric couple that occurs will increase in consequence
of a supposed slight displacement of the needle, so that the torsion
couple with a sufficiently high state of charge will finally be
unable to compensate this electric conple any more. Of course the
stable conditions are used for measurements; yet it is practically
possible to approximate the unstable equilibrium with torsion, in
which case interesting phenomena occur; if e.g. under otherwise
equal conditions one increases the state of charge continually, it
will in the end be impossible to give a fixed position to the needle
in or near the equilibrium (untwisted); seemingly the needle is at
rest, yet it gradually approaches the lower needle, at first with
slight velocity, but steadily increasing so that the image of the scale
will shortly disappear from the field of the telescope; the parallel
position of the needles is almost reached. Such conditions are meant
in communication I, when we say that the needle “turns”.

As the behaviour of the electrometer has been accounted for in
the various states of charge, there still remains to examine the
behaviour in the various states of measurement where we shall also
be able to observe the importance of the ‘‘Labilisierungsprinzip”’.
To the conception of capacity, which is connected with it, I should
like to give the meaning of what Purear and Wotrr') call the
“total” capacity of the conduetor, which conception is used by them
for cases similar to those considered here and for which the con-
ception of capacity, as MAXWELL gives it, is not sufficient; for the
conductors a and c are not at O Volt, nor does the angle of the
needles remain constant.

Further I wish to distinguish between (cf. communication I p. 29):
1: useful, and 2: injurious capacity ; meaning by useful capacity that

1) J. DEL Putgar and Tu. Wourr. Ann. d. Phys. 30, p. 700, 1909.

693

part which influences the motion of a; by injurious capacity that part that
lacks this influence and therefore means only disadvantage here. As
to the measurements, the sensibility will rise together with the increased
states of charge. In order to bring this out, we have to compare e.g. the
measurements of the two following states: State 1 : (+ 8, 0, —4) Volt
and state II: (+ 14, 0, —6) Volt, and suppose 5 to be insulated, so that a
supposed ionisation current gives a positive charge to the lower needle
b+d(f at +80 Volt e.q.). What then will be the effect with in both
cases a definite equal increase of charge? The potential value of 6
will rise, the number of lines of force between a and 5 decrease
at the same time so that the upper needle recedes from the lower.
In consequence of the fact that a, which is positively charged, recedes
from 6, part of the negative charge induced on 6 by a in the state
of equilibrium, will be set free and therefore will be spread over
the now insulated system 6+ d. The influence of this will, first of
all, consist in a decrease of potential of 6, causing the potential
value of 5 to increase less than would follow from the addition of
charge considered by itself (ionisationcurrent). This influence is felt
strongest in the case of II, where in consequence of greater potential
difference between « and 6, a greater quantity of induced charge is
set free, so that the potential decrease, caused by this will be greater.
But from yet another point of view we shall have to look at the
part played by the induced charge: as soon as the latter spreads
from 6 over bd, this im vtself means again a decrease of lines of
force between a and 5, i.e. a cause of motion on the needle. The
result of this consideration therefore is that the displacement of the
upper needle a will only partly be the consequence of a direct
addition of charge from the ionisation current, but at the same time
must be partly considered as the consequence of the displacement of
induced charge in the system.

Where, therefore, this displacement is greatest i.e. in case II, the
motion on the needle will be strongest and consequently the sensi-
bility of charge greatest.

In communication | the above mentioned explanation has been
worded somewhat differently ; it was namely said there, that the
greater sensibility in I! would be the consequence of the fact that
the increase of the capacity of 6 + d would especially mean increase
of useful capacity in the system, by which the sensibility of charge
will increase. In order to elucidate this more clearly, I shall return
to what was communicated above; that, namely, by displacement
of induced charge, owing to the motion of a, the rise in potential
of 4 turns out smaller than might follow from the addition of charge

| 5,0,

Proceedings Royal Acad. Amsterdam. Vol. XX.

694

considered by itself. When, however, through this influence a definite
addition of charge causes a smaller rise in potential than would be
the case without it, this in itself means that the capacity of 6 +d
has been increased by it. It is this increase of capacity that is of
great advantage to tbe sensibility of charge in the system, and that
because this increase of capacity means increase of the useful capacity
of h 4d. Let us first imagine the phenomenon in two phases to take
place the one after the other (practically they act at the same time).

I. the positive increased charge is distributed over 6+ d; the
upper needle describes the corresponding angle.

Il the negative induced charge which is set free by this move-
ment near 6 spreads over 4 + d.

The ‘effect of I and IL together then comes to the same, as if I
had only taken place, but at the same time a greater part of the
added charge goes to those places of 6, where the induced charge
of case Il was to be set free. In my opinion it is clearly shown
in this way that the influence of phase Il really consists in an in-
crease of the useful capacity of 5. In the state of charge (+14,0, —6)
that useful capacity is yet more increased by the movement than
in the state of charge (+ 8,0, — 4); from which follows that in
that state the sensibility of charge will also be greater, because, as
was already said in Communication I, the sensibility of charge will
of course be all the greater according as a greater part of the added
charge causes a change in the lines of force between « and 6, which
is attended by motion. Ultimately there are limits to the use of an
ever increasing state of charge; when e.g. the case of unstability as
described above, sets in. An approximation as closely as possible to
this unstable equilibrium is of course the most favourable condition
for the sensibility, because then (see above) the motion of the needle
will chiefly be the consequence of displaced induced charge and for
a small part only of the increase of charge itself.

The “Labilisierungsprinzip’ also occurs in some other electro-
meters, a.o. in those of Werrr, Wison (Kipsystem), whereas the
electrometer of Horrmann aims at such a favourable variation of
the binant-electrometer that the mentioned system was introduced
into the system for that very reason, for which purpose the shape
of the needle was chosen in a particular way. Yet the conception
that the ‘Labilisierungsprinzip” in itself would guarantee the greatest
possible sensibility in a system, is not correct according to my
opinion; with the application of this principle the ratio of useful to
injurious capacity will also remain of the greatest importance. If
e.g. one just imagines that in the system 6+ d, d possesses a great

695

capacity (i.e. injurious) there will be wanted near the unstable state
of charge a very slight increase of charge for the variation in the
course of the lines of force between a and 6 considered by them-
selves; consequently for the motion of the needle; yet at the same
time the needle d will yet require much charge for itself; or: though
the useful capacity strongly increases in the unstable state, yet the
injurious capacity must be seriously taken into account. This draw-
back makes itself felt especially when that injurious capacity in the
system undergoes the influence of the unstability as well as the
useful capacity. If it is supposed e.g. that a consists of a horizontal
disc, then part of the capacity of 6 will relate to lines of
force going from 6 to the horizontal plane of a (i.e. injurious
capacity). Also these Hnes of foree will then undergo a change
in consequence of the movement; that is to say, that also this
injurious capacity will constantly increase while passing to
higher states of charge, which in itself is disadvantageous. From
this consideration it follows that the advantage of unstability is still
bound up with another condition; the optimum is implied in the
following rule: the greatest sensibility of charge in a system will be
obtained by a maximal approximation to the unsiable state; at the
same time the amount of the injurious capacity will have to be as small
as possible and by no means to undergo the influence of unstability.
As regards further the capacity of the whole system together,
peculiar relations may crop up in the case of change of the latter.
We have noticed already that with a positive increase of charge the
induced charge, which is displaced by the movement of the needle
arrests the increase of potential in the system. Thus it may occur
that the increase of potential is compensated by that very influence,
Le. the system would then possess an infinitely great capacity ; this
will occur among others when the injurious capacity of the system
possesses a small amount of capacity. If one passes on to higher
states of charge then a positive increase of charge will even bring
about a decrease of potential i.e. a negative capacity for the system.
As to the electrometer described here, I think I have obtained
favourable results in relation to the consideration given here. Ex-
periments are arranged for in order to become more acquainted
yet with the ratio of useful to injurious capacity in the various
states of charge in this system, which cannot yet however, be con-
sidered as being put an end to; also about the influence of the
thickness of the suspension wire and modification in the shape of

the needle a closer investigation is still in preparation.
Physical. Laboratory of the University of Groningen (Holland),

DO

Physiology. — ‘On the ventricle-electrogram of the Frogs’
heart.” By Dr. S. pe Boer. (Communicated by Prof. G. van
RIJNBERK. *)

(Communicated in the meeting of October 27, 1917).

It has appeared from my former investigations that we can admit
the width of the R-oscillation of the ventricle-electrogram as a
measure for the velocity, with which the impulse is transmitted
through the ventricle.*) A decrease of the velocity of impulse-
transmission is expressed by a widening of the R-oscillation, whilst.
a narrower R-oscillation betrays an increase of the velocity of
impulse-transmission through the ventricle.

If now we want to trace the influence that the velocity of
impulse-transmission has on the shape of the ventricle-electrogram,
then we can consequently conclude from the width of the R-
oscillation, whether the impulse is transmitted with more or less
velocity through the ventricle. Now we can make the velocity
of impulse-transmission decrease by exciting an extra-systole of
the ventricle at an anticipated moment of the heart-period. If
we wish however to compare the electrogram of such an anti-
cipated extra-systole of the ventricle with those of the normal
periodical systoles of the ventricle, then the requirement must be
satisfied, that this anticipated ventricle-systole is brought about by
an impulse, that reaches the ventricle along the atrio-ventricular
systems of connection. The place where the impulse hits the ven-
tricle at such an anticipated ventricle-systole must be the same
as at the normal periodical ventricle-systoles. Only then we can
make a comparison. Otherwise the modification of the shape
of the ventricle-electrogram might be attributed to the fact,
that the impulse proceeded from another place of the ventricle (e.g.
at the surface of the ventricle as at extra-stimulation of this part of
the heart). We apply consequently an extrastimnlus to the auricle
at an anticipated moment of the heart-period. After the extrasystole

1) These investigations were likewise communicated in the meeting of the
Biological section of the Genootschap ter bevordering van Natuur- Genees- en
Heelkunde (Physiologendag) held on the 20th of December 1917.

% Zeitschrift für Biologie Bd. 65 Seite 128 and Journal of Physiology. Vol 49
page 310.

697

of the auricle, obtained in this way, the impulse continues along
the atrio-ventricular systems, and causes an anticipated ventricle-systole.
We can compare the electrogram of this systole with those of the
normal periodical ventricle-systoles. It had appeared to me already
during my investigations into this subject in 1914), that the electro-
grams of those anticipated ventricle-systoles showed R-oscillations, of
which the width, compared with those of the normal periodical
ventricle-systoles, had increased. At the same time I stated that
the T-oscillations of these anticipated ventricle-systoles had changed in
a negative sense. *)

We may expect, that after the compensatory ‘pause during the
post-compensatory systole the velocity of impulse-transmission had
increased, and a decrease of the width of the R-oscillation of the
electrograms belonging to it proved indeed, that this was the case.
The T-oscillation of these electrograms has changed in a positive
sense (a negative T-oscillation of the periodical ventricle-systoles
had decreased, a positive T-oscillation bad increased).

Modifications of the velocity with which the impulse was trans-
mitted through the ventricle were consequently expressed by the
width of the R-oscillation and by the dimension and the direction
„of the T-oscillation. During the last year I continued these
investigations and systematically observed the changes that occurred
in the ventricle-electrogram, when I modified the velocity of
impulse-transmission. This continued investigation consists of
3 parts.

1. In the first place I caused the velocity of impulse-transmission
to decrease by poisoning with digitalis or antiarine. Before the
poisoning first a photogram was made, and then, whilst the
poisoning continued, constantly with definite pauses a photogram
was made, till the halving of the ventricle-rhythm set in. As
the velocity of impulse-transmission suddenly increased again
after the halving of the ventricle-rhythm, directly an other photo-
gram was made. In this way I could compare the electrograms of
the frogs’ hearts before the poisoning with those that were made |
after the poisoning, and even before disturbances of rhythm set in,

1) Zeitschr. für Biologie, Bd. 65, Seite 428, 1915.

2) By the change of the T-oscillation in a negative sense is meant, that a
positive T-oscillation of the electrograms of the periodical ventricle-systoles decreases
during an anticipated ventricle-systole or changes into a negative T-oscillation.
If however the T-oscillation of the periodical ventricle-systoles is already negative,
then an increase of it during an anticipated ventricle-systole means also a change
in a negative sense,

698

The ventricle-electrograms produced after the halving of the ventricle-
rhythm were compared in the first place with the electrograms
taken immediately before the halving, and at the same time with
all the previously registered ventricle-electrograms.

2. A second series of experiments was made with frogs’ hearts
in which, after the application of the mentioned poisons, halving of
the ventricle-rhythm had set in already. This halved ventricle-rhythm
was thereupon converted into the normal twice as quick one by
an induction-stimulus, as was mentioned in my previous essays ’).
The normal ventricle-rhythm was converted again into the halved
one. In this way [| obtained in one photogram the ventricle-electro-
grams of the normal ventricle-vhythm and those of the halved one.
During the normal ventricle-rhythm the impulse is transmitted
much slower through the ventricle than during the halved one,
because the number of ventricle pulsations in the first rhythm is
twice as great as in the second. Occasionally a spontaneous modifi-
cation of rhythin of the not-poisoned heart was registered.

3. In a third series of experiments anticipated ventricle-systoles
were excited in the not-poisoned frog’s heart by applying extra-
stimula to the auricle. | caused then in the beginning of the irritable
ventricle-period and at a later period anticipated ventricle-systoles.
As I explained already before, the electrograms of these anticipated
ventricle-systoles could be compared with those of the periodical
ventricle-systoles. The electograms of the anticipated ventricle-systoles
were also mutually compared. During the ventricle-systoles that
were excited in the beginning of the irritable ventricle-period the
impulse was transmitted slower through the ventricle than during
the ventricle-systoles that were excited at a later period of the irri-
table ventricle-period.

These 3 series of experiments produced me a rich material for
the study of the influence of the velocity of impulse-transmission
on the shape of the ventricle-electrogram. I shall first discuss these
3 series of experiments successively, guided by some photograms,
and afterwards communicate my conclusions in a theoretical expla-
nation, and add to these conclusions a few consideration concern-
ing the signification of the views obtained for the electrophysiology
of the heart.

I. Comparison of the ventricle-electrograms of frogs’ hearts before
and after the poisoning with digitalis.

The experiments were made in the following manner. The frog

1) Archives Néerlandaises de Physiologie de l'homme et des animaux Tome I,
p. 271 et 502,

699

was extended on a cork-plate, and thereupon the heart was laid
bare in the usual way and suspended at the point. The oscillations
of the lever were photographed on the sensitive plate beside the
electrograms that were obtained after placing one unpolisarable
electrode on the heart point and one on the auricle. The time was
likewise indicated in all photograms in */, second. The experiments
‘of the second and the third series were arranged in the same way,
but for these a stimulator was moreover placed against one of the
parts of the heart, and the moment at which the stimulus was
applied, was indicated on the sensitive plate by a signal that was
linked in the primary current-circuit of the induction-apparatus *).

In this series of-photograms as well as in the two following ones
we shall trace in the first place the width (duration) of the R-oscil-
lation, then the extent and the direction of the T-oscillation. We
call a T positive, when its direction is equal to that of the R-oscil-
lation, and negative when it is opposite to the latter. Thereupon
we consider the line of connection between the R- and the T-
oscillation. When this line of connection is removed in the direct-
ion of the R-oscillation, it rises, it lowers, when it is removed in
a direction opposite to that of the R-oscillation. In this communi-
eation I shall consequently explain more accurately of the photo-.
grams represented only these 3 parts of the ventricle-electrograms.

In order to avoid occupying too much room for the figures, I
shall restrict myself to reproduce five photograms, one taken before
and four after the poisoning with digitalis dialysate (Gouaz).

In Fig. 1 the suspension-curves of a frog’s heart are reproduced before
the poisoning and likewise the electrograms (deduction auricle- point).
The T-oscillation is positive, the line of connection between the R- and
the. T-oscillation is above the line indicating the position of rest of
the string. Then I inject under the skin of the thigh 12 drops of
digitalis dialysatum. Another photogram is taken under equal condi-
tions fifteen minutes after the injection (Fig. 2). If we compare the
width of the R-oscillation of this photogram with that in Fig. 1,
we see that it has considerably increased. This teaches us that
the velocity of impulse-transmission through the ventricle has
decreased. The T-oscillation is still positive, but has become very
little and the line of connection between the R-oscillation and the
T-oscillation coincides now almost with the position of rest of the
string.

1) When the primary current-circuit was closed the signal moved downward,
when opened upward. The closing induction shocks were blended off, the opening
induction shocks directed towards the frog’s heart,

700

Fig. 3 has been taken 15 minutes after Fig. 2. The width of the

increased very much'). The T-oscillation has

has still

R-oscillation

gative and the line of connection between

now become strongly ne

1) The time has not been reproduced in this photogram, but the velocity of the

fall of the plate was the same as in the former photograms,

701

below the position

T-oscillation lies now

r

the

and

he R-oscillation
of rest of the string.

The R-

The T-oscillation is

3.

ig. 4 was again photographed 15 minutes after Fig.

Fi

tion has now become exceedingly wide.

a

oscill

702

now very strongly negative, and the line of connection between
the R-oscillation and the T-oscillation has descended more than in
the former photogram.

Fig. 5 was taken 15 min. after Fig. 4. In the mean time the
rhythm of the ventricle has halved, but after every large
ventricle-systole still an abortive systole of the ventricle occurs.
This abortive ventricle-systole gives a little nearly triangular electro-
gram (a). During the halved ventricle-rhythm the velocity of
impulse-transmission through the ventricle: has again considerably
increased. In accordance with this fact the R-oscillation has become
again much narrower. The T-oscillation is still negative, but has
become considerably smaller than in the former photogram. The
line of connection between the R-oscillation and the T-oscillation
lies for a part somewhat above the position of rest of the string.
If we compare Fig. 5 and Fig. 3, then in Fig. 5 the R-oscillation is
narrower than in Fig. 3. Im accordance with this fact the T of _
Fig. 5 is likewise smaller than that of Fig. 3, and the line of
connection between the R-oscillation and the T-oscillation in Fig. 5
lies at a higher level than in Fig. 3. After the application of the
two poisons mentioned above these results were constantly obtained
by me. As long as the poisoning continues, and still before the
halving of the ventricle-rhythm has set in, the velocity of the
impulse-transmission through the ventricle decreases. The width
of the R-oscillation increases accordingly, the T-oscillation changes
in a negative sense and the line of connection between the R-oscil-
lation and the T-oscillation descends’). As soon as halving of the
ventricle-rhythm has set in the velocity of impulse-transmission
increases again; the width of the R-oscillation decreases, the T-
oscillation changes in a positive sense, and the line of connection
between the R-oscillation and the T-oscillation rises ®).

Il. Artificial and spontaneous modifications of rhythm.

If we poison a frog’s heart with veratrine, digitalis or antiarine

1!) If after the. peisoning ventricle alternation appears. then the proportions
through the partial ventricle-systole during the little ventricle-systoles are of course
different (vide report of the Physiologendag, 20 Dec. 1917). More extensively about
this subject afterwards.

2) From Fig. 1 to Fig. 5 included the width of the P-oscillation increases through
the poisoning, whilst the auricle-rhythm remains constant. The width of the R-
oscillation has consequently decreased in Fig. 5 in consequence of the halving of
the ventricle-rhythm, but the width of the P-oscillation has increased, as tbe rhythm
of the auricle has remained unaltered.

703

the rhythm of the ventricle halves after some lime, because the

duration of the refractory stage of the ventric

then

We can

le increases.
the halved ventricle-rhythim into the normal twice as

convert

an extra-stimulus to the ventricle at the

quick rhythm by applying

h nk

EEL

ETET

i

-

Fig. 6

Fig, 5.

704

end of the diastole or the pause. This was already explicitly discussed
by me with regard to veratrine and digitalis to which I refer here’).
The same holds likewise for antiarine, about which I intend to
publish a more extensive communication. We can then convert again
the normal ventricle-rhythm into the halved one by applying an
extra-stimulus to the auricle or to the ventricle-basis in the beginning
of the ventricle-systole.

During the normal rhythm the impulse is transmitted slower
through the ventricle than during the halved rhythm of the ventricle.
It is of course clear that the conductivity inside the ventricle
during the normal ventricle-rhythm, in which in the same time
twice as many systoles of the ventricle take place than during the
halved ventricle-rhythm, is worse than after the halving of the
ventricle-rhythm.

In Fig. 6 Ll reproduce an example of such an artificial modifi-
cation of the rhythm. In the beginning of the fig. (the first two
ventricie-systoles) the rhythm of the ventricle is halved. After every
large ventricle-systole occurs another extremely little abortive
ventricle-systole, the little triangular electrograms of which are
indicated by an a. Both these little ventricle-systoles of the halved
rhythm show little negative T-oscillations, and the line of connection
between the R- and the T-oscillation lies just below the position of
rest of the string. At the rising of the signal the ventricle-basis
receives an extra-stimulus towards the end of the pause, after which
a great ventricle-systole follows. Thereupon the normal rhythm of
the ventricle is restored. The first ventricle-systole of this normal
ventricle-rhythm succeeds still after a rather long pause, so that the
impulse-transmission through the ventricle is now only unim-
portantly retarded (compare the width of the R-oscillation of this
systole with that of the two preceding systoles of the halved ventricle-
rhythm). This slight retardation is however already expressed in an
enlargement of the negative T-oscillation and in a descent of the
line of connection between the R-oscillation and the T-oscillation.
The pauses between the succeeding ventricle-systoles are considerably
shortened, and now the width of the R-oscillations has remarkably
increased. The ventricle-electrograms show likewise large negative
T-oscillations, and the lines of connection between the R- and the
T-oscillations have descended considerably, and are gradually con-
verted into the T-oscillations.

In Fig. 7 the halved rhythm of the ventricle was by an

1) Arch. Néerl. de Physiol. loc. cit.

pee a pn

705

extra-stimulus on the basis ventriculi converted into the
normal one, and the latter again into the halved ventricle-rhythm.
The halved ventricle-rhythm was here also obtained by poisoning
with antiarine.

The first ventricle-systole of the figure belongs still to the halved
rhythm. A short time after the end of the diastole the ventricle-basis
receives an extra-stimulus, through which the halved ventricle-rhytbm
is converted into the normal twice as quick rhythm.

When we compare now the ventricle-electrograms of these two
rhythms, we are immediately struck by the fact that the R-oscillations
during the normal ventricle-rhythm are wider than those during
the halved rhythm. |

In the halved ventricle-rhythm the T-oscillations are negative but
very little, and the line of connection between the R and the T lies
just above the position of rest of the string. In the normal ventricle-
rhythm the T-oscillations are likewise negative but rather large,
and the line of connection between the R and the T lies now below
the position of rest of the string.

The basis ventriculi receives another stimulus at the 2rd rising
of the signal, which gives rise to a little abortive systole. After the
compensatory pause the ventricle is fixed again into the halved
rhythm through the post-compensafory systole. The ventricle electro-
grams have likewise again obtained the same shape as in the
beginning of the figure. |

In Fig. 8 the ventricle pulsates in the beginning likewise in the
halved rhythm (after poisoning with antiarine). At 1 the ventricle-
basis receives an extra-stimulus, causing an extra-systole of the
ventricle. After this the halved ventricle-rhythm continues however.
When thereupon at 2 the extra-stimulus is repeated a little earlier
in the ventricle-period, the conversion into the normal ventricle-
rhythm succeeds, but after 3 systoles it changes again into the halved
one. During the halved rhythm again little abortive ventricle-systoles
occur, the triangular electrograms of which are indicated by an a.
During the halved ventricle-rhythm the T-oscillation is positive, and
the line of connection between the R and the T is above the
position of rest of the string. At the quicker normal ventricle-rhythm
the R-oscillations are considerably widened, the T is strongly
negative and the line of connection between the R and the T has
descended far below the position of rest of the string.

We find these proportions not only with poisoned hearts, but not
poisoned frogs’ hearts show the same phenomena. Fig. 1 of one of

706

= HEET
eth

= Se

E rei /

En 5 t mT

RESP REE

i

|

LEET IL

Fig. 8.

‘) teaches us so. In this figure we see

communications

former

my

rbythm into the

the normal ventricle

of

eonversion

spontaneous

a

Bb Koninkl. Akademie van Weten
vergadering der Wis- en Natuurk. Afdeeling van 30 Juni 1917 Deel XX VI blz. 424

and Proceedings. Vol. XX, page 404.

chappen at Amsterdam. Verslag van de gewone

S

707

halved one in a not-poisoned heart. In both rhythms the T-oscillations

are positive but those of the halved rhythm are larger than those

of the twice as quick normal one.

this

in

illation

«

During

the halved rbythm the width of the R-osc

il df

a
Bo

708

figure decreases from the first systole to the third included, and the
height of the T-oscillation increases. A line of connection between
the R and the T is not to be observed in this figure, because the
T is immediately connected with the R.

Fig. 9 may still find a place here. This figure shows bigeminus-
groups, after poisoning with veratrine, resulting from the falling
away of every third auricle- and ventricle-systole. During the second
ventricle-systole of each group the impulse-transmission through
the ventricle is retarded more considerably than during the first.
This appears from the wider R-oscillation of the second ventricle-
systole. But the T-oscillation is much more negative, and the line
of connection between the R and the T has descended much lower.

The bigeminusgroups, which I published in 1915,*) show likewise
analogous proportions. The R-oscillation of the 2"¢ systole is here
wider, and the positive T-oscillation smaller than that of the 1**
systole of each group.

Ill. The electrograms of the anticipated ventricle-systoles.

With regard to this series of experiments a short communication
will be sufficient. In a former communication *) these were already
mentioned and explained with figures. The anticipated ventricle-
systoles were excited by extra-stimulation of the auricle, which
brought about extra-systoles of these parts of the heart. After such
an extra-systole the impulse proceeded along the atrio-ventricular systems
of connection towards the ventricle, which consequently was brought
to contraction at an earlier moment of the ventricle-period.

The place where the impulse enters. into the ventricle at these anti-
cipated ventricle-systoles, was consequently the same as for the normal
periodical ventricle-systoles. For this reason there was no objection to
compare the electrograms of these anticipated ventricle-systoles with
those of the periodical ventricle-systoles. It is obvious, that the velo-
city of impulse-transmission through the ventricle during the
anticipated ventricle-systoles was inferior to that of the periodical
ventricle-systoles and the retardation was the more considerable
according to a ventricle-systole being more anticipated. In accordance
herewith the R-oscillation of the ventricle-electrograms of the anti-
cipated ventricle-systoles was wider than that of the periodical

1) S. pe Boer: Die Folgen der Extrareizung für das Elektrogramm des Frosch-
herzens. Zeitschrift fiir Biologie, Bd 65, 1915, Seite 440, Fig. 8.

4) Koninklyke Akademie van Wetenschappen, Verslag van de gewone Vergade-
ring der Wis- en Natuurk. afdeeling van 30 Juni 1917, Deel XXVI bldz. 422, and
Proceedings Vol. XX page 404.

709

ventricle-systoles, and the wider in proportion as the ventricle-
systole was more inticipated. The T-oscillation of an anticipated
ventricle-systole changed in a negative sense, and the more so in
proportion as the ventricle-systole was more anticipated. The line
of connection between the R and the T had descended at an anti-
cipated ventricle-systole, this descent was the more considerable in
proportion as the ventricle-systole was more anticipated. At the
post-compensatory systole these proportions were exactly the reverse.
Then the velocity of impulse-transmission had improved, the
R-oscillation was narrower, the T-oscillation had changed in a positive
sense, and the line of connection between the R and the T had risen.

These short indications may be sufficient for the present. For
further particulars one must compare the figures 6, 7, 8, 9 and 10
of my communication. *) |

IV. Theoretical explanation.

It has appeared most clearly from the three series of experiments
described above, that there is a constantly occurring connection between
the width (duration) of the R-oscillation (velocity of impulse-trans-
mission through the ventricle) on one side and the dimension and
direction of the T-oscillation and the level, on which the line of con-
nection between the R and the T extends itself, on the other side.
When the duration of the R-oscillation increases, then the T-oscillation
changes in a negative sense, and the line of connection between the
R and the T descends If on the contrary the duration of the R-
oscillation decreases, then the T changes in a positive sense, and the
line of connection between the R and the T rises. The modifications
that the T-oscillation is subject to, had already distinctly displayed
themselves to me by the investigations I made in 1914. I think I
am now likewise able to explain more explicitly the modifications,
that the line of connection between the R and the T undergoes, and
to bring in this way the above mentioned experiments under one
point of view.

The normal ventricle-electrogram consists chiefly of an R- and a
T-oscillation. Consequently we do not discuss here the Q- and S-
oscillation, because the occurrence of these-is of no importance what-
ever for our considerations. These R- and T-oscillations are caused

1) I intend to explain iu a more circumstantial communication more elaborately
the electrograms obtained after extra stimulation of the ventricle-basis and point.
We can for this purpose compare the electrograms of the more and less anticipated
systoles with each other, and not with those of the periodical ventricle-systoles
(Vide Fig. 6, 7 and 10 of the former communication.)

51
Proceedings Royal Acad. Amsterdam. XX.

110

by interference of the basal with the apical negativity. The upward-
oscillation by which the ventricle-electrogram . begins, originates,
because the negativity of the basis begins or domineers in the be-
ginning. A short time afterward the apical negativity begins (or the
apical negativity increases) and brings the string back to the position
of rest. Then there is for some time equilibrium between the basal
and the apical negativity, and the string remains in the position
of rest. | |

Thereupon the T-oscillation comes into existence; if this T-oscilla-
tion is positive, consequently in the same direction as the R-oscilla-
tion, this is caused by the fact, that the basal negativity lasts
longer than the apical negativity, or because in the end the basal
negativity domineers over the apical negativity. If the T-oscilla-
tion is negative, consequently in a direction opposite to that of the
R-oscillation, then the apical negativity lasts longer than the basal
negativity or then, in the end, the apical negativity domineers over
the basal one. In Fig. 10 I have represented the origin of the R
and the positive T by interference of the basa! negativity a—b—e
with the apical negativity e—f—g. When now the velocity of
impulse-transmission decreases, then the apical negativity will
begin (or increase) later after the beginning of the basal negativity,
and bring the string back to the position of rest. On account of
the retardation of the transmission the position of rest is now reached
at a later period. 9

The width of the R-oscillation increases thereby. But the other part
of the ventricle-electrogram is likewise greatly influenced by the

a
Fete i) el a en Terie
.

711

retardation of the transmission. The scheme of Fig. 11 may explain
this fact. The basal and the apical negativity consist in this scheme
of the same curves as those of Fig. 10, but the apical negativity
has now removed more backward.

Point e is now much farther removed from a than in Fig. 10.
What is now the consequence of this removal of the apical negativity ?
In the first place that at the end of the electrogram the apical
negativity begins to domineer, and consequently the T becomes
negative. If the retardation of the transmission had been less
important, then this would only have reduced the positive T somewhat.
But the line of connection between the R and the T has likewise
descended. This is also to be understood. Whilst in Fig. 10 at a
given moment the basal negativity » interferes with an equally strong
apical negativity 7’ the string remains thereby in the position of rest.

When now, on account of the retardation of the transmission, the
apical negativity is removed to ‘the end of the electrogram, then
the basal point x no longer interferes with n’ but with m’,
which is removed farther from the position of rest. This holds now
for all points of the basal negativity after retardation of transmission.
These interfere consequently all with stronger apical negativities
than before the retardation. This is the reason why the line of
connection between the R and the T descends. This simple construction .
teaches us, why at retardation of transmission not only the
R-oscillation widens, but also the T changes in a negative sense,

and the line of connection between the R
and the T descends.
6 A clear illustration of the experimental
wite data.

io At an acceleration of impulse-transmis-

TA sion the R-oscillation will on the contrary

become narrower and remove the apical

‚\e negativity in a contrary direction i.e. to the
di front. Then each point of the curve of the
: basal negativity will interfere with a less
‘ important apical negativity than before the

pi acceleration. The result is then a rising of
het the line of connection between the R and

4 the T and an enlargement of the T as the
: scheme of Fig. 12 indicates. *)

Fig. 12. Still a few words about the height or

JOS

PE
es
‘

ar

i

‘

1) | have used in the scheme of Fig. 11 and 12 for the basal and the apical
Bd

712

the R. When the velocity of the impulse-transmission of the ventricle is
so great, that apical negativity has already brought back the
initial oscillation of the string to the position of rest, before the
full basal negativity has developed itself, then the height of the R-
oscillation possibly increases at a retardation of the transmission. These
proportions are also reproduced in this way in the scheme Fig. 10.

If on the contrary the maximal basis-negativity has already been
reached, before the apical negativity brings the string back to the
position of rest, then a retardation of the conductivity will no more
increase the height of R, but only widen its top. We find these
proportions in the frog’s heart after the hemorrhage. *) I hope to
come back to this subject more elaborately in a more extensive
communication.

It stands to reason that the shape of the ventricle-electrogram is
not only determined by the velocity of impulse-transmission.
In a former communication of mine | indicated the partial asystole
of the ventricle as the cause of the modification of the shape. *)
Its shape can likewise change besides by the more or less mono-
phasical deduction (by killing the heart-tissue under one deducting
electrode).

I shall restrict myself in this short communication to a few
remarks concerning the consequences of the views developed above.

In the first place about the atypical ventricle-electrograms. In
these the proportions are as in Fig. 11 viz. a high, wide R, descent
of the line of connection between R and T and a negative T.
In such an electrogram the apical negativity has consequently been
removed backward *). This can be caused by retardation of velocity,
but in casu the longer distance that the impulse has to cover,
will most likely be the cause. In the light of these experiments
the shape of the atypical electrograms is conspicuous to us.

components the same as in Fig. 10. It is obvious, that at modification of the
velocity of impulse-transmission these two components are likewise modified:
As these modifications are for in an equal sense, the results are after all as
reproduced in Fig. 11 and 12.

1) Zeitschrift fiir Biologie, Bd. 65, Seite 428.

*) Archives Néerlandaises de Physiologie de l'homme et des animaux, Tome I,
p. 29, 1916 and Zentralblatt für Biologie, Bd. 30, Seite 149, 1915.

3) For the question under consideration it is of no importance, whether we
have to do here either with the basal and apical negativity or with the negativity
of the left- and the right-ventricle. When the two negativities coincide less, because
one of the two commences later, then the atypical shape of the ventricle-electrogram
sets In.

713

In the second place vigorous hearts have a large positive T-oscil- —
lation. This is likewise easy to understand, as vigorous hearts possess
a good conductivity.

There exists moreover a strong overlapping of the basal and the
apical-negativity, and consequently the basal negativity is at the
end strongly expressed.

Finally the considerable variability of the T-oscillation is deter-
mined in this sense by the velocity of impulse-transmission as has
been explained above.

These short remarks may suffice here. I intend to explain these
and further results of these experiments in an ulterior more elaborate
communication. —

Amsterdam, Oct. 1917. Physiological Laboratory.

Physics. — ‘“Kesearches into the Radio-Actwity of the Lake of
Rockanje”’. By Miss H. J. Former and Dr. A. H. Briaauw.
(Communicated by Prof. Haga).

(Communicated in the meeting of September 29, 1917).
§ is Fntroductron.

In a treatise on the Lake of Rockanje')*) | have communicated
that an inquiry into the radio-activity of the samples collected by
me was being made by an expert. Now that this inquiry, carried
out by Miss H. J. Former in the Physical Laboratory at Groningen,
has led to fixed results, we wish to bring them before the public.
I shall begin with a short explanation from what considerations this
inquiry has been planned and elaborated so amply.

When in July 1915 I began my researches into the flora and
history of the origin of the Lake of Rockanje I got acquainted
with Dr. Béicuner’s researches, which showed what a surprisingly
strong radio-activity the latter had found in the Rockanje mud
according to his publication in the Chemical Weekly. *) Subsequently
Dr. B. G. Escukr communicated to the Geological Mining Society
a report*) made in April 1914. Escrer discerned in the Rockanje
mud organic and mineral parts and came to the hypothesis that the
mineral parts and together with these the radio-active substance,
had been blown into it from the dunes. In the said meeting and in
the above mentioned treatise I have already pointed out that the
large mass of mineral substance is nothing but clay, which has
settled on the bottom of salt and brackish water in the form of
fine slime and mixed with many organic remains.

If, therefore, this mud were so radio-active there would be a far
greater chance that radio-active parts had been carried there as
slime by the rivers from the mountains of Central-Europe, than that

') The Lake of Rockanje in the Isle of Voorne, which forms part of the province
of “Zuid-Holland”. It is situated near Hook of Holland, separated from it by
the Isle of Rozenburg.

2) A. H. BraAuw (1917) Treatise. Royal Acad. of Sciences Amst. 2nd Section XIX
NO. 3.

5) E. H. BicuNnerR (1913) Chem. Weekly. Part 10 N°. 35.

1) B. G. EscHer (1915) Report Geol. Mining Soc. Gen. Geol. Section Part 2,

715

they should belong to the small quantity of dust from the dunes
carried into the lake by the wind. But as it must be very well
possible to make out by radio-active measurements of samples from
the soil, whether the radio-active substance present in that soil had
entered into it from the side of the dunes or rather from the side
of the slime-carrying rivers, I thought that it was better experi-
mentally to settle a question which was geologically so important.

For that purpose I have collected a great number of samples as
will be mentioned below. After Dr. Btcuner informed me, in
answer to my letter, that he did not intend to continue these
researches himself, I was happy enough to find Miss H. J. Former
at Groningen willing to undertake them. Consequently this communi-
cation is chiefly the result of researches made by her at Groningen.

First samples 2, 4, 6, 8, 9, 10, 11, 12, 13, 17, 20, 23 were
destined for inquiry, taken from the boring made at Rockanje. From
these it was to appear in what layer the activity was strongest,
after that the geological constitution of those layers and the conditions
under which they had been thrown down, had already been
discussed before in detail. Samples 20 and 23 are, however, from
layers which existed long before the origin of the Lake and of the
isle of Voorne'). These samples are called: boring 2, 4, ete.

To these others were added: |

R. Taken 25 m. to the North East of the Grotto at 40 em.
depth, consisting of brown, entirely organic mud mixed with some
grains of quartz of at most 150—300 u length.

A. Taken from the bottom of the water, on the side of the
Noorddijk, behind the Windgat, where aen the dike often burst,
on the North West side ®).

B. Taken from the bottom of a ditch close to the Molendijk in
the Strijpepolder, in order to examine whether the mud at a short
distance from the Lake is still strongly active.

C. Sand taken from the dunes nearest the Lake (at + 800 m.
distance) whereabout formerly the Swyn was situated.

D. Light-grey clay, taken at 25 em. under the meadow of the
Drenkeling behind the Vleerdam Dike of the Lake.

M. Slime, freshly. settled, from the Meuse at Grave, which Mr.
J. peN Doop from Grave kindly provided me with.

1) Cf. treatise pp. 50—55.
4) Cf. treatise pp. 90—93,

716

W. Slime, freshly settled, from the Waal at Nymegen, sent to

me by the kind interference of Dr. P. Terscnr.

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“OFS/, oppas = *sbursog mojjnys ayn Jo YNYS

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These seven samples enable us to compare:
1 the mud of the Lake with that of the environs.

2 the Rockanje mud with neighbouring dunesands and with the
slime from the Meuse, the Waal, consequently from the Rhine and
Meuse basins.

For purposes of comparison Miss Former still added mould from
the garden round the laboratory at Groningen and Fango of Batta-
glia and of an unknown find-spot.

Afier this series had been investigated it appeared from reasons
to be mentioned below, that it was indeed necessary to extend the
number. First Mr. Trouw at Rockanje kindly sent us three samples
taken from the same spots where Mr. Bicuner collected the samples
in 1913, which he found so extremely radio-active.

These samples are called:

I (about 50 m. to the South-West of the Grotto at a depth of 150 cm.)
Il (from the South-East Corner of Betjenskelder 60 cm. deep)
III (ibidem 150 cm. deep).

In conclusion we were obliged to collect yet another series of
samples scattered over the whole area and at various depths. To
that end I have chosen the spots so as to include entirely the area
of the spots from which samples had been taken already, which
were said to have shown radio-activity by means of radiograms or
emanation (cf. fig. I). They were collected 1 from 10 spots at and
about those places where the radio-active samples were collected in
1913 (see above) and besides from 5 spots from the bottom of the
ditch that skirts the area of the Lake and Betjenskelder on the North.

From these 15 spots 46 samples were taken at different depths
varying from 30, 60, 90, 120, 150, and 180 em. All this is sufficiently
indicated in the figure together with the numbers of the samples.
We have taken these samples by means of a boring machine 2 m.
long, consisting of a tube of galvanized iron. Its stock can be pushed
through these layers of clay and peat. To the bottom is attached a
box in the shape of a hollow cylinder, on the flattened side of which
is a slide, which can be opened by means of a thin rod, running
upwards through the iron tube; this slide, therefore, can be opened
at the top and fastened by a screw, after the boring machine has
been placed at the required depth. With the crank at the top of
the tube one turns the whole thing round a couple of times, unscrews
the slide-rod, closes and fastens the slide again and draws the whole
upwards. In this way we made perfectly sure that we drew up only
mud from the depth as wanted. These samples Miss Former put

718

away into small tins shut off by insulating tape, and took them
to Groningen for immediate investigation.

Finally a couple of samples the radio-activity of which was said
to have manifested itself by means of radiograms, were added to
these; and yet: two other samples 1. dregs of sulphur-bacteria 2.
very wet mud from the ditch that skirts the lake. All these samples
were investigated by Miss Former, who will describe below more
fully her mode of working and the results.

§ 2. Measurements.

The researches, the desirability of which was demonstrated by
Dr. Braauw in his Introduction and which in connection with the
experience obtained in the course of the experiment were constantly
enlarging, can principally be reduced to the two following parts:

I. To this belongs the investigation (ef. Introduction) concerning
the 12 samples of boring, the 7 samples for comparison, the samples
I, II, and III, the Fango of Battaglia, already long known to be
radio-active and finally the sample: mould. All these have been
investigated according to the ‘emanation method”, of which further
details will follow below.

II. In eontradistinction with the preceding the ‘direct’ measuring
method defined more closely below, was yet applied to the second
series of experiments in which the emanation obtained from the
mud was not investigated, but the mud itself with regard to radio-
activity. This second series of experiments principally extended over
the 48 mud samples (cf. Introduction), all of them taken from the
neighbourhood of the Betjenskelder or from the Betjenskelder itself.

A closer consideration of researches 1 and II may set forth the
motive for these two series of experiments performed at such various
times, as well as the different choice of methods of investigation.

Pare

Description of the investigation according to the “Emanation
method”.

Here the first part certainly is the laborious dissolving of the
various samples for which I used the method as indicated in outline
by various investigators.) 150 grammes were taken from each
sample after having been dried by a slight heat, then sifted minutely
and shaken forcibly for five minutes; then 5 grammes of this were

1) Cf. for this: B. B. Bourwoop, Phil. Mag. 1905. J. Jory, Radioactivity and
Geology, 1910. E. H. BücHaNer, Chem. Weekly 1913.

719

glowed carefully; after having been weighed the remainder was
extracted by hydrochloric acid, which was first distilled over chemic-
ally pure salt. After this treatment it was filtered, through which
a part of the solution to be made, viz.: the acid solution had been
obtained. The filter was burnt, the ashes with the dried residue
mixed in a platinum crucible with 5 grammes of Na,Co, and
5 grammes of K,CO,.') The amalgamation of this mixture took
place in a small electric furnace heated for this purpose to about
800 degrees for at least three hours.

Subsequently this melt of molten mass was extracted by water
and filtered in such a way that a perfectly pure alkaline solution
was obtained. The remainder was dissolved in strong hydrochloric
acid and added to the acid solution obtained first, the alkaline solution
being preserved separately. If, however, not everything was dissolved
completely, it was filtered once more, the residue amalgamated with
only alittle carbonate aud treated again as mentioned above; finally
these acid and alkaline solutions were added separately to those
prepared first. The solutions obtained i. e. of each sample an acid
and an alkaline, were shut off air-tight in a one-litre boiling flask
of Jenaglass and put away in order to await the emanation equilibrium,
which is almost perfect after 50 days. Even after this time the
solutions were still found in an entirely clear state and consequently
the chance of occlusion for the formed emanation into a gelatinous
precipitate, was absent.

As a second part of the investigation now followed the expulsion
of the emanation, which was brought about with the assistance of
an arrangement with which the various solutions were made to
boil and from which in consequence the emanation could develop
itself. This together with the escaped gases — the vapour was
condensed in a cooler — were caught into a reservoir over a
saturated salt solution. From this the emanation was transferred to a
jar communicating with a second in order to make the absorption
of gas possible, then all the tubes were rinsed again with air and
at last the top of the jar shut off. The emanation which was again
present here over a saturated salt solution, was then ready to be
investigated electrically.

Hereupon followed the third and last phase of the investigation,
viz: the measurement of the ionisation current, caused by the ema-
nation described above. For these electric measurements as well as
for those according to the “direct method’, the electrometer was

1) Provided by KAHLBAUM “pro analyse” with ‘“Garantieschein.”’

720

used by me the principle of which was published in 1914,') while
the description of the apparatus followed in a second article. *)

The method of measuring I arranged as follows:

Before making the solution boil the electrometer was given a
definite not over-sensitive state of charge and that: a +12 Volt,
6 0 Volt, ce upwards of —4 Volt. The ionisation cylinder was
charged to -+ 80 Volt.*) Having attained this, the natural leak in
the measuring cylinder was examined a couple of times. The time
was so fixed as to make 10 scale-divisions move under the eross
wire until the total of one hundred divisions were passed through ;
this happened in order to be able to include into the calculation
the natural leak, when measuring the samples. After this the ioni-
sation space of the apparatus was exhausted, making use of GAEDE's
new single barrel airpump; with this it was possible to exhaust the
space of them easuring cylinder of a volume of 1 litre with the indispens-
able rubber tubing and manometerspace down to two mm. pressure.
The emanation present in the bottle, after having been conducted
through drying-tubes with CaCl,, P,O,, and a tube with cotton wool,
was transferred to the evacuated apparatus. After having awaited
the equilibrium of the emanation with the radio-active products,
radium A. B. C., so after about 3—4 hours, the measurement proper
took place as follows: the time was fixed in which the image of
the scale moved under the cross wire, but now, in contradistinction
with the determinations just given, over 150 scale-divisions; as this
measurement did not last for more than 5 minutes for a sample
that is but little active, and also the preceding, viz: that of the
natural leak took comparatively little time, the advantage had been
obtained by repeating the measurement a couple of times, to be
able to include into the average calculation a more exact final
value. Herewith the determination proper had come to an end and
the radio-activity of the substance could be computed by taking an
experiment of the same kind with a normal solution of radium.
This I obtained from a quantity of radiumbariumbromide supplied
by Messrs. pr Haren in Hannover, containing 0.0126 m.grammes of
radium according to notification.

Here follows, by way of explanation, a calculation of one of the
samples examined, viz: N°. I from Betjenskelder.

a. Determination of the natural leak.

1) Royal Acad. of Sciences. Amsterdam. Proceedings of June 1914. .
*) Royal Acad. of Sciences. Amsterdam. Proceedings of September 1917.
3) Royal Acad. of Sciences. Amsterdam. Proceedings of June 1914.

724

Number of seconds wanted for every 10 mm. displacement:

from scale-division 750—700 from scale-division 700—650
41 40
50 42
45 57
48 49
45 52

sum total 3 min. 49 seconds. sum total 4 minutes.
repeated once more:

750—700 700—650
38 44
40 53
48 43
44 46
4] 53

sum total 3 min. 31 seconds. sum total 3 min. 59 seconds.

hb. Determination of the ionisation with the emanation N°. | in
the apparatus after equilibrium has set in between this and its
radio-active products.
1. time, wanted for the displacement from 750—700:2 m. 2 sec.
55 shat tok 5: a 10-630: Ain
4 :; ater 2 » 6506002, 25

sum total 6 m. 34 sec.

9?

2. when this is repeated once more we obtain:
a displacement from 750—700:2 m. 2 sec.

if ly toe 100 650625 IT
a zi 06002 aes
sum total 7 m. 34 sec.
Pieomean of. i:and 2 as Arsen ERN (eet iel OF Gee

From aand 5 will now follow the correction of the natural leak :

This ionisation together with that of the emanation amounted to
6 min. 56 seconds, measured from scale-division 750—600; it can
then be derived from a, that the first mentioned ionisation alone
displaces the scale in 6 min. 56 sec. or 416 seconds from 750—658.
Consequently this result has the same effect as if first only the
natural leak acts and that in 6 min. 56 seconds a displacement
from scale-division 750—658 and then the emanation alone from
658—600.

For further conclusions we then want to know the ionisation,
only caused by the emanation of the normal solution and this also

722

taken from 658—600. The latter, obtained in the same way, indi-
cated a displacement from 750—600 in exactly 23.0 seconds.

In order to derive in how much time this normal solution will
ionise from, 658—600, the ratio of the average sensibility of charge
of the distance 658—600 of the scale to that of 750—600 had to
be still defined. For this determination | made use of a special
experiment taken very accurately and which was of equal import-
ance for the calculation of all the other samples. This consisted in
examining the ionisation of a substance of which the ionisation was
not too small, from scale-division 750—600, and that in such a way,
that the time was continually checked which was wanted for passing
through 10 seale-divisions. The values in seconds amounted from 750 to:

first. determination: 21, 23, 24, 20, 22, 31, 24, 20, 24, 29; 32,

27, 31, 33, 34.
” second determination: 31, -27, 29, 21, 25,25, 29, 26; 130, 25; 36
34, 34, 28, 28.

From this follows that the charge wanted to displace the needle
from 658—600 amounted to 180/411 of the value, which applies to
parts of the scale 750-600. Then the real ionisation time of the
normal solution for 658—600 was: 180/411 23 sec. = 10.1 sec.
Consequently the sample was 416/10.1 = 41.2 times weaker than
the other, so that every gramme of the sample contained:
1.25 X 10—10
en = 0.61 « 10-!? grammes of radium. In this way the

5 X 41.2
values for the radio-activity of every sample of part 1 have been
derived ').

1) The calculation might yet be executed somewhat differently and that with the
aid of ionisation values of only 100 scale-divisions, e.g. of 750—650, and this
for a natural leak, emanation as well as for normal solutions, for from the data
as given above follows (cf. 6) that the substance ionises from 750—650 on an
average in 4 min. 16 sec. or 256 sec., whereas the natural leak (cf. a) ionises
over the same space in 7 min. 40 sec. of 460 sec. From which follows that the

natural leak together with that of the emanation would ionise in 460 sec.

460

256 1.8 time over that part of the scale, (only suppose for the calculation of
the sensibility of charge constant from 750—659), so only the emanation 0.8 time
over that part mentioned in that time. For the final calculation we then derive

from c, that the normal solution, that ionises in 23.0 sec. from 750—600, displaces

… 258
the needle from 750—650 in M1 a 25 sec. = 14.1 sec., consequently in 460 sec.

7, — 32.6 times ionises over the division of the scale mentioned. Finally there

En]

mK B20
follows from all this that the activity of the normal solution is = 3 = 40.8. times

723

The results obtained are communicated in the subjoined table.

TA Bel Bi:
Indication — | Depth in | Radio-activity ex-
of the Soil Met ‚pressed in 10—12 gr.
samples | Be | rad. pr. gr.
boring 2 | lightgrey mud with remains of peat 2.70 0.43
s 4 grey mud and dark clay .…....…. | 4.40 | 0.54
ge Gee ECeN-Dlacki clay, Jas beg lee snie | 6.05 | 0.55
Bie Ob blaerblack? tag Lv ae er 0.61
RADE black fat clay... Jr 8.30 | 0.78
ONM beblaelndnuds oe! tank | 9.30 | 0.66
Spee | | bluish-grey clay with grains of sand 9.55 0.62
ee 17 bine-Dlack “clay rs ob a ted 10.50 (2.16)
„ 13 black clay with remains of peat. 12.90 0.64
PAD | sand with little slime........... +.13.50 | 0.1
ve sn 20 enteren Clay inte! aud fected dl 21.20 0:15
Reen | | | | 0.36
DANE xa he of eit! ero ete us ei va OE | 24.80
„ 231 0.37
A brown-black peaty mud......... — 0.39
B TORE USAR Tate is Sa Oa oP eo ante — 0.78
C ‘sand ‘Of the dunes (03100. Aas 24 | — | 0.23
D | light-erey. ClAWE:, oe are ceerde en | — | 0.59
M slime of the Meuse ….…......…. | — 1.09
Ww slime of the Waal.............. | — 1.49
R | brown organic mud......... Lae 0.40 | 0.24
SOU DN res ws erage. ss RER am pie: | — 0.96
No. 1. |: blue-black clay :.........s26s.04 | 1.50 | 0.61
E | | |
Netllke oehrownsarey. clay... „caat dane | 0.60 | 0.49
No. II | blue-black clay … AAS | 0.57
Fangovof. Battaglia’..2% Jesi | es! | 34.0

stronger than the substance investigated. The first calculation is preferable as in
it the ionisations could be examined over a larger number ie. over 150 scale-
divisions. Even then the second calculation could be applied, if also the natural
leak had been examined down to 600. This, however, is difficult in connection
with the length of time of observation, if one makes several and very accurate
observations.

724

From this table it is clear amongst other things that the investi-
gated samples of the boring, as well as those of the Betjenskelder
or its neighbourhood only show a radio-activity in the order of
10-12 grammes of radium pro gramme; i.e. no larger quantity than
is normally found in most roek *). The activity of mould is of the
same order, whereas only the sample of comparison : Fango exhibits
an effect considerably higher. Notwithstanding this small radio-activity,
however, it shows in relation to the depth and the nature of the
layers, a certain regular course as may appear more distinctly yet
from the geological interpretation of the results by Dr. Braauw as
mentioned below ’). The two values for N°. 23 which agree within
the limits of errors of observation, relate to the values of activity
of two solutions, made and investigated at quite different periods.
Having arrived at the end of these experiments we could not but
conclude that our results did indeed conirast greatly with the values
for samples of mud from the Lake as indicated by Dr. BücaNer in
1913; he even found a value of 462 X 10 1? grammes of radium
pro gramme in the blue mud from the Betjenskelder.

At the very moment that these researches had proceeded thus
far, a circumstance intervened, which especially has been the occasion
of our resolving not yet to consider the experiments as being finished ;
on the contrary: to continue them in a vet more extensive way.
From Utrecht Prof. ZWAARDEMAKER sent us two samples of Rockanje
mud; from one of these, which. Prof. ZWAARDEMAKER had received
in 1913 from Mr. Trouw at Roekanje, the radio-activity had been
clearly shown in the Utrecht Laboratory, because they had succeeded
there in obtaining radiograms through this sample. After all that
preceded it was certainly remarkable that, according to the “direct
method” of investigation (Cf. Part II) I could determine that this
sample certainly possessed radio-activity 1'/, times that of Fango of
Battaglia. In order to find out whether we might have been mistaken
with other samples, Dr. BLAauw investigated this sample with regard

1) In the table the values for activity only apply to those of acid solution; in
the alkaline solutions, even in the Fango I could find no activity. An experiment
with only chemical substances was made to check this with a solution which
contained the quantity of chemical substances only necessary for each experiment;
also this had a negative result.

2) It is of importance to remark here that very little information has been
given up to now on the relation of the quantity of radium to the depth under
the surface of the earth. However, experiments showed no relation. Cf. A. F. Eve
and D. Mc. Intoscu, Trans. Roy. Soc. Canada 1910; E. H. Biicuner, Jahrbuch
Rad. u. El. 10, 1913; H. E. Watson and G. Pat, Phil. Mag. 28, 1914.

725

to Diatoms after which he could ascertain that it was certainly mud
from Rockanje and that from a layer agreeing. with samples 4 to 6
from the deep boring. Tbus the Fango of Battaglia was investigated
and it appeared that this is also recognisable by a definite kind of
Diatoms. The Diatoms as “characteristic fossils’? could again be of
use here. When, therefore, the strong radio-activity of the mentioned
sample had also appeared in my case, we passed on to a second
enlarged investigation with the purpose of obtaining yet closer
indications about the question that concerns us here. This research
contains the second series of experiments arranged higher up under
Part II (Cf. also Introduction).

Part II.

Description of the method of inquiry, viz.: the ‘direct method’.
This is principally to the effect that a part of the dried mud is
weighed and placed into the ionisation-space; the radio-active sub-
stances, so "not only the emanations present in it, cause ionisation
of the air, which is again measured. This method, introduced already
by Ester and Geirer for definitions of the radio-activity of rock and
soils and at present still used for these, among others by Prof. Gockun
in Freiburg, has, however, been disputed on various grounds’) and
it unquestionably has considerable drawbacks as a measuring method ;
among’ others this, that when applied as above mentioned it is not possible
to obtain an absolute definition or mutual comparison of the quantity
of radium, nay a mutual comparison of the activity as a whole
cannot even strictly be carried out. For the ionisation may be the
consequence of radium and its radio-active products, but at the same
time of thorium and other active substances, the rays of which
ionise in a very unequal degree. Then, the absorption of the rays
is also disturbing, because it gives differences, for the various substances
as well as for the divers radiations. Overagainst this we may,
however, mention as a very important advantage of the ‘‘direct
method”, that we can get on with it so much quicker; no protracted
chemical operations are required as in the case of the emanation
method; with the electric measurement we need not await the
formation of the equilibrium of the substance employed with the
radio-active products to be formed by it. Thus the measurement
will only take Prof. GockEL an hour, without mentioning the preceding,
necessary determination of the natural ionisation. With the use of
a sensitive electrometer, however, this advantage is shown more
favourably yet. Thus it was possible for me to examine 40 samples

1) Cf. E. H. Bücanrer. Jahrbuch Rad. u. El. 1913.
D2
Proceedings Royal Acad. Amsterdam. Vol. XX.

726

quite accurately in 5 hours, among which occur several measurements
which were repeated sometimes for the sake of a greater certainty *).
Nor was it necessary for me to glow the substance before measuring
its ionisation. Prof. Gocker, namely, experienced the disadvantage
that during the hour of the measurement emanation escaped, the
radio-active products of which settled on the sides of the vessel, thus
causing the ionisation to grow stronger during the measurement;
as several circumstances, a.o. the nature of the substance, are of
great influence on this so-called “emanating”, GocKEL tries to avoid
this difficulty by glowing the substance beforehand, or: to deprive
it of all emanation present. This is an uncertain procedure, which
BicHner has already pointed out (ef. note p. 725) and I think it is
also preferable not to shut out its activity, if the substance should
contain some emanation. This is possible if the measurement is but
short, as in that time the quantity of escaped emanation need not
be taken into account. In my opinion both methods have great
value and the importance of the “direct method” should not
be underrated; first because of the advantages mentioned in order to
determine the order of radio-activity, as in the case of the samples
of Part II, by a speedy inquiry, but secondly to penetrate deeper
into the radio-active phenomena of the substances itself. It will
never be possible to say beforehand what radio-active products,
perhaps new ones, one may happen to meet, or whether, as Prof.
Gocket also remarks, all the products are indeed present, which
arise from a series of active substances the emanation of which is
found according to the emanation method. With the help of absorption
experiments and especially in very accurate measurements it seems
to me that especially in the future this method may become of
very great value.

In order to complete the above brief description of the method,
I may add ‘the following: first of all the placing of the samples
into the ionisation space was done in a particular way. Prof. GockEL
describes the difficulty occurring in his case, viz: that in the space
of experimentation a fresh supply of air will always penetrate, which
in itself will modify the ionisation. In order to avoid this disturbance
the electrometer employed by me was provided with a special

1) It is important to remark here that a sensitive measuring method, “direct”
as well as indirect, possesses advantages over an insensitive one; where up to now,
many investigators use the latter, this will explain, though only partly, the
remarkable fact that as regards the measuring results of various observers one
meets with so much contradiction, where determinations of the radio-activity of
similar material are concerned.

727

closure.*) When in one of the basins of diameter 11'/, em., 50
grammes of the dried substance were strewn equably, it was possible
in a very simple way to exchange the basins without allowing the
air to enter the ionisation space. As to the electric measurement,
for this the ionisation time was only determined for 50 scale divisions
over the same part of the scale. This was subsequently done with
all the samples, the results of which are given in the table below;
only the ionisation time is mentioned in it as, because of the above-
mentioned reasons, a proper calculation of activity will not be
possible. Yet the various ionisation times will approximately denote
the different values of activity. For a closer meaning of the numbers
elegy “te

In order to- be able to form a somewhat more accurate notion of
the order of radio-activity with which such ionisations agree, I have,
while supposing among others that the absorptions of the various samples
are the same and, besides, that they only contain radium, traced
the calculation of activity for sample I (cf. table 2) namely by com-
paring the ionisation time of 1 with that of the Fango of Battaglia.
It should be taken into consideration here, that the natural leak will
also give its share to attain these effects, so that in reality the ratio
of the activity of the Rockanje mud to that of the Fango is still less
than one might think one could derive directly from these figures.
Thus I found N°.1 to contain 1.1 10-12 grammes of radium pro
gramme; i.e. 1.8 times a larger value for this sample than follows
for this sample from the emanation method (cf. Table 1). That these
values do not agree, might partly be caused by the fact that the
sample in question contains at the same time thorium. Many minerals
are even exempt from radium, whereas they contain a large amount
of thorium.

So also this second comprehensive series of experiments had not
disclosed an appreciable radio-activity for any of the samples. This
notwithstanding, we did not yet consider our experiments as having
come to an end and that in connection with the assertion uttered
on various sides that radiograms had been obtained by means of
Rockanje mud; this had been the matter with the already mentioned
sample 1913 and with two samples forwarded by Dr. Revs from
the Hague. These samples were investigated, both electrically and pho-
tographically ; it was already mentioned that according to the first

!) Cf. also for the drawing: These Proceedings, p. 684. The closure principally
consists in this, that instead of the circular bottom plate a ring is fastened to the
ionisation cylinder carrying two ways over which 2 metal basins can slide, which
alternately form the bottom of the ionisation space.

52%

728

TABLE 4
pei ene ER,
Number |Depth in cm. eren Number Depth in cm.) en

| | time in sec. | | time in sec.

90 178 24 150 vee = ais

2 120 Fie Ve 15 RE Pane

3 150 88 26 90 152

4 180 125 1:21 120 104

5 90 150 TN 150 115

6 990 Ais te soothe ren see ag

1 Si ir (hade hh Ae 161

8 180 IO |E | 90 121

9 | 60 150 one: cd tarten 90

robstis® Tent dv eloggh 4’ kinde onsen

11 TD | 90° || 34 | 90 | 95

12 150 «100 35 120 va RVS

13 10 | 100. | KET 95

14 Ma ted Cs e Tataren pee e 173

(5 le (2 bg NE et oo] Mert

16 | 150 oo | 39 | 60 143

17 180 123. || 40 60 125

18 90 159 | 41 90 82

19 120 ee ae bie > 120 78

20 | 150 71 nee 150 74

21 180 96 44 180 127

22 | 90 105 45 | 30 209

fe 28 2 e205 COTON eG DONRSUE ican Brake

Sample | 130 Sample II 136

Sample II 120 Mould 400
Fango of Battaglia 18

method sample 1913 showed a stronger radio-activity than the Fango.
Tbe two other samples, however, could again be ranged under the
order of 10—!2 grammes of radium pro gramme.

But also photographically these and other samples were investigated

729

with regard to their action. Prof. H. Haca was willing to under-
take the radiogram experiments at our request, of which the results
were as follows: The experiments were made in the usual way in
which the photographic plate in a black envelope is exposed to the
action of the substance to be investigated; in order to be able to
state an eventual action some parts of the plate were protected by
placing some small metal discs on the envelope, or metal pieces
with figures cut into them. In using “Schleussner Röntgen-plates”’
only a very small action was obtained after a nineteen days’ ex-
posure from Fango. Dried mud of Rockanje N°. 12 and mould
would have no action at all. In using “extra speedy Wellington”
plates a very strong effect was obtained after a 30 hours’ exposure
with Uranium-pitchblend, a very weak one with a strip of Uraninm
glass and no effect at all with a quantity of Rockanje mud 1913.
No more was any action obtained after 44 hours by strewing the
black envelope with the whole quantity of the last mentioned samples
or after a week’s exposure with the above-mentioned samples from
Dr. Reys. The same negative result was arrived at in experiments
in which for a week Schleussner-Réntgen-plates” without the wrapper
had been placed over a layer of Fango or mud at + 1 mm. distance
or had also been strewn with these substances.

These results agree entirely with the photographie action’ on
radio-active substances as described in the literature on the subject.
In such photographic experiments one trial experiment should always
be taken with a plate from the same pack without any substance
strewn upon it, and one should always be careful with the so-called
black paper, some kinds of which transmit a sufficient quantity of
daylight to obtain a misleading effect. As the photographic experiments
had again given a negative result, there only remained to us to set
forth more clearly one side of the question. It has namely been
thought that there might exist a relation between organisms and
accumulation of radio-activity, when BücHNer’s definitions of the
radio-active intensity of the Rockanje mud were known. As, accord-
ing to this investigator, radium probably occurs in mud as RaSO,
and so many sulphur-bacteria live there, whose sulphur-reserve is
oxydated by respiration, so that sulphates again are dissolved into
the water, it seemed desirable to us to examine bacteria-dregs and
watery mud from the ditch as to their activity For this purpose
Dr. Braauw once more took two samples and sent them to me,
mixed with much water. The research tookplace both according to
the “direct” as well as to the emanation method. As regards the
latter, no emanation equilibrium was awaited, but only the quantity

730

of emanation present extracted by boiling. If Fango was treated
thus, very strong ionisations could again be observed. 1 shall not
mention all the values obtained according to the various methods;

RR DEERD EER GS
Ean

2 4 6 8 9 10 4 12 13 17 20 25
Nwmber of the samples.

let it suffice to mention here, that also in this case the figures
pointed to a radio-activity not exceeding that of 10-12? grammes of
radium pro gramme.

Finally I will mention one very important experiment taken with
the various samples of boring (ef. Part I). From the “conclusions”
which Dr. Braauw has derived from my data one may notice how
he could entirely account for the course of the radio-activity in the
series of the borings. | thought it desirable also to apply the “direct
method” to these samples of boring to wind up my researches. The
result will be found represented graphically in curve Il of Fig. 2,
whereas curve I fixes the values mentioned in table 1 according
to the emanation method. The conformity in the course will
strike one: with the exception in the case of N°. 12 there occur
in both curves corresponding fallings and risings. As not only the
quantity of radium influences the values of the “direct method”

731

(ef. page 725) it is certainly remarkable to have to gather from
these curves that either these substances contain only the active
radium or that also the other active substances in it must show the
same course as that element.

Besides, this conformity in the results of the two methods may
be looked upon as a valuable check of the one upon the other.

But to what cause do the different values for sample 12 point? One
could notice already that this sample is much more active than any of
the others in series I, but again looking up the notes taken when I
investigated these various samples electrically,’ I found that it was
just for this sample that a very strange course was mentioned of
its behaviour with regard to the radio-activity during the day of
its measurement. Though, however, N°. 12 only contains 2.16 « 10-12
gr. of radium pro gramme yet the above mentioned fact considered
by itself will be the motive that this sample will be subject to
further investigation.

If then, having come to the end of all the mentioned experiments,
we recapitulate results, we will see that neither the electric research,
according to various methods, nor the photographic research have
disclosed any appreciable values for the radio-activity of the Rockanje
mud. The only exception to the large quantity of samples is sample
1913, but, as it did not come to us straight from Rockanje, but
was kept in the Utrecht-laboratory for 4 years, we need not take it
into serious account.

Our final conclusions from these researches can only be
that in our opinion the Lake of Rockanje possesses no radio-
activity of any importance.

§ 3. Conclusions.

After the detailed researches described above by Miss Former
there is no occasion left to me to seek geological explanation for
an especially strong, local radio-activity, for not one of the samples
shows an action which would be + 100 times stronger than those
of igneous rocks (Cf. Bicuner 1913, Escner 1915) and all the samples
have an activity as has generally been found in soils of the kind.

But though the occasion for further inquiry into an hypothesis
for the explanation of a strong local activity may be disregarded by
me, I yet want to point out that it is now that this question becomes
generally of importance: where do the radio-active parts come
from which are found in the alluvium (or in a more general sense
in the ground)? In this I will not venture too far on a territory
with which others are so much more familiar and better entrusted

732

and [ will only point out some conclusions to be drawn from the
values found by Miss Former. This is all the more important,
because we are concerned here for the first time with a great
number of determinations of radio-activity of a geologically amply
described country.

As has been fully described in the treatise (pp. 29—31) we are
concerned here with clay, organic material, and sand. In the
microscopic investigation into the organisms of the samples I have
always mentioned the greater or smaller presence of mineral dust,
organic remains and grains of sand, together with their size. Of
course, it may not be lost sight of that mineral dust need not
exclusively be slime from the rivers, but that also part of it may
come from the dunes. And the more we meet with grains of quartz
in the layer, the greater the chance will be that also part of the
finer mineral dust originates in the dunes or the bottom of the sea.
Reversely, very small grains of quartz are thrown down in the
slime, for in the freshly settled slime of the Meuse and the Waal
they are to be found in a small quantity: I shall leave undecided
whether they have been blown into it, carried along or rooted up.
Yet it is very well possible to conclude from all the figures Miss
Former has given, to which of the three mentioned elements: clay,
organic remains, or sea-and dunesand the radio-active parts belong.

The quantity of radio-active substance varies in the 24 samples:
boring 2, 4 ete. A.B.C.D.M.W.R. I. II. III in 14 of the 24 cases
between rather narrow limits: 0.49 and 0.78 >» 10-12 gr. of radium
pro gramme dried substance.

The following samples possess a lower number:

Boring 2 with 0.43 >< 10-12 grammes. Besides fine mineral dust
and a few grains of sand of 200—300 u length at most, especially
much organic material.

Boring 17 with only 0.1 X 10-!? grammes. Consists of blue-grey
sand (sand with little slime).

Boring 23 with 0.86 and 0.387 x 10-1 grammes of sand; the
grains of sand in this sample are at most 450 u long. Mixed with
very fine mineral dust.

A with 0.39 X 10+? grammes. According to notes in 1915:
excessively rich in Diatoms; a great many organic remains, little
mineral dust”.

C with only. 0.23 x 101? grammes. According to notes in 1915:
“very regular sand without any organic mixtures.

Rk with only 0.24 x 10—'? grammes. This sample is the same as
that described in the treatise p. 17 under No. 2: “brown mud”

733

where no admixture with the blue-grey mud had as yet taken
place; it consists chiefly of remains from plants and animals with a
few grains of quartz of at most 150—300 u length.

A higher number was found for the following samples:

Boring 12 with as many as 2.17 >< 10-!? grammes. This is the
highest amount that has been found. “Blue-black” clay; “consists
of very fine mineral dust, mixed with very few grains of sand, at
most 80—120 u long; few organic remains. (Treatise p. 38). Neither
is there any sample in this series in which the very fine mineral
dust is mixed with so little sand and organic material. It was also
pointed out that there was a resemblance between the Diatoms to
those from the slime of the Meuse near den Briel. (Treatise p. 51).
M with 1.09 x 10 grammes. W with 1.49 X 10—!? grammes.

I think that from this will appear that the quantity of radio-
active substance is smallest in samples of soil with much sand or
with much organic substance and that this quantity will be the
higher as the settled slime of the river lacks these elements. This
is clearest in the case of boring 12 and the only samples where
the quantity also exceeds 1.00 x 10—!° grammes, are these very
slimes. from Meuse and Waal. If we have to choose between an
origin from the dunes or slime of the river then I think I can
safely conclude that the radio-actiwe substance in the alluvium of
Rockanje comes especially from the side of the rivers and in a far
less degree from the dunes.

Apparently there is also radio-active substance in the sand of the
dunes (cf. sample C), though in a far less degree. Whether this is
the same substance as in the sfronger active clay will perhaps be
settled later on by others. Moreover it remains possible that also
that slight quantity in the dunes yet exists of minerals originally
carried along, together with slime of the river and which have settled
on the bottom of the sea on the shore. I must still add that, of course
independent of the quantity of sand and organic remains, the slime
settlements may possess different radio-active intensity, even if they
come from the same river. The motion of the water, either by the
current or the whirl of an eddy, may cause by fractionated settlement
on various spots a varying quantity of active minerals, altering with
-the strength of that motion.

We will now consider the numbers of the samples of boring
(2, 4, 6, 8, 9, 10, 11, 12, 13) a little more closely. They belong to
that part where in a deep basin hollowed in the sand, slime of the
river was thrown down in salt and later on in brackish water,
while in later times the slime of the uppermost Jayers was mixed

734

with ever increasing organic remains. We now see that the radio-
activity increases chiefly from the top to the bottom, i.e. with the
increase of slime of the river and with the decrease of organic
material; boring 2 with 0.43, boring 4 with 0,54 boring 6 with
0.55, boring 8 with 0.61, boring 9 with 0.78, boring 12 with
2.16 > 10-12! But why that slight fall in borings 10 and 11? It
is exactly in these two samples that there is a larger quantity of
sand. Sample 9 still consists of very black, fat clay with much
fine mineral dust, the few grains of sand being 100—150 u long
at most. Sample 10: watery mud of very fine mineral dust (slime),
mixed with more grains of sand, somewhat coarser, at most 350 u
long; Sample 11: Bluish dark-grey, because the slime is mixed with
a few more grains of sand, very much particularly fine mineral
dust, grains of sand at most 300 qu long; small remains of peat and
wood. (Treatise p. 38). While in going from 9 —12 one might expect
a further rise of radio-activity, this rise is temporarily suppressed,
because layers 10 and 11 again contain more and larger grains of
sand, especially 11 a few more remains of peat. Only when this has
entirely passed in 12, there appears a rather high number. But in
sample 13 the activity falls again to the value of 10 and 11 and
concerning this was noted down at the time: fat, black clay, many
remains of peat, very much fine mineral dust mixed with somewhat
larger grains of sand at most 200—400 u long (Treatise p. 39).
Then again a fall, if mixed with more organic substance and larger
grains of sand. So in its details it also tallies entirely with the
conclusion drawn higher up, but at the same time it appears from
this of how much geological value accurate sensitive determinations
of radio-activity can be.

Finally we may yet draw the attention to the numbers occurring
in table 2 as collected by Miss Former. There the radio-activity is
the stronger as the given number of seconds is smaller. We refer
to the table and to Fig. 1. Where 4 or 5 samples were taken from
one spot, one higher than the other, it is again striking that the
smallest action is always found in the highest layers that were
bored; that first the activity increases in the downward direction,
is strongest at 150 cm..depth generally and often decreases again
at 180 em. The same may apply here as to the great boring, that
the upper layers contain a larger quantity of organic material. The
layer of clay goes down to the depth of 13 m. on the spot of
boring, on the spots of these shallow gaugings to 5 m. at most,
generally far less deep, so that for the rest one can hardly compare
the same depths. Moreover, it will be a good thing to pay attention

i es eee

735

to numbers 38, 45, and 46. These are the most weakly active, the
only ones where the value rose beyond 3 minutes and it is exactly
these samples that consist principally of sand. Besides these 3 samples
there are only two left (87 and 39) which also chiefly contain sand,
but these too belong to the least active ones. In 37, 38, 39, 45,
and 46 the layer of sand was already present at 60, 60, 60, 30,
and 30 cm. below the level of the sea.

So here it is once more corroborated, that radio-active action
grows less according as the sample of the soil contains more sand
or organic substance. The curves in Fig. 2, when compared mutually
or with the above-mentioned particularities of the soils may demon-
strate clearly, that two very different methods have led to this same
conclusion. That sample 12 shows a surprising difference in the two
methods is very striking indeed. May be that it points out that the
very high number of sample 12 in this series is yet of particular
importance. .

Mathematics. — “On Elementary Surfaces of the third order”.
(Third communication). By Dr. B. P. HAALMEYER. (Commu-
nicated by Prof. L. E. J. Brouwer).

(Communicated in the meeting of September 29, 1917).

It has been proved that F* cannot exist if that surface does not
contain at least one straight line. It will now be shown that if £*
contains a line’), this surface still cannot exist if in no plane through
that line the section consists of three lines.

We start from a line a on #* and assume that in no plane
through a the resteurve consists of two lines. It will be shown
that this assumption leads to contradictory results.

Theorem 1: Every point of line a has a tangent plane.

Let A be an arbitrary point of a and 8 a plane through A not
containing a. A cannot be isolated in p because there are points of
F* on both sides of 8 inside any vicinity of A. Hence in 8 a curve
passes through A. On this curve we choose a sequence of points
A,, A,... converging towards A from only one side. Let @,,a,...
be the planes passing through a and through A,, A,... respectively
and let « be their limiting plane (obviously «a is the plane through
a and the tangent at A to the curve in 8). In every one of the
planes «,,a,... is situated a curve of the second order, passing
respectively through A,, A,...

Three possibilities are to be considered:

1. The curves of the second order contract towards a or part of a.

2. The curve in the limiting plane « consists of a and an oval
which intersects a at A.

3. The curve in the limiting plane « consists of @ and an oval
which has a for tangent at A.

1. The curves of the second order contract towards a or part of
a. This: part of a anyway contains the point A. Each curve of
the second order divides the corresponding plane a, in two regions ’).

1) Again line will be used for straight line.

*) An cannot be isolated in a, because the curve in @ intersects the plane ap.

Neither can the restcurve in zj consist of a line counting double, as we assumed
that no second line of F3 intersects line a.

737

We call internal region that one which contracts towards a or part
of a only. Now if A continued to belong to the eternal regions,
the curve in plane 3 would show a cusp in A with both branches
arriving from tbe same side of the tangent. and this is excluded.
The possibility might be put forward that for every m the oval in
a, has the line of intersection 6, with 8 for tangent in A„. in other
words that the two points of #'*, which 5, carries besides A, coincide.

Ay

Fig: 11.

As follows can be shown that this possibility is excluded. In the
second communication (Second part, theorem 1) we proved:

If a line a in a plane « intersects the curve in that plane at an
ordinary point A, then lines which converge towards a end up by
carrying points of /* converging towards A. The demonstration
we used there, also holds if A lies on one or more lines of F?,
provided A is not situated on such a line in plane a. We proceed
to apply this theorem to the case of fig. 11. In plane 2 the line 5,
intersects the curve (which is no straight line) at the ordinary point
A,. In plane @, however it would be possible to find a sequence
of lines converging towards 6, but carrying no points of #* which |
converge towards A,: a contradiction.

Hence A will end up by belonging to the internal regions of the
ovals ') and considering this region together with its boundary con-
tracts towards a or part of a it follows that every plane through A
not containing line a has a point of. inflerion at A with tangent in a.

}) We exclude the possibility that A continues to lie on the ovals themselves.
The cases in which A belongs to an oval in a plane through a will be dealt with
sub 2 and 3.

738

Sections in planes through a will be dealt with later on.

2. The resteurve in @ is an oval which intersects a at A. In «
four branches depart from A: AB and AC on a and AE and AD
on the oval. Regarding the connection of these branches the JORDAN
theorem for space leaves only two possibilities.

Fig 12.
First possibility: AC and AD are connected by I, AD and AB
by I], AB and AF by III and lastly AM and AC by IV. If land
IV were situated on the same side of « then a parallel linesegment
converging from that side towards £’ D’ would end up by having
two points in common with I and two with IV: a contradiction.
If I and II were situated on the same side of «, then a parallel
linesegment, converging from that side towards A’ D’’ would end
up by having two points in common with [ and also two with II:
a contradiction.
In the same way it can be shown that III and IV cannot lie on
the same side of a. Combining these results it appears that the
connecting sets of points are situated alternately above and below a.

Fig 13.

Second possibility. The following is a representative case: I con-
nects AB and AC above « and III connects AZ with AD below a.
AC is connected with AD above or below « by II and lastly AB
with AH above or below a by IV. Let parallel linesegments con-
verge towards D’C’ from that side on which II is situated. This

739

line ends up by carrying two points of II. Besides it has a point
in common with either I or III converging towards C" or D" and
lastly it carries a point of F° converging towards the second point
of intersection of D/C’ and the oval in a. Altogether four points.

It thus appears that the second possibility is excluded and we
need only consider the first.

In § 3 of the first communication we proved: /f A is double
point in a plane a, and cusp in not more than ene plane, then a is
tangent plane, assuming that no line of #* passes through A. Here
however one of the branches passing through A, is a straight line.
This is the only one, as we assumed that no second line of #'°
intersects the lie a on which A is situated. Hence in no plane through
A except those passing through a, can the curve contain a line through A
and the demonstration of § 3 still holds. The results obtained for
planes through the tangents at A in a also remain valid for the
planes through the tangent at A to the oval in a. Regarding
the curves in planes through the Zine in « however (which line
corresponds to the second tangent of the former case) the demon-
stration says nothing. These shall be dealt with later on. Also the
first part of § 3 where the connection of the branches is examined,
has to be slightly altered, but this has been done already above.

In order to be able to use the former results here, it remains to
prove the following theorem:

Tf a line of F* passes through A, which line is not intersected
by a second one, then A cannot be cusp in more than one plane
(we give a fresh demonstration as the former one must be altered
a good deal).

A is situated on the line a of #* and is cusp in a plane 3 which
of course does not contain a. Let @ be an arbitrary plane through
a not containing the cuspidal tangent in 8 and 6 the line of inter-
section of « and 8. Line 6 carries except A only one point B of
F*, In plane a the point B cannot be isolated, as the curve in 8
crosses a. Neither can the restcurve in a (that is: the curve minus
a), according to our assumption, consist of two lines and the only
remaining possibility is that the resteurve is an oval through Bb.
This oval also passes through A, because 5 has only the points A
and B in common with /* (that the oval cannot have 6 for tangent
in B follows from the same reasoning which shows that fig. 11
represents an impossibility.) *)

1) Here we are not entitled to use the theorem given at the end of the first
communication, because this was only proved for points not situated on a line of
F3, and it is not excluded that B lies on ‚such a line.

740

Hence in every plane through a, not containing the cuspidal
tangent in 8, the restcurve is an oval through A. Passing on the
limiting case, it appears that in the plane through a and the cuspidal
tangent in 2 the curve consists of a only, or a together with an
oval through A.

Furthermore it appears that an arbitrary line through A (== a) >

carries at the utmost one point of #° different from A. But in no
plane can A be isolated (because a furnishes points of /* inside
any vicinity of A, on both sides of every plane not containing a),
hence in any plane which does not contain a the point A is either
cusp or double point. Concerning the planes through a it appeared
that A is double point in every one of these with the possible ex-
ception of the one through a and the cuspidal tangent in 8, in which
case the curve in that plane consists of a only.

So far we only assumed A to be cusp in a single plane 8. Now
let A be cusp in two planes 8 and y. We shall consider separately
the cases that only one or more than one line can be cuspidal
tangent at A.

First case. A is situated on the line a of F*. Let 6 denote the
only line through A which can be cuspidal tangent and let « be
the plane through a and 6. The foregoing results show that there
are only two possibilities :

I. The curve in «a consists of a and an oval through A.

I]. The curve in a consists of a only.

I. Let c be a line through A in a, not being tangent to the oval
and not coinciding with a or b. Only line 6 can be cuspidal tangent
at A hence in every plane throngh c (|= a) A is double point,
but A is double point in « also, hence A would be double point
in every plane through c, and ec cannot be tangent in any of these
planes because c carries besides A a second point of #*. This how-
ever cannot be, as may be shown in the same way as in é 3 of
the first communication. The fact that here A lies on a line of #?
makes no difference as the demonstration merely depended on the
connection of the branches dictated by the assumption that F° is
a twodimensional continuum. .

Il. Again let c be a line through A in « not coinciding with a
or 6. In every plane through c (== «) A is ordinary double point
and in « the curve consists of a only.

Let d be an arbitrary plane through c (== «). In this plane d
the line c is tangent at the double point A, hence in d on both
sides of c at least one branch joins A with the line at infinity (on

741

one side there even can be three, when the loop reaches the line
at infinity, but in any case there is at least one on either side). -
Now let d revolve round c. The curve in a limiting plane is the
limiting set of the curves in the converging planes (no isolated points
are possible here). Besides a sequence of infinite branches has an
infinite limiting branch. Hence it follows that in every plane through
ec (+= «) we can choose on both sides of c an infinite branch such
that they merge in each other in continuous fashion when d revolves
round c. If we add the line a in a these branches are just sufficient
to give #* the character of a twodimensional continuum in the
neighbourhood of A and the branches departing from A, which we
have left out, cannot be fitted in anymore. This contradicts our
assumption that /’* is a twodimensional continuum (of course the
neighbourhood of a point on a twodimensial continuum can in an
infinite number of ways be represented on the neighbourhood of a
point in a plane, but the neighbourhood of a point in a plane can
by (1,1) continuous transformation in the plane never be transformed
in anything but the neighbourhood of a point).

Second case. A is situated on the line a of F* and is cusp in 8
and y. The cuspidal tangents do not coincide, hence the line of
intersection 6 of 8 and y cannot be cuspidal tangent in either of
these planes. It follows that 6 carries besides A a second point 5
_ of F* and the curve in the plane @ through a and 6 consists of a
and an oval through A and 5. The line 5 divides 8 in two semi-
planes: in the one the cuspidal branches depart from A, hence in
the other A is isolated. In the same way A is isolated in one of the
semiplanes in which 5 divides y.

Now a foregoing demonstration (§ 5, second communication) shows
that in this case A is isolated inside the entire angle (< 180°)
between these semiplanes. Hence the line « belonging to F* cannot
pass through this angle and it follows that the semiplanes of B and
y in which the cuspidal branches depart from A, are situated on
the same side of the plane a through a and 5, let us say below a.
In a four branches arrive at A, consecutively AP, AQ, AR and ©
AS (two on a and two on the oval). Suppose above «, AP is
connected with AQ and AR with AS. Then line 4 must lie inside
the angles QAR and PAS, because planes pass through 5 in which
A is isolated above a. Let c be a line in « through A inside the
angles PAQ and SAR (this is impossible when the oval in a has
a for tangent at A, which case we shall consider separately). The fore-
going results show that A is double point or cusp in every plane

53

Proceedings Royal Acad. Amsterdam. Vol. XX.

742

through c. In any such plane however two branches arrive at A
from above «a, one on the set which joins AP to AQ and the other
on the set by which AR is connected with AS. Now if A were
cusp in a plane through c then, considering the branches arrive
from above a, at the utmost one plane could pass through 5 in
which A is isolated above « and this contradicts the above results.
Hence A must be double point in every plane through c: a contra-
diction. |

It now remains to consider the case that the oval in @ has a
for tangent at A. We shall consider separately the following possi-
bilities :

I. There exists a semiplane through 6 above «a in which A is
not isolated.

II. No such semiplane exists.

I. A is not isolated above @ in a plane J through 4. Then in
the semiplane of Jd above « two branches depart from A, because
A is cusp or double point in every plane not containing « and line
6 has a second point B in common with #*. From the way in
which the branches meeting at A in a are connected, follows that
in every plane through 5 two branches arrive from below «a, hence
A is ordinary double point in d. Here, there. is no danger of aline
as we assumed that no second line of #° intersects a. If d revolves
round 6 then in one of the two directions A will remain ordinary
double point till d coincides with «.

From this it follows that the semiplanes of 8 and y in which the
cuspidal branches depart from A, are situated on the same side of
d, let us say below d. In d we now choose a line d through A
separated from 5 by the tangents at A. The same reasoning used
before shows that A would have to be double point in every plane
through d. Only for the plane through a and d a slight alteration
is required, which however is selfevident. The impossibility of
assumption I has thus been proved.

Il. A is isolated in every semiplane through 5 above a. In every
_plane through A not containing a the point A is double point or
cusp, hence in every plane through 6 (=a) A is cusp and all the
branches arrive at A from below a. It follows that A must be cusp
in every plane except a, all branches arriving at A from below a.
This however is only possible if the cuspidal tangents form one
plane « through a, which plane has nothing but line a in common
with #’*. Let a sequence of planes ¢,,¢,..... all passing through
a converge towards e. In each of these an oval passes through A.

Now suppose the oval in & crosses the line a at A. Then four

743

branches arrive at A in e,, forming finite angles. These branches
are connected alternately on different sides of &,. Through A in gs,
we can at once find a line through which pass two planes having
a cusp in A and such that in both the cuspidal branches arrive
from the same side of «,. Then in the same way as before we can
once more obtain a contradiction.

It now remains to consider the possibility that for every n the
oval in «, has a for tangent at A. For increasing 7 these ovals
contract either towards A only or towards a connected part of a
containing A.

If A is the only limiting point, then the contracting ovals would
give to A the character of a point of a twodimensional continuum
and a sequence of points on a having A for limiting point, could
not be fitted in anymore.

If on the other hand the limiting set is an interval on line a then
the internal points of this interval would, in planes not containing
a, be cusps with both branches arriving from the same oe of the
tangent and this is also excluded.

We thus have proved that every plane through A, containing
neither line a, nor the tangent at A to the oval in a, has an
ordinary point in A with tangent in a. The planes through the tangent
at A to the oval in a have point of inflezion in A with tangent in a

There remain to be considered the curves in planes through a.
„These shall be dealt with presently.

3. We now come to the third possibility mentioned on page 736.
The resteurve in « consists of an oval having a for tangent in A.
In « there depart from A two branches AB and AC on a and AF
and AD on the oval. In almost the same way as before it appears
that here AC is connected with AD, AD with AZ, AE with AB
and lastly AB with AC. The connecting sets of points are again
situated alternately above and below @. This being established the
further reasoning used for case 2 holds here without any alteration
(again we remind the reader of the assumption that no second line
of F* intersects a). Results: Jn every plane which does not contain
line a, the point A is ordinary point with tangent in a (in all these
planes the branches depart from A to the same side of «).

The curves in planes through a must be considered still.

In each of the three above cases, « was found to possess the

744

character of tangent plane, only we had no certainty with regard
to the curves in planes through a. Now all possibilities have been
considered it appears that for no point A two different planes can
pass through a both possessing one of the examined characters (we
obtain an immediate contradiction by considering a plane through
A not containing a). It follows that in the three above cases no
plane through a (== «) can contain branches departing from A (except
a itself). This completes the demonstration that « is tangent plane. |

Theorem 2: Lf A moves continuously along a, then the tangent
plane also changes in continuous fashion.

Let the points 4,,A,.... on a converge towards A. Tangent
planes a,,a,....a@ all passing through a. We assume that @,,a@,....
have a limiting plane «’ different from @ and shall prove that this
leads to a contradiction. Let 8,,8,....8 be planes respectively
passing through A,, A,....A and all 1 a. The line of intersection
of a,, and 8, we denote by 6,, the one of a, and 9, by b, ete.
Lastly let 6 be the line of intersection of « and 8 and 6’ the one
of a’ and g. According to our assumptions 6’ and 6 do not coincide
anid.’ is the limiting line of 5,,.b, ...- |

Now 6 is tangent at A to the curve in 8 and in the converging
planes #8; the curves have for: tangents at A, A, uo
respectively the lines b,,b,.... converging towards 6’ in 8.

According to theorem 2 of the second
communication the curve in gis the limiting
set of the curves in @,,8,.... (with.the
possible exception of an isolated point).

Let c and d be lines through A in 8
separating 6 from 6’. The corresponding
planes through a shall be denoted by y and d.

For n large enough a branch departs
in 8, from A, in both directions inside
those opposite angles between y and din
which 6’ is situated. Loops contracting
towards A are evidently excluded, hence in order that in 8 no branch
departs from A inside those angles of c and d¢ which contain 0’, it
is unavoidable that in the converging planes the above mentioned
branches leave these angles via points of the planes y and d (or
one of these) converging towards A. Hence in at least one of the
planes y and d the point A would be limiting point of points of
f* not situated on a. This means that in one of those planes a
branch deparis from A different from a, but this is a contradiction

745

considering that only in the tangent plane « a second branch can
pass through A. This completes the demonstration of theorem 2.

Let an oval in « cross the line a at A. This oval and the linea
have a second point of intersection B and the points A and B have
the tangent plane @ in common. If A moves continuously along a
then, according to ¢theorem 2, the tangent plane « also changes in
continuous fashion and the point B also moves continuously. *) From
this follows that a point A at which an oval crosses a, can only
be limiting point of points of a possessing the same character.
Besides it is easy to prove that the tangent to the oval at A also
changes continuously. ‘This result however will not be needed, but
we do want the following:

Let A,, A,... on a converge towards A. Tangent planes a,, a,...a.
If the oval in a crosses a at A it follows from the above that for
n larger than some finite value the plane a, also shows an oval
which crosses a at 4,

Now suppose all these ovals in «, turn at A, their concave sides
to the left. The oval in a is the limiting set of the ovals in @, and
considering a sequence of finite concave branches cannot converge
towards a finite convex branch, it follows that the branch in «
through A also turns its concave side to the left.

Taking these results together we obtain:

Theorem 3: A point of line a in the tangent plane of which an
oval crosses a, can only be limiting point on a of points having the
same kind of tangent plane also with regard to the side to which
the ovals through those points are concave or convex.

Theorem +: F* cannot ewist if the resteurve does not degenerate
in any plane through a.

We consider the case in which the curves of the second order
in the planes a,,a,.... (passing through a and converging towards @),
contract towards part of a. We call internal region of these ovals
that region which contracts to nothing but a. We found that the
points of a belonging to this limiting part must be situated in the
internal region of the oval in a, for n larger than some finite
number. From this follows that the part of a belonging to the

1) This theorem and some others which shall be formulated presently concerning
the directions in which A and B move, have already been given by Juve, Math. Ann. 76,
p. 552. The existence and continuous changing of tangent planes is simply
postulated by that author.

746

internal region of the oval in @, must diminish for increasing n,
because if the oval in a, crosses a at A, and B, then a, is tangent
plane at A, and in case A, ended up by being situated inside the
ovals, « also would be tangent plane at A,: a contradiction.

Hence if the ovals have the entire line a for limiting set, none
can bave points in common with a. An idea of this case may be
got by imagining a sequence of hyperbolas of which the angle of
the asymptotes (inside which the hyperbola is situated) tends towards
180° and such that the centre is situated on « and both asymptotes
converge towards a.

In this case everything is in favour of counting a as a triple line
in @. In no plane through a a branch would depart from any point
of a, and except a, #* would contain no straight tine.

A second possibility we wish to consider separately is that in the
tangent plane of every point of a the oval has a for tangent. Again
let A be a point of a with tangent plane a. The line a divides a into
two semiplanes, in the one 4 is isolated and ia the other an oval has
a for tangent at 4.

Now let A move along a. The plane « turns round a. If A moves
on in the same direetion the plane « goes on turning in the same
direction, for otherwise two points of a might be found with the
same tangent plane and this cannot be as in either point an oval
in the tangent plane must have « for tangent.

It A goes round the entire line a the tangent plane meanwhile
turns 180° round a. The ovals in the tangent planes merge conti-
nuously into each other, hence after turning 180° the branch having
a for tangent is situated in the wrong semiplane. This means that
on the way the branch must change from the one semiplane into
the other and this is only possible either via a tangent plane in
which the resteurve consists of two lines through A, or via a tangent
plane in which the oval has contracted to nothing but point A. The
first possibility is excluded according to our assumption and the last
would mean that a sequence of ovals in the converging planes
contract towards a point of a not belonging to the internal regions
of the converging ovals. This was found to be impossible hence the
assumption that every point of a has a tangent plane with oval
having a for tangent, leads to contradictory results.

Leaving apart both cases treated above, there certainly exists a
plane throngh a in which an oval has two different points A and
B in common with a. Let this plane a revolve continuously in a

747

certain direction round a. The points A and B then also move along a
continuously '). Two assumptions are possible: A and B can move
in the same or in opposite directions. Let the direction be the
same. In the time that 6 has described the original segment BA, the
point A has gone further, hence ail this time we keep tangent planes
with ovals having two different points in common with a.

When B arrives at the original place of A the plane « must
have turned an angle of 180°, but if the branch through B has
originally been concave to the left, it must now be concave to the
righthand side, and this is not possible as on the way the concave
side in B cannot jump round and no: change from concave to convex
can have taken place via a degeneration of the oval in two straight
lines (according to our assumptions).

The second possibility was that A and B move in opposite direc-
tions. Let the tangent plane successively turn round a in opposite
directions, then we obtain two different points in which A and B
meet. Such a meeting takes place either when the two points of
intersection of a and the oval converge to one point or when the
entire oval contracts to nothing but a single point on a. In both
cases the concave sides of the branches through A and 5 face each
other. A priori it seems possible that before the meeting the convex
side of the branches through A and B face each other, but then
these branches would be connected on both sides via the line at
infinity and in the limiting plane the oval would degenerate in two
straight lines’) through the point where A and 5 meet and this
contradicts our assumptions. |

Now we start from the original position of A and B and we
observe A only. Let the branch through A turn its concave side to
the left. If we turn the tangent plane in such a way that A moves
to the right, then the concave side goes on being turned to the left.
But before the meeting with B takes place the concave side must
be turned to the right (that is in the direction in which A moves)
and this means a contradiction because the curvature cannot change
its sign Ciscontinuously, neither can it change via a degeneration of
the oval in two straight lines (according to our assumption). This
completes our demonstration.

Remark. Above we spoke about the possibility that the oval

1) If « goes on turning in the same direction A and B obviously cannot change
the direction in which they move for then points of a would exist with two
different tangent planes.

*) The oval does not converge towards a.

748

through A and B contracts to the point where A and B meet. The
most rational thing to do is to consider this meeting point as a
special kind of oval in the tangent plane. We can also imagine
that the oval through A and B contracts to a segment of a. All
points of this segment would have the same tangent plane (tangent
plane of the first kind, examined at the beginning). Now the admis-
sion of this possibility has the disadvantage that we should be more
or less forced to consider the linesegment in the tangent plane as
a special sort of oval and going back to the definition of elementary
curves we should not only have to admit isolated points, but line-
segments also. This would eause the development of the theory to
become a good deal more complicated but the enlargement of the
results would probably remain very trivial. To mention an example;
to the elementary surfaces of the second order would be added the
plane convex regions including the boundary and the linesegment.

Far greater would the changes become if we also dropped the
condition that the convex are is not to contain linesegments. This
however would mean an entirely different problem.

a"
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En ee be KLIJKE A
_ VAN _ WETEN

| VOLUME XX
“— (187 PART).
(N°. 1-5)

JOHANNES MULLER :—: AMSTERDAM —
: MARCH 1918 — : See

MUR A

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REAL Ye ted

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ranslated from: Verslagen van de Gewone Vergaderingen der Wis- en |
Natuurkundige Afdeeling DI. XXIV, XXV and XXVD.

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