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Koninklijke Akademie van Wetenschappen
te Amsterdam,

PROCEEDINGS

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Afdeeling van 28 Mei 1904 tot 22 April 1905. Dl. XIII.)

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Koninklijke Akademie van Wetenschappen
te Amsterdam.

PROCEEDINGS

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AMSTERDAM,
JOHANNES MULLER.
December 1904.

(Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige
Afdeeling van 28 Mei 1904 tot 26 November 1904. DI. XIII.)

Digitized by the Internet Archive
in 2009 with funding from
University of Toronto

http://www.archive.org/details/p1proceedingsofsO7akad

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1904
June 25 >
egicualies 24 >
October 29 »
Noyember 26 >
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= 1
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- 263
Sod

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING

of Saturday May 26, 1904.

Doce

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 26 Mei 1904, Dl. XII).

Cleo WE ING EES.

H. P. Barenprecnr: “Enzyme-action.” (Communicated by Prof. J. M. van BemMecen), p. 2

J. P. van per Srox: “On a twenty-six-day period in daily means of the barometric height”, p. 18.

J. W. Lancevaan: “On the form of the Trunk-myotome”. (Communicated by Prof. T. Piacr),
p. 34. (With one plate).

Eve. Dusors: “On the direction and the starting point of the diluvial ice motion over the
Netherlands.” (Communicated by Prof. J. M. van BemMeven), p. 40.

Frep. Scuvu: “On an expression for the class of an algebraic plane curve with higher
singularities.” (Communicated by Prof. D. J. Korrrwee), p. 42.

H. E. pe Bruyn: “Some considerations on the conclusions arrived at in the communication
made by Prof. Eve. Dvsois in the meeting of June 27, 1903, entitled: Some facts leading to
trace out the motion and the origin of the underground water of our seaprovinces”, p. 45.

Eve. Dusois: “On the origin of the fresh-water in the subsoil of a few shallow polders.”
(Communicated by Prof. H. W. Baxucis Roozesoom), p. 53.

C. A. Losry pe Bruynand S. Tymsrra Bz.: “The mechanism of the salicylacid synthese”, p. 63.

J. J. Brayksma: “On the intramolecular oxydation of a SH-group bound to benzol by an
orthostanding NO,-group.” (Communicated by Prof. C. A. Lopry pe Bruyy,, p. 63.

J. M. M. Dormaar: “The inversion of carvon and eucarvon in carvacrol and its velocity.”

(Communicated by Prof. C. A. Lopry pe Bruyn), p. 63,

The following papers were read:

Proceedings Royal Acad. Amsterdam. Vol. VII.

(2)

Chemistry. -— “/nzyme Action.” By Dr. H. P. Barenprecur.
(Communicated by Prof. J. M. vax BremMe.en).

(Communicated in the meeting of April 28, 1904.)

The following is a- preliminary communication of the writer's
researches on enzyme actions during the last two years.

From the commencement it has been the writer’s object to ascer-
tain in how far a continued research of simple enzyme actions might
confirm the hypothesis that the enzymes exert their catalytic action

eeriot by radiation. This hypothesis ori-
canesugar. ginated in the peculiarity of the
action of the enzymes which
distinguishes this action so sharply
from that of the acids. A graphic
representation of the action of the
same quantity of acid or enzyme

in the same time on sugar solut-

Initial concentration of the canesugar. ions of different concentrations

Fig. 1. ; aA
= renders this difference very per-
ceptible.
Inverted canesugar. Fig. 1 gives a scheme of the

inversion by acids. The line which
remains straight indicates that the
quantity of inverted canesugar
remains proportionate to the initial
concentration.

In the case of enzymes the
Initial concentration of the canesugar. general course is represented by
See fig. 2. In the inversion of cane-
sugar for instance, the line remains straight up to an initial con-
centration of O.1°/,; it then inflects towards the « axis and runs
henceforth parallel to this.
This characteristic behaviour of the enzymes is now at once
explained by the radiation theory which will be further developed.
Let us, for the sake of convenience, confine ourselves to the action
of invertin and let us suppose we have two solutions containing,
respectively, 20°/, and 10°/, of canesugar. In the 20°/, solution the
radiation from each enzyme particle will be comparatively soon
absorbed by the surrounding molecules of canesugar; in the 10°/,
solution the sphere to which the enzyme action can extend will
be larger. So long as the solution is sufficiently concentrated to
finally absorb by a sugar molecule all radiation emanating from an

|

55:5)

enzyme particle, before the distance has become so great that the
radiation fails to cause inversion, each enzyme particle is bownd to
exert the same action. One might compare an enzyme particle in
concentrated sugar solutions with a source of light in a foe of
varying density; the denser the fog the smaller the region around
the source of light which absorbs all the light.

If, however, canesugar absorbs the radiation from invertin, we
must expect the same to a greater or smaller extent from the products
of inversion, On account of this power of absorbing the active rays
these products must retard the inversion.

The result of my often-repeated experiments showed that the inversion
of canesugar by invertin prepared from carefully dried yeast (we
shall see, presently, that the method of preparing the invertin is of
the greatest importance) is retarded equally by glucose, laevulose and
invert sugar,

For instance, the same amount of yeast-extract inverted under the
same conditions *) from

10°/, canesugar 49.3°/,
10°/, canesugar + 5°/, elucose Boron

10°/, canesugar + 5°/, laevulose 38ta0/

10°/, canesugar + 5°/, invert sugar 38.3°/,

0
From the similarity of the last three figures it is already evident
that we are nof dealing here with a retardation due to a reversed
reaction.
It was further ascertained that the other hexoses cause exactly
twice as much retardation as glucose or laevulose :

8°/, canesugar 45.6"/, inverted
8°/, canesugar + 2"/, galactose 35.5°/, *
8°/, canesugar + 2°/, mannose 36.1°/, r
8°/, canesugar + 4°/, glucose 36.1°/, 3

From these results it is evident that the inversion phenomena
behave as if there are emitted by an invertin particle two radiations
in equal quantity which we may call, provisionally, glucose and
laevulose radiations. Each radiation by itself is capable of inverting
a canesugar molecule; the glucose radiation is not absorbed by the
glucose but by the laevulose; the laevulose behaves, conversely,
in the same way. In accordance with this both radiations are
absorbed by any other hexose. We may, therefore, regard invertin

1) All sugar determinations have been made by KseLpauu’s accurate gravimetric
process,

(4)

as being, probably, a proteid containing glucose and laevulose groups
in a peculiar radiating condition.

At all events, the radiation hypothesis will be found to lead to
further quantitative research, to explain results already obtained and
to predict future results.

Let us take a sufficiently concentrated solution of @ grams of
ranesugar in LOO ¢.c. of water. A given amount of invertin then
yields in the first minute a quantity of invert sugar i, independent
of a. As soon, however, as a little invert sugar has been formed, the
condition changes. The enzyme rays are now not only absorbed by
the canesugar but also by the invert sugar. If we call x the absorption
power of the invert sugar in regard to that of canesugar, then after
the lapse of a time ¢ when « is the remaining canesugar, the inversion
velocity —dzx will be no longer m, but

dt.

= du — m

a+n(a = av)

a—w

If we substitute =y and integrate, we obtain the equation

a

( 1 ) l—n m
tj —— ]+ y= —t
1—y n * na

or, using ordinary logarithms

1 (—n m
- 0,434 y = — 0,434 t.

—y n na

log

In this equation two constants occur. The first im may be at
once determined experimentally from the initial velocity. If we take,

m :
for instance the experiments of A. J. Brown*), then — or the fraction
a

0,130

inverted per minute at the commencement is . If in our equation

we substitute this value of m, we find during the whole series of
Brown’s figures a value for « of about 0.5.

It is, therefore, evident that the absorption powers of the cane-
sugar and glucose, or laevulose molecules are in the proportion of
about 2 to 1, that is to say in the proportion of their masses or,
perhaps, surfaces.

Then we found the retarding influence of glucose and laevulose
to be equal to that of invert sugar. Per unit of weight the number
of molecules in glucose for instance and in canesugar are in the

1) Journ. Chem. Soc. 1902 pag. 377.

(5)

342
proportion of 180° If now the absorption power of a molecule of
glucose stands to that of a molecule of canesugar in the proportion
of 180:342 and if we consider that glucose transmits without
hindrance 50°/, of the total invertin radiation, the proportion between
the absorption power of one part of glucose and that of one part
of canesugar becomes:
Seen SON i 11
180° 342°2 2°
fn : 2 ; 360
After inversion, one part of canesugar yields 5
e a
sugar. Therefore n, the relative absorption power of the produets
ze ; c : 360 1 ie of
of inversion of one part of canesugar, becomes 575 — 10,020

42 2

parts of invert

The formula for the inversion velocity thus becomes
1 MAC
log —— + 0,393 y = 0,827 —¢.
1l—y a

Browy’s experiments conform still better to this formula than to
Ske lte
Henri’s empirical formula 2/, = P log ae

—— 1

The correctness of our deduction may further be proved experiment-
ally in the following way,

If in addition to the a grams of canesugar 6 grams of glucose,
or laevulose are dissolved per 100 ce., the inversion velocity will be
represented by :

uv
— dz = —____ —— dt.
vt+n(a—er)+hb
, , 3 a—wv , A *
By again substituting -=y we obtain, when using ordinary
a =
: : m :
logarithms and calling — 0,434 =:
na
1 ’ 1 1
log ani + 0,893 moo — ere kt
—y »
14+s—- 14
2na 2na

A same enzyme quantity acting under the same conditions in a
solution containing canesugar only and in one containing canesugar
plus glucose or laevulose gave the following figures :

(6)

10°/, canesugar. 10°), canesngar + 5°/, laevulose.
reas +0,393 :
og— woe =
1 “l-y b*
log +0,393y : 1+——
l-y 3 2na
t y hk t y K=—_____————
t : t
91 0.412 0.00432 85 0.297 0.00273
191 O72 0.00437 181 0.555 0.00276
2514 0.72 0.00296
306 0.763 0.0027
360 0.824 0.00271

According to our formula we have:

Jot = fet. k = 0.68 & = 0.00295
i ee
2na
10°), canesugar. 10°/, canesugar + 10°/, glucose.
t y k t y Ki
57 0.191 0.00298 57 0.124 0.00145
118 0.364 0.00287 116 0.237 0.00142
242 0.642 0.00288 237 0.485 0.00142

295 0.522 0.00144
358 0.604 0.00146

k = 0.00149.

calculated 4’ = ——
eee
2na
A further control is given by the determination of the initial
velocity after previous addition of glucose or laevulose.
From
a(1—y)

= 1 = dt
a(l—y) 4-22 y--— b

ady =m

follows that the initial velocity in a solution of a canesugar +

glucose, or laevulose is:

dy im 1
dt a) a 16

1
1 21a
: , dy m
whereas without such addition {| — ==,
lt y=0 a

These are the experimental results;

10"), canesugar 10°/, canesugar + 2.5°/, laevulose
1
inverted 14°/, (PAB) Sea Lee — te
)
Tie ae
1g 2 a
10°/, canesugar 10°/, canesugar + 5°/, glucose
é . us 1
inverted 12.4°/, et ee 12 4 99
)
| ce a a Na
+ 2a
10°/, canesugar 10°/, canesugar + 10°/, glucose
1 ;
inverted 19.1°/, 12.4°/, — Na = Pet
; Aga
1 Dae
8°/, canesugar 8°/, canesugar + 16°, glucose
1 :
inverted 16.6° a Gone (a ie IGG S 3!
,
ieee pan
2 a

It was now to be expected that many other neutral substances
would also retard the inversion according to their capacity of
absorbing the enzyme radiation.

These are some of the figures obtained:

Under the same conditions the same enzyme-quantities inverted of

10°/, canesugar 38.5°/,
10°/, canesugar + urea 28.5°/,
10°/, canesugar + 5°/, mannitol 33.—"/,
10°/, canesugar + 5°/, erythrite 28.—°/,
10°/, canesugar + 5°/, glucose 29.8"/,

In another series :

10°/, canesugar 58.1°/,
10°/, canesugar + 5°/, dulcitol 50.4°/,

10°/, canesugar + 5°/, glucose 47.8°/

There seems to be some kind of relation between the asymmetric
carbon atoms and the absorption.

In the case of inversion of more diluted solutions of canesugar
the above-mentioned simple relations will no longer exist. If we
diminish the initial concentration, a dilution will soon be reached
where a part of the radiation does not reach a sugar molecule
in time, but is either finally absorbed by the water or when arriving
has, in any case, become too weakened to cause inversion. The

(8)

quantity of canesugar inverted by a given amount of invertin will,
therefore, go on decreasing. In the end, however, we shall arrive
at an initial concentration where, within the sphere of action of an
enzyme particle, two canesugar or invert sugar molecules can no
longer shade each other. From this point, the inversion caused by
the given enzyme-quantity will be just proportionate to the canesugar
concentration. Then, during the whole of the process, the reaction
velocity merely depends on the average number of canesugar mole-
cules present within the active radiation sphere of an enzyme particle.

The following are some of the figures obtained which always
exhibited the same regularity.

Concentration Inverted Inversion
canesugar in grms. in grms. per 100 ce. in**/§.
per 100 ce.
0.05 0.022 44.—°),
O.1 0.0448 44.8"),
0.125 0.0545 43.8°/,
0.25 0,097 39.—*/,
0.5 O.174 34.7°/,
1.— 0.240 24.— °/,
2.— 0.317 Gye

Another series gave

Concentration of Inverted
canesugar in gr. per 100 ce. in grms, per 100 ce.

3 0.86
4 0.95
5 0.96
7 0.93

The fact that, in very dilute solutions, the enzyme action really
proceeds as a unimolecular reaction according to the formula

1 : : .
k =-— log was further again confirmed by experimenting with a
ce h 5

l-y
solution containing 0,096 "/, of canesugar.

Up to the present we have for the sake of convenience disregarded
the synthetical action of the enzyme rays. Light, being a catalyzer, can
act either as a synthetical or decomposing agent, so we must expect the
same from the enzyme rays. That we often do not notice such
action is due in the first place to the secondary change of the

(9)

decomposition products, at least im the case of invertin. It has
already been stated by O'Suntivan and Tompson ') that invertin
separates glucose from canesugar in a birotatory condition. Taxrer’)
and Sion *) have afterwards elucidated this birotation question.

The birotatory «a-glucose is the sugar of the «-glucosides and _ the
semi-rotatory y-glucose that of the j3-glucosides; according to the
said authors they are the stereoisomeric lactones :

HOH HOH H HOH

, C — C — C — C — C — CH,OH.

|
| ats |

The form which in solution is stable, the S-glucose conforms to
the aldehyde-formula CH,OH (CHOH),COH.,

This conclusion is opposed by other investigators such as ARMSTRONG‘)
and Lowry’), who look upon the stable form not as an aldehyde
but as a condition of equilibrium between a- and y-glucose. My
investigation goes in favour of the first opinion.

Invertin is, generally speaking, the enzyme of the a@-glucosides.
Canesugar may also be considered as an a-glucoside in accordance
with the facet that on inversion, the glucose is always separated in
the a-modification. This a-glucose is now, however, gradually con-
verted into B-glucose and, therefore, prevents the reconversion into
canesugar. Owing to this, all the canesugar is always finally inverted
by invertin.

A continued research, however, showed that there may be still
another reason for the non-appearance of reversal phenomena, The
method of preparing the invertin, that is of the yeast extract was
found to greatly affect the properties of the enzyme.

At first, I used for the preparation of a powerful invertin a yeast
cultivated in a solution of canesugar. This yeast after being mixed with
“kieselguhr’, was first dried in vacuum at a low temperature and
then for half an hour at 100° in an ordinary oven.

The addition of “kieselguhr” facilitates very much the subsequent
extraction and filtration. The above experiments have been made
each time with a freshly prepared filtrate.

Afterwards it was found that ordinary yeast also gives an enzyme
with the same properties, provided it has not been dried at too high

1) Journ. Chem. Soe. 1890 pag. S61.

*) Zie Dictionnaire de Chimie de Wiirtz. 2e suppl. p. 764.
3) G. R. 1901 pag. 487.

4) Journ. Chem. Soc. 1903 pag. 1
») Journ, Chem, Soc. 1903 pag. 1

305.
314,

(10)

a temperature. Ordinary laevulose in solution is as a rule less stable
than glucose. In invertin a difference in the same direction is
also usually revealed. Drying at too high a temperature, heating the
“kieselguhr” mixture above 100°, or precipitation with alcohol and
redrying the precipitated enzyme, repeatedly gave invertin, the action
of which is retarded considerably more by laevulose than by glucose.
Active laevulose therefore generally becomes inert sooner than
active glucose.

This explains the difference between my results and those of
Vicror Henri ') who states:

“Pour laddition dune meme quantité de sucre interverti, le ralen-
tissement est d’autant plus faible que la concentration en saccharose
est plus grande. Ce ralentissement est produit presque uniquement
par le lévulose contenu dans le sucre interverti.”

Probably, Henrt has obtained his invertin from yeast dried at more
elevated temperatures or has used commercial invertin, prepared by
precipitation with alcohol. That the retardation of a same quantity
of invert sugar becomes smaller when the canesugar concentration
becomes greater is quite in harmony with the radiation theory. The
fact that the laevulose contributed most to that retardation was only
a pathological phenomenon of the invertin.

We must further bear in mind the possibility that, owing to those
harmful actions, the radiation gets so weakened that the reversion
can no longer take place, or that the radiating «glucose has been
converted into radiating ?-glncose and also that only the latter is
capable of inverting. It is certainly to be expected that, if only active
glucose or active laevulose is left behind, the power of causing
reversion has either decreased or been destroyed,

In order to counteract the first cause of the non-appearance of
the reversal, namely, the secondary conversion of a-glucose into
B-glucose, we may apply much enzyme and so accelerate the con-
version. A larger quantity of extract of the above yeast, which had
been finally dried for half an hour at LOO’, caused indeed a slower
inversion of the last remaining percentages of canesugar.

The reversed action was afterwards noticed more distinctly
with ordinary yeast, merely dried in vacuum at about 380°. We
will first give a mathematical formulation of the phenomena to
be expected.

Let us imagine an aqueous solution of invert sugar, liable to
reversal and consequently containing the glucose in the «-form, in

1). R. 1902 Noy. S4. 917.

(11)

the presence of invertin particles, which render this invert sugar
active. The velocity of synthesis will then be proportionate first to
the product of the concentration of glucose and laevulose and further
to the extent of the active radiation sphere surrounding each enzyme
particle. The latter is inversely proportionate to the joint concen-
tration of the invert sugar and the other dissolved matters eventually
present, each with thea own absorption coefficient,

The synthetical action of invertin in a solution containing (a—.)
erams of invert sugar and w grams of canesugar in 100 ce. is
therefore :

Ohh = [Ill] dt
w+tin(a—)

The complete formula for the inversion velocity of canesugar, in
case the original products of inversion suffered no change, would
then be‘):

x 1 t—u)?
— dx =m ie ——— p ( ) dt.
latnr(a—x) 4° x4+n(a—2)
The point of equilibrium would then be determined by the equation :

1 : 0 =
gy a2) = (0),

Or returning to relative fractions by substituting —— = y:

1 aa 0
ty Uh ea =

If now we have introduced into the solution such a quantity of
enzyme that this equilibrium point is attained before the birotatory
glucose has been converted to any great extent into ordinary glucose,
the inversion will not actually come to a standstill, but the line,
indicating its progressive course, will exhibit a characteristic pecu-
liarity in that place. Then, starting from that point, the formation
of fresh a-glucose by inversion will be dominated by the velocity
of the inversion of the total a-glucose present into B-glucose. This
velocity is proportionate to the concentration of the «@-glucose. If
-this is continually replenished by fresh formation of @-glucose from
canesugar, both the said velocity and the inversion velocity will be

1) The small inerease in weight when Cy, Hy: Oj; changes into 2 Cy Hy, O, is here
neglected; we might also suppose that it is taken into account in the coeflicient p.

By substituting the variable y this factor in p would in any case disappear again.

( 12)

constant. From this point the line,
representing the inversion as a function
of the time, will of course be a straight
one until the concentration of the cane-
sugar has become too small to make
the inversion keep pace with the glu-
cose transformation.

50/, canesugar With 5°/, canesugar (see fig. 3)
ihe line commences its straight course
at about y= 0,385. This substituted
into the equation

10
minutes. 1— y—— pay?=0
4 :
40 20 30 40 : 0,65 “all 4
Pig. s1ves 7 = —— or, practically = «4.
Fig. 3. 2 1 0.153 > | 4

In the 10°

solution (fig. 4) the equilibrium must then become

10°, canesugar.

minutes,

Fig. 4

perceptible at the value of y to be calculated from
i y— 107? —()

therefore, at 7 = 0,27.

Those equilibria phenomena are observed more readily in the
inversion of maltose by yeast-extract.

The enzyme which converts maltose into glucose is generally called
maltase so as to distinguish it from invertin. It seems to me that there
is no valid reason for making such a distinction. A yeast-extract,
which inverts maltose, has always been found to also behave actively

(13)

towards canesugar'), but not the reverse. This is in harmony with
the radiation theory. Maltose, like canesugar, is an a-elucoside. The
connecting point of the laevulose in the canesugar molecule with the
a-glucose is the C' of the carbonyl group of the laevulose. In maltose
the «glucose is attached to the CH, of the otherwise uncombined
molecule of glucose. Canesugar is also much more readily inverted by
acids than maltose. Both radiating «glucose and laevulose (probably
also radiating 8-glucose *)) are liable to invert canesugar. Maltose
is only converted by active «glucose, but as may be expected from
its behaviour towards acids and its constitutional formula it requires
& more powerful radiation than canesugar. If, therefore, yeast-extract
has been weakened by elevation of temperature or by precipitation,
its power of inverting maltose may have been lost or much diminished,
although canesugar is still fairly rapidly inverted.

The preparation of a yeast-extract with a powerful inverting action
on maltose proved to me no more difficult than when inversion of
canesugar was intended. The above-described yeast, derived from a
canesugar solution, and which had been actually heated at 100° for
half an hour, yielded after a year and a half an extract which
readily inverted maltose. In this case, I used, of course, by preference
a yeast which had been mixed with ‘“kieselguhr’ and dried at a
low temperature. On extracting the dried mixture, the solution never
contains zymase as experiment repeatedly showed.

Figures 5, 6, 7, 8 and 9 now
clearly show the phenomena of
equilibrium. If@-glucose in solution
were stable, only 15°,

60

5 Ot tine

1) maltose, . .
Dea total might be inverted by yeast-

extract in a 1O°/, maltose solution.
The fact that maltose is decom-
posed by yeast-extraet so much

minutes, ¢
slower than canesugar is partly

S 10 15 20 25

Nea

due to the circumstance that the
pomt of equilibrium is reached
so much earlier, The further decomposition then again merely keeps
pace with the transformation of «glucose into 3-elucese.

1) Porreviy, Annales Inst. Pasreun 1903. p. 31.

*) Separate experiments showed that glucose, previously heated and therefore in
the B-form, and glucose, dissolved immediately before adding the enzyme and there-
fore in the z-form, both retard the canesugar inversion to the same extent. B-Glucose
therefore transmits the glucose rays (then perhaps converted into @-glucose rays)
quite as well as the z-glucose.

(14 )

3° 0 maltose.
10

minutes.

3° /) maltose

10

minutes,

Sy
7°/) maltose.

minutes,
10 20 30 40 SO Pr - =
Fig. 8.

20
10°/) maltose,
10

minutes.

@15)

The points where the transformation lines commence to run straight,
so the points of equilibrium, also conform in this case to an equation
similar to that used for canesugar. Glucose is here the only product
of inversion; we might, therefore, expect that the undisturbed trans-
formation velocity would be here:

wv (a—.w)?
— dx =m 5} —— — — gq— aes dt

vt n (a — 2x) d w+ n(a- a)

and the equilibrium equation, therefore :

l—y—qay’=0.
For 10 grams of maltose in 100 ce. we experimentally found
g=OA5.
0.85 ‘ o55
This gives g= 0.995 OD practically, 4, therefore the same coefficient
as for canesugar ’).
If in the above equation we substitute ¢= 4; or

1—y--—4ay?°=0,

the calculated points of equilibrium become for

a Y

10 0.146
7 0.172
5 0.20
3 0.25
dl 0.39

This is therefore in accordance with the experiment.

The well-known researches of Crorr Hinn?) on the reversal of
maltose gave points of equilibrium which were situated at a more
advanced transformation; for instance in a 10°/, solution y= 0.945.
These equilibria were attained only after days and weeks; the
above cited after a few minutes.

Afterwards *) Hiri himself demonstrated that the resulting biose
Was not maltose but an isomer, which he called revertose. In Hirt’s
numerous experiments, all the glucose was no doubt in the #-form.
The synthesis found by Hint was therefore a combination of two
molecules of 3-glucose to a new biose, isomeric with maltose *).

1) The diffusion velocities on which depends the velocity of meeting of two
molecules, cannot differ much for glucose and laevulose.

*) Journ. Chem. Soc. 1898 p. 634.

5) Journ. Chem. Soc. 1903 p. 578. Exwertine (Ber. 34, p. 600) had found
isomaltose as reversal product.

') Most of the natural glucosides appear to be compounds of bi- or seml-rotatory
hexoses. When endeavouring to prepare lactose from galactose and glucose by

(16)

The retarding action of another added hexose is not studied so
readily in the case of maltose inversion as in the transformation of
canesugar, on account of the immediately occurring reversion. Still it
was found that both laevulose and galactose cause the same retarding
action, as might be expected from our theory now we are dealing
with glucose-radiation only.

For instance, a same amount of yeast-extract gave under the same
conditions and in the same time:

in: inversion

6". maltose 18.9°
0 io
6°/, maltose .5°/, galactose 15.5°/,
0 0 é 0

++

6°’, maltose .0°/, laevulose 15.5°/,
Another series gave
in: inversion
6°/, maltose 26.8°/,
6°/, maltose + 1.5°/, laevulose 24.8°/,
6°/, maltose + 1.5°/, galactose 25.-"/,
=

6°/, maltose 1.5°/, glucose 13.5°/,

This last figure, verified by other experiments, requires a further
explanation. This 1.5°/, glucose was undoubtedly 3-glucose. Before
mixing it with the maltose, the glucose was dissolved separately
and completely converted into the stable form *) by placing the flask
for some time in boiling water. The observed order of retardation
shows that the 3-glucose also takes part in the process of reversion.
Now it is possible that in the maltose molecule the glucose with the
-still free carbonyl group is present in the 3-modification and it is
even probable that this free glucose group, when in solution, will be
converted into the same stable form as glucose itself. In Hin1’s
investigations, veast-extract appeared capable of uniting two molecules
of B-glucose; so, probably, also two molecules of «glucose. The
elucose formed in the enzyme-inversion of maltose may, therefore
be called homogeneous. Each molecule of that glucose can unite
itself under the influence of the enzyme radiation with any other
molecule of that glucose to a biose. Therefore, the equation of
equilibrium was here

means of laclase, Emm Viscner and Franktann Arwstrone only obtained an
isolactose.

The synthesis of canesugar has not yet succeeded because we can only add
B-glucose and not z-glucose to laevulose.

2) Separate experiments showed that unheated glucose causes the same retar-
dation as heated, #-glucose therefore the same as B-glucose.

(27)

1l—y—day’?=0.

It should also be mentioned that the inverting enzyme of yeast
appears to be always the same whether canesugar or maltose has
been present as a carbohydrate food. In an ordinary cereal extract
a little canesugar occurs along with the maltose. A pure yeast-
culture, cultivated by myself in a solution of pure maltose (plus
the necessary salts and nitrogenous food) gave an enzyme extract
which was retarded in its action equally much by glucose and
laevulose, and twice as much by galactose. In the enzyme formation,
therefore a partial conversion of glucose into laevulose seems to take
place. Lopry bE Bruyn and ALBerRDA vAN Exunstwin') have shown
that these two hexoses may be converted into each other in an
alkaline solution.

The investigations of O’SunLivan and THompson?) have rendered
it probable that the invertin-molecule (if we may use this expression)
contains a carbohydrate group. These investigators have attempted
to purify invertin and found that a constant component of the
resulting proteid-complex, their so-called j-invertan, contained 18
parts of carbohydrate to one part of albuminoid.

A further development of the electron theory will probably eluci-
date the nature of those enzyme radiations. As LopGr*) observed,
it is nof the occurrence of radiations in matter which need cause
astonishment but rather the fact that not a great many more radia-
tion phenomena have already been discovered.

Many other catalytic phenomena such as the action of hydrogen-ions
and those of Brepie’s anorganic ferments may, after all, be due to
radiations. For hydrogen-ions, carriers of loose electrons and dispersed
platinum cathodes probably also emit radiations owing to the motion
of the electrons in or around the material particle. During the course
of a same reaction, BrepiG often noticed an increase of the constant

1 1
— : 08 just as that shown by the invertin action. A retar-
=i]

dation of the catalysis by indifferent matters has also been frequently
noticed, for instance, by KNorvenacen and Tomacszewskr‘') in the
action of finely divided palladium or platinum on benzoin.

If the statements of the French investigators on the physiological

1) Rec. Trav. Chim. 1895 p. 201.

2) Journ. Chem. Soc. 1890 p. 834.

5) “On Electrons” Journ. Electr. Engineers 1903 Vol. 32 p. 45.

#) Ber. 1903. 2829.

Proceedings Royal Acad. Amsterdam, Vol. VIL.

(18 )

n-rays should be promoted to objective truth, our hypothesis would
receive a direct experimental support.

At all events, the above has demonstrated that the principal mea-
surable phenomena, noticed in the enzyme action are in harmony
with our hypothesis.

Meteorology. — “On «a tiwenty-sie-day period in daily means of
the barometric height.” By Dr. J. P. vax pEr Stok.

1. A few years ago') Prof. A. Scuusrer investigated the problem,
how to detect the presence of a periodical oscillation, the amplitude
of which is small in comparison with large superposed fluctuations
which may be considered as fortuitous with respect to the purely
periodical motion.

Starting from an analogy which may be seen between this question

and the problem of disturbances by vibrations in the aether — a
problem treated hy Lord Rayieicn*) in 1880 — Prof. ScuusterR has

endeavoured to apply the theory of probability to the determination
of the first couple of coefficients of a Fourter series, and the method
he arrives at, and strongly advocates, is applied to records of
magnetic declination observed at Greenwich during a period of
25 years.

The choice of this material, in Prof. Scuuster’s opinion not favour-
able for the discovery of small effects, is justified by the remark
that “‘the only real pieces of evidence so far (1899) produced in
favour of a period approximately coincident with that of solar
rotation were derived from magnetic declination and the occurrence
of thunderstorms.”

In this and in an earlier paper *) the author emphasizes that, in
inquiries of this kind, it is not at all sufficient to come to some
result, but that it is necessary to apply. a_ reliable criterion by
which a judgment may be formed about the value to be attached
to the result arrived at.

His mathematical investigation, however, does not, lead to an out-
come which in every respect can be regarded as satisfactory, in so far
that a method of determining the mean and probable error of the
result from the series of observations themselves is not given and,

1) Trans. Cambr. Phil. Soc. Vol. XVILL. 1899.

2) Phil. Mag. Vol. X. Il, 1880.

3) Terrestrial Magnetism Vol. Ill, 189s.

(19)

as a surrogate, the author suggests the repeated rearrangement of
the records according to different periods, not much differing from
the period in question. Thus, in a purely empirical way (“by trial”),
a standard may be obtained by which a correct estimate can be
formed in how far the outcome arrived at must be considered as
a merely accidental one.

Now the same problem was treated some 15 years ago") by the
author of this paper after a different method, applied not only to
magnetical but as well to meteorological data of different description,
and Prof. Scuustur’s important investigation gives a ready occasion
for taking this problem in hand again.

It is only natural to choose in the first place for this inquiry the
series of barometric observations made at Batavia which now extends
over a period of 36 years (1866—1901).

An investigation into a possible synchronism between the frequency
of sunspots and atmospheric temperature, commenced in 1873 *) and
recently conducted up to date"), gives some ground to the expectation
that, for inquiries of this kind, observations made at tropical stations
are of more value than those made in regions where the atmospheric
disturbances are such as experienced in higher latitudes.

In the second place it seems desirable to look for a shorter way
for coming to a reliable criterion than the tedious process of the
calculation of ScuustER’s periodograph.

2. An arrangement of quantities according to a given period 7
may be executed by measuring out the successive data from a point
” taken as origin and along straight lines drawn through this point

276
ae

If we assume the unity of mass attached to the ends of those
radii, it is evident that a judgment may be formed about the degree

at equal angular distances

of symmetry in the distribution of the masses with respect to point
O, by simply calculating the average value of all these vectors, or
in other words to determine the situation of the centre of parallel
forces supposed acting on the masses.

If the quantities 7’ show a well marked periodicity as e.g. tidal

1) Observ. Magn. Meteor. Observ. Batavia. X
Tijdschr. XLVIIL. 1889.

Verh. Kon. Akad. v. Wet. Amsterdam. XXVIII, 1890.

2) Képpen. Zeitschr. Oesterr. Gesellsch. f. Meteor. VIII, 1873.

5) CG. Norpmann. Essai sur le role des ondes Hertziennes and: Astron. Phys. et
sur diverses questions qui sy rattachent, These, Paris, 1903.

1888, Append. IL also Natuurk.

’

2*

(20 )

observations do, it will be possible to draw a line through ( in such
a manner, that on the one side all the vectors are greater than the
corresponding opposite vectors on the other side of the line.

If, therefore, we arrange in this Way a great number of quantities
which show a slight tendency to asymmetry, the radial momentum

will steadily increase, as the mass concentrated in the centre of

parallel forces is equal to the total number of observations, whilst
the distribution of accidental quantities will tend to a symmetrical
distribution,

Assuming two rectangular axes going through QO, we find for the
coordinates, by which the centre of gravity is determined, .V being
the number of observations :

1
= v= 0 vos 4 ¥, === osin Oty. Fes. oe

The calculation, therefore, comes to the same as the determination
of the first couple of Fourtmr coefficients:

» »
et SS crite ey 8 9
a, = a — GUS 4 b, = V — Osi oO. . . . (2)

and, if the periodical movement is represented by the expression:
A cos (nt — C),
b 2%

AA = ab," tang C= — n= — <2 or aN
a zl

1

This way of representing the arrangement seems preferable to the
development in a Fourier series: firstly because the development of
a function in a series, as a representation of the function, derives
its value from the composition of a great number of terms, so that,
in calculating one term only, we are hardly justified in speaking of
a Fourierisation of the function.

In the second place, because by this way it becomes at once
evident that the problem is fully equivalent .to that of the determin-
ation of a poimt in a plane by means of a great many inaccurate
observations.

This problem has been treated by several mathematicians, but
certainly in the most complete manner by the late Prof. Scnoxs '),
whose original conception of the question leads to the detection of
some laws, which are independent of the assumption of any law
1) Over de theorie der fouten in de ruimte en in het platte vlak, Amsterdam,
Verh. K. Akad. v. Wet Le Sect. XV, 1875, and: Théorie des erreurs dans le plan
et dans l'espace, Delft, Ann. UH, 1886.

( 21 )

of errors and to a remarkable analogy between this problem and
that of the moments of inertia in dynamics.

If we take NV, the number of observations, equal to unity, the
relative frequency of the ends or representative points of the vectors
may be represented by the density of these points per unity of
surface. This function of probability is called by Scions “the module”,
the “specific probability” or the “facilité de Perreur’.

We thus obtain a mechanical image of a surface of probability,
the density of which will, in general, be a funetion of the length
and direction of the vectors.

The determination of what Scnons calls the constant part of the
error — the probability of which is N =1 — is identical with the
determination of the situation of the centre of gravity, and the
calculation of the mean (not average) error:

Yo?
v= 3
N

with that of the moment of inertia, which leads to the determination
of two (in the plane) principal axes of inertia, which, in our case,
may be called axes of probability.

Assuming that these errors in the plane are due to the cooperation
of a great number of elementary errors, ScHons has proved that the
projections of the errors on an arbitrary axis follow the exponential
law of errors in a line and that the law of the resulting error ean
be found by supposing the error to originate in the coincidence of
projections of the error upon the axes of probability, these projec-
tions being regarded as independent of each other.

The application of this theory to our case can be reduced to very
simple caleulations.

Errors arising from individual or instrumental causes are always
distributed in a more or less systematical way, but there is no
reason to suppose that the fluctuations e.g. of barometric heights
within an arbitrary length of time, and cleared from their constant
part, will show any tendency to systematic distribution when arranged
around a point in the way deseribed above.

Scuois’ specific probability of an error in the plane is given by

laa y
wien M,?

r= =

the expression:

id ae = Eada eee!
20M,M, es

in which « and y are the coordinates of the error (polar coord. 9 and 4)

( 22 )
and MW, and MV, denote the principal axes of probability, so. that:
M? = M,? + M,? = M2 + Mi.

The mean error, therefore, can be calculated without any knowledge
of the situation of the principal axes, when the mean error of the
components relative to arbitrary rectangular axes is known.

If # is independent of 6:

M
M, = M, = 2
or, putting:
1
M?

—

h?

| deepsea aad
1

ween ,

The specific probability of an error, independent of the direction, is :

)-

a bal

=| pea? tpo—2h oe? 0S . eee

-

0
From this it appears that the probability of an error zero is not,
as in the ease of linear errors, a maximum, but a minimum, that
the curve of the spec. prob. (6) (given in Scnons’ paper) shews a
maximum for the value of 9:

04 Me. ns 8 ee

and, also, that the computation of the probable error will lead to a
coefficient of J/ considerably different from that found for linear errors.
We have then to ask for what value + of e:

r
.

a 1
2 ie fo ele do =5

0

OOS 206 My ene ial oe) i ee ae
This value of the coefficient of the probable error, considerably
greater than is found for linear errors, 0.6745, clearly shows that
and to what degree results, obtained in investigations of this kind, have
to be put to an unusual severe test, and also that there is some
reason to adhere to the use of the probable error, which of late

years has been somewhat neglected.
A reduction of the mean error has no sense if this reduction is

always in the same proportion, but if becomes important if this
proportion depends on the nature of the problem.
If the distribution of errors is not independent of the direction,
the coefficient of J7 is determined by the quantity :
M,* — M,’
M,? + M,?

The coefficient of the probable error, for which ScHoLs gives the

approximate value :
Pes Osprey — Welles ee 8 hoe) f())

is & maximum for errors independent of the direction and a minimum
for linear errors, when V = 1.

By the assumption, therefore, that all directions are equally probable
the most unfavourable case is chosen, which, in doubtful cases, is,
of course, the safest way of forming a judgment.

Whether the operations, which are to be applied to the data, are
considered as a determination of the first couple of constants of a
FourimR series (the very first, 4 4,, is left out of consideration), or as
a calculation of the average or most probable position of the end-
points of the veetors, or as a determination of the situation of the
centre of gravity — in all cases the result is a quantity determined
by two coordinates and the operations we have to perform are:

Istly, to separate the constant part ;

2ndly, if necessary to determine the situation of the axes of
probability ;

3rly to ealeulate the mean and probable error, in this case better
ealled incertitude.

The same method can, of course, be applied to groups of periods,
Which gives a considerable saving of labour, but also leaves some
want of clearness in the result.

3. The investigation of the series of daily means of barometric
observations made at Batavia has been conducted in the same manner
as it was commenced in 1888. The arrangement has been performed
according to a period of 25.8 days, and groups of 30 rows have
been taken together so that, out of the 510 periods, 17 groups have
been formed.
~The result of this operation is given in Table I.

If, therefore, an oscillation, periodic in 25.8 days, really exists,
its amplitude is not more than:

1.66
—— = ().055 mm.
30

( 24°)

Pens: Te 1.

||Amplitude, Argument. Components | Differences.
wan, mn, mn, wm, mia.

1 0.69 249° —() 32 ~0.61 —0.86 0.96
2 7.63 287° 2.417 7.32 1.63 —5.75
3 3.45 Q85° O87 —3.34 0.33 —1.77
4 3.44 68° 1.46 2.91 0.62 4 48
D Way) DI52 —|.24 —0.88 —4.78 0.69
6 2.08 204° —1.90 —0).84 —2.44 0.73
7 4.52 =| «345° 4.38 —1,14 3.84 0.43
8 hy 104° —) 30 seat —0.84 2.74
9 1.78 270° O.O1 —1.78 —0.53 —( 9]
10 6.31 318° 4.71 —4.20 4.47 —2.63
i 5.00 194° —4.86 —1.A7 —5.40 0.40
1 fees] 266° —0.20 —3.94 —(.74 —1 .67
13 6.00 A G}Y —1.54 —5.80 —2.08 —4.23
14 YAS 317° 1.59 —-1.49 1.05 0.08
i5 || 3.34 195° = 9108 —0.8% = 977 0.73
16 2.60 83 O34 2257 —0.90 4.14
17 7.62 4° 7 58 0.76 7.04 0.81

|

Mean || 1.66 289° 0.04 1.57
{I

sy subtracting this restant, which has to be regarded as a constant
part, from the corresponding values, the differences exhibited in the
last columns have been found, which are to be regarded as fortuitous
disturbances.

4. The value to be attached to the result may be estimated in
different ways.

The first and most simple manner is to split up the series into
two or more groups. From the data given in Table I we easily find:

( 25 )
Number of
A (@ periods.
Group 1— 6 1.68 mm. 274° 1S0O
a (14 1.62 299 150
i Ea 1.76 296 1SO
FA 9 1.42 292° 270
~ KOs 07 1.95 286 240
2 (L—Aly 1.66 289° 510

From this it appears that there is certainly some indication for
the existence of a periodical oscillation, and also that the arrange-
ment has been made according to a period which practically leads
to a maximum value of the amplitude.

The probability that three points, taken successively at random,
Ly ee
) and the probability

12

are situated within an angular space of 30° is(
of mere chance would have been even less if we had taken into

account that the amplitudes too are in good accordance.

5. A second, equally simple method is afforded by a direct view
of the outcome of the arrangement itself, split up into two or more
eroups.

Fig. 1 gives a graphical representation of the differences given in
the three last columns of Table II.

Fig. 1 shows that the curves of the two series agree satisfactorily
and. also that a tendeney to a double period, with a maximum
on the 8—9 day, which in the first group is still well marked,
vanishes when the arrangement is continued.

If these results are considered as fairly conclusive, so as to justify
a more exact determination of the length of the period, this may
be easily done by varying the arguments C of Table IT successively
by 7/2, “/,@, *;,@ ete., « -denoting the variation: of each group-
argument which leads to the most constant value of (.

In this way 17 equations are obtained from which the most
probable values of © and the period 7 can be calculated. If to each
equation the weight is given of the corresponding amplitude, the
equations will assume the form:

0.69 (— 118° == UY, @ = 0169''C

631@ — (3°13 / mn) Ga C. pie:

( 26 )

TABLE

Results of the arrangement. Average values, the general mean value being subtracted.

+H

mm.

0

=O

—(),

—t)

036

008
O09
OM
.030

270

mm.
—0.012

—0.033

—0 025

0.001 |

Three subsequent values taken together.

II +11
240 510
mn. mi.
0.025 O.014
0.08 0.006
0.014 —0.010

—0.016 —0.017
—0.029 —0.023

033 —0.029

042 —0.028
—) 066 —0.032
—.086 —0.052
—(0.098 —0.074
—0 076 —0.077
—0.049 —0.060
—0.031 — 0.035
—(0.025 —0.020

0.010 —0.011
0.024 0.005
0.047 0.024
0.050 0 043
0.052 0.057
0.068 0.060
0.080 0.061
O08) 0.055
0.064 0.048
0.030 0.041
0 003 0.028

0.026

— J

As might have been expected the result of this caleulation shews
little or no difference from that of the arrangement itself.
z= — 1°,09 C= 291- T= 25.8034.
As one day corresponds to *"°°/,,.,, a variation of w degrees for
each group is equivalent to:
1.09 258

Se (OO RE
30 3600

6. By application of the method discussed in § 2 to the differ-
ences A, and A, of Table I we find:
eS ie 1505 SM, — O12 0a Sa.) — 0.44.

From the well known formula :
IA RD em ty (10)

for the situation of the principal axes of inertia, deduced from the

( 28 )

condition that, when the axes of coordinates coincide with the
principal axes, the moment of deviation or centrifugal force My,
will vanish, we find :

and further
Mi 645 I) ins ehes N = O55
and from formula (9)
r= 0.829.

It appears, therefore, that, in this case, all directions of the
accidental quantities are equally probable, so that we are fully
justified in putting:

Osos

The mean and the probable error for each group are then :

M = 3.89 W = 3.24
and the final result for each group:
1.76 mM... . probable error 0.810
and for each row :
0.055 mM... . probable error 0.027

so that the probable incertitude of the tinal outcome amounts to
amost exactly half the amplitude.

7. The question may also be put, what will happen if the
arguments of Table [ are varied in such a manner, that the varia-
tions are equivalent to arrangements according to other periods
slightly different from 25.8 days.

The amount of the variation is limited by the number of rows
taken together in one group, which can be shifted only as a whole,
and the variation ceases to have any sense as soon as the sums of
each group would be sensibly affected by the actual arrangement
according to the new period.

If quantities, periodical within a length of time 7 are arranged
according to a period 7” in m columns, the value at the origin of
time being represented by :

A cos C,
the record to be inscribed in the # column of the p™ row (¢and p
counted from nought) will be :

Qa 1 : * 7]
A cos —— ( oh 2% p—
m on nt

( 29 )

The sum of F& rows is then :
Td
Ul
ii i) = a) ==
9
sin Ret 2 Can . f
{ — COs T + T—C +-(R—--l)e
Sin ; ML mt
When J is small this expression can be simplified by putting :
ry ryy ryy m
fhe or (i 7

in the second term under the cosine. The sum of the first, second
ete. group of R rows is then:

sin Ra 2% rer
A ———— tos — JT — ( S5 Vaid) VN Re Se Mey teens, (8 Lab)
sin @ m
sin Reo 2
{00s Se ——( +35 Ra ) ete.
sin at m ,

If the oscillation is of a purely periodical description and of equal
amplitudes the sum will show a principal maximum, / A, for «= 0,
and further secondary maxima for all values of @ which. satisfy the
equation:

KR tang a = tany Ka

Le., when A =510, for values of @ corresponding with periods of:

| 25.8724 ( 25.9254
| 25.728 } 25.675

but the amplitudes of these maxima will be resp. 5 and §& times
smaller than the principal maximum.
The amplitude will vanish whenever
Ra= nm, 2n, 3% etc.
i.e. for periods of
{ 25.8504 {
1 25.750
The upper curve of fig. 2 gives an image of the fluctuations of
these theoretical amplitudes.
If we put:

5.9004
5.790

»”
»

Tie IP = UN 8) SS ch
the amount of shifting to be given to each group corresponding
with O.OL day, is:
2S au
30 ¢ = ——_ IEA:

v in the denominator being neglected.
The variation has been carried on, as utmost allowable limit, to

(30 )

x=+ 0.15, corresponding with a group-variation of about 31°.
The group-amplitude is only slightly affected by this variation as:
sin Ra
“sina
instead of 30.
When RF, the total number of periods, increases, the secondary
maxima will become smaller and smaller, and at the same time
maxima and minima will approach nearer to the principal maximum.

TABLE iil.

Results of the arrangement according to different periods by the shifting-process.

Period A C Period A C
| EE eee eee ee
Nik, win.

d. d.
95 .65 13 4 vale 95.81 OTA 247°
9 66 6.4 29R0 22 93.7 917°
I .67 4.8% | 148° 83 15.0 169°
925 G8 16.4 + 76° s4 7.6* 84°
25 69 94S 34° 85 12.6 s44
25.70 30.0 345° 86 49.2 286
— i

a) Fpl 97.9 Sipe 87 24.0 | 244°
95 72, 22.5 965° 88 93.8 907°
95.73 14.6 931° 89 16.5 174°
25.74 7.8 171° 90 STH li {32°
95.75 6.7% 88° 94 4.9 01°
25.76 10.1 akt 92 11.9 962°
25.77 14.7 Sh 93 45 0 eye
25.78 49.8 | 346° 04 13.6 | 219°
ahah (4) 26.4 322° 95 | estes so00°
95.80 98.3 9S9°

Table IIL exhibits the outcome of this shifting of the groups and
fig. 2 shews both the theoretical and actual curves.

For the sake of comparison the data of Table IIT have been
multiplied by 2 and the amplitude of the theoretical period has been
taken equal to O.1.

25.65 25.70 25.75 25.80 25.85 25.90 25.95

It appears then that, in fact, secondary maxima and minima oeeur
and, at least as regards the first minima, in the right places, but
that the secondary maxima, instead of being small as compared
with the principal maximum, as might have been expected, are of
about equal intensity and, most so on the left side, not at all
agreeing with the theoretical lengths of period.

This result may be interpreted in three ways:

a. We may assume that every one of the three periods 25.80,
25.70 and 25.87 is due to a purely accidental distribution of the
quantities under consideration.

4. We may concede that at least for the period 25.80 there is some
indication, but that the two adventitious periods are the consequence
of the unequal distribution of the group-amplitudes so that they will
disappear when the arrangement is continued over a longer series
of observations.

( 32)

c. We can assume that the evidence is equally good for the three
periods, and will be enhanced by continued arrangement.

Of course only an actual continuation of arrangement for another
series of twenty years will enable us to answer these questions.
The only test which at present can be applied is to form two or
more groups as has been done above for the arrangement according
to 25.8 days.

Period 25.704 25.874
al Gi A C
Group 1—6 1.60 mM. 6 1.40 mM. 252
cs 7—11 330) 4° 292 Ov sue.,) 160
12—17 2.48) ~ 5; 27 2.02 4, 200.
= 1—9 eb OS) isp oy 12 ee LO
. 10—17 Uppy ta) ath 1:88) ,, 26

So far as this criterion allows a conclusion to be drawn, it appears
from this result that the evidence for real existence of the periods
25.70 and 25.87 is considerably less than of the period 25.80.

In the latter case the arguments for three groups did not differ
more than 25°, against differences of resp. 88° and 95° for periods
of 25.70 and 25.87 days. The probabilities of mere chance, therefore,
are, taking 30° and 90° :

1

and —
16

i.e. more than 8 times as great. If we take also into account that
the amplitudes of the three groups are accordant for 25.80 and
widely different for the adventitious periods, we can estimate the
probability of chance at 10 times as great.

The computation of the probable error (incertitude) of the result
for each group also gives an indication for this greater probability,
but not in the same degree.

Amplitude. | Probable error.
25.80 1.76 mm. 0.810
24.70 176 «5 0.916
25.87 Ee its: 0.830

-

7. If we apply, in so far as possible, the different criteria to the
data published by Prof Scuuster concerning daily means of magnetic
declination for Greenwich, arranged according to 26 and 27 days,
we find for the sums of groups, each of which contains resp. 14 and

13.5 rows,

( 33 )

264. 274.
A, é A, (
Group 1— 5 6.19 267° a8) 351
_ 6—10 4.08 243° PAS, 88
its 3.09 Soils 4.75 354
16—20 1.45 ils) 7.05 203°
229 2.88 229° 7.56 298°
The probability, therefore, of mere chance is :
283\4
2 Ope CEAVISS Mice thal: é ) = 0:38
3560
DU OAGS Spe oe a ele 0,20.
: 360
and the final outcome
26 days 0’.480 ........ prob. error 0’.343
Hee VEO PAU eee ee ee a OL AIS

If we vary the arguments given in Table VIII of Scuuster’s paper
for a period of 26 days in such a way that the result is equivalent
fo an arrangement according to 25.80 days we find:

A, C

Group 1— 5 10.83 54°
6—10 5.72 104°
11—15 4.88 (ie
16—20 4.61 44°
21—25 4.08 89°

As these arguments do not differ more than 60 degrees, the probability
of chance is in this case:

1

64
The final result, caleulated for a group of 14 rows,
IAANS> 5-2 eptob..<error 00292;

The accurate length of the period and the most probable value

of (, caleulated after the method discussed sub 5, are then:
Dorso4? dave = . eee 2 C= *552°6.

It appears from these calculations that an arrangement according
to 26 and 27 days leads to results the probable incertitude of which
has about the same value as the amplitude itself. On arranging
according to 25.8 days we find a probable incertitude about four
times less than the amplitude.

Further, from this investigation, as compared with ScuvstEr’s
inquiry, we may draw the conclusion, that elements of terrestrial
magnetism, as observed in higher latitudes, allow a more decided
judgment to be formed concerning the real existence of periodical

3

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 34 )

oscillations of this kind than meteorological observations made at
tropical stations.

If the outcome arrived at by the arrangement of barometric daily
means for Batavia is considered to afford some evidence or indica-
tion for an oscillation periodic in 25.80 days, a much greater
probability must be attached to the real existence of this fluctuation

in the observations of magnetic declination made at Greenwich.

Anatomy. “On the Form of the Trunk-inyotome.” (First Com-
munication). By Prof. J. W. Langenaax. (Communicated by
Prot. T: Pace).

The segmented plan of construction of the vertebrate animals,
most marked in the muscular system, has led to the conception of
the myotome.

Two methods are chiefly employed in establishing the form of
this myotome. The first method is based on the hypothesis of the
primary connection between muscle and nerve; the second, a more
direct one, is based on the dissection of the interseemental tissue.
Both methods seem equally restricted in their application, as can be
concluded from the researches of Barprern '); moreover there is
reason to believe, that they will not always yield concordant results.

The second method is followed in this research.

I. Lrunk-myotome of Petromyzon fluviatilis. (Fig. 1.

The trunk-myotome of the
adult animal has in general
the form of a crescent, the
cornua beme directed to-
wards the cranial end of the
body and slightly imelined
to each other. The dorsal
cornu (fig. Il CD) reaches
to the mid-dorsal line, while
the ventral cornu (fig. II
C'V’) ends at the mid-ventral
line of the body. Both cornua
differ in length, the dorsal

1

being about longer than

the ventral, and while both

reach to the mid-plane of
Fig Il. the body, they are slightly
torquated in respect to each other.

1) Anat. Anz. Bd. XXIII, N°. 10/11.

( 35 )

The corpus of the myotome shows a kneelike inflection (fig. IL A’),
which is always situated nearer the mid-ventral line of the body than
the mid-dorsal line. In transverse section (along the line /’/, fig. 1)

Fig. If. the corpus of the myotome is rhomboidal, this
rhombus being more and more flattened towards
the cornua; consequently the cornua appear in
transverse section as lamellae in juxtaposition
(fig. Il “.4.). These lamellae are slightly incurved.

One side of the rhombus lies in the body sur-
face (black in fig. 11). This surface is cylindrical
in the middle region of the body, the transverse
section being perfectly elliptical. The black-coloured
surface of the myotome, must therefore be con-
sidered as cut out of this cylindrical surface.

The opposite side of the rhombus is turned

towards the skeletal-axis and the abdominal cavity.
te _ In general it has the same form as the outer side,
lransverse section
through the trunk of j ; i f ; : ,
Petromyzon: the inter- Cavity, which in this part of the body is cylin-

being only slightly excavated by the abdominal

segmental tissue being drical, the transverse section being a perfect circle.
black, Both the other sides of the rhombus are con-
eruent and bound respectively, a more cranial and a more caudal
myotome,

The position of the myotome as a whole in respect to the sagittal
plane, passing through the mid-lines of the body, is such, that the
corpus shows an inclination towards the caudal end of the body.
Seen in transverse section (along the line /’/’ fig. Il) the longest
axis of the rhombus makes an acute angle with the sagittal axis of
the body, the vertex of the angle being turned towards the head.
This caudal inclination of the myotome diminishes towards the cor-
nua, so that the cornua are nearly normal to the surface of the
hody. In consequence of this caudal inclination the myotomes over-
lap to some extent. This muscular overlapping varies between ‘/, and
*/, in the neighbourhood of the knee, diminishing towards the cornua
on account of the decrease of the caudal inclination of that part of
the myotome.

The position of the myotome in respect to the dorsoventral axis
is variable along the body. If A (fig. Il) is a dorsoventral axis, at
right angle to the sagittal axis, and / a line tangent to the dorsum
of the myotome, then we have in the angle A/4 a measure for the
ante- or retroversion of the myotome in respect to the dorsoventral axis.

The first myotomes behind the last branchial cleft, show a little

3*

i 30.)

anteversion, which quickly decreases, so that the 4° and 5 myo-
tomes are strictly vertically situated. The following myotomes (as in
figure Il) are retroversed, this retroversion reaching a maximum of
LO Towards the caudal end of the body this retroversion decreases
and is again reversed behind the anal aperture, where the myotomes
are again anteversed.

The description given here of the myotome applies only to the
trunk-myotome in the middle region of the body, the branchial
apparatus as well as the appearance of the dorsal fins bringing

about notable changes in this form
ll. Trunk-myotome of Acanthias vulgaris. (Fig. lV and V).

The myotome described in this paper was situated in that region

of the body which lies between the thoracic fin and the first dorsal

Cc~

Fig. VI.

fin. In its most general features the trunk myotome of Acanthias
shows a great resemblance with that of Petromyzon, though at first
view a considerable difference seems to exist.

Looking at that surface of the myotome, which forms part of the
surface of the body, we see it interrupted in two places.

The lines of interruption are nearly parallel to the sagittal axis
of the trunk. The first line (1Z fig. VI) coincides with the linea
lateralis, the second (L'L' fig. VI) les nearer the mid-ventral line

of the body. At the place of interruption septa of connective tissue
descend and seem to divide the myotome into three parts. One
part situated between the mid-dorsal line and the line LL is the
dorsal part of the myotome; between the lines L and LL" lies
the lateral and between the latter and the mid-ventral line, the
ventral part of the myotome is situated,

Considered at the line of interruption, the surface of the lateral
part of the myotome seems to be cranially displaced in respect. to
the dorsal part; this displacement amounts to one half of the breadth
of the myotome. The same can be observed between the surface
of the lateral and the ventral part of the myotome, the lateral part
being also displaced cranially in’ respect to the ventral part; this
displacement does not surpass '/, of the breadth of the myotome.

If we follow the septa of connective tissue at the line of inter-
ruption LL, it is easily seen, that the myotome is rolled in towards
the axis of the body and then reversed till if reaches again the body
surface. In most cases the continuity of the muscular tissue at the
bottom of the fold is broken off, but the interseemental tissue which
covers the myotome is always continuous. If we now try to unroll
the myotome as much as possible, we find that the dorsal part
makes an angle with the lateral part, so that a true knee is formed.
The top of the knee is directed towards the head as in Petromyzon.
The line along which the myotome is folded in, is parallel to the
sagittal axis of the body, and seems to run over the knee, so that
the top of the knee lies at the point A’ of figure VI.

The same can be observed on following the line “’'L’ (fig. V1),
the line of folding being also parallel to the sagittal axis of the
trunk.

The differentiation of the myotome into three parts, ensues there-
fore from a process of infolding, the lines of folding being parallel
to the sagittal axis of the body.

If a model of the myotome of Petromyzon is cut out of paper,
and this myotome folded in along the lines FF and /” Fk”
(fig. II), we get a precise illustration of the displacements seen
in the myotome of Acanthias. The direction in which the outer-
surfaces are displaced in respect to each other is a direct conse-
quence of the form and the curvature of the myotome at the
places of infolding. The difference in the extent of the displacement
of the surfaces along the lines LL and L/L’ (tig. VI) is due
to the fact, that the fold along the line #/’ is longer in the direction
from outwards to inwards in correspondence with the dimensions
of the myotome. This can be seen in a transverse section through

t
|

Fig. VII. the same region of the trunk
of Acanthias, where LF (fig.
VII) is the intersegmental tissue
that divides the dorsal from the
lateral part, while 1’/” is the
septum, that the latter separates
from the ventral part of the
myotome.

The further differentiation of
the dorsal part of the myotome

takes places by the same process;
ql the lines of folding instead of
being parallel to the sagittal
axis of the body are in general
at right angles to this axis.
There are three of these lines,
agreeing with the number of
peaks which the dorsal part of
the myotome shows. These lines
considered from outwards to

—S3---¢¥ inwards, are originally normal
aes to the surface of the body, then

Transverse section through the trunk curved with the convex side

of Acanthias; the intersegmental tissue turned ventrally and towards

black. Natural size. the body surfaces. This curva-
ture is most marked in the third line of folding (A fig. VI), reckoned
from the mid-dorsal line, the first one being nearly a straight line
normal to the body surface and to the sagittal axis. These lines
of folding are visible on a transverse section, because septa of inter-
segmental tissue stretch out into the fold. The curved lines (A fig. VID)
are the transverse sections of these septa.

The lateral part of the myotome shows no further differentiations.

The ventral part has only one line along which the folding of the
myotome is well marked; consequently this part of the myotome
shows only one peak turned to the caudal end of the body.

The myotome considered as a whole, as in Petromyzon, has a
caudal inclination. This inclination is most marked at the knee.
Considering only the dorsal part, we see this inclination diminish
towards the mid-dorsal line, so that the most dorsal part of the
myotome is about normal to the surface of the body. This most
dorsal part is elongated into a dorsal cornu (CD fig. VI). In trans-

( 39 )

verse section we find therefore these dorsal cornua as lamellae in
juxtaposition (fig. VIL CY).

When the direction of the myotome is reversed at a line of fol-
ding, the same happens with the sense of the inclination, so that
these parts show a slight cranial inclination. The folding of the
myotome together with the inclination produces the elongated and
peakshaped form of the myotome at these lines of folding. The
myotomes thus cover each other as hollow, pointed tubes telescoped
into each other. In the transverse section (fig. VII) 1 indicates the
section of the first peak directed caudally, 2 the peak turned towards
the head and 38 the second peak turned to the caudal end of the
hody (fig. VI resp. 1, 2, 3). In consequence of the inclination of the
dorsal part of the myotome, these lines of folding are not quite at
right angles to the sagittal axis, but are also slightly inclined. This
is the reason why we find only part of these lines in a transverse
section, which is normal to the sagittal axis.

From a transverse section we can judge the extent of the muscular
overlapse. Conecordant with the increasing inclination from the mid-
dorsal line to the first lateral line, we see the overlapse increase to
about */,. At the knee the inclination rapidly decreases. In the
lateral part of the myotome the inclination is insignificant and the
overlapse less than ‘/,. In the ventral part the inclination inereases
at first and then decreases towards the mid-ventral line; the museular
overlapse does not surpass '/,. The ventral part terminates at the
mid-ventral line in a ventral cornu (('J” fig. VII, fig. V) turned
eranially. This ventral cornu is much shorter than the dorsal cornu.

In order fo get some idea of the dimensions of the myotome |
have measured the length of each of the three parts into whieh the
myotome is divided up. These measurements have been made over
the surface of the myotome and this surface was also followed where
it is folded in. In this way | have found for the myotome described :

Length of the dorsal part 350 mm.; the lateral part 90 mm.; the
ventral part 190 mm. The lengih of the whole myotome is therefore
630 mm. and of this **/,,,

I have made the same measurements in the myotome of Petro-
myzon. If it be conceded that the points A’ (fig. TT and fig. VI)
where the knee is located in both myotomes, are corresponding points,

belong to the dorsal region.

I have found: Length of the dorsal part, from the mid-dorsal line
to the knee 33 mm.; the latero-ventral part, from the knee to the
mid-ventral line, 26 mm. The whole length of the myotome was

therefore 59 mm., and of this also °° is contributed to the

100

dorsal region.

(805

If we compare the dorsal region in Petromyzon and in Acanthias,
it is evident that this region is strongly reduced in the latter ;
notwithstanding this, the same part of the whole myotome belongs
in both cases to the dorsal region. If this be true in general, it
seems to me, that the reduction of the dorsal region is the principal
moment which has led to the folding of the dorsal part of the
myotome.

In figures IV and V the position of the rib, in relation to the
myotome, is indicated by a crossed line. The rib is located in the
intersegmental tissue that divides the dorsal from the lateral part
of the myotome. The junction of the rib with the skeleton, lies a
little caudally in respect to the knee of the myotome ; the rib itself
is turned to the caudal end of the body concordant with the caudal
inclination of the myotome. In a transverse section, three successive
ribs are cut through.

Geology. — “On the direction and the starting point of the diluvial
ice motion over the Netherlands.’ By Prof. Eve. Dvsors.
(Communicated by Prof. J. M. vax BrmMMe.en).

Referring to the “Beschrijving van eenige nieuwe grondboringen,’ V,
by Dr. J. Lor, recently published in the ‘“Verhandelingen der
Koninklijke Akademie van Wetenschappen, 2" Sectie, Deel 10, N°. 5”,
to which, on p. 20 and 21, the author has added a critique of some
conclusions in my communication to the Academie: “The geological
structure of the Hondsrug in Drenthe and the origin of that ridge”
(Proceedings of the meeting of Saturday, June 28, 1902, Vol. V,
p. 93 sqq.), I beg leave to make the following remarks.

The critique of that eminent student of the geology of the Nether-
lands is based on such an incorrect and incomplete statement of
my conclusions and of the facts, that the reader cannot but regard
those conclusions as being of a rash character, which in fact they
have not.

Indeed, having said that he does well agree with my opinion
regarding the structure of the Hondsrug, Dr. Lori continues as
follows *):

“Another case if is with a particularity mentioned on p. + and
5”. (This refers to 12 lines on p. 96 and 97 of the Proceedings.)
“In pit ALI there was found a boulder of quartzite, having a
diameter of 0.85 M., cleft into two yieces,-in- such a manner

1) This quotation has been translated from the Dutch of Dr. Lor by me.

( 42)

that the upper piece, with respect to the lower one, has been pushed
on 1*/, em. in south-easterly direction. So this is a facet! The author
however bases thereupon the hypothesis that the ice motion has taken
place, as a whole, not from N.E. to S.W., sueh as is still generally
admitted, but from N.W. to 5... in such a way. that the starting
point is not to be sought for in Scandinavia but in Seotland. Now
it appears to me that here is a strong disproportion between the
importance of the observed fact and that of the hypothesis.”

Further, at the end, he says:

“So to find the explication ot the shifting of the quartzite boulder
at Eksloo, over a distanee of one centimeter and a half, we have not
fo admit Scotland as starting point for the ice motion, but can persist
in our old opinion.”

Thus far Dr. Lori.

Now I wish to remind those who take an interest in the matter
that it was by no means that one fact, referred to by Dr. Loris,
on which T based “the hypothesis” that the ice motion ‘took place”
from N.W. to S.E., nor did I assert at all that the starting point
of the ice motion is not to be sought for in Seandinavia but in
Scotland. In the quoted Proceedings, to which Dr. Lorm refers, I do
not speak of a hypothesis, but of a supposition, and this, clearly, is
based upon the whole consideration of the strueture and the origin
of the Hondsrug ridge. Particularly this supposition is related to
2'/, pages of my communication, (the whole text being 10 pages),
viz. from p. 99 (in the middle) to p. LOL (below). The only sentence
bringing in relation the fact of the shifting of the pieces of the quartzite
boulder, with respeet to each other, to the direction of the ice motion,
(p. 100, at the end of the second alinea), occurs half way the
explanation of 2'/, pages and runs as follows: “Now with this
supposition perfectly agrees the at first sight paradoxical direction of
motion as derived from the shifted boulder of quartzite.”

And concerning the starting point of the ice motion, on p. 104
of the Proceedings I most unhesitatingly admit Seandinavia to be
the starting point of the ice motion, whereas IT only speak of the
possibility (“it might be possible, at least’) of a deviation of the
Scandinavian ice stream in’ south-easterly direction, caused by the
Seottish ice stream.

It will be superfluous to argue that the distance over which the
two pieces of the quartzite boulder are separated from one another
in the soil ought not to be in any proportion to the large motion
of the ice over 7.

Haarlem, May 26, 1904.

|

( 42)

Mathematics. “On an varpression for the class of an algebraic
plane curve with higher singularities.” By Mr. Freep. Scuun.
(Communicated by Prof. D. J. Korrmwra.)

If an algebraic plane curve is given by an equation in Cartesian
point-coordinates, its order n can be immediately read from the
equation. The class / of the curve is best defined as the order of
the equation in /ine-coordinates. However, in the following it is
my intention to restrict myself exclusively to point-coordinates and
then the class can be defined as the number of morable points
of intersection of the curve with the first polar or as the number
of proper tangents to be drawn from an arbitrary point P to
the curve. To obtain exclusively different points of contact not
situated in manifold points of the curve we must understand by an
arbitrary point a point that

dst does not le on the curve,

2°¢ does not lie on one of the tangents in a manifold point of
the curve,

3° does not lie on a tangent in a unifold point having with the
curve a contact of a higher order than the first.

A manifold point of the curve is a curvepoint which cannot
be a single point of intersection with a straight line. The lines
comecting P with the manifold) points must not be counted as
proper tangents,

From the above mentioned detinition of the class another one can
be deduced, where no single restriction is made with respeet to the
situation of the point 72, which thus holds good for any point P.

To begin with, we make the restriction that P? may not Le on the
curve. Suppose P to lie on the tangent in a point S of the curve,
where SS may be a manifold point or a single point with a tangent
intersecting in more than two points. If the straight line PS cuts the
curve in iw coinciding points S, whilst an arbitrary straight line through
S cuts the curve in ¢ coinciding points .S,-then WS counts for w— tof
the & proper points of contact with tangents from ? to the curve,
in other words w#—+ points of contact approach S when ? approaches
the tangent in S. It is of no importance whether the curve has
one or more branches through S, touching SP, neither whether the
curve has branches through SS not touching SP or not.

The above mentioned follows immediately from the following:

Turorem. Let R be a point of an algebraic curve where all branches
through R have the same tangent l which intersects the curve in t+-v
coinciding points R, whilst every other straight line through R intersects

( 43 )

the curve in t points Ry then R absorbs v proper points of contact
with tangents from P when P les on 1 outside R,and t+ v proper
points of contact when P coincides with R.")

Now, if S is such a point, where all branches have the same
tangent SP, then #=¢-+- vr, whilst according to the above theorem
S counts for v=i—? points of contact with tangents from 72. If besides
the branches touching S/? still more branches pass through S, then
these latter do not give rise fo any new points of contact coinciding
with oS; they cause however the same increase of the numbers ¢ and
w, so that they leave w—f unchanged. Then too 7—/ represents the
number of points of contact, which in consequence of the singular
situation of P coincide with S.

If P is not situated on one of the tangents in S, then w—t, so

4 For a branch which can be represented by one single Putsevx-development
this theorem can be proved i. a. out of the relation existing between the
developments in point- and in line-coordinates. By addition follows immediately
the same theorem for more branches having the same tangent. In a paper
entitled: “An equation of reality for real and imaginary plane curves with higher
singularities” (These Proceedings of April 23" 1904, p. 764) I made use of the
same theorem (p. 765) and referred for the deduction to Srotz, Zeurnen and
SrePpHeN Sairn. I omitted however to mention G. Hatrnen, “Mémoire sur les
points singuliers des courbes algébriques planes”, Mémoires préz. par divers
savants a ? Académie des Sciences (2), t. 26, (L879), n°. 2 (112 p.). This extensive
paper was already offered to the Paris Academy in April 1874 (see: Comptes
Rendus de lAcademie des Sciences de Paris, t. 78, p. 1105—1108, where the
aulor communicates some of his results) so that this paper has the priority,
Hatpuen formulates the theorem somewhat differently, namely (l. ec. Théoréme IIL,
p. 42 or Théoréme II, p. 50):

Tntoréme. La somme des ordres des contacts des branches @une courbe avec
une de ses tangentes est egale a la multiplicité du point correspondant a celle
tangente dans la courbe correélative.

The relation between the developments of a branch in point- and in line-
coordinates was first considered by A. Caytey (*On the Higher Singularities of a
Plane Curve”. Quart. Journ. of Math., Vol. 7, (A866), p. 212, Collected Math.
Pup., Vol. 5, p. 520 or “Note sur les singularités supérieures des courbes planes”

Crelle's Journal, Bd. 64, (1865), p. 369, Coll. Math. Pup., Vol. 5, p. 424). If

y=Aur+Bar+....(p>1) is the development in poit-coordinates then
meat AG
Caytey gives for the development in line-coordinates Z = A’ Yr—! + B' X p—! +...
APTHO+:-

where the general form of the exponents is “; here A, w,.... are posi-

pei
tive integers. Implicitly the Hanenen-theoreim is included in this. Caytey, however,
does not enter farther into the relation between the developments and does not
state the theorem.

Let me finally notice, that the theorem has also been stated by M. Néruer,
“Ueber die singuliiren Werthsysteme einer algebraischen Function und die singuliren
Punkte einer algebraischen Curve’, Math. Annalen, Bd. 9, (1876), p. 166 (sp. p, 182).

( 44)

that then #— 7? continues to represent the number (namely zero) of the
points of contact coinciding with JS.

If S is a point of contact with a tangent out of ? where nothing
remarkable takes place, then for that ¢ = 1 and w = 2, so that
w—t=1, whilst S now also counts for exe point of contact.

By summing up the values #—/ for all the points |S of the curve
for which w >? we keep getting for sum / with respect to every
point P not lying on the curve, thus

00 tS eS «es i
We formulate this in the following way :
TurorrM I. Let P be a point not situated on an algebraic

curve and S an arbitrary point of that curve. Let us suppose that
the curve cuts the straight line PS inw, an arbitrary straight line through
S however in t points coinciding with S; then the class of the curve
is equal to w—t summed up for all the points S of the curve for which
wi>t and for as many other curvepomts as one likes.

To continue we suppose that P lies on the curve namely in a
point of the order ?, i.e. ¢ is the smallest number of coinciding
points of intersection of the curve with a straight line through P.
For a point S of the curve not coinciding with P the number of
points of contact coinciding with S is still indicated by w—t.
Moreover a certain number of points of contact coincides with P,
namely according to the Hatpnen-theorem to the number of ¢ + > v,,
where 7’, represents a summation with respect to the different

curvetangents intersecting the curve in? + 7,,f/—+7,,.... coinciding
points P. So we get

kf 2" 2 (uw, — #4), = 2. ev Qe)
where (w, — ¢,) represents a summation with respect to all the

points S of the curve outside P.

However we can aiso include the point P among the points S. The
line connecting P and S becomes in that case indefinite. If we take
for PS a line which is not a tangent in /?, then we get w=f?.
If however we take for PS a tangent intersecting in / + 7,’ points
coinciding with P, then we get w=?f+7,', so w—t =v’. So for
(2) we can write

=? e SiQo Stee we ee
if only we extend the summation also to the point S lying in P itself,
in which case we have but to take for ? S those straight lines through
P contributing to S (w,—+,) (thus the tangents in P) and as many
other straight lines through P as one likes.

The equation (1) is a special case of (3). If namely P is not on

( 45 )

the curve then each straight line through P has zero points of inter-
section ? with the curve; in other words 7? is a point of the order zero
of the curve, so #0. So the result of all our considerations is
included in equation (3).

We formulate this in the following way :

TurorEeM II. Let P be a pot of the order t of an alyebraic curve
(where t may also be zero) and S an arbitrary point of the order tof
that curve. Suppose the straight line PS intersects the curve in w points
coinciding with S, then the class of the curve is equal to t increased by
the sum of w—t over all the points S of the curve. If S is in P
we have to regard all straight lines through P as the line connecting
P and S.

When speaking of all poimts S or, when S is in P, of all
straight lines through P, we mean that we take those points or lines
contributing to Y (w,—?,) and as many other points or tines as
one likes.

Theorem I is a special case (=O) of this theorem II. The
theorem always holds good for any singularities the curve may have.

Sneek, May 1904.

Geology. — ‘“Sume considerations on the conclusions arrived at in
the communication made by Prof. Eve. Dusois in the meeting
of June 27, 1908, entitled: Some acts leading to trace out
the motion and the origin of the underground water of our
sea-provinces.” By H. E. pe Bruyn.

(Communicated in the meeting of September 26, 1903).

In the meeting of June 27, 1905 Prof. Duzois made a communication
dealing with a problem of great general importance, namely the presence
of proper drinking-water in the province of Holland. Although
readily acknowledging the many points of merit of this communica:
tion and entirely agreeing with many of its conclusions, I differ
from the author on a principal point which indeed is essential,
namely the origin of the fresh water in our polderland. Soa speedy
refutation of the author's opinion on this point seemed to me to be
desirable.

In his communication Prof. Dusois speaks of our sea-provinces;
this in iIny opinion ought to be Holland, since the conditions prevail-
ing in Friesland and Zeeland are different, so that considerations
which are valid for Holland cannot be applied there. So I will
only consider the tract of country chiefly dealt with in the above-
mentioned communication, which is bounded by the dykes of the Y

( 46 )

and the Zuiderzee at the north, by the river Vecht at the east, the
Rhine from Harmelen to Katwijk at the south and the North-sea at
the west.

The geological conditions of this tract inside the dunes are such
as are mentioned in the Communication: uppermost alluvium, then
pretty generally a layer of fen (partly disappeared) under which a
laver known as “old sea-clay”. Under this latter the diluvium,
consisting to a great depth of sand, coarser and finer, with here
and there banks of clay which are not continuous however. The
“old sea-clay’ mentioned is called in the paper clay-containing sand
and although in my opinion also that layer is permeable to water,
yet | think its permeability is smaller than Mr. Dusots assumes and
that it is exactly here that the cause of our difference of opinion
has to be sought. In some places this laver of old sea-clay is wanting;
in special cases this makes an investigation very difficult, for the
general condition however, which is here dealt with, this circumstance
can be neglected.

The communication consists chiefly of two parts, of facts and of
conclusions drawn therefrom. The facts I will pass without comment-
ing on them, although occasionally objections might be raised against
them or rather against the remarks that accompany them.

I perfectly agree with a great many of the conclusions, e.g. with
the following:

that in the diluvium fresh water is present to a certain depth;

that in deeper polders the deep groundwater moves vertically
upward, in shallow polders downward;

that also in the depth a current exists from the dunes to the
polders and trom the shallower polders to the deeper ones;

that no important continuous subterranean current exists from the
higher grounds from the east to the west.

But I cannot accept the conclusion that the fresh groundwater
present in the diluvium also in our polders, owes its origin to rain
fallen locally or at a relatively short distance during the wet season.

In the following refutation of this opinion I shall speak of fresh
and of brackish or salt water. Of course there is no sharp division
between these, but in order to avoid cumbrous definitions I shall
make this distinction for simplicity’s sake. I base my considerations
on quantities, but since only their relative amount concerns us here,
I have rounded my figures as much as possible.

I intend to show the incorrectness of the conclusion mentioned
from the amount of the afflux to in the Haarlemmermeer polder.
Now a paper on this amount by the member of this Academy

eS ee ee

Car)

van Diusen is found in the Versl. en Meded. der Kon. Akad. 1885").
I can by no means accept the amount found there. The chief reason
why Mr. van Dimsen arrived at an erroneous figure is that he
assumes that the groundwater in the Haarlemmermeer polder which
is situated at a depth of about a metre below the surface evaporates
as much as water at the surface on account of the interstices between
the particles of ground. This, I think, is entirely wrong; ground-
water at a depth of a metre does not evaporate at all.

Now if we assume that the groundwater does not evaporate, the
figures given in the paper would lead to a negative afflux, which
certainly is wrong too. This is a consequence of another reason
why an erroneous figure is found, namely the method of derivation.
Mr. van Drmsen, namely, calculates the amount of the afflux from
two periods of six years, for each of which he derives the equation:

— t=

in which / is the amount of the flow, « the ratio of the evaporation
at the surface and the rain fallen; @ and 6 constants, derived from
the other data. So he has two equations 4 = a — b and hk, = a,v, —b,.
Now he determines the ratio of 4 and /, from the difference in level
of the water in the ‘bosom’ *) and of the polderwater, a ratio
naturally little differing from unity; he further assumes that the wv
of each period has the same average value, so he puts = .x,. The
two unknown quantities, / and wv, can then be found from these two
equations.

But © and wv, are not exactly equal. Mr. van Diesen himself says:
“evidently this value must change according to circumstances.” If
wv, be equal to 1+ 4 we have:

h k
—b, —b a,
= ky 1
ee eS Be
k kh
—tb —— (1) — Tw == [i/
2 1 k, 1

ee : k
Now it will entirely depend on the value of aa and « whether
1
4 has an appreciable influence on the value of .. Since from the

ke
nature of the case pm and « are great values, differing little between
= 1
each other, however, a small value of 4 has a great influence on
v and hence on /.

1) Versl. en Meded. van de Kon, Akademie van Wetenschappen, Afd. Natunrk.
Be Reeks, DI. 1, p. 359—374.
2) “Bosom” is called an intermediate discharge canal or basin.

|

( 48 )

The best estimate of the afflux in the Haarlemmermeer polder is
that by Mr. Euink Sterk'). This author bases his calculation on the
assumption that the value of rain minus evaporation, averaged over
many years, is practically the same for Rijnland and for the Haar-
lemmermeer polder, Rain is here tacitly assumed to be rain plus
surface condensation, and evaporation, evaporation plus the water
withdrawn by plants. From the quantity of water discharged and
let in over an average of 14 years Mr. ELink Srerk then derives
with the aid of the assumption mentioned that the afflux in the
Haarlemmermeer polder is equal to a quantity of water corresponding
to a height of 185 mm. -+ A (A being the afflux in Rijnland) over
the whole surface.

Now he puts A=15 mm. i.e. '/,, of the afflux in the Haarlemmer-
meer polder which he calls an ample estimate as I think it is; so
he finds for the amount of the afflux in the Haarlemmermeer polder
150 mm.

The assumption mentioned that rain minus evaporation is equal
for Rijnland and for the Haarlemmermeer polder is not quite correct
of course. The rain may be taken equal, but not the evaporation. |
The rainfall is in my opinion more regular on the average than is
indicated by our rain-gauges. Under equal meteorological conditions
the rate of evaporation depends principally on water for evaporation
being or not being present. In the polders having a high summer-
level with regard to the land which mostly consists of meadows,
evaporation will be greater than in the Haarlemmermeer polder,
since water will always be present at the surface; in the dunes on
the other hand it will be less. Considering the character of the grounds
in Rijnland we may assume that evaporation there will be slightly
greater than in the Haarlemmermeer polder. So if we apply to our
figure a correction £,, making it 150+ 4,, 2, will be negative.

Mr. Extyk Srerk has left out of consideration through lack of
data: 1. the quantity of water admitted into the Groot Waterschap
van Woerden (having the same bosom as Rijnland); 2. The quantity
of water let in by locks into Rijnland and the Groot Waterschap
van Woerden. Calling these respectively 4, and 4,, the afflux in
the Haarlemmermeer polder is

k= 150+ A, + A, + A, mn.

Now the quantities 4, and A,, are both small and certainly

| 2 3 ,

ositive; probably they are together smaller than 4,. So if we omit
* « e 1

the three corrections 4,, 4, and 4,, the error can not be large and

1) Verhandelingen van het Kon. Instituut van Ingenieurs. 1897—1898. p. 683 —75,

( 49")

the figure for % probably becomes too great as it also becomes by

putting A = 15 mm.

Putting the afflux of water af 150 mm. this gives for a surface
of 18000 H. A. 27 million M*. This amount consists of three
parts: 1. of what is let in for the higher lands behind the ringdyke
through valves and syphons; 2. of the afflux through the ringdyke
above the old sea-clay which I shall call the afflux ¢hrough the alluvium ;
3. of the afflux over the whole surface. of the polder, moving upward
through the old sea-clay on account of the greater pressure, which
I shall call the afflux from the diluvium,

The first part which is no proper affliw, is estimated by Mr. ELink
SrerK at 5 to 7 million M* per year. Subtracting this and taking
the smallest figure the afflux mentioned sub 2 and 3 becomes. 22
million M* per year. How much of this is due to each of the parts
sub 2 and 8 cannot be made out, while in those places. where the
“old sea-clay” is absent no separation takes place. Probably 2 is
the greater part, therefore 1 assume for the part sub 3 an amount
of 10 million M* per year; possibly it is much smaller.

These 10 million M*. the afilux from the diluvium must consequently
either flow in as fresh water along the circumference of the polder
under the old sea-clay through the upper layers of the diluvium, or
rise from below as salt water. Let us for the present assume that
it all flows to in the former manner. In fifty years 500 million M*.
of fresh water would in this way have flowed into the diluvium.
Now the quantity of fresh water present in the diluvium under the
Haarlemmermeer polder is greater; assuming */, to */, space between
the grains of sand this quantity would only correspond to a thickness
of 10 metres containing fresh water, whereas this thickness is greater
on the average, as is proved e.g. by borings near Sloten.

The circumference of the ringdyke being about 60.000 metres, if

we assume the afflux to take place over a height of only 20 metres
and the interstices between the erains of sand to be the same as
above, this will give a velocity of motion of 30 metres per year and
the water flowed to would, even if we neglect the loss of speed
further in the polder, have penetrated into the polder only 1500
metres in 50 years and so not have reached the middle.

Moreover the assumption that all the water streaming to is fresh,
is not probable, if we bear in mind the amount of salt in the Wil-
helmina spring which is over 3000 mg. chiovine per litre. Hence it
is certain that with a flow of 10 million M* from the diluvium,
part of the fresh water nowadays present in the diluvium under the

4

Proceedings Royal Acad. Amsterdam. Vol, VIL.

( 50 )
Haarlemmermeer polder was present there already 50 years ago.

Before that time conditions were very different from what they
are now. Instead of the deep drainage there was bosomwater. How
the fresh water then present, especially in the eastern part, had come
under the Haarlemmermeer, is difficult to tell for want of data. Was
the water of the Haarlem lake always so rich in chlorine as some
old observations show? Certainly the difference in pressure of the
deep groundwater was smaller than it is now and accordingly the
quantity of water moved was also smaller, while the direction of
the current in the deep groundwater e.g. near Sloten must have
been exactly the reverse.

A thousand years ago when there were no dykes yet to keep out
the water of rivers and of the sea, and no mills vet to drain the
polders, when the dunes were so much broader at the seaside than
they are now, when there were no canals in the dunes yet for sand
transport and other purposes, I imagine the state of affairs in the
tract of land we are considering, must have been such that fresh
water was also present in the diluvium and probably more than
nowadays, that in the dunes there existed a high level of ground-
water by which water was driven to the diluvinm, the pressure at
the west side under the “old sea-clay” being greater than that of the
groundwater above it. The level of the groundwater in the alluvium
of the polderland was then probably much more regular and slightly
higher than the average sea-level. How these conditions became pre-
valent I must leave to geologists to explain.

By making dykes, by enclosing polders, by draining, the level of
the groundwater has gradually been lowered, now in one place, then
in another. The currents in the layers of fresh water in the diluvium
also had their directions changed by this; they certainly were very
small, however, before the great drainages were made. In the traet
we are dealing with, 3000 H.A. were drained before 1750, 10.000
H.A. between 1750 and 1850 and 26.000 H.A. between 1850
and 1900.

The dunes gradually decreased in breadth, while also the flow of
water towards the land increased by canals for sand-transport ete.
The height of the groundwater in the dunes will consequently also
have steadily been decreasing.

Bearing in mind the figure for the amount afflux to in- the
Haarlemmermeer polder, we may safely assume the quantities of
water which before the drainages were made, penetrated vertically

(51 )

downward through the old sea-clay, to have been very small com-
pared ‘with the amount of fresh water present. Consequently the
only source of supply of fresh water to the diluvium has been the
afflux from the dunes. At the same time I venture the supposition
that part of the fresh water which a thousand years ago was present
in the diluvium under the polderland is still present there now.

Another question arising here is whether salt water rises upward
from below. About former times nothing can be stated with certainty
in this respect; for the present time it is rendered probable by the
circumstance that the water of the Haarlemmermeer polder contains
more chlorine than can be derived from the aftlux if no salt water
from below is added to it. Therefore I have tried to estimate, though
roughly, the quantity of chlorine, discharged by Rijnland and_ the
quantities entering Rijnland in another way than from below, assuming
that the quantity of chlorine withdrawn from the ground by plants
is equal to the quantity furnished by manuring, which supposition
is reasonable.

Rijnland discharges annually on the average 476 million M®*. of
water; how much chlorine this contains is not known, but from the
data for the percentage of chlorine of the water of the bosom?) a
figure may be derived which is too small and another which is too
large, which figures I take to be 105 and 315 me. per litre, giving
an annual discharge of 50.000 or 150.000 tons of chlorine.

The quantities of chlorine arriving into Rijnland are, besides that
from the groundwater below 1. the sea-spray ; assuming that this
chiefly falls on the dunes we can estimate it; the dunes that discharge
water into Rijnland will supply about 20 million M®. of water
annually ; this water contains 40 mg. chlorine per litre, making
800 tons of chlorine; adding to this what is blown over the dunes
we obtain a total of 1500 tons; 2. the fresh water which is let in.
amounting on the average to 125 million M*. per year, containing
40 mg. per litre, which makes 5000 tons; 8. the water through
locks; it is difficult to estimate an average percentage of chlorine
here, since one lock (Gouda) admits fresh water to the bosom, others
(Spaarndam, Overtoom, et¢.) water with a high percentage of chlo-
rine; I think 2000 mg. per litre a sufficiently high estimate; putting
the water let in through locks at 5 million M*. this makes 10.000 tons:
4. what human society discharges into the bosom; this amount is
difficult to estimate ; putting it at 3500 tons, the total amount becomes

") Mededeelingen omtrent de Geologie van Nederland, no. 26, by Dr. J. Lorri,
pp. 8—1ll.

(52 )

20.000 tons of chlorine. Assuming that what is supplied to the bosom
by all these causes is more than half this rough estimate and less
than its twofold we get 10.000 or 40.000 tons of chlorine. In any
case there is a deficiency amounting to something between 10.000
and 140.000 tons which has to be ascribed to a supply of salt from
a greater depth. When we bear in mind that this will chiefly come
from the Haarlemmermeer polder and that this latter discharges on
the average about 30.000 tons of chlorine and that the supplies
mentioned sub 1—4 occur there in a small degree, the supply of
salt from below at the present time is pretty certain. A quantity of
35.000 tons of chlorine corresponds to that contained in two mil-
lion M*. of water from the North Sea.

The motion of the deep groundwater is generally very slow.

If e.g. we consider how long it would take water to travel ina
layer of sand between two impervious layers from the sea to the
Haarlemmermeer polder, which is a distance of 9000 metres, the
difference of pressure being 5 M., if the permeability is the same as
that of dune-sand, we find that it would travel ina year (31.557.000
seconds) through a distance of

31.557.000 «K 5 & 0.0006

— 10 MM.
9000

(0.0006 being the rate of filtration through 1 M. of dune-sand with
a pressure of 1 M.’). A distance of 9000 M. would consequently
require 900 years.

As to the rate with which the salt water can rise from below we
find what follows. Assuming that the rise is constant and that under
the Haarlemmermerer polder 5 million M*. rises annually, this gives
over a surface of 18000 H.A. with a space of '/, to ‘’, between the
grains of sand, a rise of about 100 mm. per year; a rise of 50 metres
would then require a period of 500 years.

The question now naturally arises: since a large quantity of fresh
water is present in the diluvium under our polderland and the salt
water flows slowly, is it possible to withdraw this fresh water for
drinking-water? The part we are considering has, after taking off
the littoral margin and the country round Amsterdam which has a
different formation, a surface of about LOO.000 H.A.; not counting
the drainages and other less suitable tracts, half of this territory
contains 5000 million M*. of fresh water, if we assume an average

1) Report of the Committee for investigating the supply of water from the
dunes to Amsterdam, 1891, supp'ement 16, p. 77.

wal

ib BA

( 53)

of 10 M® of fresh water per M?*, corresponding to '/, to ‘/, space
between the grains of sand over a thickness of 30 to 40 metres. A
consumption of 50° million M* per year being sufficient with the
existing dune-water conduit for the need of the population, taking
its increase into account, this quantity would be able to supply water
for a hundred years; now we may presume that ina hundred years
science will have so much advanced that it will be practicable then
to convert any water into suitable drinking-water.

The answer to our question must in my oOpimion be affirmative as
well as negative. Affirmative with respect to supplying water to single
dwellings, to a village, or to a temporary supply in war-time such
as the Engineering Corps has made at Sloten; negative with respect
to a lasting demand on a large scale and this because in practice
pecuniary considerations would force us to withdraw the water from
a limited surface which would be impossible without causing such a
diminution in pressure that certainly with a lateral afflux also water
from below would flow to, so that after some time brackish water
would be obtained.

Hence Prof. Dupots’ assertion, that a sufficient quantity of drinking-
water is and remains available in the ground under the shallower
polders, is in my opinion entirely wrong.

Geology. — “On the origin of the fresh-water in the subsoil of a
few shallow polders’. By Prof. Eve. Dusois. Communicated

by Prof. Bakuuis RoozeBoom.

(Communicated in the Meeting of November 28, 1903),

In the meeting of the Academy of September 26 ult. Mr. H. E.
bE Brvyn, although he agreed with most of the principal conclusions
about the origin and the direction of motion of the groundwater in
part of our lowland, contained in my communication to the Academy
of June 27, gave an elaborate exposition of the grounds on account
of which he cannot accept my conclusion concerning the origin of
the fresh-water in the subsoil of afew shallow polders. In my opinion
this has to be sought in rain, fallen on the spot or at a relatively
short distance, which Mr. pr Bruyn thinks impossible on account of
considerations about the amount of the afflux in the Haarlemmermeer
polder which, in his opinion proves that the layers above the diluvium,
especially the “old sea-clay” transmit water to a much smaller extent
than is necessary in my representation. He also supposes that part
of the fresh-water which was present under our polder-land a thou-

(54)

sand years ago, is still there at the present day and that the only
source from which fresh-water has been supplied to the diluvium
(the subsoil) has been the dunes.

About the velocity with which water can move through our always
very impure clay; which Mr. pr Breyy rightly considers to be a
cause of our difference of opinion, I will now state a few facts and
at the same time point out the arguments which led me to my
conception of a different origin of the deep groundwater mentioned.
First however I wish to point out another possible origin which has
not yet been suggested and which cannot be at once rejected, and
especially a difficulty of a more serious nature even than the one
objected to my representation by Mr. pr Bruyn.

If we assume the extremely slow motion ascribed to the ground-
water by Mr. pr Brvyy, it might namely be that the deep fresh-water
under consideration has to be considered as a remainder of what
sank away there centuries ago. For not always these polders have
been surrounded by brackish water only.

According to descriptions from the Roman period, Lake Flevo
undoubtedly contained water from the Rhine and no salt water as
the Zuiderzee does nowadays. Also the IJ was a freshwater lake
communicating with the freshwater lakes Purmer, Wormer and
Schermer. Moreover it is well known that the Haarlem Lake
(Haarlemmermeer) arose by the union of at least four lakes: the
Old Haarlem Lake, the Old Leyden Lake, the Old Lake and the
Spiering Lake whieh were fed at least partially by one or more
branches of the Katwijk Rhine. The map by Bolstra, the able
land-surveyor of Rijnland. published in 1745 and incorporated in
“Present state of the Mnited Netherlands” '), gives us an idea of the
situation of these lakes in 1531 and of their gradual union and the
enlargement of the Haarlem Lake, originated in this way, down till
1740. The waves of this large lake could easily erode the steep banks,
consisting of fen, as low as the same layer of clay which already
formed its bottom, the circumstances for this process becoming more
and more favourable, chiefly on account of the “sinking of the lands”
in these parts with respect to the sea, described already a century
before the draining. This erosion of land oceurred at a tremendous
rate at the north-east side, where the polders are situated which now
have fresh-water in their underground. Le Fraxcg van Berkiky *)

1) Tegenwoordige staat der Vercenigde Nederlanden. Vol. 6 p. 163. Amsterdam,
I. Tirion, 1746.

2) J. Le Franeg van Berkuey. Natuulijke Historie van Holland. Vol. Lp. 227.
Amsterdam 1769.

(55)

about the middle of the 18 century deseribes the water of the
Haarlem Lake as “fresh, but in some places, where the grounds
become brackish, as near Slooten and towards Amsteldam, the water
of the lake is sometimes of a saltish taste. But the abundance
of water from the Rhine and the supply from so many small
lakes and waters which discharge themselves into it, bring about
that the brackish water can by no means get the upper hand,
and so the lake has on the whole good fresh-water.’ Meanwhile
a quantity of salt amounting to 800 milligrammes per litre is
according to the latest investigations not unpalatable. An analysis
by G, J. Munprr') of water taken from the lake near Sloten
in November 1825, shows that it contained 393 mg. chlorine per
live. Now this is the season during which it will probably have
been least brackish. Hence it is improbable that the water of the
Haarlem Lake was on the whole really fresh. Indeed, the lake had
ample opportunity to receive salt from the IJ (which had already
become salt towards the middle of the 13 century) through the
upper ground of the polder-land which consisted chiefly of fen and
which separated the two waters in places (near Halfweg) like a
true isthmus. It is also known that at any rate towards the middle
of the 15 century those grounds wnder which fresh-water is found,
were brackish. Yet fresh-water of a much earlier period might in
places have remained in the underground. Water derived not only
from the north and west sides, but also from the east, may have
filtered into the polders mentioned at the north-east of the present
Haarlem Lake. The Amstel certainly contained for centuries perfectly
freshwater, derived from the Rhine. As late as 1530 the Amsterdam
canals, fed by this river, had drinkable water, but soon this supply
was gradually more and more reduced by natural causes.

Is now the motion of the groundwater, not only in the finer
alluvium, containing much sand, but even in the coarse and gravelly
diluvial sand, which transmits water much more easily, really so
slow, as Mr. pe Bruyn believes, that in three or four centuries the
influence of the altered circumstances as to level and composition
of superficial waters on the deep groundwater will scarcely be
perceptible? I believe that numerous facts, of which IT will mention
a few in this communication, are at variance with this opinion.

1) G. J. Munper. Verhandeling over de wateren en lucht der stad Amsterdam.
p. 66. Amsterdam 1827. Lorm, quoting from second hand, wrongly mentions this
same analysis under two different headings and with different amounts of Cl.
(Onze brakke, izerhoudende en alkalische bodemwateren, Verhandelingen der
Kon, Akad. 2e Sectie, DI. 6. N°. 8. 1899. p. 9).

(56 )

In the first. place the actual facts are incompatible with Mr.
pe Breyy’s idea that before the draining of the Haarlem Lake,
some 50 years ago, “the direction of the current of the deep
groundwater e.g. at Sloten, must have been exactly the reverse”
of what it is now (These Proceedings VI, p. 291) and still less I
can assume this for an earlier period. For near Sloten the country
was not lower, but even a litthe higher than the level of the Haarlem
Lake and not dyked in, so that the incessant washing away of the
steep fenny bank of the lake could be enormously great. According
to an accurate investigation, made in 17438, it amounted yearly on
the average to as much as 5 to 10 Rijnland rods (about 19 to
38 metres). It is true that the upper side of the layer of fen which
now forms the Rieker polder near Sloten, lies at 1.35 metres below
A. P. (Amsterdam level) but its lower side is still on a level with
the bottom of the Haarlemmermeer polder, as it formerly was with
the bottom of the lake, and it rests on the ‘old sea-clay’. If we
now bear in mind that fen, such as that of the Rieker polder
consists, when it is completely saturated, for */,, of water, as I have
found to be actually the ease, and that moreover the ‘looseness and
shiftiness’ of these grounds which, as it were, rose and sank with
ihe water, were well known in the time of the lake, it is clear,
how, after the draining, in half a century, by losing over */, of
their water, they might shrink so far below A. P. and that there
can be no question of an earlier current of the deep groundwater
under the Haarlem lake towards the country near Sloten. The lake
certainly did not allow such a current from the dunes to pass under
its bottom.

Though the dunes were broader and the level of the water in
them higher than nowadays, the hydrostatic pressure imparted there
io the deep groundwater must have been exhausted and the horizontal
current stopped by the water rising in the alluvial cover, which
forms an imperfect screen, long before the opposite side had been
reached. For at present the difference in pressure, owing to the
water-pressure being now 5 metres less in the polder than it formerly
was in the lake, is certainly not less and yet already in the middle
of it only a slight upward pressure remains, although pressure is
directed from all sides to the middle.

Hence in the uncergeound we only meet water that soaked the
soil from above without any considerable horizontal movement in
the underground.

I quite agree with Mr. pr Bruyn (p. 291) that: ‘part of the fresh-
water now present in the diluvium under the Haarlemmermeer

polder was present in if already fifty years ago’, if [ may consider

water that contains 200 to 3800 and even more milligrammes of

chlorine per litre as fresh-water, as I understand he does. Only with
this latter not common qualification one is entitled to say that under
the Haarlemmermeer polder the diluvium has on an average more
than 10 metres fresh-water, for the greater part of that polder has

no fresher water in its underground than with these amounts of

chlorine. Only where the higher grounds are clearly of recent origin
this is different, for the rest the water in the upper diluvial layers

of the Haarlemmermeer polder contains about the same amount of

chlorine as the water of the former lake. Of the Wilhelmina spring,
the amount of chlorine of which is over 38000 milligrammes, the
depth is unknown; undoubtedly however it goes down as far as
the salt water which in most places of this polder is to be found
below 40.50 metres.

The most serious difficulty opposing my view of the origin of
the fresh-water in the subsoil of some shallow polders is not men-
tioned by Mr. pre Bruyn. It is that the fresh-water in question in
all seasons not only is surrounded by, but also rests on and is
covered by brackish water. How can the fresh-water under these
circumstances owe its origin to the rain fallen on the brackish
upper ground

The explanation of this paradoxical phenomenon I mean to have
found in the peculiar hydrological condition of those polders which,
like those between Amsterdam and the Haarlemmermeer polder, are
themselves at a level only little below A.P. and are situated near
deeper drainings. In a similar condition are the shallow polders near
Purmerend and Schermerhorn. Like here towards the Haarlemmer-
meer polder, so yonder towards the deep polders Purmer, Wijde
Wormer, Beemster and Schermer, a considerable flow exists in the
coarser diluvium under the more compact alluvial cover and at
the same time a vertical downward movement, while in those deeper
polders the water tries to rise through the alluvial cover which forms
only an imperfect screen. Consequently in boring-tubes the ground-
water from the diluvium in these latter rises higher than the field,
whereas in the shallow polders it remains far under it and below
the level of the groundwater. These circumstances and the geological
condition of the soil form in my opinion the solution of the riddle
of the presence and the permanence under some shallow polders
of fresh-water which on all sides is surrounded by brackish water.
I arrived at this conclusion especially by studies in the Rieker polder
near Sloten, in which the source for the military Water-supply for

(58 )

the position of Amsterdam is situated. To this IT was enabled by a
few experimental wells which Mr. G. van Royrn, 1s* lieutenant-
engineer, charged with the execution of the works there, was kind
enough to have made for the purpose of this investigation. At my
proposal seven of these experimental wells were bored to various
depths in the middle of a meadow, about 300 metres south of the
Sloten road and the farm ‘Rustvrede”, within a square of three
metres side. Under the lower end of the iron tubes which were
open below, gravel had been poured to a depth of half a metre.
By examining the water that had risen in those tubes as to level
and composition one can obtain information about the state of affairs
at various depths in relation with the condition of the soil.

Letter Depth of the Layer in which the water
of the layer of water is found.

well. below the field *).

A 1 io. 2 te> <M: fen.

A i hy Onder” ea “A

B BAO ah ce0 %,, Upper clay.

C Se eo! y, Layer of sand in the clay.

C' eo" coe: 0F 5 Lower part of the clay.

D tO} 43, RODD 3; Between deep fen and sand

E AS Ge peal. 55 Sand under the denser alluvium.

A well situated 30 metres to the west, 44 metres deep, IV, 13,
was compared with these experimental wells.

Most of these wells were ready at the end of August ult. A’ was
finished about the middle of September, C' the 19% of October. The
level of the water and the amount of chlorine were repeatedly
examined by me. After all the wells had been left undisturbed as
long as November 20, the following state of affairs was found.

Water level in M. + A.P. Chlorine
in mg. per L.
A 1.465 | 1385
yi 1.495 202
B 1.495 617
C 1.493 780
C 1.677 145
D 2.720 124
E Drie 65
L¥SIS 2.747 92

1) The level of the field is 1.33 ~ A.P.

(59)

Of these wells C' may for the present be left ont of account
since a stationary condition has not yet established itselfin it; having
its lower end in the clay, the water still rises in it continually
Well IV, 13 again has a higher and varying amount of chlorine ;
determinations at different times gave 114,92,190 mg. per litre. I
believe that this increase and variation of the amount of chlorine
has to be aseribed to the neighbourhood of the deep salt water in
relation with fluctuations in atmospheric pressure and also with a
proper motion of ebb and flood of the groundwater ').

In spite of the very considerable rainfall of the latest months,
great variations in the percentage of chlorine did not occur; only
in and near the clay, hence especially in C, the amount of chlorine
decreased considerably, in C from 850 to 780 mg. per litre. This
result does not verify my formerly stated supposition that perhaps
during the wet season a continuous freshening of the water might
take place.

Yet I could not agree with the idea that the fresh-water in the
subsoil should have stayed there undisturbed for at least a few
centuries. For why then is fresh-water in the diluvium under the
Haarlemmermeer polder and the Lutkemeer polder only found at a
distance not too far removed from the shallow polders near Sloten
and Osdorp? Why does this layer of fresh-water end already before
Halfwee, before the Great IJpolder is reached, south of Sloterdijk
and also soon eastward of the Amstel? Why does the layer of
Purmerend not extend further than a short distance under the Purmer
and Beemster polders? Does not this limitation point to an autochthonous
origin of the fresh-water in the underground of the shallow polders ?

I think to have found the key of the riddle in the stated sudden
fall in pressure, amounting to more than 1,20 metre, under the clay
and the deep fen which is a consequence of the fact that the level
of the groundwater in the Haarlemmermeer polder is almost 3.5 M.
lower than in the Rieker polder. So this compressed deep fen,
acting as a semi-permeable wall, can transmit to the deeper layers
water, but no salt.

That fen in a compressed state and under similar conditions of

1) On these influences, especially on the proper ebb and flood of the ground-
water, see: F. Weype, Die Abhiingigkeit des Grundwasserstandes yon dem Luft-
dricke, dessen Steigen und Fallen wiihrend eines Tages (Flut und Ebbe), in
Meteorologische Zeitschrift of August 1903. The influence of atmospheric
pressure was already pointed out in my former communication. These influences
become perceptible only in deep wells, because in them the water follows more
easily the changing pressure of the atmosphere and gravity than it does in the
neighbourhood and so is raised or depressed,

SO

ee ee eee

( 60 )

pressure as prevail in the Rieker polder, can cause osmosis, | could
prove experimentally with apparatus which Mr. A. J. Sronn Jr. at
Haarlem was kind enough to make for me in his workshop and
with other apparatus kindly put at my disposal by Dr. Hrrinea
of Haarlem. The most important of these experiments is the follow-
ing one.

In an iron tube of 1 M. length and 154 mm. internal diameter
newly dug fen from the superficial layer in the Rieker polder was
compressed by means of a lever until no further compression was
observed. The pressure was gradually raised to 2,8 kilogrammes
per square centimetre, a pressure equal to that which is found ina
soil of sand or clay at a depth of 14 metres; the layer of fen was
three centim. thick. Of water, containing a quantity of sodium
chloride corresponding to an amount of chlorine of 1000 mg. per
litre, this layer of fen, which on account of its slight thickness, can
by no means be so perfect a semi-permeable wall as the deep layer
of fen in the Rieker polder which has an average thickness of a
inetre, water was transmitted which contained temporarily at the
utmost 750 mg. chlorine per litre; hence at least 250 milligrammes
were retained.

Now the deep fen in the Rieker polder occurs as an almost
coherent layer, extending from Haarlem, right through the Haar-
lemmermeer polder as far as Mijdrecht and from Sloten by Amster-
dam as far as Zaandam and Uitdam. This layer is missed in the
north-western corner of the Haarlemmermeer polder, i.e. in the place
of the former Lake Spiering and farther south. The lower side of
this deep layer of fen lies at about 11 to 13 metres below A.P.
Still deeper at Sloten in some three borings, parts of a second old
layer of fen were found and also repeatedly at Amsterdam and
Zaandam. This layer must be distinguished from the former with
which it was formerly identified. As a fairly coherent layer this
deeper fen can be traced above the diluvium, to the north by
Purmerend as far as Hoorn and Enkhuizen, to the west by Wormer-
veer, Beverwijk, and Velzen to IJmuiden. The upper one of these
deep layers of fen can reach a thickness of about 1 M., the lower

1/
4/32

one is rarely metre thick.

So we may understand how the underground may have derived
in former times and may still derive its fresh-water from the upper
ground although this latter always carries brackish water itself.

But will the “layer of clay” which is 7 M. thick be permeable
enough to render it possible that in the half century, elapsed after

the draining of the Haarlem lake, under the polders to the north-

( 64 )

east of this drainage a layer of fresh-water of at least 50 M. thick
may have accumulated? This means a yearly increase of at least
one metre, or, if we take into account the interstices between the
erains of sand, of about 0.80 M. pure water a year. Now this
amount is pretty much the same as of the rain that can penetrate
into the earth, while also all the other surface-water can furnish
fresh-water to the underground, by which also the fresh-water,
flown off to the deep polder, can be accounted for. In my former
communication 1 already pointed out that the power of clay to
transmit water is commonly underrated. The clay in our alluvial
grounds is generally very impure, consists mostly of very fine sand
and according as the percentage of this increases, its “permeability”
becomes greater. The fattest clay of the Rieker polder at Sloten lies
as a thin bank immediately under the superficial layer of fen and
contains 30°/

» real clay. From the 7 M. “clay” in the Rieker polder,
one has to subtract first a couple of metres of sand, the rest is also
much richer in sand than the fat upper layer mentioned. Now
Sprinc has proved that a layer of Hesbay’s loam of 7 M. thickness
admitted in 24 hours a movement of water of at least 0.036 to
0.045 M.") which is ten to fifteen times more than the velocity
calculated for the Rieker polder. A sample of loam, kindly sent me
by that scientist, proved, on analysis by Dr. N. Scnoorn, to contain
21.5°/, clay i.e. about as much as our ordinary, pretty fat alluvial
clay contains on an average. Experiments with fatter clay under
pressure, as if is in nature, give me a much smaller velocity which
however is still sutficient to explain the hydrological condition of
the Rieker polder. Of these experiments I intend to give an account
on a future occasion.

I wish to draw attention to a result of the experiments of Spring
already mentioned in my former communication, according to which,
when the thickness of a layer of sand becomes very great with
respect to the pressure-column of the water, the rate of filtration
may by no means be taken inversely proportional to the thickness
of the filter. On the contrary, Sprinc found in this case the rate
independent of the thickness of the filter. 1 can confirm this for
clay and for this substance the pressure may even be relatively
great and the thickness of the layer hundreds of times smaller on
account of the so much greater resistance ef clay than of sand. A
layer of the fattest clay, obtained in the same way as the com-
pressed fen, by squeezing out the water, having a thickness of

1) In this time namely a layer of waler of 12 to 15 mm. thickness was trans-
milled,

( 62)

15 em., under a pressure of 80 cm. transmitted no less water than
a layer of the same compressed clay of only 4 cm. thickness. Cal-
culations like those on page 294 of Mr. pr Breyy’s communication,
in which the rate of filtration through 9000 M. sand from the dunes
is simply assumed to be */,,,, of the rate found in an experiment
with 1 M. of the same sand lead consequently to erroneous con-
clusions. I also want to point out that in the coarse diluvial sand
which forms the principal way for the horizontal movements of the
water, the velocity of motion is about ten times as great as in
sand from the dunes.

Under these circumstances I believe to be justified in maintaining
my opinion that the water in the underground of some shallow
polders is of autochthonous origin.

It is also clear now, how in many places in the Haarlemmermeer
polder water can spring up which is as fresh as water from the
dunes. So in the farm ‘het Botervat’” on the Y road near the Kruis-
weg; in it I found as well in the driest periods as after much rain a
quantity of chlorine of 35 to 37 mg. per litre, whereas in the same
farm a well has been bored reaching just below the deep fen, in
which the water contains an amount of chlorine of 235 mg. per
litre. At numerous other spots of the Haarlemmermeer polder the
presence of fresh-water in wells (which proved to be no rainwater,
as I believed for some time) could be stated; it is also found at
about one kilometre east of the just mentioned farm, besides on the
Kruisweg between the Sloten road and the Sloter Tocht, on the
Sloten road near the Slaperdijk. On the other hand the water in
wells in the north-western part of that polder, in which the deep
layer of fen is entirely absent, is brackish everywhere.

The water flowing under the compact alluvial cover from the
higher environs of the Haarlemmermeer polder has there, as I showed
in my former communication, a tendency to rise and so the salt-
retaining property of the fen can here act in an opposite direction
as in the shallower polders in which the vertical component of the
water is directed downward.

As the old fen forms, as it were, a filter for sodium chloride,
so in the shallower polders the ‘old sea-clay”” by its high percentage
of iron, keeps the water in the underground relatively free from
sulphuric acid. The superficial fen in the Rieker polder contains so
much compounds of sulphur that it has a very strong smell of
sulphuretted hydrogen, when freshly dug. Water squeezed out from
it proved on analysis by Dr. Scuoor. to contain no less than 408 mg.

ax

( 63 )

SO, per litre, while that from a well, 44 M. deep, near the place
where the fen had been taken, contained only 17 mg. SO, per litre.
Already immediately below the clay, at a depth of ten metres below
the meadow, the amount of SO, has become so small. In the iron-
containing layer of clay, pyrites is namely formed by the well-known
minerogenetic process, with previous reduction of the SO, compounds
in the fen, which reduction takes place here with the help of sulphur-
bacteria by which the freshly denuded fen is coloured yellowish.

Pyrites can indeed be shown to occur in the clay. And so this
difference in the amount of SO, between the upper water and the
deep groundwater is a proof for the origin of the latter from above
instead of against it, as has been supposed.

Chemistry. — Prof. C. A. Losry pr Bruyn also in the name of
Dr. S. Tymstra Bz. read a paper: “The mechanism of the
salicylacid synthese.”

(This paper will not be published in these Proceedings).

Chemistry. — Prof. C. A. Losry pr Bruyn presents a paper of
Dr. J. J. Buanksma: “On the intramolecular oxydation of a
SH-qroup bound to benzol by an orthostanding NO,-group.”

(This paper will not be published in these Proceedings).
Chemistry. — Prof. C. A. Losry bE Bruyn presents a paper of
J. M. M. Dormaar: “The inversion of carvon and eucarvon
in carvacrol and its velocity.”

(This paper will not be published in these Proceedings).

(June 24, 1904).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Saturday June 25, 1904.

DCGe

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 25 Juni 1904, Dl. XII).

CO Ne Ean Ss

Epmunp Lanpav: “Remarks on the communication of Mr. Kiuyver: “Series derived from the
4 fz(m Q ~
series py ate) (Communicated by Prof. J. C. Krvyvrr), p. 66.
m

¥. M. Jarcer: “On Benzylphtalimide and Benzylphtal-isoimide.” (Communicated by Prof.
A. P. N. Francurmont), p. 77.

H. P. Kuyrer: “On the development of the perithecium of Monascus purpureus Went and
Monascus Barkeri Dang.” (Communicated by Prof. F. A. F. C. Went), p. 83.

A. J. P. van DEN Broek: “On the genital cords of Phalangista vulpina.” (Communicated
by Prof. L. Bork), p. 87.

P. P. C. Hork: “An interesting case of reversion”, p. 90. (With one plate).

RB. M. van Danrsen: “On the function 7 for multiple mixtures.” (Communicated by Prof.
J. D. van DER WAALS), p. 94.

Frep. Scuun: “On an expression for the genus of an algebraic plane curve with higher
singularities.” (Communicated by Prof. D. J. Korrewee), p. 107.

Frep. Scuun: “On the curves of a pencil touching a algebraic plane curve with higher
singularities.” (Communicated by Prof. D. J. Korrewee), p. 112.

C. Easton: “On the apparent distribution of the nebulae.” (Communicated by Prof. H. G
VAN DE SanpE Bakuwvyzen), p. I 17.

C. Easton: “The nebulae considered in relation to the galactic system.” (Communicated by
Prof. H. G. van DE SanpE BakuuyzeEn), p. 125.

W. H. Junivs: “Dispersion bands in absorption spectra”, p. 134.

W. H. Juxius: “Spectroheliographie results explained by anomalous dispersion”, p. 140.

H. Zwaarpemaker: “On artificial and natural nerve-stimulation and the quantity of energy
involved”, p. 147.

The following papers were read :

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 66)
Mathematics. — Remarks on the paper of Mr. Kuivyver on

Neen Ser . a: - 2 ; _ a (m)”
page 305 of Vol VI. “Series derived from the series = & (m)

m
by Epmunp Lanpav in Berlin.

(Communicated in the meeting of May 28, 1904).

In a paper recently published’) Mr. Kivyver treats the infinite series

Ko mb +h) _ wh) u(b+)) W262) Sean
fod om +h h b+h 2b+h

m=—0

where } and / are two positive integers and where / can be regarded

as <b without limiting the generality. However, this research has
to be taken only in a heuristic respect ; it does not furnish the proof
that series (1) converges, i.e. that for every pair of values b,h

, —~ u(mb+h)
awe a » mb+h (2)

m=0

exists. *)

The methods applied by me in the paper*) “On the prime num-
bers of an arithmetical progression” (“Ueber die Primzahlen einer
arithmetischen Progression”) however allow such a proof to be made
which will be shown in § 2—7 of this paper, after I have reminded
my readers in § 1 of some theorems still known; at the same time
we shall find an expression for the limit (2) i.e. the sum of the
infinite series (1) in finite form. In §§ 8—9 a conclusion is drawn
on the distribution of the numbers of an arithmetic progression
for which u (n) = + 1 resp. — 1; this justifies a supposition expressed
by Mr. Kivyver at the end of his paper.

§ 1. Let x, (7), x, (”),---, x0) (m) be the (6) characters of the
group of the classes of residues modulo 6 prime to 6, of which x,(7)
be the principal character (Hauptcharakter). If and 6 have a common

divisor we may understand by x, (2), %5(7), « - - 5 %(2) (m2) the value zero.
Let moreover J,(s) for » = 1, 2,..., (6) denote the analytic

function determined by the Diricuier series

1) These Proc. VI p. 305.

2) Only for the case b=1, h=1 this was already known and from this ensues
directly, as Mr. Kuvyver states at the beginning of his paper, the correctness of
the statement for any b and h=b, but not for the general case.

3) Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien,
mathematisch-naturwissenschaftliche Klasse, Bd. 112, Abt. 2%, 1903, S. 4983—535,

ns
n=]
As is known this series converges, in case »=1, for X (s) >1,
and in case »=2,..., 9(b), for X(s) > 0.
The equation holding good for Xt (s) > 14 and every v(—1,..., g(0) )

L,(s) = — ~L : ee Ade water ° (3)

aR
p*

where » passes through all prime numbers, shows that no Z, (s)
possesses a zero with real part > 1. The equation (3) gives for »y = 1

ri enti el (= EE) 1 Has mC)

where p passes through all prime factors of 6. From (4) it follows that
L,(s) may be continued across the right line X%(s)—=1 and that it
possesses in s=1 a pole, so that

lim

sel ©
Further Diricnit*) has expressed the quantities

< 44(n) =. ¥q0){n)
i= bea Aaah Ly = y

I n=1

(5)

in finite form by logarithms and trigonometric functions and proved
moreover — what did not at all ensue from it — that each of
the afore mentioned gy(>)—1 quantities is different from zero. So
the limits

il il F il i

lim —— = eee
s=1 L,(s) L,(1) s=1 Ly) (s) Lye) (1)

(6)

do exist.

1) “Proof of the theorem that every unlimited arithmetical progression of which
the first term and difference are integers without common factor, contains an
infinile number of prime numbers,” Transactions of the Royal Prussian Academy of
Sciences at Berlin, 1837, p. 45—71; Works, Vol. 1, 1889, p. 313—342.

(,,Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren
erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind,
unendlich viele Primzahlen enthalt”, Abhandlungen der Kéniglich Preussischen
Akademie der Wissenschaften zu Berlin, 1837, 8. 45—71; Werke, Bd. 1, 1889,
S$. 313—342).

~

5*

( 68 )

Mrs. Hapamarp and pr LA VaALike Poussin have proved that no

Le
Le is
regular for every » on the right line N(s)=1. In the quoted paper
I have proved") the more general theorem: There is a positive number
a so that, when s=o- #, in the region

1
t>3, 1—- o<2

NS a

L,(s) possesses on the right line X(s)=1 a zero, so that

each of the g(%) functions Z, (s) differs from zero and fulfills the
inequality

—_ log*
Lars es

§ 2. Now I denote by J%;,(s) that analytic function which is
determined by the Drricu.rT series
as p(mb +h)
ps (mb+-h)s’
m=0
convergent .at least for o—=2X(s)>1 and I shall show that Jf, (s)
can be brought in a very simple connexion with the functions JZ, (s).
Let the greatest common factor (/,/) of ) and h be put equal to
d. Without limiting the generality ¢@ can be regarded as being with-
out quadratic factor; for in the other case «(mb+)=0, so every
member of the infinite series (1) is equal to zero.
I. Let then be d=41, so h prime tod. Then there is an integer /,,
(determinate modulo 4) for which
h, h =1 (mod. 6).
Now ensues from (3) for 6=X(s) >1

oe

n=1

If we multiply (7) - y (h,) and sum up with respect to all
values of » we get

Sy Sees yaar nes
yh, Yo (2) a(n x u(r
= 4%, ) = / 8
L{s) Dae (H4) a ns = 2S seme Se Alyn). i)
— | 1 n—! [| i
Now according to the fundamental property of the characters the
x

sum = = he (J) differs from zero only, and then is equal to y (), when

=1 Pett b); hence

1) l.e., page 521. Here I put the greater of the two numbers ¢a, and cs; =a.

—

De =&«as lee

(Gates 19)

9(2)
= s (0), if hyn=1, 1. e. n=h (mod. b)
= x (hy, a a :
0, ifh, n=—1, i.e. n= h (mod. b)

So (8) eee into

(0) oo

yh — u(mb+h)
Sig OD. coy 20H
I m=—0

when with the aid of
ths) (2) = rhyh) = 4%) = 1
we eliminate /,, we get
1)
My, (s) = : a : bie. pres ae)
OO GQ) Hat HALAS)

Il. Let d be >1 and let & be put equal to dB, h to dH, so
that B and H are prime to each other. Evidently

= u(d) u(mB + H
und +) =a d(mB + 1) e(d) aC -++ #)

or'== 0;

according to er being prime to d or not. Hence

My,n (8) = (10)

w(mb+-h) wd) 1u(mB + HH)
(mb + h)s Os. (mB+ H)s ’
m—0 m—0
where the sign ’ denotes that m assumes only those values, for
which mB-+ H is prime to d. If m,—m, (mod. d), then it is
evident that m,B-+ H and m,B+-H are simultaneously prime to d
or not. So those m distribute themselves in certain arithmetical
progressions modulo d; i.e. among the d progressions m—0,1,...,
d—1 (mod. d) m has in certain progressions, let the number be g,
to pass through all numbers > 0. This @ is the number of those
among the d numbers in mB-+ H, (m=0,1,...,d—1), which are
(B,d)g(d)
g(Bil)
responding values of m are denoted by m,,...,7),...,™m, and
m, B+ H is put equal to hj, (A= 1,...,0), then every /; is prime
to d and — on account of (B, HW) =1 — to B, so to b=dBand
situated between 0 (excl.) and 6 (exel.) (as

0CHLh=m B+ H<d-1)B+H=)b-B+H<b
and 6 itself is not prime to &). The corresponding values of mB + + Hare
(m + ld) B+ H=lb+h,

prime to d; this number is known to be When the cor-

- =
>

( 70)

where / assumes all integer values > 0. In other words on the
rightside of (10), when it is written short

ud) ee uh)
dS dam ks’
a

i assumes all positive numbers belonging to certain 9 progressions
of the form /b+/, (82=1,2,...,0), where 0<h,<b and
(0,3) == 2 = 50

Mia = HOS aha, 0. «a

—!
and from the result (9) of the case (I) follows after application to
the single members on the right side of (11)
Pp ab)
d 1
Pad mG Dy ys
g(b) a> saaw Lom 4(h;)L4(5)

eo
Of (12) the equation (9) is a special case, as for d=1
Bei io Nh he

In the following the equation (12) may always be taken as basis.

§ 3. The equation (12) proved above for 6 >1 furnishes in con-
nection with the properties of the functions L,(s) quoted in § 1
firstly the analytic continuation of J/;,(s) across the right line
o = 1. It teaches us that all points of the right line, s = 1 included, are
regular places. From the theorem sab at the end of § 1 follows more

Mon Gg) = (12)

accurately that for ‘> 3 and 1 — make 6 <2 the function My (s)

is regular and satisfies there the inequality
Pp Ady

| Min (s)| <> pe

ca

1
——| < —.. 1.0. g(b) log t = @ log*t.
[Z.6)| ~9@)" °° :
Let now a number a >e be chosen in such a way that in the
first place for any t>3 we have
@ log*t < logtt

1
and in the second place log is smaller than the distance of the
right line 6=1 from any singular point of J/,);(s), the imaginary
part of which lies between — 3/ and 37. If moreover the equation
| Min (6 + ti)| = |My. (6 — ti)}
is paid attention to, then ensues from the preceding :

— . —— —— _ _ SS ST, DmhUL eC e

(Cd)

The funetion J7/,),(s) is regular in that part of the plane lying on
the right side of the continuous curve (incl. the curve itself)

oe for t > 3,
log@ t =
1 :
Cee = for FESS,
log 3 1
il
es for t<—3
| log*(—t)
and for t >3,1— (S53 it is

ig

a (O-+ ti)| = |Man (6 — t)| <logtt.. . . . (18)

§ 4. In § 2 of my paper') “On the function p(/) occurring in
the theory of numbers” (“Ueber die zahlentheoretische Funktion u(ky’)
the relation

x
&

NS alk) log ie 227 ule 2 as +0())

somal a 2701,

k=1 2—2i
is developed, where the integral is ata along the straight line. In
like manner we find here, when the sign ~’ means that / has but
to assume all those numbers of the interval of summation which are

=h (mod. »,

2-02

Slt
Ss u(t) log = = gla Sa Hl) a 4 0(1)
=
= = +f “Mop (s)ds+O(1). . . (14)
atl s

. as
§ 5. We now apply Cavcny’s theorem to the integral of — Mo, (8) ds
s

taken along ba closed 4 ae where A = 2—7%i, B=2 + 2%,

1

6=1— Dist = Pas eS 8) Sand

nae *) aoa sat pee log@3 ee

1

F—=1———— — 2%, where the segments AB, BC, DE, FA are

log?(a*)
rectilinear, CD denotes the are of the curve

1
ary (a? >t > 3)
log%t —————

1) Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, math.-
naturw. Klasse, Bd. 112, Abt. 24, 1903, S. 537—570.

and EF the are of the curve
1
s=1 — Elle. <_<)
log(— ————

As according to § 3 in ei closed region and on the boundary
the function to be integrated is regular, the Cavcuy theorem furnishes
for the integral appearing in (14)

of Baie
S eae = vets ae | +f + f- .- (15)
223i AF EDS DO .SGR

Now I have proved in § 10 of my paper on the arithmetic
: ‘ apt ne ‘ , . ee
progression for a certain function A‘(s) that, when integrating — A(s)
$
along the same given path, we have

fafa fe fe fae)

AR FE ED DC CB
where ¢ denotes a positive constant). Concerning that function A(s)
I have made use |. ¢. only of the fact that it is regular on the path

of integration and: satisfies for pas 2 the inequality

|K (6 + ti)| = |K (6 — ti)| <log*t.
As now according to (13) the function J/,;,(s) has exactly these
properties, we have for the present case the expression (15)

ec -
—V log s
= 0 (« M maa

This gives after substitution in (14)
* c
—~! ; & =e nee, ‘log
Ss u(h) log = = (xe vee | Naver A) (Cla
k=1 :
§ 6. Just as in § 4 of my paper on the function a(t) it could be
concluded from
x Wy
v —V log x
Sy u (k) log - ==) é ae )
| r——}

that

1) May be c= 3a.

73)

12

x

LE — Vlog
> u(k) = Of we

—4

we find here out of (16)

2 7
—" —V log x
> ue (k) = O| ae

k=1
where y denotes a certain positive constant *).
Thus for every 2 we find

log” x é
lim fe) ON ees et eae ie ee (1G
hare: 3 we (4) (17)
—"

§ 7. In § 5 Le. we found out of

x
log” a
li k)=0
ae ae

—||

that the infinite series
= 1 (k) log k
pats
—|
converges for every pair of real values m, ¢; in the same way we
find out of (17): the infinite series
y w(K) logn bk y (nb + h) log (mb + hi)

kit (mb + hye
sl m=0

converges for every pair of real values n, ¢. Here is proved in particular
for n=0, t=O the convergence supposed by Mr. Kivyver of the
infinite series

@o

(mb + h)
eas mb+h *

m=0
i.e. the existence of the quantity designated by him as 7%,;.
As now its existence is proved, it is easy to express its value in finite

ao
4 E Uu
form. As is known it follows from the convergence of » —, that
n

n=1

1) May be y=c+1. }

pws 2%

|
|
|
|
|
|

: : s . pun ; ;

approaching from the right side dim NO — exists and is equal to
s—] a 7
n=1
@
Un E : 3 . ; "

‘ —. Making use of (12), (5) and (6) we find therefore
mt)!

—!)

2 2)
b h = bh
i Gs (mb + A) =limS es Vai = lim Mp, (s)

fot «Som +h c=] sommd (mb + h)s aa
m=—0 m=

p #5)
ee eo) Ib 3 Raviealh
re (5
§ 8. In this paragraph must be proved the lemma:
When Q,,,(z) denotes the number of the integers without quadratic
factors <2 belonging to the linear form m+ (where (b,h)=d
is supposed to be without quadratic factors *) ), then

i; Qs, (7)
wm =
Fs) ;

exists and has a value differing from zero.
I. Let us suppose (6,)=d—=—1 first. Let Ayzn(v) denote the
number of those integers m which fulfill the congruence
mb +- h — 0 (mod. n)

and the unequalities

(Gye. 4. < mb + h<2). Then it is evident when 7 possesses with
6 a common factor that on account of (4,/4)=1 the congruence
cannot be solved; so

Ay hin Gt) == 10. 2 SS.

When however (7,/)=1 the congruence has modulo 7» exactly
fh

h,
7 8 equal to

one root; so the number of roots between O and

a—h+b a—h+b ;
| ae or | a + 1,i.e.

Asin (2) = — + 9 (— 19 2 2). | 2) ae

Now it is evident that when & passes through every number the

1) In the opposite case Qs, (x) = 0.

square of which divides /
= i(k) = lor 0,
according to / being without or with quadratic factor; for those
ks are the divisors of the greatest number g the square of which
divides /, so that
E(k) (4) or 0
klg

according to g=1,i.e./ being with or without quadratic factors.

Hence
[=]
b

Qin (@) = a Sa u(h),

m=0
where / passes through every integer the square of which divides
mb +h. If we invert the order of summation, & passes from 1 to
[Vv], and every «(k) appears as often as there are multiples of 4?
in the progression m)-+ A between 1 and a, i.e. Ajjz2 (a) times.
So according to (18) and (19)

val ive)
Ql) = Sw W) Anne @) = Ve W (Fs £ *)
ah i

where the sign +' indicates that / has but to pass through the
numbers < Vw prime to 6. Hence

ve]
a u(k k
Qu) == AP + OV =: one H+ 0(F2))+0v)
PJ! a
reg gi oIT 1—)+ (Ve)
we ie
ed

where p passes through every prime number not dividing ); on

account of
1 1 6
T] € l= a
P G(2) x”
Pp

we. find

6 1
Qn (7) = # 3 Tx + OV. - - (20)
b inte ue =)
p/d

where p passes through the prime factors of 6, so

. Qr(x) 6 1
lim =

ie sae
Tenis! Gees
plb

It is worth noticing that this limit is independent of A.

Il. When d>1, then 4 being equal to dB and h to dH the
numbers mb +h <u without a quadratic factor are identical with
the numbers d (mB + H) <2, where mB + H is without quadratic
factor and prime to d. Now the numbers mb-—-+ H (comp. § 2)
(Bg). : ; :
——— arithmetic progressions with
g(B,d)
the difference 4 and the first term /;, (2=1,2,...,0), , being

(21)

prime to d break up into 9 =

. . . . z
prime to 6; the number of the integers without quadratic factors << 5

in each of these progressions amounts according to (20) to

z a 6 1
(ae Sale eee

d x? b Oe: a. =)
plb x

hence
Byd d
gS Le + )=e.2 ( Soe
= sae H(- *)y (B,d)
plo
lim Qui (v7) 6 (B,d) @p (d) > 0. Jee

erie ae pee (B,d)

pb
§ 9. We now denote by 2),(z) resp. by S;,, (7) the number of
integers mb+h< a containing an even. resp. an odd number of
distinct prime factors and we put

z= <
Pri (e) = Y* ul,
k=l
where S' denotes that / passes through every number mb-+ h <a.
Then it is evident that 7

Reale) — Srale) = Ya) = Pol)
—"
and

(AUT ®)
Ron() + Son (ze) = poh w(k) = Qn (2),
k=!1
so
Reale) _ : (* (2), Pos =) (28)
“ 2 ae a
and
Sin (2) ks Z (= (v) Poy 2) (24)
BE 9 a“ a“

As now according to (17) (after substitution of n= 0)
. Pon (z)
tim ———
7) v

and on the other hand according to (21) and (22) when the limit

is put equal to Loz, ;

—-()

lim Roh ( (2) == Dan O,
120 &
we find from (23) and (24)
Ry p(x 1
lim Ron (@) == 5 Jip.
20 av 2 ;
Son (é 1
lim bh (2) = = Ibi
x00 ve 2 :
sO
n Ron (x)
lim - —

I= Soh (x) ie

This confirms the supposition expressed by Mr. Knvyver at the
conclusion of his paper:

If we divide the integers m+ h without quadratic factors into
two classes according to their consisting of an even or an odd
number of prime factors, the ratio of the numbers of integers minor
to « of both classes converges for «=o towards the limit unity.

Berlin, May 234 1904.

Crystallography. — “On SBenzylphtalimide and Benzylphtal-iso™
imide.” By Dr. F. M. Jancer. (Communicated by Professor
A. P. N. FRancaront.
’ (Communicated in the meeting of May 28, 1904).

Some time ago a preparation was forwarded to me by Prof. Dr.
Gasriit of Berlin, of a Benzylphtalimide*): C,H,.(CO),.N(CH,.C,H,)
melting pomt 115°,5, which he had obtained by synthesis. A short
time afterwards, through the kindness of Prof. Hoogrwerrr, at Delft,

1) Berl. Ber. 20. 2227.

eee eee | eee eS ee ee, ee ee ee ea

—— SS |hOU

i on ge ei BT! ee

(78 )

I obtained some erystals of Denzylphtal-iso-imide*) melting point
82°,.5. As both isomeric compounds are, structurally, closely related,
I thought it a matter of importance to investigate their crystallogra-
phical symmetry.

a). Benzylphtalimide. (m.p. 115*/,°).

From a mixture of ether and alcohol I obtained this compound
in large, very transparent and lustrous crystals of a somewhat
rhombohedric habitus. In many cases they are more flattened. The
power of crystallisating from the said solvent is extraordinarily
great.

The symmetry is ¢riclino-pinacoidal; the axial elements are:

a:b:c=0,8448 : 1: 1,3600
A017 SL @ = 108° 24’.
B==AIG ole) p= 41205 Te
C= 80° 457/,'. y=! 1s) 8G

Forms observed are c= {001}, predo-
minating; a—{100} and ’=}010}, about
equally much developed; m= {110}, narrow;
r’ = $101} small and narrow, sometimes a
little broader; «= {111}, small and much
less conspicuous.

The crystals are represented in fig. 1; the
Fig. 1. measured and calculated angles are annexed:

Measured : Calculated :
20 —— 11.00). (OL0) == 99°14 /,” st
7a, 6 (100): (001) = 63" °8}/, =
7h G— (010) - 4001) 78: 9 —
Mi == (LO). (O10) = '58.197/: =

*¢ : r'= (001) : (101) = 78 59 © —

a :m= (100) : (110) = 40 55 40° 55’
c :m = (001) : (410) = 58 227/, 58 17
as — (00) = (101) 37 55 SYP aot}
boa O10) 2 (O01) — T1135 71 28
m:r' = (110) : (101) = 61 538 Ge el

The crystals cleave quite parallel {100} and {O10}.

1) Receuil d, Trav. Chim. d. Pays-Bas, T. XIII. 99,

7 peo

(79 )

When recrystallised from ether, very strongly refractive, hexagonally
bounded little plates are obtained. On investigation it appears that
these are twin crystals: {O01} is the twin-plane with a twin-axis
standing normally on it.

In addition to ¢ = {001}, b = {010}, r' = {101} and w' = 1111}, I
noticed a form g= {011}; ¢ is strongly developed, 6 and q are
narrow, 7” and w' are equally broad and well-formed. The planes of
cleavage are the same as above.

There were also measured :

G20 — (011)@) : (010)) = 23°58'; calculated : 23°42’.

6: 6b = (010), : (010)2) = — 19°23’; calculated : — 19°28’.
7:7 = (101)q): 101); = 21°45’ ; calculated : 22°2'.
c:q=(001) : (011) =58°21'; calculated : 58°9’.

The nature of the twin-formation has, therefore, been sufficiently
explained.

On {001}, the direction of the optical elasticity-axis is orientated
nearly perpendicularly on the sides (O01): (010); an axial image
could not be observed.

The specific gravity of the crystals was determined by means of
a THoutnr’s solution; d=1,343, at 16° and consequently the topical
axes, calculated according to the formule:

277 Lf 1
eres a* V Ey ¥ /s
Ao = hy eT ’ a . . . ’
csin B sin y sin A ac sin B sin y sin A

2 7 1/
4 al (3
. . . ?
a sin B sin y sin A

in whieh v=—, if J/ represents the molecular
F

weight, are equal to:

%: 0:0 =4,8513': 5,7458 : 17,8145.

6.) During a cold winter night, I once obtained
from a solution of the compound in benzene a
second, Jess stable modification.

There were formed transparent hexagonal bounded
little plates; the crystals were single and not twin
crystals as in the former case. The melting point
was situated at 115°; the little crystals, however,
soon became opaque so that on warming a mole-
cular transformation probably takes place. But I

have only once succeeded in obtaining this modification,

( 80 )

It is monoclino-prismatic; the crystal parameters were calculated to be:
a:b:c=0,8476:1: 0,5092
B = 70°42’.
Forms observed :- a= {100}, strongly predominating; m = {110}
and 4 = {010}, very narrow; c= {001}, very small and less con-
spicuous; g = {O11} and s= {021} equally broadly developed.

Fon (100) 490) a6 aot)” —

*a> = g:= (100) 21011) = 72.40 —

*c 2g (001), (011) = 25 40 —
Shea — "O24 (Old) — 25.018 18° 12’
bm (010)- > (190) = 51 20 51 20'/,.

A distinct cleavability was not found.

Therefore, benzylphtalimide is dimorphous. As regards the remarkable
connection of this $-modification with the crystalline form of the
iso-imide and with that of the a@-modification, see the end of this
article.

c). Benzylphtal-iso-imide. (m.p. 82*/,°).

The crystals represented in Fig. 3 have been
obtained from anhydrous ether; they are long-
prismatic and here and there of a_ porcelain
white. After some time, even if kept in the dark,
they become quite opaque and look as if they
were effloresced. The crystals, however, do not
contain any co-crystallised solvent; probably some
transformation as yet unknown, has taken place
here.

The symmetry is monoclino-prismatic; the axial
elements, when choosing the following elementary
forms, are calculated to’ be:

Fig. 3. ol) = C=O Rey giles WS RY
B= T4267.

Forms observed: m= {110}, strongly predominating; c= {201},
also well developed; += {O01}, fairly strongly; g = {011}, very
plain; a= {100} narrower; 4 = {010} very narrow; between 7 and
¢ there is still an orthodoma {hOk}, which was very little developed
and could not be measured. I have retained the above letters for
the notation of the forms, because the habitus of the crystal renders,

( 81 )

at first sight, the use of c= {001}, r= {201} and C= (21.1), in many
respects more obvious.

Measured : Calculated :
*a.; m = (100) : (110) = 49° 267/,' =
- Ca (201) 3 (0) == 68" 25/3 —
Fog, — (Old) (081) = 58° 477), —
Ga: 7 == (201): (001) = 52 321/. 52° 40%)
Cay ——(A0N) (LOO) —— 1551 44/8 Sh BRU
r: @ =(001):(100) = 71 44 71 46
Carga (201): (O11) == 580 9%), 5S MET
gq : 7 = (011): (001) —29 24 29 24
m: b = (4110): (010) = 40 49 40 33
m: m= (110): (410) = 81 7 Sie Sve
q : m= (011): 410) = 53 34 53 28

——— EEE EEE

A distinct cleavability was not found; perhaps q is a plane of
cleavage. The symmetrical extinction on the planes of {110} amounts
to about 3° with regard to the vertical axis; a further optical
investigation was excluded.

The density of the crystals is approximately’) 1,145; by putting
in the above formula y and A= 90° we obtain for the proportion
of the topical axes:

%:W: ow = 8.2234 : 6.6840 : 3.9650.

If we compare the crystal parameters of the two heteromorphous
modifications of benzylphthalimide with the parameters of benzylphtal-
iso-imide we at once notice an interesting relation between the
crystalline forms of these substances.

Symmetry : Crystal-parameters :
a:b:c=0,8443: 1:1,3600.(,6,7; s.a.)

Monoclino-prismatic. ja: b:e=0,8476 : 1:0,5092. gB=70°42!

Compound :

z-Benzylphtalimide. | Triclino-pinacoidal.

6-Benzylphtalimide.

Benzylphtal-iso-imide. | Monoclino-prismatic. |a:b:c=1,23032): 1:0,5932. 6B=71°46'

1) Approximately, because the opaque erystals did not give a sufficient guarantee
of homogeneity and because the substance is atlacked on the surface by contact
with THoutet’s solution.

*) It may be observed that the proportion a:b would become for m $230
0,8202 : 1.

6

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 82)

First of all the proportion a: is the same in both the e«- and p-
modification; a proof that these erystallographical forms are closely
related to each other as regards their internal structure. I have
noticed more than once this equality of two parameters in different
modifications of a same compound; I met lately with a striking instance
in the case of the red «- and the less stable yellow 8-modification
of the 1-3-4-Dinitrodiethylaniline notwithstanding the difference in
the degree of symmetry. From a crystallographical point of view
such modifications must always be dependent on each other, although
that dependence may not always be immediately noticed.

But then the very close relationship between the less stable @-
modification and the crystalline form of the dso-imide, as is plainly
shown from the analogous values for the angle 8 and the proportion
b:c is surprising in a high degree.

According to Drs. Hoocewerrr and van Dorp *) the isomerism of
imides and iso-imides is based on a difference in the way of combi-
nation of the N-atom on the one side and the O-atom on the other:

CO C=N-ACHe 7 Cane
CHK N ROHPSCAHY and CHC yo
CO CO
Benzylphtalimide. Benzy|phtal-iso-imide.

The heteromorphous ?-modification of the first substance now
appears to be extremely closely related to the crystalline form of
the first one owing to desmotropical change. Both phenomena,
dimorphism and desmotropism therefore cause, respectively, an ana-
logous change in the crystal-symmetry of a-benzylphthalimide.

The presumption raised by me some time ago that crystallographic
polymorphism in a number of organic compounds might be caused
by a chemical isomerism *), which would then account for the tem-
porary existence of more stable atom-configurations is again a little
more justified by the fact that above connection has been found.

1) loco cit.

®) Kristallografische en Molekulaire Symmetrie van plaatsings-isomere Benzol-
derivaten. Proefschrift, Leiden, 1904, pag. 120, 121. Z. f. Kryst. 38. 600. (1904).

( 83

Botany. — “On the development of the perithecium of Monascus
purpureus Went and Monascus Barkeri Dang.” By Mr. H. P.
Kuyprr. Communicated by Prof. F. A. F. C. Wenr.

(Communicated in the meeting of May 28, 1904).

With a view to the conflicting results obtained by Wenr’),
Uyrpa *), Barker *), [kno *) and Danerarp *) in their investigations
concerning the development of the perithecium in the genus Monascus,
it seemed desirable to study once more the forms investigated by them.

The results of an investigation of Monascus purpureus and Monascus
Barkeri will be briefly communicated here ; a more extensive article
on the same subject will soon be published elsewhere.

Monascus purpureus was obtained by placing externally sterilised
ang-cac grains in boxes contaiming a sterile nutrient, after which the
mycelium developed in a few days.

Mr. Barker had the very great kindness to send me a culture of
the Samsu fungus, studied by him.

Both moulds were cultivated on sterilised bread or on thin plates
of malt-gelatine. In the latter case the gelatine was dissolved in
water of about 30° C. and the remaining mycelium, as well as the
bread, fixed in Ketsmr’s bichloride of mercury-acetic acid.

Microtomie sections 2—5 w thick, of the material melted in paraffine,
were stained with Herennarn’s ferro-haematoxilin*) and partly re-
stained for 1 or 2 minutes with a saturated aqueous solution of
orange-G. In this case the slides were at once washed in absolute
alcohol and then inclosed in canada balsam, as in the former case.

Monascus purpureus.

The two hyphae whose appearance precedes the development of
a perithecium’), the pollinodium (le premier filament couvrant) *)
and the ascogonium, do not seem to me to enter into open com-
munication. The ascogonium divides into two cells, the anterior one
of which soon becomes irrecognisable in later stages of development.
This division of the ascogonium in some cases takes place at a

1) Ann. des sc. nat. Bot. Ser. 8 T. I. 1895, p. 1.

*) The Bot. Magaz. vol. XV, 1902.

8) Ann. of Bot. vol. XVII, 1903, p. 167.

4) Ber. d. deutschen bot. Ges. Band XXI, 1903, p. 259.
*) Comptes rendus, Acad. Sc. T. CXXXVI, 1903, p. 1281.
6) 48—60 hours.

7) Went 1. c.

8) Went |. c. pag. 3.

( 84)

time, when it is entirely free from the pollinodium, whereas in other
eases the division only occurs when the somewhat tapering and
curved tip of the ascogonium lies alongside the pollinodium.

The posterior cell of the divided ascogonium, the final ascogonium,
increases in size and is surrounded by a number of hyphae, having
their origin under the ascogonium. The number of nuclei of the asco-
gonium increases at this stage ; they lie spread in the foamy protoplasm.

Round a number of pairs of nuclei protoplasm accumulates so that
free cells arise, having two nuclei each. The remaining protoplasm
with the nuclei that have formed no cells, appears chiefly as a
wall-layer. The nuclei of the free cells are bigger than those in the
remaining protoplasm. The optic section of the free cells is often
elliptic, sometimes also slightly bent in the middle of the major
axis. The nucleus of the free cells consists of a grain which stains
very strongly and which in some cases is surrounded by a zone of
a lighter tint than the dense protoplasm of the cell.

Sometimes one finds in one cell more than two nuclei, mostly
three, two of which are then bigger than the third.

From the examination of a number of preparations of various
stages, I infer that the two nuclei of the free cells fuse, thus giving
rise to one nucleus which is bigger than its components.

The development now proceeds fairly quickly, asa result of which
one finds the uninucleate stage of the free cells pretty rarely in
comparison with the binucleate. The development of all cells in an
ascogonium is by no means simultaneous.

In a following stage the cell has grown and its protoplasm is less
dense. It now contains a great number of very small nuclei. ‘Next,
some parts in the cell differentiate which apparently have no nuclei
and consist of a homogeneous substance. These spots occupy the
greater part of the cell-space and the small nuclei, which I would
call chromatine grains, are pushed back into the layers of protoplasm
between the unnucleate parts. ;

After this one sees one nucleus in the middle of the homogeneous,
elliptic spots, which themselves are divided into a central part with
the nucleus and a margin, having a lighter tint. This stage represents
the first occurrence of spores in the free cells, mostly numbering
from 6 to 8.

Between the spores one finds what IkENo calis the “Wabenwande” *),
chokeful of chromatine grains, which can be distinctly seen separately.

These chromatine grains, now, soon disappear, while the nucleus
of the spore divides, so that we find spores with 2, 4 and 8 nuclei.

1) Ikexs lc. p, 265.

en to i te

( 85 )

Finally, probably by further division, the spore is eutirely filled up
with a granular, strongly staining mass.

In the meantime the wall of the spore has become more prominent.
The number of spores, as was said above, is not constant and some-
times one finds only a single spore in one cell; in some cases however
about 16 are counted, which then are smaller than the normal ones.
When the spores are ripe, the free cell, as a unit, falls asunder
and the spores become free in the ascogonium, the contents of which
are further formed by the epiplasm of the free cells and the remainders
of the protoplasm which in the beginning did not take part in the
formation of free cells.

The spores do not fill up the ascogonium entirely but form together
a peripheral layer. between the spores one finds an intermediate
substance, which is strongly stained by orange-G. So IkrNo’s opinion
that the polygonal appearance of the spores is an optic illusion, is
correct *)

Monascus Barkeri.

The first stages of the development of the perithecium of this
form agree with those of the first species dealt with. Here also I
have not been able to state an open communication between polli-
nodium and ascogonium. The latter, which here les more parellel
to the pollinodium, is also divided by a cell-wall into two cells, the
posterior of which forms the final ascogonium. Only after it has
become surrounded, at first rather loosely, by hyphae, it greatly
increases in size. As long as it is small, sections of the whole

perithecium ascogonium with enveloping hyphae — show much
resemblance with figures 17, 18 and 25 by Barker’). Together with
the volume of the ascogonium the number of nuclei also increases.
The ascogonium is now rich in protoplasm with many vacuoles.
A first indication of a division of the protoplasm into parts — especi-
ally concerning the central part — is observed by the occasional
appearance of long-stretched vacuoles. The nuclei are here and there
seen in couples and also a few bigger nuclei are seen, which I
presume to have been formed by the fusion of two of the smaller ones.

In a following stage free cells have formed, each with a single
pretty large nucleus. This stage strongly reminds us of the corre-
sponding one with M. purpureus. The protoplasm of the free cells
is in most cases much denser than in former stages; in few cases

1) Ikeno |. c. pag. 267.
?) Barker I. ¢.

( 86 )

only it is still foamy. It very often occurs that the free cells have
assembled on one side of the ascogonium. Besides this one finds then
in the ascogonium a fairly strongly developed lining layer of proto-
plasm and one or a few very large vacuoles. This condition reminds
us of figs. 22 and 29 of Barker. The protoplasm which has taken
no part in the formation of free cells contains very few or no nuclei,
so that it is probable that several nuclei soon degenerate.

The single nucleus of a free cell now successively undergoes three
divisions, so that at last eight nuclei are present. The two nuclei
after the first division and the four after the second, do not always
divide at the same moment, so that also cells with 3, 5, 6 and 7
nuclei are found, but then in the first case e.g. one of the three
nuclei is bigger than the other two. ‘

When two divisional nuclei — here again consisting of a strongly
staining grain — have arrived at a certain distance from each other,

one often finds between them a more or less complete band of a
somewhat lighter colour than the nuclei themselves but darker than
the protoplasm of the free cell. These divisional figures agree pretty
well with IkENo’s representations for Taphrina Cerasi ‘).

After the eight nuclei have formed, spores arise with the nuclei
as centres. These spores become free in the ascogonium and further
behave like those of Monascus purpureus.

The manner in which in the two forms described, the nuclei for
the spores are developed, differs in important points, but agrees in
a remarkable degree with what has been described by IkreNo for
Taphrina Kusanoi and T. Cerasi *).

Both forms agree in this respect that in the ascogonium free cells
are formed with originally two nuclei, which fuse into one, from
which single nucleus the nuclei of the spores arise. This has induced
me to see in the free cells in the ascogonium of the genus Monascus
the homologon of an ascus, especially on account of DaNne@rarp’s
investigations on the Ascomycetes *).

With the remaining Ascomyceies, e.g. Pyronema confluens *) and
Ascobolus *) the fusing of the nuclei would, according to this repre-
sentation, only be shifted in time and place compared with Monascus

1) Flora Band 92 p. 1 (fig. 24 and 27).
2) Flora Band 92 p. 1 (fig. 24 and 27).
*) Le Botaniste 4 Serie, pag. 21—58, 1894—1895.
4) Harper, Ann. of Bot. vol. XIV, 1900.
5) Hareer, Jahrb. f. wiss. Bot. Band XXIX, 1896,

( 87 )

and take place in hyphae, growing out from the ascogonium and
into which part of the contents of the ascogonium would be trans-
formed.

In accordance with this representation I would place Monascus
in a new order, that of the Endascineae.

Anatomy. — “On the genital cords of Phalangista vulpina.” By
Mr. A. J. P. van pen Brork. (Communicated by Prof. L. Bonk).

(Communicated in the meeting of May 28, 1904.)

In the course of an investigation concerning the structure and
development of the female genital organs of marsupials, the result
of which will be more extensively published elsewhere, 1 had an
opportunity of studying a series of transverse sections through a young
female specimen taken from the marsupium of Phalangista vulpina,
measuring 16.7 mm.

An examination of the genital cords and of the Wolffian and
Miillerian ducts enclosed in them, revealed a relation differing
from what is noticed in the genital cords and the above-mentioned
duets both in Monotremes and in monodelphic mammals and which
appears to be connected with the peculiar anatomical details of the
genital system of the marsupials.

In following up the genital cords, after they have issued from
the tissue of the primitive kidney, in a
caudal direction, it is noticed that the
cords from both sides approach each
other, unite into a single cord along a
short distance, then separate again and
are continued in a caudo-lateral direction
as far as the wall of the uro-genital
sinus. In this manner a short bridge is
formed, connecting the two genital cords.
On reconstructing the successive cross-
sections, we obtain what is schematically
represented in fig. 1, where the genital
apparatus is represented as seen from

Fig. 1. Genital cords and ureters the dorsal side.
of a marsupial young one of The course of the ureters is very
Phalangista vulpina, seen from the peculiar. After having passed behind
dorsal pe the connecting bridge just-mentioned

ee no ai, Ureter: «tick, Eeonnectanthion. two genital cords

8. U. g. Sinus uro-genitalis. 2 i

they bend round the caudal edge of this

( 88 )

and, continuing their course again in a cranial direction, pierce
the posterior wall of the bladder in an oblique, caudo-cranial direction
and open into this organ on two adjacent papillae with ostia, which
are turned towards the fundus of the bladder. It is very remarkable
that the course of the ureters at this stage already agrees completely
with the adult condition. The question as to the cause of the hook-
shaped bend in the ureters of marsupials, which has already long been
known, and which, in my opinion, must exactly be sought in the
above-mentioned bridge between the two genital cords, will be
briefly discussed later on.

With regard to the mutual relation of the Wolffian and Miillerian
ducts, we have to offer the following remarks. At the level of the
caudal pole of the primitive kidney, the Miillerian duct lies ventrally
and a little laterally of the Wolffian. In a caudal direction this
relation is changed, the Miillerian duct being gradually shifted towards
the ventro-medial side of the Wolffian duct. This topographical
relation exists until near the place where they enter the uro-genital
sinus. In their course they follow the genital cords and, in doing so,
bend medially, then caudo-laterally and finally, in the last part of
their course, show very peculiar characteristics.

The Miillerian duet suddenly bends ventrally and medially,
describes a caudally slightly convex are and then opens into the
uro-genital sinus.

The Wolffian duct, at first situated dorso-laterally of that of
Mixer, describes like this latter in its terminal portion a caudally
convex are and so becomes placed caudally of the Miillerian duet.
Next it bends medially and lays itself against the medial wall of
the Miillerian duet, after which it opens into the uro-genital sinus,
cranially of the latter.

Hence it appears that the Wolffian duct in the last part of its
course describes almost a complete spiral revolution round the Miillerian
being successively dorso-lateral, caudal (dorsal), medial and cranial.
This course I have schematically represented in figure 2, seen from
the front and somewhat from above. The bladder has in the figure
been imagined cut off exactly at the level where the Wolffian
duet enters.

F. Keren’) has suggested that in marsupials the ureters enter
into communication with the bladder already at the stage in which
they appear as sprouts of the dorso-medial wall of the Wolffian

1, I. Ketser. Zur Entwicklungsgeschichte des menschlichen Uro-geniialapparates

Archiy fiir Anatomie und Entwicklungsgeschichte. 1896. p. 55.

( 89 )

Fig, 2. Course of Wolffian and Miillerian ducts in a marsupial young
one of Phalangista vulpina.
w. g. Wolffian duct. m. g. Miillerian duct. s. a. g. Sinus uro-genitalis.

duct i. e. at a stage, which oceurs as a temporary condition in the
monodelphic mammals, at any rate in so far as these have been
examined, while in the latter they move, before the terminal piece
of the Wolffian duet has been incorporated in the wall of the
bladder, towards the dorso-lateral wall of that passage, so that in
the adult condition they are found laterally of the Wolffian and
the Miillerian duct. Although I have not been able to test this
theoretical consideration by observations of my own, yet it seems to

me that besides the cause, postulated by Kerser for the course of

the ureters medially of the Miillerian duct, the spiral course of the
Wolffian duet and the consequent torsion of this canal, must
certainly have an influence and probably a not inconsiderable one
on the origin of this course.

A notable fact in this respect is that in male marsupials a masculine
uterus is absent, that no remnants have been found of the caudal
terminals of the Miillerian ducts, as is expressly stated by Waser’)
and DisseLHorst*). Only Hypsiprymnus would be an exception to this,
according to OWEN.

At the stage, observed by me, the terminals of the Miillerian

1) M. Weper. Die Siiugetiere. Jena. G. Fiscuer 1904.

2) R. Dissernorst. Ausfiihrapparat und Anhangsdriisen der miinnlichen Geschlechts-
organen in: Lehrbuch der vergleichend mikroskopischen Anatomie der Wirbeltiere,
herausgegeben von A. Oppet. 4er Teil.

(90)

ducts are lateral (and caudal) of the Wolffian. Now it is obvious
that if my observation holds for marsupials generally, no masculine
uterus can arise, because between the two Miillerian ducts those
of Wot.rr are found. Eventual remnants of Miillerian ducts will
have to be sought- for in the male sex laterally of the terminal
opening of the vasa deferentia. I have not yet been able to make
observations of my own concerning this point.

As I have already stated, the ureters, at the stage I observed, lie
at the medial side of the genital cords. Only those parts of the
female genital apparatus of the marsupials which lie at the lateral
side of the ureters, can, as I shall try to prove more fully elsewhere,
have their homologa in the female genital system of the monodelphie
mammals. The vaginal caecal sac developing phylogenetically in the
marsupial group, has no homologon in the female sexual organ of
the Monodelphia.

Finally, I think, my observation contains an explanation of the
peculiar hook-shaped course of the terminals of the ureters of
marsupials.

Either as a consequence of the spiral course of the Wolffian
ducis, or for some other cause, the ureters at a certain stage of
development lie medially of the Wolffian ducts (and of the genital
cords) in a dorso-cranial direction towards the primitive kidney.

Marsupials possess a milk-nutrition (intestinal) at such an early
stage of development as is known of no other mammal. This milk-
nutrition will have a great influence on the development of the
bladder which I found as a very voluminous organ in the marsupial
young one described, as well as in other specimens (Didelphis)
examined by me. With the rapid growth of the bladder the orifices
of the ureters are at the same time displaced cranially. The above-
mentioned ecross-connection between the two genital cords is an
obstacle to the cranial displacement of the ureters, the natural conse-
quence of which is that the ureters have always to go round the
caudal end of this bridge, while their orifices are further displaced
eranially, the result of which is the pronounced hook-shaped course.

Zoology. — “An interesting Case of Reversion.’ By Dr. P. P.
C. Hork.

Pollicipes and Scalpellum are two nearly related genera of pedun-
culate Cirripedes mainly differing from one another, by the one having
in its capitulum a restricted number of valves (Sca/pellum) and by
the other having a much larger number of such calcareous parts

(Pollicipes).

( 91 )

Darwin pointed out the resemblance of these two genera already
in 1851"). That resemblance is greatest in those species of Scalpellum
in which the carina is not bowed or angularly bent, but straight or
nearly straight. Darwin gave as an interesting case of such a species
a description of Sc. villosum (fig. 2 of the accompanying plate). He
was so struck by its general likeness to Pollicipes, that he wrote
(le. p. 278): “Sc. villoswm most closely resembles or rather is identical
with Pollicipes. Had it not been for the fewness of the valves
forming the capitulum, and from the presence of complemental males,
I should have placed this species alongside of Pollicipes spinosus and
sertus.” And under Pollicipes (l.c. p. 294): “We have seen under
Scalpellum villosum that the addition of a few small valves to the
lower whorl, would convert it into a Pollicipes” ete. Compare fig. 3
of the accompanying plate.

The genus Sca/pellum is represented under the deepsea animals by
numerous species. Those of Pollicipes are shallow water forms only.
The English “Challenger” Expedition collected during a four years’
cruise over all the oceans of the world, specimens of 42 different
species of Scalpelium, 41 of which were new to science. Only two
of these were found in depths of less than 200 m.: all the others
were true deep-sea species. H.M. ‘“Siboga” collected in the Malay
Archipelago, during a cruise of one year’s duration, specimens of
38 different species of Scalpellum. Of these, 32 must be considered
as new to science; 34 of these species are deepsea animals, 4 shallow-
water forms.

The genus Pollicipes was not represented under the Cirripedia of the
Challenger and by one species only under those of the Siboga:
Pollicipes mitella a common littoral form of tropical seas. Whereas
the number of known living species of Scalpellum was 6 in 1851
and is at least 125 at the present time (so far as 1 know °*)), of
the genus Pollicipes which figures in Darwin’s book with 6 species
also, only a seventh species has been described since the appearance
of the said monograph. When Darwin wrote that book the mysteries
of the oceanic abysses were not unveiled to him of course, but his

1) Darwin, C., A Monograph on the Subclass Cirripedia. 1. The Lepadidae or
pedunculated Cirripedes. 1851.

2) Including the species collected by the Siboga. A. Gruver, who described the
species collected by the French expeditions with the Travailleur and the Talisman,
and C. Aurivimuus, who studied the Cirripedia of Swedish collections and published
provisional descriptions of the Cirripedia collected by the Prince of Monaco during
his numerous cruises have, with the present author contributed most to our know-
ledge of the species of this genus.

( 92)

knowledge of the existing shallow-water forms of Cirripedia was
fairly complete.

Returning to the interesting species Sc. villosum, the near affinity
of which to Pollicipes was pointed out by Darwin, I may mention
first, that in my Report on the Cirripedia of the Challenger I was
able to describe a species (Sc. trispinosum) belonging to the same
division of the genus. It was collected in the Philippine Archipelago
at a depth of 150 (perhaps 180 meters). Next, that the Siboga was
suecessful in finding two more of them and that these will bear
the names of Sc. pollicipedoides and Sc. aries. Though the capitulum
of both species resembles the common form of Sca/pel/um (compare
fig. 1) more than that of either Sc. trispimosum or Se. villosum, the
shape of the carina shows their affinity to the last named species
at once.

These species were also taken in rather shallow water: in depths
varying between 57 and 94 meters. The depth at which Se. vil/osum
lives is not so well known, but it cannot be important. Darwin
says that the specimens were found attached to shells and rocks:
they were taken no doubt during shore-exploration.

So we can say that those species of Sca/pellum which resemble
Pollicipes most closely, like all known species of that genus itself,
are inhabitants of shallow water. Pollicipes as is well known,
embraces the oldest known Cirripeds and the genus Sca/pellwm is the
second in age. The structure of Se. vi//oswin makes it highly probable
that the genus Scalpellum descended from Pollicipes. This supposition
finds very striking confirmation in a peculiarity of one of the
specimens of Sc. pollicipedoides, which peculiarity I shall briefly
describe here.

The said species is represented in the Siboga-collection by six
specimens. It was found at Station 274 near the Jedan Islands,
south of New Guinea, at a depth of 57 meter. Its capitulum has
15 valves: two scuta of triangular, two terga of rhomboid shape;
a nearly straight carina, two rather small upper latera and eight
valves of the lower whorl. Of these the rostrum and the sub-
carina have the umbo pointing transversely outwards; of the three
pair of latera which, with rostrum and subcarina form the lower
whorl, those of the middle pair are by far the smallest. The shape
of all these valves is triangular with the umbo at the apex.
Whereas the scutum and the tergum stand rather close together,
the otber valves are far apart, being separated from one another
by chitinous membrane.

Though, as I pointed out above, the general shape of the capitulum

oe

ee eee ee

( 93 )

is more like a normal Sca/pel/um, im having one pair of latera more
than Se. villosum or Se. trispinosum, Sc. pollicipedoides comes nearer
to Pollicipes even than those species.

Looking over the six specimens of this new species, I was struck
by finding that one of the specimens, though in other regards similar
to the other five, differed from them by having in the lower whorl
of valves two latera in addition to the three which all the specimens
possess. In fig. 4 the left side of a normal, in fig. 5 the same side
of the abnormal specimen is represented. (At the right side only one of
the additional valves is developed.) In fact, the few small valves
which according to Darwin were wanting in the lower whorl of
Se. villosum to convert it into a Pollicipes occur in one of the
specimens of this new species. By calling it a case of reversion I
would indicate the high importance, which from an evolutionary
point of view I attach to this abnormality. We need not go so far
as to consider this species as representing exactly the “missing link”
between the genera Pollicipes and Scalpellum, but 1 think the case shows
clearly that a form with more numerous calcareous parts in its
eapitulum (like Pollicipes) is the older, the form with fewer (like
Scalpellum) the younger one; moreover, that the Sca/pellum-species
with straight carinae, inhabitants of shallow water, must be considered
as the oldest, i. e. the species most resembling the primitive form
of Scalpellum.

(It is perhaps not quite superfluous to remark here, that 80 of
the 125 species of Sca/pelluim have been studied and described by
myself and I never saw before such an augmentation of valves in
the lower whorl (or in the capitulum in general). Nor can |
remember to have met with descriptions of such cases in literature.
Darwin for the classification of the genera of Cirripedia made use
of the number, the shape and the mode of growth of the valves.
That he was right in doing so is proved by the fact that later
authors never put in doubt the value of these characteristics).

A full description of this and of the other new species of Scal-
pellum will be given in the forthcoming report on the Cirripedia of
the Siboga-Expedition. I wish however to point out here, that (Sv.
polhicipedoides by the presence of a complemental male (Fig. 6) shows
itself to be a true Sca/pellwm and that it has rudimentary caudal
appendages which occur in Pollicipes also, but which curiously
enough according to Darwin are wanting in Scalpellum villosum.

Cee ee ee 0.0.0

( 94)
EXPLANATION OF FIGURES.

Fig. 1. Scalpellum rostratum, Darwin. Seen from the right side.

» 2. Scalpellum villosum, Leach. Seen from the right side.

. 3. Pollicipes sertus, Darwin. Seen from the right side.

(Fig. 1—3 after Darwin, Monograph on the Cirripedia. Lepadidae. 1851).

. 4. Scalpellum pollicipedoides, n. sp. Seen from the left side. Magnified
11 diameters.

. ». Same species, abnormal specimen. @ the additional values. b the com-
plemental male. Magnified 11 diameters.

. 6. The complemental male of Sc. pollicipedoides. Magnified 180 diameters.

. . . . es . . ”
Physics. — “On the function : for multiple mixtures.” By Mr. B. M.
)

vAN Datrsen. (Communicated by Prof. J. D. van per WaAats).

The quantities @ and } appearing in this quotient are the constant
quantities of the equation of condition of vAN per Waats, applied

. . . a .
to a multiple mixture. The quantity ; then represents an expression

proportional to the critical temperature of the undivided mixture.
We imagine the mixture determined by the molecular fractions
&,,2,,---,@,, Where z,-| #,-...-+-2z,=1 and all 2's are positive
quantities. Further we assume for @ and 6 homogeneous quadratic
functions of the z’s, so that

p=n g=n

= Ga)
C= S » Ang Lp vq

p=1 gq=1
and

p=n g=n
— +o ae byg Xp Lq-")
p= g=l
For the quantity a we must arrive, it is clear, at a quadratic
function, as we have to do with attraction of the molecules two by
two; for 4 we ean suffice *) with a quadratic function as long as
simultaneous collisions of more molecules are neglected.
Our particular business now is to find out whether there are

. . a . .
mixtures for which 5 8 stationary.
The constitutions of those mixtures we find out of
1) Here Gp, gp and Dpg= gp. For Gpp and bpp we put in the sequel ordinarily

Gp and Dy.
2) Comp. H. A, Lorentz, Wied. Ann. 12, p. 134,

P. P. C. HOEK. “An interesting case of reversion.”

Proceedings Royal Acad. Amsterdam. Vol. VIL. |

xe aly

(95 )
0a Oa Oa
a 0a, de, Lee
amon IR LObj ob
thus out of the system of linear equations
ol
payee oe) ko
du, 02,
ob
eu a4—=0, besides 2#,--a,--+...+%=1
Ow, 0 Ve
joy
Oxy O2'n
or explicitly
{(a, — ab.) x, + (@,, — 4b,,) % +--+ sss + (din — Abin) &n = 9
(V) \G@,, — 40,,) 2, + @, — 4b, )%, + + --:-- + (dan — Abon) am = 9
(Gareth ia 1 (Ghal eae) eg rts SN (Gp) —= 2B) ta = 0
and

2 +a,+....+a=1.
In these equations },,=6,, and dq =; they are the same
quantities appearing in the theory of the binary mixtures. All a’s
and 6’s are essentially positive.

Out of the system (V’) can be deduced that — for the mixtures
sought for — = or 2 can be found out of an equation of degree n:
Og —— AD Ota —— TAU ae) (aoe Ain — Adin
NG = Ae Noes A ab, aan — Aba == ()),

Ge LES Na te ea ee aes Gig Fh

Algebraically there are thus 7 solutions — the values of « belonging
to the obtained values of 2 are found out of

M,, MM, Min
CS a owe. 9 tn == —————. .
1 sn 3 sn 7)
‘ Mis ‘ Ms Ms;
s=1 s=l »»

Here M,, represents the coefficient of a, —4b,q in the develop-
ment of A, according to the elements of the p™ row whilst in these
expressions we must substitute successively for 2 the roots of A, = 0.

If such a mixture is to be realized we must find for the mole-
cular fractions positive values so that one of the two following
systems of inequalities must be satisfied

( 96 )

(A) Mr 0 oe 2. el Min > 0
(B) a el Min < 0.

We have just now made use of the last (m-—1) equations of
system (V) to calculate the z’s. As we might have chosen (2—1) other
equations it is evident that for the existence of a stationary point
M,, must always be either >0 or <0. The set of inequalities
given above is however sufficient to judge the possibility of the mixture.

At most one of these n mixtures can be realized. Suppose that two
different possible mixtures were to be found

[GG - =~ (),) and -[(@,), "Gj. 22 - Ga
a = "
for which ;, Were stationary. Now as in consequence of the set of
J

equations (V) equal roots 2 lead to equal constitutions, different
values 2 belong to different mixtures. If we call the roots 2 belonging
to the above mentioned mixtures 2, and 4, we arrive at the following

sets of equations
0b
z Ow, -

0a A < | ( 9a

Ge) * Ge) or
0a ; Ob ney
ae ‘ G.)=

ae -) 3 (Ge) =? and V, (
@-O- |@-@-
(wv

Multiplying the equations V, resp. by (,),, (®).»--+(%n), and those
of the system V, resp. by (—a,),, (—#,),,---(—®@,), and summing
these up we find in connection with the identity holding good for
homogeneous quadratic functions

p—=n (a
Loo(e)s ¥ on.(F, =)
ab ob
G7) Joos) + (z,), (5) +: : ten( 5) | =
2

or, as we supposed 4, ==
sn

NX (xs)q [ost (ei), + bs2 (@a)y = + + Bsn (@n).] = 0-

s—
All },,’s however being positive and all 2’s also for possible mix-
tures, the above mentioned equation cannot be satisfied.

B a
So there is at most but one mixture possible for which ; becomes

(97)

stationary ; the 4 of that mixture satisfies either the inequalities (A)
or the inequalities (2).

Before passing to the investigation of the nature of the stationary
point we shall first prove the theorem

\ pr qen
0A, =
aa — \ \ (— bn) Moy)
0 cece ecm
D—lg—
This theorem can be easily verified for n= 2; it holds still good

when 6,,,==6,,, i.e. for asymmetric determinants.

The general proof is supplied by showing that if the theorem is
correct for determinants of (#—1) rows and columns (also asym-
metric ones), it also holds good for determinants of 2 rows and columns.

Now we have
An = (a,—4b,) M,, + (4,,—4b,,) My, + - +» + (tin—Abin) Min.

Let us now make use of the following notation :
4), is the determinant derived from A, by omitting the pth row
PY “ to)
and the g' column.
4A,, is the determinant derived from A, by omitting the rows
PY . t

rs

with numbers p and 7 and the columns with numbers gq and s.
We now find

0A), a OM.
ei = — b, M,,—),, M,,.--— bin Min + Bae i) Will

Further we find
My, = (—1)s— Ai.
Ais is an (asymmetric) determinant with (7—1) rows and columns,
for which we have supposed the theorem to hold good, so that

oA 7—r sn
SS = ih, HN (1) Gn dp)
UN ary =

p=nr q—l!,2;--(8) on
omy SY bp) (p21) A. 4)

Pq

p—2
where the positive sign must be used for g>s and the negative
one for qs.

Performing the summation according to s first, we find

1) By placing s between brackets we indicate that for the summation all values
from 1 to 7” except s must be assigned to q.

i=
‘

Proceedings Royal Acad. Amsterdam, Vol. VIL.

( 98 )

dA, rn pen ial
yo a —byh,+ YY (—bpq) Apg (— 1)? (— 1)
5 dt Srey

or
0A, p=n q=n
4 = 2D ba (— bpq) Mpg -
; p=| q=!

a
For a possible mixture for which : becomes stationary all quan-

)

tities VW, have the same sign as is proved. It is now evident that

ATAPY

: ‘ : ; OL, ;

for such points, whatever the sign of .J/,,, may be, >) Min is always
O04 :

negative. With the help of this theorem we can investigate the con-
ditions to distinguish an absolute maximum or minimum.

Let us now write Ff — a— 4b.

This expression fF regarded as function of v,,.7,...0, and 2 is
zero for every given set of values w,,.7,,...0,, if we assign to 4 the

a . . 7
value of 7 belonging to that constitution. If inversely we start from

a given 2(=4%,) then the solutions of the equation F — a@— 4,b= 0,

regarded as an equation in 7, .,,.-., furnish all the mixtures for
: a 2 : .
which pp nO seen the given value 4,. If moreover that value 2, is

D
an absolute maximum or minimum, then only a single set of possible
values @,,.2,,-.-U, may satisfy that equation.

As fF is a homogeneous quadratic expression in the ” ws we can
write it down as the sum of 7 squares.

Let us again call 4, the determinant forming the first member of
the equation in 2 of order 1, 4,—1 the determinant derived from the
former by the omission of the #'* row and the n column, Ap»
the determinant obtained by the omission of the last two rows and
columns, so that finally 4, —a,— a,. —

The transformation into a sum of squares is brought about in sueh
a manner that the first. square contains ail the terms with .,, the
second all the remaining terms with «,, ete., until finally «*, is
our last square. In order to evade surds we must every time
multiply our function by definite coefficients. It is now evident that
by executing the development, if we represent the successive linear
homogeneous expressions to be squared by L,, da... Ing We havek:

( 99 )

Pir ono yt Ag F = Ay APA, 353 Arno On—1 Ag iL", +
A, 4, A*,..- A*, 2 An—1 A, L?, + A? AAA’, . .. A*n—2Ay 1,0", +
cee AeA A, 9 Ap Ay AX. - At 2 Ay An D4)...

Bee AeA BAN 6 4, 4):

Here is

pn
we
— N (a1p — Abiy) Xp 7)
p=
p=n | ay —Ab, d1p—Abip |
L,= VY |
2) | 5 Ney
= | 19, —Abo| 2x)—Abo, |
@, Ab, G,,—4b,, ~~ + Ui, s=i —Abj s—1 Ap —Abip |
Hee ler db,, a, —2b, 22. do,5- Abo 1 yp — Aba |
L.—N ies , Ups
pas |" -1,1—Abs Sylvie use @s—1,s—1 —Abs—1,s—1 ds—1, »—Abs—1,p
Us,1 Vien odode dis,s—1 —Abs,s—1 asp —Absy |
i —— eine
So we find
lof 4 Gp. 4 bie ke lie. "
— —— a oS)
4, 4,4, A, 4) An—A,,
For a stationary point 4, becomes equal to 0, and so the last
. . . LS, 2 4
term disappears as it is ——— 2’, ‘).
n—l

1) For the deduction a continual use has been made of the theorem :
AA=AA—BAA

Pq P q rs ps qr
rs

Weser: Lehrbuch der Algebra [, 2"4 edition, p. 115.

2) In connection with the following it is easy to find back out of this the system (V).
Comp. for a ternary system: van ver Waats, Proceedings Royal Acad. of Se.
1902, Vol. 5. p. 235.

8) Without looking more closely at the breaking up into squares, as is performed
above, Rourn shows in his Rigid Dynamics, Advanced Part, 5th ed., p. 426

with the help of invariants that the coefficient of Lp has the sign of -, which
1

agrees to the above,

4) There is a difficulty however when for A= Ay also An—1 becomes 0, so when
a stationary point in the mixture (1, 2,....—1) becomes at the same time a statio-
nary point for the total mixture. When breaking up into squares we then shall
change the order of succession in order to avoid r,2 presenting itself as last term.
If all first minors belonging to elements out of the principal diagonal An are 0
then all minors must be 0, if there are to be stationary points. The equations
(VY) have then a higher degree of dependence and there are an infinite number of

. a .
mixtures for which 7 becomes stationary.

( 100 )

Suppose the stationary point to be an absolute maximum or mini-
. . a rn . +
mum. Let 4, be the corresponding value of 7 Chen tor 2 = 4, there
)
is only one constitution; thus for w,,.2,,...0 only one set of values
may be found. Now this is only possible when all the coefficients of
is.
case then it would be possible for 4—= A, (tor which the last coefficient

L,,--+-Ly—1 have the same sign for 2 = 4,. For, if this were not the

has already disappeared) to satisfy the equation # =O without the
necessity of L,, £,,.--2,—1 being individually zero and then many
sets of adjacent values «,,.,,..., might be found for whieh
)
had that absolutely maximal or minimal supposed value; which is
absurd.
For a stationary point to be an absolute minimum or maximum
it is therefore required

A,>0 A>0 _(A.<0 A,<0

(NS) (7,) A,>0 4,4,<0 (—1)4,<0
8 PESO: Ola: SA) = ’
[Er14,>0 je |<” By A) ae
\ INN CSSD An ) De Tel) (—1)"2A, 420
Let 2, be a root of A, =O indicating an attainable absolute

maximum or minimum, then for 2A, the coefficient of the last
square (v,7) in the development of © becomes zero. For 4= 4, + &
; : An : : ee :
the sien of A,—1 - determines the sign of that coefficient, whilst
= 02
04),

OA”

for 4 = 4, —« the sign is determined by — A,-\

Now however, as we just before indicated, we find, fora possible
stationary point
04,
On
So it is evident that for 2=4,-+¢ the last term is always negative
and for 4= 4, —e always positive.

0A),
Ae — Mann ate 0.
Tc) Sa aS

From this ensues that in the case of an absolute minimum the
inequalities (7) must be satisfied, whilst for an absolute maximum
the inequalities (7) must be fulfilled. In the first case the conditions of
possibility (A) are still to be added, in the second case the conditions (4).

It is clear that by a different numbering of the components other
inequalities would have been obtained — evidently however the
system (7') or (7) is sufficient to indicate an absolute minimum or
maximum,

( 101 )

Suppose an attainable minimum presents itself in the mixture of
some components say 1, 2,... p.
Within the limits of possibility for that mixture a second set of

Pp

values of the «’s for which 4, — 0 cannot be found. As farther

is negative for that minimum we can draw the following conclusions
from the system of inequalities ( 7’):
a
An attainable absolute minimum lies lower than for the com-
)
ponents and lower than eventually appearing minima in any mixture
to be compounded of the given components.
If the original mixture has a maximum and if there is also a
maximum in the p-fold mixture (1, 2,..p), then for the maximum

; 04,
in the p-fold mixture 4, is equal to O and = A,—-1 negative, so

OA ’ : :

7h has there the sign of — A,1. Now (—1)p-? Ap, is <0, so
; ; . 9A 4

A,-1 has the sign of (—1)P—!. So a2 has the sign of (—1)r. Let

a represent that maximum, then as A, becomes but once O for
possible mixtures 4, is furnished for every value of 4 > 4,, with the
sign (—1)r, but for every value of 4< 4, with the sign (—1)e-!. For
the maximum in the #-fold mixture (—1)e—! A, is <0, and so for
Ay the sign is indicated by (—1)r. From this ensues that the set of
inequalities (7',) can be expressed as follows:

An attainable absolute maximum lies higher than the ; for the

)
components and also higher than eventual maxima in mixtures to
be formed of the given components.

The question now arises whether a maximum or minimum in the
n-fold mixture implies anything about maxima or minima of the
binary mixtures to be formed of the 7-fold mixture. :

Suppose the -fold mixture to show a minimum for 2= 2, and
the constitution of that mixture to be indicated by [(y)ns (@y)ne(@n)m]s

2

then we find
(a, i. an b, )(@.)m =F (4,, — Am by) (@)n =| eh nats —+ (@:n — 4m bin) (@n)m =0
(4, — Anbs1) (&)m + (4, — An b, )(@)m + + (Gan — Am ban) (@n)m = 0
\ (on, eG Cen (ng — am One) (es) + (4n — am On ) (@n)m = 0-

Now we know that for a possible absolute minimum a,—z.,, 6,>0
whilst of course also (ws)m > 0.

( 102 )

If the above equations are to be satisfied then in every equation
at least one of the coefficients must become negative.
This is most profitably attained for m= 2¢ if

@,,—Anb, <9, @5,—Amb, <9, and a21—1, 2t — Am bor—1, 94 <0.
So then we have e.g.

ds — Ay, bs = 0 as, st+1 — Am bs, s+l < LU s+} am bs > 0
so
ecetek <Uam and Ap ae and An Pg oi
Ds sH1 bs bs41
so that a fortiori
ds, 541 Zia : Qs, s+1 As4|
bs, s-+1 bs bs sa a ;

The mixture {s,s 1] then possesses a minimum.

So at least minima are wanted in ¢ binary mixtures if the whole
mixture of 2¢ components is to show a minimum.

If n=2¢-+1 then there are at least ¢+1 or 4 (+1) minima
wanted in the binary mixtures if the total mixture is to show a
minimum. Let us take e.g.

O45 — Ae b,. < Q = 7k DS << UO ogae Qn—2, 21 — a Dro 11 <e 0
and
ahi — re by} < 0.

For # even the case is taken that each component shows with
but one other one a minimum, whilst for 7 odd one of the compo-
nents gives a minimum with two other ones.

If a component forms with more other ones minima then more
conditions are absolutely necessary; if a.o. we assume that — 1 of
the components give mutually no minima, then certainly the last
component -must give a minimum with each of the (7 — 7) remaining
ones, if the total mixture is to show an absolute minimum.

Of course the above theorems may not be reversed; so at least
Ln (resp. $ (2+ 1)) minima are wanted, but these do not in the
least guaranty the existence of an absolute minimum in the #-fold
mixture.

In case of an attainable absolute maximum a,—Ayb, is <Oand
so in our set of equations at least ove positive term is required to
present itself in each member. From

ads — Aw bs <0, dys — Ay b,s > 0 and a,— Ayb,< 0

follows

“8 Say, whilst ay >>“ and Ay >,
bys bs %

( 103 )

so a fortiori
Ars hs Arg

i aR and he. :

rs Is

so that the binary mixture (7, s) then gives a maximum.

In the same way as for a minimum we reason here, that to have
an absolute maximum in a mixture of 2¢ (resp. 2/-- 1) components
we must require at least maxima in ¢ (resp. ¢-- 1) binary mixtures.

For a further investigation we shall have to look more closely
into the quantities 4,, and a

Pg Pq"
For 4,, we shall use the formula given by Lorenz
Ore =e 32 (Tp oa To) ly,
where 7, represents the radius of the molecule of the p'® component.
This formula holds good for p= q too.
The coefficient of 2" in the equation A, =O is
erence 6 Uiyn

(SNe Oy ce etary | (ov aye

: 2
but bna--- dn |
Now we have
Ayan = — **/, 2? (7, —7,)? [727.5 +30 7? 727 +8 72 7, +8 7, 7]
Ay=s = — (*/;)* x 6 (7, —7,)? (7273)? ("3 —7,)? (7, (7275)? +
Fs sys an : )
Ari (*/, *)* 9 (7, —7,)" ©, — 75)" 4 —7,)? (7)? (2-74)? Oa)”
For five components the pene on the 6’s vanishes identically ;
for we find

Avs) As) = Avs) 45) — eS 6) =
44

44 55
i ("/,20)* 9 (7, —r,)* (, Fs) (a= (”, ee (, meh ) ("; ge x
x C/s%)? 9 (7, rae (7, =n eae = lee ae AY ( te) eal

he (°/37)* {97 —7)" (Cha un))e Cita) wey. Ae

) (CA) (7,—7;)} ==

So 4¢)=0 as Ag) is not identical equal to 0.
44
59

For the determinant of the order 6 not only all minors belonging

1) Wied. Ann, 12 p, 134. In reality still another constant factor NV presents
itself here.

2) These results have been obtained by remarking that Ay is always divisible
by (p—1)?; the coefficients of the remaining factor have been found by means
of the method of indeterminate coetticients.

( 104 )

to elements of the principal diagonal become O but also all other
minors of degree 5°).

Then however it is clear that for mixtures with 6 and more
components the determinants on the 4's disappear identically.

BertuHeLot and before him Ganitzing have made about the quantity
a,, the simple supposition @*,, =, da . Although this formula may
not be strictly true, as has already been clearly proved by experiments
on binary mixtures, it is worth while to consider to what conelu-
sions the afore mentioned rule leads us for multiple mixtures.

For ap, = Vay a, all minors of degree 2 and higher of the deter-
minant on the as become 0, from which ensues that the equation
An =O can be reduced to

p=n g=n
An—l a y Bog Ang — AYN 10)
p=! q=1
where when developing the determinant 4, the coefficient of b,,
is By. So there is an (7—1) fold root A=0, which cannot of course
indicate an attainable maximum or minimum, not even another
stationary point.

Let us now consider the different mixtures assuming the rule
of GALITZINE-BERTHELOT.

e's

Beside the root 420 a second root appears which can certainly

a a a a.
. . . . 12 1 12 2
not point fo a maximum, for from 5 a and ; La would ensue
) ) ) ,

12 1 12 2
ae 2. lj 7 wi itl b? } ij -ertainly
—— and in connection with 6°,, > 6, 6,, certainly

1) So we find for the asymmetric determinant

Dy Dyg Dys Ory Ors
|

Day Dg bag boy Vo;
D=| Os Ges Ds. DsuDa,
| Dy, D4g Dag Dg Oy;

|
| Dey Deo Yes Des Yes

DD =(8/37)*9(7) as rs)20y a 13)2(72—Ts)2(7) 24 1)" = ry)2(7's &: ry)-X
44

XKEF/34) 90 — 7a) — 137003 = 137207. — 153) —T M12 = 15)(72 — M6 )(1'3 — M573 — VQ)
—(8/a%)'9(7) —19)7(7) —73)2('2 = 7'3)7(7] — 1") — 75) (2 — TH 2 — 15 )(773 — 173 —TZ)X
XE/s2)4 7 = 7a) = 13) 2 = V3) — PV — Pe M2 = TYMa = MGN'3 — M3 = Me) =
=0.

\

(105 )

(i Sy Wrc
The appearance of a minimum is not a priori excluded; the general
conditions now pass into”:

Abi
Di. > aes b, and bys >
t

V«

Va,

1 a,

Dee

So it is not evident, why the appearance of a minimum should
coincide with -a*,, <C a, a, ').

Even for a*,, >a, a, & minimum can very well present itself.

In fact there are objections to assuming that a*,, << a,q,, for this
supposition leads when the temperature becomes lower to partial
mixability *) and this phenomenon probably does not appear for
normal substances *).

The following however holds good for a possible minimum :

tae) UN»
@ > yo and ae b; pe sO

bis 12
9 9 1s Ma
a, + hes = (2D ei > (b, -l- b, — 2655) —— , SO

bi,
a, +a, — 2a,, > 0.
For all temperatures excluding partial mixability, we find thus
when a minimum appears

Va, a, Kile << “/, (4 + 4,)-
T= "3:
The equation 4, =0 has a double root 2—=0, so we obtain the
following series of signs :

4 As j 4;
te + fo) +
eee ox tonne I
i+ | — — | +
0 0 | 0) |)
07 | = | 00 | =

1) van per Waats. Proceedings, Amsterdam, Oct. 1902, p. 244.

2) van perk Waats. Continuitiit I], p. 48. However in the deduction it has been
supposed there that bpg = 1/2 (bp + bg).

3) VAN DER Waats. Slatique des Fluides, p. 32.

( 106 )

Case I.
There is a positive root 2 making A, = 0, which can however
ae sk _ dh,
never indicate a possible absolute maximum or minimum, for ah
ah

is positive there, denoting a negative value of the minors of degree
2 and these must be (see inequalities (7',) and (7,)) positive for a
maximum as well as for a minimum.

Case II.

One root making 4,0 is negative, the other O, so neither of the
two can indicate a possible mixture.

So for a ternary mixture the rule of GALITZINE-BERTHELOT cannot
be united with the appearance of a maximum or a minimum, A
stationary point, however, being neither maximum nor minimum,
is not excluded in case I.

4A, = 0 has for n= 4 a threefold root 2—= 0.
So the series of signs becomes:

Lene an | [base aa
eet Peale,

I | 0 | 0 | a ae 0 | I
ie bes
bee 0 0 0
|

In case I the simple root 2 belongs to a negative value of
a

5? 0 it cannot represent a possible mixture.
)

,

van ae. ©
In table II vi is positive for the simple root; thus the minors of

degree 3 must be negative in case the mixture is to be possible. So it is

3

evident, that as soon as we put «,,.° equal to a, a quaternary
a
a
this would lead to maxima in at least 2 binary mixtures to be
formed out of the components and these are not possible if da, * is

mixtures cannot show a minimum in A maximum is excluded, as

put equal to a dq.

. . . . a . . .
For mixtures where 7>4 a mixture for which 7 is stationary is

( 107 )

certainly excluded as soon as we put y° = ap A, for there is then
an (n—1) fold root 20 and a root A=o.

Resuming :

By assuming the rule of Gaxirzine-BerTHELOT, we find:

n—=2. No maximum; a minimum is possible.

n=3. No maximum.or minimum — a stationary point, but no

maximum or minimum, is possible.

n=4. No maximum or minimum; other stationary points are

possible.

n—=5 and higher. All stationary points are excluded.

If we assume for / a linear function of 7, thus Gog |a(Op-1- O5)s
then already for 73 the determinant on the /’s becomes identical
to O, so that then for ternary and higher mixtures no stationary
points are any more possible as soon as we put dpg* equal to a, dy.

Mathematics. — “On an expression for the genus of an algebraic
plane curve with lagher singularities.’ By My. Frep. Scnun.
(Communicated by Prof. D. J. Korrrwne.)

Lately I gave the following theorem in these Proceedings‘) :

Let P be a point of order t’ of an algebraic plane curve (where
t’ can also be zero, namely when P does not le on the curve) and
San arbitrary point of order tof that curve. Suppose the straight line
PS intersects the curve in w points coinciding with S, then t’+- (w,—t)
(summed up over all points S for which w is >t) is independent
of the situation of point P and equal to the class of the curve.
If S les in P we have to regard all straight lines through P as the
line connecting P and S or if one likes only those which furnish a
contribution to XS (w,—t,) ¢.e. the tangents in P.

From this a corresponding and as far as | can see a more
important theorem for the genus of an algebraic curve can be
deduced, where moreover the straight line connecting P and Scan be
replaced by an alyebraic curve. Lateron I hope to connect this with
problems of contact (numbers of algebraic curves determined by
conditions of contact) in particular with respect to the number of
normals on a curve with higher singularities (also in connection
with the circle points and the line at infinity) let down from a point
(which can also have a_ particular situation with respect to the

1) On an expression for the class of an algebraic plane curve with higher
singularities. These Proc. VIL, p. 42.

————

( 108 )

curve) and the question annex to it after the singularities of the
evolute. It seems to me that in no other way known to me these
and suchlike questions can be so simply answered.

The genus has been introduced in the theory of functions by
Riemann and is defined out of the connection of the 7’-leaved RieMann-
surface on which an w-valued algebraic function is univalent. If s
is the number of branchpoints of the function, gy the genus, then
we have the following relation given by Ripmann*) (l.¢. p. 129)

s— 2n' =2(q—1),
for which a branchpoint where ¢ leaves of the Rimmany-surface are
connected is to be regarded as #—1L branchpoints.

For the theory of the algebraic curves the notion and also the name of
genus (“Geschlecht’’) has been introduced by CLesscn*), whilst HALPHEN*)
has given for the genus of a curve of order » and class % with
higher singularities the equation

2 (g—1) = k—2 n + J (t,—1),

in which 2 (¢,—1) represents a summation over the origins of the
separate branches of the curve (which can be represented by one
Purseux-development) whose order ¢ differs from 1 and over as many
other origins of branches as one likes. If a branch of the curve is
represented by the development

y—y=a, (ew — §) +a, (*@—§) 4+...,

according to integral ascending powers of (iE) I call the point
(§,7]) the origin, the line y—a=a, (w—§) the tangent and the
numbers ¢ and v the order and the c/ass of the branch, where thus any
point of a curve can be regarded as the origin of at least one
branch, for which however if the point is a common point of the
curve ¢ will be equal to 1. If one and the same point of the curve
is the origin of more branches we shall regard this point successively
as if belonging to the different branches.

The Ha.pnen-relation is an immediate result of the Rremann-
relation if only one decomposes the branchpoints into those which

1) B. Riemann. Theorie der Apet’schen Functionen. Cre//e’s Journal, vol. 54,
(1857), p. 115—155.

2) A. Cresscu. Ueber die Singularititen algebraischer Curven. Credle’s Journal,
vol. 64, (1865), p. 98—L00.

5) G. H. Hatenen. Sur la conservation du genre des courbes algébriques dans
les transformations uniformes. Bulletin de la Soc. Math. de France, vol. 4, (1875),
p. 29—41.

fe 109")

do and into those which do not depend upon the choice of the
system of coordinates.

If ¢',,¢,,-.- are the orders, v',,v',,... the classes of the separate
branches having their origin in P? (so that + /, = 7) then

2 (g—1) = k—2n 4+ = (¢,—1) + (¢,—1),
whilst according to the quoted theorem
>= S(¢,+',) + = (w,—t,). (equation (2) 1. ec. p. 44)

In the two last equations the first -sign refers to the origins in
P, the second S-sien to the origins outside P.

From these equations follows

2 (g—1) = — 2 (n—#t’) + Y(',—- 1) + Sw — 1)... . . (A)

Here n—t'=n' represents the number of movable points of intersection

of the curve with straight lines through ?. If one draws through 7? an

arbitrary straight line / which is nota tangent in P, then the points of

intersection of that straight line with the curve furnish to Y (7e,—1l)a
contribution equal to 7/— .V;, where .V; represents the number of origins

of branches lying outside / on the straight line /;*), thus the number of

branches over which the # movable points of intersection with the
straight line /; distribute themselves. If then we draw through P a
straight line 7; touching NN’; branches through P and if we let V;
be the number of branches over which the 7 movable points of
intersection with the straight line UG distribute themselves, then NG == NG
of these .V; branches have their origin outside ?. The points of
intersection outside P with these straight lines give a contribution to
(w,—1) -equal to
(v’— = vj) — (N;—N';) = (n'— N;) —=2 (vj —1),
in which Sv'; and (v'j;—1) are taken only over the branches
touching the straight line /; in P. From this ensues:
= (w,—1) = = (r'— N)) + = @'—N;j) — J (v',—1),
or
= (vw,—1) = J (v’'—N,) — J (',— 1),

in which now (v',—1) denotes a summation over all the branches
with P as origin, Y (v’— N,) a summation over a// straight lines through
P for which w' is > N, and over as many other straight lines as one likes.

If we substitute in equation (1) for (w,—1) the obtained value
and replace n—? by 1’ we find

Gl ne aes) as ee. C2)
We can sum up what has been found in the following theorem :
THrorem I. If an algebraic plane curve is intersected by the straight

1) Counting here also each of the points of intersection as origin of a branch.

( 110 )

lines of a pencil with P as vertex inw movable points, which distribute
themselves for the various straight lines of the pencil over N,, N,....
branches of the curve, then 1—n' + 4 & (n'—N,), (where X (n-—N,)
is taken over all the straight lines through P), has for every point P
the same value equal to the genus of the curve.

This theorem however can be considerably extended by making
use of the property that the genus of the curve does not change
by a one-to-one transformation. If namely we apply to the whole
figure a Cremona-transformation, then remains the same, but 7’ too.
The straight lines of the pencil are turned by the transformation into
rational curves having multiple points in the fundamental points of
the transformation. Through every point lying outside those fundamental
points only one of the rational curves passes, so that we treat a@ pencil
of rational curves; the manifold points here making the curves to
rational curves are not present as movable points but as fixed
points, which gives rise to linear relations between the coefficients
of the curves. The movable points of intersection with a straight line
are now transformed into movable points of intersection with a rational
curve, and they remain the same in number on account of the one-
to-one character of the transformation.

By a Cremona-transformation a branch is furthermore always
transformed into one single branch (where we always understand by
a branch the whole of the’ curvepoints whose coordinates satisfy
the same Puisevx-development). If thus the 7’ movable points
of intersection with a straight line distribute themselves over
branches then in the transformed tigure the ’ movable points of
intersection with the rational curve originated by transformation of
the straight line distribute themselves also over .V_ branches.

From this it is evident that all quantities of equation (2) are
invariant with respect to rational transformation, so that the equation
(2) holds good unchanged if the pencil of straight lines is replaced
by a pencil of rational curves.

This gives rise to the following theorem:

Tueorem Il. Lf an algebraic plane curve is intersected by a pencil
of rational curves inn’ movable points distributing themselves for the
different curves of the pencil over N,, N,,.... branches of the fined
curve, then 1—n’ +342 (n’ — N\), (where S (n’ —N,) is taken
over all curves of the pencil), has for every pencil of rational curves
the same value equal to the genus of the fixed curve.

This theorem can then be extended froma pencil of rational curves
io an arbitrary algebraic pencil of curves by means of the following
considerations which ave however less rigorous than the preceding ones,

G1it’)

If we investigate which rational curves of the theorem IL contri-
bute to S(n’ — N,) then we find 1s* those rational curves which
pass through an origin (lying outside the basepoints of the pencil) of a
superlinear branch (denomination of Cayniey for a branch with order
more than one) of the fixed curve, 2°¢ those rational curves touching
the fixed curve outside the basepoints, 38" those rational curves
where two or more of the movable points of intersection have
approached one of the basepoints along the same branch of the fixed
curve. In the main theorem Il comes to the determination of the
number of rational curves of a pencil touching a given curve and
the change which this number undergoes on account of higher singu-
larities of the given curve and the particular situation of the basepoimts
with respect to that curve. Here however it is difficult to imagine
how it would cause a difference whether we are working with a pencil
of rational curves or with an arbitrary pencil of curves; for in both
‘ases the coefficients of the equation of the movable curve satisfy
some linear conditions amounting to one less than is necessary for
the definition of the movable curve ').

Let us explain the preceding by an example. Suppose the number of

cubic curves through eight points touching a given curve were required,
suppose further that the given curve has a singular point S with a
singular tangent / and that the C, of the pencil through S also

touches /. Point WS will then absorb a certain number of points of

contact proper with curves of the pencil, and this number will depend
on the nature of the singular point SS and on the order of contact
of (C, with the given curve, but in no wise on the fact
whether of the basepoints three have coincided somewhere outside S,
either in such a way that in one of the basepoints tangent and
curvature are given, or in such a way that the coinciding basepoints
form a triangle with finite angles, in which case the condition of the
passage through the three basepoints includes the curve having a
double point in a given point and being thus rational.

Led by the above considerations I think I may. state the following
theorem :

THeoreM Ill. Lf an algebraic plane curve ts intersected bya pencil of

curves in nv’ movable points distributing themselves for the different
curves of the pencil over N,, Ny... branches of the fixed curve,
then 7" 4 > (n’ — N,), (= (n’ —.N,) taken over all the curves

1) Of course it would be different if the movable curve had to be rational without
the singular points reducing the genus to zero being given; if thus e.g. the question
were of cubic curves through seven given points, and furnished with a double point
not given in position.

( 112 )

of the pencil), has for every pencil of curves the same value equal
to the genus of the fixed curve.

The remarkable thing here of the obtained expression for the
genus is that the genus which is invariant with respect to rational
transformations is -really exclusively expressed in quantities each
invariant in itself over against rational transformations.

1 feel the more justified in stating the above theorem having found
the theorem confirmed in various simpler cases, e.g. for the case
that the given curve admits of double points and cusps only whilst
the basepoints can assume any particular position with respect to the
given curve, also for the case that the given curve is provided with
higher singularities where however only the simplest particular cases
with reference to the position of the basepoints have been considered,
e.g. the case that one or two basepoints fall in a higher singular
point.

But all the same a rigorous and simple proof which renders a
subtle distinction of the great number of particular cases which can
present themselves superfluous, is very desirable.

Sneek, July 1904.

Mathematics. — “On the curves of a pencil touching an algebraic
plane curve with higher singularities’. By Mr. Frev. Scnvn.
(Communicated by Prof. Korrewnc.)

In the previous paper I have stated the following theorem:

If an algebraic plane curve is cut by a pencil of curves in n
movable points distributing themselves for the various curves of the
pencil over N,, N,,... branches of the fined curve, then

1—n'+42(n — N,)
2X (n' — N,) taken for all curves of the pencil) has for every pencil
of curves the same value which is equal to the genus g of the fived curve.

Expressed in a formula this runs:

2 (gn — 1) = Sr — N,). « . ee

With the aid of this theorem the following problem of contact
‘an be solved :

To determine the number of curves of a pencil touching a plane
curve C, of order n, class k and genus 4.

To this end we first substitute in equation (1) for (n' — N,)
a summation over the points of (, or better (as we always
count a point of C, through which more branches pass as more
than one point) over the origins of branches of C,. Let S be

as

Ml
-
/

’
)

lg nd

(113 )

an origin of a branch of ©, whilst the curve of the pencil through
S intersects the branch under consideration in w points JS, then

= (n' — N,) = = (w, — 1), so that equation (1) becomes
2! (gee! 1) SSeS aD) ee 8 ot ee ei (2)
Here (av, — 1) represents a summation over «a// origins S of

branches of (C,, i. e. only over those origins for which w > 1.
If one or more basepoints of the pencil lie on C, the summation
must also be extended to those origins coinciding with a basepoint B.
We must then regard as movable curve through that origin the
limiting position of the movable curve through P if we allow P
to approach to 6 along the corresponding branch of C;,, in other words
that curve of the pencil intersecting the branch at least in one
point 4 more than any curve of the pencil. For such an origin
in B the number of movable points of intersection, approaching B
along the branch under consideration when P approaches 6 along
the same branch, is represented by w.
The following well known relation
21(g) = — ht) (ea si a wee ()
exists between order, class and genus of C},.

Here (¢,— 1) is a summation over all the origins of branches
of (, whilst ¢ represents the order of the branch, i. e. the number
of points of intersection with an arbitrary straight line through that
origin coinciding with that origin.

From (2) and (3) then follows :

22 (6, 6) = h(a) ey ee (4)

TueorEeM I. Lf a pencil of curves cuts an algebraic plane curve Cy
of order n and class k in n' movable points of which w fall in
the origin S of a branch of order t of Cy, we have the relation
= (w, —t,) =k + 2 (n' —n), where XY (w, —t,) must be taken over
all the origins of branches of C,, also over those couciding with
basepoints of the pencil.

With the equation (4) the given problem of contact for every Cy
and every particular situation of the basepoints with respect to C;,
ean, as will be shown, be regarded as solved.

We have but to discuss the found equation.

If m is the order of the curves of the pencil, then 7’ = mn for
arbitrary situation of the basepoints with respect to the given curve

an sO:

= (wo; 1, Sk a ene Vie ey nt 1D)

Here we understand by an arbitrary situation with respect to Cy

in the first place that the basepoints do not lie on C,. Let us
5
Proceedings Royal Acad. Amsterdam. Vol. VII.

(114 )

suppose moreover that the basepoints are situated in such a way
that not a single curve C,, of the pencil passing through a singular
point S of C, touches one of the branches through S and that not
a single C), has with the fixed curve a contact of higher order then the
first; then only the curves C,, showing a common contact with C, (v=2,
t= 1), furnish to S(w,—4,) a contribution equal to the number
of those touching curves C,. Here however a restriction ought to
be made. It may appear namely that a (,, of the pencil has a double
or multiple point lying on C, which then furnishes a contribution
to S(w,—t,). We can avoid this case by requiring that for an
arbitrary situation of the basepoints with respect to C, no singular point
of C,°>may lie on C,. This is however no longer possible when by
mutual coincidence of basepoints the pencil admits of curves showing
an infinite number of double or multiple points, in other words when the
pencil contains curves, which consist of or contain two or more coinci-
ding parts. Though the equation (3) can still be applied to these cases we
shall exclude them for simplicity’s sake from our discussion. With these
suppositions we find that > (#w,—¢?,) is equal to the number of
curves of the pencil touching C,. So we find the following theorem:

Tueorem Il. For a pencil of curves of order m, none of which
contains two or more coinciding parts, the number of curves touching
an algebraic plane curve C, ef class k with respect to which the base-
points of the pencil have no particular situation is represented by
k+2n(m— 1). “

If C, is a general curve in point-coordinates then / = n(n — 1),
and the required number is 7 (m+ 2 m— 3). If we compare this to
the number given in the above theorem we find :

Turorem III. Every singular point S of C, diminishes the number
of curves which belong to the pencil mentioned in theorem IT and
which properly touch C,, by the same number as that with which S
diminishes the class of Ch.

We now investigate which particularities can present themselves
in consequence of a particular situation of the basepoints with respect

to C,. To this end we consider in the first place an origin WS of-

branch 7’ of order ¢ of C,, supposing S not to be situated in one of the
basepoints; further we suppose that the curve C,, through S touches the
branch 7’ and intersects it in w—=¢-+ y coinciding points (so in y
points more than when the curve through .S of the pencil were not to
touch the branch). Then this point S counts (as far as the branch 7’
is concerned) according to (5) for w—?¢= y points of contact proper.
If we restore by a small displacement of the basepoints their arbi-
trary position with respect to (,; the curve through S of the pencil

(115 )

intersects the branch 7’ in ¢ points S and in y points lying near S.
The preceding holds good invariably when the pencil does contain
curves containing two or more coinciding parts if only those do
not pass through the considered point S.

This gives rise to the following theorem :

Tuvorem IV. Let S be the origin of a branch T of an algebraic
plane curve Cy. Lf now the basepoints of a pencil of curves pass

from an arbitrary position to a particular one so that no basepoint

approaches S whilst y points of intersection of C, with the curve
through S of that pencil approach S along the branch T, then as
many (so y) points of contact of Cy with curves of the pencil approach
S along the same branch. Here has been supposed that tf the pencil
contains curves containing two or inore coinciding parts these parts
do not pass through S.

If the basepoints have the particular position described in this
theorem, then S counts for y points of contact proper. So we can
formulate the theorem as follows:

If S is a point of Cy not coinciding with one of the basepoints
of the pencil, whilst the curve through S of the pencil cuts a branch
of order t of C, with S as origin in t+-y points S, then S absorbs
as far as that branch is concerned y points of contact proper.

Theorem IV is an extension of a theorem of HaLpHEN and Srepann
Smirn, which I discussed in a paper in a previous number of these
Proceedings ').

The indicated theorem can be expressed as follows:

Let S be the origin of a branch T of a curve, 1 the tangent of
that branch in S. If a point P approaches l but not S, then as
many points of intersection with PS as points of contact of tangents
through P approach the point S along the branch T.

This is no other than our theorem IV where the pencil of curves
is replaced by a pencil of straight lines.

Let us further consider the case of a singular point S of C,, with
which coincide one or more of the basepoints. As our only business
is to determine the number of points of contact proper coinciding
with S we can assume for simplicity’s sake that no basepoints
coincide with other points of C;.

Further we exclude the case that the pencil contains curves
admitting of coinciding parts.

Let ¢,,¢,,... be the orders of the different branches 7",, 7",,...
of C, having S for origin, whilst an arbitrary curve of the pencil
1) On an expression for the class of an algebraic plane curve with higher
singularities. These Proceedings VI, p. 42,

8*

( 116 )
cuts those branches successively in 2,,2,,-.. points S. Then we
have
n’ =mn— = z,’.

Further we can break up (w,—?,) into the share Y (w,’ — t,’)
of the point S and the share of the other points of C,. The
meaning of w', is here, that the curve of the pencil cutting the
branch 7”, in more then 2’, points S does this in 2',-+2', points.

For equation (4) we can then write

> (w, — t,) =k + 2n(m — 1) — TS (w', + 22', —#,),- - (
where (w,—t,) must now be taken only for the curvepoints
outside S. If we represent by w, the number 2', + 2’, of the points S
in which the branch 7”, is cut by the osculating curve of the peneil,
equation (6) becomes :

= (w, — t,) =k + 2n(n — 1) — (wv, 4+ 2, — @)).

1

It is evident from this equation that the point .S, as far as
branch 7”, is concerned, absorbs w', + 2', —?’, points of contact proper.
This can be formulated in the following theorem:

Tueorem V. Jf a single or multiple basepoint of a pencil of
curves coincides with the origin S of a branch of order t of an
algebraic curve C,, whilst that branch cuts an arbitrary curve of
the pencil in z, the osculating curve of the pencil on the contrary in
u points S, then the point S absorbs u-+-2—t points of contact of
curves of the pencil with C,, in other words for an arbitrary displace-
ment of the basepoints coinciding with S the point S furnishes
u+2—t points of contact, which are then situated on the considered
branch.

By allowing the basepoints to undergo not an arbitrary displace-
ment but a particular one, another theorem in connection with
theorem IV can still be deduced from this. We can namely make the
basepoints change their places slightly along the osculating curve of the
pencil in such a way, that no more basepoints coincide with S. In that
case the point S continues to absorb after the displacement of the base-

points a certain number of points of contact proper, and according to _

theorem IV to the amount of w— ¢; in fact after that displacement
the curve of the pencil through S intersects the branch in w points S,
so that for the point S now w is equal to wv. If we compare the
number «—¢ of the absorbed points of contact to the amount given
in the theorem V_ we find:

Turorem VI. Jf the curves of a pencil cut the branch T of an
algebraic plane curve in 2 fived points coinciding with its origin S,
then point S gives to that branch z points of contact with curves of the

———————— ee =

—

pencil, when the basepoints falling in S move away from S along
the osculating curve of the peneil.

It is clear that the theorems V and VI invariably hold good when
the pencil contains curves degenerated in coinciding parts if only
they do not pass through point 5S.

Theorem VI is like theorem IV an extension of the above mentioned
theorem of HanpHyn and Situ. If namely we substitute for the
pencil of curves the pencil of straight lines with Sas vertex, theorem VI
passes into:

Let S be the origin of a branch T of a curve whilst that branch
is intersected in t points S by an arbitrary straight line through S. If
now a point P moves away from S along the tangent in S, that point
S gives to that branch t points of contact with tangents through P.

It is not difficult to see that this is correlative to the formulation
given above of the SmrrH-Hatpnen-theorem. However when the pencil
of curves is not a pencil of straight lines the theorems IV and VI are
not correlative and so we have to regard them as entirely different
theorems.

Sneek, June 1904.

Astronomy. — “On the apparent distribution of the nebulae.” By
C. Easton. (Communicated by Prof. H. G. vAN DE SANDE

BAKHUYZEN).

It being admitted that the results of the visual observations must
be kept separated from those obtained by photography, the systematic
investigation of the distribution of the nebulae by means of photo-
graphy begun by Max Wotr should not prevent us from carefully
examining the very extensive material regarding nebulae which has
been formerly obtained by direct observation and laid down in
catalogues; and the less so because it is highly improbable that even
in future a visual “‘Durchmusterung”’ of this kind, which for the rest
is very desirable, will be carried out on account of the different
method which now is being followed at Heidelberg.

It is noways unimaginable that the distribution of the nebulae as shown
by photography will differ greatly from that of the visually observed
nebulae; yet it is certain that the latter distribution shows remarkable
features which call for an explanation. Witu1Am Herscuen has found
that in the main the nebulae are numerous where the stars are
sparse. In a certain sense we have here the reverse phenomenon
from that of the stars; Newcoms (The Stars, p. 187) expresses it thus :

(118 )

“The latter (the stars) ave vastly more numerous in the regions near
the Milky Way, and fewer in number near the poles of that belt.
But the reverse is the case with the nebulae proper. They are least
numerous in the Milky Way and increase in number as we go from
it in either direction.”

CLEVELAND ABBE who after the publication of Joux Hrrscnen’s
catalogue in 1864 statistically investigated this peculiarity (Month. Not.
R. A. 5S. XXVII, p. 257) rightly put the question whether the paucity
of nebulae in the Galaxy did not rest upon a mere optical delusion
due to the luminous background of the Milky Way. He thought
himself justified, however, in answering this question in the negative,
because the regions on either side of the Galaxy proper did not show
a considerable increase in the number of nebulae; nor was this the
ease with increasing optical power.

With a much more extensive material — 9264 objects — STRATONOFF
(Publ. Tachkent N°. 2) arrived at chiefly the same result. With some
reservation, however, For in fact if we develop Srravonorr’s data in
a somewhat different way we find that, as we go from the galactie
plane, the faint nebulae increase more rapidly than the bright ones. This
fact, also because it contradicts a preliminary result of Max Wo tr:
that the (photographed) small nebulae are in general distributed more
equally over the sky, raises the surmise that in visual observation
the light of the smallest and faintest nebulae in the galactic region
is to a certain amount extinguished (table I).

18/0) se Dial ee 6

Increase of the mean density of bright and faint nebulae in the
direction from the Galaxy.

: Bright nebulae Faint nebulae
Mean gal. lntinde (N. brillantes) (N. faibles)
py 7.5 10.2
Ee iy 3.8 7.0
415° 2.3 3.4
ve 1.9 1.5
2258 2.3 1.8
a5 ay 54
Soe 6.2 11.0
Se a5e 6.5 12.5

In order to investigate this problem more fully I have compared
the density of nebulae in the different parts of the northern Galaxy

(219: )

with the intensity of the galactic light in those same portions. For
if really the luminous background of the Milky Way had a highly
disturbing influence, a certain parallelism between the distribution
of the galactic light over that girdle (which distribution is very
irregular) and the distribution of the nebulae in the same region
would manifest itself in this sense that the nebulae, especially the
faint ones, are least numerous in those patches where the galactic
light is strongest. Table I] gives the galactic zone proper between
—10° and + 10° gal. latitude (northern hemisphere) divided into
areas of 30 degs. in galactic longitude ; the two upper lines show
for each area the mean density of nebulae derived from STRATONOFR’s
data; the lower line shows the mean intensity of the galactic light
derived from the table on p. 18 of my “Distribution de la lumiére
galactique’ (Verhand. Kon. Akad. v. Wet. VIII, 3).

PAB i EO.
The density of the nebulae and the galactic light in the
Milky Way compared.

180° 90° 0°

Bright nebulae | 1.2 0.7 | 13 0.9 1.0 | 09

Faint nebulae | 0.8 0.7 1.8 0.7 1.0 1.0
Galactic be | 1.03 | 0.72 0.78 | 1.09 | 1.31 4.08

No parallelism is to be detected.

Other causes which might influence in the same way as the
“extinction” due to the luminous background of the Milky Way, must
be disregarded. Up to now the investigation has yielded nothing in
contradiction to the view that the peculiar distribution of the nebulae
in the sky results, at least in the main, from their particular real
distribution in space.

CLEVELAND ABBE tried to explain the paucity of nebulae in the
galactic region and the (supposed) increase of their number towards
the two Poles of the Milky Way by the supposition that the visible
universe consists of systems of which our Milky Way, the two
Nubeculae and the nebulae are the individuals, which in their turn
are composed of stars and (or) nebulae ; that moreover the galactic
plane is nearly at right angles to the major axis of “a prolate ellip-
soid” in which all visible nebulae are equally scattered.

This theory is founded on the supposed symmetrical distribution

( 120 )

of the nebulae with regard to the Galaxy. Also Bauscninerr (V. J. 5S.
24, p. 43) and Srratonorr adopted this symmetry. STRATONOFF sup-
poses first that the sky from the North Pole to é— 20° has been
surveyed uniformly with a view to nebulae (I. ¢. p. 43); secondly
— like his predecessors — that the actually observed decrease in
the number of nebulae between about — 50° and the South Pole
of the Galaxy must be ascribed to the incompleteness of the obser-
vations made in the southern hemisphere.

I shall now proceed to demonstrate that StratoNorr’s first surmise
is certainly erroneous and that the second, considering the present
state of our knowledge, is not justifiable and moreover improbable.

That the nebulae-material from the North Pole to d — 20° would
be collected with equal completeness throughout, as would follow
from SrravTonorr’s first supposition, cannot, apart from the lack ofa
systematic “Durchmusterung’, be the case on account of the great
difficulty to detect faint objects like the nebulae, which arises from the
atmospheric absorption in regions far from the zenith. In Lord Rosse’s
telescope, for instance, it has never been possible to observe properly
the Omega nebula, though it lies only at — 16°. (Dreyer, Trans.
Dublin Soe., N. S. Il p. 151).

Besides, the circumstance that the summer nights are never totally
dark in the relatively high latitudes of the observatories of the northern
hemisphere where the nebulae observations have for the greater part
been made, must render the number of catalogued nebulae in the
southern regions which then rise above our horizon much too small
in proportion to the opposite equatorial region.

With regard to this I have investigated the tables of Bauscninerr.
I have compared two equally large areas, occupying the same position
with respect to the celestial equator and the Milky Way, A: « 5" to
9", d + 15° — 30°. B: a 17" to 21", d + 15° — 30°,

The number of bright and faint nebulae in those areas A and B
may be seen on table IIT:

TABLE III.

Numbers of bright and faint nebulae in opposite equatorial regions.

A B
bright nebulae. .. . . 26 :, as eee
faint nebulae 5) ee er al Cif Act’ sb aL Od

There appears indeed to exist a difference at the expense of area

eee

(121)

B and this difference, as might be expected, is much greater for the
faint than for the bright nebulae.

It may be remarked that such a difference is not found — rather
the contrary — in areas which in the northern galactic zone border

upon those mentioned, where therefore the influence meant cannot
manifest itself so strongly. It remains possible of course that the
number of nebulae in the direction 4 is indeed smaller than in the
direction A; but we cannot consider such a large difference as
established where the disturbing influence is undoubtedly at hand.

If we consider only the northern galactic hemisphere, then the
nebulae seem indeed to increase fairly gradually though not regularly
towards the Pole of the Milky Way. For the southern hemisphere
such an increase is also visible in the table of SrraTonorr as far
as about —60° galactic latitude. Straronorr, however, constructed
his table omitting the two Nubeculae. Now it seems to me that for
such a statistics this omission is not justifiable. The Magrinanic
Choups must not at all be considered as patches torn off from the
Milky Way, which also appears from the fact that the nebulae
proper’), which are searece in the Milky Way are four times more
numerous than the star clusters in the MaGr.ianic CLoups (JoHn
Hurscuen, Results Obs. Cape). As to their composition the Nubeculae
keep the middle between the Milky Way and the accumulations of
nebulae (sometimes intermingled with star clusters and stars) in Coma,
Pisces, ete. — and though the latter agglomerations are less dense
than the C1ovps it is not allowed, in my opinion, to include these
agglomerations in our table and to exclude the CrLoups. Especially
also because, as may be seen clearly on STRATONOFE’s Own maps
(Publ. Tachkent, N°. 2, Atlas, pl. 16 et 18; comp. Smnry Waters,
M.N. XXXII, p. 558) the Nubecuiae are connected with streams of
nebulae and not with the Milky Way.

If, however, we include the Nubeculae into the statistics then
we must substitute my table 4 (table IV) for Srraronorr’s own
table A for the southern gal. hemisphere :

1) Where in this paper we speak of nebulae proper or nebulae without more,
we mean the relatively regular and well-defined nebulae (“white nebulae’), while
with diffused nebulae are meant the extensive, formless and according to their
spectra gaseous masses (“green nebulae’).

(122 )
1 ACB Dy iV,

Mean density of nebulae in the southern gal. hemisphere,
according to STRATONOFF.
(A including, B omitting the Nubeculae).

A B

B gal. — 85° 19 19
Sas 24 24
65 29 29
055 36 36

— 45 25 26

— 35 22 31
AE 19 19

Pasay 10 10

pes 4 4

If we consider what has been said above on the incompleteness
of the observations also in the zone between O° and — 20° gal.
latitude, little remains of the systematic increase as far as about
— 60° which, according to Straronorr’s table A, seems to exist.

One more remark. An eminent observer as JoHN HERSCHEL,
for the very reason that he naturally avoided the South Pole in
his “sweeps”, is sure to have accounted for the incompleteness arising
from that circumstance and yet he says emphatically (Outlines, Ed.
1851, pg. 596): “In the southern hemisphere a much greater uni-
formity of distribution prevails, and with exception of two very
remarkable centers of accumulation, called the MagrLianic C1oups,
there is no very decided tendency to their assemblage in any parti-
cular region.”

We have, however, a means to find indirectly whether the real
distribution of the nebulae with regard to the galactic plane is in the
main symmetrical, and consequently that the greater incompleteness
of the observations in the southern hemisphere is the cause that the

tables do not show a similar progression towards the galactic South»

Pole as they do towards the galactic North Pole.

The galactic equator and the celestial equator form a_ large
angle: 60°. A considerable segment of the northern galactic region
occurs south of the celestial equator — hence within the less thoroughly
investigated portion of the sky —,; on the other hand an equally
large segment of the southern galactic hemisphere lies within the
well-investigated region nerth of the equator. The strong influence of
the incompleteness of the observations (the real distribution of the

a . -F

ss

{ 123 )

nebulae being about the same in the two galactic hemispheres) would
then reveal itself in those two segments, so that the southern galactic
segment B (lying on the northern hemisphere) is found to be richer
in nebulae than the northern galactic segment A (lying on the
southern hemisphere), and the difference would be most marked in
the small and faint nebulae.

i Nie ia cd Oe Se ie

Nebulae in the segments between the celestial equator and the
galactic equator.

A, segm. north, gal. hemisphere 6, segm. south. gal. hemisphere

(southern hemisphere) (northern hemisphere)
pvoinigemenalste: at. 924s ROAM Mc to Roe Ms. ak 6S a
pabipnentiac. a. >. 12) Code e oS ey. ee. 1045

In table V the difference meant is indeed very great for the faint
nebulae; it is remarkable, however, that for the bright ones there
is a large difference in the opposite sense. This raises the surmise
that the possible influence mentioned above does not play a prepon-
derating part.

If we compare the structure of segment A with that of the remaining
part of the northern gal. hemisphere it appears that the density of
nebulae in the segment is only 0.6 — which is by no means surprising

as it borders upon the galactic plane — but it is very remarkable

that the proportion ae in the segment, 4.96, is exactly the same

in the remaining part of the northern gal. hemisphere, viz. se == 4:95;

The relation in the segment 5, the southern galactic segment, is

on the other hand quite different: — = 14.7. This points very
oht

markedly to a surplus of faint nebulae in the southern galactic
segment situated on the northern hemisphere, and this circumstance
together with the presence of the Nubeculae and their relation with
the accumulations of nebulae in the southern gal. hemisphere makes
it very probable that the structure of the southern galactic sky with
regard to the nebulae differs entirely from that of the northern
galactic sky.

In the very improbable case that by a more accurate survey the
northern galactic segment would be enriched so considerably by faint
nebulae that the proportion of the numbers of bright and faint

( 124 )

nebulae agrees more or less with that in segments B, i.e. 1:15,
we should count there 2500 nebulae against 3200 in the remaining */,
of the northern galactic hemisphere, and the increase towards the
Pole, at least for the faint nebulae, would then almost disappear in
the northern galactic hemisphere.

But whether the number of faint nebulae remains 754 or increases
to 2500, in either case a symmetrical increase is out of the question
and hence there is no reason why we should adopt CLEVELAND ABBR’S
prolate ellipsoid which moreover was not very probable.

Several considerations plead against the view that the nebulae
must be considered as distant galactic systems; the most important,
which has been expressed half a century ago, is the occurring together,
in most eases even in streams, of nebulae and stars, and also the
existence of stars and nebulae in the Nubeculae; (it is obvious that in
such cases we have to do with objects of the same order of magnitude).
Moreover we have the well-established fact that a star may pass into
a nebula. (Comp. Vatentiner, Handworterbuch d. Astron. III, 2,
p. 524; also Mounroy, Astrophys. Journal XI, 2, ScHAEBERLE, Nature,
Vol 69, No 1785, Srenicer, Abh. bayer. Akad. XIX, p. 572).
Should on the other hand a “distant galactic system’ be visible to
us it ean only appear to us as a nebula. But the scarcity of nebulae
in the galactic region, if that phenomenon is real, points to an un-
mistakable organic connection between the great mass of the nebulae
and our system of stars.

If then we may accept that the basis on which CLEVELAND ABBE
built his theory of the ellipsoidal figure does not hold and if moreover
we need not consider the nebulae as being situated at enormously vast
distances on either side of the Milky Way, but if, on the contrary,
it is far more likely that these distances are comparable with those
of the stars, it becomes probable that the greater part of the nebulae
are contained in a space similar to the oblate spheroid in which
SretigeR places our whole stellar system, in other words: we may

begin by adopting that the great mass of the nebulae belongs to .

our stellar system and that they are asymmetrically scattered on
either side of the chief plane.

Moreover, if we supposed that the great mass of the nebulae were
systems outside our own, the problem would not be capable of further
development. We must therefore not start from such an hypothesis.

a ee ee

(125 )

Astronomy. — “The nebulae considered in relation to the galactic
system.” By C. Easton. (Communicated by Prof. H. G. VAN DE
SANDE BAKHUYZEN.

If we consider the nebulae as forming part of our galactic system
(comp. my previous paper “On the apparent distribution of nebulae’)
their distribution must be considered in connection with that of the
other classes of objects in this system. And then, speaking generally,
we not only find a contrast between the apparent distribution of the
stars (and clusters) and the nebulae, but also between the distribution
of the large diffused nebulae (which, as far as we know, occur almost
exclusively in the region of the Milky Way) and that of the nebulae
proper; on the other hand it is probable that the galactic agglome-
rations for the greater part consist of stars of the first spectral type.

Starting from the consideration that the nebular and the star-like
conditions are phases in the development of matter and no invariable
final phases, it is obvious that as the distribution of the star-like
matter in some parts of the system differs from that in the others
— which is accompanied by a different constitution of most of the
centers themselves, as appears from the spectral differences — so the
distribution and the constitution of the matter which exists in the
nebular condition will not be the same throughout the system.

In the galactic region of the system we find a great number of
star-like objects probably placed for the greater part at (relatively)
small distances from each other, mostly of the first spectral type.
In the “extra-galactic” portions of the system we observe a smaller
number of suns, separated by vast distances and belonging for a
great part to the second spectral type.

In the same way we find in and outside the galactic region proper
two forms of nebulae. In it are found the “green” nebulae with a
spectrum formed of lines, sometimes round and fairly well-defined
(planetary nebulae), but mostly extending over immense regions which
they cover as with a veil (large diffused nebulae). In the extra-
galactic regions we find the ‘white’ nebulae with a continuous or
— more probably — a mixed spectrum; isolated and_ generally
widely separated objects, probably as a rule of a spiral form, and
certainly more condensed than those of the other kind.

As little as we can sharply distinguish between star-like and
nebular objects, is the above mentioned distinction intended as a
precise classification. But thus taken in connection with the “galacto-
phily” of the “green” nebulae, the “galactophoby” of the nebulae
proper does not appear as something exceptional; this principal

( 126 )

peculiarity of the distribution of the different kinds of nebulae may
then be considered as a natural sequence of the same cause as
in the agglomerations of the Galaxy has given rise to a type of stars
differing in constitution and distribution from those outside the
Milky Way.

For the investigation of the problem mentioned at the head of
this paper we must take into consideration :

a. The place of the sun in the solar system.

b. The great difficulty to detect a nebula as compared with a star
at the same distance from us which has the same quantity of matter.

Further I adopt as established: ~

1. That besides a gradual increase in the number of stars towards
the galactic plane there exist real agglomerations in the galactic region.

2. That the sun is situated in a region of the Milky Way which
is relatively poor in stars, hence as to the type of distribution in a
region that must be considered as “extra-galactic’, or perhaps in a
transition layer. (Comp. Kaprryy, Vers/. A. A. v. W. 1892/93, Publ.
Groningen n° 11, p. 32; Easton, Astr. Nachr. 3270, SEELIGER, Betracht.
p. 627; Newcoms, The Stars, Chap. XX).

As probable:

3. That the system of stars is contained within a spheroid with
the galactic plane as its principal plane (SEELIGER).

And from my own investigations (Versl. K. A. v. W. 1897/98;
Astrophys. Journal XU, 2; Verhand. K. A. v. W. VIII, 3) it seems
to follow :

4. That in the Cygnus-Aquila region of the Milky Way the
preceding branch is much nearer to us than the following.

5. That the brightest galactic portion in Cygnus occupies almost
a central position in the system of the stars.

Now, while discussing the distribution of the nebulae catalogued after
visual observation, [ shall try to give an explanation founded on
the supposition that, the dim light of the nebulae in general taken

into consideration, the very distant nebulae escape visual observa--

tion (though they for the greater part perhaps may be registered photo-
graphically), and hence that, according to what has been said above
especially sub 4, the very distant part of the system in which the
Aquila branch of the Galaxy is situated, must be disregarded.

If the sun lies about in the central plane of the system in a poor
region amidst galactic agglomerations, and if that poor region is
comparable with the ‘“extra-galactic’ regions on either side of the
galactic plane, the thus connected “extra-galactic” regions acquire more

or less the shape of dumb-bells (comp. fig. 1, A), the sun lying in
its bar n—z. If the nebulae proper are in the main confined to that
extra-galactic region, the galactophoby of these nebulae and their
accumulation towards the two Poles would be explained by it.

If, however, we leave the Aquila branch of the Galaxy *) out of
consideration, we must bear in mind that in the remaining and
nearer region of the Milky Way the galactic agglomerations on the Cygnus
side have a northern galactic latitude; i.e. that the sun now lies
south of the plane passing through those agglomerations (C) and
the opposite ones in Argo, N; hence it is no longer in the middle
but at the bottom of the bar of the dumb-bells, and consequently
the apparent crowding of the nebulae towards the North galactic

1) Comp. Fig. 1, B. The section of the Aquila branch is A, that of the Cygnus
agglomeration is C, that of the Galaxy in Argo is N. The circle round the sun is
the area within which are situated all nebulae of which the luminosity exceeds
a certain minimum. The region ¢r is considered as a transition region between the
(dotted) agglomerations of stars in the galactic plane and the extra-galactic
region contained within the spheroid. The bar of the dumb-bells is represented
much thicker.

( 128 )

Pole will increase and that towards the South galactic Pole will
diminish or disappear. From the circumstance that the nearest
galactic region in the direction of XIX h. (towards C) has a northern
galactic latitude it moreover follows that the maximum of nebulae of
the northern gal. hemisphere lies within more than 90 degs. of the
galactic equator in a great circle passing through C and M, whereas
in the southern galactic hemisphere the maximum of nebulae is
situated on the Cygnus side of the Pole (towards m,).

This theoretical consideration has been tested by observation in
the following way. On the maps for the distribution of the nebulae
by Srratonorr (Publ. Tachkent 2, pl. 18 and 19) we have estimated
in a zone 15° wide in the direction of the galactic meridian of
45°, which crosses the Cygnus region the densities in areas from
10 to 10. degrees galactic latitude. The result is given in table VI.
(For tables I—V see the paper “On the apparent distribution of
nebulae’).

The much. smaller densities of the southern galactic hemisphere
could be expected owing to the greater incompleteness of the obser-
vations; for the rest the displacements of the maxima from the Poles
are in the same sense as they should be according to theory. In
how far the hypotheses made are thus supported cannot be decided
because of the incompleteness of the data, at any rate they are not
at variance with the results of the observations.

If, with the now available data, particulars are to be detected
concerning the true distribution of the nebulae, we shall have to
look for traces of them in those regions where the galactic and
the extra-galactic regions meet. In a mean galactic latitude we shall
have to search for the great irregularities in the apparent distribution
of the nebulae and compare the places where they oceur with the
places where the irregularities in the distribution of the stars are
found, especially the irregularities in brightness and in width of the
diffused light of the Milky Way.

To this end the following tables have been constructed (VII, VIII, LX).

Over Srratonorr’s maps of the distribution of the bright (nm. bril-

lantes) and faint (n. faibles) nebulae, a galactic system of coordinates
was laid. For each area of 15° in gal. longitude and 10° in gal.
latitude we have then estimated as carefully as possible the density
of bright and faint nebulae. These values were combined with due
regard to the unequal surfaces of the areas in order to obtain the
mean densities for zones of 15° gal. longitude between 0° and 50°
gal. latitude on either side of the galactic equator (table VII). For
the galactic zone proper, between + 10° and — 10° gal. latitude,

4

EEE

( 129 )

TABLE VI.

Density of the nebulae in the direction of the galactic meridian of 45°.

Northern galactic hemisphere. Southern galactic hemisphere.
a gal. @ A Neb. 2 gal. 2 A Neb.
45 + BPs ee «Dib A220, 3 — Digby tre, OF
15 3.5 15 0.2
25 8.0 25 3.7
35 13.8 35 5.7
45 12.5 45 9.3
55 11.8 55 9.2
65 18.2 65 4.2
75 21.0 75 9.5
ADEEEeenSOe 2 ot 41.0 225 5} Se ao. teull
Pome eGo nr. . 43:0 45 85 10.3
75 40.3 ja 12.0
65 22.2 I 65 11.8
55 17.8 55 16.5
45 13.0 45 15.0
35 11.4 35 11.3
25 12.0 25 5.8
15 6.0 15 3.3
225 + Dr vcs wey MURS 450 — 5 1.9

we have combined in table VIII the regions north and south of
the gal. equator, as it was of no use to consider these regions
separately. On the construction of the two tables the compensation
method used by Srratonorr has probably influence, but in my
opinion it is of no consequence that the details of the distribution
are more or less obliterated and hence the contrasts are less distinct.
For an investigation of the main features this is rather an advantage;
the method is only inconvenient with the Nubeculae. In these two
tables the data relating to the southern hemisphere have been added,
=
Proceedings Royal Acad. Amsterdam. Vol. VII.

( 130 )

TABLE VIL

Density of nebulae in the zones 0° to 50? galactic latitude.

A. Bright nebulae.

(Values of the upper line north, values of the lower line south of the gal. equator).

«
3

4 = eS

S S Ss = Sg = D 1D a o =) ° os
oD cD oD MM a i) se ae oa © oO ron)
Copa ene 12.1/9.2| is ie 6.3/4.1}3.1|3.7 ali 4 5/8.9 BPE EE |: 1.9}1.7 |

| | | |
s7laclealesis 3 saleoleadioola 5|5.2/3.8,9.0)2.7|2.4/2.5|7 s| 8.8/4.1]4.7/ 4.2/3.1] 4.5/3.7)

B. Faint nebulae.

(Values of the upper line north, values of the lower line south of the gal. equator).

Sis a a A 4.5|8.2/6.6|6.5| 7.2|6.8/7.214.9/4.5| 4.5] 5.2) 4.0/8.6 |8.7/6.4)6.9| 7.1
Fels |

'3.7\2.513.0\2.5 !o.4|1.6/1.8|10.0/4.1|1.9)2.4/4.6/9.47.6/4.5|5.8|15.5

llr algole. slaalaalo 5.2

TABLE VIII.

Density of nebulae of the galactic-equatorial areas (— 10 to + 10° gal. latitude).

A. Bright nebulae.
}1.9]4.1|2.0)1.7|1.9|4.4/3.1|4.1]4.4|2.7|3.4|1.6|4.3/2.0)2.4|1.1|3.1|3.4|2.1|2.5|2.2/2.8/2.4/ 2.2]

B. Faint nebulae.

|1.1/8.0]2.8|1.2|0.9|0.91.3/1.4/0.4|1.0|1.5|1.8|3.2|2.6|1.9/2.4/8.5]2. als 2|2. 08S
S & s S nS x a to s 3
ee) 3 3S S ° S 3 =

TABLE IX.

Intensity of the light of the Galaay in gal. equat. areas, — 14° to + 14° gal. latitude,

(Values of the upper line north, values of the lower line south of the gal. equator).

|
0.97 | 1.10

1.20 | 1.89 | 1.86 | 1.07 | 1.37) 1.19
Ne) =) Ne} So 212 2
~ Ne) a oO os °

| }
1.14 | 1.29 | J.19 | 0.63 | 0.57 | 0.64 | 0.80 | 1.14 | 1.34 | 1.74

0.82 | 1.00 | 0.37 | 0.67 | 0.81 1.14
eo S 3 8 = 8 S
ri to | r= bo ct re

( 131 )

but only for completeness, for especially for the faint nebulae these
data have but litthe value and especially from the curves of table VIII
it may be easily seen that the material relating to the southern hemi-
sphere cannot be compared with that for the northern hemisphere.

Table 1X gives for areas of 15° gal. longitude and 28° gal. latitude,
14 degrees on either side of the gal. equator, the brightness of the
light of the Milky Way expressed in terms of the mean brightness
of the northern hemisphere derived from the table of p. 18—19 of

Fig. 2.

Monoc.

Graphical representation of table IX.

( 132 )

La distribution de la lumitre galactique by the author, (1903); there-
for we had only to accept that the diffused light of the Milky Way
in the middle areas (—2° to 4+ 2° gal. 8) is equally distributed.

A detailed discussion of these tables and of the curves which
are constructed by. means of them (fig. 2) would lead us too far.
We shall only draw the attention to the following points :

The general smoothness and the small values of the ordinates of
the curve belonging to table VIII B, southern hemisphere, can cer-
tainly not lead to the conclusion that the nebulae in the southern
galaxy proper are indeed much sparser and more uniformly scattered.
It is remarkable that about 110° gal. longitude, even within the
galaxy proper the density of faint nebulae nearly equals their average
number in the entire northern hemisphere (8.5 and 10.0).

If we consider the general shape of the curves for the density of
the nebulae we perceive a certain contrast with the curves of table
IX for the galactic light as compared with the minimum of IX the
maximum of VII seems somewhat displaced towards 90° — but in
the details no complementary shape can be detected.

The most remarkable feature of the nebulaecurves is a strong
maximum about 100° to 110° in the northern and in the southern
hemisphere, which seems to find its counterpart, at least in the
northern gal. hemisphere, 180 degrees further on at about 280°.

If in the neighbourhood of the galactic zone the space occupied
by nebulae extended equally far in all directions from the sun
(which, with the suppositions we have made, would mean that
the nearest agglomerations of stars in that plane do not lie at greatly
varying distances) and if within that space the distribution of those
nebulae were almost uniform, there would be no reason why the
eurves of VII and VIII show considerable maxima and minima.

We know, however, that the nebulae show a strong tendency to
oceur in “streams” and ‘nests’, hence in the details their distribution
must be very irregular. Consequently the peculiar positions of the
principal maxima of the curves might be explained by accepting one
or more nebular streams, running from 100° to 280° gal. longitude
somewhat obliquely towards the galactic plane. Obviously this
supposition is arbitrary.

Another acceptable explanation is that the region of the nebulae,
viz. the extra-galactic region, extends in and near the galactic plane
farther in the direction 100° towards 280° than in other directions;
in other words: that in the direction mentioned the galactic aggrega-
tions of stars are lying at greater distances.

It deserves attention that the line which in space connects the

(453 )

points represented by the greatest maxima of the nebulaecurve
(table VID), (excluding the maximum at 280° in the southern hemisphere,
due to the Nubeculae) is almost at right angles to the direction of
the great agglomeration of stars in Cygnus (comp. table IX, 30° to

45° north). If, as it has been argued in Ap. J. XII, 2 — comp.
Versl. K. A. v. W. 1897/98 — the agglomeration of stars in the

direction of Cygnus forms the center of the mainly spirally arranged
galactic system, of which the unequally dense windings surround
the sun, this result (if we also accept the other suppositions mentioned
in this paper) was to be expected.

For then (comp. fig. 3) the poorer region between the principal

Fig. 3 eee

star-windings in the galactic plane — which, as regards the type
of condensation of the matter, has been identified with the “extra-
galatic’”’ regions (rich in nebulae proper) extending on either side
of the galactic plane — will extend farthest out in the direction
AB, at vight angles to a line passing through the sun and the central
condensation of the system. This will especially be the case at 90 degs.
from the central condensation or rather a little nearer to it in the
direction @ where the great gap between the windings (Perseus

( 134 )

region) is to be found. And indeed we find in table VII fig. 2 the
best marked maxima of the nebulaecurves at about 105° gal. longi-
tude at less than 90 degs. from the Cygnus region and a less strongly
pronounced one in the northern hemisphere at 280°.

The available material is certainly not sufficient for us to decide
with any probability whether the secondary maxima (at 165° ete.)
of the curves of the tables VIL and VIII result indeed from the
arrangement of the galactic agglomerations or whether they are
produced by a merely local accumulation of nebulae. Among such
“Joeal deviations” from the uniform distribution we ought then also
to reckon the Nubeculae, which apparently have such a great in-
fluence on the distribution of the nebulae in the southern galactic
hemisphere. It should be borne in mind that the Nubeculae are not
connected by streams of nebulae with the southern Milky Way and
neither probably by streams of stars. Nor is the influence of the
vast nest of nebulae, which constitutes the Nubecula Maior percep-
tible in table VIII.

Finally, if the nebulae in the very distant regions of the system
remain in general invisible and hence are not included in the statis-
tical data given here,‘ while they can be more easily photographed,
this would in connection with our preceding remarks explain the
fact observed by Max Wo xr (Sitewnysber. Miinchen XXX1, II, p. 126)
that the mass of the very faint nebulae photographed by him are
scattered more uniformly over the sky than those observed visually.

Physics. — “Dispersion bands in absorption spectra.” By Prof.
W. H. Juxius.

(Communicated in the meeting of May 28, 1904).

The appearance of absorption lines depends on various circum-
stances. As to the absorption phenomena in gases and vapours, such
conditions as temperature, density, pressure, velocity in the line of
sight, intensity and direction of magnetic field, have been fully
studied and discussed. In the present paper we purpose to show
that anomalous dispersion in the absorbing gas is also, to a great
extent, accountable for certain typical features of the dark lines.

An originally parallel beam of light, when passing through a
mass of matter, the density of which is unequally distributed, will not

( 135 )

remain parallel and, generally speaking, the greatest incurvations
will be noticed in those rays, for which the medium has refractive
indices differing most from unity, i.e. in those which, in the spectrum,
lie closest to the absorption lines on either side. These particular
kinds of light, while diverging into space, will spread in many more
different directions than the average waves, and, as a rule, a smaller
portion of them will fall into the spectroscope, than of waves with
refractive indices nearer to unity.

Accordingly, there must always be certain places in the absorp-
tion spectrum, from which light is absent owing to dispersion in
the absorbing vapour, for it may be taken for granted that the latter
is never absolutely homogeneous. These darker parts in the spectrum
we shall call dispersion bands. It stands to reason that these bands
will overlap the regions of real absorption; so they might easily be
mistaken for strengthened absorption lines, which no doubt has

-often been done.

We will now look somewhat closer into the characteristics
by which dispersion bands may be distinguished from absorption
bands.

The curvature of a ray of light of a definite wave-length, at any
point of a non-homogeneous medium, not only depends on the gradient
of optical density at that particular spot, but also on the angle
which the beam makes with the levels of equal density. Its diver-
gence will be greatest when this angle is zero.

Strong ray-curving through anomalous dispersion in vapours may,
therefore, be artificially produced in two ways: first, by using masses
of absorbing vapour, presenting in a small space considerable dif-
ferences in density, such as e.g. occur in the electric arc‘); secondly
in larger spaces where the density varies but moderately, by making the
light travel over a considerable distance under small angles with
the levels of equal density.

I have chosen the latter method of investigation, especially on
account of the extensive use, which may be made of the phenomena
presenting themselves, by applying them to the interpretation of
numerous peculiarities of the spectra of celestial bodies *).

The absorbing medium was a Bunsen flame, of a peculiar shape,
containing sodium vapour and so arranged, that the introduction of
the salt could be easily regulated.

1) H. Epert, Wirkung der anomalen Dispersion von Metalldimpfen, Bourzmann
Festschrift, S. 443.

( 136 )

Fig. I represents a section
of the burner. A is a cop-
per trough, 80 eM. long,
8 cM. wide and 5 eM. deep,
thickly coated with varnish
and having a broad flange.
The planed brass plate 6
is firmly screwed upon
the flange and a leather
packing makes the joint
air-tight. On this cover,
which has a rectangular opening 75 ¢.M. long and 2 eM. wide, are
fixed two brass rulers, C and C’, 75 eM. long. They are so adjusted
that at O they form a slit, having an exactly uniform width of about
0,1 eM. over the whole length. The prismatic space between C and
C" is closed at each end by a small triangular brass plate. The
trough is filled to a certain height with a saturated solution of soda,
and into the remaining space a mixture of illuminating gas and air
is conveyed by means of tubes, entering at both ends. These tubes
are fed from a mixing bottle in which the gas and the air are being
driven through two separate regulating taps.

If now the flame were left to burn without any further precautions,
the slit O would soon be closed in consequence of the onesided
heating of the rulers. It was therefore found necessary to place the
trough in a vessel with running water, reaching up to the burner.
In this way a uniform and steady flame was obtained.

A few millimeters below the level of the salt solution a platinum
wire P is stretched over the whole length of the lamp. Its ends are
soldered to insulated copper wires, which pass through the walls of
the trough, and are connected to the negative pole of a storage
battery of 20 volts. From the positive pole two insulated wires
lead to the ends of a long strip of platinum P’, which rests on a
glass plate at the bottom of the trough. As soon as the circuit is
closed, innumerable minute particles of the fluid rise into the space

1) The abnormal solar spectrum of Hate; the peculiar distribution of light in
several of the Fraunnorer lines, even in normal conditions; the variations in the
average appearance of the spot spectrum accompanying the eleven year period,
all these phenomena have been easily explained from the considerations here
alluded to (See W. H. Jurius, Proc. Roy. Acad. Amst. IV, p. 589—602; 662—666;
V, p. 270—302).

The present investigation is a continuation of the experiments with the long
sodium flame, a short account of which has already been given on those former
occasions in support of our theory,

EEE ——— ve

( 437 )

R, and cause the flame to emit a beautiful, clear and constant sodium
light, the intensity of which can be controlled and regulated by means
of an ampéremeter and a variable resistance.

L

L

In Fig. 2, @ and 4 are shown two dif-

ferent ways in which the light travels
through this long sodium flame. L repre-
sents the crater of an electric are of 20
amperes. The lens A throws an image of
the crater on the slit .S,, which, in its turn,
is depicted by the lens 4 on the slit S, of

a grating spectroscope. About half of the
conical beam of light which leaves A is
intercepted by the screen P, and the part,
which the slit S, allows to pass, falls almost
entirely on the screen Q, which has been
shifted so close to the optical axis of both
lenses, that only a narrow streak of light
can reach the slit JS,, through the middle of
4. The large gas burner stands on a hori-
zontal slide, which is movable up and down
and round a vertical axis; thus, by means
of screws, if can easily be put in any
position required.

When the axis of the flame (which we
assume to be in its most luminous part,
i.e. a little above the blue-green core) coin-
cides with the optical axis of the system
of lenses, both the JD-lines will be seen
symmetrically widened in the spectroscope.
If not perfect, the symmetry will easily be
corrected by slightly shifting the screens
P and Q.

Fig. 2 a. Fig. 2 b. Picture 1 Fig. 3 (see Plate) refers to the
case when the flame NV is not burning; the narrow absorption lines
are due to traces of sodium surrounding the carbon points. When
the flame is burning, a very weak current passing through the sodium
solution will produce the effect shown in 2. The photographs 3, 4
and 5 were obtained with currents of about 1, 3 and 6 amperes,
the flame always being in the symmetrical position.

We will now examine the case represented by Fig. 2, a. Here
the axis of the flame has been shifted 3 m.M. towards the right.
The narrow beam of light which reaches S,, only penetrates that

( 138 )

part of the flame, where the density of the sodium vapour zncreases
from left to right. In a strueture of this kind, waves, for which the
vapour has a great index of refraction, deviate towards the right,
e.g. S,G. They are not intercepted by Q and consequently reach the
slit S,. In fact, the presence of the sodium vapour allows similar
waves to enter that slit even in larger quantity than they would do
without it, for rays of this kind, issuing from the uncovered half
of A, which if travelling in a straight line would be intercepted by
( can, when refracted, penetrate the lens B.

The ease is entirely different for those kinds of rays for which
sodium vapour has refractive indices that are smaller than unity.
Such rays deviating towards the left (as shown in S, A), are intercepted
by @ and consequently will be absent from the spectrum.

Nos 6, 8 and 10 are reproductions of photographs taken under
these conditions. On the left are seen the smaller, on the right the
greater wave-lengths (in fact, in the whole series of photographs the
stronger D-line appears on the left side); so it is obvious that really
the waves lying on the red-facing side of the D-lines, i.e. those for
which the vapour has high refractive indices, are strengthened by
anomalous dispersion; and that, on the other hand, the waves on
the violet side have been considerably weakened.

Alternately with 6, 8 and J0 the photographs 7, 9 and 11 were
taken. The position of the flame was now as indicated in Fig. 2, 4,
i. e. its axis had been shifted 3 mM. to the left, so that the central
beam had to traverse that part of the flame where the density of
the sodium vapour decreases from left to right. Here we notice that
the rays with low refractive indices deviate towards the right and
that a larger number of them reach the slit .S,, e.g. S.A, whilst the
rays with high refractive indices, such as S,G, are intercepted by Q.
“ Nos 6 to 14 show the effect of a gradual increase in the density
of the sodium vapour. In No. 12 we again notice the sharply defined
sodium lines after the flame has been extinguished at the end of
the series of experiments; they are somewhat stronger than those
at the beginning of the series, because much sodium vapour had.
spread through the room during the operations.

When carefully examining the original negatives it is possible in
most of them to distinguish the rather sharp central absorption lines
from the overlaying dispersion bands (especially in the photographs,
obtained when the position of the flame was symmetrical; the repro-
ductions fail to bring out this peculiarity). Advantage has been taken
of this fact in so arranging the twelve photographs here reproduced,
that equal wave-lengths occupy corresponding places. Then it is seen

ae

W. H. JULIUS: “Dispersion bands in absorption spectra.”

9

10

Ut

Proceediigs Royal Acad. Amsterdam. Vol. VII.

Fig. 4.

{ 239 )

that the “centres of gravity” of the two dark bands, as well as the
brighter space between them, have been alternately shifted to the left
and to the right a phenomenon which needs no further explanation.

As a matter of course the interposed flame causes the illumination
in the plane of the slit S, to be very irregular, especially with
regard to those radiations undergoing anomalous dispersion in the
vapour. It is evident that some kinds of rays which are absent
from one part of that plane, will be found in excess at another.
The distribution of light in this irregular field of radiation might be
explored by moving S,, together with the spectroscope, within it. The
same object can be obtained with less trouble by means of a thick piece
of plate glass, mounted vertically between 6 and S, in such a manner,
that it may be moved round a vertical axis. When turning it a little
we make the whole radiation-field beyond the plate glass shift parallel
to itself, thus causing other parts to cover the slit. This influences
the aspect of the dispersion bands very materially. In certain positions
apparent emission lines of sodium vapour may happen to be seen,
which disappear as soon as the are-light at S, is intercepted *).

In conclusion we wish to draw attention to a peculiarity we
repeatedly observed in the dispersion bands. The dark shading in a
dispersion band does not become deeper in proportion as we approach
nearer to the central absorption line, but seems to reach its maximum
obscurity at certain (though not always equal) distances on both
sides of the centre; whilst in the space between, the light appears
somewhat intensified just as if a wide absorption band had been
partly covered by a narrower emission band, the centre of which
is again occupied by the fine absorption line. This phenomenon
cannot, however, be attributed to radiation emitted by the absorbing
sodium flame; for in our arrangement the intensity of the emission
from the flame could bear no comparison with that of the are for
corresponding waves. In order to make sure we tried to photograph
the emission spectrum of the flame, exposing the plate during the
same leneth of time and under the same conditions as had been
done for obtaining the absorption spectrum; but not a trace of any
impression could be detected on the photographic plate.

The light on both sides of the central line therefore originates
in the carbon points and this we explain on the principle of ray-
eurving. The kinds of rays which are most strongly refracted in the
flame may, under certain conditions, be curved twice or even more

1) These bright lines originate in the same manner as the light of the chromo-
sphere. The chromospheric lines are not emission lines, but “bright dispersion bands”,

( 140 )

times, when passing nearly parallel to the system of the levels of
equal density (in the manner described on a former occasion')) and
will therefore have a greater chance of reaching the slit S,, than
rays which are less strongly curved. The relative intensity with
which the waves, belonging to those central parts of the dispersion
bands, appear in the spectrum increases with the distance over which
the light has travelled along such a lamellar or tubular structure.
Should the true absorption line happen to be exceedingly narrow,
the dispersion band may give the impression of a double absorption
band, which need not be symmetrical *).

We hold that the dispersion bands play an important part in many
of the well known spectral phenomena, such as the widening, shifting,
reversal and doubling of lines. In a subsequent communication I
purpose to examine from this premise various phenomena observed
in the spectra of variable stars and other celestial bodies.

Physics. — “Spectroheliographic results explained by anomalous
dispersion.” By Prot. W. H. Juris.

It is not surprising that the scientific world should be highly interested
in the beautiful results, obtained by Hate and ELLerman with the
spectroheliograph *). The brilliant method elaborated and applied by
these investigators enables us to see at a glance as well as to
study in minute details how the light of any selected wavelength
was distributed on the total solar disk at any given moment. W.S.
Lockyer, in giving an abstract from the paper here alluded to °in
Nature N°. 1800, rightly entitles it: ‘A new epoch in solar physics.”
Indeed, the spectroheliograph proves capable of providing us with an
abundance of new information, which other existing methods could
never give and the value of which will remain, whatever may
be the ideas on the Sun’s constitution derived from it.

But, nevertheless, even the most splendid collection of new facts
is useless so long as we have no theoretical ideas connecting them

with achieved knowledge. Hate and ELierman, accordingly, in

1) Proc. Roy. Acad. Amst. IV, p. 596.

2) In Fig. 4 on the plate is given an enlargement of one of the photographs
obtained by an almost symmetrical position of the flame. It has been somewhat
spoiled in the reproduction. The original is less blotchy and the transition of the
dispersion bands to the bright background of the spectrum is there much more
gradual.

5) G. E. Hae and F. Extermay, “The Rumford Spectroheliograph of the Yerkes
Observatory,” Publications of the Yerkes Obseryatory, Vol. III. Part. I, (1903).

( 144 )

describing the observed phenomena, lay down quite definite conceptions
regarding certain conditions and configurations of matter in the solar
atmosphere, by which the observed distribution of the light in the
image of the Sun is assumed to be produced. In the cited publication they
put forth the working hypothesis that the “caleium-floceuli” or bright
regions showing themselves all over the image of the Sun when if
is photographed in so-called calcium light, are columns of calcium
vapour rising above the columns of condensed vapours of which the
photospheric “grains” are the summits (1.e., p. 15). This hypothesis,
though at first proposed mainly as a guide to further research (L.e.,
p. 13), has been subsequently’) employed by the same authors with much
less restriction as the basis on which the photographs ought to be
interpreted,

The great authority of Hane and of such critics as W.S. Lockyrr,
J. Eversuep and others who, in abstracts from the work of Hae and
ELLeURMAN, concur in most of the interpretations there given, might
cause the value of those ideas to become overestimated and extended
beyond the original intention of the authors.

It is not superfluous, therefore, to show how we may quite as well
account for all the new phenomena thus far revealed by the spectro-
heliograph, if we start from the entirely different conceptions of
the Sun’s constitution, which the consequences of ray-curving in non
homogeneous media and of anomalous dispersion of light in absorbing
vapours have suggested to us.

Both these circumstances are left absolutely out of consideration
by Hatn and Enierman. Their conclusions are all founded on the
erroneous supposition that the monochromatic light by which their
images of the Sun are photographed, has travelled from the source
in straight lines, and that they are right, accordingly, in supposing
light-emitting masses of calcium vapour to exist in the exact directions,
along which calecium-radiations seem to reach us. In making this
supposition they fall into the same error as one who would assume
the refracting facets of the crystal globe of a burning lamp to be
independent sources of light.

Our new explanation of the spectroheliographic results will be founded
on the hypothesis that the Sun is an unlimited mass of gas in which
convection currents, surfaces of discontinuity and vortices are conti-
nually forming under the influence of radiation and rotation, so that
the various composing elements are mingled as completely as nitrogen

1G. KE, Hare and F. Exrerman, “Calcium and Hydrogen Flocculi,” Astro-
physical Journal XIX, p. 41—52,

( 142 )

and oxygen in the Earth’s atmosphere‘). This hypothesis too will,
of course, want modification in the light of future results; but for
the present it seems, so far as the visible phenomena are considered,
not to clash with any observation or physical law.

The irregular motion of electrons in the deeper layers of the
Sun, where the density is very great, gives rise to the radiation
with a continuous spectrum. We shall only take ‘his radiation
into account. Peculiar radiations, emitted by the more rarefied
outer parts of the gaseous body and giving a bright-line spectrum,
may perhaps add a perceptible quantity of light to the bulk, but
this selective emission, if present, does not play any part in our
explanations. So we behold the brilliant core of the Sun through
an extensive envelope, consisting of a transparent but selectively
absorbing mixture of gases, into which the core gradually spreads.
It stands to reason that the average density of this envelope slowly
decreases in the direction from Sun to Earth; but at right angles to
that direction the density must be in some places much more variable.
For it is a minimum in the axes of vortices; and the average
direction of the whirl-cores, lying between the Earth and the central
parts of the Sun in the surfaces of discontinuity, differs but little
from our line of sight. The rays of the Sun thus reach us after
having travelled a great distance along lines, making small angles
with the levels of slowest density-variation in a lamellar, partly
tubular, structure *).

Under these circumstances the solar rays will be sensibly incurvated
on their way through the envelope, especially those suffering ano-
malous dispersion. As a rule, beams consisting of the latter kinds
of rays will show an increased divergence; they will reach the
Earth with less intensity than the normally refracted light and so
will give rise to dark dispersion bands *) in the solar spectrum. And
the degree of divergence will not only be different with waves, which
in the spectrum are found at different distances from the absorption
lines, but it is also clear that the divergence with which various

beams of any dejinite kind of light arrive at the Earth must differ.

largely according to the dioptrical properties, exhibited along the

1) A sketch of a solar theory, based on this hypothesis, is to be found in the
Revue générale des Sciences, 15, p. 480—495, 30 May 1904.

2) For considerations which have induced us to hold that a similar structure of
the Sun is very probable, I refer to former publications: Proc. Roy. Acad Amst.
IV, p. 162—171; 589—602; V, p. 270—302.

5) W. H. Jutrus, Dispersion bands in absorption spectra, Proc. Roy. Acad. at Amst.
Vol. VII, p. 134.

——— ae

—

( 143 )

paths of those beams by the system of surfaces of discontinuity.

The foregoing inferences really imply the whole of our inter-
pretation of the results, thus far obtained with the spectroheliograph.
This we shall show by amply discussing some of their main features.

The broad dark bands, designated by Hane and ELimrman as H,
and Ay, are not absorption bands, but dispersion bands. Real absorp-
tion by the solar calcium vapour we hold to be restricted to the
central dark lines H, and A,. The bright bands //, and ,, predo-
minating in the spectrum of the “floceuli” and attributed by Hane
and EniurmMan to strongly radiating caletum vapour, result in our
theory from the fact, that with beams of light the wavelength of
which is very near to that of the central absorption lines, the
divergence may be diminished or even changed to convergence by
the tubular structure. Indeed, such rays deviate more strongly than
those standing farther from the absorption lines ; and as soon as they
undergo more than one incurvation, they have a chance of reaching the
Earth with increased intensity. This chance improves in proportion
as the index of refraction departs from unity, be it in a positive or
in a negative sense*). We conclude from it, that the brightness of the
calcium flocculi must, as a rule, increase as the monochromatic light
in which the Sun is photographed approaches the true absorption line.

This consequence of our theory exactly corresponds to one of the
chief peculiarities, which immediately struck Hate and ELLerman on
inspecting sets of photographs taken at short intervals of time with
the second slit in different positions within the H and K bands. In
order to account for the same fact, those investigators are obliged,
by their working hypothesis, to suppose that in higher regions of
the Sun’s atmosphere the calcium vapour radiates more strongly than
in lower levels. This cannot be called a very satisfactory inference ;
and less so, as the supposition is added that the incandescent vapour
is rising from much deeper layers and, therefore, considerably
expanding a process during which, according to our physical
notions, the temperature must fall. Here we meet with a serious
difficulty; Hate and Enieman try to get rid of it by means of the
rather vague assumption, that some electrical or chemical effect may
be responsible for the bright radiation emitted by this calcium layer,
which is intermediate between two absorbing layers *).

8) In the experimental investigation on dispersion bands, before mentioned, this
brightening in the middle of the dark bands has been distinctly observed. Cf. also:
Proc. Roy. Acad. Amst. Vol. IV, p. 596.

*) Have and Exzerman, Astrophysical Journal XIX, p. 44,

( 144 )
Our theory can dispense with such additional hypotheses.

Another characteristic peculiarity, observed in every series of
photographs taken at short intervals with the slit set at various
points on the broad H and K bands, is the following. When the
slit is set, e.g., at a remote point of K,, the structure of the solar
image appears relatively fine, sharp and detailed; approaching the
central line, we see some of the brilliant spots vanish, others grow
more extensive, especially those lying in the vicinity of sun-spots ;
at the same time their outlines become less sharp, so that finally
the whole image gives us the impression of a coarser and at the
same time a more woolly structure‘).

Hate and Exierman hold that the successive photographs refer to
gradually higher levels and conclude that the masses of calcium
vapour must have a tree-like shape. W. S. Lockyrr, in Natur
No. 1800, draws a scheme showing this conception.

Against this interpretation we propose the following one.

The amount by which the divergence of a beam of light is altered
in consequence of the presence of calcium vapour in the streaming
and whirling mass depends, of course, on the proportion of caleium
in the mixture, and besides on two other circumstances, viz. 18* on
the position occupied in the spectrum by the selected kind of light
with regard to the absorption lines, and 2" on the steepness of the
density gradients in the mixture along directions perpendicular to the
path of the beam.

Let us suppose the selected light to correspond to the extreme
edge of H, or K,, then its index of refraction differs but little from
unity. Accordingly, very considerable inequalities of density are required
to cause a perceptible change in the divergence of such beams.
Similar great inequalities may indeed occur at many separate places,
but at each of them they cannot, of course, extend very widely.
This accounts for the fine and rather sharply defined reticulation
shown by the so called “low-level” photographs.

If the second slit were set a little nearer to the centre of the line, —

the distribution of the light in the solar image would at all events
differ considerably from that of the former ease; for the indices of
refraction being very different for neighbouring waves within a dis-
persion band, the divergence of beams, starting from the same point
of the Sun, must vary largely with the wave-length. So it is clear

1) Such series of photographs ave reproduced in: Publications of the Yerkes
Obervatory, Vol, Ill, Part I, Pl. V, VI, X, XI, XII, XIII.

a ee eee

(145 )

that bright or dark spots, visible on one photograph, may be wanting
in the other.

Moreover, the general character of the image must change as we
approach the central line. For in proportion as the indices of refrac-
tion depart from unity, slower variations of density suffice for pro-
ducing sensible differences of divergence. And, as a matter of course,
in any whirling region slightly inclined density-gradients will take up
larger spaces than very steep ones. Besides, when the second slit of
the spectroheliograph, having a given width, is set near to the central
absorption line, the wave-complex which it allows to pass, covers a
greater variety of refractive indices, than when it is set farther from
the central line. In the former case the distribution of the light in
the solar image must, therefore, be less differentiated. Both circum-
stances cooperate in causing the bright-and-dark structure generally
to appear coarser and more woolly in proportion as the spectro-
heliograph is adjusted for kinds of rays that are more liable to
anomalous dispersion.

From the same point of view it is not surprising that on photo-
graphs, taken in H, or K, light, the calcium flocculi are parti-
cularly bright and extensive in spot regions, for in such regions
the “tubular” structure of the gaseous mass, by which the strongly
curved rays are kept together and conducted, is most developed.

Hate and ELiermann also mention “dark calcium flocculi’” *),
which they describe as special objects, visible in so-called “high-level
photographs” and not to be confounded with the general dark back-
ground, produced by the absorbing calcium vapour of deeper layers.
Dark floceuli often surround the large bright flocculi of spot regions,
as is shown e.g. in Fig. 4, Plate V of the cited publication. The
explanation given by them is, that we might have here some indications
of the cooler K, calcium vapour, which rises to a considerably greater
height than the K, vapour of the bright flocculi.

In our theory the presence of these darker regions is a direct
consequence of the fact, that the particular distribution of the light
in the solar image is not produced by local absorption and emission,
but by irregular ray-curving. The rays are only caused to change their
places; so an excess of light in the bright flocculi must necessarily
be counterbalanced by a deficit in the surroundings.

H and K are by far the broadest bands of the visible solar spec-
trum ; even with moderate dispersion the second slit of the spectro-
1) Publications Yerkes Observatory, l.c. p. 19.
10
Proceedings Royal Acad. Amsterdam. Vol. VII.

( 146 )

heliograph could easily be set at different points within these bands.
When the dispersion of the instrument was increased by means of
a grating, photographs of the Sun could be obtained with light falling
entirely within a widened line of hydrogen or of iron.

Photographs made with Hz or Hy, light showed also a flocky
structure, differing, however, materially from that obtained with H
and K. Hare and Eiermany therefore assume dark and bright clouds
of hydrogen to exist in the solar atmosphere. Upon the whole, but
not in the details, the hydrogen flocculi correspond in form and
position to the calcium flocculi photographed with H, or K, light ;
the general aspect of the photographs is fainter, they show less
contrast, and the detailed structure observed in H, or K, light is
wanting. The most striking fact, however, is that the bright
calcium flocculi of the H, or K, photographs are
replaced on the Hz photograph by dark struc-
tures of similar form. Only in a few places in the vicinity
of sunspots small bright hydrogen floceuli occur which coincide
with parts of bright calcium flocculi.

Hate and Enierman hardly make an attempt to explain these facts
which, in the light of their working hypothesis, are really puzzling.

We get a much clearer view of the matter as soon as we suppose
the widening of the hydrogen lines also to be produced by anomalous
dispersion, instead of by absorption only.

Indeed, the ray-curving in the solar gases must generally be less
with waves belonging to those narrower dispersion bands than with
waves lying near the centres of the broad H and K bands. Even
in the powerful whirls of spot regions there will only sporadically
be found places where the tubular structure is sufficiently marked
to keep together rays belonging to the dispersion bands of hydrogen
in the same way, as it does gather the strongly curved H, and K,
light in the large, bright calcium floceuli. Accordingly, we shall
meet with:very few places in bright calcium flocculi, where the
photographs in Hz or H, light also exhibit brilliant pots. All the

rest of the bright H, and Kk, regions correspond to those parts

of the gaseous mass where the differences of density — though not
so excessive -— are nevertheless very considerable; but whereas in

that structure the H, rays are repeatedly curved and may be made
to converge, the less strongly incurvated Hz rays will in the same
regions diverge and be dissipated in a considerable degree, thus giving
rise to dark places in the photographs. Outside the bright calcium
flocculi, finally, where the H, and K, photographs are dark in con-
sequence of increased divergence of the beams, no strong incurvation

a i a BB

EE Eee eee

( 147 )
is given to the Hs or H, light; at those places the image of the Sun,
photographed in hydrogen lines, must therefore be less dark,

The rather faint character of the hydrogen floceuli, the absence
of sharp outlines and of strong contrasts in the structural elements,
we ascribe to the dispersion bands of hydrogen being relatively
narrow and so allowing rays with a great variety of refractive indices
to pass simultaneously through the second slit of the spectrohelio-
graph. The hydrogen photographs too would show finer details, like
those in K, light, if the dispersion of the apparatus were still greater
and the second slit still narrower.

We believe that we have shown that every peculiarity, thus far
noticed in the photographs obtained with the spectroheliograph, may
easily be deduced from the same fundamental hypothesis regarding
the constitution of the Sun, which has already proved capable of
giving a coherent interpretation of the solar phenomena known
before. Not a single new hypothesis was required.

Physiology. — “On artificial and natural nerve-stimulation and the
quantity of energy involved.” By Prof. H. ZWAARDEMAKER.

A living nerve, laid bare, can be stimulated artificially in a number
of ways; there are but two kinds of stimuli, however, the effect of
which is instantaneous and whose strength can be accurately regulated,
namely mechanical and electrical stimuli. Mechanical stimulation has
been considerably improved by an artifice of ScuArer'), who used
falling drops of mercury instead of electrically driven little hammers.
When using droplets the size of which is about equal to the breadth of
the nerve, even with a very small height of fall distinct effects we
obtain, manifesting themselves by contraction of the muscle which
has remamed connected with the nerve. Scnirer himself obtained
this result with a drop weighing 100 mer. falling through 10 mm.
In our laboratory his method gave still better results; an effect was
noticed already with a drop of 50 mgr. and a height of fall of 5 mm.
Such a drop possesses at the end of its fall an energy of 24.5 ergs.
Not the energy as such is a measure of the stimulus, however.
Apparently the energy has also in this respect to be considered
as consisting of an intensity-factor and a capacity-factor *), and this

1) Proc. Physiol. Soc. 26 Jan. 1901.

2) W. Osrwatp. Ber. d. k. Siichs. Ges. d. Wissenschaften 1892, math. physik.
Ql. $. 915.

G. Heim. Die Energetik in ihrer geschichtl. Entwicklung 1898 S. 277.

10*

( 148 )

after its being transferred to the nerve. That it is not the intensity-
factor before the transfer which must be regarded here as the phy-
siologically ‘‘auslOsende” (liberating) force, appears from some curious
results obtained on varying the height of fall. A fall of 5 mm.
gives an excellent lifting distance when the muscle is loaded with
50 grams, which is reduced to */, with a fall, of 15 mm. and to’/,
with a fall of 80 mm. Consequently a smaller effect is found when
the velocity with which the drop comes down is increased, instead of
a larger effect. This would be impossible if the intensity-factor of the
kinetic energy of the falling drops had been decisive. On diminishing
the height of fall again the original lifting distance returns ‘).

The second kind of artificial stimuli can be measured in a very
simple manner if at the instance of Cyputsky, Mares, Hoorwre and
others, condensers are used. A preliminary trial in the laboratory
showed that the best results are obtained with a condenser of a
capacity of 0.004 microfarad. This has only to be charged to a
potential of 0.012 volts to show already a contraction of the nerve-
muscle preparation. The available energy in this case amounts to
0.00029 ergs, which is much less than what was found for mecha-
nical stimulation *)

But it is clear that the one as well as the other is a most unsatis-
factory way of stimulating. The mechanical stimulus only reaches
part of the axial cylinders constituting the nerve-bundle and it is
questionable whether the softness of the mass does not to a great
extent obviate the suddenness of the pressure. The electrical discharge,
although more instantaneous, spreads in no small measure, besides
over the axial cylinders, also over the sheaths and the septa between
the separate fibres. A means of diminishing the resistance and so
making the time of discharge shorter is hitherto lacking. It is always
the full resistance of the nerve, measured across, which one is obliged
fo put in.

These difficulties will never be completely overcome with artificial
stimuli; in order to find the real minimum one must have recourse

fo natural stimuli. It is only with sensory nerves that we have these

at our command for the present.

There are two organs of sense in which the nerve-cells themselves
(or their immediate prolongations) receive the stimulus, viz. the
organs of sight and of smell. In the former case it is the rods and

4) An analogous phenomenon has been observed by Wepensky for faradic
stimuli (of a frequence of 100 per second).

2) J. Cruzer. Journal de Physiologie et de Pathologie générale 1904 p. 210 gives
as smallest values 8, 5 and 7, 2 milli-ergs.

.

(149 )

cones, in the latter the fine smelling-hairs which both form part of
the terminal neuron. We will consider the stimulation of these senses
a little more closely.

A. Organ of sight.

If one looks through an artificial orifice of 2 mm. at a small
Hefner lamp at a distance of 6 metres, a definite amount of the
light emitted by the lamp will enter the eye. This energy is con-
centrated on a small circumscript field of the retina, where cones and
rods lie ready to receive the light-stimulus. When the room is made
dark and the light is made feebler by a system of more or less
erossed Nichols, the much less sensitive cones will at last cease to
be active and the feeble glimmer that remains, will be perceived by
means of the rods only. For this the visual axis will have to deviate
a little, as in the fovea centralis itself no rods occur, so that the
point-shaped feeble little star seems to be displaced a little upward,
at any rate for my eye.

It is clear that with such an arrangement of the experiment those
rays only will be effective that are absorbed by the purple of the
rods. According to A. K6niG') these are the rays the wave-length
of which ranges from 600 to 420 micra. From K. Anesrrém’s recent
determinations *) we know the energy of this part of the spectrum
for the Hefner lamp. It is 2.61 % 10 ~* gram-calories per second
and per square centim. at a distance of one metre. At a distance
of 6 metres this becomes 0.05 ergs per second and per square centim.

In this experiment we supposed the Hefner lamp to be looked at
through a system of more or less crossed Nichols, leaving only a
feeble glimmer. Besides this we will also insert an instantaneous
shutter the time of exposition of which has been adjusted so as to
give the most favourable results for feeble visual impressions. From
a series of experiments by Messrs. Grins and Noyons, the results of
which will be communicated later by these gentlemen themselves, |
knew that a time of exposition of the order of a milli-second is most
suitable. An instrument formerly used by Dr. Laan and deseribed in
his doctor-dissertation *), with slits moving in opposite directions, gave
expositions of about 0.6 milli-second, repeated every 0.06 second.

By these two means in addition to the narrowness of orifice, hence
in all by three circumstances :

1) A. Konia. Stzber. d. Berliner Akademie 1894 p. 585.
*) K. Anastrém, the Physical Review, Vol. XVIIL p. 302, 1903.
8) H. A. Laan, Onderz. Physiol. Lab, Utrechtsche Hoogeschool. (5) Ill p. 182.

( 150 )

1. crossing of the Nichols

2. shorter time of exposition

3. narrow orifice
the stimulus which otherwise would have amounted to 0.03 ergs per
second and per square centim. was still considerably enfeebled.

The Nichols were completely crossed at 28° 12' of the scale.
Hence full light was obtained at 118° 12" In this position, whichis
most favourable for the transmission of light, something is lost on
and in the Nichols as well as on and in the media of the eye, but
we shall neglect this amount, since it is small compared with the
uncertainty of the coefficient of absorption of the retinal purple, which
we will take into account presently, and of the disturbing influence
of adaptation, which cannot be entirely eliminated.

When the instantaneous shutter is moving and gives a flash every
0.06 sec. one of the Nichols is slowly turned. The sharp image of the
flame disappears and instead of it we see a dot-shaped glimmer, which
at last seems to move a little upward. A minimum glimmer I found
without previous adaptation to the dark

coming from the right at 36°. 6° of the scale

coming from the left at 20° ord ics) Ge
which means a_ rotation of the Nichols of 82°.6' and 81°.48' or ¢
mean rotation of 81°.57' reckoned from the position for full light.

On account of the crossing of the Nichols we may assume that
the intensity in the ratio of cos*. 81°.57' to cos’.0° or as 0.0196 to 1.

desides the time of exposition was only 0.6 milli-seconds every time.

Finally the artificial pupil had a surface of no more than 0.0314 em?.

By all these circumstances the original quantity of energy of 0.08
eres, contained in the rays that can be absorbed by the rods, has
been reduced to

0.0196 >< 0,0006 X 0.0314 >< 0.03 ergs = 1.1 10-5 ergs.

From the measurements of absorption by A. K6énic it follows that
‘y,, of the light of these rays is retained in the retinal purple and
since only the really absorbed light must be taken into account for
the stimulation, this amount is still further reduced to

0.02 >< 11.10 eres’ == 232) ale ercs.

In a second series of experiments with a time of exposition of
0,00062 sec. repeated every 0.64 sec., and an angle of the Nichols
of 84°42’ from the position of full light, T found

0,00854 >< 0.00062 X 0.314 0.03 ergs = 4.9 x 10-* ergs.

And '/,, part of this is again

= cued)

————

( 51.)

0,02 K 4.9 10-9 = 1 << 10-9 'Bires.

So the energy capable of causing the impression of an extremely
feeble glimmer is, in the case of eccentric vision, of the order of
1.10—” eres. This quantity would have been found: still 100 times
less if I had completely adapted myself to darkness. By a corre-
sponding method Messrs. Grins and Noyons made such experiments,
which will be published later.

The transformation of the retinal purple into retinal yellow under
the influence of light is a reversible process. Hence it must depend
on a displacement of the equilibrium, which towards the end of the
adaptation is complete, in a direction opposed to the chemical forces
which are the cause of it. The small variation of thermodynamic
potential, brought about in the appendix of the retinal nerve-cell by
a quantity of energy of the order mentioned, is sufficient to cause
a state of stimulation in this nerve-cell. Hence the natural nerve-
stimulus ean be taken ever so much smaller than the artificial stimulus,
even in its most favourable form.

B. Organ of smell.

If one wishes to make an attempt at calculating the value of the
energy of the natural stimulus in the case of the organ of smell,
this can be done in the following manner.

In the so-called smelling-box *) (a closed space of 64 litres, having
glass walls on all sides), let a smelling substance be diluted to the
utmost degree at which it is still perceptible. Let this be done by
completely evaporating a few drops of the substance itself or of an
aqueous solution of it. Let then a little air from this space be sniffed
in. The quantity of air inhaled in a single sniff is estimated by
Vanentin at 50 ce. which T believe to be correct, sinee in sniffing
at my olfacto-meter (which is done unilaterally) 80 ec. ave inhaled
on the average. If we now suppose that part of this air is directly
conveyed to the olfactory fissure, in the most favourable case the air
there present will be replaced by fresh air perfumed in the manner
indicated. In this case + 0.2 ce. of this air is in contact with the
olfactory mucous membrane on both sides of the narrow fissure.

We shall now indicate for a few substances the quantity that must
be present in 0.2 ce. in order to be exactly perceived by the smell.
This quantity is for

1) Physiologie des Geruchs. Leipzig 1895 p. 34,

( 152 )

methyl alcohol 0,000.06 mer.

formic acid 0.000.12 mer.

acetone 0.000.008 mer.
camphor 0.000.000,009.6° mer.
ionon 0.000.000.000.019 > mer.

Let us restrict ourselves to these substances for the present. When
fully oxidised they are converted into H,O and CO,. Hence their
products of combustion may be considered as entirely indifferent
additions to the cell-substance if they are produced gradually and
in small quantities. If we regard smell as an intra-molecular property,
which is probable for these substances (leaving if an open question
whether for substances containing atoms like S, As, ete. smell
may perhaps depend on intra-atomic conditions), the quantity of
energy involved in an olfactory stimulus will never exceed the heat
of combustion.

For the four first-mentioned substances the heat of combustion is
known and amounts to 5.7, 1.5, 7.5 and 9.3 gram-calories per milli-
gram respectively; that of ionon is unknown, but may be estimated
at 9.6 gram-calory. Then the quantities of energy involved here
(for the amount of smelling substance contained in 0.2 ce.) are for:

methyl-aleohol 14443. ergs

formie acid 7460 a,
acetone PSA) 5.
camphor BH)
ionon 0.008 ,,

Attempts made in order to find out how much of these smelling
substances is absorbed in the olfactory mucous membrane were
hitherto unsuccessful. Neither in oil, nor in nerve-substance we were
able to detect an appreciable quantity of ionon, after they had
been left in contact for some time with an atmosphere of ionon.
Presumably very little is absorbed. Moreover it follows from the
remarkably different degree in which cliemically related substances
show smelling power‘) that only an extremely small part of the
intra-molecular energy displays any olfacto-chemical effect. So we
are justified in assuming that here also the minimum stimulus in
ihe nerve-terminal will later appear to be of the order of the light-
stimulus or even smaller.

1) Onderzoekingen Physiol. Lab. d. Utrechtsche Hoogeschool (5) IV. p. 282.
Chloroform, bromoform, iodoform have a specific smelling-power of which the
mutual ralio is as 1:69: 153 24.

( 153 )

So of the direct natural nerve-stimuli moments only the light-stimulus
has for the present been quantitatively calewlated. Bearing in mind
that the stimulus we found in what precedes to be of the order
107" eres, is a minimum stimulus on a small field of the retina
scarcely measuring 0.002 square millimetres, whereas the total surface
containing retinal purple is put by KéxiG at 700 sq. mm. ; moreover
bearing in mind that the light of the sun at Marseilles is estimated
by Fapry') at 100.000 candles and that this enters the eye not
under an angle of 23’, under which the Hefner lamp was seen, but
from all parts of the field of view, it will be clear that the light-
stimuli of ordinary life can by no means be called immeasurably
small. Nor do they act for a single millisecond but all day long.
The energy entering the nervous system in this way will not be a
fraction of an erg but a number of ergs. It is difficult to make an
estimate in this respect as with strong light not the rods but the
cones serve as terminal apparatus and in these teleneurons only by
analogy a photochemical process is assumed, which for the rest is
unknown to us. Therefore we must refrain from such an estimate,
but at the same time it seems to us to be undoubtable that the
nervous system receives relatively large quantities of energy in a
different way from that by metabolism.

1, Compt. Rend. 7 Dev. 1903.

(August 26, 1904).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Saturday September 24, 1904.

DC

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 24 September 1904, Dl. XIILD.

ereyagp ae aa} aNp ae Se

J. D. van perk Waars: “The derivation of the formula which gives the relation between the
cencentration of coexisting phases for binary mixtures”, p. 156.

G. C. Gerrits: “On Px-curves of mixtures of acetone and ethylether and of carbontetra-
chloride and acetone at 0° C.”. (Communicated by Prof. J. D. van per Waats), p. 162. (With
one plate).

J. J. van Laan: “On the latent heat of mixing for associating solvents”. (Communicated by
Prof. H. W. Baxuuis RoozEroom), p. 174.

A. Smirs: “On the phenomena appearing when in a binary system the plaitpointcurve meets
the solubilityeurve”. 3rd Communication. (Communicated by Prof. H. W. Bakuuis Roozesoom),
p- 177. (With 2 plates).

A. F. Horteman: “The preparation of silicon and its chloride’, p. 189,

F. M. Jarcer: “On the preservation of the crystallographical symmetry in the substitution
of position isomeric derivatives of the benzene series”. (Communicated by Prof. A. P. N.
Francuimont), p. 191.

J. C. Kruyver: “Evaluation of two definite integrals”, p. 201.

C. A. J. A. Oupemans: “On Leptostroma austriacum Oud., a hitherto unknown Leptostro-
macea living on the needles of Pinus austriaca, and on Hymenopsis Typhae (Fuck.) Sace. a
hitherto insufficiently described Tuberculariacea, occurring on the withered leafsheaths of Typha
latifolia”, p. 206. (With one plate).

C. A. J. A. OupEmans: “On Sclerotiopsis pityophila (Corda) Oud., a Sphaeropsidae occurring
on the needles of Pinus silvestris”, p. 211. (With one plate).

Eve. Dvusors: “On an equivalent of the Cromer Forest-Bed in the Netherlands”. (Communicated
by Prof. K. Marri), p. 214.

H. Kamertincu Onnes and C. Zakrzewski: “Contributions to the knowledge of vay per
Waats’ /-surface. IX. The condition of coexistence of binary mixtures of normal substances
according to the law of corresponding states”, p. 222. (With 2 plates).

H. Kameriincu Onnes and C. Zakrzewski: “The determination of the conditions of coexis-
tence of vapour and liquid phases of mixtures of gases at low temperatures”, p. 233. (With one
plate).

J. Werper: “A new method of interpolation with compensation applied to the reduction of
the corrections and the rates of the standardelock of the Observatory at Leyden. Honwii 17,
determined by the observations with the transitcircle in 1903”. (Communicated by Prof. H. G.
VAN DE SANDE BakuvyzeEN), p. 241.

The following papers were read:
HAL
Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 156 )

Physics. — “The derivation of the formula which gives the relation
between the concentration of coexisting phases for binary
miatures.” By Prof. J. D. van DER Waats.

(Communicated in the meeting of June 25, 1904).
Already in my molecular theory (Cont. II, p. 10) I derived a

formula for the concentration in coexisting phases of binary mixtures.
This formula has the following form:

| db da db . da

, “ a da da > eh OH de dat

MRT lt — + MRT - — — | = | WRT 1—-4+ MRT —- —
l—« v—b al) 1—wz v—b vl,

In the case that the second phase is a rarefied gasphase, the

L

second member is simplified to MRP Us __ and we find:
— Ww
da db
ims iF 29 “de. 2
MRT 1+ = — 3 = |— wer (1)
l—w, ay a) y— iy) :

From this | have drawn the conclusion that the circumstance that
two coexisting phases have the same concentration can only oceur

5 ; 5 ; pee 5 2 Uy
for mixtures, for which a minimum value of the quantity 5, occurs,

a
and so a minimum value for the critical temperature. For the
limiting case, with exceedingly low values of 7’, the mixture for
Ay ee ‘
which — has a minimum value, would be exactly the mixture, for
x
which the value of w is the same in the two phases; but for in-
creasing values of 7’ this concentration shifts to the side of the
substance with the lowest value of the size of the molecules. (Cont.
II, p. 19 and p. 120).
Afterwards | have derived in “Ternary Systems” for equation (1)
the following equation:

z, l—« 7 dT). 1 dp:
l = eS — 0 ae
2, T du ae pk de u (2)

which also holds only approximately for the case that the second

phase is a rarefied gas-phase. For the derivation of (2) I have not

directly used the equation of state, but I have considered the well-
P LL

known formula for the vapour-pressure — /— = / —a «8s suffi-
Pk

ciently accurate for liquid volumes which are not much smaller

( 157 )

than that of the pressure of coincidence (pressure of the saturated
vapour for the wnsplit mixture).

Equation (2) however, can also be found directly from the equa-
tion of state. It was to be expected that this was possible, because
as I have shown in “The liquid state and equation of condition” !)
the formula for the vapour may be derived from this equation. If
we want to find also for the factor / the real value of about 7,
it is necessary to consider / as function of the volume. This not
only renders the derivation very complicate, but it places us before
the unsolved question: in how far is the decrease of 4 with the
volume to be ascribed to real or quasi diminution ?

I have therefore confined myself for the moment to examining
what follows for the form of (2) from the equation of state, when
6 is put independent of the volume.

We have then to reduce:

db
MRE — —

da du:

v—b v
We write for this successively :
dla ha

de du ae a db 1 da
reer ics Bena

‘je
ib b G@ aNaba fl 1\da
==h7) — = 0
I du da: als vy b? ) dx v b)/du
db (1 1 1 1) da :
Now for a _ — we may write:
der 6? v b} de :

a(v—b) \l da v+b1 db —— da 2db i 1 db
ade bde 7 “\F =

by ade v bde
Oa
is equal to

5 i av
and as according to the equation of state

ii a5}
COS!) aan 4 anne)
v
we find after some reductions :
LO a
MRT ie aoe al
da: du db b MRT 6? mig
Marmite da <8 die =
—b1 db
a See POO) ee ey ww 8)

1) These Proc. VI. p, 123.
ii

( 158 )

We may also write the second member of (3) as follows:

| diz
db v ; b? (vw—b)? db
cj Wi Oe das ke
a a“ a
a— dl 58 b d log ri 4
epee MRP A. MRE, ae
da la: v da

In order to examine the general value of the quantity which is
to be reduced, we have to distinguish two cases. The first case,
that v — > is small and p(v— >) may be neglected compared with
MRT. In this case (4) may be simplified to:

a a

d— dl—

db b RT b?
P dx < dx = aye dx

The second would hold for high pressures; then the value of
p(v—4) approaches to MRT, when v approaches to 6. In this
case (4) is simplified to:

a
db b
ae ha alee

As we assume coexistence with a rarefied gasphase we have only
to deal with the first case. In the second there would not even be
question of coexistence with a second phase. We find now for the
formula, giving the relation for the concentration of the two phases :

a a
jh dl 75
| z, ee cn He b Pe iersiecer «= (5)
ae ~MRT da da ,

1—2, ‘

db. dl ia
in. which p,, is neglected, or rather where it is cancelled by an
av

almost equal value, which would occur in the second member of
the equation given at the beginning of this paper.
Let us put:

5

uRT, Se
27 6b
and
i
ie a ane

then (5) assumes the form of:

j ee 27 1 aT dpk
= ge N-Gage peda

hee eee ee See

(159 )

2
The factor 3 is in perfect concordance with the factor which

occurs in the formula of the vapour pressure, when we put the

quantity 4 independent of the volume. | have shown before that. it

must be about doubled, when we assume variability for / — or rather
27

the factor 5 is not increased, but the assumption of the smaller

value of 4 comes to doubling the factor, when we substitute the

value of 7), for ra Without carrying out the elaborate calculations,

)
which in our case might be the consequence of assumption of the
variability of 6, 1 think to be justified in concluding to the doubling
of that factor as a sufficiently approximated value. Then we find
back exactly the same value as | had found in ‘Ternary Systems’, viz.

zw l—a«, f dT}. 1 dp,
1 B Sa (oe — Like pom fo oe (a)
l—e, w, T dz jh Gl Jf
in which formula / may put about 7.
4 Ty / ‘ dlp).
s Bx273p, », We may put for Fin
dlogT;, dlogh
de dw
Hence (6) becomes :
@ ilar, if W\ che Al lo)
ih —— = Se + re oe aod (7)
lag ty afi Te) dex b dav
db

= () Sande so

From the form (7) we derive, that only when
av

when the molecules of the mixed substances are of the same size,
the concentration of the coexisting phases is the same for the mixture
with minimum critical temperature. If the size of the molecules is
not the same v7, =. for the mixture for which

ft Nagle 1 db
i = 3 eu etc a= Toei)
A Die) ae b da
db. si ;
If qa 18 positive, as is the case for mixtures of acetone and ether
aan

(ether as second component), then «=, for a mixture for which .
dTy -, F :

eS negative. Then the concentration where wv, and a, are equal
av

and therefore also the maximum pressure in the p, a line has shifted
to the side of the component with the smallest molecule. If we

multiply both members of (8) by 7’ the shifting proves to increase

( 160)

for increasing value of 7; and so we arrive at a conclusion, to
which I came already before, viz. that the concentration of the
maximum value of p in the p, curve is sufficiently the same as
that of the mixture with minimum critical temperature only for the
very lowest values of 7. It only appears that already at ordinary
temperatures the shifting mentioned above may be rather considerable.
A consequence of this is, that the shifting between the ordinary
temperatures and 7’= 7), may be only slight. This shifting is however
the greater as the difference in the size of the molecules is the more
considerable, and as the decrease in critical.temperature takes place
the more slowly.

Now that we have found an approximate value for u',, we can
immediately derive from it an approximate value for w’,, a quantity
which must be known, if in the equation :

ra dp (en) (52) es + ear
02?) pP 1

the factor of dz, is to be considered as known. We have viz.

0% 1
— MRT ts
Gea) aa

We find then:

a a
a? — 2 (—
FP 1 b b?
qt = —— ay
i MRT da? da:*
or
re fF Cle aly
x= =

7 dir* da?

So for small vapour pressure the equation:

1 (dp 1 if eT, d?lpy,
are ) = (7,- a) | Rae yor aH oF Po
Pp \dt,) 7 v,(1—z2,) T dz, da

holds approximately.

ep

In general the quantity ——~ will be positive, and this is certainly
av,” =

so when there is a minimum value for 7); the value of the other
term may change this of course. But as a rule g”,, will be found
negative for normal substances.
f < T (dp :
In the value of the quantity 7 (F): only one of the two parts
1
of w,, oceurs, viz. — and not the other part ve
pst |

part depending on 7'is kept. In different ways the value of this quantity

So only the

4

(461 )

may be found. If is easily found from the equation, occurring in
Cont. II p. 146, slightly reduced, viz:

p , tf}
ap = Whe) tay etnjetn itn!

From this form we derive, keeping «, constant:
hg Uhr,
ve ee

i ae al 5 CN Noms,
Bea ple pe dT
aelha ak

for which we may write:
dp i Fe du'z, | d(Ux,—2, W'x,)

pol, EP a eee dT
or
dp 1 du',, du,
5 == a (v7,—a,) = alr = :
pat ch rdT
du', a elie: duty,
For re we find re ; s » and) for a we get the value
a 4 at a
if a
T, — :
T? Gp
Hence :

dT;,,
da

Ldp Ui N\
Se |) Se S(O)
jx ie all
Multiplying the second member by WRT, we find w,,. For wy,
we find then 2 terms, the first IR 7 7), representing the heat of
evaporation, when the mixture «, evaporated as an unsplit substance,
and therefore the vapour phase would have the same concentration

1

is Tr.
— (x,—#«,) — denotes the mo-
7 da,

dification, which is the consequence of the circumstance, that the
vapour phase has another concentration than the liquid phase. This
modification can be very considerable in certain cases, viz. when
x,—w, is very large. If 7; should depend linearly on x, then

dT,

Ty, +(e,—4,) = = 7), and for 7), we might write in that case,
val

as the liquid phase. The second part

denoting the components of the mixture by @ and 6: T;,= Ty,
: T (d T dpe
G2.) -- 2, %- (Cont. I, p. 155) or ( Hee zs
P a

ffi pdt

ee oe
« 2 mm
pod

Then w,, = Mara (1—a,) + Mi ry x,, and the process

of mixing in the liquid state will take place without heat of mixing.

If the graphical representation of 7), as function of w, is a curve
(Cont. Il, p. 45), lying everywhere above the tangent, which is the
a, , 4a, 2a,. \ a = ‘ dT;,,
case when +—-— is positive, then 7), -+ (7, —.7,)
Vi he b.? b, hb. 1 "1

1

is

smaller than 7%...

If we draw a tangent to the curve in the point «,, this tangent
cuts the ordinate of v, in a point which lies lower than the curve,
and the distance from that point of intersection to the curve is a
measure for the quantity of heat required for mixing the condensed
vapour with the liquid phase considered. As u”, consists of two
terms, the latter of which is only negative, when the mixing in the
liquid state is attended by absorption of heat, we are not justified
in expecting that this latent heat of mixing alone determines the

sign of wy.

Physics. — “On Pr-curves of miatures of acetone and ethyl-ether
and of carbon tetrachloride and acetone at O° C.’ By G. C.
Gerrits. (Communicated by Prof. J. D. VAN DER Waa1s).

(Communicated in the meeting of June 25, 1904).

The imperfect concordance found by Cunanus ‘) between the rela-
tion deduced by van per Waats?) in his theory between the vapour
tension over a mixture of two liquids, the molecular concentration
of the vapour and that of the liquid, induced us to take up the
investigation once more according to the same method as had been
used by Cunarus and with the same substances, acetone and ethyl-ether.
Afterwards also mixtures of carbon tetrachloride and acetone were
examined.

It had viz. appeared, that improvements might be applied to the
method of investigation.

By means of the determination of the refractivity of the vapour,
both of the simple substances and of the mixtures, the molecular
concentration of the vapour was determined by means of the law
of Bror and AraGo.

This determination of the refractivity was made according to the
method of Lord Rayiricn*) also followed by Cunakus *).

1) Gunaeus, Proefschrift, Amsterdam, 1900, blz. 47—51.

2) van per Waats, Arch. Néerl. 24, blz. 44; Continuiliit des gasf. und fliiss.
Zustandes Il, blz. 137.

8) Rayteiau, Proc. Roy. Institution, Vol. XV, Part. 1, pag. 1; Proc. Roy. Soe.
Vol. 59, blz. 201.

4) Cunagus, Proefschrift, blz. 4—6. Proe.

—— S

( 163 )

Rayieien always compares the gas, the refractivity of which he
wishes to determine, with dry air free from carbonic acid. He takes
care that the gas and the air are both in sueh circumstances that
their refractive indices are the same in these circumstances, which
may be tested by means of a Fraunhofers’ diffraction phenomenon.
This equality of refraction indices is obtained by regulating the
pressure of the two gases. With the aid of a cathetometer and the
barometer the pressure is determined, and from the two pressures
the refractivity of the gas.

This is applied to mixtures of vapours coexisting with mixtures.
The pressure of the air, the refractivity of which is compared with
that of the vapour, is regulated in such a way that the vapour and
the air are in such circumstances that their refractive indices are the
same.

From the relation between the pressures the molecular refractivity
of the vapour (or of the vapour mixture) may be determined, the
law of Bror and Araco gives then the concentration of the vapour
mixture .,.

The vapour pressure is of course, also known. The concentration
of the liquid was obtained by weighing the original quantities of
liquid brought into the apparatus, and diminishing each with the
weight of the quantity of that substance in the vapour.

The arrangement agreed on the whole with that of Cunagus. It was
a modification of one given by FrauNnnorEeR based on the interference
of light, which fell through two vertical parallel slits, after having
passed through two tubes of equal length, shut off by two plates of
plate-glass. Then it was converged by the object-glass of a telescope
and the diffraction phenomenon brought about in this way, was
observed by means of an eye-glass consisting of two cylindric lenses.
In one of the tubes mentioned the vapour is found, in the other
air or dry carbonic acid, the refractivity of which was known, and
the pressure of which was now regulated in such a way that the
diffraction phenomenon occupied the place which it would also have
taken, when the two tubes had been filled with the same gas under
the same circumstances.

The two tubes were connected with open manometers and an
apparatus to increase or decrease the pressure of the air.

The improvements which were applied, were chiefly the following :

1st. The two substances which were used for the determination
were sealed in two separate small pieces of glass tube, so that the
air could be removed from the liquid as much as possible by boiling
the liquid under decreased pressure.

( 164 )

Kouxstamm ‘) has proved that the way in which these little tubes
are filled, may give rise to errors in the determination of the vapour
pressure, even when they are filled with a simple substance, and
not as with Cunagus with a mixture. *)

2»d. By adjusting a glass spiral between the globe in which the
liquid was brought and the experimental tube the globe could be
violently shaken, which prevented insufficient mixing of the layers of
the liquid.

3. The difference in appearance between successive fringes was
very slight with Cunarus, which rendered the adjustment of the
movable interference phenomena compared with the firm, pretty un-
certain. By taking a more favourable relation of the distance between
two corresponding edges of the slits to the width of each of those
slits, spectra were distinctly visible on the sides of the bands, all
turning their violet rim to the middle fringe. So there was a means
of ascertaining the latter.

Moreover it was investigated whether possible absorption of vapour
on the plates of plate glass would have influence on the course of
the pa-curves. This appeared not to be the case, however, so that
this was no longer done in the examination of the second mixture.

The temperature at which the substances were examined, was 0°C.

The substances were obtained from Kan.Bacm at Berlin and were
purified before use by distilling them a few times by means of
a Youne and Txomas’ *) dephlegmator with twelve constrictions.

The mixture acetone-ethylether.

The results of the experiment are stated in Table I. (p. 165). For the
refractivity that of dry air free from carbonic acid has been taken
as unity, the pressure is given in mm. of mercury at 0°C. Ethylether
must be considered as admixture.

The per,v, diagram obtained in this way is graphically repre-
sented in fig. 1.

From the course of the curves appéars: There is a maximum
pressure, just on the side of the ether.

The point of inflection found by Cunarus is also found here, and
at about 2, = 0,66.

1) Konxstamm, Proefschrift, Amsterdam, 1901, blz. 170—180.

2) How difficult it is°to get the substance pure and free from air has been lately
observed once more by Tricuyer, Ann. de Phys. 13 p. 603.

3) Sypvey Youne and Tuomas, A dephlegmator for fractional distillation. Chem.
News, 71, blz. 177.

( 165 )

TAY Bi Bol

Pressure. ete Xp xy
69.08 3.7658 0. 0.
80.27 4. 0110 0.173 0.024
98.03 4.3097 0.342 0.083

107.03 4 4353 0.421 0.12%
127 01 4.6458 0.553 0.232
134.40 4 7036 0.590 0.295
164.72 4.9548 0.748 0.561
180 15 5.1112 0.846 0.796
185.60 5.3562 il, Ae

itiie 7 — 7 (@,)) curve is convex froma — 0) to’ 2 = 0,66.

The curve p=//(v,) has a very simple course: it turns every-
where its concave side to the «-axis.

From the course of the curves foilows finally, that the vapour is
always richer in admixture than the liquid, the greatest difference
between «, and wv, amounting to about 0,33,

These curves were now compared with the differential equations
derived by van per WAALS:

1

1 dp u,—a

p des #2.)

3
which the curve p= /(v,), must satisfy throughout its course, when
the volume of the liquid may be neglected by the side of the volume
of the vapour and when the vapour phase may be considered as
being rarefied, and:
UD eS

p ; dz, —-«, (1—z,)

which must hold chiefly for the borders of the curve p= /(r,) under
the same conditions.
E f dp ,
From the figure p and —— have been found for the values
to) l 1
av,

of x indicated in Table II, then v,—., is derived from the equation
and from this 7, and these values are compared with the observed
values :

( 166 )

PA DsL Beh:

j
Z_— 2 2,

2 cale. observed. | cale. | observed. |
01 | 7 | 71 | 0.086 | 0.088 | Oots | 0.016
ODS 8225" a 88.5 | 0.172 0.170 0.028 | 0.030
0.3 | 93 | 109 «| 0.246 0.238 0.054 0.062
0.4 | 104 | 133 0.307 0 292 0.093 | 0.108
05 119 158 0.332 | 0.315 | 0168 | 0.185
o6 | 136 | 188 | 0.392 | 0.314 | 0.968 | 0.986
0.7 | 156 | 190 | (0.258 0.250 0.42 | 0 450
| os | 17 140 (0.129 0.122 0.674 0.678
0.9 | 183 | 73 0.004 | 0.004 0.896 0.896
|

The differences between the observed and the calculated values of
wv, are evidently slight, so that the observations may be considered
as satisfying the equation in question.

When testing the observations by means of the second equation,
we got the following result :

T AB Goh: ae

| Te — ty | Ts

dp ns 2
| ie ? ot | calc. | observed cale. observed
| oos | 9 | 385 | om | 0.998 | 0.953 | 0.97
01 102 232 | 09222 | 0.280 | 0322 | 0.380
02 | 19 198 0.260 | 0.322 | 0.460 0.522
0.3 138 156 0.239 | 0.314 0.539 0.614
0.4 149 5 119 | 0.191 | 0.976 | 0.591 0 676
0.5 189 | 102 0.160 | 0.22% | 0.660 | 0.724
0.6 168 | 83 0.119 | 0168 | 0.719 | 0.768
O71) 6 | °79 009% | 0.120 | 0.794 | 0.820
0.8 180.5 | 46 0.041 0.032 | 0.841 0.852
09 | 183.5 | 12 | 0.004 0.006 0.904 0.906

( 167 )

The closest concordance between the observed and the calculated
values of wv, seems to be found at the edges of the curves.

Ow
Van pur WAAts derived from the equation of (=) for both phases:
uD

Ou
ay 2 M vy
1—w, l—«,
in which :
; duty 1 ote dy
[Ne al) and (ty, = MRT len — MRT log (v — bz,) — es

With the aid of the observed w, and v, we may derive wx, as
function of w, from this equation. The values found in this way are
given in Table IV. For 2, smaller than 1 and larger than 0,9 the
values of ws, are too inaccurate, so that they are not given for the
borders of the curves :

ie Ava Eh.) al V'.

f .
vy vg Ce

0.4 0.380 5.516 1.71
0.2 0.522 4.368 1.47
0.3 0.614 3 712 1.31
0.4 0.676 3.130 1.14

0.5 0.724 2.623 0.96
0.6 0.768 2.207 0.79
0.7 0 820 1 952 0.67

0.8 0.852 1,439 0.36
0.9 0.906 1.071 0.07

a

The function wy, as function of x, has been graphically represented
in fig. 2. Evidently the points give a pretty continuous curve.

It appears from the table that w’,, approaches to O for higher
values of wv. This is in accordance with the fact that the p-line has
a maximum on the ether side, so for v, = 1.

Van per WaAats has given ') an explicit expression for p as function

1) van per Waats, Versl. K. A. v. W. Amsterdam, Januari 1891, bl. 409;
Continuitét I, p. 146.

( 168 )

dp
of v,, from which —— may be derived :
di, -
¥S.
dp (e * —1)f1 + x, (1 — 2,) pn" z,}
—= — » ———_—_- — — i bd
de } FE ,.,
1 1 a ! . - bao |
‘ “, T ey €
SO) 10k an —
pe r,
dp Ca
a
du, “ X,
e

If p is a maximum for «,=1, then ws, = 0.

Van per WaAALs has also given an approximate value for p’y, in
“Ternary Systems” ') :
if: id he : d log Per
Today * day
where 7, and p,, represent the critical temperature and pressure
of the unsplit mixture and / the known constant, about 7, which
inter alia also occurs in the formula, given much earlier by vAN DER

. P _L.—T
Waats for a pure substance: Nep. log — = — f ———.
Pk 7

This formula for «wz, has been derived by van per Waats also
directly from the equation of state *).

If it only contained the first term, then a minimum critical tem-
perature would be attended by a maximum pressure, the minimum
would therefore lie near the pure ether. Now however this minimum
will not be found there, as appears from the occurrence of the

'
by =

second term.
If we now assume that 7’, depends linearly on . and that this is

iif d Ve.
es about — 15,

Pi ge :
also the case for p,,, then —“ is about — 43 and
¢ vy t Ly

in consequence of which we find for g, about 0,8, a value which
really lies between the values found.

Also w’,,, the differential quotient of w’,, with respect to v7, may
be determined. The accurate relation derived by vax per Waats for
the dependence of the vapour pressure on the molecular concentra-
tion of the liquid mixture, where it is only assumed that the liquid
volume may be neglected by the side of that of the vapour, and that
the vapour phase may be considered as a rarefied one, is

1, dp ] Fr

p : de, = (w, — «) Pie (bea c= w,) + xl:
1) van DER Waats, These Proc. V, p. 9.
2) van DER Waats, These Proc. VII p. 156.

( 169 )

From this «,, may be found. These values, calenlated, are given

in Table V;

Tab bE VY.

ot | 102 252 0.280 8.8 — 2.3
0.2 | 492 198 0.322 5.0 Bie:
0.3 | 138 156 0.314 3.6 edhe
o4 | 149.5 | 419 0.276 29 BRE
0.5 | 49 102 0.234 | 29 i
0.6 | 168 83 0.168 29 aie
0.7 | 176 19 0.120 3.7 ae
OLS emrtsoe |). 46 0.052 re eo ee
0.9 | 183.5 | 412 0.006 10.9 ie

The values given for w,=0,7, 0,8 and 0,9, however, are not
very reliable, on account of the small value which «,—., then has,
by which the first member is to be divided. For the same
reason m’,, cannot be given for v,, smaller than 0,1.

Evidently all the values of w",, are negative and as to their
absolute value, they are smaller than 4. This in accordance with the

org
fact, that for stable phases, (; ) must be larger than 0.
vy p

From the course of the curve w., = /(«,) appears that the diffe-
rential quotient of about w= 0.2 to «=0.6 must have a constant
value. And this is really found for w'y,.

With regard to w',, we may still remark, that its numerical value
may be found also for the x, of the inflection point and the corre-
sponding «,. Van per Waats has namely derived : *)

5 re Py
a*p dp, dx, fy 2(e “—1) Ti 2 ae
dx?, dv, da, Os ba Bl ian pa
, 1- x, +a,e 1—z, +2,
cara : Sala dp du
is being O for the point of inflection, the factor of des fae
dx, du, dx,
must be 0.

1) These Proc. Ill p. 172.

- & l—«

With the aid of the relatione “’ = ——*— . ———* we find from this:

ac % 2, (1—«’,)
tL uy =——_ z a

2 ( a, @, (1 —w,)

For the inflection point we have about «, = 0,66 and x, = 0,40.

For «’,, we find then about — 1,5. Taking into consideration that
the place of the point of inflection cannot be determined accurately,
and that with change of the «, of this point at the same time also
its w, is changed, the agreement may be called satisfactory.

The miature carbontetrachlorid-acetone.

These two substances, carbontetrachloride and acetone were chosen
with a view to the critical pressure, which is about the same for them.

Now Van ber Waats’*) has derived, that for «’,, = constant the
relation between p and wz, is represented by a straight line, that
between p and «, by a hyperbola.

On the supposition that at low temperature for ~’,, may be written :

i
z1

~ MRT dz,
GaALitziNER—BERTHELOT @,, =Va,a, holds good, the condition ma’, =
constant involves, that the critical pressures of the components are

the same.

Now the above mentioned improved approximation for w, has
already been given by Van per Waats himself instead of the one

for 4,,: 4 (6, + 6,) and that also the relation of

mentioned here; moreover the relation a,,—=(Va,a, has already
been discussed and rejected by Konystamm.*) It is therefore not
surprising, that a strong deviation from the straight line and the
hyperbola was found for this mixture *).

The px,«, diagram is given in Table VI. Acetone is to be consi-
dered as admixture :

1) These Proce. Ill p. 168— 169.
2) Konnstamm, Proefschrift, blz. 99.

3) From the survey of the investigated mixtures given by Kounsramm, Zeitschr.
f. Phys. Chem. XXXVI. 1, it appears also, that equality of critical pressure ef
the components does not necessarily involve the existence of a straight line for the
p=f (a) line.

a.

eee ee a) oe ey Oe

(274) )
T Ay Be Ly VI.

Pressure. ph oes a 2)
34.20 | 6.1407 0. 0.
M46 | 5.7352 | 0.172 0.044
46.43 | 5.4772 0.281 | 0.122
$2.27 | 5.1253 0.430 | 0.234
55.47 £.9261 | 0.515 | 0.333
60.43 | 4.6135 | 0.647 | 0.477
6449 | 4.3219 0.771 | 0.696
60.32. | se78i34 || 7, ly aa

In fig. 8 it is graphically represented.

Both the curves have a very simple shape. They turn their con-
cave side to the v-axis. The vapour is always richer in the admixture
than the liquid. The greatest difference between wv, and wv, is about 0,190.

Both the curves are now compared with the differential equations:

1 dp t,—2,
Pp ; div, a wv, (1—z2,)
1 dp v,—e,

peda’ we n(n)
The results are given in table VIL and table VIII:

TABLE VII.

dp Tg — 2} a,
i Z dy Eaieaintonl| Thoeedie henlentatea obseeved:
ot | 39 39 | v.09 | 0.090 | v.00 | o.o10
02 | 43 37 5 0.439 O44 0.061 0.089
(hie ee © Gam | 0.165 | 0.178 0.135 0 122
Osi Si gh a6) | * O68 | 0.190 | 0.931 0.210
Oe 1 55 35.3 O.NGE 0.18% 0.339 0.316
oe | so) 88 0.142 | o.t6s | 0.488 | 0.435
0.7 | 62.3 | 99 0.097 0122 | 0 603 | 0.378
0.8 65 | 24.5 | 0.060 | 0.080 0.740 | 0.720 |
0.9°| 67.5 | 23 =| 0.031 | 0.083 | (0.869 0.865

12
Proceedings Royal Acad. Amsterdam, Vol. VIL.

(172 )
TABLE VIII

| | Sar rs

ly a pees Ow ae Pe

e 4 ae; j calculated | observed | caleulated observed
0.05 | 41.5 2 | o1t7 | 0.190 | 0.167 | 0.470
04 45.5 72 | 0.143 | 0.160 | 0.243 | 0.260
0.2 50.5 48 i) OE52 0.180 0.352 | 0.380
03 | 54.5 39 | 0130 | 0.182 | 0.430 | 0.482
04 58 33°) 1) 0487 OAT 0.837. || ser
0.5 | Gt 24 | 0.098 0.154 0.598 | 0.656
os | 6 | 2 | 0.076 | 0.120 0.676 | 0.720
0.7 64.5 | 15.5 0.030 | 0.078 0.730 0.778
0.8 66 15 | 0.036 0.040 0.836 0.840
0.9 68 14.5 0.019 0.020 0.919 | 0.920

|

Evidently the curves satisfy the first differential equation much
better than the second. For the second the closest agreement is obtained,
here too, on the borders of the curves.

Also for this mixture w, as function of x, was determined by
means of the relation :

wv Lv

1 2

'
— ex =

1—u, 1—a, :

The values, found in this way, are given in Table IX:

Df AVB EG 1X

a Xg | Pe Px
0.1 0.260 3 162 1.15
0.2 0.380 2.452 0.90
0.3 0.482 2471 0.78
0.4 0.572 2.005 0.70
0.5 0.654 1.890 0.64
0.6 0.720 1.714, 0.54
0.7 0.778 1.502 0.41
0.8 0.840 1.313 0.27
0.9 | 0.920 1.278 0.25

ils

Here too the values of «’,, are not accurate on the borders of the
curve. In fig. 4 the results have been represented. Here too the
points give a continuous curve,

The critical pressures of the components being the same for this
mixture, the second term in the equation :
dcr d log Per

!
Le = — ryy ; =
= T de, de,

will be small compared to the first. If we take for w, only the

ie AT oy oy “ ey
ierm — *.—— and if we assume linear dependence of 7. on ,,

T de,
then is here about — 40, so ws, about 1. In this neighbourhood

aw
1

also the values of w,, are found.
Also in this case uw", is determined from the equation :

thes he) fs
5 ‘ din = (ca) ies | Se hs
The values found from this are given in Table X.
TAG Bele bi Xe

z, P ze Sec lear 2 Paine Gin meee

Onaele4s 5 | 72 0.160 9.9 —1.2
0.2 | 50.5 | 48 0.180 ou —1.0
ea S459) | 39 0.182 3.9 —09
Creel) 58.) sae | O72 ed —0.9
Gis | 64 26 | 0154 2.6 —1.4
Or6" |, 163 20 0.120 2.6 -15
OF |, 64/5) 15.5 0078 | 3.4 | 4.7
08 | 66 1s | 0.040 St —0.6
0.9 | 68 14.5 0.020 10.7 —0.4

|

Here too the value of «w",, for r,=0,7, 0,8 and 0,9 is not very
reliable on account of the small value of v,—.,.

The quantity «’,, proves again to be negative and in absolute
vatue smaller than 4.

From the course of the curve @.,—=/(r,) appears that it seems
to have two points of inflection, one at about 7, = 0,3, the other
at about v,= 0,7. It is remarkable that this also follows from the
change of the «’,, in Table X; just as we had to expect from the
course of the curve a, = /(,) wv’, gets a maximum at about 2, = 0,3,
and a minimum at v2, = 0,7.

12*

( 174 )

Chemistry. — “On the latent heat of miving for associating solvents.”
By J. J. van Laar. (Communicated by Prof. H. W. Baknuts
RoozEBOOM).

(Communieated in the meeting of June 25, 1904).

1. When some substance is solved in an associating liquid, as

e.g. water, and we try to find an expression for the latent heat of

mixing of these two substances, we shall in the first place have
to take into account, besides the change of the potential energy,
the heat of éonisation of the solved substance, if this substance is
an electrolyte. The fact, however, that the state of association is
changed by the solving, is nearly always overlooked. We are inclined
to reason, that in much diluted solutions the influence of the addition
of a few molecules of the solved substance must necessarily be
exceedingly slight, with regard to the degree of association of the
solvent; but in doing so it is overlooked that the z2mber of molecules
of the solvent which each undergo a very slight change in their
state of association, is very great. For infinitely diluted solutions
therefore, a value is obtained approaching to OX a, and I shall
demonstrate in what follows, that the absorbed /eat in consequence
of the change in the state of association, approaches to a definite
value, which is jinite and even comparatively high.

2. In diluted solutions — which we solely have in view in the
following pages — the state of equilibrium of the associating mole-

cules of the solvent may be expressed as follows:

(= By
N

= — = K,
‘7, (1—a) (1—B)
Aer at x
1.€. :
pela
ip Ws a eerie aso! (i!)

For, given 1—w mol. H,O, normally reckoned, « mol. salt
(calling the solved substance sa/t for convenience’ sake), then there
are '/, (1—w«) mol. H,O, if all are double. Therefore if the degree
of dissociation of these double-molecules is 3, then there are:

1/, (1—«) (1 — p) double mol. ; */, (1—-x) 28 = (1 — a2) 8 single mol.

The total number of particles is .V. If the degree of dissociation
of the saltmolecules is @, then there are (in binary electrolytes):

C175.)
v (1 — a) neutral mol. ; 2wa Tons.
We have therefore :
N= '*/, (1—«)(1 + 8) +a + a),
or with */,(1+8)=y , 1+ a@=7, where therefore 7 has the usual

meaning, and y is the reverse of the so-called association-coeflicient:

N=y(l—a2)+ie=y(1—2) |) a =

Our equation (1) becomes therefore:

Grey being = */, (1 + 8):

Bp? =a gues 7 s
1 noo 4 = 9
From this follows, putting e a, Seg
y ie

r= [| ae S| ltd
o IPE A 14S, Ke ey

Now evidently

1. e. the value of 8, if z or d=0O, so that we have got the pure

. By”
solvent, for which the equation —~"—

> =, holds. Therefore we
1—8,?

leiee ye
=!3, ——_—_— =f, ee
‘ ee (: i );
if, d bemg very small and approaching to 0, we content ourselves

with a first approximation.
Substituting for d its value, and taking into account that y= '/, (14-8),

we obtain :
aye Hee co i ae
tere | esti) 9 fen (eer) jaan ’

or with 1+ B=1-468,:
e=e(1+d—=ayij-)... = Se3)

obtain :

(v6?)

So this is the sought-for expression for the change in ,, caused
by the addition of « gr.mol. salt.

1—2:
3. Now on 1 saltmol. there are —— er.mol. H,O (normally
ve a:
; ; 2 1—#
reckoned), among which there are evidently */, | —— ]28=- B
wv v

single mol. In consequence of the fact, that the state of dissociation
of the watermolecules is changed by the solved substance, this number
according to (3) will amount to

l—-wx 1—a: 1—wx c

Be Pats ae a (1—6,)* ’

zc x a l—«

that is to say an increase of

wv

1—wr
5 | x By (1 Be) 1

a —wWw
And now it is clear that, as was already observed above, the

l—w
one factor of this product, viz. ——, approaches to », while the other
wv

v
factor, viz. 8, (1—3,)’——, approaches to 0. The product however is
=e

i
evidently jinite, viz.:
A =p 2an o>. 3 eee
Now if Q is the heat, absorbed when 1 gr.mol. (18 Gr.) H,O
changes from the state of double molecules to that of single molecules,
then the heat, absorbed in consequence of the state of association
being changed by 1 mol. of the solved subsiance, is:
W= 6. I—8)4Q. 24a « ee
And this heat it is, which we have to take into account for
associating solvents.
For H,O at 18° 30,21"), so that the factor 8,(1—~,) becomes
=0,17. Further Q (as J calculated some time ago *)) = + 1920
gramecalories, so for water (at 18°) will be:

Wi 32608 2. ce ee ce a

If the solved substance is no electrolyte, then (= 1, so for much
diluted solutions about 825 gr.cal are absorbed with every concen-
tration, if 1 gr.mol. is solved in the water, only in consequence of
the change in the degree of association of the water; for salts, acids

1) Zeitschr. fiir Phys. Ch., 31, p. 4 (1899); Lehrbuch der. math. Chemie, p. 36

(1901).
2) Z. f. Ph. Ch., 81, p. 5 (1899); Lehrb. der math, Chem,, p.87 (1901).

and bases, where 7 nearly 2, this number becomes 650 gr. cal.

So e. g. for KCl, of which the heat of ionisation of 1 gr.mol.
= — 720 er. cal."), the total heat of mixing with much H,O,
(excluded the change in potential energy) will therefore be not —720
er. cal., but only — 720 -+ 650 = — 70 er. cal.

So it is seen, that the order of magnitude of the heat to be expected,
‘an be totally modified, and that in general a great mistake would
be committed, when we neglected the above caleulated 3267 er. cal.
in the caleulation of the heat of mixing.

Therefore, with diluted solutions of non-electrolytes in associating sol-
vents, 325 er.eal. on each gr. mol. of the solved substance must
always be subtracted from the absorbed heat determined by experiment,
in order to calculate the pure (absorbed) heat of mixing, that is to
say that heat, which is caused solely by the change in potential energy.

Physics. — Prof. Baknuis Roozesoom, in the name of Dr. A. Sits,
presents a paper, entitled: “On the phenomena appearing when
in a binary system the plaitpointcurve meets the solubility curve.”
(Third communication). *)

(Communicated in the Meeting of June 25. 1904).

The previous qualitative examination of the binary system ether-
anthraquinone showed that a good survey of the whole could only
be obtained by continuing the examination in quantitative direction
with the aid of the pump of CaiLunrer.

Some difficulties were to be foreseen; the investigation would have
to be extended over a range of temperature from + 170° to + 300°,
in which the pressure might be expected to reach a pretty conside-
rable amount —— and the combination of high temperature and high
pressure being exactly the thing against which glass is but seldom
proof, it seemed at first that we should meet with great experimental
difficulties in the quantitative examination. The experiment however
showed that the pressures were not exceedingly high; it appeared
& maximum pressure of 100 atm. would suffice, and this pressure
Jena-glass could withstand up to more than 800° *).

1) Z. f. Ph. Ch., 24, p. 611 (1897); Lehrb. der math. Chem., p. 53 (1901).

*) This paper is a continuation of the two preceding ones on the system ether-
anthraquinone. The title ehosen first seemed to me undesirable and was therefore
modified.

5) With pleasure I avail myself of this opportunity to thank professor Kamertincu
Onyes for his kindness towards me in procuring the necessary information and in
lending me some instruments wanted.

(178 )

The object of the experiment was to determine the p-v-sections
of the p-v-t-surface at different temperatures, and if possible also the
v-w-sections of the v-v-f-surface. At the same time I should get to the
knowledge of some projections already spoken of in the previous
paper, viz. the projections of the p-w-f-surface on the p-¢ and the
tv-plane. I shall briefly state the result here.

In order to have the same succession as was chosen in the prece-
ding communications, the p-/-projection will be treated first.

10

170% jd0 190 Lop Lio 210 L380 140 250 gbo ayo L200 29¢ 30
T
Fig. 1.

In fig. 1, ea represents the vapour pressure curve of pure ether
with the critical point in @ (193° and 36 atm.). ¢p and gd represent
the portions of the three phase curve which can be realized. Up to
193° the three phase curve practically coincides with the vapour
pressure curve ea of pure ether, in consequence of the very small
solubility of anthraquinone in ether. On the curve ap lie the plait-
points of the unsaturated solutions of anthraquinone in ether, and
p denotes the jirst plaitpoint of a saturated solution (208° and 48 atm.).

The second plaitpoint of a saturated solution of another concen-
tration lies in g (247° and 64 atm.) and on the curve q/ lie the
plaitpoints of the second series of unsaturated solutions. Probably
this curve, which runs on to the eritical point of anthraquinone,
has a& maximum.

( 179) )

The line fd, whieh partly coincides with the 7"axis, is the vapour
pressure curve of solid anthraquinone, and dy that of liquid anthra-
quinone. dh is the meltingpoint-curve, which (as vAN per Waats ‘)
has proved) marks the direction of the three phase curve near
the meltingpoint ¢. These last three curves are drawn here schema-
tically.

The main result represented by this p-éfigure is this that by the
meeting of plaitpointeurve and three phase curve a part of the latter
has vanished or rather has become imaginary, and in the examined
system that part that contains the maximum.

The plaitpointeurve is metastable between p and g and therefore
still to be realized, but this is not the case with the three phase
curve. However it appeared to me that at temperatures between p
and g, with concentrations greater than those of point q, three phases
could temporarily appear together, if they had originated at a tempe-
rature above 247°
quickly cooled down to less than 247°. The three phases however
were not in equilibrium now, for at a constant temperature a slight

and if afterwards the system in equilibrium had

change in volume proves to cause a great change in pressure.

The liquid therefore, thongh in contact with solid anthraquinone,
Was supersaturated ; if was very viscous and passed very slowly,
at times not until after an hour, to the stable condition of solid fluid,
under secretion of solid anthraquinone.

Fig. 2 gives a number of p-v-sections for different temperatures,
the pressure being given in atmospheres and the concentration in
1 mol. total of the mixture.

We may immediately point out here that all the lines in this
figure joining points of equal value, as plaitpoints (4), liquids coexis-
ting with vapour and solid anthraquinone (c), vapour coexisting with
liquid and solid anthraquinone (7), are all projections on the p-v-
plane of curves, which occur in the p-e-/-surface *).

The branches ¢, p and e, p, which pass into each other conti-
nuously, represent the series of liquids and vapours which if we
come from a lower temperature, coexist with solid anthraquinone.
In p, the point of confluence of the two branches, we have the jirst
point, where a saturated solution reaches its critical condition. This
takes place at a concentration 0,015, temperature 203° and pressure
43 atm. If we pass on to higher temperatures a stable solution is
impossible over the range of temperature 203°—247°, and instead we

1) These Proc. VI p. 230.

*) If the plaitpointcurve has a maximum, it must possess a maximum also in
fig, 2. In the @x-projection on the contrary no maximum occurs.

( 180 )

get fluid phases coexisting with solid anthraquinone. Above 247° liquids
can again exist and the continuous curve dc, c, ¢, Cy Cy 7 Cs C4 Cs Cn es
consisting of two branches then represents the series of liquids and
vapours which coexist with solid anthraquinone above 247°. The
point of confluence here is g, in which therefore for the second time
a saturated solution reaches its critical condition. This occurs witha
concentration 0,13, temperature 247° and pressure 64 atm.

The liquid branch ¢, p of the first loop and the liquid branch q e, d
of the second loop are what we are accustomed to call two parts of
the solubility curve. As however the two liquid branches pass conti-
nuously into their vapourbranches, there is no objection to calling
the two continuous loops solubility curve.

Branch c,p of the first solubiliy curve and branch de, q of the
second show here a particularity. The circumstance that these branches
pass continuously into the branches e, p and dc, q and that the point
of confluence coincides with the highest pressure involves the pheno-
menon of retrograde solubility.

Cop points to retrograde solubility in the liquid branch (ef. also fig. 4)
and de,q to retrograde solubility in the vapour branch. The extent of
these phenomena however surpassed all expectations. It was known
that the liquid and the vapour branch of the curve dc, qe, d from q
to a higher temperature have to separate first in order to come
together again afterwards, but it was not to be foreseen that the
distance would be so large as to make the vapour branch extend to
the concentration 0,01. From this particular situation results the very
interesting phenomenon that, after we have reached point p, with
a concentration 0,015 or in other words after the saturated solution
has reached its critical condition, at a higher temperature there may
again occur three phases. The vapour branch ge, d extends namely
as already mentioned, to the concentration 0,01, and the concen-
tration of point p is 0,015; therefore we get from point p at a
higher temperature into the region on the right of the vapour
branch de,qg, in which three phases may occur. This phenomenon
was observed at a temperature nearly 60° above the plaitpoint-
temperature of the concentration 0,015 (p), that is at 260°. After
the formation of the three phases, first the solid and then the liquid
might be pressed away by raising the pressure, so that finally only
a fluid phase was left.

Fig. 2 shows further the p-a-sections at temperatures above that
of point qg, beginning at 250°. The p-v-section corresponding with
this temperature is separately drawn in figure 2a. The continuous
curve ¢,/,¢, which represents the coexisting unsaturated liquids and

( 181 )

vapours, has a peculiar shape and shows that retograde condensation
is wanting.

1
/00 F;, +Anthraquinone
go H (solid)
'
G0
we

bt

Cs

3
1? GY)
\,

v0 7 0

Jo £50. #, + Anthra-

lo quinone (solid),

10

0 01 02 O03 G4 G5 7 7

x

Ether. x Anthraquinone.

Fig. 2a.

The curve g,e, on the contrary indicates a tolerably strong retro-
grade solidification. The curve ¢,s, shows that here the solubility of
anthraquinone in ether decreases with increased pressure. The curves
Jy e, ad c,s, are two portions of a continuous curve, of which the
partly not realizable intermediate part is schematically represented
by a dotted curve. | propose to call this continuous curve henceforth
isotherm of solubility.

Passing on to a higher temperature, we see that the p-v-section at
255° is still of the same type as that at 250°. At 260° (fig. 26)
however, the situation is already considerably changed; not only the
p-«-loop c,h, e, has become much larger, because the points e, and
c, have become more widely separated and /, has moved to higher
pressure, but also it is
which is. still wanting
temperature.

clearly visible that the retrograde condensation,
here, will have appeared at a slightly higher

In the part g,e,, though in a smaller degree than at 250°, the

( 182 )

Ether. xX Anthraquinone.

Kig. 2d.

isotherm of solubility shows. still clearly the phenomenon of retrograde
solidification.

At a still higher temperature the region of retrograde condensation
becomes greater and greater, so that at 270? we get a p-r-section
like that drawn in fig. 2c.

The retrograde condensation is here very strong and undoubtedly
ranges over more than 40 atm. A retrograde condensation of such
strength, however, could not be observed because the volume of the
compression tube was too small; the strongest retrograde condensation
observed by me covered a range of pressure from 55 to 39, of 16
atmospheres therefore. The small volume of the tube prevented us from
observing whether any retrograde solidification still existed at 270°.
As, however, it is not very probable that we still should have retro-
erade solidification here, it is not represented in the figure.

Above the melting-point of anthraquinone (283°) the retrograde
condensation is enormous, so that 1 could observe it at 290° over a
pressure-range of 83 to 40 atmospheres.

Further 1 mention that most of the p-a-sections are crossed in

( 183 )

100

G
OWI TS Op IS ab ap as ag 4

Ether. x Anthraquinone.

Vig. 2c.

different ways. In fig. 2a the regions passed are marked with arrows.

1 indicates the transition from the region for /’, + solid anthraquinone
into the region for /’;, + solid antraquinone, the three phases appearing
intermedially. /, denotes here a fluid phase which in ordinary cir-
cumstances, that is to say below the critical temperature of ether,
would be called gas-phase; and /’; denotes a fluid phase which in
ordinary circumstances would be styled liquid phase. It is evident
that the difference between /, and /;, exists solely in their foregoing
history.

2 marks the transition from the region /, + anthraquinone into
the region #7, the three phases appearing intermedially.

3 indicates essentially the same as 2, but yet the phenomenon is
somewhat different, because now we do not in the end pass the
liquid branch, as in 2, but the vapour-branch; this is marked by the
sign /, over the branch ¢, /:,.

4 is a very remarkable transition, as here we pass directly from
the region for /, - solid anthraquinone into the region for /%).

As to the lack of retrograde condensation at temperatures between
247° and + 260° and its appearance at higher temperatures, | want

( 184 )

to say a few words about it in connection with the appearance of
retrograde solidification.

If in fig. 3 the p-r-loop
dckeR represents the liquids
and vapours which may
coexist at a given tempera-
ture, but of which a series
of liquids and vapours are
not to be realized in a stable
state because of the appear-
ance of the threephase pres-
sure curve’), then several
cases are possible. If the
threephase pressure curve,
as drawn in fig. 4 lies above
the critical point of contact
Rk, then no retrograde con-
densation will oceur, not-
withstanding its possibility is

A xX B

strongly pronounced in the
character of the p-z-loop, because the part giving rise to the retrograde
condensation lies in the metastable region. Now this occurs in the
system ether-anthraquinone from 247° to 260°.

The dotted vapour and liquid curves below the threephase-pressure
curve ec f are metastable; the stable state here is solid B by the
side of a fluid phase, and now the question was raised: “how is
this part of the isotherm of solubility situated?” Evidently this stable
curve must le left of the metastable curve ¢ Re or in other
words towards smaller 5-concentrations. This conclusion is of great
importance for us, for from it follows that, (7 the threephase-pressure-
curve lies above the critical point of contact of the vapour curve
coevisting with liquid and for that reason the retrograde condensation

falls in the metastable region, retrograde solidijication must occur

instead of retrograde condensation, and this vetrograde solidification
must be stronger than the retrograde condensation would have been.
If the threephase-pressurecurve passes exactly through the critical
point of contact, retrograde solidification is no longer necessary.
Resuming, we conclude that, given the case that the plaitpoint-

1) I propose to give this name to the curve that in a p-a-section denotes the
pressure at which the three phases coexist. This curve refers therefore to one

temperature, whilst the threephasecurve embraces a series of temperatures.

( 185 )

curve meets the solubility curve, if is possible to prove in a very
simple way the necessity of the appearance of retrograde solidification
in p and q.

Here however we must at once point out that, as will be discussed
presently, retrograde solidification also occurs between pq. The fact
that theory requires this, can only be proved mathematically *).

Returning to fig. 2, we must still state that the curve q / uniting
the plaitpoints of the different p-x-loops, is very steep and, as far as
it has been observed, parallel to the first part of the plaitpointcurve
ap. This course however will probably change towards a higher
temperature, for if the plaitpointeurve possesses a maximum, which
is probably the case, then the projection of the plaitpointeurve on
the p-w-plane must also show a maximum.

The p-«-sections below the
temperature 203° are not
drawn in fig. 2,as the scale
is too small to render the par-
ticulars conspicuous. There-
fore this part of fig. 2 is
separately reproduced on a
larger scale in fig. 4.

In accordance with the pre-
ceding we see that, though
at 200° no retrograde con-
densation occurs, instead of
it there appears retrograde
solidification. Soon however
the situation changes here,
for already at 196° retro-

X grade condensation could be
Fig. 4a. observed.

What was observed when going from point q to a higher tempe-
rature, is naturally also found in point p, but here towards a lower
temperature. This is illustrated by figures 4a and 44; tig. 4a applies
to temperatures above point qg and fig. 44 applies to temperatures
below point p. In both figures three p-x-sections are represented
schematically ; the sections 1 and 2 differ but slightly in temperature,
and 3 applies to a temperature considerably different from that with
which 2 corresponds.

In fig. 4a section 1 apples to the lowest and 35 to the highest of

1) vaAN peER Waats, l. c.

|
|

a

( 186 )

the observed temperatures; in fig. 4d) the reverse is seen, but all
the same it is seen that in the two figures the same things are to
be met with in the succession of sections 1, 2 and 3. The three

x
Fig. 40.
pressure curve lies highest in 1 and

§ lowest in 3. In 1 and 2 we do not
find any retrograde condensation, but
retrograde solidification, and in 3 we
find retrograde condensation only. In
fig. 4a however the plaitpointpressure
increases in the order 1, 2, 3, and

a decreases in fig. 44, but this is due
to the fact that in the first case the
order 1, 2, 3+ means towards higher
temperatures, and in the second to-
wards lower.

Concerning the course of the iso-
therms of solubility above the three
phase pressure curve, VAN DER WAALS
has shown the probability of a course

4 as given in fig. 5, from which results
that the branch cs also shows retro-
evade solidification. This case, in which

@187 )

the whole isotherm of solubility points to a double retrograde solidi-
fication, has not been ascertained as yet. What has been found, is
1 that below 240° the upper part of the isotherm of solubility runs
foward the right, which points to an increase of solubility of anthra-
quinone in ‘the fluid phase with increase of pressure '), whereas
above 250° a reversed course was found. Between 240° and 250° a
change of direction seems to have taken place, and in this range it
might be possible to ascertain the course foretold by vAN per WaAats.
As however the small range of temperature 240°—250° corresponds
with a great difference in concentration, the point when the change
of direction takes place is not easily ascertained,

The results obtained at temperatures between 203° and 247° are
represented in fig. 6. Here the isotherms of solubility for the fluid
phases at 210°, 220°, 230° and 240° are drawn. All these isotherms
show, as predicted by van per Waats’), the phenomenon of retro-
grade solidification, and the nearer we get to point ¢, in other words
the nearer to 247°, the larger the region of this retrograde solidification.

Ether. xX Anthraquinone.

1) This influence of pressure has been examined up to more than *0) almospheres.
*) loc. cit.

Proceedings Royal Acad, Amsterdam. Vol. VII.

( 188 )

This isotherm of 210° has the steepest course; with increase of
temperature the course becomes at first less steep, but at 240° a steeper
course seems to reappear, which is probably connected with the
change of direction which appears above 240°.

The projection of the solubility curve and the plaitpoint curve on
the ¢w-plane is represented in fig. 7, where the dotted curves represent
the vapourbranches. The projections of the two parts ap and gb of
the plaitpointeurve are almost straight lines. If we examine the
course of the line g, in order to see at what temperature this line
will meet the line for pure anthraquinone, we shall find + 800°,

Lastly we find in fig. 8 the course of the molecular volumina of

300) |

100

ad

Ether. Xx Anthraquinone.

Fig. 8.

the saturated solutions. Here too we have two continuous branches,
each of them consisting of a liquid and a vapour branch. dq and ep
ave the liquid branches and qe, and pe are the vapour branches. p and
q denote the molecular volumina of the two critical saturated solutions.
The dotted vapourcurve ge runs on to the concentration 0.015, so that
from this figure also directly follows that at higher temperatures and

larger volumina three phases may again be obtained with the con-
centration with which point » may be realized. Here too the curves
| cp and ge indicate clearly the phenomenon of retrograde solubility.
| | So the investigation described here has furnished proof positive of
| | the general points of view which were prominent in the qualitative

( 189 )

investigation, and of which the theory of VAN pur Waatrs could
give a closer description.

The peculiarity of the examined system, which lies in the fact
that the vapour pressure of the one substance (ether) far exceeds
that of the other (anthraquinone), caused some wholly unexpected
phenomena, and made it on the other hand possible to realize retro-
grade solidification on a much larger scale than had been thought
possible till now.

Laboratory for Anorganic Chemistry of the University.

Amsterdam, June 1904.

Chemistry. — “Zhe preparation of silicon and its chloride.” By
Prof. A. F. Honiemay.

(Communicated in the meeting of June 24, 1904).

The numerous proposals which have been made for the prepara-
tion of the element silicon in both the amorphous and crystallised
form prove that a simple method has not as yet been found. W.
Hemprn. and von Haasy’) have published in 1899 an additional
process consisting in the decomposition of silicon fluoride with sodium.
They melt this metal in small portions at a time in an iron
apparatus and then pass over the mass a current of silicon fluoride,
which is then very readily decomposed. The brown porous mass,
which has been brought to a faint red heat is allowed to cool for
two or three hours in the current of silicon fluoride. An attempt to
convert it into silicon chloride by heating the mass without previous
purification in a current of chlorine was unsuccessful. It was
impossible to remove the Na Fl and Na, SiFl, by boiling with water ;
so in order to obtain pure silicon it was necessary to fuse the mass
with sodium and aluminium. The latter dissolves the silicon which
is then left insoluble on treating the regulus with dilute hydro-
chlorie acid.

Mr. H. J. Stuprr who has repeated these experiments in my
laboratory showed (1) that by a small modification of the process
the crude product may be purified to such an extent by boiling
with water that it may be used for preparing silicon chloride ;
(2) the reason why the crude product on being treated with chlorine
does not yield silicon chloride.

1. It is known that sodium fluoride readily absorbs Si Fl, and

') Zeitschr. f. anorg. Ch. 28, 32.

( 190 )

passes into Na, SiFl,. By allowing their apparatus to cool for 2
to 3 hours whilst transmitting this gas Hrmprn and von Haasy
practically converted the sodium fluoride, which had been formed
according to the equation: 4 Na + Si Fl, = +NaFl-+Si, into sodium
fluosilicate, which is soluble in water with great difficulty. If, however
the action of Si Fl, is stopped as soon as all the sodium has been
introduced into the apparatus, it is easy to almost completely avoid
the formation of Na, SiFl,. 100 grams of sodium yielded to Mr.
SLuUPER 219 grams of crude material (4 Na Fl + Si) instead of 213.6
the quantity caleulated; 55 grams of the Na gave 119 grams,
theory 117.2, and in some further experiments the theoretical
quantity was but little exceeded. By washing and boiling with water
and with dilute hydrochloric acid the 119 grams were reduced to
20 grams whilst the product may contain 16.7 grams of silicon.
The product so obtained is not, however, pure amorphous silicon,
only about 40 per cent is volatilised in a current of chlorine and
may be condensed as silicon chloride, and a residue is obtained,
which is only to a slight extent soluble in water and principally
consists of silicon dioxide.

This must have been formed during the washing; for if the crude
product is heated in a current of chlorine there remains besides
sodium chloride only a very small quantity of insoluble residue. As
the crude product when immersed in water causes a visible evolution
of gas with the odour of SiH, it is probable that the SiO, has
been formed by decomposition of SiH, which may have been pro-
duced by the action of water on some sodium silicide. Motssan has
recently shown that on treating silicon with boiling water the dioxide
of that element is formed.

2. In accordance with Hemprn and von Haasy, Mr. Suvprr found
that on heating the crude product in a current of chlorine not a
trace of silicon chloride is obtained. As the said product consists
mainly of 4 Na FI -+ Si, it was surmised that this must be attributed
to the fact that the primary formed silicon chloride reacts with
sodium fluoride according to the equation

Si Cl, + 4 Na Fl= Si Fl, + 4 Na Cl

It appeared indeed that on heating sodium fluoride or sodium
tluosilicate in the vapour of silicon chloride the said decomposition
takes place. If, therefore, chlorine is passed over a mixture of Si
and NaFl as is present in the crude product the reaction must
proceed in this manner :

Si-+ 4 Cl + 4.Na Fl=SiFl, +4 NaCl.

( 191 )

That such is practically the case was shown by the fact that the
gas evolved consisted of Sill, and that the substance left behind
in the boat was found to be almost pure sodium chloride.

A better method of preparing amorphous silicon seemed to be
the decomposition of silicon chloride by sodium. When boiled in
benzene-solution with sodium or potassium no action took place.
A reaction, however, took place on heating sodium in the vapour
of silicon chloride, but it became very violent; the brown powder
obtained could certainly be readily freed from sodium chloride by
means of water, but on heating in a current of chlorine a large
amount of SiO, (about 30°/,) was left behind showing that even
this process does not lead to pure amorphous silicon.

Much more simple is the preparation of crystallised silicon accor-
ding to the method recently published by R. A. Kine (Chem.
Centr. 1904, I. 64) if we introduce a slight modification. A mixture
of 200 grams of aluminium shavings or powder, 250 grams of
sulphur and 180 grams of fine sand is put into a Hessian erucible
placed in a bucket with sand. Upon the mixture is sprinkled a thin
layer of magnesium powder and this is ignited by means of a
Gotpscumipt cartridge. The mass burns with a beautiful light and
the contents of the crucible become white hot. When cold, the mass-
is treated with dilute hydrochloric acid, which dissolves the aluminium
sulphide and leaves the silicon in a beautifully crystallised state.
The yield amounts to about 30 grams. On heating in a current of
chlorine SiCl, is very readily formed, only 3°/, remaining as non-
volatile products. It is a material eminently suited for the prepa-
ration of SiCl,, but Mr. Sturer did not sueceed in converting it into
silicon sulphide by heating with sulphur.

Groningen, Lab. Univers. March 1904.

Crystallography. — “On the preservation of the erystallographical
symmetry in the substitution of position isomeric derivatives of
the benzene series’. By Dr. F. M. Janaur. (Communicated by
Prof. A. P. N. Francuimonr).

(Communicated in the meeting of June 24, 1904).

Some time ago when engaged in a research as to the connection
between molecular and erystallographical symmetry of position-isomeric
benzene derivatives‘), | demonstrated, that of the six possible tribromo-

1) Jarcer, Crystallographic and Molecular Symmetry of position-isomerie Benzene-
derivatives. Dissertation, Leiden 1903. (Dutch).

( 192 )

toluenes, the 1-2-4-6 and the 1-2-3-5 derivatives exhibit an isomorphy
bordering on identity. I then endeavoured to explain the similar
molecular structure of these two substances by referring to the
analogy of the (CH,)-group and Br-atom in the positions 1 and 2,
particularly in a spacial respect. The small differences in crystalline
form, melting point ete., are then caused by the difference which of
course exists between CH, and Br.

We may now inguire how matters will be in both molecules as
regards the substitution of the two remaining H-atoms of the core
for instance. It is interesting to notice that as regards the substitution
by (VO,) the two H-atoms in each of the two isomers are quite equi-
valent and — what is still more striking — that this substitution does
not perceptibly alter the molecular symmetry of the two molecules,
so that the erystallographical relation of the initial products is pre-
served in the substitution derivatives.

If we nitrate the 1-2-4-6-tribromotoluene with nitric acid of 1,52
sp. gr. a dinitro-product is obtained, as shown by Nevite and
WINTHER *).

Wrosiewsky *) had previously obtained a mono-nitro-derivative
which differs in its melting point but little from the dinitro-product.

But notwithstanding many efforts, I have never succeeded in
obtainng a mononitro-compound not even as a bye-product, when
using fuming nitrie acid.

The mere formula of 1-2-3-5-tribromotoluene does not at once raise
a suspicion that a dinitro-product will be formed in this ease. If,
however, the analogy of (CH;) in the position 1 and Sr in the
position 2 is really so great that the difference amounts almost to
nothing, we may surmise that the 1-2-3-5-tribromotolwene will behave
on nitration also nearly quite analogously to the other isomer. The
experiment shows that in this case also not a trace of any mononitro-
derivative is obtained; we obtain exclusively a dinitro-product, which
is in all respects quite analogous with the above named dinitro-
derivative.

After nitration by red fuming nitric acid (sp. gr. 1,516 at 16°) in
the cold, separation by adding an excess of water, agitation with
benzene and ether and recrystallisation from benzene, in which both
isomers are very soluble, the two substances are at once obtained
pure in large colorless or pale sherry-colored crystals, whose bromine-
amount corresponds with that of the dinitro-derivative.

1) Nevite and Winruer, Journ. Chem. Soc. Vol. 37. 438; Berl. Ber. 18. 974.
2) Wrostewsky, Ann. d. Chemie 168. 147.

re -

; ( 193 )

The 1-2-4-6-tribromo-3-5-dinitro-toluene melts at 220°; the 1-2-3-5-
tribvomo-4-6-dinitro-toluene at 210°.

Like the two ¢ribromotoluenes themselves, these substances are again
quite dsomorphous and owing to peculiar twin-formation, they so
resemble each other, that at first sight we cannot distinguish the
two kinds of crystals from each other.

a. 1-2-4-6-tribromo-3-5-dinitro-toluene.

C, Br. Br. Br. (NO,) . (NO,) . (CH,); melting point 220°.
(6) @) (2) (5) (3) (1)

From benzene this substance crystallises in large apparently qua-
dratic, colorless crystals which are nearly all twins, — which may be
recognised at once by a very fine diagonal on two of the broadest
planes. We also may obtain needle shaped or very elongated thick
pillar shaped crystals. The planes are generally angular and give plural

—
So SESE Sey

( 194 )

reflexes. The crystals are also frequently bordered by curved planes
and by vicinal forms in the prism-zone. These circumstances render
an accurate investigation very difficult; occasionally, however, I
obtained betier formed crystals, which gave very sharp reflexes and
served for the following accurate measurements.

They are monoclino-prismatic with the axial relation :

a:b:c=0,5217: 1: 0,7803
B= 85°12".

Forms observed are: m = {110} and p = {120}, broad and lustrous;
a = {100} and n={130}, very narrow; ais generally hazy; b={010}, a
little broader, but is generally absent; c={O0O01}, large and very lustrous;
r= {101}, well developed and lustrous; ¢= {104}, narrower and
is ofien absent; o={112!, generally small and dull, oceasionally a
little broader and better reflecting; s = {132!, large and lustrous, but
generally there are only two parallel planes present.

Combinations of a// the forms rarely occur. Generally such with
M,p,c,7; m,p,b,c,T,t,s and 0; m,p,a,6 and c, ete. The more
typical crystals are shown in figures 1-3.

Fig. 2. Fig. 3.

In the properly formed erystals, the angular values are very
constant; the reflexes are as sharp as possible. The substance has
a decided tendency to twin-formation; in this, one form {102} is a

twin-plane with a twin-axis
plane of p may

found.

The following are the calculated and observed angular values.

arn —= (110)
m
Mra —— (1 O)\:;
Res n> == )(O01):
r:a =(101):
e:'d-— (001):
ee m= (OL):
aupe=—= (OOK):
e7e — (O01):
€2 0+ (001):
pe: 1 (120):
nip = (AL0)::
p:r = (420):
Gertie (OOD:
E79 = (O01):
Gass (O0N)::
m:s = (110):

A distinct cleavability was not observed.

On c, 7, and a the position of the optical elasticity-axis is orien-
tated perpendicularly to the direction of the orthodiagonal ;
symmetrical angle of extinction on m amounts to 23° with regard
An axial image could not be observed.

The sp. gr. of the crystals is 2, ele at 15°; the equivalent volume

to the vertical axis.

is, therefore :

170,6.

The topical axes are:

2a: @ = 38,9087 : 7,4921 : 58461.

: (110) =
:@ = (110) : (100) = 27 281/,

(195 )

Observed:
54° 561/,"

normally
be often observed a delicate line parallel p :c¢;
in this vertical zone the most important geometrical anomalies are

(f01) — 43 20

(101) = 59
(100) = 35
(100) = 85
(110) = 85
(120) = 86
(130) = 87
(010) = 89
Geo) aa
(120) = 18
(01) 55
(104) = 21
(i) et
G22)
(132) 47

39
20
15
44'/,
40
30
58
a?
35
25)
alt,
CONE
49
36"),

6b. 1-2-3-5-tribromo-4-6-dinitrotoluene.

Co Be. Br, Br. m0); pO). (CH); goeleng
()

(3) (2)

standing

‘aleulated:

on it. On the

27° 281/,!
43 13
35 33
85 12

BS 54
47 351/,

the

point: 210°C,

( 196 )

SS This compound crystallises from
—o N ~ |! benzene in very large, colorless, iso-
metrical-developed crystals, which are
always twins and of exactly the same
form as that of the previous compound
|| with which this substance is isomor-
| phous.
et The geometrical anomalies caused
| by the curvature or angulation of the
planes are more considerable with this
derivative, than with the previous one;
| 3 the development of the crystals is less
| perfect and they also exhibit a smaller

a’
¢ Pe al |; number of combining forms. From
ee = y i :
“—____._ | ———"_ ether and acetone we obtain besides
Fig. 4. iwin-crystals also single needles which

can be measured more accurately.
The symmetry is monoclino-prismatic; the axial relation is:

@ 000 Doo al Oso ee
ff = 86°28".

Forms observed are: m = {110} and p = {120}, broad and lustrous;
c= {001}, very lustrous and well developed; 7 = {101}, smaller
but properly measurable; 4 = {010}, narrow and often absent.
Angular values:

Observed : Calculated :
+p tp a0) (120) == S$) 3) —
+n ¢—1 20) (O01) S iasorf —
*y : 7 = (120) : (101) = 56 53'/, =

Dt —(taG) RO) —— 18 50 18°49’
m:m==(110):(410) = 5654 56 36
m: 7 == (110): (101) = 44 57 45 2
c: r=(001): (101) = 57 6 56 55
m: ¢ == (110) : (001) = 86 59 86 52"/,

A distinct cleavability was not observed.

This substance also has a decided tendency to twin-formation
towards {102} as twin-plane; single crystals are rare. Owing to this
peculiarity, the external resemblance of this isomer with the former
is increased in a high degree. It should be observed that these twins

and also those of the former substance often also show {010} and

symmetrical vertical lines on the prism p, with regard to the twin-
section. The sp. gr. of a perfectly homogenous fragment of a crystal
was found to be 2,465 at 15°; the equivalent-volume is therefore
169,98 and the topical axes become:

X:W: wo = 4,0286: 7,4713 : 5,6580.

I have tried, of course, to gain some insight into the progressive
course of the binary meltingpoint-line of these two isomorphous
derivatives, although the want of sufficient supply of material proved
a great obstacle.

This investigation had, however, soon to be abandoned because the
mixtures of the two substances assuming a much darker colour are
decomposed at their melting point with violent evolution of gas; the
temperatures are situated between 210° and 220°. The more the mixture
contains of the compound with the higher melting point, the more
readily the decomposition takes place, and therefore the only mixtures
of which I could sharply determine the melting point, were those
containing from 0°/, to 44'/,°/, of the derivative with the higher
melting point; this melted at 214°, the others between 210° and 214°.
I, therefore, suspect that we have here a continuous melting curve ;
whether an absolute maximum occurs in the meltingpoint-line, such
as happens with the two non-nitrated initial products, could not
be decided. The mutual behaviour of these two isomorphous deri-
vatives probably corresponds entirely with that of the two tribro-
motoluenes.

From mixed solutions of the two nitro-derivatives in benzene we
obtain large curve-planed and badly formed mixed crystals which
exhibit the same typical twin formation as the two components, but
which generally possess only {110}, {120} and {001}. They are unsuitable
for measuring purposes.

ec. 1-3-4-5-tribromo-2-6-dinitro-toluene.

C,. Br. Br. Br.(NO,).(NO,).(CH,) ; melting point: 216°C.
6) @) @) (6) (2) (1)

When recrystallised from benzene, in which this compound, which
was prepared in the same manner as the preceding ones, is less
soluble at the ordinary temperature than the two other isomers, the
substance is obtained in thick, quadratically-bounded, crystalline
plates, which at first sight appear tetragonal, an occurrence to be
expected, when taking into account the slight morphotropical force
of the (NO,)-group, and the previously found tetragonal symmetry
of the corresponding tribromotoluene.

(198 )

The investigation, however, soon showed that the symmetry is
not tetragonal, and I first thought I had before me a monoclinic
compound, — certainly with plenty of geometrical anomalies, — but still
reasonably conforming to all the requirements of the monoclinic
symmetry. This would also have agreed with everything that I
have formerly communicated as to the formrelation of tetragonal
and monoclinic crystals.

I hesitated a long time before I could give up this last idea,
particularly because the measurement of a very large number of
crystals taught me, that in this case just as with the two former
compounds, the peculiar softness of the substance causes geometrical
disturbances of the crystals during the crystallisation, and because
the axial elements of this compound when assuming the monoclinic
symmetry exhibited an analogy bordering on isomorphy with those
of the two investigated isomers.

The deviations of the angles from the values required for the
monoclinic symmetry appeared however, to be so systematical, that
there could be no longer a doubt as to the ¢riclinic symmetry of
these remarkable crystals, even though they were found to represent
a monoclinic limit-form.

Afterwards, I have also been able to find exceedingly small devia-
tions in the optical orientation of the directions of extinction which
again corroborated my belief, that triclinic erystals of a pseudo-
monoclinic form are present here. This makes us feel convinced that
even if the morphotropical force of the single (NO,) group is gene-
rally feeble fvo such groups may cause a comparatively strong
deformation of the crystal-molecule ; and also that the spacial structure
of the benzene derivatives is not fully explained by the usual formu-
lation of these substances. For the ordinary schematical manner of
writing alone does not explain why a compound of the type 1 should
have another symmetry than one of the type I, — even though, owing
to the occurrence of the pseudo-monoclinic /imit-form, the close
relation of the two kinds of symmetry is in each case very plainly

perceptible :
CH, Te
ane co.) 7 \qno.)
oe, => lin
Br\ f° Br Bry Br
Br Br

I Il

( 199 )

The erystals of the 1-3-4-5-tribromo-
2-6-dinitro-toluene ave represented in
Fig. 5 in the form in which they
crystallise from a mixture of carbon
disulphide and benzene. The quadra-
tical looking plate-like crystals, deposited
from benzene only, exhibit nearly the
same forms; in their case, however, the
form a is very predominating ; then
follows c, whilst the other forms are
narrow and little developed; moreover
the form gq is always and o often
absent. I also occasionally obtained
from benzene, thick prismatic crystals
with a, 6 and © and even o.

Triclino-pinacoidal. The axial rela-
tion is:

a@:6:¢= 0,5322 :1:0,9581

a = 88°21’. A = 88°26’.
@=95° 4’. B=95° 2’.
y = 90°58’. C = 90°50’.

Forms observed are: a= {100}, and 6 = {010} equally strongly
developed, @ generally more lustrous than 4; ¢ = {OO1} very lustrous;
o = {122! well developed and lustrous; @ = f112} smaller than o
but very lustrous; g = {012}, small and less conspicuous; m = {110},
narrow, but broader than p = {110}, which form is often present
with a single plane; s’ = {344} very narrow and° not properly
measurable *).

Observed: Calculated :
e000) (O10) ==" 89° 10" _
7a) C= (00). OL) — 84 58 —
ae —= (O10) 2 (O01). 91-34 —
*9 : a@ = (122) : (100) = 59 35 =
*9 : ¢= (122):(001)= 55 25'/, —
i p— (Oo) say 27.32 27° 451/,’
a :m= (100): (410) = 28 7 28 7
1) If, in view of the symbol of s' we rather choose 0 = {243}, » = {223}

Si {343}, whilst retaining the other symbols, we get @:b:¢=0,5322; 1: 0,7186,
which agrees better with the other two isomers,

( 200 )

Observed. Calculated.
b : o= (010): (122) = 54 33 54 34
b :m= (010): (110) ti 8) 62 51
Hh: — (OL0) 3070)" 61-35 61 24 i
eo: 7 (001): (012) == 26) 5 25 50
6b: 7— (O20) (01.2): == 65.39 65 49
o :m= (122):(110)= 48 21 43 39
o : s’ = (122):(844)—= 11 52(cirea) 12 8
w: 6 = (112): (010) = 108 49/, 108 47
Dy? VS (112) : (122) == ihe ail 16 39
w: c=(112):(001)= 48 17, 48 0'/,
w :m= (112);(110) = 47 30 Ay Sty
a : w= (100): (112) = 53 35 53 351/,
a): igi —14 00) (012) == Sa Bi 85 55
oc — (112) : (012) = 40 35 40 30°/,
q : 0= (012): (122)= 38 56 38 54
m: c= (110): (001) = 84 47 84 45°/,
pc 410)-(001) ==. 66724 86 19

The position of the optical elasticity-axis on @ was determined by
means of a very thin lamelle, with the aid of a Bertrand Quartz-
Ocular, and found to be 4° with the side (100): (010); on 4 the
angle amounts to more than 45° with respect to the side (110) : (O10).
An axial image could not be observed.

By means of a THourEt’s solution, the sp. gr. of the crystals was
found to be 2,459 at 17°; the equivalent volume is, therefore,
170,39, and the topical axial relation becomes :

ZY: W: w = 3,6982 : 6,9496 : 6,6584.

The close relation of this triclinic derivative with the two mono-
clinie isomers is therefore obvious. That there is still an essential
difference in the nature of the molecular orientation, which in these
three cases determines the crystalline structure, is also shown by
the fact, that the first two derivatives exhibit a decided tendency to
a similar and typical twin-formation, which is utterly absent in the
latter isomer.

I will give lateron some information as to the other nitro-produets
of the isomeric tribromotoluenes, as these have not as yet been

obtainable in measurable crystals.

ee

( 201 )

Mathematics. — Prof. Kivyver presents a paper: “valuation of
two dejinite integrals.”

Supposing w to be real the integrals

* 608 aut sin vt
Te, iene (pepe dt and (a, n=fo G-paym @

will have a Rete meaning, if only the real part of the parameter
m be positive. In what follows we will show how to expand these
integrals into rapidly converging power series.

The first of them, the integral /(#,m), is a particular solution
of the linear equation

dy
a(n — 1) ay == 0;
dx*

the primitive of which, involving two arbitrary constants A and 5,
may be written in the form

y =AL (a, m) + Bu! M (a, m),
where

a \2h
2
(a ping !n(—

m+ 3+ h)’
h=0

\2h
h=o i
2
UA. epi Ne
Mies) De h! Dim+4-+h)
h=0
and the constants 4 and £6 must now be determined so that y
: 3 if
represents the function / (7, im). To tind A, we suppose m > 3 and

put w equal to zero. In that case we have

a dt r(4) P(m—3) A
FOm= [i = = A L (0, m) = ——____

(1+ #7)" 2 Dm) T'(— m + 3)
0

and hence
‘ sg r(3)

2coeam P (m)

A=

For the deduction of the constant 6 it is convenient to consider
first the function /(v,m) in another form, Let the real part of m
still be positive, then we have

— .

( 202 )
x
I’(m) fo i+) m—l

—————— ne u du

(Ife)
0

and hence
* 2 a 3
‘age —ul? so Ty
T'(m) 7 (em) =o ‘infe cos xt dt = } vafe ee aes
0 0 ,

From the latter integral a simple functional relation is derivable.
r

Changing the variable w into 4, We may write
=

aw \2n—1 ree =a
T'(m) 7 («, my =5V =(= ‘i fe be) [a0

0

» \2m—1
= (5) I'(1 — m) 7 (#, 1 — m)

=

‘
.

and so it follows that the function

T(m) f(x, m) fa 4 i z\—m
== —T(-)(-) Le,
5)" 2Zeosxm \2/\2 Soe.

a \n—l
+ 22¢—1 B I'(m) (5) M (x, m)

remains unaltered, if m is replaced by 1 — m.
Now obviously the series 1 and J/ are connected by the relation
L («, 1 — m) =-M (a, m),
hence we must have

x il
22a—1 B U(m) = + ——_ TL 6!

2cosam

B a (3) 1 \2m—1
~ Qeosam” I (m) (5)
and therefore ;

= cos xt 2 ee T(4) C¥y, i 2m Vv
less) =f ciee! ~~ eos am” I(m) en) ee

Now it will be observed, that the series LZ (w,m) and A (x, m)
converge for all values of « and mm, and so we must conclude, that
the function .f (7,1) exists over the whole .x-plane, that its only
singularities are #«=0O and «=a, and that therefore the integral,
we started with, represents the function ina very incomplete manner,

( 203 )

Numerical evaluation of the integral for not too large values of 7
offers” no difficulties, as the series L (ain) and M (a,m) converge
rapidly. Because of the equation

a 2 3

] ‘4 Gene ue
P'(m) f(a, m) = = vr{ e du 2 du

0
the result will always be a positive number and the integral will
not vanish for any real value of
A few further remarks may be made. Firstly we may state, that
f(a, m) is intimately connected with Brssrn’s function /, (7). In fact,
by means of the usual expansion of -/, (7) we may verify the relation

1 ; “A
: = ray (#\"—2| ~Fm-3) ae
ere ~ Qvos um > Fa ; oon aon me \ °° =

From this we infer, that for positive integer values of mm the origin
v=Q0 ceases to be a singular point, and that / (7, 7) ean be expressed
in finite terms. We shall find by actual substitution of the finite

me a
expressions for ./ 1 (« ") and J (w |
=m i=

h=m—1

: a e-% aw \e—-l ~~ (m—1-+h)! 1\4
J (es m) 2 <a 2; ) comet m—i—h)! 2a:

h=0

pol

However this result may be obtained in a simpler way as follows.
It can be shewn, that /(w, 1) obeys the relation

= m4 vite (—1) hD(m+h)

a

u Va, m) her ———_ a) m +h),

I
T(m)

al

and since we have
Sle Va, 1) ={2 sacks “lt — Ea ee 5
148 2
we gef for all positive values of im
. are met (a2 |
a “(m—1)! = | Va Ve

a result that can be identified with that obtained before.

The singularity in the origin «=O becomes logarithmic, when
2m is an odd integer 2/--+- 1. The expression of f(@,m) is in this
14

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 204 )

case somewhat intricate. By repeated differentiations it is derived

1
from i(« >) for we have
, 1 —1EM(k+3 1
Dp rfl (ea ear = eee — ( ra) vs k + — }.
2S 2 Mie 2

1 1
ro evaluate 7 é >) we put m= - > -+ 4 in the general expres-

sion for /(v,7), and make d tend to zero. In this way we get

v 2h an 2h
ies & r 59 ae 4) |
7 («. = = rim — a NS = (; y Vv 2

s— 0 Zsin ad |— i! T(h+1—9)

Cp N\2h )

= h=o{
: L\ ( cos at ae 2 1 Ae 1 C—1 i |
1s ae ee |

0 h=v

We shall now pass on to consider the second integral

sin ut
gp (uyn= aeye “

and it will appear that its character is quite similar to that of the first.

Again we transform the integral by the aid of the identity

a
P(m
( is =| e— UCL) yn —1 du

(pe
0
and obtain
a na
I'(m) — (we, m) =| e—% yn—1 iw | e—ul sn at dt.
0 0
A further transformation gives
l
a 2 “2;
ei os if CL?) =
e sin atidt = e 3u dt = — du (l—w) | = ae
; Qu
0 0
and therefore
is I ae atw
= wv 3 ean hee
I (m) @ («, m) = “i (l—w) 2 dwfe du y du.
0 0

Comparing this equation with the equality obtained before

=— th 2 m :
Bn) Fala 7) — 5 Va | e Au y, 2 lu,

it follows, that we may write

1

ih ]
EF (m) @ («, m) = a fa-w 2 dw Vn — vy (or W, mM — — ) Aye
oY *% a

0

; ]
We now expand P(m— $)/ (« Yw,m — —

) and make the sub-

stitution

‘ SE Eee es !
(m—4)f | «© Vw,m— 9) > Win am © M(3) lee BY wm — 9 15
w\2n—2 1 }
45 a M| a Vw, m — 2 (

Then integrating with respeet to iw, we find the desired expansion
of g(v,m) in the form

ier aut oe mu I(%) | Vv ae i
ec) =Ja 1+? ym Qsinam 1 T(m) eT) ) ae PNG

where .V (a, 7m) represents the new series

Zz 2h-+1
h=oa (=)
9
= =

N(«,m) = N°- = - =F aeanTanTTe :
GM (ae + 3) P(—m+2+h)
feo

The same remarks as were made concerning the first integral
f(v,m), can here be made again. The integral has only a meaning
for real values of « and for positive values of m, but from the
expansion is inferred, that the integral incompletely represents a
function of w which exists over the whole «-plane, quite indepen-
dently of the values assigned to the parameter im. Again the origin
vO and «=o are the only singularities of the function. The
singularities are logarithmic, when 77 is an integer and the origin
becomes a regular point, when 27 is equal to an odd integer

1) It is possible to invert this relation. It may be shewn that we have also
1
I (m) 7 (w, m) — 5 Va T(m—})=

G 1

Ey 5 1
foo 2 du rim — a (« Vir,m——).

4
0

14%

( 206 )

2h-+-1, but in no case is a finite expression by means of elementary
funetions obtainable.
The function g (wv, im) as well as f(w, 7) satisfies the relation

|

h mf —i)k '(m h)
2 su gp (e,m + h)

\
D4) a = 9p (uw Ya, m) T(m)

9

]
and by means of this rule expansions for g (v,/) and ¢ (« k+ )

may be deduced from the equations

h=o
sin wt ~ ah+l 1 ] ] 1
t—N ——— —-—— =| + ...——_ — C—loge |=
(wt) =f" La Qh i 22 gt gooey ae
h=0
1 .; ye }
= = \ —< Li (er) = et Lh et,
and
+ h=oa (-3)
1 s at fd ae 4 2
> ss um a= Bes Beek
2 Vite ey reer 2
0 =o Fi> +1
2
Botany. — On “Leptostroma austriacum Ovp., a hitherto

unknown Leptostromacea living on the needles of Pinus austriaca ;
and on Hymenopsis Typhae (Fuck.) Sace., a hitherto
insufyiciently described Tuberculariacea, occurring on the wi-
thered leafsheaths of Typha latifolia.” By Prof. C. A. J. A.
OuDEMANS.

1. LEPTOSTROMA AUSTRIACUM Ovo.
(Plate 1)

On the 13% of June 1904 1 received trom Dr. J. Rirzema Bos,
Professor at Amsterdam, a number of specimens of transplanted seedlings
of Pinus austriaca, originating from Schoorl, all dead and of which
the accompanying letter informed me that the roots showed here
and there cushionlike prominences, the surface of which was covered
with shuttle-shaped conidia, divided into cells, and the microscopic
properties of which resembled most those of conidia of the genus
Fusariun.

Besides | found, without my attention having been directed to
it, that most needles of the dead plantlets were spotted on both

( 207 )

sides with small black specks and streaks, the external appearance
of which showed most resemblance with the perithecia peculiar to
Leptostroma ov Leptothyrinun.

The plants sent to me, provided with a here and there ramified
tap-root of about 1 decimetre length and 1—38 millimetres thickness,
proved on closer inspection to have much suffered, since in various
places the bark was loose from the wooden kernel, if it was
not entirely lacking. These circumstances justified the supposition
that the young pine-trees had succumbed under the attack of the
Fusarium-plantlets and that the Leptostroma- or Leptothyrium-
individuals had chosen the sickly, lingering or dying needles as the
seat of their fatal activity.

The /usarivm-cushions that had remained were little numerous,
1—3 mm. in diameter and had a light rosy tint. Lacking suitable
objects for investigation, I had to restrict my answer to the com-
munication that here very likely Musarium rosewm had been active,
and I left the further elucidation of the devastation caused by that
fungus to the care of Prof. Rirzmma bos.

A closer examination of the very numerous specks and streaks
found on the needles of Pinus austriaca, induced me, on account
of their generally elongated, sometimes more, sometimes less hysterium-
like shape, their little tendency to loosen at the circumference
and to fall off, the fact that nowhere a parenchymatic structure
of the perithecium-wall could be distinguished and that the basidia
had not developed, to think rather of the genus Leptostroma than
of Leptothyrium, and besides to mark the fungus as non-deseribed
and to give the name Leptostroma austriacum to it in order to
distinguish it from other fungi.

One of the characteristics of Leplostroma austriacum is that the
perithecia are never united to continuous series, but rather form
greater or smaller groups of streaks or small shields, which differ
greatly among each other in size, and are rather dull than glossy.
Their length varies from '/, to 1 mm. and their breadth from */, to
“, mm. Their perithecium-wall is ‘thalved*t, as the term is, does
not reach further than the epidermis of the leaf, and consequently
has the shape of a cupola. This wall has no foundation or basis.
Moreover it is black, carbonaceous and structureless, so that there
can be no doubt that we have here a cuticle (Fig. 2 and 3), from
which follows that the space, occupied by spores, rests on the epidermis,
as is clearly shown by Figs. 2 and 3. By reasoning more even than
by observation, one is lead to the conclusion that the spores are
produced by a very thin layer of threads extending over the epidermis.

( 208 )

Above this layer the spores form two layers or storeys. A third
layer does not exist, as the space, required for it, is occupied by the
spores which have loosened themselves and have become entangled.

The spores have an elongated (cylindrical?) shape and are colour-
less and undivided. Their foot is rounded and encloses (Fig. 4 and 5)
a circular or oval, glossy vacuole; their top is more pointed and
empty. They measure 7—S¢ in length and 1'/, u in breadth.

The difference between Leptostroma austriacum and other Lepto-
stromata, peculiar to pine-needles, like L. Pimorum, L. Pinastri and
others, is: that in the latter the perithecia form mostly narrow parallel
series; that the spores are not broader than 0.5 4, and finally, that
no vacuoles are found.

The Latin diagnosis of the new species is as follows:

“Peritheciis cuticulam inter et epidermidem occultatis, amphigenis,
irregulariter distributis, majoribus et minoribus, item longioribus et
brevioribus intermixtis, dimidiatis, nigris, opacis, diu clausis, tandem
irregulariter ruptis, persistentibus neque decedentibus nec cireumeires
a substrato solutis. Sporulis sessilibus, cylindraceis, hyalinis, continuis,
vulgo 7.51, basi rotundatis guttulaque sphaerica vel ovali, mieante,

>

praeditis, apice acutiuseulis, vacuis.’

EXPLANATION OF THE FIGURES OF PLATE I.
Fig 1. A piece of a needle of Pinus austriaca with small heaps of perithecia
(p.) on them. (°/;).
, 2. Vertical section of a not yet fully mature perithecium.
a. Cuticle.
b. Epidermis.
c. The two layers of rod-shaped colourless spores. (°0°/,).
, 3. Vertical section of a ripe perithecium which has burst open.
a. Cuticle.
b. Epidermis.
c. Spores, partly undamaged, partly in a displaced position. (5°°/)).
, 4. Spores, with a rounded foot and a sharper top. At the foot a vacuole.
(1000/,).
» v. The same (*900/,),

2. HYMENOPSIS TYPHAE (Fuck.) Sacc:
(Plate II).
This fungus, found for the first time at Nunspeet in July 1904 on

the withered leaves of Typha latifolia, was sent to me among many
others by Mr. C. A. G. Bens.

( 209 )

Unlike the Sphaerellae and Leptothyria it has not the appearance
of small specks but of raised black spots (Figs. 1, 2 and 3) which
are spread in the grooves between the nerves and have a length
of 1—4 and a breadth of '/, mm.

Fvuexen deseribed the fungus first under the name of JM/yrothecium
Typhae (Symb. 364), in the following words : “Peridiis hemisphaericis,
oblongis, '/, lineam longis, aterrimis; conidiis oblongo-ovatis, utrimque
obtusis, simplicibus, biguttulatis, 15 >5< 64, pallide fuscis,” and gave
a not quite satisfactory picture of a conidium in Fig. 21 of Plate 1.

He was succeeded by Saccarpo (Syllabus IV, 745), who agreed
with his predecessor that the fungus belongs to the Tuberculariaceae,
but nevertheless removed it to the genus Hymenopsis, on account
of the spore-bed (sporodochium) of Myrotheerum being surrounded
by a circle of fringes, which is not the case with MHymenopsis.

In a very successful drawing by Mr. C. J. Kontne of a vertical
section of Hymenopsis Typhae, (Plate 2), the structure of the fungus
is excellently seen, much better than in other pictures, also of other
species of the same genus.

Where the black disks or specks rise above the surface of the
leaf-sheaths (Figs. 1, 2 and 3), one does not find, as FuckrL writes,
a “perithecium” (i.e. a more or less completely occluded fruit-body),
but a globular assemblage of reproductive cells or conidia (Fig. 4 s.s.),
covered by the cuticle and produced by a layer of peculiarly shaped
sporophores (Fig. 4 «.), collectively called stroma or fruit-bed. Under
this stroma the epidermis is found (Fig. 4 0.): a layer easily recog-
nisable by the width of its cells. It deserves to be mentioned that
the black colour of the prominent little disks (Figs. 1, 2 and 3) must
not be ascribed to the colourless cuticle (¢), nor to the colourless
epidermis (0), but only to the conidia (Fig. 5 y) which have been
left uncoloured, however, in Fig. 5, in order not to make the picture
too full.

One of the most important Figures of Plate 2 is Fig. 5. At a it
shows the favoured club-shaped threads or basidia, whose task is
the production of the conidia; these latter, let free by their bearers,
being seen in their neighbourhood in a free condition (y). The conidia
have an elongated, cylindrical shape, are more or less asymmetrical
or curved, rounded at both ends, somewhat more transparent at the
base and of the grey colour of mice. (Sacc. Chromotaxia, pl. I, Fig. 3).
They contain 2—4 consecutive vacuoles each and have a length of
about 10 and a breadth of about 4 uw.

Comparing the Figures of Plates ! and II, one might get the
impression that in the Figures 2 and 3 of Plate I a perithecium is

( 210 )

lacking as well as in those of Plate II, althongh this term is usual
in descriptions of the Leptostromaceae. Therefore we remark that this
latter family of the Sphaeropsideae forms a transition between the
perithecium-bearing and the peritheciumless forms and that in judging
these two cases weight has been attached to the black colour of the
upper half of the shields, which sometimes consists of the cuticle
only, sometimes of a combination of the cuticle with the epidermis.

In addition to this the Leptostromaceae do not produce well-
developed basidia and have remarkably small spores.

The Latin diagnosis of Hymenopsis Typhae is as follows:

“Sporodochiis amphigenis, hemisphaericis, inaequaliter in vaginarum
suleis distributis, majoribus et minoribus, item longioribus et orbicu-
laribus intermixtis, primo cuticulam inter et epidermidem caelatis,
1—18 «w in diam., aterrimis; denique expositis, calvis, thalamo basidio-
phoro basilari: praeditis ; basidiis dense fasciculatis, clongato-clavatis,
hyalinis, continuis; conidiis oblongis, rectis vel paullo curvatis et
utplurimum inaequilateralibus, utrimque obtusis, basi vulgo clarioribus,
104, murinis (Sace. Chromotaxia Tab. I, f. 3), 2—4-guttulatis,
guttulis hyalinis, nune binis sibi oppositis, tance iterum ternis (aut
quaternis) in seriem dispositis.””

EXPLANATION OF THE FIGURES OF PLATE IL.

Fig. 1. Piece of a leafsheath of Typha latifolia, studded in the grooves between
the nerves with sporodochia (spora = spore; docheion = receptacle),
(natural size); p.p. perithecia.

, 2. Piece of a leafsheath with two sporodochia, of which one is opened, the
other closed (#°/)).

, 3 Piece of a leafsheath with two sporodochia, of which one has a groove
on the dorsal side (#°/;).

> 4. Vertical section of a ripe sporodochium. — c¢.c. cuticle; 0.0. epidermis;
s.s. conidia; 7b. vb. vascular bundles ; 2. club-shaped basidia.

>» 9 A bundle of club-shaped basidia (2...) with some conidia (y. y.), in which
two or three vacuoles. The end of the conidia resting on the basidia
or turned towards them is always somewhat more transparent than

the other.

Plate II.

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J. BiuTeL Lith., P. J. MuLpER Impr. Leiden.

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(20 |}

Botany. — “On Sclerotiopsis pityophila (Corba) Ovp., a
Sphaeropsidea occurring on the needles of Pinus silvestris.”
By Prof. C. A. J. A. OupEmans.

In the “Nederlandseh Kruidkundig Archief’’, 3¢ series, vol. I, pag.
247, | mentioned a fungus found in 1901 by Mr. C. A. G. Berns
at Nunspeet on the needles of Pinus silvestris, which fungus, dis-
covered in 1840 on the same host near Prague by the botanist
A. J. C. Corpa, was deseribed in vol. IV of his “Icones Fungorum”
on page 40, under the name of Sphaeronema pythiophilun *)

The same fungus received a place in Saccarpo’s “Sylloge Fun-
gorum’, vol. II. (A° 1884), p. 101, this time under the name of
Phoma pityophila, whereas on account of a new investigation of
fresh specimens I thought it necessary myself, in the article quoted
above, to change the name Phoma again and to replace it by that
of Sclerotiopsis.

Besides Saccarpo, also ALLESCHER, in the 6% vol. of WurytrErR’s
Kryptogamen-Flora (1901), page 199, uses the name Phoma pityophila
for this fungus, which name is changed into Sclerotiopsis, by way
of improvement, in vol. VI, p. 847 of the same work.

Having been enabled through the kindness of Mr. Bers in January
1904, to examine again some fresh specimens of Sce/erotiopsis
pityophila, | availed myself of this opportunity of testing once more
my former experience by facts and had the advantage of having at
my disposal the drawings by Mr. C. J. Koning, chemist at Bussum,
which accompany this article. | have to thank Mr. Koning for the
kindness which he has repeatedly shown in assisting me on former
oceasions as well as on this. Some particulars supplementing former
communications may be mentioned here.

The reason that induced Saccarpo in 1884 to change the name
Sphaeronema into Phoma was that some very characteristic properties
of the former genus had been passed over silently by Corba, viz.
that in his paper no mention is made either of a beak- or brush-
shaped prolongation of the pevitheciumwall or of spores which,
conglomerated to a ball, should have been found at the surface of
the perithecia.

The generic name chosen by Corba could not be retained and so
no other name seemed more appropriate to the Italian mycologist
to replace it than that of Phoma, which judgment has not been
doubted by any subsequent writer.

') The Greek for pine being zétu¢, in what follows Corpa’s wrong orthography
has been corrected.

( 212 )

Meanwhile it was evident as well from the very brief deseription
of Phoma pityophila in Saccarvo’s Sylloge as from his silence on
the microscopic properties of the fungus, that this author had not
been able to examine freshly collected specimens, so that mycologists
working after him under more favourable conditions might possibly
find something to improve.

Having had this opportunity myself it may not be superfluous to
return once more to my Sclerotiopsis pityophila and to consider more
fully the difference between Sclerotiopsis and Phoma.

First of all it must be mentioned that the perithecia of Phoma,
when produced by leaves, although they le concealed below the
epidermis, yet are by no means buried deep in the tissue as is the

ease with Sc/erotiopsis (Fig. 3—5) and probably on account of this
are much more irregularly shaped, sometimes coalesce and come
forth with a stronger and less rounded appearance.

Secondly any one who has examined many specimens of Phoma
must have noticed that with Sc/lerotiopsis stronger and denser peri-
thecia are found which are carbonaceous at the surface, whereas
those of Phoma belong to the forms that offer little resistance, and
are tender and light-coloured ; finally that the perithecia of Sc/eroti-
opsis have no orifice but decay or burst, whereas with Phoma the
rule is that a small round ostiolum is found through which the
spores are discharged.

In addition to this we remark that the spores of Sclerotiopsis do
not lie loosely together like those of Phoma, but remain long con-
nected by means of a sticky substance (fig. 3 and 4), the consequence
of which is that a few drops of water are sufficient to cause Phoma-
spores to diverge in all directions whereas with Sc/erotiopsis a slight
pressure or friction is required to make them fit for a closer exa-
mination.

This latter peculiarity was exactly the reason why Corba imagined
to have found a Sphaeronema, overlooking that the beak- or brush-
shaped prolongation of the peritheciummouth was absent and that
consequently no cluster of spores could be formed at the top of
such a prolongation.

The question whether the spores of Se/erotiopsis are produced on
the top of sporophores is difficult to answer, although analogy pleads
for it, since there is no distinct division between the wall of the
perithecium and the gleba (the cluster of spores) but a gradual
transition of one into the other. Yet not far from the surface of the
perithecia ((Fig. 6) a segmentation seems to take place and the
formed spores seem to be slowly pushed to the centre.

Sclerotiopsis pityophila (Corda) Oup., a saprophyte, appears
as black, fleshy grains (Fig. 1), '/,—-2 mm. broad, which are expelled
from the tissue of the needles. They consist of polygonal parenchym-
cells which at the circumference are larger, harder and darker but
in the interior become smaller, softer and colourless and seem to
border on a small cavity, which is soon filled with spores. These
latter are oval or egg-shaped, straight or slightly curved (Fig. 7),
unicellular and undivided and have rounded tops. They vary from
7—8X38—-4u, have no polar drops and no appendices. Germi-
nating spores were not found.

The first Sclerotiopsis was found by Spweazzini in the Argentine
republic on rotting leaves of Lucalyptus Globulus and was called
S. australasiaca. A second and third species (Se/. Cheiri Ovp. and
Sel. Potentillae Ovv.) were found by myself and Mr. Berrys, the
former on the stems of Cheiranthus Cheiri in the Botanic Garden at
Amsterdam, the latter on the leaves of Potentilla procumbens at Nun-
speet. Finally Corba first mentions Se/. pityophila (Corda) Ovv. which
was collected in 1840 on pine-needles at Prague and 60 years later
at Nunspeet.

EXPLANATION OF THE FIGURES.

Pig. 1. A few needles of Pinus silvestris studded with perithecia of Sclerotiopsis
pityophila (Corda) Oup. — Natural size.

, 2. A needle of Scl. pityophila loaded with some perithecia. Cross-Section.
Magnification 100.

» 3, 4 and 5. Vertical sections of Scl. pityophila, magnified 400 times. The
carbonaceous wall of the perithecium is clearly visible here everywhere.
In 3 and 5 the perithecia have broken through the epidermis, in 4 not

yet; in the former two also the conglomerated spores are discerned.

a
oz

A piece of a peripheral part of the wall of the perithecium with some
stalked spores. Magn. 1000.

» 7. Single spores, 1000 times enlarged,

( 214)

Geology. — “On an equivalent of the Cromer Forest-Bed, in the
Netherlands.” By Prof. Eve. Dusois. (Communicated by Prof.
Kk. Martin).

On the eastern frontier of the Netherlands, along the middle third
part of the province of Limburg, there is the steep west border of
a plateau, made up of gravels and sands, which, for the greater part,
is enclosed between the valleys of the Meuse, the Niers and the Roer
and rises to an average height of about thirty meters above the adjoining
low land. That border is falling within the Dutch frontier opposite
Venlo, Tegelen and Belfeld, further, east of Swalmen and of Herken-
bosch. The plateau is a piece of a formerly coherent, much larger
plateau, extending to Nimeguen and Cleves, of which, according to
Dr. Lori¥, the Veluwe is also a part. This author is inclined to suppose,
that the large mass was still entire at the time of the principal
extension of the Scandinavian Ice-sheet and that only after the retreat
of that ice-sheet, by melting, the great eroding process began, which
divided it into a number of pieces and also assailed each of these ;
so that under consideration was partly divided by the Swalm and
the Nette with their affluents. Dr. Lorn showed that the northern
and eastern parts of the platean do not merely consist of ‘Rhine-
diluvium”, as SvTarinG supposed for the whole till Nimeguen, but
that these northern and eastern parts expose traces of having
been reached by the Seandinavian Ice-sheet of the great Glacial
Epoch, in consequence of which they consist, at least at the surface,
of “Mixed Diluvium”. This is not the case with the western piece
of the plateau of gravel, which we consider now more particularly.
In this only stones are found, which have been transported by the
Rhine and its large tributary, the Meuse; further, the horizontal
stratification has not been disturbed by an ice-sheet having moved
over the plateau.

Nevertheless here too the characteristics of the co-operation of ice
in the transport of the sand and the gravel, out which the mass
has been made up, are not wanting, but these occur to great depth
in it, till 80 metres and more below the surface, and, as already
stated above, the horizontal structure has not been disturbed by
an ice-sheet having moved over the surface. This stratification, with
fluviatile carrent-bedding, can be observed in a number of places
where gravel is dug. At the same time there are repeatedly found,
among sand and finer gravel, big angular stones.

So 1 observed in a gravel-pit in the Jammerdaalsche Heide, opposite
Tegelen, the following boulders, which were found on an area of

( 215 )

about one heetare, and 2 or 38 M. above the basis of the sand and

eravel deposit: A large boulder of Thonschiefer, of 1.85 % 0.75 > 0.35
M., and three smaller ones, of about 0.5 M. greatest dimension, a
boulder of veined gray quartzite, of 0.80 >< 0.75 >< 0.50 oy
another gray quartzite of 0.67 x 0.36 x 0.20 M., a flint nodule ¢
0.60 >< 0.85 0.15 M. Other large stones were knocked into pieces.
East of Belfeld boulders are not so frequent in the gravel. Amongst
others I observed there a basalt of about 0.40 M. largest dimension.
From these observations we are led to suppose a transport on a
large scale by floating ice, and we can imagine that ice having
had its origin, in the upper-course of the Rhine and the Meuse, from
bottom-ice. The basal part of the deposit, 2 M. thick east of Tegelen,
5 M. thick east of Belfeld, is, however, entirely devoid of pebbles,
it consists of rather fine sate

All this induces us to consider this ‘“Rhine-diluvium” as a glacio-
fluvial formation of the first Pleistocene Glacial epoch, chronologi-
cally the equivalent of the thuvioglacial Deckenschotter of the Dilu-
vium of the Middle-Rhine.

This interpretation is now affirmed by the character of the bed
underlying the gravels and sands in the plateau in consideration. Save
eravel and sand there is dug clay, which furnishes the material for
the many tileries and stone-factories, in a great number of places,
of the Netherland province of Limburg and of the adjoining region
of the Rhine-Province of Prussia, chiefly on the borders and along
the transverse valleys, of the Swalm, ete. That clay is lying confor-
mably and with not eroded, rather well horizontal separating plane
under the “Rhine-diluvium’, the equivalent of the Dechenschotter.
Her own planes of stratification ave also generally horizontal. In the
clay-pit of the well-known stone-factory of the firm Canoy-HERFKENS,
on the western border of the Jammerdaalsche Heide, her upper surface
is at 27 M.+ A-P. East of Belfeld, near Maalbeek, 4.5 K.M. 5.S.W.,
| found that surface at 35 M.-+- A.P. East of Reuver and 8.5 K.M.
S.5.W. of the pit opposite Tegelen, it is at 43 M. + A.P. East
of Swalmen, near the Duteh Custom-house on the frontier, 14 K.M.
south-west of the pit in the Jammerdaalsche Heide, it is at 50.5 + A.P.*)
The same clay is also dug roundabout Briiggen, on the Swalm, in
the Khine-Province, 5 to 8 K.M. east of the pit near the Custom-
house. It is probably also the same clay, which is met with, at the
surface, east of the Zwartwater, (north of Venlo), and west of the
plateau, in the communes of Tegelen, Belfeld Reuver.

Evidently this clay constitutes a continuous bed underlying the

') In the Dutch text of this communication the altitudes were only estimated,
this english version they are given from exact determinations by levelling.

( 216 )

“Rhine-diluvium”’, which has a regular gentle upward slope to the
south and probably also to the east, and of which rests appear in
the low country bounding the plateau to the west, where the ‘“Rhine-
diluvium” has been removed.

There is not much known about the total thickness of the bed, by
reason of the under layer of it having not yet been attained in any
pit. In the Jammerdaalsche Heide the clay is dug out 6 M. deep,
and it has further been ascertained by means of bore holes, that even
2 M. deeper, so at about 19 M. + A.P., the clay makes place for
sand. It is however probable that another layer of clay is underlying
that sand. In bore holes put down on several places in and about
Venlo, some of which approached the last mentioned pit to 2*/, K.M.,
they met with similar clay, in a layer of 8 M. thickness, resting, at
4M.—A.P., on coarse white sand and gravel with much mica, and
covered at 4M.-+ A.P. by 3 M. sand and about 12 M. of gravel’).
Along the right bank of the Meuse, south of Venlo, the edge of this
very ferrugineous and somewhat consolidated gravel appears, covered
by loam, at about 14 M.-+ A.P. Even at very low watermark, ofa
few decimeter above 8 M. + A.P., generally the underlying of the
gravel cannot be seen in this outcrop. On a few spots however,
about 1 K.M. south of the Meuse-bridge and 2 K.M. north-west of
the mentioned clay-pit, I observed similar clay as that of Tegelen
in the original situation, over 7 M. in horizontal connection, under
the gravel. It reaches there upward to 11 M. + A.P. Evidently this
clay in the bank of the Meuse belongs to the same bed as that which
was met with in the bore holes at Venlo, the bed having been unevenly
eroded a long time before the development of the present river channel.
In such a way a difference of 7 M. could arise in the upper sur-
face of the clay. In that clay on the right bank of the Mense I
found a tibia of Rhinoceros, which is only assimilable with that bone
of R. etruscus and R. Merchki. The bone was still a little fastened in
the clay, for the greater part enveloped with the consolidated gravel.
This clay is thereby characterised as interglacial or preglacial (pre-
pleistocene). If belonging to the same bed as the Clay of Tegelen, the
whole thickness of the latter, including sandlayers, may be estimated
at about 30 M. In this computation it has been supposed that its
under surface, from Venlo to Tegelen, is horizontal, which seems

1) According to a communication of Mr. pe Waau Materur the top of the clay
was 5 M. lower in a bore hole put down on the right Meuse-bank at Venlo. Pro-
bably the clay which has been met with in bore holes, 24 K.M. south of Venlo,
on the east side of the Meuse in the neighbourhood of Roermond, at about
3 M.-+-A.P., under as much gravel, is geologically identical with that under Venlo«

( 217)

allowable over such a small distance, in comparison with the great
extension of the bed, and with regard to the horizontal structure of
the clay. Then we have to regard it as prepleistocene, an inter-
pretation entirely confirmed by the following palacontological facts.
Much more improbable I hold it that the clay under Venlo and
on the Meuse near that own, was deposited after the first Pleisto-
cene Glacial epoch, that of the “Rhine-diluvium’. In that case we
should be obliged to suppose two periods of erosion. In_ the first
one the “Rhine-diluvium”, with the underlying Clay of Tegelen in
the valley of the Meuse, should have been eroded, afterwards (during
an interglacial period) clay should again have been deposited in it,
which was attacked in a second period of erosion, on which than
in the Second or Great Pleistocene Glacial epoch a deposit of gravel
accumulated.

For the chronology of the different beds of the Dutch Pleistocene
formations now it is of great importance to ascertain, by means of
enclosed fossils, the age of the Clay of Tegelen, which was deposited
in the time preceding the accumulation of the “Rhine-diluvium”’.

I am much obliged to Mr. L. Stuns, at the time medical student,
now physician at Roermond, for having shown me, already in 1897,
fossil remains of Mammals (especially Zroyontherium and Deer) and
of Molluses, together with such of plants, which he had found in
the clay-pit of Messrs. Canoy, Hrrrkens and Smunpers, a number of
which he has yielded to me for a closer examination. I have further
to acknowledge the benevolence of the last named gentlemen for
the opportunity of collecting some fine and characteristic fossil remains
of Mammals, especially of Cervus, Rhinoceros, Equus, Hippopotamus and
Trogontherium (now in Teyler Museum at Haarlem), by the aid of
which the fixing of the geological horizon has been arrived at. The
shells and plant remains (especially seeds and wood) and many bones are
found at about 5 M. below the upper surface of the clay bed, where this
is rather sandy, another, more abundant, ossiferous niveau is at nearly
3M. below that upper surface, in stiff clay '). Opposite Belfeld bones

') To an average of 2.70 M. below the upper surface, from below which a very stiff
clay begins, the clay in ihis pit has a yellow colour, caused by the action of the
atmospherilia on the ferrugineous compounds in the clay, which action is lower down
shut off by that stiff clay. The latter itself is of a bluish colour and at the bottom
of the pit it is nearly black. Excepted near the upper surface, the yellow clay is
on the whole sandy, only at a few places in the pit it is rather stiff. In those
places the blue colour continues up to a higher level and the limiting line is not
at ail right and horizontal, on the contrary the yellow clay, there, is sinking
down, in that blue clay, which continues to a relatively higher level. Agatiform
wrinkling brown parallel lines, in those yellow insinkings, imitate then contortions,

( 218 )

are mostly found at a depth of 4 M. in the clay; at 1.25 M. below its
upper surface it there encloses a layer of sand, 0.30 M. thick. The outside
of the bones is always absolutely uningured, they do not look rolled
worn at all. The following enumeration of fossil forms will suffice
for the determination of the geological horizon. I hope to be able
to work out and to complete the list on a later occasion.

As regarding the Molluscs, it is in the first place noteworthy,
that these for the most part belong to forms proper to fresh water,
and especially to stagnant or very slowly running water; a few land-
snails belong to species which may have lived upon the vegetation on
the shore. Till now I have recognised :

Fig. 1. — Cervus teguliensis, sp. n. Left antler, lateral aspect. (1/s).

The figured specimen belongs to the collection of Mr. Stuns. Several other
specimen do not possess the strong, curvature of the beam at the
origin of the tres-tine, in such a manner that the beam is
on the whole straighter.

Paludina, 2 Sp., Planorbis Sp., Helix: hispida | WES Helix arbustorum L.,
Helix sp., Limnaeus sp., Pisidium, 2 sp.,° Unto sp.

Of the Mammals the following species are well determinable :

Trogontherium Curvieri Owen, Cervus Sedguwickit Fale. (= Cervus
dicranius Nesti), Cervus teguliensis, sp. n., Cervus (eis) rhenanus, sp. 0.,
Cervus (Avis) sp., Hippopotamus amphibius L., Equus Stenonis Cocchi,
Rhinoceros etruscus Fale.
such as they have been produced elsewhere by the motion of the Pleistocene
ice-sheet, but here we have indeed only before us a result of the process of the
blue clay to yellow. Elsewhere, as opposite Belfeld, where still the original thick

gravel bed covers the clay, and consequently the underground water is at a higher
level, has preserved the greyish blue colour up to its upper surface.

( 219 )

There are found too remains of: Cstudo lutaria Mavyrsili, of a
Frog and of Fishes.

Of great importance are the remains of
plants, from which especially seeds were
carefully collected by Mr. Stns. They
enable us to form a conclusion concerning
the climate and thereby concerning the
time in which the plants lived. And that
is so much the more desirable as remains
of Elephants have not yet been encoun-
tered, among the mammalian remains. The
species already determined are :

Viburnum sp., Prunus sp., Trapa natans
L., Cornus mas l., Vitis vinifera L., Sta-
phylea pinnata L., Juglans tephrodes Ung.,
Nuphar luteum \.., Stratiotes Websteri Pot.,
Abies pectinata DC., Chara sp.

That is an assemblage of animals and
plants which can only be preglacial in
the sense of prepleistocene. The group of
Mammals is distinguished from that of the

Sands of Moosbach, which are now gene-

Fig. 2. — Cervus (Avis) rhena-
nus, sp. n. Right antler, y , :
lateral aspect. (1/s). glacial period before the great or second

Pleistocene Glacial epoch, by the possession of Hquus Stenonis*),

rally regarded as a deposit of the inter-

of two species of Deer of the Acs type and an other species
belonging to a type not represented by any living Deer. They give
to the whole a Tertiary character and make, for themselves, the
equivalence of the Clay of Tegelen with the Cromer Forest-Bed
probable. From the last mentioned deposit we know one species Deer
of the Avis-group (C. etueriarum, probably nearly related to the new
species from Tegelen), a second species has been described from the
somewhat older Norwich Crag; from the Pliocene of central France
there are described as many as six species. Cervus teguliensis closely
resembles C. fetraceros, Boyd Dawkins, from the youngest beds of that
Pliocene, characterised by Llephas meridionalis, and from the Cromer
Foresi-Bed, but the antler of the large Deer of Tegelen never obtained
more than three tines. The other Mammals of the Clay of Tegelen
awe all known from the Cromer Bed. The presence of Lquas Stenonis
and Rhinoceros etruscus together with Trogontherium Cuvieri and
Hippopotamus amphibius major leaves no doubt on that equivalence.

') Of this species it is the variety distinguished by Mr. M. Bouts as the one with
great dimensions, which he believes to be the immediate ancestor of EB. cuballus.

15
Proceedings Royal Acad. Amsterdam. Vol. VIL.

What is known from the species of plants in the Clay of Tegelen,
points even to a somewhat warmer climate than that indicated by
the flora of the Cromer Bed, and so, seemingly, to a somewhat older
age. The Prunus-species is certainly different from P. spinosa L.,
which belongs to the Cromer fossil flora, and which is now also
indigenous in northern Europe. The seed is only assimilable with
that of species, which now appear to be spontaneous in Turkey,
south of the Caucasus, in Armenia and northern Persia. Amongst the
plants of the Cromer Bed Vitis vinifera is also wanting, which grows
now spontaneously in temperate West-Asia, especially in Armenia
and south of the Caucasus and the Caspian Sea, also in southern
Europe, Algeria and Marocco. Remains are also found in Pleistocene
iravertines of Tuscany and southern France, where the species, with
Ficus carica \.., is considered to be a remnant of the Tertiary flora,
further in Italian lake-dwellings. Amongst the numerous species of
plants of the Cromer Bed is wanting too Staphylea pinnata. In wild
state this species does not grow now more northerly than southern
Germany, it is especially indignous in the Pontine countries and
also in Asia Minor. No species of /uglans has been found in the
Cromer Bed, Juglans tephrodes, the nut of which, like that of some
nearly related forms, is hardly discernable from the present American
Juglans cinerea I., is a Tertiary species of Italia and the middle of
Germany. The seed of the Stratiotes is very different from that of
S. aloides, on the contrary strikingly similar to that of S. Webstert
from the Upper Miocene of the Wetterau. In the Cromer Bed as yet no
Viburnum was found. The seed of the species from the Clay of
Tegelen closely resembles that of V. Opulus L., it is only larger
and a little less flat. The circumstance, that the genus Viburnum
played an important role among the Tertiary flora seems to me not
to be without bearing, in connexion with the above mentioned facts.
A similar consideration applies to the genus Cornus, of which
another species, C. sanguinea lL. is found in fossil state on the coast
of Norfolk. C. mas appears to grow, besides in Asia, only in southern
and central Europe. C. sanguinea, on the other hand is also indige-
nous in northern Europe.

It wants no demonstration that the flora from the Travertines of
Taubach, to which, on conclusive grounds, the same age as that of
the Sands of Moosbach is now attributed, is a much younger one
than that of the Clay of Tegelen. The former contains arctic and
alpine forms, which are wanting here, on the other hand the fossil
flora of Taubach lacks the mentioned Tertiary forms and_ those
pointing to a warmer climate. From the fact that the flora of Tegelen

apparently lived in a somewhat warmer climate than that of Cromer
we are, however, not obliged to conclude that the former is older
than the latter. For we have to consider, that the situation of Tegelen
is about 2° of latitude more south than Cromer, but especially, as
Prestwich and Crement Rew have shown, that local circumstances
must have made the climate of Cromer relatively a less genial one.
Of the Mammals, by which the older Pliocene deposits of Norfolk
are distinguished from the Cromer Bed none are found in the Clay
of Tegelen.

Taking all these facts into serious consideration, there seems to me
hardly to remain any room for doubting the equivalence of the latter
with the Cromer Forest-Bed. Like this celebrated fluviatile and
estuarine deposit and like the undermost gravel beds of Saint-Prest
near Chartres, the alluvia characterised by Llephas meridionalis in
central-France and the lignite beds of Leffe near Gaudino, not far
from Bergamo, they must be placed at the top of the Pliocene.

On good reasons it is generally accepted, that at the end of the
Pliocene period the continual subsidence during that period, the
unmistakable proofs of which have been found as well in the Netherlands
and Belgium as in England, has been interrupted by an uprising
of the region, properly a flattening of that great concave zone or
geosynclinal, in which the marine Pliocene sediments were deposited.
In consequence the southern half of the North Sea was converted
into land and England united with the continent. The great river
of that sedimentation basin, the Rhine, as has been shown by
Ciement Rew and by Harmer, then poured its waters over the east
of England into the North Sea, and in Norfolk the Cromer Forest-Bed
is a deposit of that river. Also Harmer rightly remarked, already in
1896, that this river, before it reached England must have passed
somewhere over the Netherlands; so we should perhaps one day find
the equivalent of the ossiferous beds of Cromer in our Country. In
the Clay of Tegelen we now have really met with such a bed,
which evidently accumulated in a shallow fresh-water lake, flow
through by the Rhine.

On good reasons it is also accepted that with the beginning of
the Pleistocene period the geosynclinal became steeper, in conse-
quence of which, over the greater part of the present Netherlands,
sand, gravel and clay could accumulate, attaining a thickness, in
Holland, up to more than 150 M. But, at the same time, on the
border of the steeper basin, in consequence of its larger angle of slope
and the increased transporting capacity of the running waters there,
first deposition of coarser material, the “Rhine-diluvium’’, took place

A55

( 222 )

and, as the slope increased still more, an important erosion. By
this erosion the Meuse could excavate a broad valley through the
Rhine-diluvium and deep in the Upper Pliocene clay, in which, the
slope having somewhat decreased, probably already in the second
or great Pleistocene Glacial Epoch, there accumulated a mighty
deposit of gravel.

Physics. — Contributions to the knowledge of vax per WAALS’ w-
surface. IX. The conditions of coexistence of binary mixtures
of normal substances according to the law of corresponding
states. By Dr. H. KameruincH Onnes and Dr. C. ZAKRZEWSKI.
Supplement N°. 8 to the communications from the Physical
Laboratory at Leiden.

(Communicated in the meeting of February 27, 1904).

1. The graphical treatment of the conditions of coexistence. In this
paper where the theory of mixtures of VAN DER WaALs is illustrated,
as in the former contributions we have placed in the foreground
the law of corresponding states.

The data required calculating vaN DER WaaLs’ w-surfaces for all
temperatures may be defined in the following way from the point
of view of this law:

1°. An equation of state agreeing with reality, must be given for
one normal substance over the whole range of temperatures and
pressures to be obtained, (comp. § 2).

2°. For the different mixtures of the two substances considered, as
well as for these substances themselves, the deviations from the law
of corresponding states must be known (comp. § 3).

3°. We must know the critical temperature 7’, and pressure
pe of each mixture taken as homogeneous’) with the molecular
proportion z of one of the components, derived from the law of cor-
responding states, as functions of those of the simple substances and
of a (comp. § 4).

With these data at our disposal vAN Der WaAAts’ theory will teach
us all possible cases of coexisting phases of those substances if we
roll tangent planes over the y-surfaces of each pair of substances for
different temperatures.

In the treatment of the problems concerning conditions of coexistence

1) Whenever we speak of critical temperature, a maximum vapour tension ete.
of a mixture without more, we always mean “laken as homogeneous’’,

(993°)

which have been solved in general by van per WaAats, the following
simplifications have been made:

1°. The first form with two constants @ and 6, in which van per
Waats has written the equation of state, is used instead of the real
equation of state ;

2°. The supposition has been made that also the equation of state
of each mixture with the proportion « has the same form with
two constants a and 6,, whereby the law of corresponding: states is
rigorously satisfied ;

3°. it has been supposed that the critical quantities determined by
ad, and b, are related to those of the simple substances determined
by a,, and 6,,, a,, and 6,, by means of the relations

dy = a,, 2? + 24,,¢ (1 — 2) +4,, (1 — a)’

by = b,, a7 + 26,,07(1 — «) + d,, (1 — a)’,
so that the entire behaviour of the mixtures of two known substances
is determined by two additional constants a,, and 0,, ;

and 4°. it has often been assumed that the vapour phase satisfies
the laws of ideal gases.

In this way vAN per WaAAts has obtained important approximation
formulae. Though they do not always represent numerically accurately
the behaviour of the mixtures, most of the particularities of the con-
ditions of coexistence are sufficiently explained by these in general
valid formulae *).

If in the treatment of these problems we want fo use equations
of state which over the whole range of temperatures and pressures
agree accurately with the observations, if therefore we do not require
the simplifications ad 1°, 2° and 3°, and if finally we want to
consider other than rarefied vapour phases, so that the neglection of
the deviations from the law of BoyLe-Gay Lussac-AvoGapRo mentioned
ad 4° is not allowed, an analytical treatment of the conditions of
coexistence in general becomes impossible.

Such problems we come across, for instance, when we derive
the conditions of coexistence for mixtures of oxygen and nitrogen
(critical state of air and relation between composition and pressure
in the boiling off liquid air) from equations of state which also at
the’ ordinary temperature accurately represent the compressibility of
these substances and of their mixtures. An instance of an entirely
different kind is given by the following group of problems : determine
at ordinary temperature the absorption of hydrogen in ether and the

1) van peR Waats, Die Continuitiit ete. Il p. 52.

( 224 )

deviations from the law of Henry for this pair of substances, investigate
the variations of this absorption with a small variation of temperature
and finally find for the same temperature the pressure to which
Kunpt ought to have gone in his experiments*) on the removal of
the capillary ascension of liquid by pressing gas on it in order to
see the meniscus of ether disappear under the pressure of hydrogen,
which, as has been remarked in van Enprk’s thesis for the doctorate
p- 7, comes to a determination of the plaitpoint pressure of the
mixture of ether and hydrogen of which the plaitpoint lies at this
temperature *).

In such cases we can only obtain solutions by means of the
graphical treatment described in Comm. N°. 59a (Sept. 1900). It
is true that the graphical method lacks the general character of
the approximate solutions just mentioned, yet by means of it a better
numerical agreement may be obtained for each special case. *) By
a proper choice of special cases some data may also be derived
for the qualitative characters of phenomena in mixtures *).

If we only aim at such qualitative results we may simplify the
graphical treatment as well as the analytical by introducing different
approximations according to the nature of the problem, while it lies
at hand to derive wanting experimental data or results of calcula-
tions from empirical formulae on a larger scale than in the analytical
treatment. For instance, everything that may be neglected in the
analytical treatment may also be neglected in the graphical method.
Occasion for this exists only, however, with problems which neither
qualitatively are solved by means of the analytical method.

If we do not want to neglect to such an extent as in the
analytical treatment, we might for instance retain the neglection of
the deviations mentioned sub 2° of different normal substances and
mixtures of normal substances from the law of corresponding states,
which first occurs in Comm. N°. 59a and is kept up in this whole
series of contributions.

1) Repeated | by van Etpik (thesis for the doctorate; Leiden 1898) for hydrogen-
ether and ethylene-methylchloride.

2) Comp. also van peR Waats, Die Continuitat ete. Il, p. 136.

3) For instance, one of us and Remyeanum have derived (Gomm. N?. 59D Sept. 1900)
a numerical fairly approximate representation of the retrograde condensation
observed by Kuenen with mixtures of methylchloride and carbon dioxide from the
isothermals observed by him at other temperatures.

+) So for instance the character of the retrograde condensation (comp. note 2),
and also the peculiarity in the conditions of coexistence in mixtures of which the
critical temperature varies almost linearly with the composition, far below the

critical temperature. Comp. these contributions Comm. N°. 59a § 8 at the end
and III, Comm. N°. 64. Harrman, Livre jubil. Lorentz, p. 640.

The simplifications relating to the case that one of the two
coexisting phases as compared with the other has a very small
density, will be considered in the sections 7 and 8 of this Commu-
nication.

§ 2. Empirical reduced equation of state. As to the supposition
mentioned in § 1, we are now under much more favourable cireum-
stances than at the time when Comm. N°. 59@ was written.

The beauty of van per Waats’ theory lies above all in the fact
that it brings under one point of view phenomena in mixtures which are
distributed over a large range of temperatures and densities. Hence
for a satisfactory illustration of this theory we require first of all
an equation of state which holds true over a large range of tempe-
ratures and densities. Now most of the equations of state — this has
been made clear especially by D. BErraeLot — hold only for a limited
range. For considerations as are meant here probably only those
equations of state can serve which are developed in series and made
to agree with the observations over a very large range. Such equa-
tions of state which are very suitable for the calculation have been
obtained in Comms. N°. 71 (June ’O1) and N°. 74 (Livre jubil.
Bosscua p. 874) by combining as well as possible the known pieces
of reduced equations of state for substances with different critical
temperatures. As now we neglect the deviations from the law of
corresponding states in the different substances and in their mixtures,
we may without more base our considerations on a similar empirical
reduced equation of state.

We have used a form which does not differ much from the more
preliminary one given in Comm. N°. 74, which was indicated by
1’ 2. We obtained it by making it agree with hydrogen 0°C."), oxygen
and nitrogen 0° C. (all of Amacar) and ether 0°C., 100° C., 195°C.
(Amacat, Ramsay and Youne). This polynomial, which contains for
instance all the reduced temperatures which occur on the y-surface
for ether and hydrogen at O° C., will be designated by VJ 1. As in
Comms. N’. 71 and 74 we have for a substance with the critical
temperature 7), and pressure pr, if v is expressed in the theoretical
normal volume,

Bare SUD A CE, LE
Parana tin eatin etsy. 3 yom boepede 1)

hi
Where at an absolute temperature of ¢ above freezing-point
A= 1 + 0.0036625 ¢

1) For hydrogen the critical quantities of Orszewskt are still used in the calculation,
*) Comp. Comm. No. 71 form. (10).

( 226 )

% fb f big H hy Al Be ie
%, C=—G& D=—=~5, r= €& —* §
Pk Pk pk Pk Pk

op ; : : : vf
and the reduced virial coefficients $, €, D, €, § with t= 7, are
k

determined by

Reduced virial coefficients VI. 4.
? Pa Amie
108 3 | 189.523¢ —465.197 — 21.291-- —184.410- 5
|
8 il vel 1
10Uu ¢ | 58.598 ¢ + 23.55 — 14.451 a +159.936 3 =F 21.692—
1 ] 1
a, 482.544 ¢ —379.527 — 562.94 3 +203.384 5 —158.21575
Ml 1 1
10% © | —1910.43¢ +6797.37 — 5322— +1143.47 =
| 1 i
10°? ¥ | 2052.16 —7742.41 +7204.66 t —1848.03, + 192.5975

BP

The calculations which show the systematical deviations of different
normal substances from a similar equation of state are progressing.

§ 3. Validity of the law of corresponding states for mixtures. We
can judge of this much better now than when Comm. No. 59a was
written. The different applications given’ in this series of Contributions
to the knowledge of vAN per WaAALs’ y-surface ') seem to indicate
that the deviations in mixtures of normal substances are not much
larger than those which occur in normal substances inter se.

This is very striking, especially if we look for the basis of the
law of corresponding states in the mechanical similarity of the move-
ments, as a mixture even geometrically is not at all similar to a simple
substance. It would seem to follow from this, that the linear quan-
tity which determines the geometrical scale of similarity is a mean
value of very different linear quantities, which play parts in different
collisions. In that case we should have to attach to the “volume of
a molecule” not so much a physical meaning and rather the geome-
trical meaning of a sphere drawn with this linear quantity as a radius.

To establish a systematical relation in the deviations of the mix-
tures of normal substances from the law of corresponding states
is not likely to be feasible until this has been done for the devia-

1) Comm. No. 59) KamertincH Onnes and Reteanum. Comm. No. 64, Harrman.
Comms. Nos. 65, 81, Suppls. Nos. 5, 6, 7 Verscuarrerr, Comms. No. 75 and
No. 79 Kersom.

( 227 )

tions of the normal substances themselves. Still a beginning has been
made with calculations which aim at a representation of those deviations.

§ 4. Determination of the critical quantities of the mixtures taken
as homogeneous. As we neglect the deviations from the law of cor-
responding states, we may derive these quantities (occurring in the
unstable region and hence not to be determined directly) from any
observed range of the equation of state of any mixture.

The most obvious means is the shifting of logarithmical or partly
invariant diagrams of isothermals in the area near the critical state.

v
In Comm. N°. 594 it was applied to P" isothermals with regard to

)
7
logv, in Comm. N°. 65 to log p-isothermals with regard to /og v, in

wv,
Comm. N°. 88 to log *-isothermals with regard to log p and to

Ov ; F :
log 7 -isothermals with regard to /ogv. We may also imagine,

however, that we have at our disposal a sufficient number of obser-
vations of another range. Thus, to give a simple instance, we know
the critical temperature of a mixture if the temperature is found, at
which under relatively small pressures it does not deviate from the
law of Boyie. And it is also possible that we may derive the data
from observed conditions of coexistence. If the critical quantities for
some mixtures are found, it will in graphical solutions be preferable
to derive the 7°, and p,, as a function of « also graphically. For
the experiments of Kurnen have made us doubt whether the suppo-
sition made ad 3° is in general possible, and this doubt is strengthened
by Krrsom’s experiments.

If on the other hand we confine ourselves to qualitative investi-
gations of mixtures of substances about which all the data which
belong to the mixing are lacking, the supposition first lies at hand that
@,=V4,,4,,') and 6,, =3(6,, + 4,,).

§ 5. The reduced y-curves. In Comm. N°. 59a is briefly set forth
how the different y,-curves can be derived from those that have
been calculated once for all for a simple substance. If we write a
little more extensively, and if v, indicates such a large volume that
with this the mixtures are in the ideal gaseous state,

v

Wry = — | pdv + RT ja log « + (1—2) log (1--2)}
omitting a temperature function linear in 2. This with

1) Gauirzine and D, BertHetor.

Pk vk 1 Pp
aA PH S>—;
RT; Ce Pk Ui:
may be easily transformed into:
»
‘xe 1 1 N Pak
RT ———— C, = feds — log Ton + wx log « + (1—2«) log (1—z) . (1)

nH
if we put 2 a certain large number and neglect other temperature
functions.
»
For convenience we may call {pas as function of v the curve of
n

reduced free energy for t. In the construction of each given ¥-surface
occurs the group of curves of reduced w that lie between the extreme
values of reduced temperature which occur on this surface.

On the planes =O and «=1 of the y-model we can draw
the y-curves of higher and lower temperature. In passing over to
a y-surface of higher temperature the w-curves (at least in the most
common case) are moved on the surface from the side of the
highest reduced temperature to that of the lowest, while the linear
dimensions in the two directions yw and v undergo a certain variation.

§ 6. The w-surface for mixtures of methylchloride and carbon
diovide at — 25°C. As an example of the application of the graphical
method and the empirical reduced equation of state we have now
chosen the prediction of the composition of the coexisting phases
and the coexistence pressure for mixtures of methylchloride and
carbon dioxide at — 25°C. We were led to this choice by the
following considerations :

1°. we can derive the critical quantities from the experiments of
Kurnen'), they lie tolerably far from each other, are given in
Comm. N°. 594, while for + 9.°5 C. a model has been constructed
by HarrMan;

2°. —25° C. corresponds to the lowest-reduced temperature for
methylehloride for which the empirical reduced equation of state has
been calculated ;

1) As critical quantities are used, comp. Gomm. N°. 590,

x Trk Pk
carbon dioxide 0 303 72.2
If, 336 73
1, 363 71.8
3/4 391 68.9

methylchloride 1 416 64.8

H. KAMERLINGH ONNES and C. ZAKRZEWSEI.

Contributions to the
knowledge of VAN DER WAALS’ J-surface. IX The conditions of
coexistence of binary mixtures of normal substances according to the
law of corresponding states.”

Fig iL

Proceedings Royal Acad. Amsterdam. Vol. VII

3°. with no other equation of state suitable for a calculation we can
reach such a low temperature ;

4°, van ppr Waats, for similar circumstances as are found on
this surface, has also derived analytically the particularities in the
coexistence-phenomena (Contin. II, p. 146, sqq.), and the agreement
between these results and the observations of carbon dioxide and
methylehloride, found partly already by Hartman at + 9.°5 C., will
probably appear even more clearly at — 25° C.,

5°. finally that a paper on an experimental determination of these
conditions of coexistence, which also vAN per Waats thinks very
desirable (Contin. I, p. 154) will, as we hope, soon be published.

The numerical agreement becomes less accurate because methy|-chlo-
ride is not similar to ether, with which substance the empirical reduced
equation of state for the reduced temperature of the methylchloride
on this y-surface is made to agree whereas this is the case with carbon
dioxide at the reduced temperature, which it has on this y-surface. An
empirical reduced equation of state in good harmony for the reduced
temperature of 0.6 with methylehloride of — 25° C., for the reduced
temperature of 0.8 with carbon dioxide of — 25° C., would have
been more favourable for the obtainment of a numerical agreement.

The plaster model obtained is represented on the annexed plate
fig. 1, it is 0,7 m. long (v-axis), 0.4 m. high (y-axis), 0,38 m. broad
(z-axis). The large dimension in length was made necessary by the
great difference in density of the vapour phase between carbon dioxide
and methylehloride.

The binodal curve and the tangents which connect two coexisting
phases, the nodal lines, are found by rolling a piece of plate-glass.
Fig. 2 shows the binodal curve with nodal lines, and also sections
v =const. projected on the «y-plane, fig. 3 the same on the zv-plane;
fig. 4 shows the values of the pressure as function of the composi-
tion of the coexisting phases.

It is obvious that:

1°. the liquid ridge, i.e. that part of the surface which lies on
the side of the small volumes, when the dimensions of the surface
are not taken extraordinarily large, becomes -very thin and the con-
struction is practically possible only if we take for it a plate of
uniform thickness (for instance a sheet of tin);

2°. while the tangent plane is rolled over the ridge of the liquid
part and over the convex vapour surface, as the ridge near the pure
methylchloride rapidly changes its direction, the point of contact
moves a long way on the vapour branch of the binodal line,
while the node on the liquid branch moves only a little. Hence in

( 230 )

the directions of the tangents the double fan-shape is very prominent.

3°. the vapour branch of the binodal line in the projection is
almost a straight line, and in agreement with this, according to
VAN peR Waats, the law of Henry holds over the whole area of
composition variation, while the vapour branch on the pa-diagram
is again almost a hyperbola.

§ 7. Simplification of the determination of conditions of coexistence
when the liquid phase is far below its *) critical temperature. In order
to determine the binodal line of the transverse plait we need only
know two zones on either side of the plait. Let the border curve of
the homogeneous mixtures be that curve which on the y-surface con-
nects the vapour and the liquid phases in which the mixture, with
the composition « taken as homogeneous, would be in equilibrium,
then the binodal line wanted lies beyond *) this border curve, which
it meets for the compositions 0 and «. Therefore it is more or less
indicated, which zone on the vapour side we have to calculate.

If, as in the case of methylchloride and carbon dioxide at — 25° C.
(or with ether and hydrogen at the ordinary temperature for the part
of the ether side) the zone on the liquid side is shrunk to a plate,
we need only calculate a single curve on the liquid side. For then
the point of contact is so near the rim curve — that for which y is
a minimum —- that this curve may be substituted for the y-surface.
The v of this curve may be easily derived for each « from the
equation of state because @ = — p= 0. With the value v,~o or

e
‘yo we then find by integration w,—o, while in the way as described
in Comm. n° 66 § 5,*) the tangent through the point y—o, v)=o
to the curve w, on the vapour surface is drawn graphically and the
vapour tension of the homogeneous mixture is obtained.

If we only wish to determine the pressure and the composition of
the coexisting phases we have also a neglection of little importance
in the supposition that ryge = Vig, @ + Vig, (A—r), in other words
that the rim curve lies in a plane, while in many cases we may
suppose that this plane coincides with the wr-plane.

In order to find the conditions of coexistence we roll a plane (piece
of plate-glass) over the rim of a thin plate cut after the calculated

: : : ate
rim curve { better still the rim curve for RT and over a model

1) Comp. § 1 footnote.
2) Van per Waats Continuitiit II p. 100.
3) Arch. Néerl. Livre jubil. Lorentz, p. 660,

( 231 )

(made for instance of plaster) of the vapour surface (which if we
y MEET

choose —— and if we neglect the deviations from the law of Boyne
RT ;

is always the same).

Instead of executing this process, which directly expresses the

theory of vAN DER Waats, on the plastercast, it may also be done
entirely by a drawing on the plane as indicated in Comm. 59a § 6.
The given construction now becomes much simpler if only we
neglect the thickness of the ridge. In fig. 2 is drawn the rim
curve of the liquid plate and the different sections » = const.
Then we combine the points where the sections have the same
dw Pe ae ns : eee :
Fe (or s comp. Comm. 59a § 8) as certain points of the rim curve
to the substitution curves (comp. Comm. 59a § 5) which therefore
; x A a LU gt
always belong to the point with the given a (or s) value on the
rim curve (with mixtures taken as perfectly gaseous the substitution
curves on the «yw projection are straight lines at right angles to the
w-axis), connect the point on the rimeurve with different points ot
the substitution curve in the vapour zone, rotate the section with a
plane at right angles with the xv-plane passing through the connecting
line with the latter on the ywv-plane. The point where the section
touches the connecting line gives the phase wanted, which is in
equilibrium with the first.

The construction becomes still simpler if we neglect the distance
from the rim curve to the plane »=0. Then we need only

: B
transfer in fig. 2 the sections v= const. by A +— + .... lower,
=

and draw a common tangent to the rim curves and each of these
transferred lines in order to find aya, and ayy.

If finally we neglect the deviations from the law of Bortz, we
have only to move down unvaried the entire sections » = const. of
the vapour phase in fig. 2 in order to obtain the just mentioned
system of curves, to which with the rim curves common tangents
must be drawn in order to find P yap and wig.

§ 8. Application of the empirical lav of reduced vapour tension
of pure substances to the phenomena of coexistence in mixtures. In
§ 7 we drew the attention to the circumstance that the liquid branch
of the binodal line is obtained sufficiently accurately by substituting the

dw

rim curve, the points with (‘ :) = 0, for the points with
dv / >

( 232 )

dw ; : ‘
(7) = — Prot; M the same way we can put instead of these the
PD
: : é dw eS:
points of the border curve with (5 ’) —= — Prmax, Where px maz indi-
av /x

cates the maximum vapour tension of the mixture x, called by van
pER Waats the coincidence pressure.

i as func-
RI

According to (1) the liquid branch of the binodal (
tion of 2 is obtained as the sum of three terms.
The first
Yur =aclgr+ (l—2)log(1—2x). . . . . (2)
is only determined by the molecular composition.
The second term is derived from the function

b

lars
ta =—| se {rae| Petey eign ere ee (2)

n Y= DV jig. t
if as the higher limit in the integral we take the reduced liquid
volume vj, of a simple substance at the reduced temperature t.
We imagine that the function + a has been graphically represented

once for all as a function of f.
The properties of mixtures of a given pair of substances are
determined by

xk
i log Te > E a > ' 3 ‘ - : (4)

as a function of wv and by = as a function of «. By combining the gra-
rk
phical representation of the latter with the graphical one of gst we
obtain gz, the graphical representation of the value which the
quantity gz adopts with the value of t, which belongs to 7. Then the
liquid branch of the connodal line, and hence also the rim curve
and the border curve are satisfactorily given by :
wy ¥
Pea 1 Pin Pps pen Sos, ay On
The quantity «, which plays such an important part in the theory
of van per Waats about the coexisting phases is, while neglecting
and the
TRIE
S of van pER Waats also with the same neglection (for which
moreover a correction may easily be applied) = y.

pe, determined for the liquid pbase by 72 + yx =

H. KAMERLINGH ONNES and C. ZAKRZEWSKI. ,,Contributions to the knowledge
of VAN DER WAALS’ :-surface. IX. The conditions of coexistence of binary
mixtures of normal substances according to the law of corresponding states.”

= i6

/ 1.4

tides

1 0.8

| 0.6

ROP... 0.4

18 X 0.94

fig. 5.

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 233 )
The considerations, constructions and figures given seem therefore

suited to illustrate this part of the theory of van per Waats.

4 Al

v
is in the case of carbon dioxide and methylehloride a curve slightly
bent downwards. Fig. 5 shows how the liquid branch of the binodal

w
line (the rim curve of the liquid part) 5 (comp. also the other
L

ul
figures) follows from the curve Ui and the curve @,,.

As to the calculation of jt it should be remarked that pyra:
as function of t is known through the vapour tension law, and
hence also from the empirical equation of state »%,.,, and that

n n
[vw =| Pdv + Pnaz (Yvap — Yig)- Especially if »j, is small and
vliq Yeap
hence Yyq,) large, (so that at the utmost — comes into consideration
=
for the deviation from the ideal gas laws) this, when at the same
time neglecting %,;,, leads to an important shortening of the calculations.
Neglecting entirely the deviation of the vapour phases from the ideal
1—t
gaseous state and accepting for a simple substance log pmax = — f ——
4 are
we return to the developments given by VAN per WAALS in his theory
of the ternary mixtures’) in which theory many problems about
the binary mixtures are developed more in detail.

Physics. — “The determination of the conditions of coexistence of
vapour and liquid phases of mixtures of gases at low tempe-

ratures.” By Dr. H. KameruincH Onnes and Dr. C. ZAkrzewskl.

Communication N°. 92 from the Physical Laboratory of Leiden
by Prof. H. Kamernincu Onngs.

(Communicated in the meeting of June 25, 1904).

§ 1. Introduction. The two methods for the determination of the
molecular coexistence compositions 7, and «, of the liquid and vapour
phases of substances which are gaseous in normal conditions, it is
known, ean be described as follows. Following the first method we
separate small quantities from the two phases at a series of coexistence

') Proceedings, May 31, 1902, p. 1, sqq.

( 234 )

pressures p and determine each time the composition of those two
quantities either by chemical or by physical means.

Fig. 1. Fig. 2.

On the v.w-projection of the binodal curve of the transverse plait
on VAN DER WAALS’ wavy-surface for a given temperature 7, fig. 1,
and also on the pw diagram of that binodal line, fig. 2, two sueh
phases are indicated by a and /, for instance. The determination
of several pairs of values a, a b' ete. gives then the whole course
of p, & and v over the transverse plait for 7”.

If we follow the second method we observe in a series of mix-
tures with a known composition «, the beginning- and endcondensation
phase and determine for them p and v, hence per and pye7, and
ver, and v7. This investigation comprises each time the phases
represented in fig. 1 and 2 by 4 and ec. By combining the results
be, b’c’ ete. we can derive again the binodal line and hence the
values for 27, %pT, VpT and v7. The application of this last
method to low temperatures especially under moderate pressures,
forms the subject of this paper.

It is possible to follow also the first method in the case of low
temperatures as it has been applied by Hartman in Comm. N°. 48
(June °98) for ordinary temperatures. Yet as a rule the analysis of
a gaseous mixture is much more difficult than the preparation of a
mixture of a definite composition (among others, by means of the
mixing apparatus of Comm. N°. 84, Dee. ’02) and it is difficult to
obtain certainty whether the quantities of vapour and liquid run off
have the same composition as the phases which are brought to
equilibrium by stirring.

Therefore it is important to solve the difficulty which accompanies

( 235 )

the second method for temperatures below 45° (melting-point of
mercury). When we made a first trial in this direction, a high degree
of accuracy was not aimed at. In order to answer several questions
the accuracy we have reached is sufficient and for the calculation
of corrections for more accurate determinations it is quite sufficient.

In our measurements we have used the
apparatus which is represented schematically
in fig. 3. It is in principle a twice bent tube
of Camuerer, of which one end is immersed
in the cryostat of the temperature 7’ and

|

eS filled with a known quantity of a mixture

of known composition «, which by forcing
Fie 3 up mercury at the ordinary (or higher)
Fig. 3. ‘ A

temperature 7” is brought to condensation
at 7.

The mixture of which the quantity and the composition « are
known, is then only partly contained in the vessel 7’ at the tem-
perature of investigation. Another part is necessarily in a tube at
the ordinary or higher temperature 7" which is connected with the
vessel by means of a capillary tube. This involves complications,
not with regard to the measurement of the begin condensation
pressure p,.. for the composition «; for until the condensation of the
first traces of liquid, with respect to which the vapour is a phase
of equilibrium, the composition of the vapour phase in the space at
low temperature remains as it was originally, and hence the com-
position of the vapour phase of equilibrium is perfectly known; but
with regard to the determination of the end condensation pressure
Pu of the mixture with the composition w. For we cannot condense
the whole quantity of the mixture at low temperature. Hence
the composition of the liquid phase at 7 above which there is a
vapour phase of a composition differing in the main from this or
the original composition, is no longer indicated by the latter and
therefore unknown.

We can, however, find this composition by applying a correction
fo the original composition « which, as long as the vapour phase
occupies only a small volume and remains under a moderate pressure,
is not very large (comp. § 5).

§ 2. General arrangement of the measurements.

A schematical representation of it will be found on PI. I, fig. 1.
The letters are the same as those used on the plates to which we
shall refer.

The volumenometer / (with manometer JZ, comp. Comm. N°. 84,

16
Proceedings Royal Acad. Amsterdam. Voi. VIL.

( 236 )

March °03, Pl. Il, figs. 1 and 2) contains the gaseous mixture whieh
has been prepared in it by means of the apparatus connected at
r, and 7, (comp. Comm. n°. 84). Through the cock +,, the steel
capillary y,' and the steel three-way stopcock /: (see Comm. N°. 84,
Pl. I, fig. 2) it is led to the pressure tube 4 with calibrated stem
(see Comm. N*. 69, April 01, PI. Il) placed in the pressure cylinder 4,'
(Comm. N°. 84, Pl. 1) in order to be compressed by means of
mercury from the pressure reservoir C,' (comp. Comm. N°. 84).
Thence through the three-way stopstock / and the steel capillary
y, it ean be brought under pressure into one of the two apparatus
Dor PY. For the present, according as we wanted to determine either the
begin or end condensation pressure p..7 Or Pur, We have connected
either the capillary of D,D), or that of P Y,, to the capillary g,”
by means of sealing wax.

Different from Comm. N°. 84 Pl. I fig. 1 the mercury of the
pressure cylinder (see our plate fig. 1) is also connected by means
of n with that of a manometer to determine the pressure in D
or Pp.

By means of the three-way stopcock / the apparatus Dor P may
also be connected directly with the volumenometer and then the
pressure is measured by means of JZ.

A detailed representation of the apparatus D and is given in
figs. 2 and 3 of PI. 1 and a description in §§ 4 and 5. In both cases
the glass tube to which the mixture which is to be investigated is
conducted by means of the capillary y,", p in P and a@ in Y, is
immersed in a cryostat where the desired temperature is kept in the
same way as described in Comm. N°. 83 (Feb. and March ’03).

The regulation is brought about for ) through the exhaust-tube
7,, and for 2 through the tubes 7, and m,. In our measurements
we brought liquid methylehloride into the cryostat and the tempe-
rature was regulated according to the indications of an alcohol ther-
mometer.

By means of the cock 7, the volumenometer / is connected not only
with the gasreservoirs but also with the mercury airpump so that also
the pressure-tube and the test-tube in D and P can be exhausted.
We shall not expatiate on the process of filling the apparatus with
a definite quantity of the desired mixture and of measuring any
quantity of gas which we allow to escape from them. For the
rest we refer to Comm. N°. 84 for the volumenometer and the
mixing apparatus, to Comms. N°. 69 and N°. 84 for the pressure
cylinder and its appurtenances, to §§4 and 5 and Comm. N°. 83 for

the cryostats.

§ 3. Determination of the molecular volume of the coexisting phases.
As we did not aim at a very high degree of accuracy a few remarks
will be sufficient. As to the liquid phase: with the limitations and
conditions to be treated in § 5, part of the gaseous mixture in the
apparatus ), called for shortness the piezometer, may by conden-
sation attain a liquid phase of a composition which, as said in § 1,
is derived from « of the original gaseous mixture by a correction.
Measurements with the volumenometer yield the volume that the
liquid phase would occupy in its gaseous state. And by means of the
divisions at p, and p, Pl. | fig. 3 we read the volume of the liquid
phase itself. The molecular volume of the gaseous phase may best
be derived from the coexistence pressure and from isothermals which
with not too small pressures can be determined after the method
of Comm. N°. 78, April °02, if necessary with the dew-point appa-
‘atus D itself. With not too large pressure it will in most cases be
possible to apply the correction for the deviation from the law of
Boy.e-Gay-Lussac-AvoGapro by means of the empirical reduced equa-
tion of state (Comm. N°. 71, June “O1) according to the law of cor-
responding states.

§ 4. Beginning of the condensation. In order to determine this we
have applied the principle of ReeNauir’s hygrometer '). To this end
we observe the first condensation which is formed in a part of the
apparatus of which the temperature is a little below that of the
surrounding gaseous mixture when the gaseous mixture is brought
to the begin condensation pressure. In order to be able to observe
a very small condensation we have taken for the place of the lowest
temperature a shining mirror, and next to it is placed another mirror
which is not cooled.

he apparatus is blown of one piece in the same way as an ice
calorimeter of Bunsen. The outer space @ has a capacity of about
20 ¢.c. and is provided with a capillary to which the steel capillary
g',, through which the gaseous mixture is led, may be sealed on.
One of the mirrors ¢ is sealed on to the bottom. The inner tube c,
carries at c, the second reflecting surface and serves at the same time
as a cryostat to keep the temperature of this surface constant, a
little below the temperature of the gaseous mixture.

To this end we have devised in the same way as for the outer
cryostat the cover m,, fitting hermetically on the tube c,, the small
stirrer h, of which the rod 4, projects through an india-rubber tube

1) Barrett, (Ann. de Chim. et de Phys. ser. 6, vol. 25, p.59, 1892), has found
that for pure substances the deposition of liquid on a not cooled mirror placed in
the vapour is a suitable means for the determination of the condensation point.

16*

( 238 )

as in Comm. N®. 83, and the thermometer 7 sealed on to m, while
the vapour of the liquid gas in c, is exhausted through m,.

On the tube ¢ at c, the glass cap /, is sealed on, through which
with an india-rubber stopper the capillary connected to c¢ passes at
J, This cap forms the cover of the larger cryostat. Through this the
apparatus has become very firm and very easy to handle. The
capillary 4, is protected against the stirrer by a metal frame 7.

Through the cover at /, pass the wires by which the stirrer is
suspended. At 7, the vapour of the bath of liquetied gas is exhausted.

The surfaces of d and c, are made reflectors by platinizing them
at a redhot temperature with platinum chloride in camomileoil.
Though the platinum surface is not so shining as that of a silver
mirror, yet the advantages of platinum for this purpose are evident.

The regulation of the temperature in the two cryostats of the appa-
ratus is performed by means of the same exhaust-pump that keeps
the pressure constant by means of a pressure regulator (according
to the principle of Comm. N° 87, § 3, March “04) in a main tube
which branches off into two exhaust-tubes, each shut with a cock.

By adjusting these cocks we can take care that the temperature
in the inner cryostat is a little below that of the outer cryostat. What
difference of temperature can be kept constant in the two depends
on the temperature itself and on the liquid gas.

In measurements to be described in the continuation of this paper
the temperature of the large cryostat was — 25°.0 C., that of the
small one — 25°.1 C. The required decrease of pressure in the main
exhaust-tube could be easily kept up (boiling-point methylehloride
— 23° C.) with a water airpump.

The pressure regulator was adjusted at about the pressure belonging
to — 25° C. By means of the regulating cocks an assistant took
care, as signs were given by another assistant according to the
thermometer readings, that both temperatures, hence also the diffe-
rence between them, remained constant. This could be attained to
within 0°.05 C.

The accuracy which we can reach in the determination of the
dew-point with our apparatus depends in the first place on the
difference of temperature in the two eryostats. As a matter of course
it is smaller as the temperature coefficient of the begin condensation
pressure is larger. At a given difference of temperature it increases
according to the difference of the pressures at which the conden-
sation appears or disappears. The amount of this difference is also
determined by the illumination and this is much impaired by the two
walls of the cryostat. In our experiments, observations made with the

(e239)

naked eye with side-illumination of the mirror proved to be the best.
of the
yressure. The optical part of the method may certainly be much
| | | ; A

The difference ranged between limits which amounted to 2°/,
improved. The accuracy attained will, however, be sufficient in
many cases.

Adiabatic pressure-variation must naturally be avoided. Yet all
such difficulties arise also with measurements at ordinary tempera-
ture. With a view to the large dimensions of the vapour space a
we have not made use of a stirrer and have tried as much as
possible to surmount the difficulties by operating slowly.

§ 5. Determination of the end condensation pressure. For this
determination the mixture, after the begim condensation pressure
is measured, is led back to the volumenometer, the dew-point
apparatus is disconnected from the steel capillary and in its place
the piezometer p in P Pl. I figs. 1 and 3 is connected with the steel
eapillary y",. The piezometer consists of a wide glass tube p, Pl. I
fig. 3 fastened to a capillary, both graduated and calibrated. The
dimensions are chosen with regard to the quantities of e@as that
may be intended for the measurements. If these are decided upon,
the exact quantity of gas, necessary for filling the piezometer at a
suitable position of the mercury in the pressure tube 4 with liquid
to near the end of the capillary, must be determined before each
determination by means of a preliminary experiment.

The equilibrium of the phases in p, is reached by means of a
magnetic stirrer gq moved by the coil S. The immediate effect of
this coil is not sufficient to move the stirrer forcibly through the
liquid meniseus. Therefore a soft iron tube 2, with a groove 2, which
enables us to read on p,, is moved up and down at the same time
with the coil.

This movement ought to be independent from that of the stirrer
in the cryostat. But as we did not require a very high degree of
accuracy in our experiments we have for simplicity devised the iron
tube 2 as a connection between the upper and the lower part x,,
and x,, Of the ringshaped valved stirrer (comp. Comm. N°. 83).
This is moved up and down mechanically and with the hand in
turns, one time to stir the liquid bath, the other time to establish
equilibrium between the liquid and the vapour while we simultane-
ously move the magnetizing coil S.

The essential difference between a determination of the end con-
densation pressure in ow apparatus and that in a Carmierer tube
does not lie so much in the circumstance that we do not liquety
the whole quantity of gas, as in the fact that, as remarked in

( 240 )

§ 1, the temperature is different in the different portions of the
gaseous mixture. The portion (see fig. 3 § 1) at the temperature of
investigation 7’ is separated from that at ordinary (or higher) tem-
perature 7” by a_ series of layers (in the capillary) of which the
temperatures range from the highest 7” to the lowest 7. One of
these temperatures we shall eall 7”.

This circumstance — involves
several restrictions and conditions
in the application of the method..

In order to make measurements
possible for all compositions . the
temperature 7” must be taken or
raised so high above 7. that on
the pr-diagram (fig. 4) the vapour
branch of the binodal curve of 7”
does not intersect the liquid binodal
curve 7. Then we need not fear that
condensation begins at 7” while

condensation takes place during
the transfer of the mixture from
Fig. 4. the compression tube (or perhaps
we then always have

x fe)

ws

the volumenometer) to the piezometer. At 7
a gaseous mixture of the original composition. When the above-
mentioned curves intersect, as is represented in fig. 4 for the case of
7", we can make measurements only for the compositions represented
by points outside the region included approximately between d and e.

Even if, without condensation taking place in the compression
apparatus (or perhaps volumenometer) the gaseous mixture can be
transferred to the piezometer, the part of the capillary where the
temperature falls from 7” to 7’ still offers another difficulty of the
same kind. Here we necessarily find temperatures 7” at which the
vapour branch of the binodal curve of 7” intersects the liquid branch
of 7". If drops ave formed at 7”, the composition ey7r Of the liquid
belonging to the observed coexistence pressure can no longer be
indicated. By flowing down and by distillation (the effect of capilla-
rity exceeds that of gravitation) the drops gradually pass over into
the liquid phase at 7. if care is taken by means of the cock / (see
Pl. I fig. 1) and by adjustment of the pressure in 4% (see Pl. I fig. 1))
that gas streams only into and not out of the piezometer, until finally
when we stir with open cock & it appears that equilibrium is
reached and the capillary contains vapour only.

In order to further the disiillation and the disappearance of the

( 241 )

condensation, if is desirable that the capillary at the place where the
temperature is between 7’ and 7" should not be too narrow. Moreover
the capillary is surrounded by an air jacket p,, made ofa glass tube
tightly closed with india-rubber rings p, and fish-glue. To avoid
diffusion the other portion of the capillary g", must be narrow.

If by previous determinations with the dew-point apparatus we
have determined 7 (as a first approximation it will in some cases
be possible to use a preliminary w-surface as constructed in Supple-
ment N’. 8 see p. 222), it is easy to apply the correction necessary to
derive the composition of the investigated liquidphase at 7’ from w,,
the original composition. On the piezometer divisions the volume of
the vapour is read. Let V, be this volume reduced to normal cir-
cumstances and corrected for the first virial coefficient L (comp. for
instance the continuation of this paper), let J” be the entire volume
of vapour and liquid, measured and corrected in the same way,
then V,)=V—J, is the volume of gas, measured and corrected in
the same way, which would form the liquid phase.

Hence :

X, V— Xr Vi Vi
bee Se Sa Oe

If we operate under moderate pressures, the correction will be
always small and even if v,,7 is not very accurately known, it can
be applied in a satisfactory way.

Astronomy. — “A new method of interpolation with compensation
applied to the reduction of the corrections and the rates of the
standard clock of the Observatory at Leyden, Houwi 17,
determined by the observations with the transit circle in 1908.”
By J. Wreper. Communicated by Dr. H. G. van pr Sanpr

BAKHUYZEN.

§ 4. In the Proceedings of Nov. 29, 1902 oceurs a paper ‘On
interpolation based on a supposed condition of minimum,” of which
the present paper is to be considered as a continuation; this explains
the numbering of the sections. In order to interpolate between the
ordinates S belonging to the abscissae ¢— a, 6...y,2, I have there
determined the interpolating curve for which the total sinuosity

CS?
Is =a z) dt has the least value. I found that between two sue-
t

a

( 242 )

cessive points which are well defined by observations this curve
satisties an equation of the third degree S, = S,-+ gq, 7’+ ¢, T? +e, T°,
where 7=t— q and ¢<¢<yr, and has moreover the properties
dS Ss
that besides S also 7 and 7p “are continuous, while at the ends of
2S
this curve aa is zero; these data appeared to be suflicient for a
detinite determination of such an interpolating curve.

In the same paper in § 3 I have referred to the problem of com-
pensation for the case that the results of the observations S are
affected with errors.

Now I shall describe how that problem was solved and I shall
apply this method of compensation to the corrections and the rates
of the clock Honwi 17 during the period Jan. 14, 1903—Jan. 14, 1904.

For the sake of the compensation I have used instead of the

supposition on which my interpolation was based, another hypo-
thesis which covers more, and for this purpose I have accepted that the
probability of a group of corrections |S is proportional to e~ 7s,
where the factor 42 is independent of the intervals between the obser-
vations; the original condition of minimum is a result of this hypo-
thesis, if we accept as interpolated for the time ¢ that value which
(considered as result of observation) makes the expression e—*4s as
large as possible.

As formerly I have called the error of observation in S,: fq and
the real clock correction L,, so that L, = S,— jj, meanwhile,
I venture to use the same letters 7, and 4, in the sense of most
probable values of the errors and measured quantities, although
there is naturally a difference between the latter quantities and the
real values.

As soon as we have to do with errors of observation we must
substitute /;, for /s in the expression for the probability of a group.

The probability that the errors fa,..- J); Tas Frye fe We
spectively, occur in the observations, must in those cases where
the continuity of the observed quantity agrees with the hypothesis

= Wy = Is

: . S—f j
made above, be proportional to the product of e and ¢ 9s
in the latter expression my, represents the value of the mean error
in S,. The system of errors of greatest probability satisfies the

condition :

- ple aft
Afs¢+ 7). = a? = minimum,
q

(92434)

whence for each of those most probable errors the relation :
OLS _¢ Ja
Oy ey”

S=f ' a
—— contains all the unknown quantities /, practically,

Of

od
however, besides /, only the 3 or 4 preceding and following ones,
J4 : t

Theoretically

since, owing to the small coefficients, the other terms may be neglected.
The most probable errors occur in it in a linear form, and hence
the complete system of error-equations is suflicient to determine
them all.

§ 5. In this manner | have treated 61 clock corrections determined
by observations made from Jan. 14, 1903 to Jan. 14, 1904 by
KE. F. van pe Sanpe Baknuyzen and A. Pannekork. This problem
was somewhat complicated because the observed rates depend on
the pressure and the temperature of the air in which the pendulum
moves. An increase of atmospheric pressure of 1 mm. mereury gives
to the clock rate a retardation of 0.015 seconds a day, while an
increase of temperature of 1 deg. C. accelerates the rate by about
0.030 seconds a day.

I have found that also the differences of temperature observed in
the clockease have a perceptible influence on the rate’).

1) This temperature gradient forms a new element in the reduction of the clock rates,
in so far as at the Leyden Observatory it was not accounted for until 1903.
Although in the clockcase of Hohwii 17 two thermometers have been suspended
at different heights since 1866, the influence of the temperature gradient did
not clearly appear from the earlier observations owing to the inaccuracy of
those readings. Although Dr. E. F. van pe Sanpe Baknuyzen detected a yearly
periodicity in the differences between those thermometers, which might explain the
phenomenon of the yearly periodicity in the rates (Proc. Kon. Ak. vy. W. 1902)
yet at that time he still doubted of their reality, because the thermometers were
graduated according to Réaumur and the accuracy of the reader in estimating
the tenth parts was insuflicient to warrant an inequality of 0.1° R. in the monthly
means of the difference in temperature.

In order to obtain certainty on this point, two thermometers were placed into
the clockcase in January 1903, which were divided into tenth parts of a centigrade
and had been compared with each other beforehand by the director of the Obser-
vatory; since that time they have heen read with accurate estimation to a hundredth
part of a centigrade.

The vertical distance between these thermometers is 63 em. It soon appeared
that in the clockease there is a variable temperature gradient, which cannot but
cause a considerable variation of rate.

In July 1903 it was derived from observations that a difference between the

( 244 )

A temperature gradient in vertical direction of 1 centigrade per
meter causes, roughly calculated, a variation in rate of O0%.25.

thermometers (upper—lower) of one centigrade corresponds with a retardation of
03.30 a day.

This difference of temperature, apart from variations in connection with mete-
orological conditions, clearly shows a yearly periodicity; in winter it is small, in
summer on some days it increases to above 0.50 C. It can hardly be doubted
that, at least to begin with December 1898, the yearly periodicity must be ascribed
to the temperature gradient; for it is since that date that the standard clock has
been placed in the vestibule in a niche cut out from the pier of the 10-inches
refractor. Since about March 1899 the place on the Eastern pier in the transit
room where the standard clock hung during the former periods 1862—’74 and
1877—"98, treated by E. F. van pe Sanne Bakuvyzen, has been occupied by the
clock Hohwii 46 with a Rieffler-pendulum.

In order to obtain aceurate data about the temperature gradient also for this
place, two thermometers, which had been compared with each other and were
graduated to tenth parts of a centigrade were suspended in the case of Hohwii 46
at a vertical distance of 65 ems. on February 27, 1903. It appeared that here the
differences in temperature were in general greater than in the pier-niche, and in
July 1903 it was derived from the observations that a difference of temperature
between the thermometers of 1 centigrade corresponded with a variation of 05.40
in the daily rate of this clock.

A yearly periodicity in the temperature gradient appeared also distinctly in
Hohwii 46, as will be seen from the fo'lowing monthly means, which are given
by the side of those of Hohwii 17.

Hohwii 17. Hohwii 46.

1903 March + 0°14 +0°.25
> April + 0.09 + 0.16

» May St + 0.42

, June + 0.28 + 0.37

» duly + 0.29 + 0.32

» August + 0.18 + 0.22

> september -+ 0.16 + 0.26

» October + 0.06 + 0.12

- November + 0.02 + 0.10

= December +. 0.01 + 0.03
1904 January — 0.02 =) 110108
» February + 0.05 + 0.12

Moreover in the temperature gradient in the clockease in the transit room a
daily inequality was observed, which was hardly perceptible in the niche.

In the case of Hohwii 46 the mean values of the temperature gradient from
0» to 12" mean time, are regularly greater than the mean values from 12" to 24°
mean time. From the differences between these mean values in connection with the
differences in the relative rates of the two clocks for the corresponding half days
1 conld derive the influence of a variation in the temperature gradient on the
rate of Hohwii 46 for a shorter period.

The investigation is not yet finished, but in connection with the theoretical

( 245 )

A 4th inequality in the rates was due to the difference between
the personal equations of the observers. From direct determinations

we have obtained the following differences in the clock Corrections :

B—P
L April 1901 (900
18 Mareh 19038 —— {1 Fi
15 Mareh 1904 - 0,224

For the time being I have adopted for this difference during the
period Jan. 14, 1903—Jan. 14, 1904 : 03.250 and have reduced
the clock corrections determined by P. to B.’s system. Besides the
corrections for atmospheric pressure, temperature and difference in
temperature I introduced a correction for the assumed personal equa-
tion; these 4 unknown quantities are introduced into the equations
and will be derived from the observations. I represented them by
wv, y, 2 and w and chose the following units so that these quantities
should have about the same values: for atmospheric pressure (/3)
the unit =
= '/,, degree Celsius, for difference in temperature (}") the unit

‘/, mm. mercury of 0° C., for temperature (%) the unit

== */,,, degree Celsius, for clock rates per day the unit = '
second of time; «, y and z represent the influence of each of the
three units on the daily rate, expressed in thousandth parts of a

second of time, while w is the 10™ part of the correction for the
9

/1000

assumed difference in the personal equations of the observers during
the same unit of time, viz. 6— P= — 250 — 10 w.

§ 6. The observed clock corrections were reduced beforehand to
midnight of the data of observation. This was done with clock rates
according to a preliminary formula with due regard to atmospheric
pressure, temperature and temperature difference in the period from
the instant for which the clock correction was determined to mid-
night. This formula is derived from observations during: the first half
of 1908, it has been regularly tested by each new determination of
the clock error, and has proved very satisfactory for the purpose
set here.

Constans = daily rate —0.0157 (bar. — 760) + 0.032 (temp. — 10°C.)

— 0.31 (temp. difference).

development of B. Wanacu of Potsdam on the influence of the temperature-gradient
on clocks (A. N. Nrs. 3967—68) I think it worth mentioning that in this way |
found that per 1 centigrade difference in temperature between the thermometers,
there was a variation in rate of about O%.80, twice the amount derived from the
before mentioned observations.

( 246 )

From readings of the barometer and the thermometers, daily means
of atinospheric pressure, temperature and difference of temperature
from midnight to midnight have been derived. Let the clock corree-
tions at midnight be S, and the mean daily rates in the intervals
between the successive determinations of the clock corrections be Q,
in accordance with the letters previously used by me. From the said
daily means, the mean values of the atmospheric pressure, of the
temperature and of the difference in temperature were derived for
the same intervals. In the interpolation these quantities correspond
with Q, hence I call them Q, Q*, Q'. For the barometer readings I
used the deviations from 76 ems. The temperatures # are those of
the upper thermometer.

In this paper I shall also use the letters S?, S=, S' to indicate
quantities that can be computed for each observation by taking the
sum of the daily means of B, ® and J’, to begin with a certain
date, say Jan. 14, 1903 till midnight of the day for which the
clock correction is determined. S? denotes a value which is + 10 for
each observation of Pannekork, and zero for each observation of
Baknuyzen, and the series Q’? relates to the series S? in the same
way as each series Q relates to the series S according to the formula

S, — S,

Q, = ———.. S, and 5S, are two successive quantities of the series,
it

and 2 is the interval between them expressed in days.

>

Lg Se aS PS eS aS ear
must then be considered as a formula for a reduced and compen-
sated clock correction.

Hach of the letters 1, S, 7 represents a series of discrete values,
one for each observed clock correction, but by means of the inter-
polation along the least sinuous line they can also be taken as con-
{inuous variable quantities of which the derivatives of the first and
second order also vary continuously, but of which the derivatives
of the third order vary abruptly. To determine such a variable, say
S’, for the instant ¢=q+7' between the epochs q and 7 of the
determinations of the clock correction, we use according to § 4 the
formule

B =e: B B B

SLT — S, + Ig T + Cy Tie fe e fs
where q is the epoch of the observation which immediately precedes ¢.
The coefficients g? c? ce? can be derived from the series Q? if
we use the formulae C in § 2 of the previous paper on this
method of interpolation.

Taking the quantities L, S, 7 in the sense as explained above

( 247 )

we can develop the total sinuosity of the reduced and compensated
correction as follows:

(EEN. a(S—f) @SB @S? @SY ase?
i — d= tt —_ — 4 —_—2 a dt.
dé dt? dt? dt dt? de

a
In order to ae the greatest probability to the series 4 we must
choose «2, y, 2 and w so, that the partial derivatives of /;, with
regard to each of these are zero; ie. that they satisfy the following
relation :

*Ta(S—f ) Sb dS SV SP | Sb
=a 2S) SSS SE u —— = =) ¢
dt? dt? mia dt? dt? dt?

a

and 3 others, which we obtain by substituting for the last factor

PSE i ES? d?SV i BSP
——., successively —— ant ,
dt?’ Sa ae dt?
a. a SB as :
Definite integrals, such as aa a dt, which here occur as
C at”

a

coefficients, can be computed in the following way:

es d? 8B dS* (dS? PSB
7 = 5 —
dt? : dt dt dt ah Gh ;

a a a

1) esti ey Sees ae vs 7h ee Ae Mees
=2 [eB y| —6§= en (S; = S,)* =2 [ce g | — On Cn Qi
a

If in the last term we also substitute for
B B
3 ne, , (¢r — Cg) , we have

fees es ‘ s
ie Fe EE dt = [cB fa 255 Q. Ga

a

and after interchanging the indices a and 3 in the second member also

[2] + > fe = 0).

I have computed these coefficients according to both formulae, and
thus obtained a rigorous test.

At the beginning and the end of the interpolation the quantities
¢ are always zero, so that products such as ¢? g= would be zero at
either end. With a view, however, to the continuation of this com-
putation for next vear, I have not closed the interpolation-compu-

( 248 )

tation for the atmospheric pressure, the temperature, etc. on January 14,
1904. Hence on Jan. 14, 1904 c®, c*, cl’, c? differ from zero and
in the formula I had to retain the term for the end.

“ad? SB d? (S—f)

— dt | first put all errors equal

In integrals such as = es Ew
= 7 dt? det*®

a
to zero and computed the known terms of the four equations accord-
ing to formulae of the form :

“2 SBS Saet ene “a
le) Ga ae = A+ Fy 0) @
t

By solving these equations, I have obtained preliminary values
for v, y, 2 and uw, and in the way to be explained in § 7, also for
the most probable errors /; from the mean daily rates computed
with these preliminary values, using the formula
Q@,=0Q—2 G2 —7@ =—20"— uQP BP Lite

a
nt

| have derived as a second approximation the corrections ., y, 2
and w to these preliminary values. The influence of those corrections
on the most probable errors was of little importance.

° d* S .
Integrals such as | (=) dt have been computed twice, first
- dt

a

ESB GS
according to the formula for '/, | —,> at
a 2) de de

a

~ r? iy 4
1. | (=) d= leg) + D0, (y— 1)
a

a

and secondly according to the formula, deduced in § 2

(22 eer ee)
a= mn (C, y Cr + Cp
‘| dt id 3 Vw Cq 77] c

a

The following are the 4 equations expressed in numbers :
5460 ¢—252y— 142+ 59u=- 21664 — 145

— B%e4+952y+5122— 8u>— 1757— 88
— 14745127+5712-— 2u=—+ 383-4 87

4+ 592— Sy— 22z24+101u=— 101— 21

( 249 )

The second members are written in 2. parts. The first part is
derived from the observed rates Q, the second from the prelimi-
narily reduced compensated rates Q,. The solution of the two sets
vielded :

— + 3.90 — 0.04 = + 3.86

y = — 2.30 — athe —— Big
z= + 2.72 + 0.46 = + 3.18
uw = — 2.86 — 0.09 = — 2.95

According to these results the influence on the daily clock rates
of the atmospheric pressure per 1 mm. mercury of 0° C.is ++ 080154,
of the temperature per 1 centigrade (upper thermometer) — 0.0264,
and of the difference in temperature per 1 centigrade

(upper—lower) + 0,318,
while for the personal difference BakHuyzeN— PANNEKORK in the clock
corrections is found — 08.220.

Table I contains the values required for the said computations;
for each interval between two consecutive determinations of the clock
correction : the number of days 7, the observed clock rate Q, the values
Q8, Q, Q’ and Q”, and also the values (c, — c,)® , (¢,—7,,), (Cy—er)",
(¢,—c,)?. To this must be added in order to render the computation
possible the values for the last epoch Jan. 14, 1904, :

gP eee) ge = + 87.7 g V — —_ 3,1 gP =+ 2.4

coo 1209 20 Co 086 cP + 0.54
y =— 146, adopted in case we use the series of the observed clock rates Q.

G=— 194, i = a A .» preliminarily reduced Q,.

§ 7. Here follows the reduction of the relations between the most
probable errors and the determination of the latter according to these

relations.
Sub § 5 I have derived for each error 7, the relation :
Oly, I; OL 1 :
,—— ~=0 or see FO
Of, My Ofg Atty’

I have thought myself justified in adopting the same value for the
mean errors in the determinations of the clock correction, although
for 1 of those corrections only 2 stars, for 183 corrections 4 stars,
and for all the others 3 stars were observed, since the difference in
accuracy resulting from the different numbers of stars is relatively

TAB iy ay i

( 250 )

bs arena Atmospheric Temperatures Temperature | Personal

3 ‘, eae pressure (8B). perature (3).) difference (V). | equation (P).

ES a a Be bps ate] a lq—=9"| a eae
3 | 40-230) 459.9 | 4 9.45) 49.3 | 44.88 | 446.0 | 43.62 | —3.3 | 1.78
3 | + 196] 440.5 | + 0.13) 42.0 | 2 Agee tes ~3.46 | 43.3 | 49.33
8 | + .036) +-23.3 | — 4.4 , 49.6 | —2.69 ) + 4.4) —1.89 |) 0.0 —0.14
3 | + '.050] 295.5 | + 3.98] 71.3 | +3.93 | 419.3 | 49.83 | —3.3 | 4.57
6 | — .034] 444.9 | — 2 74.7 | —1.44 | + 7.7} 1.92 | 44.7] 44/94
10 | + 024 Seo ao a 80.9 | 42.76 | + 7.9 | —0.20 | =40 | Se
3 + 164 +60.8 | 11.39, 72.0 | —2.95 | +107 | +0.72 | 43-3 | 44.77
7 | — .067| + 4.9| — 2.99] 80.4} +-0.84 | + 8.9 | —0.65 | —1.4 | —1.06
8 | — 235] —31.0 | —14.05] 88.3 | 40.54 | 4410.6 | +0.18| 0.0 | —0.54
3 | .000] 493.2 | 444.99} 87.7 | 44.48 | +412.3 | —0 45 | 13.3) +2.81
a cos +H10.7 | — 0.90) 81.7 | 4.73 | $13.0 | 1.95 3.3) —3.44
5 |= 417] 2.9 | —9.97| 81.8} —0.34| 47.5 | —1.91| 0.0 | 44090
G | — .042] 117.4) +44 57) 89.5 | 1.36 | 441.8 |—1.00| 00] —0.95
5 | — .201) —23 1 | —42.98) 105.0 | 441.62 | 499.8 | 41.99 | 42.0 | 44.46
7 | — 139) — 2.7 | 4 6.05) 108.3 | 44-19 | 419-1 | 4.0.38 | —1.4 | —0.98
7 | — 150) + 0.6 | — 3.54 98.9 | —0.21 | 440.2 | 1.47] 0.0 | +0.50
10 = 106, + 4.2 | + 7.51) 85.3 | +0.38 | + 7.8 | +1 .32 0.0) —0.39

|

7 | — .291) —46.1 | — 6.63) 77.0 | —3.77 | + 2.9 | —3.58 | 41.4 | 40.63
10 | — 245) —37.9 | — 0.51] 407.8 | 441.81 | 497.2 | +3.45 | —1.0 | —0.82
HON) )— 192) — 207 | + 0.67] 126.6 | 41.63 | 495.3 | —0.24 | +0.8 | 41.22
3 | — 104 422.41 | — 5.49) 195.7 | 3.82 | 491.7 | —2.84 | —3.3 | —2.91
3 | — .052| +40 1 | 444.67] 138.3 | 441.92 | 432.3 | 41.41 | +3.3 | 43 98
5 | — .190 + 3.4 es 9.29) 153.8 | —2.61 44.2 | —1.20 | 0.0 | —1.38
5 | — .295) + 5.0 | — 4.61) 174.0 +7.85 | 53.2 | 46.93 0 0 | +0.29
4 | — .139| 130.8 | +19.40) 157.3 | —5.25 | 430.2 | —5.55 | 0.0 | 40.30
10 | — .314] —14.8 | —10.09| 156.7 | 42.97 | +97.0 | +2.44 | —1.0 | —0.90
4 278 ier 1.00) 144.2 | —2.24 | +16.2 | 947 | 42.5 | 44.95
4 | — Ans) 244 | ,.99| 147.8! —9.51 | 490.8} 1.9%} 0.0 | —0.84
4 | — 158) +26.4 | + 0.40 467.0 | +0.57 | 436.2 | 44.84] 0.0 | +0.09
4 | — ,209] 420.3 | +0 sl 480.8 | $4.27 | 4+40.5 | 42.88 | 0.0 | +0.42

( 251 )

|
| clock | Aimesobels ftemeratare cy] femperatars | Persone
Es 2 Jen el NS SUES SS
Ba) @ | @ Mera? ed lla —ny lo lames
|
6 |—o°e1s! + 9.1! + 0.491 171.0 | 1.93 | 497.7 | —9.55 | 4.7 | 0.65
7 | — .324] ~ 4.9 — 2.98] 168.6 | —0.63 | +97.4| +0.05| 0.0] +0.36
g | — .344| —5.4 | + 9.45] 175.0 | +0.61 | +99.6 | 44.07] 0.0 | —0.09
11 | — .379| 12.0 | — 3.95] 175 9 | 40.99 | +21.7| 0.88] 0.0 | 40.02
g | — .338/ + 4.8/] + 7.33] 173.3 | —9.56 | +20.8 0.12 | 0.0 | +0.02
6 | — 436] —30.0 | — 8.03] 173.5 | 44.16 | +19.8| 40.80] 0.0 | —0.143
42 | — .365| — 2.1 | + 3.54) 467.4 | 1.73 | +44.6 | 1.81 | 40.8 | 40.93
3 | — 288) 410.5 — 2.83 174.0 | 0.8%) 422.7} 40.02) 0.0 | —0.41
4 | — .o71| 443.9 | + 8.68] 180.5 | 43.45 | 497.0 | 42.10] 0.0 | —0.16
8 | — .363| 12.2 | —13.53| 163.4 | 10.99 | 447.4 | —0.43| 0.0 | +0.64
4 | — .196| +28.2 | + 9.44] 143.8 | —2.04| + 6.8 | 4.10 | —2.5 | —1.05
5 | — .209| +26.9 | — 4.18] 146.2 | —0.55| 4+ 7.6] —0.56| 0.0| +0.81
4 | — .937| 116.4} + 9.75] 152.8 | —0.70 | +43.0 | —0.04| 0.0 | —0.49
7 | — .373| 43.5 | — 4.44] 460.7 | 42.46 | 447.4 | 44.53 | 44.4 | 40.42
8 | — .429| 35.0 | — 5.03] 149.5 | —0.06 | 440.0 | —0.34| 0.0 | +£0.09
1 | — .333| 16.4 | + 6.18| 131.4] 0.72 | + 2.2 | 9.45 | 0.9 | —0.91
3 | — .447| 37.3 | — 2.68] 124.7 | 1.02 | +0.3 | 40.94 | 43.3 | 44.87
2 | — .395] —35.4| —41.48| 196.0 | 14.47] 0.0] —0.90| 0.0 | 4.31
6 | — .193| +19.6 | 442.45] 121.2 | 0.37 | + 2.2| 44.95] 0.0 | —0.01
42 | — .447| 492.9} — 0.56] 111.9] 40.15] + 0.8] —1.68] 0.0| 40.53
3 | — .185] + 8.1 | + 4.55] 104.0 | 0.34 | + 7.7| +9 82 | 3.3 | —0.70
40 | — .243| —18.0 | — 0.58] 95.2 | 44.54| +0.2| 3.65] 0.0 | —0.05
3 | — .274| -40.5| — 4.52] 77.0] +0.02 | + 9.0 | +3.96 | +3.3 | 40.96
7 | — :265| —43.7 | — 0.09] 66.7| —3.31| —1.4| —2.59| 0.0 | —0.85
3 | — .309] —4.5 | — 3.32] 73.7| 40.99] — 2.7] 40.81 | 0.0] 40.99
4 | — .935| —90.1 | — 4.541 80.5 | 41.48 |—25]|-0.86| 0.0| —0.07
5 | — .067/ 442.2] +4 5.45} 78.2] 44.42 | +0.8| +40.58|/ 0.0 | +0.03
7 | — .008| +48.4 | — 0.81] 61.3 | +0.89 | + 9.6 | —0.296| 0.0] —0.09
5 | + .060) 445.9 | + 3.98] 39.6 | 2.43] +4 3.2| 44.31] 0.0) 40.34
42 | — .101| 41.3 | — 3.¢4| 49.2 | 2.46 | — 5.0 | —1.79 |~ 0.8 | —0.84
17

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 252 )

small as compared with that depending on other causes which are
difficult to account for. But the computation would not become
much more difficult if we should assign different weights to the
determinations.

I now use the following thesis for the interpolation with regard
to the smallest sinuosity that is proved in § 3 of my previous
paper viz.: the partial derivative of the total sinuosity /s of a series
of clock corrections with regard to one of the corrections Sj, is
equal to twice the abrupt variation which occurs in the derivative
of the third order of the interpolation curve near the abscissa g and
the ordinate S,. Hence according to § 4

ols
OS,
I apply this relation to the interpolating curve after compen-
sation determined by the corrections 4, and I obtain
oly
OL,
in ease that each part of that curve is represented by an equation :
Ty = Fg + Gt Cyt + EP.

In accordance with the previous paper I have used capitals for the
interpolation coefficients belonging to the corrections which are freed
from errors. Also the meaning of > for this interpolation corre-
sponds entirely with that of o used for the interpolation without
compensation.

= 12 (e,—em).

= 12 (£,—En) = 12 =, ,

wai
we may substitute ———— , because L, + fy is invari-

For au Yi;
able, hence 0L, = — 9/q.
OL k aa y
After the substitution of eee ,, each relation 4- sae St EL =
0 q gq {lg
5 : D> =. ; .
takes the form .= Dae if

The first member of this relation depends on wv, y, 2, uw and the
errors /, and all these quantities occur in it in a linear form:
B 3 V P
Zq = % — 26, — y6, — 26, —UuG, — 6, -
If we use the approximate values obtained for 2, y, 2 and w in
the supposition #= 0, we can compute the expression:

B $ 4 P
Yq = Fy — £6, —yO, —26, —Ud, .

I have made this computation by determining the differences between

—

( 253)

the suecessive coefficients of # for the reduced clock rates
Q— 2Q8 — yQ —zQ’—uQp,

according to the new method of interpolation.
fr : ‘ :
6, may be developed in the following way:

St yO) rP p r op ra 7S p
Oy =...- + Ky fo +- Kg fp + Ko fo + Ba fr 4 Ko fs +»:

; lie es ~ (dt 2
We have also = Tr ee where [, =| Gaz

Hence:

aij 2
Pe ois Mel Ap 40 & ae) _ Se wie

Romi; mapslolop ono 12 ys

therefore the coefficient of 7, in the expression for of, is equal to the

coefficient of /, in the expression for oe.
Let us consider the case that /, = 1 and all the other values of 7=0O

= r rp rg >r ;
ier G, Ky, 6, = ie One KGa —— Ki or = GE , ete. The series
of the quantities of for that case gives us directly all the coefficients

K which are required for the development of al In this manner an
interpolation was made between the numbers..., 0,0, 100,0,0... as
mImany times as there were observations, each time moving the
number 100 further over one interval between two successive
determinations of the clock correction.

If this computation is properly arranged, it can be made in a
very short time and the accuracy can be tested by the results
themselves as the coefficients are derived twice independently of each
other.

I shall now describe in detail how | have arranged the compu-
tation of those coefficients for a determination of the clock correction.

According to § 2 of my paper “On interpolation ete.’ of Dec. 10,
1902 we have the relations ;

ow Pa Iq -}- Ot 2 (OF Ip + D0) an 2 Or,
Oy ease roe Cm Oi aan E WAT en = a=
Tic m

which allow us, once the series of the g being found, to deduce the
series of the o from the series of the y and Q.

The series of the y is determined by the equations (C) of § 2 viz.
n Qn ++ m Qn n (Qn =z Ip) m (Qn — gr) ;

m+n 2 (m + n) 2(m + n)’

Iq =

17*

( 254 )
which for the first and the last epochs @ and z become:
Qu = 96 25 —Y
ga = Qu + —— z= + ——

where uw and pr refer to the intervals at both ends.

In the example given here, also one of the latter relations occurs,
i.e. that for the observations of January 20, 1908. The successive
intervals with the values of Q belonging to them are here:

co 3 3 8 3 6 ORE wen \(Cam.au.ro)

eee 100 / j= dani 0 0 Or yaa: (Q)

The line separates what precedes this observation and what follows.
As a first approximation for the quantities g I have adopted

n Qn+ m Qn

only three of these values differ from zero, namely :

>

m+n
l
Ip = Rey hs > Iq Qn — Qn 5 — eas Qn
In my example: Q,. = + 33; Q,=— 12; 1=3, m=3, n=6&,
o=8; hence gy =+17; 97 = + 21; 9 =— 3. With these three

values begins the annexed scheme, in which the successive corrections
of the g are computed. (see p. 255 sq.)
Annexed to this paper is a table (II) (p. 256 sq.) containing the

coefficients 100 A for this computation; under the heading “Sum of

the squares....” have been given the sums of the squares of the
coefficients for each date, and to the first and second approximations
the values of 100 w which belong to the observations.

Each equation of errors is reduced to the form:

Ka fo + Ky fp +(Ky + 8) fo + Kg fr + Ky fs ++ = My
1 : :
where ® = 50a” is constant for all equations, and must be deter-
22u

mined in order to enable a solution.
In the relation:

qe: AK fh + K. D fy + Ka fy + Kole EKG is Teh

which is independent of the hypothesis of the probability of a group

of clock corrections, the quantities / represent real errors of observa-

tion, of which the mean value is a. The error of interpolation gg

in the interpolation with regard to the least sinuosity, i.e. the

difference between the real clock correction /, and the interpolated
2 >

correction for the same instant is = From the expression of pro-
q

=

ae

( 255 )

7 i

YSuoud [][VUS 91n suoNoa.
—109 ysul a1} TUuNn ‘suoyvoydyynar ayy

IVI Sutuutseq suoytiedo asaya guadaa -

soordoag Aq qySut a3 0} poaow OT Aor *
S[VAIO}UT *

udIS asuLYoO pu Aolaq
Yo 9ya OF Staqunu ayy Aq g oprAtp

“OL

S[UATOPUT DAISSADONS ON4 DOTM JO WINS AY} 9YUI"G

L pu 9 jo wns oy 0404

AMO[PG Yay AYI OF stoquinu >< s[VAdozUT *

AAOGE JYSU oY} 07 stoquinu >< spwasozUT

Gysu ayy oy Ajoubiyqo | wos; Z Jounqns) +
(s[ta.toqur) +
(yey aug 07 Ajoubryqo youuqus) -

) °

(uoyrurxotdds ys.iy +6)

8

or

aor

aL

0)
0

(é
OT+
SR jee

‘OT

6

8

Vo_

Sie Gale We ar )—
Ci Gas ie ae 0

ee Gee Oe os
BSS bel ie Coe

ip ae ae
Gale ee liaie

0

Ter

0

9

0

0

6

a ei oF
Ge Bean —

8 € Grocneh
Leah ae oe Gate de Geet er he
Cisten CG aa a
Sl epe CO) 10 )ier OSes eee
Oo Uae eile Sp
0 0 UN pee oe WAS eer hs
aa re as a Bate

S
#
S)
+
i
zi

|
|
|
|
|

6 jo sanyea any

uorsurxotddy ysuy at oF
SUOT}OALIOD IS.AY
SUOTJOA.AIOD PUOVAS

SUOTAALOD IS.ty

( 256 )

TA Bab BIT

7% | ce
Coefficients for the compensation. e . BE Bs
| 100 100 | 400 | 40 100 | | 238 = ae
| ae | KP | Ke | Ky" | kK; | i = = = =
Be

108 | S| ft ea
Jan. 44 1.0/2.4/4.2/0.2]0.1} 68]— 30/— 2
» 17 2.115.413.5]0.9|0.4]0.0| 44/4 o7/4 4
» 20 1.2/3.5/2.9]1.3/0.8/04]0.0/%2/4 30/4 4
, 28 0.4/0.9|41.3}2.5}2.4/0.6|0.2|0.4 | 44.6]— 80} — 79
» 3110.4] 0.4) 0.8] 2.4] 2.8]4.0] 0.4] 0.2] 0.0] 45.7] 4 59] 4 57
Febr. 60.0 | 0.1 | 0.6 | 1.0 | 0.71 0.6 0.4;0.0/0.0) 24/— 4;— 8
» 46 0.0/0.2) 0.4/0.6/4.9]/1.9]05/02]04] s0]/— 9/4 2
» 19} 0.4 | 0.2]0.4/41.9| 9.3] 0.9]0.4]0.3] 0.4] 40.3] — 15] — 95
> 26 0.0] 0.0] 0.5 | 0.9} 0.7/0.8] 0.6] 0.4]0.0}] 2.6]+ 61] + 57
Mreh. 6| 0.0 | 0.2 | 0.4 os|o8 3.8 | 4.8 | 0.3 | 0.4 | 26.9 | — 301 | — 320
, 9 0.1 | 0.3 | 0.6 | 3.8 | 7.2 | 5.4 | 4.5 | 0.3 | 0.4 | 95.7 | + 543 | 4+ 580
> 12) 0.1 0.2 | 4.8 | 5.4/5.6 | 2.7] 0.8 | 0.2 | 0.0 | 68.3 | — 380 | — 405
>» 16 0.0/0.3] 4.5 | 2.712.314.2105] 0.4 | 0.0] 46.8 + 191 | + 199
> 22 0.0 0.2] 0.8 |4.2/1.5]4.0/0.3]04]0.0] 5.5|/— 48|— 40
> 27/ 0.4] 0.2] 0.5] 4.0] 4.2] 0.6|0.2| 0.0] 0.0] 3.2/4 o|+ 44
Apr. 3 0.0/0.4 | 0.3} 0.6] 0.7|0.4]0.4]0.0/0.0] 4.2/4 44|-+ 44
» 410| 0.0} 0.4/0.2] 0.4] 0.4|0.3]0.4] 0.0/0.0] 05] — 44]— 42
» 20) 0.0 0.0/0.1] 0.3] 0.4]0.3/0.4/0.0/0.0/ 04/+ 9/4 8
> 27) 0.0 | 0.0 | 04] 0.4| 0.4] 0.2} 0.2/04]0.0| 0.4]— 21] — 47
May 7/0.0/0.0/04|0.2/0.2| 0.3/0.3] 0a 0.0} 0.3]+ 20/4 44
» 490.0} 0.0} 0.4/0.3} 2.4/3.2}4.6/0.2]/04/47.4|/— 3/+ 4
» 29/0.0] 0.1 | 0.3} 3.2] 6.6] 4.5] 4.2/0.3]0.4|76.2}/+ 8]/— 6
» %/0.0)0.411.6| 4.5] 44/24/08] 0.2100} 47.6} — 4|— 27
» 30, 0.0 0.2/4.2) 24/21/46) 0.6/0.4) 0.0) 13.2) 4 70] -+ 69
June 4| 0.1 | 0.3 | 0.8 | 1.6] 2.3/4.5] 0.4/0.2] 0.4] 41.2] — 4]— 60
» 8104/0.2/0.6/4.5/4.3/0.6/0.4/04/0.0} 48|/— 51+ 7
» 48) 0.0/0.4 | 0.4] 0.6 | 4.3] 4.7] 0.9/0.2|0.0] 6.3]+4+ 51]+ 46
» 2910.0] 0.2 | 0.4| 4.7] 3.3] 2.7] 4.4 | 0.3] 0.0| 92.4] — 82] — 80
>» 26) 0.1/0.4 | 0.9} 2.7/3.7] 2.7] 4.0] 0.2]0.0] 30.0} 4+ 4 | 4+ 47
» 30] 0.0] 0.2 | 4.4 | 2.7] 3.5]2.2]0.6]0.4] 0.0] 26.0] + 55|+ 48
July 4] 0.0/0.3] 4.0 | 2.2} 2.4] 4.0] 0.3] 0.4 | 0.0 | 44.4 | — 404 | — 102

|
|

Coefficients for the compensation. . g = g
Pele aet tat lerocal eon: | A uae &
100 | 100 | 100 | 100 | 100 | ae | 25
| Ss | o&
Ky? | KP | Kg | Ky" | en Ee} | Sie

Br et to lees) =|. |
July 10| 0.0 | 0.2 | 0.6 | 1.0 | 0.9 | 0.5'| 0.2 | 0.0 | 0.0 + 63|/+ 67
» 47/ 0.0} 0.4 | 0.3 0.5 | 0.6 | 0.3 CMa KOON LOGO = wie 5
» | 0.0] 0.1 | 0.2/0.3] 0.3/0.2)0.4]0.0)0.0] 0.3] — 28|— 30
Aug. 5] 0.0 | 0.0 | 0.1 | 0.2| 0.3 | 0.4 | 0.2 | 0.0 | 0.0 03/4 50/4 51
» 13} 0.0/0.0} 0.4] 0.3] 0.6/0.5] 0.2]0.2]0.0| 0.8}— 91] — 89
» 419 0.0 0.0 | 0.2 0.5|0.5|0.4] 0.4] 0.0] 0.0! 0.8]+4+ 83] 4 80
» 34] 0.0| 0.0] 0.2 | 0.4 4:9)| 2:6) 4-0:0:9 | 04) 486} = 64 | — 58
Sept. 3) 0.0 /0.2| 0.3/2.6 | 4.2) 2.2/0.6 / 0.3/0.4 | 29.6) + 95 | 4 26
» 17/0.0|0.0/1.0/2.2/4.7/0.9/05/04/00| 98|/— 6 0
» 45 0.0] 0.2/0.6] 0.9/4.6 /4.6| 0.7/0.2} 0.0) 7.0|+4+ 97} +4 95
>» 1910.4] 0.3/0.5;4.6/2.5/1.8]0.7/04]/0.0/1430/— 2]/— 2
> 0.1/0.4] 0.7) 1.8) 2.5/4.7] 0.4] 0.4 |0.0 | 43.4] — 58] — 56
» 98| 0.0] 0.21 0.7/4.7] 4.7] 0.7/0.2 | 0.4 | 0.4 6.9/4 63/+ 61
Oct. 5/ 0.0/0.1 /04|0.7| 0.6/0.3] 0.2}0.2/0.0| 12|— o5|— ox
» 43, 0.0 0.1) 0.2/0.4) 0.4/0.5 10.4) 0.4/0.0) 0.7) — 16} — 47
» 24/ 0.0] 0.4/0.2] 0.5/2.5] 4.5/2.5 | 0.4 | 0.0 | 32.8 | + 249 | + 954
> 27/ 0.4 | 0.2] 0.4] 4.5 114.6 | 8.3 | 1.0 | 0.2 | 0.2 [295.7 | — 619 | — 699
» 29) 0.0 | 0.4 | 2.5.) 8.3] 7.4] 4.4] 0.4 | 0.2 | 0.0 498.8 | + 448 | + 455
Nov. 4/0.0| 0.4 | 4.4/4.4] 0.6/0.5 | 0.3/0.0] 0.0] 3.8/— 60]/— 64
» 460.01 0.210.4/0.5/4.514.51/05/0.3/0.0| 52/— 8 + 10
pig} 024) |50;2,1 0.3) 4.5 | 4.7 |.0.9)/ 0.6} 0.4] 0.0) 6.2) 6] — 47
» 2910.0] 0.0/0.5/0.9]2.0/20/08]04/01| 9.9 + 70/+ 97
Dec. 2) 0.0 | 0.3 | 0.6 | 2.0 | 2.5 | 4.6 | 0.9 | 0.2 | 0.0 | 44.4 | — 493 | — 4149
» 90.0] 0.1/0.8 | 4.6 | 3.4 | 3.4 | 4.2] 0.2 | 0.0 | 25.7 | 4+ 446 | + 450
> 12/0.0}0.4| 4.0] 3.4| 49} 2.8] 0.8/0.2 | 0.0| 4.9] — 54]— 65
» 46] 0.4 | 0.2 | 4.2 | 2.8] 2.7/4.4] 0.4] 0.4] 0.0/ 49.0] — 95} — 90
>» 2/ 0.0/0.2! 0.8/1.4] 1.3/0.8 | 0.3 | 0.0 | 4.9 | +401 | + 101
>» 98] 0.0] 0.2/ 0.4/0.8 1.0/0.6 | 0.1 | | 2.3}— 9] — 86
Jan. 2| 0.0] 0.4 | 0.3] 0.6 | 0.5 | 0.4 | 0.8/4 64 4 55
» 44| 0.0 | 0.0 | 0.4 | 0.4 | 0.0 PRGrOnp= “gi 4a

( 258 )

bability e 7k we may deduce the mean value of g,. Let us suppose
that all the real clock corrections remain unaltered with the exception
of that for the instant g. Then J; will be a minimum for L,— gy;
and for any other clock correction adopted for the same instant,
which differs from this interpolated value by «¢, /7 is equal to its
least value augmented by 6 A; &.

Hence the probability of the group of clock corrections contains the
factor e—’4,*, whence follows that the mean value of ¢€, or also

that of is equal to —————. Then the mean value of g, VA, is
hae 122K, is

1
constant and equal to SAT this constant is called vr.

Hence the 2°4 power of the mean value of the first member of
the above relation is:

Kemer |. «eK, jo) a Keke eee

The 2.4 member yw, computed with approximative values of x, y, z
and w is known. I had no direct data for the determination of mu’.
It consists of one part which is independent of the number of stars
observed for the clock correction, and another part which is in inverse
proportion to this number. For one star observed by Bakuvyzen, the
latter part amounts to about 900 and for PANNEKOEK it is a little
less. I wished to avoid a too large value for mw’ for fear of exag-
gerating the regularity of the clock at the cost of the accuracy of
the observations, and therefore I have put for each of those parts
of vw’? 300, together uw? = 600.

According to the value given above for § we have r?:uw7=§
and the mean value of the expression:

VK Rp. KOE KP LEE KP + KE +...
will be equal to w.

With different suppositions for §, I could derive from this relation
the corresponding value w*. My result was that for @ = '*/,, the
value of gw? is 592 and therefore I have retained this value of § in
the further computation.

The hypothetical expression of the probability was tested in two
different ways.

In the first place I investigated whether indeed »* might be regarded
as equal for the different intervals of the period treated. Therefore
I have arranged the observations according to their coefticients Ay,
and have derived from the half with least Aj separately the value
of uw? belonging to § =*/,,. The result was uw? = 591,

( 259 )

In the second place I have investigated whether also for intervals
of longer duration, the constancy of »* remains the same. In that
case I could derive »? from the total sinuosity /; of the real clock
corrections.

If we imagine that from the series of those clock corrections one
is dropped, which deviates from the interpolated value by Py,
then /; is diminished by 6 A, 7. 59 times we can drop succes-
sively one observation from the 61 observations at our disposal, and
thus each time diminish the total sinuosity of the remaining clock
corrections by 6 A’g*. At the end /7 is zero and hence /;, may be
considered as consisting of 59 parts, each of a mean value of 6 v?.
Henee the mean value of /7 is 354 »*.

From the reduced rates Q—w« Q? —y Q?—z Q”—uQP I deduced
the total sinuosity of the clock corrections L-+-/ and obtained
Tp4-7 =8756.

In my previous paper on this interpolation it is demonstrated that
if the errors / and the clock corrections expressed by I are independent
of each other /74,=/7-++-/y. Instead of the formula derived there:

[p= 68 fg—r) 1 now write [>= 26f,(&—&n)==6f,0,5

and substitute for oi, its value expressed in terms of the errors //.

In this way we get:

eNO les ig fate Mog fs Naty ae Bah Bee fe sel

In the 224 member occur under + many products of real errors.
The mean value of these terms is zero. I omit them, substitute mu?
for the square of each error and find as mean value for Jy 6 uw? = Ky.

The computation of + A, yielded 1.39.

In this way I have found the following relation between »v and u:

8756 = 854 v? + 8,34 p03

whence, if uw’ is equal to 600, »? = 11, which result is in good
harmony with the value first found for §.

The set of equations by which the most probable errors are
connected is readily solved, if we use as a first approximation

Roe Pe ; erg
= 0.46 ————_. After I had found by the substitution of these
Jq K, +8 y
values that the 2"? member required still another correction A, w, I used
A, wy, ;
Meo as a correction for the first approximation.
LSS AY

When computed according to the empirical formula
0.45 p, + 0.80 A, w,
io =
i Ky + 8

( 260 )
the errors appeared to satisfy fairly well the set of equations. Where

A
it was necessary I took away the last differences by adding + ¢ a

cg +
to the errors.

Table III contains the results of the compensation which are
necessary to compute the clock correction at any instant between
the observations. Let this instant be g-+ 7, the epoch of the im-
mediately preceding clock correction being q; if moreover during
the interval from g to g+ 7:
the mean atmospheric pressure, expressed in mms. of mercury

of 0°C. is 760 + By,

the mean temperature in centigrades 10°C + 7,
the mean difference in temperature in centigrades Vr,

then we can compute the clock correction S,47 according to the
following formula:

a hl

So4-7=S8,—fa +——(Gy+15,4B7 —26,497 4818 Ve +C,T+EnT").

T7000

The values of S, 7, G, C, E, occurring in this formula can be
derived from table III.

The 5" column shows the mean rates for each interval between two
successive determinations of the clock correction after the reduction and
the compensation. From this we may judge of the constancy of the rate
of the clock. It must be remarked that the small yearly inequality
occurring in these values is very probably due to a little inaccuracy
in the coefficient of temperature obtained as described above.

The last column of table III shows the quantities Y, the diffe-
rences between the successive values of /. They give us a simple
test for the computation of the compensation, because they must be
equal to the errors / multiplied by 20, or f#, = SY, : 20. The adopted
series of errors satisfies tolerably well this relation, if we admit
small differences, which in thousandth parts of a second of time do
not exceed the intervals i or 7 expressed in days, and hence give
rise to a difference less than 0%,001 in the mean daily rates.

( 261 )
Ae a DET:

unC(O el) ee ee
Fe Compen-!

5 Clock corrections) accord- cated
| Date 5 reduced to B ea daily G C Ie p>
2 Ss sation na
1903 4
Jan, 14} P | — 3" 98'02 |0°008 | 5 171 0.0 _ [- 0.44
» ATEB 27.561 |+ .004 Sue 476 |— 4.3 pe + 0.12
» 20|P 97.453 | .010 ie ie 184 |— 4.5 rape + 0.44
» 28] P 26 864 |— .014 18h 4 4.5 — 0.21
3 |B 26.744 |4- .006 | =” 177 4. 0.7 | eae + 0.04
Febr. 6 | P 26.921 | .004 ie e 175 |— 0.2 Ae: 4+ 0.07
» 16/8 26.710 |— .004 : 1m |4 0.5 — 0.08
» 19]; P ranma steko ual Pi + 0.02
» 6|B 26.690 | 005 hs ee 474 |— 0.8 Peon +012
Mreh. 6 | B 28.568 |— 020) 473 |-- 1.0 C ee 0.50
i 9'| P 28.539 |4 .038 sel Bangi ais 4 0:80
» 12| 8B 98.757 |— .021 i ee 186 |+ 0.6 ites ae
» 16] 8B 29.295 000 | 480 1 0.9 — 0.02
» 2|B 99.475 |— .004| He 168 |4 0.9 a tare
» 27| P 30.449 |4 .002 | Ls 166 |—. 0.5 | ip + 0.05
Apr. 3] B 21.453 + 007] | 180) — 44 De 40.12
10) 5 32.506 |— .004 189 + 0.2 — 0.07
» 20} B 33.565 | .001 |) fs 193 |+ 0.4 ree pe + 0.02
Bed? | <P 35.575 |= .007 | Hy 172 4 4.4 32) a) ear
May 7| B 38.051 |- .007 | ig 475 |— 1.6 eu + 0.13
» 19] P. 10.326 | .002 197 |— 0.3 — 0.01
» 2/8 ee naling ool | L040
» %|P 40.79% |— .002 | oe 200 |— 0.6 RAL + 0.02
» 30| P 4.746 | O14 2 a 1 |— 4.4 ms + 0.2%
June 4| P w.s72 |— 00 | | mo] 44 |" 0.0
Py | P 43.497 |— .002 200 + 1.2 — 0.04
[ » 18) B 46.570 [4+ .008 | 494 |— 0.5 ae + 0.47
» 2/| P wtse | 12] | 198 |r 0.8 |" | |= 0.85
» 26| P 48.265 + .009 494 |— 0.9 + 0.20
» 30} P 48.896 |-+ .006 | Tae eal tLe 0.09
; — .196 + 0.15 |

( 262 )

£ |Clock corrections| accord- |CO™PE?-
Date | £ | reduced toB [imgtothel Gainy | c E Ss
ate 3 | reduce 0 compen-| ally 2
= S sation | ates |
° f Qr
] l
1903-04 | | =
July 4|/ P| — 3° 49'732 |-0:0200| . | 4192 |4 1.6 en
—0°190 | — 0.22
» 10| B 51.290 |4+ .016 196 |\— 2.3 + 0.29
| I— .209 | + 0.07
ee aia :y) 53.488 |4 .002 | | 919 |— 0.9 | + 0.05
I— .219 | + 0.42
>, | B | 56.936 |— .011'} | "912 |-- 4.9 | — 0.23
_ Lat i— .204 — 0.44
Aug. 5| B| — 4: 0.408 |-++ .045 | 20 |— 1.8] + 0.36
I— .249 +019
» 13] B 3.415 |— .026 | 202 |4+ 27 — 0.51
— .198 — 0.32
» 19| B 5.734 |+ .094 | 906 |— 3.4, + 0.47
— .99 + 0.45
eel Py 10.082 — .012 M5 4+ 2.3 — 0.%
|— .209 \— 0.40
Sept. 3| P 10.946 |— .004 204 |4+ 41.4 | — 0.02
— .200 \— 0.42
| P 12.0299 + .005 198 |\— 0.4 + 0.05
=n = 07
» 15] P 14.936 |+ .008 9 \— 4.7 | + 0.25
— .U6 + 0.48
, 19] B 45.752 |— .003 217 |4+ 0.4 — 0.43
— 4s + 0.05
» %&| B 16.795 |— .010 210 |+ 4.4 — 0.2
— .208 — 0.16
wart: B 17.744 |+ .013 209 |— 0.8 + 0.27
— .209 + 0.4
Oct. 5| P 90.328 — .009 904 |4+- 4.5 — Oa
— .196 — 0 06
as | (P 93.762 |+ .001 a 192 |4 0.4 0.00
ed —0
» %| B 97.460 |+ .020 2 |\— 41.9 + 0.40
eye! + 0.34
peed babel 98.771 |— .02%4 913 [4 4 — 0.69
— .212 — 0.35
» 29) P 99.561 |-+ .024 a2 \— 0.9 + 0.42
— 25 + 0.07
Nov. 4| P 30.720 |— .004 215 |4 0.4 == "0.05
— .207 + 0.02
» 16| P 32.482 |— .004 195 | 4 — 0:43
— 199 — 0.4
» 19| B 32.918 |4+ .002 190 |4+ 0.3 + 0.09
== es — 0.02
» 29|B 35.348 | .01 187 |— 0.2 + 0.16
Sy + 0.44
Dec. 2} P 36.441 |— .020 485 |+ 4.4 — 0.30
— .186 — 0.16
re 37.998 | .026 195 |— 2.3 + 0.43
— .199 + 0.27
, 12) P 38 926 |— .005 | 201 + 0.4 — 0.0
i— 4197 + 0.22
» 16| P 39.866 |— .021 190 +- 2 — 0.49
— 482 — 0.27
» “| P 40.201 |-4 .019 484 |— 4.9 + 0.39
pcs) + 0.42
Prd ha 40.254 |— .017 479 |4- 4.3 — 0.32
ealties — 0.290
Jan 2| P 39.953 |4 .015 482 |— 4.7 + 0.25
— .195 + 0.05
» 14| 8B 44,200 |— .005 204 0.0 — 9.05
(October 20, 1904).

Error

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Saturday October 29, 1904.

SE ——

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 29 October 1904, DI. XIII).

GiO Ny Gees ANG oS:

JAN pe Vrins: “The congruence of the conics situated on the cubic surfaces of a pencil”, p. 264.

A. F. Hotreman: “The nitratioa of disubstituted benzenes”, p. 266.

A. P. N. Fraxcuimonr and H. Friepmann: “On zz’-tetramethylpiperidine”, p. 27. ;

P. J. Montacne: “On intramolecular atomic rearrangements in benzpinacones”. (Communicated
by Prof. A. P. N. Francmimonr), p. 27.

A. P. N. Fraxcuionr presents the dissertation of Dr. J. Morn van Cuarante: “Sulpho-
isobutyric acid and some of its derivatives”, p. 275.

W. A. Versiuys: “The relation between the radius of curvature of a twisted curve in a
point P of the curve and the radius of curvature in P of the section of its developable with
its oseulating plane in point P”. (Communicated by Prof. P. H. Scmovre), p. 277.

L. J. J. Muskens: “Degenerations in the central nervous system after removal of the floc-
eulus cerebelli”. (Communicated by Prof. C. Winker), p. 282.

H. Kamesrincit Onnes and C. Zakrzewskt: “The validity of the law of corresponding states
for mixtures of methyl chloride and carbon dioxide” (continued), p. 285.

B. Meminx: “On the measurement of very low temperatures. VII. Comparison of the platinum
thermometer with the hydrogen thermometer”, p. 290. — VIII. “Comparison of the resistance
of gold wire with that of platinum wire”. (Communicated by Prof. H. Kameriixcu Ones),
p- 300. (With cne plate).

Jan pE Vries: “A congruence of order two and class two formed by conics’, p. 311.

W.. E:yrioven: “On a new method of damping oscillatory deflections of a galvanometer”, p.315,
(With one plate). |

W. H. Junius: “Dispersion bands in the spectra of 2 Orionis and Nova Persei”, p. 323.

J. Ove Jr.: “The transformation of the phenylpotassium sulphate into p-phenolsulphonate of
potassium”. (Communicated by Prof. C. A. Losry pr Breyy), p. 328.

J. F. Suyver: “The intramolecular transformation in the stereoisomeric ~- and 3-trithioacet and
g- and j3-trithiobenzaldehydes” (N°. 11 and 12 on intramolecular rearrangements). (Communicated
by Prof. C. A. Losry pr Brvyy), p. 329.

J. W. Diro: “The viscosity of the system hydrazine and water”. (Communicated by Prof.
C. A. Losry bE Bruyn), p. 329.

J. M. van Bemueren: “On the composition of the silicates in the soil which have been
formed from the disintegration of the minerals in the rocks”, p. 329.

Erratum, p. 329.

The following papers were read :

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 264 )

Mathematics. — “TVhe congruence of the conics situated on the cubic
surfaces of a pencil.” By Prot. Jax pe Vries.

(Communicated in the meeting of September 24, 1904).

lL. On each cubic surface S* of a pencil lie 27 systems of conics,
of which each system has one of the 27 right lines for common
chord. Through any point ? of space passes one S* of the pencil;
so P bears 27 conics C* of the congruence formed by the C* of
all (S?.

The pairs of lines of the congruence evidently form the skew sur-
face of the trisecants of the basecurve ’; for the points of inter-
section of a right line of one of the S* with a second S* belong to
all S* of the pencil. Quite independent of this consideration we
can find the degree of the above mentioned skew surface in the
following way.

Let wu be a homogeneous function of degree / in wv, y,2. If we
take a point of A’ to be the origin of the system of coordinates,
the pencil has for equation

U, t+u, tu, = 0,
where the coefficients contain a parameter 2 to the first degree.
If a right line through © must lie on S*, for all values of m
the equation

mu, + mu, + mu, = 0

must be satisfied, so we have simultaneously
0%, ele U0:

By eliminating «, y, 2 out of these three equations a relation is

found containing the coefficients, so 2 too, in degree

33h 2:4 8 Pee Se 3)
The basecurve R° is thus an e/evenfold curve on the skew surface
of the right lines lying on the surfaces S* of the pencil.

Each surface S* bearing 27 right lines; its section with the indicated _
skew surface is of degree 27 + 9X 11 = 126; the skew surface is
thus of degree 42 °*).

2. Any right line / is a chord of 42 conics, whose planes are
determined by the 42 right lines of the skew surface 9** resting on /.

Let us now consider the surface (?) formed by the C*, the planes

1) See a.o. Cuesscn, Lecons sur la géometrie, Ul, 13,

2) See a.o. Kuvyver, Kenmerkende getallen der algebraische ruimtekromme
(Characteristic numbers of the algebraic twisted curve.) Versl. K. A. v. W., 3"
series vol. VII, page 152, 1889.

( 265 )

of which pass through the point P. An arbitrary ray through LP is
intersected outside P in 84 points. As P bears 27 C* the degree
of (P) is 84+ 27 = 111.
According to the notation of Scuuserr we have thus
pa = 42 and Ty eer Fa
From the well known relations *)

By = 24 + d+ 4a and 30 = y+ 20+ 2u,

where in our case 7 is equal to 0 we deduce by symbolic multi-
plication the following system of relations :

suv—dut+4u’, s3uo=2Z2du+2u ,
3yn°=—dv+4yur, 3v0 =2dv-+ 2e?r,
€ —- 2 Ces) | 9

3vo—do+ 40, 30 =Ze0d-2No.

We have here six equations for nine characteristic numbers of
which we have already determined two.

But the number dr we ean find directly. For on the arbitrary
right line / rest 42 right lines of 9; each of these right lines is
intersected bij 10 right lines of the surface S* to which it belongs ;
so it furnishes 10 pairs of lines resting on /. Consequently

dv = 420.
We now find successively
00 du= 165, po=do4,
[po — 1387, do = 510), 0”? = 432 .

3. Out of vr? = 288 follows that the surface A formed by the conics
cutting the right line 1 is of degree 288.

Evidently / is a 27 fold right line of 4 and a chord of 42 conics
lying on A. It is evident that on A lie 462 right lines, which are
situated three by three in 210 planes.

If / is a trisecant of R’, thus a right line of a surface S,*, then
A*** breaks up into the surface S,* counted double and the loci of
the conics passing through each of the three points of intersection
of the trisecant.

The conics having a point of the basecarve R* in common, forin
thus a surface of degree 94.

The surfaces 7° belonging to the points of intersection 7’, and 7’,
of the trisecant have evidently the 10 conics in common which are
determined by the 10 trisecants through the third point of inter-
section 7’,.

1) Scuupert, Kalkiil der abzdhlenden Geometrie, p. 92.
ditsi

( 266 )

If / is a chord of R® then 4 degenerates into two surfaces 7'"*
and a surface of degree 100.

If / is a line cutting R* once then A consists of two parts, which
will be successively of degree 94 and 194.

4. The numbers d9 = 510 and d# = 165 furnish well known
results. ') The first tells us that the skew surface 0** of the trisecants
of R* possesses a double curve of degree 255 ; for each plane through
a double point of a pair of lines is to be regarded two times as
tangent plane. The second number furnishes the property that the
threefold tangent planes of the surfaces S* of a pencil envelop a
surface of class 55,

The surface (7) contains thus 165 right lines lying three by three
in 55 planes.

The numbers vo and we furnish with reference to the plane at
infinity the following properties :

The parabolae of the congruence form a surface of degree 354,
their planes envelop a surface of class 138.

Each .S* contains 108 parabolae.*). As a definite S* can cut the
parabolae on the other S* only in points of the basecurve R® the
locus of the parabolae passes (3 >< 354 — 2 108) :9 = 94 times
through R°.

So through each point of 2° pass 94 parabolae.

Chemistry. — “Vhe nitration of disubstituted benzenes.” By Prof.
A. F. Ho“ieman.

(Communicated in the meeting of September 24, 1904).

If we introduce into a benzene derivative C,H,X a second atom
or group this takes up in respect to X a position either chiefly
meta or para-ortho depending chiefly on the nature of X. The cause
of this is as yet obscure. The efforts for, elucidating this phenomenon
are totally inadequate, first of all because they are too vague, secondly -
because they do not take into account the relative quantities which
are formed from the isomers; in fact they could not do so, as these
were still unknown at the time that these ‘‘explanations” were
given. A better insight into this problem can only be rendered possible
by the quantitative study of the substitution process, which has

1) Kiuyver, page 152.

2) J. pe Vries, La configuration formeée par les vingt-sept droites @une surface
cubique, Arch. Néerl., sér. 2, t. VI, p. 148.

( 267 )

already been made in a number of cases of nitration of the sub-
stances C,H,X.

For the present we must content ourselves with accepting the
results of those quantitative studies as facts. Doing this, we may
put the following question: Given a benzene derivative C,H,XY,
in which a third group Z is introduced. If now we know the
relative quantities of the isomers of both C,H,XZ and C,H,YZ which
are formed by the introduction of Z into C,H, and into C,H,Y ;
can we then deduce from this the structure and relative quantities
of the isomers C,H,XYZ which are formed by the introduction of
Z into C,H,XY?

Suppose (by way of an example) that we have determined how
much para- and ortho-compound is formed in the nitration of chloro-
benzene and how much meta- and ortho-benzoic acid in the nitration
of benzoic acid; can we then determine beforehand which and how
much of the possible nitrochlorobenzoie acids will be formed in the
nitration of chlorobenzoic acid ¥

Qualitatively this problem has been studied rather fully, but as a
rule not very systematically. In a great many cases it has been
determined which of the possible isomers C,H,XYZ are formed by
the introduction of Z into C,H,XY and one has tried to draw con-
clusions therefrom which render it possible to prediet what may be
expected in unknown cases. Bemsterm has summarised these as
follows: “In the introduction of a group Z into a substance C,H,XY
both X and Y exert an influence but that of one of these groups
is predominant and directs Z.”

Undoubtedly, this rule is correct in a great many cases, but not
in a good many others. For instance it cannot be applied to the
nitration of m-nitroanisol, which I have investigated. In any case it
shows that the groups X and Y do not exert their directing influence
independently of each other but that this is modified by their
simultaneous presence. This has been fully confirmed by a quanti-
tative investigation in the case of a number of nitrations of the
compounds C,H,XY.

If the groups X and Y exerted a directing influence on a third
substituent independently of each other we ought to have the following:
If we call. the proportion in which the three isomers are formed
when Z is introduced into C,lI,X

Portho + meta + Vpara
and that of the three isomers when Z is introduced into C,H,Y

t a = !
Portho + Gmeta + © para

( 268 )

the quantity of the isomers, when introducing Z into C,H,XY, would
be expressed by products as pq’ etc. as shown by the subjoined
scheme: :

xX yp xX

BF ee a7 \¥ mt” Nx
N/a ee

,

q rq

In this it has been supposed that on introducing a second group
into a monosubstituted benzene derivative all the three possible
isomers are formed, which in practice will most likely be the case
even if the quantity of one of these should be so small as to be
generally overlooked. In fact, in a number of cases where at first
only two. substituents had been found, such as in the nitration of
nitrobenzene, a careful investigation also revealed the presence of a
third one.

The quantitative investigation as to the relative quantities of the
isomers which are formed in the nitration of substances C,H,XY
now showed that those quantities generally differ very considerably
from the products pq’ ete. so that a serious diversion of the directing
influence on the third substituent must be admitted. This diversion
was found to depend not only on the nature of the substituents but
also on their place in the molecule as proved by the following
example.

In the nitration of chlorobenzene at O° para-, ortho- and meta-

nitrochlorobenzene are formed in the following quauitities:
COH

Cl é ; ;
Jw: In that of benzoic acid BAC
29.8 F a 3 z 18.5
| | the nitrobenzoic acids in | |

| t

0.8 : : 80.2
the following proportion : wee
69.9 1.8

If CO,H and Cl did not modify each other’s directing influence, nitro-
derivatives obtained in the nitration of ortho- and metachlorobenzoie
acid would be formed in the quantities indicated in the subjoined

schema :
CO. CO.H

4\a ah 69.9185” \o08x 185
69.9 X 80.2 29,8 < 80.2 \ cl
4 SG Nar

of the other possible isomers only very insignificant quantities. These
were in fact so trifling that they were not found. Of the two isomers
to be expected in both cases, the relative quantity ought to be just
as large as that which is formed in the nitration of chlorobenzene
itself. Instead of this was found:

2

COH COU

a: 91.3 AN a

PRU Zee ee
In the subjoined table a number of such observations have been
collected. By “diversion” is meant the quotient of the quantity of
the byeproduct actually found and that of the quantity calculated.

Amount of byeproduet

Nitration of on 100 parts of main prod.

at —30° | at 0°
,H,Cl 36.4 42.0)
O,H,Br | BRL | loa, |
G,U,COsH 16.9 93.1 |
Diversion of the directing influence
of the halogen | of carboxyl.
a | eee
Tt —302 eat OS | at —30°} at 0°
SLA ey EE Ts ere | Ks
|
0-C,H,Cl.CO.H 16.3 AOA) V3 SOARS Oh 108455 te
0-C,U,Br.cO,H | 20.6 Wh Ons | Os pea 28
Beeu CLCOH' | 94) | 9.5 0.250 | 0.996 | 0.539 | 0.444
m-C,H,Br.CO,H | 13.4(?)| 12.9 Ose) | FOts = |) 0.601
0-0, H1,Cl, fe cde I SOA. «SUSI AUP erktss || tee 0 (ies
m-O,H,Cly DW died OO | OE | = WS

In the nitration of o-halogenbenzoie acid and of o-dichlorobenzene
the NO,-group in the byeproduets places itself adjacent to the halogen;
in that of the m-dichlorobenzene befereen the carboxyl and the
halogen. On comparing the diversion figures of the directing influence
of the halogen in these acids and dihalogen-compounds, those of the
meta-compounds amount to about half of the ortho-compounds. This
is one of the many cases which show that the introduction of a
substituent between two others meets with a particularly great
resistance.

Gronimgen, Lab. Univers. Aug. 1904.

( 270 )

Chemistry. — “On a a’-telramethylpiperidine.” By Prof. A. P.N.
Francuimont and Dr. H. FrinpMany.

(Communicated in the meeting of September 24, 1904).

This substance, which was obtained in 1885 by Canzoneri and
Sricd but in an impure condition, was prepared by us in another
manner, namely, by reduction of y-bromotetramethylpiperidine with
a copper-zine couple (Gladstone-Tribe’s method) in absolute alcoholic
solution.

It is a liquid boiling at 155,°5—156°5 at 760 m.m. pressure having
a sp. gr. of 0.8367. With water it yields a crystalline compound
which melts at 28° and loses its water totally or partially in a dry
atmosphere. The compounds with hydrogen chloride, hydrogen bro-
mide and sulphuric acid form very beautiful crystals; those with the
two first-enamed acids sublime on heating without previous fusion,
those with sulphuric acid melt: the acid one at 174’, the neutral
one at 270°.

Compared with piperidine, this amine reacts remarkably slowly
on acid chlorides such as benzoyl chloride, chloro-formic esters,
pierylehlovide ete. In aqueous solutions the reaction takes place
hardly at all, in ethereal solutions extremely slowly. However, there
were obtained: methylurethane as a liquid with a strong mint-like
odour boiling at 227° at a pressure of 760 m.m.; sp. gr. 0.9848
and the benzoyl derivative as crystals melting at 41°—42°; the piery]
derivative melts at 225°. An effort to prepare a urea from the hydrogen
chloride compound and potassium isocyanate has resulted as yet in
failure. This reminds us of experiments of Dr. K. H. VAN per Zanpr
in 1889 with di-isopropylamine, where urea could only be obtained
with difficulty and in very small amount, whereas dinormalpropylamine
presented no difficulties‘). If we compare the formulae of di-isopro-
pylamine and «@ a@’-tetramethylpiperidine we notice that they only

a Os
SiS
BC CE

Hoan ey
(CH), CC (CH), (CH,),C — C(CH,),
aes a
N N
H H

1) Some years before, I had already noticed an analogous phenomenon when
treating propyl- and isopropylmalonic acid with nitric acid; the first compound is
much more readily attacked than the second.

differ in this way that the two hydrogen atoms of the first compound
(indicated by asterisks) have been replaced in the second one by the
bivalent group CH,—CH,—CH, ; piperidine and tetramethy|piperidine
differ becanse the first one contains hydrogen atoms where the other
possesses methyl groups, namely at the @ C atoms in regard to the
nitrogen.

As piperidine reacts strongly with the above substances and tetra-
methylpiperidine does not do so and as there exists an analogous
difference between dinormalpropylamine and di-isopropylamine it is
natural to look for the cause of this in the methyl groups. As,
however, their nature does not explain this difference we are bound
to consider their mass and their position in space in regard to the
nitrogen. This is then a case of so-called sterical obstacle which is
to a certain extent comparable with a number of other cases which
have been chiefly observed in the aromatic compounds; a case which
may, perhaps, affect the views held as to the nitrogen atom.

It must be finally observed that tetramethylpiperidine yields like
di-isopropylamine a crystalline compound with nitrous acid, which is
fairly stable and is only decomposed at a higher temperature into
water and the nitroso-compound.

Chemistry. — “On intramolecular atomic rearrangements in bens-
pimacones.” By P. J. Montacxn. (Communicated by Prof.
A. P. N. FRANcuimont).

(Communicated in the meeting of Seplember 24, 1904).

The following research originated in an effort by Nnr’) to explain
the intramolecular atomic rearrangement in the conversion of benz-
pinacone into benzpinacoline by assuming the presence of an inter-
mediate product. His explanation when put into formulae is as
follows :

Benzpinacone is dissociated into water and unsaturated hydrocarbon:

OH C,H. OH C,H,
; Vong a Ve
oe Oe a PUL SY ees oe eee

| 1 | 1
Go OH 9" O,H, .H ore
OH C,H, )
ty of
then addition to: C,H,—C———C
| Xe
CH, a

1) Ann. 318 p. 38.

OH po
then dissociation: © C,H,—C-————— C—C,H,
| NOH,

me C,H

= aoa era)

then addition to: © C,H,—C———_——_C—C,H,
No,

O

Ak ee
then formation of C,H,—C——C(C,H,),.
If, however, we represent the matter by structural formulae we

o OH, OW CH,
C H,—C——C—C,H, => eo. as
|
ray ch e
wee Ee
va C,H, sj C,H,
C,H,—C— C-—C,H,’) — C.H,.—C_—_C-6. be =

ee oie

oe a
Wg

bo ea. : C,H

fe | Y / ae

=O Oe Oi ee Ore ae C=tz4
Cy 4

This shows now in the plainest manner that the core I is
attached to different C-atoms before and after the migration. If we,
therefore apply this representation to derivatives of benzpinacone,
the group or the atom in the core I of benzpinacone then occupies
a different position from that in the benzpinacoline obtained from
the same.

1) Which H-atom of core I migrates to the group CgHy, is not stated by Ner;
the ortho-placed H-atom was taken arbitrarely.

( 273 )

Some time ago’) | pointed out the possibility of the existence of
a similar intermediate product in the transformation of hydrobenzoin
into diphenylacetaldehyde

OH H OH H
ri
H—C—————_C—C,H, —~ H-—C C—C,H,

| \ou |

but at the same time I have shown experimentally the incorrectness of
that supposition. In an appended note I have already observed that
in view of my results obtained, the theory of Nrr was not acceptable.
It appeared to me therefore, to be of importance to extend my
researches also to a derivative of benzpinacone and thus to form a
definite opinion as to the correctness or incorrectness of Ner’s theo-
retical explanation.

For this research I took 44'4"4" tetrachlorobenzpinacone obtained
by reduction of 44 dichlorobenzophenone. On being heated with acety'
chloride it passes into tetrachlorobenzpinacoline. If this is boiled
with alcoholic potassium hydroxide it is resolved into trichlorotri-
phenylmethane and p. chlorobenzoic acid according to this scheme:

(4). Cl C,H,), : COH .COH : (C,H, Cl (4), —
((4).Cl. CoH, )ax

C=C0=CHACi(a) =>
wh ofl, )
> C1C,H,

—

(2. Cl. CH) ve
aS JH 44 (C8 5, OLY
2 Cl. C,H, an

This trichlorotriphenylmethane now appeared to be identical with the
44'4" trichlorotriphenylmethane obtained from 44'4" triaminotriphenyl-
methane (p. leucaniline). This explains the para-position of all chlorine
atoms in the first-named trichlorotriphenylmethane. This further shows
that the phenyl group is attached to the same C-atom before and
after the migration and that therefore the intermediate product as
suggested by Nur is an impossibility. The explanation given by Nr
for this intramolecular atomic rearrangement is, therefore, incorrect.

The views held as to the transformation of ¢-elycols into aldehydes
are two in number:

1) Ree. 21. p. 30,

(274)

1. Splitting off of HO, in such a manner that the group OH
departs with the H of a C-atom, for example:

Ou OH
CHA fF :
oS oS.

2 ea) CY XS

Nrr’s view of benzpinacone is in accordance with this.

The object of this representation is to abandon the idea of an
intramolecular atomic rearrangement and to substitute so-called nor-
mally-proceeding reactions.

2. Splitting off of H,O in’ such a manner that the group OH
departs with an H of the second OH group, for example :

CH CH CH CH
\GOH=COHS 4 0-5). ee cee
CH, CH, CH” \,/ SCH,

Ner'): “Es ist jetzt vollkommen klar, dass diese Reaction (Umwand-
lung der 1.2 Glycolen in’ Ketonen) bei welcher eine scheinbare
Verschiebung der Hydroxyle eintritt, auf eine intermediaire Bildung
von Alkylenoxyd zuriickzufiihren ist.”

H H
C,H,—_C—OH C,H,—C
We BC

iH Cc nO
C,H,—C—OH Sn

In this representation we still admit an intramolecular atomic
rearrangement; not, however, with the 1.2 glycols but with the oxides.

In the transformation of hydrobenzoin into diphenylacetaldehyde,
and now again in that of benzpinacone into benzpinacoline, I have
shown that the first theory is untenable. In view of this I consider the
existence of a trimethylene-ring also in the transformation of pinacone
into pinacoline too as being less probable.-Of course a direct proof,
as in the case of the aromatic a-glycols, cannot be produced, but,
provisionally, this theory seems to me to lack all foundation. It
looks to me as if Nev himself is abandoning this theory, because,
whilst formerly ‘) he considered the trimethylene-ring as very pro-

1) Covrurter. Ann. chim. phys. [6] 26 p. 434, Entenmeyer. Ber. 14 p. 322. Note.
Zeunsky and Zeuikow. Ber. 34 p. 3251.

2) Ervenmeyer. Ann. 316 p. 84.

3) Ann. 335 p. 243.

4) Ann. 318 p. 38,

(275)

bable, he now, to judge from the above quotation, definitely adopts
the oxide-ring *).

The results of my researches, [| may draw up in the following
rule ;

In the transformation of the 1.2 glycols into aldehydes, a real
intramolecular atomic rearrangement takes place, which cannot be
explained by any normally-proceeding intermediate reaction; if has
not, however, been decided as yet whether this atomic migration
takes place with the 1.2 glycols themselves or whether the oxides
are formed first and then undergo an intramolecular rearrangement.

IT am now making experiments in that direction with Dr. Menreura.

Chemistry. — Prof. A. P. N. Francumonr presents to the Library
of the Academy a dissertation from Dr. J. Mont van CHARANTH,
entitled: ‘“Su/pho-isobutyric acid and some of its derivatives”

and offers the following explanation.

(Communicated in the meeting of Seplember 24, 1904).

Dr. Moti van Cuarante has commenced at my instigation to tho-
roughly investigate sulpho-isobutyrie acid. He prepared it according to
the process which I had published many years ago for the preparation
of those aliphatic sulphocarboxylic acids in which the sulphonic acid
group is attached to the same carbon atom as the carboxyl group (namely
from the acid anhydrides with sulphuric acid). These acids are not
only important from the fact that they are bibasic acids, of which our
knowledge leaves generally much to be desired, but also because
the two acid functions are of themselves, and not merely on account
of their position, of different strength, and are situated together more
closely than in the case of the aromatic acids, and can therefore,
exert a greater influence on each other. The difficulties experienced
in the case of sulphoacetic acid, sulphopropionic acid ete. caused by
the mobility of the hydrogen atoms which are placed at the same
carbon atom could not present themselves here, because the atom to
which the two acid functions are linked, does not carry hydrogen.

The said method of preparing, which had never been fully eluci-
dated, in which two mols. of acid anhydride react with one mol.
of sulphuric acid to yield one mol. of sulphonic acid is thus explained
by Dr. Moni van Cuaranty: a diacylsulphuric acid is formed which

1) At least if the quotation is meant for a/Z the 1.2 glycols.

on being warmed is converted into monacylsulphonic acid, which in
contact with water yields sulphonic acid and carboxylic acid:

Cy Hongi CO.0.802.0.CO.C, Hanzi passing into Ca Hoy, CO.O.
SOo.Cy, Ho, CO.OH and then by H.O into fa, Ho,4) CO.OH and HO.SOs.
Ca Hon-CO.OH.

Specially undertaken experiments led him to this conclusion and
also taught him that when the acid chloride was used instead of the
acid anhydride also two mols. of the latter are required to one mol.
of sulphuric acid. The action of chlorosulphonic acid on carboxylic
acids, which is also given as a method of preparing sulphonic acids,
is understood by him to first vield the acid chloride and sulphuric acid,
which then react on each other with formation of the sulphonic acid.

Sulpho-isobutyric acid itself is a very hygroscopic substance con-
taining two mols. of water of crystallisation. The barium salt contains
three mols. of water, the sodium salt half a mol. The neutral silver
salt is anhydrous like the acid salt, which latter can only be obtained
in the presence of a large excess of the acid.

When acting on the sodium salt with phosphorus pentachloride
Dr. Mon. van Cuarante obtained, according to circumstances, either the
dichloride ora chloro-anhydride, which is the chloride of the carboxy lic-
and the anhydride of the sulphonic acid function. The dichloride
is a colourless liquid, which distils at about 55° under a pressure of

9
1—'/)m.m. mercury, with a sp. gr. d m = 1.4696 and a refractive

power 7p = 1.4887; it solidifies at — 10°. The sulpho-anhydride-
carboxy-chloride is solid, crystallises from ligroin and melts at 61°.

With a little water the dichloride yields sulpho-chloride-isobutyric
acid, which is crystalline and melts at 184°. With more water, sul-
pho-isobutyric acid is formed. With methyl alcohoi the ester of the
carboxylic function is generated whilst the sulpho-chloride function
remains. This ester. sulpho-chloride is a liquid, which passed over at
a pressure of 1'/, m.m. at about 60° and solidified at 21°.5; the

2

sp. gr. was d= = 1.3436, the refractive power np = 1.46658.

Treatment with sodium methoxide dissolved in methyl alcohol yielded

not the dimethyl ester but the ester sodium salt of the sulphonic acid.

The dimethyl ester prepared from the neutral silver salt with methy]

iodide was a liquid which passed over at a pressure of 1—‘/, m.m.

between 82°—78°, solidified on cooling and then melted at 4°; the
2

gr. was Se = 1.2584, the index of refraction 7p = 1.44481.

Sp.

(277)

The neutral ester is saponified by methyl alcohol and then yields
an acid one like all sulphonic esters. With ammonia it yields an
ammonium salt of the sulphonic ester funetion, which is also an
ester of the carboxylic acid.

The acid ester, namely the carboxylic ester of the sulphonic acid, was
also obtained from the sodium salt of sulpho-isobutyric acid by means
of hydrogen chloride and methyl alcohol and is hygroscopic. [ts isomer,
the carboxylic acid of the sulphonic ester, which was prepared from the
acid silver salt with methyl iodide, is not hygroscopic, it crystallises
from benzene and melts at 90°. Dr. Moni vax CHarantn’s experiences
with the esters of sulpho-isobutyric acid agree fairly well with those
of Weescuniper with metasulphobenzoic acid.

The melting points of the compounds obtained behave as might
be expected ; those of the sulphonic acid chlorides are more elevated
than those of the sulphonic esters; those of the carboxylic chlorides

are lower than those of the carboxylic esters. The melting points of

the esters as well as those of the chlorides of the carboxylic acids
are lower than those of the carboxylic acids themselves.

Mathematics. — “The relation between the radius of curvature of

a twisted curve in a pot P of the curve and the radius of

curvature in P of the section of its developable with its osculating
plane in pot P2’ By W. A. Vurstuys. (Communicated by
Prof. P. H. ScHours).

(Communicated in the meeting of September 24, 1904.)

Sah. Tuororem. Lor each twisted cubic C* the ratio ts constant
of the radius of curvature in any point P to the radius of curvature
of the section of the osculating plane in the point P with the developable
O, belonging to C’.

Proor. If we take P to be origin of coordinates and the tangent,
principal normal and binormal of the curve C* in the point 7 to
be the axes of coordinates, then C* is the cuspidal curve of the
surface O, enveloped by the plane

Me 33 CoD 0
where
Die,
C= 4,
B=b,«+b,y4+ 6,2,
A=a,e+a,y+a,z2+a,.

( 278 )
The coordinates of the points of the curve C® satisfy the conditions :
tl =-Z, fis — se. A 2.
whence
CR BEC 5 G0 tae

ge) b, e,—a, ¢, t+(a, b,—a, b,) C—a, b,c, vs N :

Cal et a,t (c,—b, t—b, ¢, t?)
) SSS eS

N N

Now the radius of curvature #, of the twisted curve C* in the
point P is the same as the radius of curvature of its orthogonal
projection on its osculating plane in P, the curve with its projection
in P having three consecutive points in common. The parameter
expressions for the coordinates of this projection are

ade Mt led, t—b, 0)

N N

—

dy
From the value of y we find = =0 for ‘=O; so for the general
cf

formula
aS (da? + dy?)'/s
dx d?y — dy dx :
giving the radius of curvature of a plane curve, can be substituted
the simpler expression :

res die _ a
URE aoe 2a, b, oe

The equation of the surface O, enveloped by the plane
At—3BC?43Ct—D=0

AD—-bABCD+4AC+LABD—-3BC’=0.
The curve of intersection with the osculating plane D = z= Ois:
C(4AC—3 B)=0.
So the equation of the conic d, lying in the osculating plane is:
4(a,xtayta)cy —3(b,¢+4 6, y? =0.
The equation of the parabola osculating this conic d, in the
origin is:
4a,¢,y —3 0," a? =0.
This parabola has in the origin the same radius of curvature 7,
as the conie d,. The radius of curvature in the vertex of the parabola

a -

|

( 279 )

is the parameter. So the radius of curvature 7, of the conic d, in

0

tl oe uy 24, 6,
1e@ Origin is ———.
= 3b?
From the values :
a, 2 a, Cy
it. = and 7, =——-
a b, 3 Die

now follows:
(0), on 10),

§ 2. The theorem can be easily expanded to a general twisted
curve C.

Let P be an ordinary point of C, the tangent and the osculating
plane in P showing no particularities. Through point P? and five
consecutive points of C a twisted cubic C* can always be laid.
The radius of curvature #, in the point /P is the same for the curves
C and C", having six consecutive points in common. The osculating
planes of the curves C and C* in the point P will coincide too.
This common osculating plane ( intersects the developables belonging
to C and C®* according to the tangent in ? counting double and
moreover according to two plane curves d@ and ¢,.

If the curves © and C* had but a three-point contact in P, the
curves @ and d, would have a common tangent in the common
point P, so that the curves d and d, would have in P at least two
consecutive points in common. If the curves C’ and C® were to
have a five-point contact, a Common generatrix of the two develop-
ables not lying in the common osculating plane ( would meet the
osculating plane @ in a third common point of the curves dand d,.
Now that the curves ( and C” have a six-point contact in P the
curves d and d, will have at least four consecutive points in common.
These two sections d and d, have thus in P the same radius of
curvature 7,. Consequently in the ordinary point P of the twisted
curve C’ we have:

Reith =e:

§ 3. When two arbitrary twisted curves have in a point P a
three-point contact, they have in that point the same radius of eur-
vature Fe. If now the common osculating plane O in P of the two
enrves cuts the two developables belonging to the curves in the plane
eurves d@ and d’ then the radii of curvature in P of these sections

4
d and d' are both 3 R and therefore equal. The curves / and d'

have thus in P also a three-point contact.
theorem :

From this follows the

19
Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 280 )

If two twisted curves have in P three consecutive points in common
this will be also the case with the plane curves forming part of the
sections of the common osculating plane with the developables belonging
to the twisted curves.

The radius of curvature of the section d in the point P being
four thirds of the radius of curvature of the cuspidal curve C in
this same point, the curves d and C' have in P but two points in
common.

From the theorem proved here, follows once again the theorem
communicated by me before, concerning the situation of the three
points which a twisted curve has in common with its osculating
plane. (see These Proc., Febr. 27%, 1904).

§ 4. By expansion of the coordinates of an arbitrary algebraic or
transcendent twisted curve in the proximity of an ordinary point
P into convergent power series of a parameter f, the theorem of
§ 1 can be proved also directly for such a twisted curve without
using the twisted cubic.

Let P be an ordinary point of the curve C; if the tangent, the
principal normal and the binormal in P are taken respectively as
X-axis, Y-axis and Z-axis, then the coordinates of the twisted curve
C become :

@ Ont ent a. eS
SO SU oo ’
BON Ee ttl ee

The point P corresponds to the value zero of the parameter ¢.
If P is an ordinary point the coefficients @,, 6, and c¢, cannot be zero.
Let &, be the radius of curvature of Cin point P, thus the value
obtained by the radius of curvature FP for ¢=0. The radius of
curvature in P of the projection of C on the osculating plane z=
is also &,, this projection having in P three consecutive points in
common with C. :

The coordinates of the points of this projection are :

eHat =a,F Se... G

yah, be eee

dy
As is equal to O for = 0 the general formula for the radius
a

of curvature
(dix? + dy?)'/2
dx d?y—dy da ”

l=

( 281 )

transforms itself into the simpler one

dav’
a —
d*y1=0
It is easy to find
Res
ae 28,

The coordinates §, 4 and § of an arbitrary point Q on the developable
belonging to C ean be expressed in the parameters ¢ and 7 where
r represents the distance from point @ (S, a, $) to point (wv, ¥, 2)
of the cuspidal curve measured along the tangent of C’ passing through
Q. The coordinates of Q are:

or plas eae ry ;
dt dy

== The ee ’
dt dz

Sean iene ;

For the points @Q situated in the osculating plane § = 0 the relation
=2+ Ae Ke
ds dt
must exist between the parameters 7 and ¢. By eliminating 7 out of
this relation and the equations for g and 4 we find expressed in
functions of ¢ the coordinates of the points @Q situated in the plane
=0. These coordinates of the points of the curve of intersection
d are
du dz

ae : =
5. dt dt °
dy — dz
YS y-—2—:-—,
Mr ek tei Bde
or
rt + et'+... +2a,t4+... 2
Sat +a,?+..— (ctl He ae (a, avlats) = —a,t-+... ,
3c,t? + 4,04... =13
(c,t* ++ ¢,t* +...) (26,24 3b,e-+... il
ee pity oo a ee) ere) ee
3¢,0)+4e,0°+... 3
Ms] 5
As here too -- is equal to O for =O we find as above that the
¢
radius of curvature 7, in point ? of the curve ¢/ is:
Gh
— a
ea 7]

19%

( 282 )

_ the value :

This formula gives for 7,

aes
a"
ie 2a,"
= ar ile
: i a? 2a,”
From the obtained values &, = — andr, =—— we get
2b, 3b,
RK, 27, 3-4
Delft, Sept. 1904.
Physiology. — “Degenerations in the central nervous system after

removal of the flocculus cerebelli”. By Dr. L. J. J. Muskxns.

(Communicated by Prof. C. Winker).

(Communicated in the meeting of September 24, 1904).

In 6 rabbits the flocculus of the right side was extirpated. This
organ lies, as is well known, in these animals in a separate bony
hole, so that we here have the possibility to remove a part of the
cerebellum without disturbing the nervous structures of the neigh-
bourhood in their conditions of nutrition as well as of pressure. The
animals were killed after 8 days to 5 weeks and complete series
stained after Marchi, were prepared.

The degenerations of fibres after this lesion in 4 of the 6 cases
were found exclusively directed upward i.e. to the superior crus-
cerebelli and to the pons.

In one case there was a fine degeneration all over the restiform
body; in this case however it could not be made out with certainty
whether we had to deal with really descending degeneration,
because firstly all through the cord fine, black spots were found,
and secondly the black spots were of so little dimensions, that there
is much doubt about the genuineness of such a fine degeneration.
In this animal the staining was insufficient, irregular and not limited
to degenerated nerve-fibres, for an unknown reason, so that we
do not think much value can be attached to this single case, in
whieh downward degeneration was found.

In another wellstained case in the restiform body a number of
degenerate fibres on the operated side was found; also in the longi-
tudinal posterior fascicle and in the field of the tecto-spinal bundle,

equally on the operated side. In the superior cervical region also a
small field with the base lying towards the margin. the point
towards the restiform body was found full of degenerate fibres.
Lower down than the upper cervical segments, these degenerate
fibres do not reach. In this case not only the flocculus and the
floceular peduncle, but also the vestibulary nucleus was severed,
so that also this experiment cannot be recognized as a clear
experiment.

Although allowance must be made for an eventual different result
after extirpation of other parts of the cerebellum of the rabbit, so

we think, that these experiments show clearly the absence of

descending degeneration after a sharply localised lesion of the floc-
cular cortex. The discussion in the literature between Marcu, Frr-
rier and Turner, Tuomas, Biepn and Risten ReussenL regards the
question, how much of the found degenerations must be ascribed to
lesion of the neighbourhood, because, as THoMmAs justly remarks, exactly
in this region of the cerebro-spinal axis it is characteristic, that also
without direct lesion by the severing instrument yet by haemorrhage
or an alteration of pressure, extensive degenerations can be caused.
As in these experiments certainly no such lesion of the neighbourhood
can have arrived and in the completely successful cases the cord
was found free of degeneration, we may be sure, that from the
eanelioncells o: this part axis-cylinders with centrifugal course to the
medulla are not found, so that for this part of the cerebellum at least,
the original data of Marci are not confirmed. Thus these observations
as also those of Propst can be regarded to agree with the Enelish
observers, after whom only after lesion of the nucleus-Derrmrs des-
cending degeneration of the anterior and lateral tracts is found. In
judging this result it is important to observe, that also in’ another
point than by its own bony capsule the rabbit must be regarded as
an abnormal form.

The flocculus of the rabbit contains viz. except its part of the
cerebellar cortical gray matter and its af- and ef-ferent fibres also a
nucleus of large multipolar ganglioncells, such as are found in the
nucleus dentatus. The study of the development of kindred animals
(squirrel) leave not the least doubt, that indeed a part of the dentate
nucleus is dislocated in the floceulus. It appears that it is not
always in connection with the principal nucleus.

Now | do not think that for the elucidation of the question, whether
there exist descending cerebellar tracts, this circumstance must be
regarded an indesirable complication, but rather we may reckon this
a useful detail, in so far as it allows to exclude at the same time,

( 284)

that such efferent fibres descending in the cord, should spring from
(this part at least of) the dentate nucleus.

Regarding the ascending degeneration in the different operated
animals the most complete accordance is found. Two bundles are found
in all successful cases, very clearly and in exactly the same place of
the cross-sections and both find in the same region of the cerebrum
their end, viz. in the regio subthalamica. In the first place the supe-
rior cerebellar peduncle being the most voluminous bundle, where
we find fibres of heavy caliber. This degeneration shows especially
gross fibres, compared with the fine degenerations, found elsewhere
in the rabbit. The degeneration is found especially in’ the middle
third part of the superior cerebellar peduncle, whereas the medial
and lateral thirds are nearly entirely free from degenerate fibres.
Arrived about at the posterior quadrigeminal body, the degenerate
fibres curve downward in a nearly rightangle, as this is repre-
sented by the authors, building the wellknown peduncular decussa-
tion. Only a few sections separate the commencement and_ the
finish of the decussation in the sections. In the substantia reticularis
the direction is again purely longitudinal to the long axis of the
cerebral stem, where as in the region of the red nucleus it becomes
clear, that especially the ventral part of the red nucleus comes in
contact with the crossed peduncle. This crossed connection is, as far
as the flocculus is concerned complete. Here it may be recalled, that
Probst has shown, that after extirpation of more dorsally situated
cerebellar parts of the cat also non-crossed fibres run to the subtha-
lamic region.

Besides this most important upward degenerating bundle, there
is another tract up to now only described as far as I am aware by
Prosst, which is constituted of finer fibres than the first bundle, takes
its course by the substantia reticularis, of the contra-lateral side,
and joins the first tract about its arrival in the red nucleus. Both
together run frontalwards, and end in the ventral part of the
nucleus ventralis thalami. The sections leave no doubt, that no fibres
from the floceulus arrive in the thalamic region uncrossed, but all
decussate either in the decussation of the superior peduncle or as far
as the second bundle is concerned in the pontine region, right near
its emergence from the flocculus. Also ‘THomas has designed this
degeneration, but he thinks, that here we have to deal with descending
collaterals of the frontal cerebellar peduncle, which leave the principal
bundle after the decussation of this peduncle. Propsr on the other
hand thinks, that these fibres arise from the dentate nucleus, pass
directly through the region of the vestibular nuclei, to the substantia

( 285 )

reticularis of the crossed side and ascending frontalwards are found
in the same region up to their junction with the superior crus
cerebelli.

My own sections suggest very strongly indeed, that these centri-
fugal (from the cerebellum, or rather from the nucleus dentatus)
fibres, take their course by the superficial layers of the middle cere-
bellar peduncle and then can be followed right through the pyrami-
dal bundles or partly winding around them to the reticular substance.

In different series if becomes clear that proceeding in the series of
sections from below upwards there, where are found the first dege-
nerate fibres in the reticular substance, also the first degenerate fibres
appear in the middle peduncle. While by Thomas no sound reasons
are given for his conception about the significance of this bundle,
it pleads against the opinion of Prost that in the region of the
vesubulary nuclei, no degenerate fibres are found.

Finely the sections show, compared with the sections gained by
other experiments, that the ventral thalamic bundle originates for
the greater part from the ventral portions of the cerebellum, especi-
ally of the flocculus. Sections of cats-brain after similar operations
leave no doubt, that after lesion of more dorsal cerebellar portions,
there exists a very marked contrast between the very pronounced
degeneration of the crus cerebelli ad corpora quadrigemina and _ the
very slight degeneration of the ventral thalamic bundle, whereas as
well in the cat as in the rabbit after exclusive lesion of the flocculus,
both bundles are affected about equally.

Physics. — “The validity of the law of corresponding states for
mivtures of methyl chloride and carbon diovide,’ by Prof. H.
KAMERLINGH OnNes and Dr. C. Zakrzewskt. Communication
N’. 92 from the Physical Laboratory at Leiden by Prof. Dr.

H. Kamerninch Oxnes (continued).
(Communicated in the meeting of June 25, 1904).

§ 1. Lntroduction. In n°. IX of the “Contributions to the knowledge
of VAN per WAALS y-surface’” we have expressed the hope of giving
an experimental contribution to the investigation of the co-existing
mixtures of methyl chloride and carbon dioxide at low temperatures
in connection with the test of the law of corresponding states for
mixtures, which for many years has formed a subject of experimen-
tation at Leiden. Of the extensive territory of reduced states, which

( 286 )

ihe mixtures of carbon dioxide and methyl chloride afford for mea-
surements on either side of the critical state, (reason why in about
1890 it was chosen for the first investigations of the w-surface) a
considerable portion round the critical state has been immediately
investigated by Kurnex (Comm, N°. 4, April 92). Harpmay in Comm.
N°. 48, June 95 has added to this area that of the coexisting phases
at 9.5 C. We have extended the area investigated in two directions,
albeit only by a few preliminary researches.

The results of some of those measurements, though a few are
only preliminary and served chiefly as a means for us to decide
upor the method of investigation, seem important enough now that
still so littke is known about the different degrees of approximation
to which the law of corresponding states holds for mixtures in diffe-

v dhe
rent fields of reduced state | » = —, t= —].
Ve diy

Our measurements refer in the first place to gaseous mixtures
under almost normal conditions, ‘in the second place to coexisting
phases at low temperatures.

For the normal gaseous phase we found the law of corresponding
states to be confirmed to a high degree of approximation. The virial
coefficient 2, which determines the deviation of mixtures of methyl
chloride and carbon dioxide from Boyur’s law at small densities, can
be sufficiently derived by means of the law of corresponding states.

Greater deviations were found when we investigated the coexisting
phases at low temperature. Here we have determined by means of the
dew-point apparatus, deseribed in the first part of this communication,
the begin condensation pressure of the mixture v= ',, at — 25° C.: the
iemperature for which we have constructed the y-surface in Suppl.
N°. 8, Sept. “04. The deviations found are rather great, they point
to an increase of the deviations from the law of corresponding states
in the mixtures at low temperatures in the liquid state. The deter-
mination of the end condensation pressure for the same mixture «= */,
at — 25°C. with the piezometer of the first part of this communication
would involve complications (comp. ibid. § 5). In order to obtain an
idea of the deviations of the liquid branch of the binodal curve at «= */,
from that according to the law of corresponding states, we have
investigated the condensation pressure for a= '/, at a lower tem-
: 38°.5 C. This corroborated the result of the investi-

perature, viz.
gation of the vapour phase at — 25° C,

I. The compressibility in the neighbourhood of the normal state.

§ 2. Determination of the second virial coefyjicient. The mixtures

( 987)

were prepared and the compressibility determined in’ the mixing
apparatus and volumenometer deseribed in’ Comm. N°. 84, March
03. The method of observation and calculation has been treated in
detail by Kursom, Comm. N°. 88, Jan. "04.

The gases were prepared by distillation first in ice, subsequently
in solid carbon dioxide. From previous Communications it will appear
that in this way pure carbon dioxide is obtained. Of methyl chloride
the same will be proved in the continuation of this paper (§ 8).

The values found at the temperature ¢ of the pressure p, volume
Vand molecular composition of methyl chloride w are given in

table I.

TABLE I. Compressibility of mixtures of |
carbon dioxide and methyl chloride.
v=1 (CH, Cl)
|
NO. pin mm Vin cc, | t
|
Als 1137.33 537 67 | 90.05
IL. 593.69 | 41043 51 | 20.07
II. 479.23 4996.33 | 20.07
|
a = 0.6945
| = 4900%32 | 537.49 20.09
Il, 624.45 | 1043.50 20 10
IIL. 503.49 | 1297.01 20.08
v — 0.5030
in 1173.08 | Soe eto
|
Il. GO8.87 1043.66 | 19.87
Ill. eZA0ESS 4296.30 | 19.87

The values for «=O may be borrowed from Comm. N°. 838.
Por the calculation of these observations we shall use the empirical
reduced equation of state of Comm. N°. 71, June °01, which is
particularly suited for the investigation of the degree of validity

( 288 )

of the law of corresponding states, in the form as laid down in
§ 4, which deviates little from that of Suppl. N°. 8, Sept. *04.

In the first place the observed pressures must be reduced to the
same temperature 20°C. For this purpose we have calculated the real
coeflicients of pressure-variation of carbon dioxide (0.003460) and
methyl chloride (0.003586) with the equation of state mentioned and
the coefficient of pressure-variation given below for ideal gases, and
we have taken linearly interpolated values for the mixtures.

Owing to the small differences in temperature the errors ensuing
from this remain below those of the observation,

Let ¢ be the volume expressed in terms of the theoretical normal
volume (introduced in Comm. N°. 47 Febr. °99), then we have
approximately

B
pe —A+—,where A=1+ a, t
zs

and «, the coeflicient of pressure-variation of an ideal gas. One
of the advantages of the empirical reduced equation of state is, that
it teaches us the degree of approximation to which the higher terms in
1

v

may be omitted. Then we have for the calculation of the second

virial coefficient as a first approximation, if B® also is neglected (for
further approximations see § 5):
Pr" 1 B
SE

Pols x A?

Yo So a Glo o om > (1)

and with @, = 0,00366195 (instead of 0,0086625 of Comm. N°. 71)
we derive from table I:

TABLE Il- Second virial coefficient for mixtures of carbon dioxide
— (v =0) and methyl chloride (2 = 1) to the first approximation.

| composi- B B ees
‘ ~» from I and I > from IT and III, > mean B

tion A A= | A*
eae
je=1 — 0.01797 — 0.01800 | — 0.01798 —0.02071
0.6945 — 0.01302 — 0 013.9 — 0.01310 / —0.01509
0 5030 — 0.01034 | — 0.01005 — 0.0i019 —0.01175
0 | Kerresom, Comm. N® 8&8. ; —0. 00654

( 289 )

§ 3. The virial coe ficient B asa quadratic function of the molecular
composition a.

According to vAN per WaAAts’ equation of state B= RT yyy — aypy,
if @ppr and bypy represent VAN purR WAALS’ constants with regard
to the theoretical normal volume. Hence, to the first approximation,
we must have for the mixture with the Composition «

(Cl Me. CO.) (CiMe} (Cl Me. UO, (COx)
)

abs — Bvt + 7) ao b «(4 —- a) -f- B { {= 2)?

By means of least squares we found"):

(Cl Me)
Booo = — 0.020772
(Cl Me. COs)
(12) Baoo —— O.OLO0067
(CO.)
Boov = — 0.006515.
The agreement appears from the following table:
ie B observed 6B computed = Obs.—Comp.
i — 0.02071 — ().02077 —- 0.00006
0.6945 — 0.01509 — 0.01490 — 0.00019
0.5030 — 0.01175 — O0.OLI9O0 + 0.00016
0 — 0.00654 — 0.00652 — 0.00002

The deviations are less than 2°/,, hence also less than the devia-

)

2
tions of the single values of —, inter se. Thus the agreement with

the quadratic form was sufficiently proved, so that for the time
being measurements with other mixtures, not exceeding this accuracy,
could be left off.

§ 4. Validity of the law of corresponding states for the virial
coesyicient B. According to the law of corresponding states the virial
coeflicients are derived from the coefficients of the reduced equation
of state through multiplication by functions of 7) and p, (comp.
Comm. N°. 71 and also Suppl. N°’. 8, Sept. ’04, the first four
sections).

As the critical data of mixtures of carbon dioxide and methy1

chloride have been derived in Comm. N°.59/ from Kurnen’s experiments,
‘we may determine / for a given temperature, for instance Bay by
Ty
ete
Pk

1) The coefficients given here have been derived from values for B which do
nol differ essentially from those given in table I.

( 290 )

where Br is the value of the function B of the reduced temperature
293.04

fly}

For 8 we have used a function of a form differing slightly from
the form VI.1, given in Suppl. N°.8, which did not only agree with

belonging to ¢=

hydrogen, oxygen and nitrogen but also with ether, viz. a form VI. 2,
which instead of agreeing with ether in the same way as VI. 1,
agrees with the average of ether and isopentane :

9 Pe a aa 1
10°. 3 = -+ 179,883 t — 374,487 — 181,324— — 110,267
t 13
The agreement appears from the following table, where we find
in the first column the values calculated according to the last formula,

and in the second column those of the quadratic formula of § 3;

according to according to difference
corresponding states quadratic formula
L — 0.021920 — ().020772 — 0.001148
de — 0.016502 — 0.015866 — 0.000636
2/: — 0.012179 —= 0): O10S5p — 0.000324
0 — 0.006485 — 0.006515 + 0.000080

The deviations on the side of the methyl chloride are larger than
those of the errors of observation and those of the quadratic formula.
Methyl chloride, therefore, does not agree so well with ether and
isopentane as carbon dioxide. This same result is also arrived at in
another way. It appears, however, that the mixtures do not deviate
more than the methyl chloride itself.

(To be continued).

Physics. “On the measurement of very low temperatures. VII.
Comparison of the platinum thermometer with the hydrogen
thermometer”. (Continuation of Comm. N°. 77. Febr. 1902). By
B. Memixk. Communication N°. 93 from the Physical Labo-

ratory at Leiden by Prof. H. Kamernincn Onnus.

(Communicated in the meeting of June 25, 1904).

§ 5. The measurements at low temperatures. The thermometers
were mounted as described by KAMERLINGH OnNEs in Comm. N°. 83
Febr. 1908 § 5. During the first preliminary measurements, the hydrogen
thermometer and the resistance thermometer (cf Comm. N°. 77 § 2)
were paced in the cryostat (described in Comm. N°. 51, Sept. 1899),

ee i de

i el a i i hn es, 4

(29)

which was modified as described in § 2. Comm. N°. 83. In this
cryostat a vacuum vessel was placed inside of 4, (Pl. 1, Comm.
N°. 51), which vacuum vessel by means of cork was pressed against
the walls of 6, (ef. end of § 2 and also Pls. I and IL of Comm.
N°. 83). The inner wall took the place of §, in PI. IH, Comm. N°. 83
(the same parts of Comms. N°*. 83 and N". 77 are marked with the
same letters).

These preliminary experiments had shown, that, after repeated mea-
surements at the lowest temperatures, the original value was again
found for the resistance at 0° C., hence that the platinum wire, though
its expansion differed from that of glass, was not lengthened and that
it also remained properly in the notches. Further it had become clear
that an accurate Comparison of the two thermometers was only pos-
sible when the temperature of the bath was kept constant with the
utmost care, and there we met with the difficulties treated in § 2 of
Comm. N°. 83. It was attained by arranging the cryostat as described
in § 5 of Comm. N°. 83. It may moreover be remarked that
the liquid gas was always kept higher than §,’ (Comm. N°. 83,
Pl. II); else, notwithstanding the level in and outside the protecting
cylinder would go up and down through the motion of the stirrer,
no circulation would be produced.

The course of a measurement was as follows. As soon as the cir-
cumstances under which we desired to make a measurement were
established, the resistances of the leads were determined, then the
resistance of the platinum wire was adjusted and, by giving signs
to the assistant charged with the regulation of the pressure, care was
taken that this resistance, and hence the same temperature, were
maintained. After about ten minutes we began, while constantly
reading the galvanometer, the measurements with the hydrogen ther-
mometer and continued them until the liquid was evaporated or
until we deemed that sufficient data were obtained. At the end the
measurement of the resistance of the leads was repeated.

The observer at the galvanometer had, therefore, only to look
after the continual closing and breaking of the currents and the
noting down of the values of the galvanometer readings and of the
time belonging to them. Afterwards the deflections were derived from
this (see Pl. IIT Comm. N°. 83) and the mean deflection during the
time of observation was found by means of a planimeter.

§ 6. Zero after the measurements. By a too rapid decantation of
liquid oxygen, numerous bursts had unfortunately come in the eylinder
of the resistance. To repeat with it-the above described operations

for the determination of the zero seemed rather dangerous, especially

( 292 )

as the refastening in the cryostat would have involved many difficult
operations.

Therefore in order to brine the resistance thermometer to a constant
temperature near 0° C., the case U of the cryostat (Pl. 1, Comm.
N°. 83) was screwed off from the cover .V, (Pl. 1, Comm. N°. 51),
while the other parts of the cryostat remained fastened to the cover,
and it was replaced by a zine cylindrical vessel, which could be managed
more easily. This vessel was provided with a rim fitting on to V,
and was placed in another larger zine vessel, so that a jacketing
space of 5 em. remained which was entirely filled with ice. Then
isopentane was distilled into ,, (Pl. UH, Comm. N°. 88) and the
apparatus was left to itself during one night. The next day the
temperature (near O°) had become constant and we determined if
(while stirring) by means of a thermoelement (@ Pl. II, Comm. N°. 88).

§ 7. Corrections. A survey of the mounting of the WHratstonr’s
bridge (cf. § 3 Comm. N°. 77) is given in fig. 5. A indicates the

he ee
Tig. 5.

resistance to be measured, A, and F, the two coils of manganin
wire, /,' and &," the resistance boxes of Hartmann and Braun and
R,?
Rae
which with 7,, forms the fourth arm of the bridge; C, and C, are
the commutators with mercury contact (Comm. N°. 27, May and June
1896), C, is the copper commutator treated in § 3 Comm. N°. 77 and

of Stzmens and Haske giving together the resistance R,= PR,’ -

represented there in fig. 2.
Putting for the factor for the inequality of the branches of the

fo]
R :
bridge —* = 1 — a, we found @ = 0.00216 (as mean value of twenty
v
2

( 293

values ranging from 0.00219 to 0.00214). As according to § 3
Comm. N°. 77 the resistance of the platinum wire FP is equal to
the difference between two measured resistances, one / (the resi-
staice of the leads), another /? + ¢, and as the branches of the bridge
are so nearly equal, 7,, is eliminated, and hence we need not know

the value of 7,,.

23

To the resistances read on the box /?,', the corrections found by)
calibration must be applied. We may easily convince ourselves that
the arrangement of box f," parallel to box /,' has no perceptible

influence on the value of the corrections at /,'. The corrections of

the errors in the nominal box values could be neglected for all
coils below 1 Ohm.

For the measurements at low temperature a correction had to be
applied, because during the regulation of the temperature the mean
deflection was not zero.

In order to express that deflection in terms of the resistance, the
platinum thermometer in the bridge, after the measurements were
made, was replaced by an equally large box resistance and for a
known modification of this resistance the deflection was observed. The
regulation of the temperature was in most cases so successful that
it was hardly necessary to take the correction into consideration.

The resistance, measured near 0°, was reduced to O° C. with the
approximate formula HW” = 110.041 (1 + 0.0088644 ¢— 0.00000 L081 /),
derived from preliminary observations.

§ 8. Survey of a measurement. The course of a measurement is
described in § 5.

The quantities which are derived directly from observation are given
in table I (p. 294). Under the head “Connection” I have recorded between
which blocks of the commutator (, a conducting connection existed.

Therefore commutator C, was not used while the measurement
lasted. This had become possible because the platinum wire was
wound free from induction, so that no induced current was observed
when the principal current was closed.

From these data we now derive for each connection the value of

R,, i.e. the resistance of the branch of the bridge in which the
resistance boxes are placed (apart from 7,,).

If the value of R, which is found when we measure the platinum
wire with the leads, is diminished by the value of 2, which is found
with the leads alone, we obtain the resistance of the platinum wire,
in the supposition 1°. that. the arms of the bridge are equal, 2°. that
during the measurement the mean deflection was zero and 3°. that the
resistance box requires no correction. For each of these suppositions

—=—_ - To ee

EE a A Et ee Re

lati mtn

TABLE 1. Calibration Platinum Thermometer in Oxygen
Boiling under Reduced Pressure (May 22, 1902).

Resistance Measurements.

( 294 )

2) we!

| Position Position of Deflection

Time. Connection. | Rk’, | R’, | commutator | equilibrium of
| | C, galvanometer. | galvanometer.

ooo
| 1-5 2-6 | 0.4 | 1.5 | 38.5

Leads. Left. 74.5
| | Right. 41.7
14.6 | Left. | 43.4
|
58.2
4392-4] 0414.6} Left. 43.8
| 58.0
41—5 2—4 |20-+-2) 3200 |
platinum
wire. + 0.3
| 45.2

3h.49" Left. 44.0

45.0

| 46.0

AG.7

47.0
45.0
50! 44.2
45.4
to Right.

4,95! and so on
| for all
minutes to
| 4b 25!

p | = oo

4n.27' — Leads.
| 4-3: 9— 4. | 034 | 455 Left. 46.0

55.7
| 1-5 9-6 | 04 | 4:5 Left. 53.8
46.0

———————— esti“

EEE ———

( 295 )

we must apply the correction mentioned in § 7 in order to find the
true resistance. The mean deflection during the measurement is found,
according to Comm. N°. 83 § 5, by means of a planimeter.

(See for the graphical representation Pl. Ill, Comm. N°. 83, which
does not, however, bear upon our case).

In the following table the corrections are combined.

TABLE Il. Calibration Platinum Thermometer in
Oxygen Boiling under Reduced Pressure
(May 22, 1902).

Resistance measurement, Corrections.

RAtONALMS ye sie tee 0.00216
| Correction to R for box values 20 and 2 -- 0.0005
| Mean deflection . heb 0, og AUR LCA I |

After what has been said above about the method of calculation the
further calculation will be sufficiently clear from the following table.

TABLE IIL. Calibration Platinum Thermometer in Oxygen
Boiling under Reduced Pressure.

Resistance at — 197°.08 C,

roe x Lite 7; a9 ny : Resistance
Time | Connection Rn, Re R"'", mean R, Pi wire
1—5 2-6 0.4 | 1.54
Ab) 0.3182
1—3 2—4]| 0.4 | 1.57
3h,.49— | 1-5 2-4 |20-+-2) 3200 22.1457
Ah 25 40.3
1—3 2—4]| 0.4 | 1.53
‘ 1.53 0.3171
1—5 2-6 0.4 | 1.53 | 21. 8281

Correction arms of the bridge. . . . . . . . --0.0485
Correction resistance box. . . . . .. .. . +0.0005
Correction to mean deflection 0 . . . . . . . +0.0002

RESISHANCEIs IGS are ie) vee ttt OI 8773

20

Proceedings Royal Acad. Amsterdam. Vol. VIL.

.

( 296 )

Determinations made at other temperatures did not vield anything
particular. Only for the zero determination the corrections are some-
what different, as that for the reduction to the mean deflection O is
no longer necessary. A new one, however, is added because the determi-
nation has not been made exactly at O° C. but at a little higher
temperature. After what has been remarked about this in § 7, it seems
supertluous to illustrate this small variation by an instance.

§ 9. Determinations of the resistance at O° C. They are made in three
series. For the first we still used leads of 0.5 m.m. (Comm. No. 77
§ 2 and fig. 3), the insulating liquid was petroleum ether or amylene ;
for the second the leads were 5 m.m. thick (1c. fig. 4), the insulating
liquid was isopentane; and the third (insulating liquid isopentane)
was that treated in § 6.

~ —_—— ~~ -— _ — )

| TABLE IV. Calibration of Platmum Thermometer.
| Zero.
| Number of Tape eee
sa al Sm Gir ae Mean value.
Series 1. June ’04 | 4 | 110.031 140.048 110.040
2. Nov.Dec.’01 7 033 57 43 |
3. Nov. 702. 3 043 Dl 48 |
| |
Mean resistance at 0° C. | 110.045 |

§ 10. Determinations at low temperatures. The measurements were
made at fairly gradually decreasing temperatures; at the lowest
temperatures the intervals are smaller.

The measurements with the hydrogen thermometer (see Comm.
N°. 77 § 2) are made by Dr. W. Heuser to whom my best thanks
are due for the trouble he has taken.

The determinations are made up of two series.

The first series was made between May 13 and July 10, the
second series between Dec. 10 and Dec. 22, 1902.

It seems desirable to consider the two series separately.

The first series has yielded results that may be derived from the
following table.

In order to judge how the values given here agree inter se, I
have first calculated the formula of the form

ww, (1+ at + br),

( 297 )

TABLE V. Calibration of Platinum Thermometer.

First Series.

ie |
Temperature Resistances Bath in which the
Date. | determined with the |

hydrogen thermometer., measured. ‘measurements were made.

reduced pressure.

0° ¢. 110.045 comp. § 9.
May 24, ’02 — 51°.43C. 87.760 methyl chloride boiling
under reduced pressure.
May 13, ’02 — 104° .66 64.956 | ethylene, |
— 1049.38 64.3714 id. |
2 SL ate earn’ ethylene boiling under
F __ 498°.88 53.379 reduced pressure.
May 16, 702 — 1619.15 38.676 methane,
A — 1619.15 38.672 id.
p — 164°.47 38.515 id.
May 22, 02 — 182°.63 98.692 | oxygen. |
| July 10, ’02 (— 195°.75)1) | (22.600) | nitrogen. |
May 22, 02 — 497°.08 | 94.877 | One oistoiiigr wider |
| f — 197° 58 21.673 | reduced pressure. |
July 10, ’02 — 209°.93 16,025 | nitrogen boiling under |
|

which agrees with the observations at 0°, at —104°.66 C. and at
—182°.63 C., the temperatures which best correspond with those which
as a rule are also used by other observers.

The formula becomes

w = 110.045 (1 + 0.0038788 ¢ — 0.000 000 9257 7).

The deviations of the observed resistances from the formula are
given in the column Obs.—Comp.7 of table VI, and are quite
appreciable. In the case of methyl chloride the deviation amounts to
65 on 87760 or a difference in temperature of 0°.15 C. For methane

these deviations are 63 on 38674 or a difference in temperature of

also 0°.15 C. In oxygen, boiling under reduced pressure, the deviation
is 90 on 21637 or about 0°.2 C.

1) This observation is less reliable because an uncertain correction to the
hydrogen thermometer attained a rather high value.
20*

———

( 298 )

To find out whether these deviations are perhaps due to irregular
errors in the measurement, it will be useful to investigate whether,
by addition of another term, the differences between these obser-
vations and the calculation might be reduced to within the limits
of the errors of observation. It succeeded indeed fairly well as may
be seen in column Obs.—Comp.z, of table VI. The calculated values
are derived by means of the formula

w = 110.045 (1 + 0.0039167 ¢ — 0.000 000 8432 & +
+ 0,000 000 002069 #).

| TABLE VI. Test of a Parabolical and of a
Third Degree Formula.

Virst series.

| Temperatures deter- | Measured |

mined with the hydro- ae : ded ag Sr ph Ba 2
gen thermometer. resistances |

0° 110.045 0 0

51°. 43 87.760 — 0.065 4- 0.012

| 104°.38 64.371 — 0.01 — 0.017
1049.66 64.256 — | 0 — 0.005
197°. 74 53.910 + 0.039 —

1289.88 53.372 + 0.030 — 0.044

| 1619.45 38.674 | + 0.063 + 0.019

| 161°.47 38.515 | + 0.054 + 0.008

182°.63 98.692 | 0 + 0.008

| 195°.75) (22.600) | (+ 0.014) | (++ 0.078)
1979.08 94.877 — 0.083: | Lso0bie" |
197°.58 21.637 — 0.090 —'0.017

| 209° 93 46.085 | +0.077 | +4 0.999

The deviations from formula II, with the exception of the last,
although they are not entirely within the limits of the errors of
observation which were expected, ave only little in excess.

In the case of methane, where the deviation is 19 on 88674, an
error in the temperature of O°.04 C, is sufficient to explain this
amount,

( 299 )

In the case of nitrogen boiling under reduced pressure, however,
the deviation has become very large, so large even that if cannot be
explained by errors of observation. Hence the circumstance that the
formula is not fit to represent the resistance so near to the absolute
zero must account for this deviation. All the same it is remarkable
that this turn appears so suddenly. At 197° ©. the formula. still
holds, at —210° C. there is a deviation of 229 on 16025, i.e. a
deviation of O°.49 in temperature. But if we take into consideration
that, according to the formula, the resistance at 243° ©. would
become zero and that we are only about thirty degrees from this
point, we need not wonder at this result.

In order to gain certainty that there was indeed a fairly rapidly
increasing variation in the shape of the curve that represents the
resistance as a function of the temperature, [ resolved to repeat
especially these measurements at very low temperature in nitrogen.
These constitute what | have called at the beginning the 284 series.
Unfortunately the result was unsatisfactory.

Though the observations indeed point in the same sense, yet one
error or another seems to have crept into them and it could no
more be detected at the time when the calculations revealed it. We
shall omit them here.

Therefore the results as to the amount of the deviations remain
more or less uncertain; yet it is very probable that even in nitrogen
boiling under reduced pressure, a beginning may be observed of the
variation in the course of the temperature function which, as follows
from Derwar’s experiments, appears so strongly at the temperature
of liquid hydrogen.

The conclusions to which the measurements lead may be summarized
as follows.

A representation of the resistance by a quadratic formula, according
to the temperature, even if we do not go below — 180° C., is only
permitted when no higher degree of accuracy than 0°.2 C. is aimed
at. When a greater precision is desired we require for the calibration
of a platinum thermometer a greater number of points of comparison.

: A Tae |
For a comparison to within 20 °C. a number of at least 6 tempera-

‘tures of comparison is considered very desirable.

Below —197°C, the deviations of the platinum thermometer
become so large that before using it for this range an investigation
must be made of the course of the resistance as a function of the
temperature.

( 300 )

Physics. — “On the measurement of very low temperatures. VILL.
Comparison of the resistance of gold wire with that of
platinum wire.’ By B. Memink. (Communication N°. 93
(continued) from the Physical Laboratory at Leiden by Prof.

H. KAMerRLINGH ONNEs).

(Communicated in the meeting of June 25, 1904.)

§ 1. The investigation described in this paper forms part of the
subject mentioned sub 2 in § 1 of Comm. N*. 77, Febr. 1902 and
had for its chief object the establishment of the method of observation.

The gold wire was made of the material kindly given us by
Dr. C. Horrsema, inspector and assay-master general of the Mint,
according to whom no impurities could be detected in it by means
of chemical processes, which, with regard to the accuracy of the gold

1 o/
‘goo /o°

»» Mm. in diameter. The great length of
the wire used, however, had rendered a soldered joint in the

analysis, excludes an impurity of more than The piece was

drawn out to a wire of */

middle necessary.

§ 2. Arrangement of the wires. The same advantages which made
us prefer a naked platinum wire to one enclosed in a glass tube
(cf. Comm. N°. 77) exist also when the wire is made of an
other metal, though the difficulties, especially with regard to the
action on the metal, are greater.

The difficulties of the arrangement increased, however, considerably
with metals of such high conductivity as gold, because then the
wire must be so much longer in order to produce a sufficient resis-
tance. With the first forms that were tried, the metal wire lay in
a serew-thread etched on a glass cylinder. But with the longer
wire the latitude for the expansion became so great that it could
slip too easily from the screw grooves and thus cause short-cireuiting,
In order to obtain deeper grooves the glass cylinder was coated
with a paste of oil and carborundum and slowly spirally moved
with a speed of ‘/, or */, mm. The cylinder grinds against an iron
or copper disk, which is kept in rapid rotation and thus by means
of the carborundum a groove is ground in the glass. This groove
proved to be much deeper than that formed by etching’); the wires
of 0.1 mm. in diameter were entirely enclosed in it. On a cylinder
of 37 mm. diameter and 55 mm. height we could wind more than
12 meters of wire.

1) Later we have again succeeded in making still deeper grooves by etching.

a

( 301 )

} ‘ ‘ ‘ 4 ‘

rhe investigated wire covered two of these cylinders c, and ¢,
(see fig. 1, Pl. 1), the one fitting into the other and leaving a jacket
of about 2 mm. for the cirenlation of the liquids. A’ third cylinder

Cs

round these two served to protect the wire.

The cylinders rest on a copper star with 38 teeth d@ in which
concentrical grooves are made to hold the glass cylinders. At the
other end each of the cylinders, by means of copper ridges ¢,, ¢,
and e,, lying on the glass rim, is pressed against the star by means
of a tightening rod f and nuts y,, g, and y,, thus forming one
tightly connected whole. As the ridges ¢, and ¢, and also the lower
star ¢ served at the same time as connective places for the wire,
they were insulated from the tightening rod and the nuts by e@lass
cylinders 4, and 4, and plates of mica 7, ¢, and /,.

The winding and the mounting was done in the following way.
We began by fixing the inner cylinder c, between the star and the
ridge ¢,, the wire was soldered on to the ridge ¢, and led downwards
along the groove of the screw-thread on the cylinder. At the bottom
it was soldered on to the star. Then the second cylinder was placed
round it, the wire was turned upwards along it and fastened to
¢,. If the two serews in the glass are wound in the same sense, the
wire is almost free from induction. (Cf. the platinum thermometer
of the previous paper where this was attained in a different manner).

As in the case of the platinum wire which was treated in the
previous paper, there were 4 leads. For the method with the
differential galvanometer (cf. § 4) it does not matter that they have
little resistance, hence wires of 1 mm. were taken, flattened over the
last 5 cms. The entire apparatus was suspended by a copper tube 4,
which was screwed on to the tightening rod 7, and which, in order
io prevent too much conduction of heat along it, had a piece of
ebonite inserted in it (not shown in the drawing).

§ 3. Determination of the zero. The zero was determined in the
same way as described in Comm. N°. 77, when thin leads were
used. Besides with the gold wire, determinations were also made
with a copper and a silver wire. A single determination never
offered any difficulties. With copper, however, the values determined
at different times did not agree. They showed a regular increase of
- the resistance at 0° C. This must probably be ascribed to a chemical
process. Copper oxydises so easily that the greatest precautions must
be taken to avoid moisture during the storage. If in distilling the
insulating liquid into the zero-vessel the vapour was passed over
phosphorous pentoxide and if care was taken that while the wire
was kept, the air could only enter over phosporous pentoxide, we

( 302 )

succeeded in stopping that process. Yet at any rate this experience
obliged us to perform measurements at low temperatures, like those
treated here for the gold wire, in a very short time in the case of
copper wire. With gold and silver this phenomenon did not appear.

§ 4. Comparison of the resistances. In order to investigate the
variation of the resistance of the wire, wound as described in § 2,
the ratio of this resistance to that of platinum had to be deter-
mined in different baths. To this end previous investigators have
always measured the resistance of the two wires alternately and
hence derived the mean ratio. The first experiments made by me
were also arranged in that way.

In order to attain a higher degree of accuracy [ have followed
the advice of Prof. KAMERLINGH ONNEs and arranged the measurements
so that at a definite moment the ratio itself can be read.

If we use the Wuerarstronr’s bridge, it seems that this may be
attained by arranging the wire to be compared in parallel to the
box of Hartmann and Bravn (/?,) instead of arranging it in parallel
to the platinum resistance.

An insuperable obstacle for this simultaneous determination is, that
at any rate a connection is required between the two wires, which
connection may be made by two of the leads from the bath, in which
we measure, to one of the angles of the bridge. Elimination of all the
resistances of the connections except that of the stops used in the
measurement, as it was obtained in the previous paper, is impossible.
Errors may then creep in of which the amount may only be estimated.
Resistances of the connections may occur to a considerable amount
and be brought about by minor causes. If we cannot constantly test
their amount the results remain uncertain.

Moreover in the Wueatstonr’s bridge the unavoidable resistances
of the connections are so large that we cannot reach the aceuracy
proposed in the outset with wires of so little specific resistance as
gold. For in that case, even though we succeeded in winding a wire
of more than 20 meters in the necessarily. small compass of the bath,
a resistance of no more than 35 Ohms at 0? C. was obtained, which
in liquid oxygen or nitrogen fell to below about 6 Ohms.

All these reasons led us to choose the method of the differential
galvanometer for the comparison of the metals inter se. It enables
us to determine at once the ratio of two resistances, while resi-
stances of the connections have no appreciable influence. It is true
that a measurement cannot be made in such a short time as
with the Wuearstonr’s bridge but this inconvenience is sufficiently
balanced by the advantages mentioned.

( 308 )

§ 5. The mounting with the differential galvanometer. The mounting
as it finally was made is represented in PI. 1, fig. 5. The commutator
serves for the element. The commutator C, enables the observer
to compare either the platinum wire with the resistance /?’,
(complete lines), or the gold wire with the platinum wire (dotted
lines). It is convenient that by a single commutation we should be able to
interchange these two mountings, because for the comparison of
gold with platinum wires it will be always desirable to know
the approximate value of the platinum resistance for the determination
of temperature (cf. § 8). C, is a commutator to interchange the
two conducts of the differential galvanometer. This is necessary for
the determination of the ratio between the currents in the two
not perfectly equivalent conducts when the deflection is zero and
moreover it seemed desirable to me to test continually whether this
ratio remained unchanged.

The galvanometer was first a thin wire THoMsoN with two pairs
of coils each with a single wire. To attain the required symmetry,
the four coils were replaced by new ones, in each of which two
wires were wound together. This was done in the workshops of
the laboratory. The sensibility was a deflection of 1 mim. on the scale
with a difference in intensity of current of 10~'® Ampere, period of
oscillation 20". Then the galvanometer was aperiodic. Like the
galvanometer of the Wubarstonn’s bridge in the previous paper,
this was protected against disturbances arising from terrestrial mag-
netism by a soft iron ring.

The resistance boxes Ff’,, R, and /,, are all of manganin wire,
having therefore an almost negligible temperature coefficient; 7,
is the carefully investigated resistance box of Hartmann and Brawn,
also used in the measurements with the Wueatstonr’s bridge; 2,
is the box wound in our own laboratory, which produced two
branches in the Waerarstonr’s bridge (cf. previous paper), of
which I also determined the absolute values. , is a resistance
box of Srempens and Haxskr tested by the Reichsanstalt. I have
determined a few times the ratio between the units /’, and /, and
found 1.00255 (11 determinations, greatest... 264, smallest... 242),

§ 6. Lhe measurements at low temperatures. After the zero of the
wire that was to be compared, had been determined a few times, and
sufficiently harmonizing results were obtained, the wire was placed
into the cryostat inside the platinum thermometer at the place occupied
by the hydrogen thermometer when the former was calibrated (ef.
Comm: N°. 77,§ 4 and N*. ‘83, Pl. Hf).

When enough liquefied gas was poured off, the ratio between the

( 304 )

resistances of the two wires was determined while we stirred. Before
and after this determination, we measured the ratio between the
platinum resistance and one of the resistances in /,. As a rule that
resistance in ?, was taken of which the value corresponded best with
that of the platinum wire. Thence the temperature could be derived.
In later measurements a thermoelement was sometimes placed inside
the cylinder C, of the wire, and after the temperature had been
determined once by the measurement of the ratio to /,, an assistant
at the thermoelement took care that the temperature remained constant
in the way deseribed in Comm. N°. 83. Consequently the measure-
ments could be made in a still shorter time.

§ 7. Caleulation and corrections. The mounting is drawn schema-
tically in Pl. I, fig. 2, where 7, and 7, are the resistances to be
compared, JI, and I, the resistances of the galvanometer conducts
provided with the resistance boxes F”, and #,. Suppose that with
certain values in the boxes there is equilibrium and that we can then
represent the ratio between the two currents in the two galvanometer
conducts by 1+ 3, where 3 may be considered as a small number;
then we have the relation

es WwW, p ie
Peer I dane BW,

W, and JW., however, are as regards the galvanometer coils
copper resistances, and if a rather high degree of accuracy is required,
W, and JW’, must be determined before each determination of the
rauio between 7, and 7,,. If the galvanometer could be placed in
a space of constant temperature the greatest difficulty of this would
be removed and one adjustment would show us the ratio.

HW", and JW, being unknown, we can proceed as follows. We add
to WW, and W,, @, and «, units of resistance so that again equili-
brium is attained, then we also have:

1 a 5 WwW, te a, fae is) wile
1+ery Wyte 148W,+4,
ul Is __ a
ence 1 ze 3 Pt — ae
My, a,
or — == (i 5c B)—.
pt a,

How large we shall choose @ depends on circumstances. The
variations in JW, and IV, with regard to the leads during the course
of one experiment, as appeared in the measurements with the Wurar-
stonr’s bridge in the previous paper, certainly never exceeded 0.01

( 305 )

Ohm. If the galvanometer is carefully packed in cotton wool if
appeared that during the time required for one measurement the
temperature was sufficiently constant and hence the total variation
in WW, and Wy, did not exceed O.L Ohm. If then we take for the
smallest of the @, and «, 1000 Ohms, the inaccuracy due to this

l
/10000*°

uncertainty need not therefore be larger than It is likely,

moreover, that the first adjustment after the measurement where a,
and a, were added to IW, and IW’, will be repeated in most cases
so that we then can more or less judge of those variations in W’,
and JV,.

5 Tr a, . . . ’ °
In the relation — = (1+ 8)—, 2 is still to be determined. To this
Tot a
I 2
end by means of commutator (, (PI. I, fig. 3), the conducts of
the galvanometer are interchanged, so that the tums which first
were parallel to 7, are now aranged in parallel to rc and reversely.

1

. . . ry " a, . 5
A new determination yields — = (1 — }) — (3 is considered small).
Tyt Gr
5 5 . Ue G ' a,
If B is unknown we find —=}(—+ — }.
Tyt a, (rs
Besides we can determine $, and find for it
'
eae
a
a Gi
a
Bi epee
'
CC

If a rapid determination of the ratio is desirable, as it was
naturally always the case with the measurements at low tempera-
tures, it is better to determine ? beforehand. We must then have
convinced ourselves by preliminary experiments that $ remains
sufficiently constant. This was the case for observations made at not
too long intervals. Thus 11 determinations from June 14 to July 12
yielded as smallest value 0.000764, as mean value 0.000780 and as
largest value 0.00083 (°?).

Thus supposing the value of @ to be known we can derive the

y

c we . . . . .
ratio — from a single determination of @, and @,; @, and «@, were
Ppt rc z

p

read on the resistance boxes #’, and /,.

§ 8. The determination of the temperature. The determination of

the temperature, now that we do not measure the resistances separa-
tely but determine the ratio at once, need not be made with the
accuracy which was required for the test of the platinum thermo-

meter. It was obtained by measuring the resistance of the platinum

1g

= oe

APR Ae Pi poe

( 306 )

wire after each complete determination of the ratio. If in’ the
arrangement of Pl. I, fig. 3, the commutator C, is placed differently,
we can, instead of comparing the gold with the platinum resistance,
compare the latter at once with the resistance in the box &,.

In determining the ratio between the resistances of the platinum
wire and the values in box R&R, we have not, however, repeated the
whole measurement as described above for the determination of the
ratio between the platinum and the gold resistances, but one measure-
ment sufficed.

For with FR, instead of 7, as in § 7 we find:

ly? eae pose
iGo ea Lee

where W, and W’, are the resistances of the circuits where the

galvanometer turns and the resistances occur.

P R . ;
If we know the resistance of the turns, —- may be derived from
vis
pt

one determination of a resistance in the boxes.
The resistances of the turns can be derived from the two deter-
minations of the ratio between the platinum and the gold resistances.
Let A and £B be the resistances of the turns round the galvano-
meter, and 7, and 7, the resistances in the boxes, then they give with
W,=A-+r,, WW = BET
1 rt, A-+?, B Tx
l+Bron B+r, 1+pB+7,
1 Tr Bor, B Tr
1—Bryx Atr, 1—~fA+4+r,’
whence A and £ may be derived and consequently W, and W, may
be found. Inaecuracies occur, that is to say we neglect the resistance of
the connecting wires between the commutator holes; but as the value
of A and £# was about 940 Ohms, an inaccuracy of 1 Ohm was
allowed and this resistance certainly remained below this amount.

3 Sertel A he
For the calculation it is important to remark that —— — is equal
1+ Tpt
a, ee. a, ns a’, )
to — and — is equal to —, while — and — may be derived
a, 1—Brot a, a, a, ‘

b

directly from observation. Besides j is small, so that in the last

term of the equations an approximate value for A and / is sufficient.
§ 9. Survey of an observation, The way in whieh the observations
were made will be seen best from an instance which at the same

B. MEILINK. “On the measurement, of very low temperatures. VIII. Com-
parison of the resistance of gold wire with that of platinum wire.”

“ty

Proceedings Royal Acad. Amsterdam. Vol. VII.

|

( 307 )

time will be developed as an instance for the calculation. It refers
to a measurement with the gold wire in oxygen boiling under
reduced pressure.

The values of the following table (p. 808) were read directly.

In this measurement the temperature was kept constant by means
of the thermoelement so that the determination of temperature need
not be repeated.

For the derivation of the results from the observations see table
Il, which does not require further explanation.

§ 10. Results. The determinations of the zero yielded the following
results.

Resistance Method of the measurement
Date. 5a eR
gold wire. of resistance.
April 23, 1902 31.5506 Wubatstonn’s bridge.
May 23, __,, 31.556 e Ae
pat Le A 31.565 Differential galvanometer.
May 26, ,, 31.555 WHbATSTONE’s bridge.

Our chief object of the determination with the differential galvano-
meter was to ascertain that the two methods gave the same results,
so that the latter could also be employed for a determination of
temperature. During this measurement we did not stir and so it is
possible that the temperature has increased a little. In connection
with the following results the agreement is sufficient.

After the measuremenis at low temperatures, which are made
between June 17 and July 12, the zero was redetermined in October
and then a deviating value was found, viz. 81.048. I have searched
in vain for the reason of this deviation. It does not probably lie in
the measurement of resistance. A known resistance determined in
the same way gave the true value. While J searched for possible
causes the gold wire broke so that I have not attained any certainty
ou this subject. It may be that during the interval between the two
determinations a short circuit has been formed between the two
ends of the gold wire, in consequence of which the resistance is
apparently so much diminished. A change in the gold wire itself
would probably always have produced an increase of resistance.

The values for the further determinations of the ratio between
the platinum resistance and the gold resistance will be sufficiently
clear from the following table (p. 310).

The measurements show that this method to determine the ratio
between the resistances is certainly a good one and preferable to
that of measuring the resistance of each wire separately. It will

( 5308 ) ;

| TABLE I. Comparison between Gold and Platinum
!
Resistances in Oxygen boiling under Reduced Pressure.

Observations.

+2 |

= Resistance in = = Fs a
Bs £.=-| 53 So
EIS) Ee) Saud zz) /3|
ae On] 26 3c |Time.
5= SreMens &}] HARTMANN | © 2 =e = 2
= — ay se, S + aS a=
Zz HarskE. | & BRAun. | 2 RS &
oy a
Temperature determination. 2h. 30!
500 140 27.00 | 33.00
= 136 23.00
Determination of the ratio.
27.50 a
0 | 40004-400-+-} — |j 31.5
300+20-+-10} — 93.90
= \| 31.50
2 28.05 7
a us 97.00 |2h. 45!
|| 28.30
1000 |3+4.1000] || 26.50
44-3. 100) = 27.70
\| 26.60
|
|, | 25.40
+3 = 29.20
\| 25.55
s< L000 3+1.1000} || 30.30
449. 100} = 28.30
| Tl 30.30
i 294
agar | 2 | 29.62
0 4000-- | = 2.00
443. 100) |j 38.20
944. 40
2.10 I 31.99
4+ 3 = 27.75
|| 31.85
0 | 1000 |
| = 14+-3. 400) = 26.70
| Peden ADH i 31.65
= | 26-80
, 4000 8+41 1000) = 2940
A+2. 100] || 27.50

( 309 )

CE6VE 0] SOV

| GEBYE"O

YOGYE 0 |

‘SW GO'RG = "V

UOTRNI[Vo aanyesoduiay Loy HOURISISEY|

VEGVE 0 || GOBPE'O

suUOTJIVIIOD

‘ONRY oy. Jo ToIL[NoTKD

paket ll os'zecy |CE® a a et |
ous 006 + 00%
) | 8G L986 °7)) ovo | cag | co'o— |]0g'998¢| O00T eB) or +0001 + 0008 _ LL
| OS BOOK” abe eota or +03, + 008 z
O feLb | 08 ¥ — series ONG 0 =
hdl geal ee L-+06-+
Se WE one-Eia + ot
YO" 9GLE
E + 0% + 008
po a et OG | eee aat y
hercanr2|| 80+ | Se°G+ | c0"0— || 99° F286] 000F Se aca
0g 0 + oy + 00 + 00% x
Bae Foot tonne
9° L6SY 7 va
= jee 006 + 00Y F
00°% + oor toooe || _900F
entaaey aioe ot ef
SHURA SKIS SINT B 003 —- OOF rive
OF 2986) Tlf eee nt lca | coto— \llea-edee' | OCU SVE | + o001-E 0008 0001
ZOOL” Vf tC =| ab DUSG cans 7) ae Ga
Se re a, 0% + 008 =
ore “E oay + 000! v
oe Ua 3" H ‘NaVUg ee
Parent “4 YESH a HY'S || ssoyouvsq 0M} |) 9 “WIV ET We ‘NAVUG NY WLIV IT s0)
Ey wals'7) : H's ayy ut ELEC) oy atees ayy jo 104V}
| payor ei eae ae ee Pea AOULISISOYT
z es Sa0ILate -inba DAME OUSE ADURYSISAY ||-N ULL’
| saouaaayiq peu peat WeLonte ws ula YO net a

‘HUSSAI PaUpoy opuN Surplog ussSAX() UL SooUV]sIsey WU], pur pox) Uwoajoq uostVdUOD “TT ATAVAL

vyevETeCr = a,
a

q

;

ooo

een Gold and Platinum |

TABLE III. ‘Comparison betw

Resistances.

| | |
Aisiciane Ratio Ratio between |
poet lecietance| ‘the resistances of a
me eel gold gold wire and a Dates
pea ld wire, resistance _| platinum wire ae
F gold wire. latinum |Which at 0°C. have
wire platinum ‘
: resistance |the same resistance.
| — oe
| | |
140.045 | 31.5505 | 0.98672 April 23, 1902.
(at 0° C.)| 31.556 0.28675 1 May 23
31.565 0. 28684 id.
31.555 0.28674 May 26.
0.295677
0.205736 ; June 17,
64.78 0.29554 in ethylene.
0.295569 0.28674 —
9.295564
0.295531 = 1.0307
O4.85 0.295562
52.34
0.301954 0.301950 June 17,
0.301952 0.28674 in ethylene
0.301949 boiling under
0.301946 = 1.0530 reduced pressure.
52.68
28.74
0.330287 June 23,
0.330314 0.330295 in oxygen.
0.330285 Moshtia aS
0.330204 Spee Te
0.330256 =
0.330289 ele
28.71
92.79
0.34490
0.34497
22.81 July 12
0.84490 0.34497" = in nitrogen.
0.34498 0. 28674
22.74
0.34464? —— | YO PLS)
0 34495
22.79 }
21.39 0.34933 June 23, in
| 0 34934 5c oxygen boilin
0.34932 le under reduce
Se :
2 AT ae —= 4.9173 Wessels

(3)

be necessary, however, at least for low temperatures, to take great
care that the temperature is kept constant during the time required
for a measurement. In these measurements the determination of the
temperature was less accurate than seemed desirable with a view
to the accuracy of the determination of the ratio during the measure-
ments on one day.

In the results there is a striking difference between the gold and
the platinum. Though the values found do not help us to fix the
temperature function for gold for want of certainty about the zero
to which they belong, yet they show that the curvature of the line
which for gold represents that temperature function is much smaller
than in the case of platinum and that the curve is bent more towards
the absolute zero. Hence a gold wire would be more suited for
extrapolation than a platinum wire, because here the deviations
which we cannot but expect, are much smaller.

Mathematics. — “A congruence of order two and class tivo formed
by conics’. By Prof. J. pe Veins.

For a twofold infinite system of conics (congruence) order is called
the number of conics through an arbitrary point, class the number
of conics with an arbitrary right line for bisecant.

The congruences of order one and class one arise from the pro-
jective coordination of a net of planes to a net of quadrics'). Under
investigation were furthermore the congruences of order one and
class two and those, the conics of which cut a fixed conie twice *).
In this communication the characteristic numbers are deduced of
the congruence determined by the tangent planes of a quadric (Q
on the planes of a net [Q*| of quadrics to which they are project-
ively conjugate.

2. To obtain this conjugation we project the points P? of Q? out
of a fixed point ?, of Q? on a plane ¢. A projectivity between
the points P’ of and the surfaces of [@Q?] furnishes then imme-
diately a projectivity between [Q?’] and the system [2], of the
tangent planes a of (Q?.

To a pencil (Q*) in [Q*]| corresponds a range of points (7) in

1) D. Monresano, Su di un sistema lineare di coniche nello spazio, Atti di
Torino, 1891—1892, t. XXVII, p. 660.
*) M. Piert, Sopra aleune congruenze di coniche, Atti di Torino, 1892—1893,
t. XXVIII, p. 135.
21
Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 312 )

¢, thus a conic on , thus the system of the tangent planes 2
passing through a fixed point 7. Through T and a point Y of the
base-curve of ((*) two planes a pass; wherefore Y bears two conics
of the congruence, which is thus of order two (P= 2).

3. To the tangent planes a through an arbitrary point T corre-
spond the points ? of a conic not passing through P,, having thus
for image a conic in %. So to this system (2r),, of index two, is
conjugate a system (f°), possessing likewise index two, having two
surfaces in common with each pencil (Q*). When considering the
ranges of points determined by the projective systems (a), and (Q*),
on an arbitrary right line we find that they generate a surface 7°
of degree six, which is the locus of the conics of the congruence
the planes of which pass through a fixed point 7 Hence we get
[ju —10.

4. Through two arbitrary pomts pass two tangent planes 2,
hence the planes of two conics; so an arbitrary right line is bisecant
of two conics, and the congruence is of class two (u* = 2). ;

The numbers P=2, we=—6 and w?=2 satisfy the well known
formula P= ur — 2 w’.

Through a right line of @Q* pass an infinite number of planes a;
the conics they bear form a cubic surface.

As each ray through meets two conics, T* has in T a double
point. If A? is one of the comes on which T is situated, T* is
touched in T by each bisecant of A* out of T. So Tis a biplanar
point.

If T is one of the eight base points of the net | Q? | then
T has in Ta fourfold pomt; for on every ray through T he but
two points besides ‘T.

5. Let us take for (Q* the paraboloid « y = ¢, then the substitution

e=ao, y= Bo, z=7o furnishes first e—y:a@ and then

ay?

Saris cst ey cuca

P= ea
So the tangent planes a are represented by
Byz+tayy—apz—y'? =0.
The above-indicated conjugation is arrived at by putting
aA+PBpB+yC=0),
where A, B, C are quadratic functions of wz, y, 2. We represent their
coefficients by ari, De, Che and we write brietly
dy = a aig + B bik + ¥ CK.

( 313

If a Q of the net is to be touched by the conjugate plane a, then

dy d., tl (ep By

d.4 de. (ips hy ay

Ce ds. ds. da, ap =——a()3
Cn d,, (hp de, y?

By ay —ap —y’ 0

must be satisfied.
We find here a relation
Der bay) =O;
which is homogeneous and of degree 7 in «, 3, y. If we regard
these parameters as homogeneous coordinates, this relation represents
a curve of degree 7 possessing nodes in the points 4 (g=0, y=0
and (a4 — 0) = 0):

6. For the conies passing through point T(v,, ,,2,) we have

the relation
M, (a, 8, y) =, By + y, ey — 2, 08 — 7? = 9.

It is represented by a conic passing through A and Bb.

Besides 4 and # the auxiliar curves D* and J have ten points
in common. So through T pass the planes of ten conics each degene-
rated into two right lines (dj = 10).

That the points A and B&B must not be taken into consideration is
shown as follows: For e=0,y=0 we find B=0O and y= 0:0,
thus the pencil of planes around OLY; of these tangent planes of
course only one is conjugate to BO and the conic determined
by it does not form a pair of lines generally.

Out of the relation *)

suv = 2yu + du + 47

2

ensues, as wy = 6, du=10 and uw’? =2,

(fie

This could be foreseen, for the cones of [ Q?] form a system co!;
the number of those cones touched by the homologous planes 2
is thus finite and all twofold symbols in which a appears have there-
fore the value zero.
ieee hie right line 0s aml) is cut by the conics for which
we have
apz -- y*> = 0 and d,, 27 - 2d,,2-+d,,—0,

1) Compare my communication in these Proceedings, p. 264.
gs, |

21*

( 314 )

thus
N, (a, 8, y) = 4,, y‘ — 2d,, aby? + d,, a? Bp? = 0.

The curve .V* representing this relation has evidently nodes in
A and B.

By connecting .V* with /? and D' we find anew ur = 6 and
farther

dp = 27.

The pairs of lines of the congruence form a skew surface of

degree 27.

8. To find the characteristic numbers containing the symbol @
we consider the pairs of pomts which the conics of the congruence
have in common with the plane z= 1. They are indicated by

pye a ayy = a? a y’,
d,,a?+2d,,wy+d,,y?+2(d,,+¢4,,)e+2(d,,+4,,)y+(d,,42d,,4+d,,) = 0.

So for the conics touching z= 1

a. dis d,+d,, By |
ae dx d,, + d,, ay | =
d,,+d,, d,, + d,, BSS RB bop Laer | == \l)-
BY ay —«asp—y? 0

This is a relation
R, (a, By) = 9,
which is represented by a curve R* having A and & for nodes.
By combining R* and D?, W* and NV‘ we find successively

iS

bo

Jo = 34, oO = 8; vo = 22.

From this ensues that the skew surface of the pairs of lines has
a double curve of degree 17 and that the conics touching a given
plane (in particular thus the parabolae of the congruence) form a
surface of degree 22.

Out of the relations

3n? = dp + dur and 30? = 2do + 2u0
we finally find for the missing characteristic numbers
D—slis anda ow.

So the conies cutting a fixed right line form a surface of order 17,

Physiology. “On a new method of damping oscillatory deflections

of a ygaleanometer”. By Prof. W. Errnoven.
(Communicated in the meeting of September 24, 1904).

In a number of investigations, requiring the use of a galvano-
meter or electrometer, it is desirable to damp the oscillatory detlect-
ions shown by most of these instruments under many circumstances.
Either mechanical damping is applied or electromagnetic damping
or both are combined in order to obtain a stronger effect.

In some instruments, e.g. the Drprez-p’ ArsonvVAL galvanometer, in
which the coil is movable in a stationary magnetic field, the electro-
magnetic damping may without any special arrangement be so great
that the deflections have lost their oscillatory character and have
become quite dead-beat. The movements are thereby retarded. This
retardation may be very considerable and so become troublesome,
even to such an extent that the instrument becomes impracticable.
Means of diminishing the damping are then applied, e.g. by increasing
the resistance in the galvanometer.

In order to apply electromagnetic damping in a needle-galvano-
meter the rotating magnetic system is to a greater or less extent
enveloped by a mass of pure copper in which during the motion
of the needles damping vortex currents are raised,

Mechanical damping is applied as liquid) or air damping, thin
plates of aluminium or mica or insect wings being often used.

The method of damping to be described in this paper is entirely
different from the methods just mentioned. It consists in inserting
a condenser between the ends of the galvanometer wire as is indic-
ated in fig. 1. In the figure
FE represents a source of current
by means of which an arbitrary
potential difference can be esta-
blished between P? and P,. G
is the galvanometer and C’ the
condenser,

The action of the condenser

Fig. 1. is most easily understood by
asstiming the mass of the moving parts of the galvanometer to be
zero and the eventual causes by which the motion is damped to
iend to zero. If under these conditions the capacity of the condenser
is zero, when a potential difference between P? and P, is suddenly
established, the galvanometer will also at once assume the corre-

( 316.)

sponding position of equilibrium. If on the other hand<there is a
certain capacity, the deflection will require some time.

The way in which the image of the mirror or in the string-
galvanometer the quartz thread then moves, is entirely determined by
the way in which a condenser is charged or discharged. Calling «@
the deflection of the galvanometer at the time ¢ after the potential
difference is established and A the permanent deflection, we have

— a
C A i —— he ze

where e is the base of natural logarithms, ¢ the capacity of the
condenser and 2’ a resistance of which it is easy to give a nearer
definition.

In the closed circuit containing the source and the galvanometer
the external resistance be II,, the resistance of the galvanometer
be IV;, then, if we neglect the resistance of the wires joining the
condenser and the galvanometer, we have

WwW; W.,
i = = teeta totsc’ (Il)
Wit We
The valne a’e is the time constant of the deflection
we ——- A hes

Expressing 7’ in Ohms and ¢ in Farads, 7’ is given in seconds.

When ithe deflection of the galvanometer is recorded on a uni-
formly moving plane, a curve will be obtained which is the ex-
pression of an exponential fanetion and which agrees entirely with
the wellknown normal or standardising curves of the capillary
electrometer’).

The constants of the curve, besides being determined by the rate
of motion of the recording plane and the amplitude of the deflection,
will depend only on the value of 7. By changing w’ and ¢ we
can regulate the value of 7’ at will. This means that we are able
to retard or damp the deflection of the galvanometer to any extent.

The reasoning given is confirmed by the observations. As an
example we reproduce three curves, figs. 1—8 of the plate, recorded
by the string-galvanometer*). The connections are schematically

1) See e. g. W. Eiytnoven. Priicer’s Arch, f. d. gesammte Physiol. Bd. 56,
p. 528. 1894, And ,Onderzoekingen” Physiol. laborat. Leyden, 2nd series I.

2) See W. Eiyrnoven, Ann. der Phys. 12. p. 1059. 1903 and 14, p. 182. 1904.
Also in Kon. Akad. v. Wetenseh. te Amsterdam, Report of the meeting of June 27,
1903 and March 380, 1904.

(¢ oie)
——— = represented in fig. 2. Here 17
| is a. battery, S a key, G the
s = SS string-ealvanometer and C' the
~ Re le condenser, A, 6 and Lf repre-

‘il | senting resistances. The sensi-
|

B ree tiveness of the galvanometer has

Se ANAAARARAR been kept about equal in the

OWE
Mo D . ‘s
Be: of 4 mm. corresponds to a

current of 2><10-7 Amp., the electromotive force /7 of the battery

and the resistances A, B and R& being so chosen that when the
current is passed a permanent deflection of 20 mm. is obtained. The

three cases so that a deflection

rate of motion of the recording plane is 500 mm. per second. Hence
in the net of square millimetres on the plates ') 1 mm. of abseiss
= 0.002 sec. and 1 mm. ordinate = 2<10-‘ Amp. The circuit
was automatically made and broken at S by an arrangement attached
to the recording plane.

For R a carbon resistance was taken with large resistance and
/& was small compared with R. W, could be put equal to PR without
an appreciable error. In figs. 1 and 2 of the plate W, was 1.11
megohm, whereas JI, in fig. 8 amounted to 117000 ohms. The
resistance of the ealvanometer |; was 8600 ohms.

In fig. 1 of the plate the capacity of the condenser is 0. The
string is seen to make oscillatory movements with a period of about
IPMN — 2LOG. >):

These movements are damped by inserting a certain capacity in
the condenser. In fig. 2 of the plate that capacity is 0.94 microfarad,
in fig. 3 of the plate 0.2 microfarad.

Calculating the value of vw’ from IW; and JW, by formula (1) and
then the time constant 7’= w'c, the time constant of fig. 2 is found
to be 8.00, and that of fig.3 1.66 and it is clear that the amount
of retardation or of damping is determined by the value of the
time constant.

For clearness’ sake we started in the above reasoning from the
simplest case and assumed that the mass m of the string and the
forces Which independently of the condenser damp its motion and
Which we will collectively indicate by 7, may be neglected. This
hypothetical case will the more closely agree with reality, the larger,

other conditions being equal, 7 is taken. Hence in this respect fig. 2

1) On the way of recording and the net of square millimetres see Annalen der
Phys. 1. c.
*) 1¢=0.001 sec.

(318)

of the plate answers better the conditions required than fig. 3, but
the great practical importance of the method is exactly that it is
possible to damp the oscillations and at the same time to retard the
deflection as little as possible. When measurements are made one
will always try to choose 7’ such that exactly the limit between
oscillatory and aperiodical motion is attained. In this case 7’ is rela-
tively small and im and + may no longer be neglected.

The question now arises how for known values of m and 7 the
value of Z must be calculated in order to obtain the limiting ease
mentioned.

In passing it be remembered that with the capillary electrometer
the damping of the motion of the mercury meniscus is also composed
of mechanical friction and of retardation by capacity.!) And from
the combination of these two results a motion which can be expressed
by a simple exponential function either quite accurately or with
only small deviations. The resistance of air or liquid damping as
well as electromagnetic damping influence the motion of a body
having mass, in exactly the same way as conductive resistance in-
fluences the motion of electricity when a condenser is charged or
discharged. ;

A simple reasoning will show, however, that adding a condenser
to the galvanometer has not always an influence on the movements
of the string of the same nature as an increase of the damping
forces which we called 7.

For the addition of the condenser has the effect of a temporary
change of the active force. And the way in which the force is
increased or decreased from moment to moment is not determined
by the motion of the string, as the mechanical and electromagnetic
damping, but by the product of the conductive resistance and the
capacity w' c= 7.

When applying the condenser method, the character of the motion
of the string near the limiting case of aperiodicity can only be
represented by a more or less complicated -formula. U have therefore
for this limiting case preferred direct experimental determination of
the value of 7’ to calculation.

Some curves have been reproduced which exemplify the motion

1) Some investigators have been of opinion that the motion in the capillary
electrometer is dependent on the charge of the mercury meniscus only. But in
reality damping by mechanical friction is much more active here. See Prxiicer’s
Arch. f. d. ges. Physiol. Bd. 79, p. L. 1900; and .Onderzoekingen” Physiol. laborat.

Leyden, 2nd series, 4.

( 319 )

of the string in the limiting case in question’). Figs 4, 5 and 6 of
the plate were taken with the same string as the former figures.
For the connections we refer to figure 2 in the text. The deviation
is now 30 mm. Again 1 mm. of absciss is 0.002 sec. and 1 mm.
of ordinate = 2X 10-7 Amp.

R = 1300 Ohms.

ee el

WW; = 8600 _,, , from which we calculate

W,= 1327 and w' = 1148 Ohms.

In fig, 4 the capacity of the condenser = 0, hence 7’= 0.
Pe dG ao 8 =a nee, eee — 0:69/0.
eee WO! "35 28s Oder 5 Onin, . —0,80 6.

One sees that the oscillatory motion, the period of which is about
2,7 6, is damped by the application of the condenser method and
that the time constants 7’ of 0.69 and 0,80 6, obtained by means
of capacities of 0.6 and 0.7 microfarad, are required in order to
reach the desired limit of aperiodicity.

In fig. 5, where a capacity of 0,6 uf. is used, the limit has not
yet fully been reached, in fig. 6 the limiting value has already been
passed with a capacity of 0.7 wf.

The two last-mentioned plates show that the motion of the string in
the neighbourhood of this limit is not very simple. In the small oscil-
lation which has remained in fig. 5 the string, after having deflected
through 50 mm., passes the new position of equilibrium by 0.5 mim.
and then returns to a point which is still 0.5 mm. lower than the
position of equilibrium, The ratio of the values of these deflections
does not agree with the laws which damped motions generally obey.
Moreover the first turning-point is reached after 2 6, the second after
1 6, whereas with damped vibrations, such as generally occur, these
times are equal.

In fig. 6 the string comes to rest after about 0.002 sec. at a
distance of 0.8 mm. from the new position of equilibrium and reaches
its equilibrium after a small movement in the opposite direction. If
in the measurement of a current one is contented with an accuracy
of 2°/, the result is known in about 1,5 o.

Another example is found in figs. 7 and 8 of the plate. These

to)

photograms were taken in the same way as those immediately

1) The process by which the photograms of the plate have been reproduced
does not reveal the minor details of the curves. | shall be pleased to send direct
photographic copies of the original negatives to those who are interested in them,

( 320 )

preceding, but the string is lighter here, has a greater conductive
resistance and is slightly more stretched.

lL mm. abseiss = 0,002 sec., 1 mm. ordinate
W; = 17800, W, = 20000, hence i’ = 9420 Ohms.
In fig. 7 the capacity is 0; in fig. 8 it is 0,05 a7, hence 7’= 0,47 o.

3X 10-7 Amp.

In this latter photogram the string shows a turning-point after about
L1do exactly on the new position of equilibrium. It moves back
through 0.9 mm. and then reaches its equilibrium again and finally.

If in the measurement of a current one is contented with an
accuracy of 3°/,, the result is obtained in 0.86. If an accuracy of
0.3°/, is wanted, the result is only obtained in 2.2 6.

These examples may suftice to see what can be expected of the
method. It is obvious that when seeking the exact value of 7’ for
reaching the limit, we were led by theoretical considerations although
we could not use a rigorous formula. One of these considerations
was that for a given string and constant resistances, the capacity
required for the limit must be the smaller the more strongly the
string is stretched. For with greater tension of the string the period
¢ of its oscillations becomes smaller and we may expect the wanted
value of the time constant 7’ to change in the same sense as the
period 7.

This consideration leads to some paradoxically sounding predictions.
So, for example, it is to be expected that the motion of a strongly
stretched string that has been made dead-beat by applying the con-
denser method, will become oscillatory again as soon as the tension
is diminished and thereby the motion is retarded. Such an expectation
seems at variance with the experience gained with other galvano-
meters, we might say, gained without exception with all instruments
in which vibratory motions are observed.

The result, which was expected with some anxiety, completely
confirmed the prediction. A quartz-thread of such a tension that a
permanent deflection of 1 mm. corresponded to a current of 2X 10—7
Amp., showed, when a current was suddenly passed or interrupted,
(see figure 2 in the text) a number of oscillations. By inserting a
capacity ¢ = 0.185 47 the motion was damped to such an extent
that the limit of aperiodicity was reached. Next the tension of the
string was exactly 4 times relaxed so that a deflection of 1 mm.
was caused by 5X LO-§ Amp. The oscillations then re-appeared,
and could not be checked again until the capacity was increased to
0.40 vf. With a 4 times smaller tension, i.e. with a 4 times greater
sensitiveness the capacity and at the same time the value of 7’ had

( 321 )

fo be increased 2.96 times in order to reach the limit of aperiodicity

Observations with other quartz-threads, the tension of which was
varied, always gave corresponding results: with strong tension a
small value, with a more feebly stretched string a larger value of
we is required in order to check oscillations.

If w' is kept unchanged, one has an easy means of accurately
regulating the desired degree of damping in a commercial condenser,
in which capacities are shunted in by means of plugs in the same
Way as the resistances of a resistance-box. And if is remarkable that
less of the means of damping is sufficient as the oscillations pass the
zero-point farther and last longer and consequently the need of
damping is greater. The phenomenon that, leaving the other circum-

stances unchanged, diminution of the tension only, i.e. — with the same
deflection —, diminution of the moving force, changes an aperiodic

motion into an oscillatory one, stands quite isolated and has, as far
as is known to me, no mechanical or electrical analogon, no more
in scientific instruments than in industry.

We shall now give some results of measurements which although
they cannot compensate the lack of a simple formula, may yet be
helpful to form an idea of the method in practical work.

1. When the damping influences already existing are increased,
for example when the electromagnetic damping is strengthened by
diminishing the resistance in the galvanometer circuit, a smaller
value of 7 will suffice in order to reach the limit of aperiodicity
when the quartz-thread is the same and the tension is not changed.

2. If the change in the electromagnetic damping which is caused
by varying the value of JI, is taken into account, it makes no
difference how the single factors 7’ and ¢ are chosen. If only their
product w'c = 7’ retains the same value, also the damping influence
will remain the same. This latter is only determined by the product 7.

3. If the motion of the quartz-thread is oscillatory it will be
observed when the condenser method is applied, beginning with small
values of 7 and gradually rising until the limit of aperiodicity is
reached, that increasing 7’ does not always cause a regular increase
of the damping. Especially with feeble tension of the quartz-thread,
when only a few small oscillations are normally produced, one sees
an irregularity appear. The addition of a very small capacity can
then even slightly enlarge the existing vibrations.

When such a value of 7 has once been taken that the limit of
aperiodicity is reached, 7’ has only little to be raised in order to
obtain a regularly shaped curve. With further raising of 7’ the motion

( 322 )

is more and more retarded, the regular form of the curve being
retained.

+. That we may form some opinion about the value of the time
constant 7’ that is required in’ various circumstances in order to
reach the limit of aperiodicity, we give the following table, containing
the results of measurements, part of which have already been men-
tioned above.

——

Ww Ww = c T= w'e t k
; : : a in thousandths in thousandths :
in micro- ae atts Damping-

in Ohms. in Ohms. in Ohms. farads. second. second. ratio.
8600 117000 8000 0.40 BED YheEl 7.6
S609 117000 S000 0.135 1.08 ey | } aps |
8600 ed >< 102 8520 0.412 1.02 2.64 sect
8600 1327 1148 0.65 0.75 2.7 | 45

17800 20000 9420 0.05 0.47 1.41 3.16

The first five columns of this table need no nearer explanation;
they give the conductive resistances, the capacities and the values
of the time constant 7 For the values of 7’ mentioned the limit
of aperiodicity was just reached.

The two last columns indicate how the string vibrates when the
capacity of the condenser and together with it 7’ is zero. In the
last column but one we find the period ¢ expressed in thousandths
of a second, while the last column gives the damping ratio /. The
observations have been arranged according to the values of 7”.

Finally some remarks may find a place here on the circumstances
under which the condenser method will be useful in practice. For
the present the applications will presumably be restricted to such
measuring instruments as possess a great internal resistance and a
short period of oscillation. A galvanometer for thermo-electric currents
with a small internal resistance and a great period of oscillation
would for damping by the condenser method require an enormous
capacity. The mica or paper condensers, which admit of easy regulat-
ion, would be out of the question here, since even the largest sizes
of the trade would turn out to be still a hundred thousand times
too small. So one would have to have recourse to another kind of
condensers, e.g. electrolytic ones, and it would require a separate
investigation how far these can indeed be rendered practicable for
the purpose in view.

( 323)

The conditions of a short period of oscillation combined owith a
relatively high internal resistance are fulfilled by only one instrument
besides the string galvanometer, as far as is known to me, namely
by the oscillograph. Here the damping is effected by means of oil
Which is heated ').

The temperature of the oil determines its viscosity and the regulation
of the degree of damping is consequently obtained in the oscillograph
by regulating the temperature of the oil. It is doubtful whether the
instrument would greatly gain in practical usefulness if the oil with
the heating arrangement were done away with and replaced by a
condenser.

In the string galvanometer the condenser method will be success-
fully applied in cases where it is desired to measure variations of
current of very short duration. Taking a very short and strongly
stretched quartz-thread, it will be possible to obtain deflections whose
quickness leaves little to be desired. Without a condenser these would
be useless for many purposes on account of the oscillations, whereas
now they may become useful for a number of physical and electro-
technical investigations by a judicious damping. In these cases the
string galvanometer will for equal quickness of deflection appear to
be a much more sensitive apparatus than the oscillograph.

Also in a number of electrophysiological investigations we can
avail ourselves of the condenser method, while the study of sounds
will be particularly facilitated by it. I hope to make a nearer com-
munication on this subject in a following paper.

Physics. — i bands in the spectra of db Orionis and
Nova Perse’. By Prof. W. H. Junius.

When light, giving a continuous spectrum, passes through a
selectively absorbing, non-homogeneous mass of gas, the spectrum of
the transmitted light contains places which, according to circumstances,
may contrast as bright or as dark regions with their surroundings *).
Though resembling emission and absorption lines, these bands have
a wholly different origin. They are due to anomalous dispersion
and, therefore, the name dispersion bands has been suggested for
them *).

1) Also a mixture of two liquids is used, of which one has a great, the other
a small viscosity. The mixture is so chosen that the desired viscosity is just obtained.

*) Proc. Roy. Acad. Amst. Il, p. 580 (1900).

5) Proc. Roy. Acad, Amst. VII, p. 184—140 (1904).

( 324 )

Dispersion bands always appear in the proximity of absorption
lines, covering them more or less symmetrically; they show great
variety in width and strength, and the distribution of the light in
them may be irregular, so as to give the impression that one is
witnessing cases of shifting or doubling or complicate reversal phe-
nomena of widened absorption lines. All these cases can be produced
almost at pleasure in the absorption spectrum of sodium vapour by
merely varying the structure of the non-homogeneous medium through
which the light is made to travel.

In the spectrum of the various paris of the solar image dispersion
bands play an important part’). We can scarcely doubt that they
are also present in stellar spectra; for the light coming from the
stars must, as a rule, have travelled through immense gaseous
envelopes and suffered ray-curving and anomalous dispersion, just
as well as the light from the Sun.

Taking for granted that most of the visible stars are rotating
gaseous bodies, with or without a solid core, we must suppose
them to have a_ structure, describable by surfaces of discontinuity
with waves and vortices, and resembling the peculiar structure of
the Sun, by which it has proved possible to explain solar phenomena’).
Consequently, the stars too give existence to “irregular fields of
radiation” rotating with them. Our line of sight continually cuts
other parts of the refracting mass; if may pass closely along sur-
faces of discontinuity, now on the one, now on the other side
of them; so the light reaching us must vary in strength and in
composition.

The variability of many stars is very likely to result from this
cause; and from the same principle it necessarily follows that their
spectral lines should be liable to every kind of change in place and
in appearance.

In many cases where the application of Dorpier’s principle leads
to very unsatisfactory conclusions, the dispersion bands afford a
plain solution. Let us consider, for instance, the spectrum of d Orionis.

In this spectrum rapid changes in the position of the lines had
been observed by Dersnanpres (1900), who concluded from them
that d Orionis was a spectroscopic binary having a revolving period
of 1.92 days. Some observations made by J. Hartmann *) did not
agree with this period. Professor Harrmayn, therefore, submitted the

1) Proc. Roy. Acad. Vil, p. 140—147 (1904).

2)-Proc. Roy. Acad. Amst. V, p. 162—171; 589—602; VI, p.270—3802 (1903).

5) J. Hanrmany. Untersuchungen iiber das Spectrum und die Bahn von 4 Orionis.
Sitzungsber. der Kén. Preuss. Akad. d. Wissenschaften, XIV, 8. 527—542, Marz 1904,

——— a ee

( 325 )

star to an extensive spectographic investigation in the winter months
of 1901—2 and 1902—3, and, from the 42 plates obtained, drew
the following conclusions.

The spectrum contains chiefly the lines of hydrogen and helium;
besides a few belonging to silicium, magnesium, calcium.

The calcium line at 4 39384 (corresponding to A’ of the solar
spectrum) is extraordinarily weak, but almost perfectly sharp; all
the other lines (nineteen in number) are very diffuse and dim, often
appear crooked and unsymmetrical, sometimes indeed double. While
every prepossession of the observer was most strictly avoided during
the measurements, if was found, that the centres of the diffuse lines
really oscillate, the period being 5,7333 days: but, owing to the
unsymmetrical appearance of many of the lines, no evidence could
be obtained that the values of the displacements were in mutual
agreement for ali the lines on one and the same plate. From the
average displacements Hartmann calculated the ‘variable velocity in
the line of sight”, and finally the elements of the orbit.

An utterly surprising result, yielded by the measurements, was
that the calcium line at 24 3934, does not share-in the
periodic displacements of the other lines, but shows a
constant shift corresponding to a velocity in the line of sight of
+ 16 km. (reduced to the Sun).

HARTMANN rejects the idea that this line should have originated in
the Earth’s atmosphere; also the assumption that it belongs to the
second component of the binary system. He is thus led to the hy-
pothesis that at some point in space in the line of sight between
the Sun and J Orionis there is a cloud of calcium vapour which
recedes with a velocity of 16 k.m. By examining the spectra of
neighbouring stars no further information as to the existence of such
a cloud was obtained.

A quite similar phenomenon, however, had been exhibited by the
spectrum of Nova Persei in 1901: the lines of hydrogen and other
elements were enormously broadened and displaced and continually
changing their appearance, but during all the time the two calcium
lines at 23954 and 23969, as well as the D-lines, were observed as
perfectly sharp absorption lines, yielding the constant velocity of
+ 7 km. Hartmann therefore assumes that also in the line of sight
between the Sun and Nova Persei there exists a nebulous mass
consisting, in this case, of calcium and sodium vapour, and moving
from the Sun at the rate of 7 kim. a second.

It must be admitted that these hypothetical clouds do not form
& satisfactory solution to the problem.

( 326 )

A much simpler explanation of the phenomena may be derived
from our conception of the irregular fields of radiation caused by
the stars.

We need only suppose that the outer. parts of d Orionis and
of Nova Persei, like those of so many other stars, contain much
hydrogen and helium, little calcium and sodium. The currents and
vortices in the gaseous mass, which produce the irregularities of the
field of the star’s radiation, bring about very broad dispersion bands
in the vicinity of the lines of hydrogen, helium, ete. The darkest
parts of these bands will be displaced when, by the star’s rotation,
masses in which the density is variously distributed, pass our line
of sight. The dispersion bands of calcium and sodium, ou the other
hand, are so narrow, that the varying position of their darkest parts
cannot be distinguished from the fixed position of the corresponding
absorption lines. The constant displacement of the latter indicates
that d Orionis recedes from the Sun with a velocity of 16 km.,
Nova Persei of 7 km. a second.

According to our opinion d Orionis, therefore, is no spectroscopic
binary.

In the spectra of a great many stars oscillations and duplications
have only been observed with diffuse lines. In those cases too the
displacements are, as usual, expressed in so many kilometers a
second, because no other interpretation than motion in the line of
sight is thought of. From the above considerations it follows, however,
that the observed oscillations are very likely to be executed by
dispersion bands and not by the absorption lines; then no sufficient
ground remains for classing such stars among spectroscopic binaries
and for calculating orbital elements.

Several difficulties to which the conclusions derived from DoppLer’s
principle lead us, will then disappear at the same time. How, for
instance, are we to realize the physical conditions of the orbital
motion in such so-called binaries as ¢ Orionis, 57 Cygni, 6 Orionis
and many others, all of which are involved in nebulous matter, but

whose motion in the lie of sight is nevertheless — according to
Frost and Apams — subject to periodical variations of 70, 90,

140 km. a second, in spite of our physical notions concerning
resistant media? When, on the other hand, the observed displacements
of spectral lines, as well as the oscillations of the brightness of
similar stars, are supposed not to result from motion in orbits, but
from irregularities in thei fields of radiation, there remains nothing
astonishing in the fact that such variations often occur with stars
involved in nebulosity.

( 837.)

In order to explain certain peculiarities in the spectra of Novae
the principle of anomalous dispersion has already been applied by
H. Esert'). A characteristic of those spectra, viz. the presence of
double lines consisting of a bright and a dark component, the bright
one being displaced towards the red, the dark one towards the violet,
is very suggestively explained by this author in connection with the
theory of Sweniger. According to this theory the appearance of a
Nova results from a dark or faintly luminous celestial body entering
at a great velocity into a cosmic nebula. During this process. the
front part of the star’s surface will become excessively heated and
luminous, and a dense gaseous atmosphere will be formed, in which,
as Epert shows, the incurvation of the rays must necessarily be such
as to cause the dispersion bands appearing in the spectrum to be
bright on the red-facing and dark on the violet-facing side of the
absorption lines.

Epert expresses the opinion that displacements and duplications
of lines in the spectra of many variables of short period might be
explained in a similar way, i.e. by admitting that the radiating power
of such bodies is very unequal in different parts of their surface,
and that they are surrounded by dense atmospheres. Their rotation
will then cause us to see, as it were, the phenomena of the Novae
periodically repeated.

In certain cases this interpretation may undoubtedly account for
the peculiarities observed in the spectra of variables; nevertheless
we cannot generalize the idea without meeting with some serious
difficulties. First, it is not easy to form a clear conception of tle
physical conditions prevailing in a star, the incandescent surface of
which is supposed to contain permanently large regions radiating
very much less than the rest. The Sun with its spots may certainly
not be adduced as an analogous case. Moreover, there are plenty of
instances that in the spectrum of a variable, bright bands appear at
the violet side, dark bands at the red side of the absorption lines,
Le. just the reverse of the phenomenon presented by the Novae:
and it happens that with one and the same star bright and dark
dispersion bands change places in course of time with respect to the
average position of the absorption lines. This occurs e.g. in the
spectrum of Mira Ceti, as will appear when comparing the obser-
vations made by VoeEL and Wiisine in 1896 (Sitzunesber. der Berl.
Akad. XVII) with those made by Camppety in 1898 (Astroph. Journ.
IX, p. 31) and by Srepsrys in 1903 (Astroph. Journ. XVII, p. 841);

1) H. Eserr, Ueber die Spektren der neuen Sterne. Astron. Nachr. Nr. 3917.
Bd. 164, p. 65, 1903.

22

Proceediugs Royal Acad. Amsterdam. Vol. Vii.

( 328 )

also in the spectrum of @ Orionis observed by HuGerns in 1894 and
1897 (An Atlas of representative Stellar Spectra, p. 140), etc. In those
cases the explanation suggested by Esrrt would require the addition
of special hypotheses.

Our fundamental hypothesis that the structure of most stars is
similar to that of the Sun (it being admitted, of course, that the
stars may greatly differ as to the extent of their respective gaseous
envelopes, the average steepness of the density gradients in them,
their chemical composition, temperature, ete.) seems to admit of the
interpretation of a greater variety of facts. It makes displacements
of the dispersion bands towards the long and the short waves almost
equally probable — if we leave the asymmetry in the form of the
dispersion curves out of question and provisionally assume the directions
of the axes of the stars to be distributed at random through space.

The direction in which we see a star may be regarded as a steady
line in space, allowance being made for aberration and parallax. If,
now, the distribution of the matter constituting that celestial body
remains nearly unchanged for a long time, then after each rotation
of the star our line of sight will again pass through the same points
of the ‘optical system”, and we shall observe an accurately perio-
dical course in the star’s brightness and in the appearance of its
spectral lines. In most cases, however, currents and vortices will
cause more or less considerable alterations to arise in the distribution
of the density of the gaseous mass, and, consequently, in the com-
position of the beam of light reaching the Earth at a given phase
of the star’s rotary motion. Thus the strictly periodical succession
of phenomena is open to any degree of disturbance. The very irregular
and sometimes rapid changes in the brightness of objects like o Ceti,
SS Cyeni, « Cephei, etc. are much more intelligible from this point
of view, than from interpretations based on the assumption of violent
eruptions, large spots, or eclipses caused by dark companions. And
it is so difficult to make a sharp distinction between variables of
long period and Novae, that we should not resent the idea of com-
paring even the appearance of a new star to the sudden gleam of
a revolving coast-light when the optical system, giving to the beam
a considerable decrease in divergence, passes our line of sight.

Chemistry. — Prof. C. A. Lopry pre Bruyn presents a paper of
J. Ovum Jr.: “The transformation of the phenylpotassium
sulphate into p-phenolsulphonate of potassium’.

(Communicated in the meeting of June 25, 1904).

(Tiis paper will not be published in these Proceedings).

a

( 329 )

Chemistry. — Prof. C. A. Losry pr Bruyn presents a paper of
J. F. Suyver: “The intramolecular transformation in the
stereoisomeric a- and 8-trithioacet and a- and B-trithiohenzal-
dehydes”. (N°. 11 and 12 on intramolecular rearrangements).

(Communicated in the meeting of June 25, 1904).
(This paper will not be published in these Proceedings).

Chemistry. — Prof. C. A. Losry pe Bruyn presents a paper of
J.W. Divo: “The viscosity of the system hydrazine and water’.

(Communicated in the meeting of June 25, 1044).

(This paper will not be published in these Proceedings).
Chemistry. — Prof. J. M. van Bemmeten read a paper: “On the
composition of the silicates in the soil which have been formed

from the disintegration of the minerals in the rocks.”.

(This paper will not be published in these Proceedings).

ERRATU M.

p. 238, 1. 5 from the bottom, for “increases” read ‘decreases’.

(November 23, 1904).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday November 26, 1904.

—_—_———=00C~ ——

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 26 November 1904, DI. XIII).

CORN ai set aNG EY Se

Tu. Zieurn: “On the development of the brain in Tarsius spectrum”. (Communicated by
Prof. A. A. W. Hunrecnt), p. 331.

P. H. Scnourr: “On the equation determining the angles of two polydimensional spaces”, p. 340.

J. Carpiysat: “The locus of the principal axes of a pencil of quadratic surfaces”, p. 341.

A. Sommerreip: “Simplified deduction of the field and the forces of an electron, moving in
any given way”. (Communicated by Prof. H. A. Lorentz), p. 346.

C, Easron: “Oscillations of the solar activity and the climate”. (Communicated by Dr. C. H.
Winpb,, p. 368. (With cne plate).

W. Karrrery: “The values of some definite integrals connected with Brssrx functions”, p. 375.

H. Kamerringn Oyyes and C. Zixrzewsx1: “The validity of the law of corresponding states
for mixtures of methyl chloride and carbon dioxide” (Continued), p. 377.

Corrigenda et addenda. p. 382.

The following papers were read :

Zoology. — “On the development of the brain in Tarsius spectrum.”
By Prof. Tu. Zikuen of Berlin (Commumeated by Prof. A. A.
W. Husrecut).

(Communicated in the Meeting of September 24, 1904).
8 i

Owing to the kindness of Prof. Husrecur seven series of embryos
of Tarsius spectrum were at my disposal, among them a sagittal
series. With regard to the central nervous system of the adult animal
I refer to two short papers published by myself, Anat. Anz. Bd. 22,
N". 24, p. 505 seq. and Mon. Sehr. f. Psychiatrie u. Neurol. Bd. 14,
p. 54 seq.

' 23

Proceedings Royal Acad. Amsterdam. Vol. VIL.

The first stages of development are only known to me from
Huprecut’s paper. In the youngest embryo the segregation of the
two hemispheres has only just commenced. The next youngest embryo
shows the hemispheres already developed, namely at the lower
posterior periphery of the anterior vesicle. They are separated from
this latter by a suleus hemisphaericus which anteriorly forms a pretty
deep and sharp groove but occipitally terminates in a shallow groove.
The segmentation of the posterior brain (5 segments) is clearly shown
especially by the youngest embryo.

The following stages are very similar to those of other mammalian
orders. In frontal sections through the hind-brain the uncommon depth
of the sulcus limitans in the distal parts is especially striking. Frontally
if soon becomes smooth. A sulcus intermedius (Groenberg) is indicated.
The inner and outer labial grooves (“Lippenfurchen”) are present. The
cerebellum consists of two symmetrical lamellae, one to the right and
one to the left, joined by a thin and narrow medial part. On the
outer surface of each lamella a broad medial longitudinal groove in
the immediate proximity of the median part and a narrower but
relatively deeper lateral groove are to be seen. Corresponding to
these two grooves we find on the ventricular surface of each lamella
two longitudinal ridges and three grooves (an unpaired sulcus
medianus dorsalis, a sulcus medialis dorsalis and a sulcus lateralis
dorsalis on each side). The roof of the mesencephalon is rather pointed
and edged like a keel. The pharyngeal part of the hypophysis shows
an almost compact appendix, extending backwards and downwards.
Also two lateral continuations ina backward and downward direction
of the ventricle of the mid-brain deserve notice. The chorioid fissure is
already developed and shows some bulgings. The sulcus hemisphaericus
has also become much more marked occipitally. The sickle-fold
(“Sichelfalte”) forms a sharp but shallow groove and is enclosed by
the bifurcating ventricle of the fore-brain. The suleus hemisphaericus
lies at the right and left in close proximity of it. The Ammon fold
(Hippocampal furrow) is still entirely absent. At a stage»which for
the rest has only little advanced, the shape of the fore-brain has
already materially progressed in development. The sickle-fold is a
deep groove. In its wall the Ammon fold is noticed to which corre-
sponds on the surface of the ventricle a distinct Ammon ridge. In
an occipito-parietal direction the sickle-fold | reaches as far as the
anterior limit of the mid-brain, basally it finally terminates smoothly
in the lamina terminalis. The sulcus Monroi is very sharply marked.
On one side it terminates smoothly in the neighbourhood of the
stalk-fold of the optic vesicle and on the other side in the neigh-

bourhood of the floor fold of the primary foramen Monroi (not in
the foramen itself).

The above-mentioned appendices of the lumen of the midbrain
have already become rudimentary. The characteristic enclosure of
the cerebral part of the hypophysis by the pharyngeal part is met
with here in a similar way as in other orders of mammals. The
lamella of the cerebellum has grown thicker. Of the longitudinal
erooves, it is only the suleus dorsalis medialis and the sulcus
medianus dorsalis that are well marked on the inner surface. On
the outer surface the ridges and grooves have been almost entirely
flattened out. The posterior longitudinal bundle, the spinal root of
the trigeminus and the lower olive form already distinct prominen-
ces. The chorioid plexus of the fourth ventricle has already invagi-
nated itself considerably. The differences in level of the fossa
thomboidea have already a little more flattened out. The suleus
intermedius is lacking, the sulcus limitans is distinet. The labial
erooves have become flatter.

The next-following changements may be briefly summarised as
follows :

a. The hemispheres show a deep groove corresponding to the thalam-
encephalon, vallis diencephalica. The medial wall of each hemisphere
shows on horizontal section three ridges projecting towards the lumen
of the ventricle, which we will denote by F, S and 7 in their
order from before backwards. Between S and 7’ we find, following
up the series in a basal direction, a great diminution in the thickness
of the ventricular wall (part (). In this thinner part and much
nearer to S than to 7’ the formation of the chorioid fissure occurs
and the invagination of the plexus chorioideus ventriculi lateralis. S
and # coalesce more and more. Meanwhile from the lower posterior
part of the wall of the hemispherical ventricle the broad ridge of
the caudate nucleus arises. The lateral ventricular wall shows only
a very slight thickening, resp. elevation .V in ils posterior part,
which at higher (i.e. more parietally situated) levels, together with
the caudate nucleus marks a narrow. slit and coalesces with the
caudate nucleus at lower levels. Between the caudate nucleus and
the ridge 7’ there is a fold, which may be denoted by r. A very
shallow prominence P is also shown by the lateral wall in its
most anterior part. The further the series is followed in a basal
direction the more conspicuously a short anterior portion is marked
off on the medial hemispherical surface of the vallis diencephalica,
Which portion is not contiguous with the thalamencephalon, but is
separated from the corresponding part of the other hemisphere by

93%

( 334 )

the primitive sickle fold only. Within reach of this portion we find
now also a slight ridge projecting into the ventricle, which we shall
here designate by Q@ for briefness’ sake. To all these just mentioned
prominences of the interior wall-surface correspond only very slight
grooves of the exterior surface or none at all. Only the ridge S
corresponds in future pretty accurately to a shallow groove S’, which
must be interpreted as the fissura hippocampi. This groove belongs
to that part of S which is nearest the chorioid fissure and finally
has an almost hook-shaped bend near the neighbouring lip of the
chorioid fissure. The more in the following sections S is arched, the
more also the fissura hippocampi is deepened, while at the same
time all other ridges are levelled. Only the caudate nucleus remains
entirely unchanged. On its surface a very slight groove is tempo-
rarily seen. The grooves » and tr become gradually obliterated for
the greater part, so that the caudate nucleus coalesces with P and
T. The thinner part ¢ of the medial wall does not coalesce with
the caudate nucleus. So the bottom of the groove t finally corresponds
exactly to the border of d and 7. When we progress still further ina
basal direction, the first connection between the thalamencephalon and
the hemispheres appears immediately below the bottom of the groove
t, ie. in the former region 7’ and rapidly increases first in an
oecipito-basal direction. At the same time the lamella d@ now seems
to originate in the bend between the caudate nucleus and the optic
thalamus.

In the following sections we find @ more and more connected
with the lateral wall of the thalamencephalon, i.e. with the optic
thalamus itself.

The insertion of the lamella d@ seems to shift more and more
towards the roof of the thalamencephalon and the lamella itself
seems to become shorter. It is one of the most difficult: questions
of cerebral development whether this shifting and = shortening of
the lamella ¢ and also the coalescence of the caudate nucleus
with 7’ and the lateral wall of the thalamencephalon must be
interpreted as a secondary coalescence of originally separated parts.
My histological investigations of Tarsivs do not allow me to give
a definite answer to this question. Still more basal sections show
the disappearance of the chorioid fissure. Since @ has in_ the
meantime also disappeared, S passes immediately into the epithelial
roof of the thalamencephalon or primary fore-brain. The groove-shaped
longitudinal depression on the surface of the corpus striatum becomes a
litthe more distinct. The fissura hippocampi resp. the fissura prima is
pointed so that in the cross-sections the well-known picture appears,

(eoounn

resembling a lance point. Also the roof of the median-mantle-slit
which was at first formed by the folded roof of the thalamen-
cephalon, now becomes sharply pointed. The fissura hippocampi or
rather its lower lip marks pretty sharply, even after the separation
of the hemispheres from each other, the point reached by the for-
mation of the pallium. It is also very remarkable that SS and |S’
no longer exactly correspond to each other topographically, but that
S’ becomes situated somewhat ventrally of S. Below WS the medial
wall of the frontal brain shows a seeond feeble prominence (7. 1
and the caudate nucleus terminate in the medial respectively lateral
wall of the ventriculus lobi olfactonii and in doing so grow smoother.
After the disappearance of the fissura hippocampi the medial wall
first remains quite undivided in) many sections, but then (almost
exactly in the corresponding place) the medial terminal part of the
rhinal lateral fissure appears as a shallow groove.

Bb. The cavity of the thalamencephalon shows in its posterior parts
two grooves, an upper one #, which is continued in the lateral
groove of the midbrain ventricle (aqueduct) and frontally very soon
smoothes out, and a lower one 4, which can be traced as far as the
region of the foramen Monroi. The latter one is accompanied during
the anterior part of its course by a parallel groove w of a slightly
more basal situation. Further folds of the surface are temporarily
observed within reach of the optic stalks and of the corpora
mammillaria. I regret not to be able to decide whether 2 or w has
to be interpreted as the suleus Monroi; to me it seems more likely
that w deserves this designation.

The picture is materially completed by studying a sagittal series
belonging to about the same stage of development (length of the
embryo 11 mm).

In a section situated somewhat laterally of the medial plane, we
find as follows:

The fossa mesodiencephalica is sharply marked. Before it and for
a smaller part also in it, lies the long-streteched cross-section of the
posterior commissure. A very shallow groove which [ designate by
corresponds approximately to the frontal plane of the epiphysis.
There is no objection to designating with v. Kuprrer (cf. v. Kuprrer
in Hprtwie’s Vergl. Entw. geschichte p. 95) as synencephalon the
root part between the fossa mesodiencephalica and /. The fossa
praediencephalica is not sharply marked. It might be sought at the
place where the invagination of the plexus chorioideus ventriculi
-fertii occurs. It must be emphasised that it remains doubtful whether
this spot corresponds to the velum transversum of lower vertebrates.

( 336 )

The region above the invagination of the plexus is the posterior
roof portion of the primary fore brain. Below the invagination of
the plexus lies the lamina reuniens, the lower part of which cor-
responds to the lamina terminalis of the adult animal. Before the
praeoptic recess we find on the basal inner surface a shallow trans-
verse prominence, which is laterally prolonged into the central mass
of the striated body.

This transverse ridge of which even in the median plane we can
recognise indications, corresponds to the crus metarhinicum corporis
striati of the human embryo, described by His *). Consequently it
passes medially into the lamina terminalis. Only in somewhat more
laterally situated sagittal planes we meet before the crus metarhinicum
with a second transverse prominence, which sinks away into the
olfactory lobe. I designate it as crus rhinicum corporis striati. If
corresponds to the crus mesorhinicum of the human embryo. A crus
epirhinicum is scarcely indicated. In my opinion it is only counter-
feited by the fold of the lateral rhinal fissure. On the outer contour
we find corresponding to the border of the crus metarhinicum and
rhinicum a shallow groove, corresponding approximately to the
posterior edge of the cappa olfactoria and also approximately to the
anterior edge of the olfactory tubercle. It by no means corresponds
exclusively, as His seems to assume, during its whole course to the
crus rhinicum or mesorhinicum. | believe that its formation is essen-
tially independent of the morphological condition of the striated corpus
and has rather to be explained by thickening of the wall of the
olfactory lobe by superposition of the olfactory ganglion (cappa
olfactoria) on one hand and of the olfactory tubercle *) on the other.
To this is added a sharp bend in the brain tube in a_ basal
direction when it passes from the hemisphere to the olfactory lobe.
Moreover we must bear in mind that the lumen of the ventricle,
when passing from the hemisphere to the olfactory lobe, at first
tapers very rapidly, but then again very slowly *). Especially at the
base this behaviour is very conspicuous. Obviously this must result,
quite independently of a thickening of the wall by the striated
corpus, in a basal transverse groove. The designation “fissura meso-
rhiniea” which His has given to this latter, does not seem to me
appropriate under these conditions. I propose to speak of a vadlis
mesorhinica. A second transverse groove is found caudally of the

1) His, Die Entwickelung des menschlichen Gehirns wiihrend der ersten Monate.
Leipzig, S. Hirzel, 1904. S. 61 (ef. also fig. 34, p. 56).

2) The two borders only coincide by chance and not accurately.

8) Die Entwickelung des menschl. Gehirns ele. Leipzig 1904, p. 54 and p, 60,

Vallis mesorhinica behind the origin of the olfactory tuberele and
before the region of the praecoptic (== optic) recess. | propose to
denote if as vallis praeoptica, whereas His seems to look upon it
as a continuation of his stalk-fold) (my suleus hemisphaericus) *).
It also is brought about independently of the striated corpus by the
bulging of the olfactory tubercle on one hand and of the region of
the chiasma on the other. The niche in the ventricle corresponding to
the olfactory tubercle is designated by His (at least in man) as
the posterior olfactory brain in opposition to the olfactory lobe
s. si, which he ealls the anterior olfactory brain. To the posterior
olfactory brain he reckons particularly also the substantia perforata
anterior. Against the designation “posterior olfactory brain’ I only
object that it favours confusion with the lobus piriformis, i. e. with
the posterior part of the rhinencephalon ’*).

1) Cf. on this point also His, die Formentwickelung der menschl. Vorderhirns,
ete. Abh. d. math. phys. Cl. d. Kgl. Siichs. Ges. d. Wiss. Bd. 15, fig. 32, p. 725.

2) Concerning the nomenclature in this region I would
make the following remarks. All cerebral parts that lie
basally of the fissuma rhinalis (ectorhinalis=rhinalis
lateralis of many authors) I denote as rhinencephalon.
As lobus olfactorius (lobus olfactorius anterior of many
authors) I designate in a purely topographical sense the
anterior part of the rhinencephalon as far as it is quite
separated from the lower surface ofthe pallium. For the
posterior part of the rhinencephalon the designation
“‘lobus piriformis” might be reserved, but it is more advis-
able to give up this name entirely. Part of the lobus ol-
factorius is covered with a microscopically sharply cha-
racterised formation, the so-called formatio bulbaris.
This cover I designate as cappa olfactoria. From the
developmental point of view it corresponds to the gang-
IOneOlincionimmm® Of Hiss hie separation of the llobus
olfactorius and the pallium is exelusively brought about
by the fissura rhinalis (lateralis). Hence we must also
regard as a part of the fissura rhinalis the separating
groove of olfactory lobe and frontal lobe which is
visible in the median plane, i.e. 1 assume that the fissura
rhinalis incises at the front as far as the median plane.
The cappa olfactoria is marked off from the free sur-
Huey ie, trom the surface of the olfactory lobe that is
not scovered by the formatio bulbaris, by a limiting
Eroove, the mango cappae ollfactoriae Laterally it is
much more clearly marked than medially. In many ani-
mals (hedgehog, Echidna) the cappa olfactoria covers
almost the whole olfactory lobe. The olfactory tuberele is
also marked off against its surroundings by a shallow

( 338 )

Finally we mention as self-evident that in exactly medial sections
all the just mentioned borders, grooves and prominences are almost
entirely lacking.

Sagittal sections further prove concerning the mid-brain that at
this stage it covers the isthmus to a relatively small extent, less e. g.
than in a human embryo of 2 to 3 months. The cerebellum shows
in sagittal sections still a horn-shaped form. Of the occipitally directed
ridge, which is so characteristic for man, nothing is to be found yet.

The oldest of the embryos at my disposal had been dissected into
a somewhat slanting frontal series. Here also nothing can be detected
of a crus epirhinicum corporis. striati. Notable is the considerable
thickening of the wall in the basal portion of the medial ventricular
wall. It only becomes distinct after the lateral ventricle and the
ventricle of the olfactory lobe have already coalesced for some time’).
The thickened lower portion and the not-thickened upper portion of
the wall are separated on the inner surface of the ventricle by a
very distinct groove which can be followed almost as far as the
frontal plane of the terminal lamina. It has nothing to do with the
marking off of the olfactory lobe, as it appears considerably later
and also lies somewhat higher than the fissura rhinalis lateralis. In
the same way it is entirely independent of the ammon fold, since
it lies considerably lower than this latter. Microscopically it forms
a pretty sharp basal limit for the pallium formation. Therefore 1
designate if as margo pallii medialis internus (v. also below). In
the section where the third ventricle is visible for the first time, it
appears as a paired structure; between the two terminal points of
the ventricle a median groove incises into the ventricular roof
(cf. above).

Special notice deserves the floor of the third ventricle. Its median
groove forms a very sharp incision. The lateral parts of the floor
rise, so to say, in. three gradations. The most lateral prominence
groove, the margo tuberculi olfactorii. This groove also
is generally not so distinetly perceptible at the medial
edge, on the other hand at the lateral and anterior edges
it is well developed and hence has here often been desig-
nated as fissura rhinalis medialis s. entorhinalis. At the
posterior edge, towards the substantia perforata anterior,
it is generally rather shallow. When the cappa olf. reaches
far backward, the anterior margo tub: olf. coincides
entirely or partly with the margo cappae olf.

1!) In what follows the series is supposed to be examined from before backward,

which at the same time is the broadest, proceeds from the crus
rhinieum of the corpus striattum and forms in future the principal
mass of this ganglion. The middle elevation corresponds to the crus
metarhinicum of the corpus striatum. It disappears with the separation
of the hemispheres. In the lamina terminalis it meets the homologous
Opposite prominence. It is interrupted by the anterior commissure.
The most medial and smallest elevation only becomes visible before
the lamina terminalis and is at first very flat; then it rises pretty
steeply frontally but remains narrow. From the homologous opposite
elevation it remains separated by the shallow median floor groove.
Following the series farther in a frontal direction, the two hemis-
pheres split finally within reach of the median groove of the floor
and the most medial elevation coalesces before the foramen Monroi
with the upper portion of the medial wall, corresponding pretty
accurately to the margo pallii medialis internus.

From this description we must conclude that also this most medial
elevation can by no means be interpreted as the crus epirhinicum
in His’ sense.

The optic thalamus projects in the following sections between the
middle and lateral ridge just mentioned. More sharply developed
than im preceding stages a longitudinal groove on the outer surface
of the thalamencephalon (sulcus fastigialis thalami) is now visible
dividing the pointed crest of the optic thalamus from the broad
basal mass of this g@anelion.

It is situated somewhat higher than the above mentioned groove
z, Which for the rest is now much less distinet. 2 and ware no longer
clearly divided. Instead of them we find a broader groove, which
doubtless must be designated as suleus Monroi.

The hind brain presents no peculiarities.

Reviewing the whole of the peculiarities in the development of
ihe brain of Tarsius that have been noted in the preceding pages,
a far-reaching agreement with the development of the brain of the
primates is obvious. The essential differences ave sufficiently explained
by the rather pronounced macrosmatic character of the brain of
Tarsius. It is much more difficult t6 determine the relations of the
Tarsius brain in the descending direction. Carnivores and Unegulates
are out of the question. The development of the brain of Chiroptera
is unfortunately too little known as yet but certain analogies are
certain. Very great is also the agreement with the development of
the brain of Rodents, only one must not consider the brain of the
rabbit as the typical representative of the brain of Rodents, as is
often done. As the rodent brain in its turn is not far distant in its

a

( 340 )

development from the brain of the Insectivores, it is clear that
the Tarsius brain shows unmistakable genetic relations with this
latter also,

A more detailed account with figures will be published in the
Handbuch d. Entwicklungsgeschichte edited by Hrrrwic

Mathematics. — “On the equation determining the angles of two
polydimensional spaces”. By Prof. P. H. Scnovutr.

The problem which we wish to solve is the following:

“In a space S, with 7 dimensions a rectangular system of coordi-
nates O (YX, NX, ... X,) has been taken and with respect to this
system a space S, passing through O has been given by the equations

Up i= aie + aogae+ . ~~ 4+ Api ty;
(=U, : =) 7% —p)

supposing this space S, to have with the space of coordinates

DA Sheen sees Xp) but one point Y in common, the p angles
(),@,,- . .@ are to be determined between these two p-dimensional

spaces.”

sy means of geometry we should set to work as follows. Suppose
in the given space S, a spherical space having ( as centre and
unity as radius and thus forming in S, the locus of the points at
distance unity from ©; if this spherical space projects itself on the

space of coordinate 0 (NX, NX, .. . X,) as a quadratic space with
the half axes a, a,,. . ..a,, we get
@, = Cos a, d, — COS a,, -- - , Gp — COS a).

In an almost equally simple way the tangents of the demanded

angles are connected analytically with the central radii-vectores of

an other quadratic space. If P is an arbitrary point of |S, and @
its projection on the space of coordinate O (XY, AX, . . . X)), then
the angle PO Q=a is also determined by the relation

rn—p

SS (ajar + ag; ve
TONE =, O12? = a

+ Uy, x))?

iia = = = ; ———
OW? ee
a
il

If we consider in S, the points P? the coordinates of which are
bound to the condition

n p
= (aii 21 + aogao+... + ayia)? ==1......-. (1),

|

ee ee

( 341 )

containing only @, a,...4, and thus expressing that the projection

Q of P on OVX, X,...X,) remains in this latter space on the
quadratic space represented by (1), then the relation holds good
|
i a= .
. OR

hie erate tee h,, are the half axes of this new quadratic space, we
shall find

; 1 j 1 1

61 SO!) ee ! t/a

viet b’ es aa h ae )

2 P
Now (1) passes into the symbolic form

(A, a -{- Ae vy, -+ Aint -} Ay, & yO) al

by the substitutions

n—p u—p
Gap AE Or Opn —— Ans
—I —l

so the well know secular equation

pa pena e ieee Bae
73\5 a Aloe —A oe tote rie Aye
hoes As» Ayo = 2
furnishes by its roots 4,,2,,---4) the coefficients of the equation
of that quadratic space reduced on the axes.
From the relations

a s060 7) = =
: [a :
1
ensues immediately that the demanded equation is arrived at by
replacing in the above mentioned determinant 2 by ty? «.

Mathematics. — “Vhe locus of the principal aves of a pencil of
quadratic surfaces’. By Prof. J. Carpixaan.

1. The envelope of the axes of a pencil of conics was investigated
among others by M. Trepirscurr'). He found that the axes of the above
mentioned conics envelop a curve of class three touching the right
line at infinity of the plane in two points conjugate to the directions
of the axes of the two parabolae of the pencil with respect to the

1) Ueber Beziehungen zwischen Kegelschniltbiischeln und rationalen Curven dritter
Classe, Sitzungsber. der Kaiserl. Akademie der Wissenschaften, Bnd, LXXXI, p. 1080,

isotropic points / and ./. So the curve is of order four, i.e. rational.
This result is mentioned, in the *Eneyklopadie der Mathematischen
Wissenschaften” II], p. 101. However, if we consult in the same

work the theory of the quadratic surfaces we find no evidence of

an attempt to determine the locus of the principal axes of the sur-
faces of a pencil. The present writer makes it his aim in the
following to publish some investigations on this locus.

2. We presuppose a simpler special case of the pencil and we

take a pencil of concentric quadratic cones, of which the locus of

the principal axes is a cone the order of which can be determined.
Let us suppose to this end the section of one of the cones with the
plane at infinity; the conic formed in this way determines with the
isotropic circle a common autopolar triangle and the vertices of that
triangle determine the directions of the principal axes of the cone.
From this follows:

The principal axes of all the cones of the pencil eut the plane
at infinity in the vertices of the common autopolar triangles of the
conics situated in this plane and the isotropic circle. These triplets
of points form the Jacobian curve of the net of conics determined
by two of the conics and the isotropic circle.

So the cone of the axes is a cone of order three cutting the plane
at infinity in the just mentioned Jacobian curve.

To realize the position of the principal axes of this cubie cone
we choose a generatrix a,. If we assume a plane through the vertex
normal to @, this will cut the cone according to three rays a, a, 4:
a, and a, are normal to each other, 4, belongs to an other trieder
of axes, obtained by assuming through the vertex a plane normal
to 4,; this plane passes through a, and cuts the cone moreover in
the two principal axes 4, and 4, normal to each other.

As a rule this cone will not have a nodal genevatrix, so it will
not be rational.

3. Suppose a pencil of quadratic surfaces be given. Out of a
point QO in space as vertex we construct the parallel cones of the

asymptotic cones of the various surfaces; in this manner a pencil of
cones is formed, with respect to which we can construct the cone of

the axes. The trieders of axes of this cone are parallel to the trieders
of axes of the surfaces of the pencil.

Let further a skew cubic gy, be constructed, which is the locus of

ihe centres of the surfaces of the pencil; if then out of each centre
a trieder is constructed parallel to the corresponding trieder of axes
of the cone, the surface is formed which is the locus of the principal
axes. From this ensues:

( 343 )

The locus of the principal axes of quadratic surfaces belonging to
a pencil is a skew surface of which one of the directrix curves is
a skew cubie @, possessing a director cone; each point of the skew
cubic is homologous to a trieder of rays of the cone.

4. The order of the surface can be determined by investigating
by how many principal axes an arbitrary right line / is cut, or how
many planes possessing a principal axis can be made to pass through
/, which comes to the same thine.

Let A be a point of y,, to which three points A’,, A’,, A’, cor-
respond on the Jacobian curve C, in the plane at infinity 7”,
Let moreover 7? be a plane through /; then this cuts gw, in three points
A, B, C, to which correspond again in P, the points A’, A’,...C’,,
so to the plane ? correspond through / nine planes ?”, ?’,... /’,.

If reversely we assume a point A’ on C, only one point 4A
on g, corresponds to it. If we now make a plane ?' pass through
/, it cuts C, in three points A’, BY, C", to which correspond three
points A, B, C; so to a plane P’ correspond three planes ?. From
this ensues :

The two coaxial pencils of planes P? and /” have a (8,9)-corre-
spondence. So the number of elements of coincidence amounts to
12. From this reasoning, however, we may not conclude that the
order of the skew surface is to be 12; this number must be dimi-
nished by the number of points common to gy, and C,. The three
points of intersection of g, and P. are namely situated on C,; if
we call one of these points S, then JS, coincides with S quite in-
dependently of the position of the assumed right line /. So of the
12 planes of coincidence 3 pass through the points of intersection of
gy, and (4; so 9 remains for the order of the skew surface.

5. A full investigation of this surface O, is a very extensive
one; however, we can consider some properties and trace some
particularities. From the plan of the problem ensues that from
each point of g, three generatrices can be drawn meeting P. in
the three corresponding points; so g, is a threefold curve of (,.

The section of P?. and QO, possesses some particularities which
we shall look into. In the very first place lie on it the three centres
S,, S,, 8, of the paraboloids of the pencil supposed to be real for
the present. Out of each of those points two principal axes can be
drawn having therefore twelve common points of intersection. More-
over each of these axes cuts Cy, in two more points, which can thus
be regarded as double points. One of these points belongs however
to a triplet of pomts corresponding to a point of intersection of gy,
and (,; so it can be regarded as a point of contact of the plane

( 344 )

P. and O,. If we combine these results, we arrive at the following
theorem:

The section of O, and /?, is a degenerated curve of order nine
broken up into a plane cubic and six right lines. On this section are
situated twelve nodes, - points of intersection of the principal axes
two by two; moreover six nodes are situated on it, formed each
time by one of the points of intersection of a principal axis with
(,, and six points of contact, which are the remaining points of
intersection. So P?, is a sixfold tangent plane of O,.

So we come to the conclusion that OU, possesses besides the three-
fold curve g, still a nodal curve of which for the present we cannot
make out how it is composed, but of which the total order is 18.

The number of points of intersection of this curve with one of
the generatrices of O, can be determined. Let « be one of the right
lines connecting a point A of g, with one of the corresponding
points 1, on (,. An arbitrary plane Q through a cuts g, in two more
points B and C to which correspond on C, two triplets of points
B',, B',, B', and C’',, C’,, C’,. In like manner a plane Q' through
w cuts the curve Cy in two more points to which correspond
two points on g,; so there exists between the pencils of planes
Q and @ a (6,2)-correspondence and the number of planes of coin-
cidence amounts to 8. So all together @ is cut by 8 principal axes.
As in the general case this number must be diminished by 3,
for now too the three points of intersection of g, and C, must be
faken into account; so @ is cut by five principal axes. Each gene-
ratvix of O, has thus five points in common with the nodal curve.

From the preceding is apparent that the general section of the
surface possesses 18 nodes and 3 triple points; if we have in mind
that the latter are equivalent to 9 nodes we see that the general
section is not rational, as a curve of order nine can have 28 nodes
and the curve under investigation possesses only 27 nodes.

6. We shall consider a single case, where the surface 0, is
simplified. We have already noticed that the cone of axes is of order
three without nodal generatrix; there would be one if the net of
conics possessed in 7, a point, common to all conics. As however
to this pencil belongs the isotropic circle this case is excluded; it
may however happen that the cone of axes breaks up into a qua-
dratic cone and a plane, or into three planes.

7. We choose an example of the first case. When the cone of
axes breaks up into a quadratic cone and a plane, then the Jacobian
curve in 2, must degenerate into a right line (, and a conie (4.
This happens :

( 345 )

a. When the conies of the net pass through two fixed points.

4. When the net possesses a double right line.

We restrict ourselves in this communication to the first of these
cases; then the base curve of the pencil of surfaces is circular,

It is in the first place necessary now that the cone is degenerated
into two parts to consider the distribution of the axes on cone and
plane. If the base curve of the pencil of surfaces is circular, there is
a system of parallel planes so that each plane is cut according to
a pencil of circles. Of each surface of the pencil one principal axis
runs parallel to these planes. From this ensues :

When in consequence of the existence of a circular base curve the
cone of axes degenerates into a quadratic cone and a plane, then
of the three points A’,, A’,, A’,, homologous to a point A on g, one
lies on the right line C, in P, and two on the conic Cy. So the
skew surface O, degenerates into two other skew surfaces intersecting
each other in their common directrix g,. For one skew surface g, is
a nodal curve, for the other it is single. This already suggests that the
former of the two skew surfaces is of order six, the latter of order
three. This can be reasoned more minutely in the following manner :

Let / be once more a right line; a plane P through / has three
points A, 6, Cin common with g, to which six points A’, Ay... 05,0;
on (,, correspond; so six planes P?’ correspond to P; if reversely
we make a plane /” to lie through /, it cuts C, in two points to
which on g, two points correspond, so that between the planes ? and
?” a (2, 6)-correspondence exists. However g, has a point in common
with €,, as C) contains the point of contact of a hyperbolic para-
boloid of the pencil with 7. ; so there remain for g, two points of
contact with C, and the order 8, which would arise on account of
the (2, 6)-correspondence, must be diminished by 2; so we get a
skew surface QO, of order six. The second skew surface is of order three.

In the general case the section of 7, and (, consisted, besides of
C,, of three pairs of right lines, to be called a,a,, 6,b,, cye,. If O,
degenerates in the manner described above these right lines will also
be distributed themselves on O, and Q,. Let A’ again be the point
where g, cuts the right line C), thus the point of contact of a
hyperbolic paraboloid of the pencil; through A’ pass the two prin-
cipal axes a,a, and these belong to O,, whilst the principal axis not
lying in P, through A’ belongs to O,. To O, belongs thus one
principal axis of each of the pairs 6,6, and c,c,, so 7. is a double
tangent plane of QO, and the section of O, and 7. consists of the
conic C,, the pair of axes a,a, and the principal axes 4, and c¢,.
Of a

and a, the point of intersection a, is the node in the curve

1 2

( 346 )

of intersection of P. and ©,, one of each of the points of inter-
section of a, and a, with C, is point of contact; so on a, as well
as on a, another node is situated. Of each of the points of inter-
section of 4, and c¢, with C@, one is point of contact, the other is
also point of intersection of g, and C,. So the points of intersection
h., ¢y, dy, @ mutually are left as nodes; these are five in number
besides the point of intersection counted already a,a, belonging to g,.
So the entire number of the nodes of the section of O, and P. not
lying on g, amounts to 7. From this ensues that (, has besides ¢,,
another double curve of order seven. If we make a plane to pass through
a generatrix QO, and if we investigate how many right lines are
situated in it, we shall find the number to be 3 corresponding to
former results.

The nodal curve of order seven is thus intersected three times
by the right lines of O,.

8. The closer investigation of the surface 0, as well as that of
QM, and the other possible forms appearing by variously assuming
the pencil of surfaces, gives rise to very extensive considerations,
which are not to be included in this communication, as for the
present its aim was but to show the general properties of the
discussed locus.

Physics. — “Simplified Deduction of the Field and the Forces of
an Electron, moving in any given way.” By Prof. A. SOMMERFELD,
(Communicated by Prof. H. A. Lorentz).

§ 1. Summary.

In the “Géttinger Nachrichten” ') 1 communicated a general repre-
sentation of the field of an electron, moving in any given way,
which seems to be simpler than the formulae, hitherto known, which
are based on the work of H. A. Lorenz. This is the difference:
My formulae express the potentials by a simple integral, extending
over the past time and containing only the varying distances of the
point in question from the centre of the electron, supposed to be
spherical, whereas the formulae hitherto known are double or triple
integrals, extending over the space, charged with electricity, and
containing the distance of the point in question from the position of
the charge ata certain former time. \t may be remarked, that

1) Nachrichten d. K. Gesellschaft d. Wissenschaften 1904 Heft 2; in the follo-
wing to be cited as “first paper”.

( 347 )

P. Herrz') has published a method, though only for special cases,
equivalent to my general representation of the field, for which he
very happily uses the figure of a sphere contracting itself with the
velocity of light.

In the “G6ttinger Nachrichten” I start from rather a toilsome
Fourimr’s integral, whereas I shall now choose a very simple way,
using only the theorem of Gruen. In this way I represent the
potential in the first place by a quadruple integral, (§ 2), one integration
extending over the time, the others over the space occupied by the
charge.

Here the road divides: Hither you can calculate the integration
over the time; this leads to Lorenvz’s representation; then the inte-
gration over the charge gets rather a complicated form, relating no
more to simultaneous positions of the elements of the charge, but
to positions occupied by each element at a certain former time, or,
as you may say, relating no more to the real shape of the electron,
but to a deformed one.

Or you can calculate the integration over the charge; this leads to
my formulae; it is then no longer the integration over the time, in
general cases of motion, that you have to evaluate (§ 3).

§ 4 applies our formulae to problems, essentially known, viz. to
the determination of the field in a great distance from the electron,
and to the case of stationary motion, especially with a velocity
exceeding that of light, in order to complete the statements of my
first note and to study in detail the behaviour of the field in the
neighbourhood of the “shadow of motion”.

In the last § I pass on to the representation of force, exerted by
the electron’s own field. This force is computed exactiy for any
motion, excluding rotations, according to the principles of H. A. Lorenz.
At first sight the general formulae I am using here, seem to be more
complicated, than the more explicit formulae, I have published in
the “GOttinger Nachrichten’ *) but in reality they are very easy of
application to the case of stationary motion. For you may derive
immediately from them the known result, that the stationary motion
with a velocity jess than that of light is in every case a possible
free motion of an electron. Moreover you deduct easily the value of
the force, necessary to maintain a motion of a bodily charge with
a velocity exceeding that of light. This force is distinctly finite, even

‘) Untersuchungen tiber unstetige Bewegungen eines Elektrons. Dissertation Gét-

‘tingen 1904, § 3.

*) Nachrichten d. K. Gesellschaft d. Wissenschaften 1904, Heft 5, in the follo-
wing to be cited as “second paper”.
24
Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 348 )

in case of infinite velocity ; its value has been calculated, for the
first time, as far as I know, in my second paper. Further you derive
from the same formulae the surprising result, stated in my second
paper not only for the case of stationary, rectilineal motion, but for
any motion you like: The motion of a surface-charge with a velocity
exceeding that of light, is actually impossible, requiring continually
an infinite supply of force. In order to make this more evident,
let me point out: the more the charge is concentrated, the more the
force will increase ; in case the charge is concentrated at one point,
the force is infinite even in the case of a velocity less than that
of light.

It may seem unsatisfactory, to be restricted to the special shape
of a sphere. Only a few of the following results are independent
of this restriction, that is those, which do not contain the radius of
the electron, e.g. the approximate formulae for the field of a charge,
stationarily moved, in the case of a velocity less than that of light,
and those in case of a velocity exceeding that of light, in the regions
I and Ill (§ 4), whereas the formulae relating to the limit of the
shadow of motion, that is to the region II, depend on the special
spherical shape. Yet it is evident, that on more general suppositions,
you could probably not proceed so far.

It is kown, that H. A. Lorentz) has lately supported the hypo-
thesis, that the shape of the eleciron is variable, conforming itself
to a ‘“Havisipr-ellipsoid’, according to its momentary velocity. As
for velocity exceeding that of light this hypothesis fails, because in
this case you can hardly speak of a “Heavistpr-hyperboloid”. So I
have not been able, to use this hypothesis.

§ 2. Green’s Theorem.

All natural philosophy proves the wonderful power of GrrEn’s
theorem. We shall use it here very much like Kircunorr’*) in his
enunciation of Hvuycens’ principle.

Let gy be the scalar potential, satisfying the differential equation:

1 0
oT Pastas Fay Le be. 2 ot

where ¢ means the velocity of light, and ¢ the density of charge of the
electron; as regards the choice of units see H. A. Lorentz, Eney-
klopadie der Mathem. Wissenschaften Bd. VY. Art. 18, N°. 7.
1) K. Akademie van Wetenschappen te Amsterdam Mei 1904. Proceedings p. 809.
*) Vorlesungen tiber Mathematische Optik, 2te Vorlesung, § 1. Leipzig 1891.

( 549 )
Let v be an auxiliary function
1
i — (r—o(@—t)), »= = «= » « « (2)
fe

r the distance of a certain point in question A from any point P,
¢ the moment, for which the value of g at the point A is required,
# a variable moment of time. Our auxiliary function » then satisfies
the differential equation :

ns re Si i sn ot (8)
c

Like Kircunorr we shall suppose, that the function /(r) is
represented by a narrow prong, enclosing the area 1, viz. that /()
vanishes for all abscissae different from «=O, but in the point
v =O increases so strongly and so suddenly, that notwithstanding

+n
[Po@dst. eee ee ce k's ie, (4)
Sire

If we apply Green’s theorem in its most common form to the

functions w and v, we have:

fose-reyus= f(y se) as ay ates 1 (O)
On

The surface integral on the right hand side is to be extended over
the border of the space S and over an infinitely small surface,
enclosing the point A, in as much as this point is contained in
S. This holds good, because we shall let S finally expand into
infinity. The part of the surface integral, relating to the surface
enclosing A, is known to give:

4xp, F(—c(t—#)).
If we use on the left side of (5) the differential equations (1)

and (3), substituting in yg as variable time /¢ and noticing, that
Ov 0% |
it follows:

on aL?”
cally aka Tuga ge ——* on) ie
ce? OF! ot!
4%, F(—e¢—?t)). . . fag (0)

The second integral on the left is extended over the are of the
electron, the first on the right no more than over the surface of S.

Multiply the last equation with cdf and integrate with respect to
# from —c to +o. Thereby the first term on the left vanishes
on account of the nature of the function #. In fact this member
relates, after the integration is performed, only to the moments
24*

( 350 )

{= +o, and certainly » =O stands for ¢ = + according to the
definition of /. Further we suppose, that the first integral on the
right vanishes also, on account of the nature of the potential g. In
order to understand this, we may perform the integration with
respect to 7, as follows:

+a +a
Og dt! OP, It’ 1 Op,
v—C ———— 2c = =
on Trae faa r On

where g, means the value of g for that 7, for which the argument

. = . . ' We xe .
of F vanishes viz. t = ¢—W—. Similarly we show :
c

If the electron was at rest originally, for mstance until the moment
f,, We can in any case expand the bordering surface 6 so far, that
the value 7? just now defined gets less than ¢,. In this case g, becomes

0
the electrostatical potential of the electron and of = 0. The surface

Ot

integral in question is then reduced to the following expression

which we know by the potential theory to vanish, if it is calculated
for a surface sufficiently distant.

So you keep in equation (6) only the second member on the left
and on the right, and you have:

+2 Ries
fe aw few as = tx fo, F (—c (t—t)) edt’.

Perform the integration on the right in the way used repeatedly
and denote for short with g the value at the point A at the time ¢.
We get conclusively :

+o
say =a [ OF Aa )as: ae
=

The scalar potential is represented here by a quadruple integral,
viz. a time-integral and a space-integral.

(oo)
‘ 7 iti INTZ'S mtial-formulae the > hand
§ 3. Transition to Lorentz’s potential-formulae on the one hand,
and to those given by myself on the other.

It is tempting, to perform in (7) the integration with respect to

,

t. As JF is different from O only for the moment ¢ = +t —

'
“

we get immediately

tag=| STIS 2) ag eo oy a Pam (5) |
.

where {0} means the density, contained in the element dS at the
z
time ¢ =t — =a

As for the proof of formula (8), H. A. Lorentz ') refers to the
expression (8) satisfying the differential equation (1).

Wircuert?) and others do not start from Green’s theorem, but
from Beirram’s, which naturally is only a transformation of Green’s
theorem and, it strikes me, not a very transparent one.

Instead of performing the integration in (7) with respect to time,
it is better, to evaluate that in respect to space. Now for this pur-
pose it is necessary to add a certain supposition as to the shape of
the electron. In the first instance we suppose the electron to be an

infinitely thin spherical shell of the radius a, on which

“the charge € is uniformly distributed. So we take :
V, = 16 instead of o dS
RK, co imstead of o d:
Nu and get from (7)

+ o
4 j At’ lp st) i
aon —— > dt — fF c— “A = 6.
te eae , ag mee
—on

Let O be the centre of the spherical shell. Round OA we count
the azimuth 4 and from OA the angle 9%, so that 4, ® mean the
geographical longitude and the complement of the latitude on the
surface of the electron. Let R be the distance OA from the centre
to the point in question, it follows :

: m— R? + a? — 2 Racos3, rdr= Rasin dds,

1) La théorie électromagnétique de Maxwett, Leiden, 1892, pag. 119.

) Elektrodynamische Elementargesetze. Lonentz—Juse.panp, pag. 560, Haag 1900,

*

Qe 73
| Po. iS a= | { BAe tee O19 =2 xa" | ... sn dy=—
0 0

0
R-+a
; r dr
2 xa - ee >
leer
|R—a
so
+a R+a
& c dt’ :
er os {Foe —)d Beer)

2a, Wage 3
— on | R—a |

The lower limit |R—a equals R—a, if R >a, but equals a—R,
if R<a. Now as follows from the definition of /, the integral
with respect to 7 is equal to 1 or O, according as the argument
y—c (t—t') vanishes for one of the values of 7, contained between
|\R—a and R-+-a, or not. Write for abbreviation :

ttt;
then our integral becomes equal to 1, if
R+a>crand |R—a|<er
that is, if we can form a triangle with the sides R,a@ and er; it
becomes equal to O in the contrary case. If 4 means the number
1 or O, aecording as that triangle is possible or not, we may also write

— 2

éc (Adt

BOE DE te R
+ a

instead of (9); reversing the limits and noting that the condition
|R—a| <er is never satisfied for t< 0. we may finally put:
ao
ec (Adt

42 ——— A a Pi 8
< aie 2a R (10)

0

This is exactly the formula (17) of my first paper. There is no
question of imagining a deformed shape of electron. The formula
(10) holds good for the exterior as well as for the interior of the
electron. The only difference between the two cases is this: The
limits r,, t,, between which 2 does not vanish, are determined at a

point of the exterior by
R, —a=cr, Pie i Peet ace es” (tt 1l))

1
R,, R, meaning the distances of the point in question from the
position of the centre occupied at the time #—t, and ¢—z,; on the
other hand for a point of the interior by

ee ( 353 )

a— Rk, =cr, Gath ores, ase (LD)

It may happen, that several pairs of values 7, 1, exist; it may

also happen, that the point in question is situated so as to be an
interior point for a certain time, an exterior point for another.

It is easy to pass from the potential for a spherical shell to

that of a solid sphere, uniformly charged, which is to be divided

into spherical shells. If ¢ is the whole charge of the solid sphere, the

U

dr. : 5 waar :
charge 3¢7'?— is contained in a shell of radius 7’ and the thick-
a

sie : : bE
ness dr’. Now it is only required to substitute 7’ for @ and — 7? dr'
ae
for « and to add an integration with respect to 7’ from O to a;
then we have :

a

Bec (Cdr et,
ed Age ie 6 oe am | (IPA)
0 0

To begin with we take an exterior point, for which Rk >a, and
a certain value +r. Equations (11), in which 7’ is to be substituted
for a, show, that 4=1, if cr lies between R—z7' and R-+7, or,
what comes to the same thing, if 7’ > | &—cr|. Now two cases
are possible: | &—czt| may either be smaller than a or larger.
In the first case a triangle with the sides (a, R, er) is possible, not
in the second case. In the first case we have :

a a
f? e dr! = fr dr = : (a? — (R — cr)’),
0 |R=cr|
in the second :
a
f r dr =0.
0

If we define a quantity x by

me NSE eee a Weeiet. ag 13
= 5( A) SS) fie y % ae Jo. eo ((0h3))

according as the triangle (a, &, cr) is possible or not, we can write
for an exterior point instead of (12)

ygaty baathg x dt vi
ey C4

( 354 )

(11') require, that ¢r must be smaller than 7’ + FR and larger than
** — R, so as not to let 4 vanish. Accordingly 7’ must be > er — R
and < er-+ R. Now three cases are possible :

a). erc— R<a, er+R> a, triangle (a, R, cr) possible.

b). er + R<a, consequently cr — R< a, triangle (a, R, cr) im-
possible, a the largest of the three sides a, R, er.

c). erc— R>a, consequently cr + R >a, triangle (a, R, er) im-
possible, a not the largest of the three sides a, R, er.

In these three cases we evidently have:

a a
% 1
iz r dr = | r dr = 3 (a? —(er— RY). . «. . . (a)
0 cz—R
a +R
farar=| ite O00 Has. cl eeee neo}
0 sR
faratso. SS tied NW A LG. Wo. | (5)
0

Now if we define the quantity z by (13) in the cases a) and e),
but by

oR
nie Sug eet 6 ANI ena oe

a?
in the case 4), the potential is given by (14) for interior points as
well, according to equation (20) of my former paper.
Notwithstanding the simplicity of our quantities 2 and x, it is
easier for further purposes, to replace them by an analytical expres-
sion, holding good for all values of rt.
As for an expression of 2, we know, that

na

: ds ba 4 Eg
foneS= Eo Measrere Mey eaeCn re, we (15)

s “ a

0
according as v is positive or negative. Now:

sins (a + R —er) + sins(a — R + er) —

sin sa sin sR sin set = re

— sins (a Rf) sins («= Ren) Mae et, Mn Gacmers! (1%)
As for the four quantities
a+ R—cr,a—R-+cr, —a—R—cer, —a+h+ecr

.
three are positive and one negative if the triangle (a, 7, cr) is possible,

two positive and two negative, if the triangle is impossible. After having

ds 4 P
multiplied equation (15') by and having integrated with respect to
8

ae
s from s=0O to s =o, you get in the first case p in the second case 0;

i.e. you have in both cases:

i)
: “ ‘ ds a J
fin sa sin sR sin ser et eae easy eae 0 (0)
8 4

e
0

That is the required expression for 2; substituting it in equation
(10), we have in the case of PG eae ae

ds
= af sa sin sR sin set ete Coe aA
2274 R s

Replacing further @ by 7’ in equation (16), we get simultaneously
for the integral, contained in (12):
a

<o a
' ! 4 ds 5 . list po. .
27 dr = — | — J sinsr'r dr sin sR-sin sex =
0 8
0 0

0

no
4 (‘sin sa — sacossa , : ds
4 - - — sin sR sin sex —

Eid 2

s s
0

Therefore we can write instead of (12) in the case of bodily-charge :

BEC Psin sa: — sa.cos.6a : : ds a
2 ssh simscu—. . . (18)

2a? me (sa)? s

It may be remarked, that in my first paper the foregoing equations
(17) and (18) appear as primary and the equations (10) and (14) are
deducted from them by performing the integration with respect to s.

Moreover it is probable, that the quantities 2 and x may be
replaced in several other ways by a uniform analytical expression.

Almost the same formulae stand for the vectorpotential, if the
motion is free from rotations. Our deduction proves immediately, that

Orie * : Dp :
it is only necessary, to multiply the integrand by ——, »,_- meaning
c 7
the value of the velocity v at the time #—r. If on the contrary
the motion is accompanied by rotations, you must add to the part
due to translation another part due to rotation, where the quantities

4, z% are toe be replaced by some quantities 2', x’ rather more com-
4 { ’

( 356 )

plicated. The expressions for this are derived in my first paper and
may be derived more easily by the present method.

§ 4. The jield of stationary motion especially with a
velocity exceeding that of light.

In my first paper I have applied the foregoing representation of
the field in order to derive the well known approximate formulae
of Lixarp and Whrecuert for the field at a great distance of an
electron moved anyhow. It strikes us in these formulae, that the
cases of velocities smaller or greater than that of light seem to
differ from each other only by the sign, whereas in reality a funda-
mental physical difference must exist between the two cases: If the
velocity is less than that of light, the whole surroundings of the
electron is seized by the effects of the moving electron, if the velocity
exceeds that of light, only those points are seized which lie in the
“shadow of motion” of the electron so to speak. This incongruency
is cleared, if the roots of equation (11) are discussed, what was not
sufficiently poimted out in my first paper.

In general we note this (details depend on the special character
of the motion). If veiocity is less than that of light, each of the
equations (11) always has a positive root; if the velocity exceeds that
of light, imaginary and negative roots are possible as well; they
appear in all those points which are situated so to speak in the
front of the electron; positive roots exist only for those points that
lie in the shadow of motion; and here for each of the equations
(11) even a pair of positive roots exist. Only for a narrow region
bordering on the shadow of motion and about equal to the diameter
of the electron we have not two but only one pair of positive roots.
It follows: The approximate formulae mentioned before, which I
have derived formerly supposing two roots 1, 7, to exist, hold good
absolutely if the velocity is less than that of light; in the opposite
case they are to be replaced by O out of the shadow of motion,
and they are to be completed by a member similarly formed within
the shadow of motion.

Fig. 2 explains, what shadow of motion
means. Here the momentary position O
of the electron and its preceding path OP
is marked. Round every point P of the
path the sphere may be constructed with
the radius cr, where +t denotes the time,
in which the electron gets from that
point to OU. The envelope of these spheres

defines the shadow of motion. Evidently it is the smaller, the more
the velocity of the electron exceeds that of light. The region bordering
on the shadow of motion, which was mentioned before, is also sketched
in the figure as a narrow strip.

The foregoing general remarks are corroborated by closer discus-
sion of stationary motion with constant velocity v. The field of
stationary motion can be found exactly by a singular process of
reciprocation ') if v<c. What happens if v > c, has been explained
by prs Coupres following the steps of Heavistpe. Compared to
pres Coupres’ treatment the following is hardly new. It may merely
be pointed out, that the infiniteness of the Hnavisrpn-pus Couprns *)
solution near the borders of the shadow of motion is not real, the
formula no longer holding good in this region. The infiniteness
mentioned just now results from pes Coupres treating the case of a
charge concentrated in one point, which is passing to the limit of
vanishing dimensions of the electron. We shall adopt in general this
simplification and thereby dispense with a rigorous solution, but at the
same time we shall point out, that this simplification is not legitimate
near the border of the shadow. We suppose bodily charge, as it
will be shown later, that in case of surface charge any motion
with v > is actually impossible.

Let the stationary motion be directed towards the positive axis
of w. The system of coordinates has its origin O at the position of
the centre of the electron at the time ¢ Let the coordinates of the
point in question be «, y, 2, let its distance from O be r = Va? + aaa
so that 7 now has a different meaning from that in § 2 and § 3.
At the time ¢—r the centre of the electron was in the point — vt
of the axis of w; the distance of the point in question from this
point is

jie) (i) aaa PaaS A oe! (19)
The conditions, under which the triangle (2, a, cr) is just possible,
are given by the equations (11)

KR, —a=ct, F IS Sia, a ere (11)
in the case of an exterior point (more correctly hk >a); we can
combine (11) into

(@ >> 0n)> > 2" = (eu a= a)?

or

) V. the summary of H. A. Lorenrz in “Encyklopidie der mathematischen
Wissenschaften”, Bd. V. Art. 14. Nr. 11.

*) Zur Theorie des Kraftfeldes electrischer Ladungen. Lorentz-Jubelband, p. 652.
Haag 1900,

(358 ) |

(? — *)r?—2(veyzachrt+a—-r=0 .. . (20)
the upper sign relating to the roots 1,, the lower one to the roots
t,. The product of the two roots rt, or the two roots rt, is:

a — 7

As 7 >a in a point of the exterior, this product is negative, if
e > v, it is positive, if ¢< v. We conclude:

If the velocity is less than that of light, the two roots of our
quadratic are real and have opposite signs. Each of the two eqqua-
tions therefore has one available positive root.

If the velocity exceeds that of light, the two roots may be con-
jugate-imaginaries; if they are real, they have equal signs, and
therefore they may both be either positive or negative. Each of the
two equations has therefore either no roots or two available positive roots.

We distinguish between real and imaginary roots by consulting
the discriminant of our equation (20). The roots are imaginary if

(v a = ac)? < (x? — a) (v' — 2),
for which we may also put
(Copal <i t+a@—e).... . @Y
We introduce the abbreviations :

v 2
ales S$ 22 eR) "or yaeen

so that 9 means the distance of the point in question from the
direction of motion, § the distance of the same point, measured in
the direction of motion, from two points P, P, (see fig. 3) of which
the coordinates are «= + a8.

Replacing in (21) the sign < by =, we get
B= 91 (P= Die ae ee

This defines a cone of revolution about the direction of motion,
of which the apex lies in the point P, or P, according to the

-

meaning of § and of which the generatrices are inclined towards

the direction of motion in the angle arctg (s* — 1) * For points in
the interior of these cones, i.e. between the conical surface and the
conical axis, the roots of (20) are real, for points in the exterior
they are imaginary.

In case of the reality of roots the distinction between positive
and negative values results from the sign of the coefficient of t in
equation (20).

( 359 )
Both the roots are positive if

ve a cS 0, - a - < a a :

c? — v? p

they are negative, if

“EE — <0, 1. @. 3 ze 3"
t

a . . ,

Evidently the planes «= + BR are polar planes of the points P, 1,
{

with respect to the surface of the electron. We distinguish a back

and a front of these planes judging from the direction of motion.

ew.
Niteeesentl
pT]

Fig. 8 gives the result of our discussion. Here the points P,, P,
and their polar planes are constructed. From P,, P, the cones
K,, K, diverge, which touch the surface of the electron at its inter-
section with 4, /,; they appear in the figure as two pairs of straight
lines. We call such points region 1, for which both pairs of roots
are either imaginary or negative. Region IT consists of such points,
for which only one pair of roots (t,) is positive. Finally region I1T
is that, in which oth pairs of roots are positive. The regions I, I
and II are distinguished in the figure by different shading. We
need not concern ourselves with the interior, where the field acts
differently.

We now proceed to the computation of the scalar potential.

If »<c, we get on account of the existence of a positive root
t, and rt, (see (13) and (14)):

4 a 1 R—cr\?\ dt 23)
imme amram mai 7

ot

Here we introduce the variable

sdf
fee, | aS eee hee ER ee
a
From (11) we have
fOr oe — 7, oc tb = — NS DO a

Further according to (19) and (24)

OR vtovr
adn =(0—F)ar=(< —v — jae.

cdt du
ak R—Ble+er)’
or, if we put in the value of R from (24)
cdt du
aR et(1—p?)—au — Bu

(25)

On the right hand we must here express t by wu. According to
(24) we have between + and w the quadratic:

(au—erp=HR=(«e#+rvry?+y?4 2’.
If we solve it with respect to cr, we get
(1 — 8°) (er)? —2(au+ Bre)ct=7r —aw,
_au+pea + VY (au+ 8a) + (1B)? — au’)

o2
1—6

Ct

(26)

What was proved before for the limits 7,7, of the interval of
integration viz. that only one of the two possible values is positive,
holds as well for the values of t in the interior of the interval.

The one positive value is obviously given by the positive sign of the
root in (26). Omitting therefore the negative sign we conclude from (26):
ev(lL—P)—au—Be=4+ VY (au+ pe) + (1 — &) (??—au’) =

+ V (# + aupy + (1 — 6) (y? + 2°) 27

This happens to be the denominator occurring on the right of
(25). Instead of (25) and (21) we therefore write :

cdt du

ak V (@ + au py + (1—6?) (y? + 2’)

+1
fergie’ Rene 2 ai. aie
tJ Vetauy +0 +2)

This is the rigorous expression of the potentiat. The integration
leads to elementary functions, but becomes rather troublesome. So we
content ourselves with an approximation putting @ = 0. Then the root
in (29) becomes independent of « and we get:

( 361 )
ai
& i (ee &

————————————— (1—wu?)du= = =
Vv? +(1—B?)(y? +27) | Va? + (1—B?)(y? +27)

4ag= (30)

So the equipotential surfaces are oblate ellipsoids of revolution about
the direction of motion.

If v >c, we have to distinguish, whether the point in question
lies in region I, IL or III.

Ill). In this case there are two positive roots rt, and two positive
roots t,. We distinguish them as 1,', 7,", t,', 7," and easily see that
they arrange themselves according to their magnitude as follows:

! ! " "
Trg, May Carine Wy

Indeed if we imagine a diagram in which for the abscissa cr the
curves y= R—a, y= RK-+a and the straight line y—=cr are
drawn, the latter intersects the former curves in four points, viz.
firstly, beginning from ©, the curve y= R—a, then the curve
y= R-+a, then the curve y= Rk -+ a for a second time, lastly the
curve y= R—a for a second time. These four points belong to the
values t,', t,', t,", t,", before mentioned. Moreover the diagram
shows immediately, that the triangle (2%, @, cr) is possible only for
those values of +v, for which either t,)< t<t,' or r,"<1< 1,".

On this account we obtain from (13) and (14):

' Si

2 wk

bo 38 (, (Bae \de 8 (7, (Raa\ Nae a
© eee z0i a R as da, fa a R : c )

7 9

We introduce by (24) the variable w.

It is to be noticed in expression (26), giving t by wu, that the
denominator 1—° is negative as well as the term a@u-—+t @wvin the
numerator, the latter being so because we are in region II1. From this
follows, that the negative sign of the root in (26) belongs to the
interval of the larger values of t(t,"<C t<1,"), the positive sign
to that of the smaller ones (t,)<1t<r,'). Accordingly we have to
put in the second integral of equation (81), which is extended
between the two largest roots 1," and t,":

ede du

whereas we have to use in the first integral the former equation
(28). It follows that

( 362 )
+1
be (* : (1 — u*) du ;
tJ Vetus +0 —Fe +2)

=!

4xg=

2 ue ; er wc) u*) du
te WG aE) ree

(32)

As the two integrals equal each other, we get the following
rigorous expression :
+1
DE (1 — u*) du
4p = — : = Oca
2) Vie + anpy + 18)
=)

We dispense with the evaluation of the integral, which is likewise
possible in an elementary way as in the case » << c, and we put
a=0O. Then

2e

Ver — ey? +2)

The equipotential surfaces are now hyperboloids of revolution about
the direction of motion with two shells. Yet only one of these two
shells concerns-us as lying in region III.

Region Ill is bordered by the cone A,; in its points we have
according to (22)

dag

(34)

(« + a8 =(@ — 1) (y? +2).
If we substitute in (34) and write «=—J/z on account of the
negative sign of w in region III, we get by neglecting a’:
2¢
~ V2apia|

This value is large of a higher order than those in the interior of LI,
because it contains Ya in the denominator, but it is nevertheless
finite. Yet you can doubt its legitimacy, -as we already omitted
members of the range @ in passing from (33) to (34). However you
confirm the value (85), if you calculate the integral (83) rigorously,
which does not give much trouble for the points of the cone A,. For

dag (35)

Re ie a3 :

if you expand it in powers of Tal and restrict yourself to the lowest
oa

power, the rigorous value becomes :
8 E

Axe (36)

~ 5 Wasi)’

( 368 )

8 :
The number —=—1,6 here takes the place of V2=1,4... in

~

equation (35).
I. If the point in question is situated in region I, there are no
positive roots t, and t,; here therefore stands:
y = 0.
II. In region II, where only the pair t,', t," is to be considered,
we have according to (13) and (14) in the first instance the following
expression;

7)

Ps 3ec re R—cr\?\ dr (37
ie ae = ; R° 5 - = 4 ol)

“1
While t goes from t,' to r,", our variable w passes from — 1 to
a certain value wv, <1 and returns to — 1.
du
At the turning-point w,, for which = = 0, the square root on
at
the right of (27) vanishes also (see (25) and (27)) and changes its

sign. So the integral, extended forwards from — 1 to w,, equals the
integral, extended backwards from w, to —1, and we get instead
of (37):
up
DE (1-—wu?)du
4x9 = (38)

2S V (efaupy+-— By? +2
=i
Here it would no longer be permitted to put @ =O, as, by ne-
electing it, the denominator would become very small in the whole
of region II and the value of the integral would become very in-
exact. We content ourselves with stating, that uv, grows continually
from —1 to +1 and therefore g increases continually from 0 to
the high value given by (35), as we traverse the region II
from its front border, the cone A,, to its back border, the cone X,,.
It is hardly worth while computing the manner of this increase
in detail, as the whole of the region II (the so called border of
shadow) is of the same thickness as the diameter of the electron.

§ 5. The force exerted on the electron by us own pield, especially
if the velocity is stationary and exceeds that of light.

Whilst in general only such parts of the field would be of interest,
that are at a great distance from the electron, viz. a distance great
in comparison with the radius of the electron, we need just the field
in the interior or at the surface of the electron in order to calculate

25

Proceedings Royal Acad, Amsterdam. Vol. VII.

( 364 )

the force caused by its motion. For this purpose the approximations,
mentioned in the beginning of the last §, are not sufficient and we
have to resort to the rigorous formulae of § 3. Using the latter
always in the form of equations (17) und (18), I have sueceeded
in my second paper in computing the foree exerted on the electron
by its own field for any given translation, excluding simultaneons
rotations ; performing the integrations extended over the charge of
the electron I merely had to use GreeN’s theorem once more.

I here put these formulae less explicitly ; whilst the final formulae
there only contain one integration with respect to the former time r,
I shall here refrain from performing the integration with respect to
s as well as that with respect to rt. This means a decided simplifi-
cation of the following calculation, at least in the simplest case of
stationary motion.

Let § be the required force for the moment ¢, © the chord of the
path, traversed by the centre of the electron during the interval from
t—r to ¢, 7 the length of this chord; in the case of surface charge
we have according to the equation (48) and (50) of my second paper :

ie +) Dp
27 a7c_ 4 = PO : , ds
— ——§ =Lim | f*—(b» )}— de] — —— sinsr sinassinest —+
& = Tot Sci aes Sie isto hd ey Sa
0 0
3) nD 4
AMO) Buus ; : ds
+ Lim — yp, dt | —— sinsrsinassmmest—. . . (39)
r—=a Ot ae w | oe
0 0

in the case of bodily charge according to equation (48) and (50')

> o i) XR . ry” . °
27 ac ; ai i y / “O sinsT (sinas—ascosas\* , ds
SS FS NS (ip dt — - - sencst — —-
t 5 Tp AW 2.2 =
0

Ve? 1 a's
0

an

“sin sT (sin as —ascosas\* , ds
al — SU CST . . (40)
il as? 3?

0

Ale
+5, f° aaiehe
0

We are not allowed to pass to the limit 7=a@ in (39) before
performing the integration because the electrical intensity behaves
discontinuously at the surface of the electron, where charge is con-
centrated. The force 8 is determined by the foregoing state of motion,

that is by the path = and the velocity »—-. It would seem from
the foregoing formulae, as if all the former states, from r= 0 to
t= », contributed to the value of &; in reality only the states

during a short interval preceding the time 7 come into account, as
is seen from the more explicit value of ® given in the equations
(54) and (54’) of my second paper.

( 365 )
If the motion is rectilineal, = and » and therefore § have the
49
same direction. In this case we may replace 7 by 1, t: by »,

dispensing with expressing the direction of *. Moreover if the motion
is stationary, we get:
(Tin AT Datel Kee

at the same time the path and the velocity will become independent
of ¢ and the differential coefficients with respect to f will disappear
in (39) and (40). By taking out the constant factors and expressing
the differentiation with respect to 7’ by that with respect to 7 and
reversing the order of integration we get :

wo

DL.
Qx7a%e e—y ds, ; O sinust .
_— = , Lam | - sinsr sin as ae — sinestdr, . (41)

t

0 0

oO fo 2)
Qa7a%e_ =e? — »?_ (ds (sinas — ascosas\? (0 sinvst , Ix . (42)
aa s=— - sin est dt , (42
9¢? v a as? Ont tee,
0 0

Integrating by parts:

Dn

co) .
*O sinvst , ra dt
= sin est dx = — cs | sin vst cos cst — =
T

T ©
0

0
re A / a 1
== G5] frome tom t+ fain on].
2 C CG
0 0

2

a

Of the two last integrals the first always equals —, the second

Ls 4 ;
equals — > or + 5, according as v << or v Se (see equation (15)
-_

2
in § 3). We therefore have:

etedie Us nooons OK E
Se sin est dt = eR ETE alse 2 he veueee (40)
0 2
The result of the equations (41)—(43) consequently is: J case
of stationary motion with a velocity less than that of light we have
&=0,; this motion is a free possible movement of the electron.
Further the equations (42) and (48) give, in the ease of bodily
charge if v > c:

ico}
4aa* x ie —c* sin as—as cos as\* ds
9¢? wv a’s? s

( 366 )

The value of the integral, still to be calculated, is a mere number,
1 : , °
namely z? a8 one sees, on introducing the new variable p= as and

transforming as follows (note, that the expressions taken in [ |
disappear) :

2
[om p—p coxpyp ee |e eo cos p) if aha p(sinp—p coy ee
0 p 0 0
% 2
_| ae eee | +— ma cos p (sin p—p cos p) -f- ps sin ye ces
Pp
0 0

1 1 sin2p pcos 2p 1 (-d sin2p 1 / sin Ip il

—f{ — ———— dp = — dp= =.

4 2 pa Di 8.) dp p 8 P 4
0 0 p=o0

Thus the force exerted by its own field on an electron bodily charged
and moving with a velocity exceeding that of light becomes:

2 9 cg?
BE aie eed lee . . . . . (44)

This foree acts contrary to the movement. The opposite force is
to be exerted in order to maintain the motion and to balance the
less of energy caused by radiation. The force is absolutely finite and
and remains so for v=o. For v=c we have § = 0, a value which
is connected continually to the case of velocity less than that of light ;
for v=o we get

a Li § a
vw =
4a 4a?
: Be.
this equals the attraction of two point charges 3 im the distance a,

according to CovLoms’s Law.

Although the stationary motion with velocity exceeding that of light
is no free possible movement of the electron, yet this motion is not
impossible from a physical point of view as requiring (even if the
velocity is infinite) im every moment only a finite expense of force
and also for every finite path only a finite expense of work.

We finish by studying the motion of a surface charge with a
velocity exceeding that of light, returning to equations (41) and (48).
These give us with v > ec:

is]

4 xa v2 —c* as
= -§ = was Lim | sinrssinas—. . . « (48)
s

ee v —i.
0

( 367 )

In order to evaluate this integral we divide if into one part
from O to a quantity ¢ to be conveniently chosen and another from
¢ to o. In the second part we express the product of the sines by
the difference of the cosines:

2) )
. .

“ £
i. ; ds ? . ds 1 ds 1 ds
sim 7s sin as — = J sin 7s sinas — +- 5 f cos (r—a) s— — cos (r+a)s —.
¢ s Y
e a e

¢
s s a 8
si a

0 0 é E
In the second and third integral we introduce the new variable
of integration p= (7—a)s and p= (7-+-a)s respectively. Then the
difference of these two integrals becomes:
2(r-a) “rta

AY Dm
1 dp 1 S dp ] ‘ dp __ 1 “dp
RON) cos p == || Cavan a5
2 ) 2 ) 2 ) 2 )
= / = } = / IE I;
r—a) (ra, (r-a) ar—a
3(7-++-a) 2(r-a)
te 1 ra ae puad
es ——— dp = — log —— — | sin? ——,
2 r—a 2 p
e(r—a) e(r—a)
or, if we sum up:
ra z Z +a)
na , ds oe : ds 1 ra “ep i
sin rs sin as — = | sin rs sin as — + — log —— | sin? —
s 8 2 “ra 2 p
tt) 0 e(r—a)

Now if we choose ¢ sufficiently small, the first and second integral
of the right-hand side may be made as small as we like. Namely in
both cases we have to integrate an entirely finite function within
two limits indefinitely close to each other. Therefore for any given
y and a (r >a) there results rigorously :

ao

; : ds a am a
fo 78 sin as — = a ats or eee (4G)
8

r—a’

0

Making r converge to a, our integral becomes positively logarithmic
infinite. It follows, that the force necessary to act on the electron
in order to maintain its uniform motion also becomes infinite.

The stationary motion of an electron, charged uniformly over
its surface, with a velocity exceeding that of light, is actually im-
possible; it would require an infinitely great expense of force and
energy.

‘

{ 368 )

Meteorology. — Oscillations of the solar activity and the climate
hy Dr. C. Easton. (Communicated by Dr. C. H. Wiyp).

The parallelism between the oscillations in the “solar activity”
and the variations of the magnetic elements of the earth is certain,
and a similar parallelism is suspected for some other terrestrial
phenomena. The meteorological elements, however, have always
seemed to be subject to so many different perturbations, as to obscure
the corresponding parallelism, which most probably does exist in
this case also. Brickner has considered this point in his well-known
investigations on oscillations of climate *), but he only reached a negative
result. It is true that KOppEn and NorpMANN *), restricting themselves
to tropical countries, have established a parallelism with the 11-year
period of the solar spots for the period from 1840 to 1900, and
KOppeN’s curve aiso shows this parallelism tolerably well for the
southern temperate zone, while for the northern temperate zone
Lanciry’s bolometric observations*) give us a right to expect much
from his method for the future. For the non-tropical zones on the
whole, however, (and therefore also for the earth as a whole) the
disturbing influence of terrestrial causes would seem to be such that
the oscillations produced by a cosmical cause are entirely obscured.

The reason of this is apparent. The direct inthuence of the solar
radiation can only be visible in the genera’ temperature-curve for
regions where the difference between the seasons is neither very
large, nor their change very irregular. This reason is already
sufficient to explain why Briickner’s so-called “temperature-curve
for the whole earth’ jon which the observations in the northern
temperate zone have a preponderating influence| differs so widely
from the curve representing Rupote Worr’s “‘Relativzahlen” for the
sunspots. It appeared to me therefore that this question must be
considered from a different point of view.

Dr. W. J. 8. Lockyer has recently published an investigation *)
in which he reaches the result that Brickner’s period of 35 years
in the climate is also found in the irregularities of the 11-year period
of solar activity. He tries to show this by comparing the variable
quantity .—m (which represents the interval of time between a
minimum of sunspots and the following maximum), with Brickner’s

1) E. Britcxner. Klimaschwankungen seit 1700., Geogr. Abh. IV, 2 (1890).

2) W. Kopren. Zeits. Oesterr. Ges. f. Met. VIII, XV, XVI. —Cu. NorpMany. Comptes
rendus T. 136, p. 1047 (1903).

3) S. P. Lanetey. Astroph. Journal XIX, p. 305 (1904).

4) W. J. S. Lockyer. Proc. Roy. Society, LXVII (1901).

( 369 )

curve for the rainfall. This meteorological clement appeared to me
to be a very unsuitable one for comparison on account of its ex-
ceedingly large local variations, it might however be of some interest to
compare Bricknyr’s curve for severe winters. Further the very few
oscillations recorded by Lockyrr proved very little, in my opinion,
unless the previous oscillations of the solar activity, though less
accurately recorded, agreed at least approximately with the result.

An investigation in this ‘direction led me to a negative result with
reference to the confirmation of Bréckxnr’s climate-period, which
was suspected by Lockynr. Another very surprising result appeared
however, viz. a parallelism (though imperfect) between the .J/—m
curve of solar activity mentioned above and the curve of the frequency
of severe winters.

I do not give these curves here, since they are of no direct further
use. This parallelism suggested however two important conclusions,
viz. 1st that the J/—m curve {or preferably the deviations of the
maxima and minima of solar activity from their normal positions
as determined by Nerwcomp')| could be of great value along with
the frequency-curve of sunspots (Relativzahlen), while it appeared at
the same time that these deviations are real, at least for the greater
part, and 2°¢ that in the records about severe winters we possess
a rough but important material from which we can derive a judgment
concerning the general course of the weather in the past.

The parallelism which | found is in this sense that the more
severe cold corresponds with the larger number of sunspots (i.e.
with the greater solar activity, to retain this term). This does not
agree with Sir Norman Lockysr’s views. It is in accordance however
with the view, which is now generally accepted, that the spots do
not represent excessively heated areas. It is also well in keeping
with the result of an experiment by Savéiiér*), and with the con-
clusions arrived at by Prof. Junius in his solar theory *).

That the inequalities in the eleven year solar period cannot be
attributed in the main to errors of observation had already been
indicated by the investigations of Fritz and Loomis on the aurora ‘)
and of Haim on corrections to the inclination of the ecliptic, on the
variations of latitude, ete. °), which show corresponding inequalities.

1) S. Newcoms. Astrophys. Journal, XIII, 2 (1901).

*) Saveur. Comptes rendus T. 118, (1894).

8) W. H. Junius. Archives néerlandaises, Série Il, Ts. VI, VIII. IX.

') H. Frivz. Das Polarlicht. Leipzig 1881; — E. Looms, quoted by Hatm, A.N.
3649.

®°) J. Hatm. Astron. Nachrichten 3619, 3649; Nature Vol. 62, 1610.

( 370 )

The records concerning severe winters naturally are a very rough
material, though the remark has already been made that they are
far more reliable than e.g. records about hot summers, the excep-
tional formation of ice being an unmistakable criterion. The data
must however be carefully and critically arranged and compared.
This has already been done by Prof. W. K6prren in his well known
investigations on the periodicity of severe winters. The data as given
by Koéprrrn have therefore been used instead of those of Brickner,
which are simply taken from PILGRam.

On comparing K6ppey’s list with Brickner’s curve I was struck
by an indication of periodicity in very long periods, different however
from what K6ppeN sought (viz. the regular occurrence of severe
winters in determined, equidistant years), and also not consisting of
regular oscillations like those suspected by Brickner, but of a recurrence
of the general character of the weather in periods of about 180 years.
The distribution of the winters within each of these periods is the
same, viz. very many severe winters in the first 60 or 70 years
of the period (e.g. the 60 years following 1220, 1400 or 1580),
very few in the next 20 years, many in the following 20, few and
irregularly distributed winters in the remaining part of the period.

In accordance with what was said above this phenomenon, if
it is real, must be attributed to a secular oscillation in the solar
activity, presenting itself to us in the form of systematic variations
of the eleven-year period. For this reason I took as the basis of my
investigation a period of 16 x 11.18 = about 178 years, 14.13 years
being the normal period according to Newcoms. The available material
covers a period of more than a thousand years, viz. from the middle
of the ninth century to our own time, including the additional data
procured by K6ppen himself. The reality of this periodicity was
made very probable by a statistical investigation in which the year
848 was taken as the first of a period of 178 years. We denote
the influence of a ‘normally severe” winter on the climate
by a ‘“cold-coefficient” unity. To an exceptionally severe winter
(‘winter of first class” of K6ppEN) the coefficient 3 is assigned, and
2 to winters of intermediate severity. It then appears — taking all
the periods since 848 together — that the four sub-periods of 67, 22,
22 and 67 years have total coeflicients of 114, 15, 39 and 62 respec-
tively, Ze. 1.70, 0.68, 1.77 and 0.93 respectively for one year. These
oscillations are of such amplitude that the proportional number of
severe winters in these cold periods of 67 years (e.g. from 1561

of 22 years, the ratio in the ease of exceptionally severe winters

( 371 )

being seven to one. The mild 22-year periods have, up to the present
time, contained altogether 12 severe winters, only one of which was
exceptionally severe.

It is now important to ascertain the character of this oscillation.
On the one hand there seemed to be some indication of a period
consisting of two consecutive 178-year periods (i.e. of 356 years),
but even our material does not cover a sufficiently long interval of
time to allow any reliable result in this direction to be derived. On
the other hand the rise and fall of the curve in the middle of the
178 year period is the most characteristic feature, and points to the
possibility of dividing the period into two. In fact the period can
be divided into two halves of 89 years, which show a remarkable
curve, somewhat different on the two sides (see diagram fig. II).
Perhaps this period of 89 years may be further divided into two
periods of 44.5 years. The depression in the middle of the 89-years
period is but indifferently indicated for all the severe winters together,
but becomes more marked, if we take into account only the excep-
tionally severe winters. In the following list (Table I) the twelve
89-year periods which are available since 848, have been entered,
each divided into intervals of 11 years, corresponding to the computed
normal maxima of solar activity (according to Newcoms). The last
interval of each period contains 12 years, which however has only

TEA ie Ee oT.
Distribution of severe winters in periods of 89 years (848—1916)

(divided into intervals of 11.1 years).

hee gi ei Mae a ses le Ge
Tt 937 Se gieeedat a NES: aha et Rae 8: + tag
Wp upOen Ne aha fhe oe wpa Geen tk a, he
IV 4145 pT Ae RC, ge
V 1204 pa den he wr: @ (4
VI 1293 aE rend er ee! tg, hs
IES PRE SN EE 7. Ta ariel an
A Gy Me A a
ewe Rie 80-6 ke ia
Miviasds Mees eroe og fe gc
Metre eeeesee; akg 5. 6 4
ie Ae Os 8 Ga at

(eae)

“in appreciable effect on the total of this interval. The totals of the
exceptionally severe winters alone are: 7, 11, 6, 2, 10, 6, 3, 0. —
In order to keep the division according to whole years | have dropped
one year — the year 1560 — near a minimum of the period. The
last interval of the last period being of course unknown, I have
taken for this interval the mean of the last column, viz: 1, but
even a much higher coefficient would not appreciably alter the
general results.

The division of the period can certainly not be continued beyond
a period of 4 > 11.18 vears. In our material there ig no indication
of a regular alternation (at least not in the majority of the cases)
of cold and mild periods of 22 years. There is even less evidence of
a regularity in the succession of the 11-vear-periods.

Taking a period of 44'/, years as a basis, we can express our
results as follows:

There exist oscillations -of climate with a period-
icity of 44'/, years and multiples thereof, chiefly
thus, that one period of 1113-years contains Jess
cold than the three preceding and the three following
ones; that at intervals of 89 years there is one period
with very little cold; that in two consecutive inter
vals of 178 years the last 5S or/6 periods, of onevog
them are colder than the corresponding periods of
the other interval. This oscillation of climate corre-
sponds to an oscillation in the “solar activity’, of
a higher order than the well known eleven-year
period.

The very doubtful difference between two consecutive 178-year
periods, which was indicated above, would be in this sense that the
greatest amplitude of the oscillations in one period occurs in the
beginning of the period, and is displaced in the next towards the
middle of the first 89-year sub-period. Nothing however can be
ascertained on this point, and still less on the existence of still longer
periods,

It seemed interesting to investigate whether the 11-year variation
of the solar activity itself is shown by this material. For this
purpose the distribution of the ‘‘cold-coefficients” over five phases
of the eleven-year sunspot period was investigated, ¢/2.: im = 2 years
on both sides of the observed minimum, J/= 2 years on both sides
of the maximum, ap = ascending phase, dp, and dp, = two halves
of the decreasing phase. The observed maxima’ and minima are
taken in accordance with Newcome (the deviations from R. Wotr’s last

( 373
list *) are of no importance). The unequal duration of the phases
has, of course, been taken into account. The periods have been
arranged in four groups, A, 5, C, D in order of decreasing cold.
We then find the following values of the frequency for one year.
The values in parentheses are derived from those periods only for
which the weights assigned by Nrwcoms (Le. p. 7) to the determ-
ination of the maximum and minimum are together at least equal to 8.
pp Ata bee obit Els

Distribution of cold winters over the phases of the 11-year sunspot period.
Groups arranged in order of coldness).
t=]

A B Cc D
m 4.50 (1.67) 0.75 (0.75) 0.42% (0.125) 0.47. (0:47)
ap | 4.43 (1.44) 4.43 (4.18) 0.60 (0.50) 0.32 (0.39)
mu | 1.08 (1.50) 1.95 (4.95) 0.33 (0.95) 0.95. (0.95)
dp, 0.98 (1.09) 021 (0.21) 0.76 (0.65) 0.57 (0.57)
dp, 1.06 (1.03) 0.48 (0.48) 0.36 (0.22) 024 (0.24)

For all the groups together, and also for the two coldest groups,
we find a curve corresponding with the sunspot curve; for the two
mild groups the cold-maximum seems to be displaced towards the
descending phase. This phenomenon however does not necessarily
depend on a really different distribution of the cold winters within
the eleven-year periods, no more than the high vaiue of the minimum
in the coldest group. The reason probably is that the variations of
temperature precede or follow those of solar activity by a certain
interval of time.

This is apparent from a comparison of the temperature variations
with the curve of the Relativzahlen for the sunspots. In the diagram
fig. I, A is the latter curve according to R. Wotr, B is the second
half of the temperature-curve the period being taken at 356 years,
C is the temperature curve, if a period of 178 years is adopted, D
is the last 178-year period alone *). In presenting these curves it is
not my intention to contend that they agree with each other in
details, as is the case with the curves of the variations of magnetic
elements and the sunspots. Such a parallelism could not be expected

1) R. Worr. Astron. Mittheilungen LXXXIL (1893).
2) In drawing the curves B, C and D no other process of smoothing was

: : a--26-Fe
applied than that given by the formula eed The cold-coefficients have

been taken for each year, only in the case of D for every two years on account
of the small number.
The scale is: coefficient unity = 40 Relativzahlen.

( 374 )

a priort having regard to the material on which the curves are based.
I only wish to show that the general character of the temperature
curve is the same as that of the curve of solar activity. It is
important to remark that the mean curve B is more similar to A
than D. The deviations in position of the variations of temperature
relatively to those of solar activity seem to be at least partly system-
atic. On comparing the temperature curve and the solar curve in
the separate eleven-year periods we must however not only consider
the greater or smaller number of spots, but also another phenomenon
(which, it is true, is certainly connected with the abundance of spots)
viz. the deviations of the minimum and maximum from the normal.

This phenomenon appears already vaguely in the correspondence
of the values J/—m with the number of cold winters, to which
I referred in the beginning of this paper. It is shown more clearly
however in fig. II] on the plate. For the last 17 periods of 11 years
the dotted line represents the acceleration (—) or retardation (+)
(expressed in years) of the observed solar minima and maxima,
compared with the normal period of 11.13 years. The corresponding
cold-coefficients are represented by the continuous line having four
different ordinates corresponding to the four degrees (4, B, C, D) of
cold. On the whole we find that to an acceleration of the solar period
corresponds a more intense cold, and to a retardation a greater heat.

This rule also holds for the individual periods, provided the
deviations are large. Combining this result with the distribution of
cold-coeflicients over the phases of one and the same eleven-year period
which was mentioned just now, and with the larger oscillations found
above, we find the following rule, which is presented as a hypothesis :

On the whole the oscillation of terrestrial tem-
peratureis accelerated relatively to the eleven-
year variation of solar activity in the colder part
ofthe larger period, and retarded inthe hotter
part. In cold 11-year periods the centre of gravity of cold is
near the minimum of sunspots, and often there are very cold years
preceding the minimum; in warm periods it is situated near the
maximum or thereafter; in periods of intermediate character it falls
between the minimum and the maximum.

For the individual cases we find that, in cases of considerable
retardation or acceleration of the solar minimum or maximum, the
centre of gravity of cold tends to be exceptionally retarded or
accelerated. If the minimum of a sunspot period occurs very early
and the subsequent maximum is retarded, the cold period also is
largely extended.

( 375 )
Mathematics. — The values of some definite integrals connected
with Bessel functions’. By Dr. W. Kaprnyn.

The integrals referred to are

* cos (w sin 8) — cos (w si sin ) 4

cos O +- cos .
oC Ea (a sin A): sin 8 — sin (a sin p) sin P dA,
css 7) + co cos gf)
0
Pa eNOS Dee EE EEE 5,
cos 4 + cos «p
0
ox
s= * sin (w cos @) cos A — sin (w cos g) cos dé.
cos + cos @

If in these integrals we insert the wellknown equations

cos (7 sin 0) = I, + 27, cos26+2T7,cos404+ .

sin (x sin @) sin 6 = I, (1 — cos 2 6) + I, (cos2A—cos4O)+...
cos (w cos 0) = I, — 21, cos2A+2F,cos4O—...

sin (a cos A) cos 0 = I, (1 + cos 2 6) — I, (cos2A+ cos4 Ot...

where /, stands for the Bessel function /, (7) of order p, and if
we write

con 2. 8 — eos 2n@
Aan a dg
cos @ + cos

it is easy to find
Bea An ek, Ae ae 2D A te
Opa le a) Me t= (= AN
[Se PEAY IO UP ea es ea
Se a(n ear A en ay of . .
In order to determine As, I notice that
. cos 2nO — cos 2ng ; :
(a) sng ace Oise Tk @ = — sin 2ngp + 2 sin (2n — 1) gy cos 6 —
— 2 sin(2n — 2)pmeos264...
+ 2 sin p cos (2n — 1) 4

This formula can be proved as follows :

( 376 )

If we multiply the 2"' member of this equation by cos 6 + cos @
we find
2n—1

| < cos @ = sin(2n- ly+ > > ts 1) |sin(2n-p-1)g4-sin(2n-p-1)y |cosp 6+
=

+ sin pcos 2nd

2° member

2n—1
| cosy =-sin2ng cosy + a 1)jp—" [sin 2n-p + 1) y+ sin(2n-p-1)@ |cos ph;

so the sum becomes
sin (2n — 1) gy — sin 2n @ cos P+ sin & cos 2n 6 = sing (cos 2n 4 — cos2ng).
From the formula (a) follows immediately

sin 2n p

Ao, = — 2% A
sin &
If we replace this value in the expressions just found, we arrive at
da
P= — —— [L, sin 29 + I, sn 4g 4+ J, sn 6p + .. |:
sin @
Aa a
= [(Z, — Z,) sin 29 + (I, — I) sin 4p +..]
sin &
= 4a [Z, ose + I, cos 3p + I, cos 5e—.. |,
4a
i [J, sin 2 p — I, sin 4— + T, sin 6p — ..],
sin Pp
sS=— [Z, sin 2g — I, (sin 2g + sin 4g) + TI, (sin 4— + sin 6@)..],
sin &
Aor cos Pp ee
=— —[ 1, sng — I, sin yp + 1, sin dy —.. |.
sin Pp

Moreover from the formula (@) we can deduce another result:
When we develop
cos 2nP — cos2np _
cos 4 -+- cos @

then we know that

a, + a, cos 6+ a, cos2A+ ...

tol

2 (cos 2nO — cos 2ngp
a; - --
5 cos G + cos @

cos pO dé
Fig

0
If we compare this to the equation (@) we arrive at
—1 . sin (2n —j
Ae (1) ee

sim

for p= 0,1, 2,., (2n—1), whilst for greater values of p we have
(1h = (0).

(5877)

‘Physics. — “The validity of the law of corresponding states for
j mivtures of methyl chloride and carbon dioride.’ By Prot.
H. Kammriincu Onnes and Dr. C. Zakrzuwskr. (Communication
N°’. 92 from the Physical Laboratory at Leiden by Prof.
Dr. H. KAmertincu Onnes) (Continued).

(Communicated in the meeting of October 29, 1904),

§ 5. Further approvimations. By second approximation we get
by taking also 5° into consideration :
22

Pi? B a
ie pe (oe pe) 1 (py — Pa) (p, + 2p,). «= (2)

Ps; At

If, however, we take ° into consideration, it is more rational to
do the same with C, and we have:
Piri
Ps Us

R3

Bie
—1 =e CPs) ae (Pi—P)

CA
p+ (1S +r

CA
According to the reduced equation of state aS is about ‘/, for

methyl chloride at 20°.
The repetition of the calculations of § 2 yields:

TABLE III. Second virial coefficients of mixtures of carbon dioxide
(v = 0) and methyl! chloride (a = 1) to the 2"¢ and 3 approximations.
approxim. com posit. x 7 from la. II 1 from Ta. Ill = mean | Bie

2nd 1 _ — 0.01698 — 0.01710 — 0 01704

a 3rd — 0.017114 | — 0.01723 | — 0.01717 | — 0.01978
2nd 0.6945 — 0.01246 — 9.01269 — 0.01258 |
3rd — 0.01260 — 0.01282 — 0.01271 — 0.01464
Qnd 0.5030 — 0.01001 — 0.00976 — 0.00988
3rd — 0.01009 — 0.00983 — 0.00596 — 0.01147
2nd | Oh Yh — 0.00572 — 0.00545 — 0.00558 |
rd = 0.00575 | — 0.00547 | — 0.00561 | — 0.00646

_ The repetition of the calculations of § 3 and § 4 with these
corrected values gives:

1) Here are two determinations of Kresom agreeing with I and II and Land IIL,
calculated according to (2) and (8) (data see Comm. N°. 88).

( 378 )

(Cl Me.)
Bae = — 0.01983

(Cl Me, COs)
(12) By 9° = — 0.01005

(C0.)

B,,° = — 0.00644

The comparison of the values found with those of the quadratic for-
mula and the law of corresponding states (using Brinkman’s')
critical data) gives:

TABLE IV. Deviations of the second virial coefficients of
/ mixtures of carbon dioxide (@ = 0) and methyl chloride (v= 1)
from the quadratic formula and from the correspondence.

|B from law of corre- B quadrat. form.| B observ. | B observ.
a | sponding states. — #corresp. | — B quadrat. | — B corresp.
1 — 00256 | +4 0.00173 | +4 0.00005 | + 0.00178
0.6945 | = 0.01870 | $-0,00027 0.00021 | — 0.00006
0.5030 | —00135 | — 0.00029 | + 0.00016 | — 0.00013. |
| 0 | — 0.00650 | + 0.00006 | — 0.00002 | + 0.00004

In the conclusions of §3 and §4 no further modification is brought
about by these further approximations. It is noteworthy that the
deviation now only applies to the methyl chloride.

§ 6. Comparison of the results with those of Lepuc and Cuarruts.
The compressibility of methyl chloride with small densities has
been examined by Lepvc*) in collaboration with Sacerporr. He does
not give the observed results but the quantity calculated from it
yn AN, Yves

Pa SCN ae OB ee eee
emerge! Laie oe mat CV ay

pe Op
(ClMe) . (Cl Be.)
In order to pass from ow 3p tO Bes we have calculated
(Cl Me.)
0B ? : A ; .
yy according to the formula given in § 4 for the reduced value of

1) Brankaan, Thesis for the doctorate, Amsterdam 1904.
2) Recherches sur les gaz, p. 82.

t 379°)

(Cl Me.)
? Cl Me

OB
BB, and found = O.0001194, so that Bs — 0.02026,
(
while oe =1+ 16¢4,. Thus we find
xz A, =0.01814 (K. O: and Z.)
while aa Si == (OWI (bp)

; me ‘ ~ les 3
was found by Lepuc (Le.). The uncertainty of + contributes
, (
probably but little to the difference. It is noteworthy that in con-
tradiction with our result the value of 4 derived from Lepuc’s

(Cl Me)
observations B,., = — 0.0215 perfectly agrees with that according
to the law of corresponding states: — 0.0216.

Measurements on the compressibility of CO, have been made by
Cnappurs ') at 20°C. He found (mean from some determinations with
about the same pressure each time):

Tp =1260.670 mm. pu = 1312485

Il Acie Siew. 1313826
Ill 992791 = 1314981
By third approximation yield
(COs)
(ae
I and IL: —— = — 0.005309
ee assumed mean — 0.005296,
I} and III os — 0.005282
T and Il Ps == (005297
(COs)
from which follows Bo = — 9.006100, whereas according to
(COs)
Knrsom B..° = — 0.00646.
From the observations of Lepuc on carbon dioxide KiEsom derived
(COs)
By, = — 0.0059.

HU. Conditions of coexistence at low temperatures.

§ 7. Determination of the begin condensation pressure at — 25°.
In the introduction we mentioned a determination of the begin
condensation pressure at — 25° for testing the results obtained
with the y-surface. This determination was made with a mixture
with the composition « = 0.5042 of methylchloride.

1) Tray. et Mém. du Bureau Intern. des Poids et Mesures, t. 13, 1903.
26

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 380 )

A condensation was seen on the mirror of the dew-point apparatus
(see § 1 and Comm. N°. 92 p. 233) at a pressure of 157.4 cm. mercury
and disappeared at 154.6 em. As mean we may take 156.0 cm.,
and the accuracy may be put at 1°).

§ 8. Determination of the end condensation pressure. In the first
place the condensation pressure (vapour tension) of pure methyl
chloride was determined with the aid of the small piezometer (§ 5
Comm. N°. 92). At — 25° we found 72.9 em. when the piezo-
meter was filled to the capillary tube, 72.8 cm. when the liquid
had nearly all evaporated. In both cases the liquid in the piezometer
was stirred. The agreement of these values speaks for the purity of
the methyl chloride. At —37°.4 the vapour tension was found to
be 42.7 em.

In order to determine the end condensation pressure of the mix-
ture with « = 0.5042 of methyl chloride the temperature of the
piezometer had to be lowered down to — 38°.5; at higher tem-
peratures condensation took place at other places in the apparatus.
(Cf. § 5 of the Comm. N*. 92 “On the determination of the condi-
tions of coexistence etc.” It may further be observed, that no
arrangement was applied to heat the press tube and the capillary
tube above the ordinary temperature). For the determination of the
conditions of coexistence (according to § 5 just mentioned), are
wanted: in the first place the observed end pressure p,_,,., for which
6.13 atms. was found. Then the volume J”, of the vapour with the
composition ry" The vapour oceupied 0.488 cc. under 6.13 atms.

at —38°.5. Finally the volume J’, which vapour and liquid phase
together would occupy in normal circumstances; by blowing off in
the volumenometer we found 347.7 ce. at 20° C. and 760 mm. (reduced
according to the law of Boyir, Gay-Lussac).. The composition lyr
was determined with van per Waats’ hyperbolic formula, which
yields «= 0.0375. From these data follows (see lc. § 5)

x = 0.5084, — 6.13, t= — 88°.5.
“eT ).5084 Pret atest ate jee)

The circumstances were not favourable for an accurate determi-
nation of the correction. It appears, however, to be so slight, that
we may safely assume the composition to be accurate down to 1 °/,.

§ 9. Comparison of the conditions of coexistence with theory. For

this purpose it is required in the first place to know the vapour tension

of methyl chloride and carbon dioxide at — 25° C. and 38°.5 C.
We have calculated the coetticients for methy} chloride in the formula
B

of Dupré and Raykine in. p= A — aa CluT with HarrMan’s value

e561 }

(8.48 aims. at 9°.5C.)'), and the two values found by us (§ 8). Extra-

polation to — 38°.5 C. gives 0.53 atms. For carbon dioxide follows
from KuUBPNEN’s observations?) at ase sp == Gc aims: ‘at
maa
- Sasa Ome p = 10.4 alms.
mar
From our model for 25° C. (Comm. Suppl. N°. 8) follows for

methyl chloride at —25° C. = 0.59 atms. and for carbon dioxide

)
i mar

Prane == 16 atms. (instead of 0.96 and 16.5). The begin condensation

pressure of the mixture « — 0.5084 is according to the model 1.30 atms.:
instead of 2.05 atms., as was found by us.

The insufficient concordance on the side of methy! chloride shows
once more clearly that for methyl chloride and ether the agreement
required for the validity of the law of corresponding states leaves
much to be desired.

In order to judge about the degree of deviation of the mixtures
from this law, we must first free ourselves as much as possible from’
the deviations of the separate components. We have tried to do so
by raising the pa-curve 0.4 atms. Then we get for Prnay 0-99 atms. for

methyl chloride, 16.4 atms. for carbon dioxide, and p= 1.7 atms. for
vr

the examined mixture. The remaining difference with the observed
P= 2.05 atms. is, no doubt, partly due to the faet that the surface,

had not yet been constructed with sufficient care, to determine’
with sufficient certainty the place of the points of contact when
the plate of glass is rolled over the ridge and over the convex part.
More points should have been calculated if we wished to render the
model sufficiently precise for an accurate determination.

This uncertainty when rolling, makes it also doubtful whether on
account of the law of corresponding states, with an accurately executed
theoretical model, a straight line would really be found for the liquid
branch of the border curve in the pa-diagram, as was derived from
our model. We got the impression, that a curve of the same character
as that derived by Hartman in his experiments and from his model,
ought to be found also by us, on our model (Comm. Suppl. N°. 8),
but we had not enough data for ascertaining the deviations from the
straight line, and had therefore to adopt the straight line as the
simplest approximation.

That the experimental liquid branch of the border curve in the

1) Comm. NY 64.
*) Phil. Mag. Jan. 1902.

( 382 )

pe-diagram at — 25°C. is not straight, could not be proved with our
mixtures and our apparatus, as has been said. The measurements at
— 38°.5 C. raised it however beyond doubt, that the line under con-
sideration is not straight at

38°.5C., but is for the greater part
convex to the v-axis. According to the straight line py would have
to be 5.388 atms. for «= 0.5084, instead of the value 6.13 atms.
found. This difference is much too large to be accounted for by errors
of observation or of the value ascribed to the composition. There is
therefore no doubt but that the liquid branch is curved in the same way
at — 25°C. In concordance with this deviation is also the fact that the
begin pressure at the same temperature does not agree with the
hyperbolic function for the composition (VAN DER Waats, Continuitat
II, p. 154). For according to this formula the begin pressure would
have to be 1.78 atms. instead of 2.05 atms., as was found (§ 6).

It appears from all this, that from the conditions of coexistence
deviations from the law of corresponding states follow for isothermals
of mixtures of methyl chloride and carbon dioxide, which become
very distinet for Liquid densities and low temperatures.

The comparison of the conditions of coexistence, derived from the
law of corresponding states by means of the y-surface, with the
really observed data, is an indirect method for judging about the
deviations from the law of corresponding states for the isothermals
of mixtures. It was our purpose to give an instance of the application
of this method and to do this at low temperatures.

In order to ascertain the accurate amount of the deviations new
measuremens at low temperatures will be required, and for the
temperature of observation a y-surface will have to be constructed
with more precision than that devised in Comm. Suppl. N°. 8 for
the sake of preliminary elucidation and for the sake of corrections.

Corrigenda et addenda to the paper “On an equivalent of the
Cromer Forest-Bed in the Netherlands,’ in this Vol. of the
Proceedings, pag. 214—222.

Page 214, line 14 from top and line 10 from bottom, for “mass”
read “massif.
215, ,,. 4 from bottom, after “Belfeld” add “and”.
SEP) ae Oo es, ie for “14? read “13.5”.
215, ,, 14 and 16 from bottom, for “her” read “its”.

216, ,, 2 from top, de/e “and probably also to the east’.

( 383 )

Page 216, line 4 from top after “removed” add “by denudation”
y» 246, 5 6 , 5, jor “under layer” read “underlying”
SLO Ole Oe ade. even”

» 218 , 3 4, &, «for “uningured” read “uninjured”

» 218, , 14 , 4, after “strong’ add “undoubtedly arti-
ficial”

yA je Ol, -,,. for “straighter” read “rather straight”.

» 218, ,, 4 from bottom, read “process of the development of”

» 218, bottom-line, after “level”, add “it”.

» 219, line 19 from bottom, after “possession” add “of Rhino-

ceros eftruscus,”’

Pee Oke Ge ° for “an” read “two”

Pe Dien ear dele vl after “species” add “of”.

pe CDG” aes Cees eee after “of” add “Cervus Sedqwichii,”

» 220, ,, 18 ,, top, for “indignous” read “indigenous”’.

(December 21, 1904).

oapnieens

a)

a

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday December 24, 1904.

ee

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 24 December 1904, DI. XIII).

C@ ay LERETANG AES:

D. J. Korrewne and D. pe Lance: “Multiple umbilies as singularities of the first order of
exception on point-general surf2ces”, p. 386.

A. F. Hotreman: “On the preparation of pure o-toluidine and a method for ascertaining its
purity”, p. 395.

Miss T. Tames: “On the influence of nutrition on the fluctuating variability of some plants”.
(Communicated by Prof. J W. Mo tt), p. 398. (With one plate).

J. W. Motz: “On the nuclear division cf Fritillaria imperialis L”. Results from Dr. B.
Sypkens’ thesis for the doctorate, p. 412.

J. M. Jayse: “An investigation on polarity and organ-formation with Caulerpa prolifera”.
‘Communicated by Prof. HuGo pr Vries), p. 420.

P. Zeeman and J. Grest: “Donble refraction near the components of absorption lines mag-
netically split into several components”, p. 435. (With one plate).

H. A. Lorentz: “The motion of electrons in metallic bodies” I, p. 438.

S. Brox: “The connection betweea the primary triangulation of South Sumatra and that of
the West Coast of Sumatra”. (Communicated by Prof. J. A. C. OupEmans), p. 453. (With one
plate).

Cu. M. van Deventer: “On the melting of floating ice”. (Communicated by Prof. J. D. yan
pER Waats!, p. 459.

J. J. Buanxsma: “On trinitroveratrol”. (Communicated by Prof. H. W. Bakuvis Roozesoom),
p- 462.

S. Trmsrra Bz.: “On W. Marckwatp’s asymmetric synthesis of optic:lly active valerie acid”.
(Communicated by Prof. H. W. Baxuvis Roozrsoom), p. 465.

A. H. W. Aten: “On the system pyridine and methyl iodide”. (Communicated by Prof.
IL. W. Baxuvis Roozesoom), p. 468.

J. Borseken: “The reaction of Frrepe, and Crarrs”. (Communicated by Prof. A. F,
_ Horieman), p. 470.

J. E. Verscnarrerr: “The influence of admixtures on the critical phenomena of simple sub-
stances and the explanation of TrE1CHNER’s experiments”. (Communicated by Prof. H. Kamer-
LINGH OnNES), p. 474. (With one plate).

J. A. C. Otpemans: “Determinations of latitude and azimuth, made in 1896—99 by Dr.
A. Payyexorx and Mr. B. Posruumus Meryses at Oirschot, Utrecht, Sambeek, Wolberg,
Harikerberg, Sleen, Schoorl, Zierikzee, ‘Terschelling (the lighthouse Brandaris), Ameland,
Leeuwarcen, Urk and Groningen”, p. 482.

Errata, p. 485.

The following papers were read :
27
Proceedings Royal Acad. Amsterdam. Vol. VIL.

_ |

( 386 )

Mathematics. — “JMultiple umbilics as singularities of the first
order of exception on point-general surfaces”. Communicated
by Prof. D. J. Korrrwre and Mr. D. pe Laner.

(Communicated in the meeling of November 26, 1904).

1. Let us suppose a point-general surface, i. e. general if considered
as a geometrical locus of points, in whose Cartesian equation parameters
appear; then for a continuous change of those parameters also the
surface will in general vary continuously in shape’). If then we fix
our attention on any kind of singular points, plaitpoints, umbilies,
etc. appearing on a point-general algebraic surface in finite number,
it may happen during the deformation that two or more of those
singular points coincide. Such a point where this takes place may
be called a twofold or multiple singular point of that kind.

Now such a coincidence may generally occur, as the results tell
us, in more than one way. For some of these ways the coincidence
depends on a single relation between the coefficients of the Cartesian
equation being satisfied, whilst for others it depends on more suchlike
relations. The former cases belong to the singularities of the first order
of exception, the latter to those of a higher order. It is only with
the former that we shall occupy ourselves in this paper’).

For plaitpoints the singularities of the first class, which must be
regarded as multiple plaitpoints, were investigated by the first
mentioned *). Two entirely different kinds of double plaitpoints were
found (the homogeneous kind and the heterogeneous one) ; furthermore
the points of osculation proved to be threefold*plaitpoints, the nodes
of the surface twentyfourfold plaitpoints.

It seemed advisable to make an investigation also for other singular
points. This we have done for the umbilics. The results obtained
are communicated in this paper. For proofs and more elaborate
considerations see the dissertation by the second mentioned Mr. D.
DE LANGE issued recently.

a. The double umbilice at finite distance.

2. If we place the origin of a rectangular system of coordinates
at an umbilic and if we use the tangent plane in this point as cy-plane

1) See for more general considerations of the same kind as follow here: ,Ueber
Singularitiiten verschiedener Ausnahmeordnung und ihre Zerlegung’, Math. Ann,
41, p. 286—307 (1893).

2) See for the reason why these are asking’ in the first place our attention the
paper just quoted, on page 287. ;

3) D. J. Korrewec, ,Ueber Faltenpunkte”, Wien. Ber. 98, p. 1144—1191, (1889)
also Arch. Néerl. 24, p. 57—98, (1890).

( 387 )

the equation of the surface can be written in the form:

z= 6, (#7 + y*) + d,a* + day + dry? + dy*® +ea'+ . (1)
By a slight deformation we arrive for the new surface at the
aie

=a + Be + By + ¢,2* + yey + (¢, + 7s) y? + de® + d,a?y . (2)
me the Greek letters represent small quantities, which can all be
regarded as of the same order, namely of the order of the small
variation which an arbitrary parameter appearing in the coefficients,
has had to undergo. Also the Latin letters must be regarded as
having been varied somewhat, which is however immaterial.

Let us now calculate by means of the wellknown conditions:

Oz 072 =

: f0z\2 dz Oz
+(55) da” dy ae

the position of the displaced umbilic ; then we shall find after neglecting
all terms which are small with respect to those which are retained,
the two linear equations:

Y,+2d,a + 2d,y=0; 7, + (d,—3d,)@ + (Bd,—d,)y =0 . (4)
from which in general we deduce without difficulty the sought for
displacement.

This however is different when the determinant

d, — 3d 3d, —d
BS? tg |= 4-8 +ad) (5)
3 ‘8

(3)

disappears. In that case no finite values satisfy the linear equations
(4). This proves, however, only that the displacement of the umbilic
has become of a lower order than the quantities indicated by the
Greek letters and that therefore the terms of the second order in «
and y must be included in the equations (4). If we do so we obtain
by comparing the two new equations and eliminating the linear
terms the new equation :

(d, —3d,)y,— 2d,y,+[12d,e, + 3(d,— 3d, Je, —2d,e,—8c,*d, x? +

+ [6d,¢, + 4(d,—3d, Je, — eee ae _
Be deaetSe,%dyly2—=0) 0 sea Ss Ss) (6)
which must be combined with one of the equations (4).

This equation (6) is of order two in x and y, from which therefore
ensues: 1 that the displacement becomes of order 4 with respect
to that of the Greek letters used in (2), 2'4 that the umbilic originally
situated at the origin of the system of coordinates on the surface (1)
27*

a

( 388 )

is broken up into two at the deformation of this surface, which two
umbilies diverge in general, real at a variation of the parameter in
one sense, and imaginary in the other. So we have to do with
a double umbilic, namely with such a one at whose effective ')
occurrence a transition takes place from the real to the imaginary.

3. Before considering the further properties of this double umbilie
we wish to observe that the condition A, =O was already known
us an important characteristic. It characterises namely the case
of transition between two of the three general kinds of umbilies
distinguished for the first time by Darsovx*) according to the manner
in which the lines of curvature bear themselves in their neighbourhood.

For the first kind, see fig. 1, lines of curvature are starting from
the umbilic in three different directions — namely in each direction a

Fig. 1. Fig. 2. Fig. 3.

single one, which we have represented by a right line because
its curvature depends on the terms of higher order of the equation
(1), to begin with those of the fourth. Those three directions have
the property that they cannot be represented in one quadrant, i. e.
each of them lies inside the obtuse angle formed by the two others.
For this kind A, > 0°).

For the second kind, see fig. 2, also lines of curvature start from
the umbilic in three different directions; these directions are however
such that one of them falls inside the acute angle formed by the
two others, so that the three can now be contained in one quadrant.
Moreover an infinite number of lines of curvature — five of which, the
right line included, are indicated in fig. 2 — start in the firstmentioned
direction which might be called the middle one. For this kind A, < 0.

For the third kind, see fig. 8, only one line of curvature starts
from the umbilic, the right line of that figure. The two other directions

1) See for the meaning of this term page 289 of the paper quoted in the first note.

2) G. Darsovx. Legons sur la théorie générale des surfaces. Quatriéme partie.
Gauthier-Villars, 1896, p. 4483—465.

8) This characteristic K,>0 means moreover as is proved in the dissertation
in a simple way, that the lines of curvature turn in the neighbourhood of O every-
where their convex side to the umbilic, but for A, <0 on the contrary their con-
cave side.

( 389 )

of departure have become imaginary. For this kind too K, < 0. To

distinguish it analytically from the preceding one we can notice the
f) ? . o

sign of the discriminant of the cubic

d,n* + (2d, — 3d,) n* + (8d, — 2d,)n—d,=0. . . (7)

which proves to serve for the determination of the three directions
of departure. If we call this discriminant A, chosen in such a way
that for A, > 0 the three roots are real, we have for the jirst kind
K, >0, Kk >0, for the second K,<0, K,>0, for the third
K, <0, K,<0. A fourth kind K, >0, K,<0 does not exist,
because as is demonstrated also algebraically A, > 0 includes A, > 0.
4. As is apparent from this explanation the double umbilie forms
the ease of transition between the first and the second kind, for which
case of transition A, must of necessity be equal to nought, and A, > 0.
The form of the lines of curvature now becomes very simple as long
as one confines oneself to the approximation which has led to the
figures 1, 2 and 3. Out of the differential equation

ly? |
[d,e + d,y] [: es. i) i [(d, — 3d,) @ + (8d, —d,) 9] : == (8)

which serves to determine the lines of curvature, a factor separates
itself namely d,v-+ d,v, which made equal to zero represents a
right line, whilst the, remaining furnishes two mutually perpendi-
cular pencils of parallel lines. In this manner, however, from each
point of the first mentioned right line three lines of curvature
would start, so that there would be an entire line of umbilics. This
is of course in general not the case, so that this representation of the
lines of curvature must undergo a considerable modification as soon
as the terms of higher order are taken into consideration. We shall
soon refer to this again.

5. We shall first mention the results of a closer investigation of
the deformation of the double umbilic. From this we were able to
prove, 1**. that for a variation of parameter in the sense in which the
two single umbilies diverge in a real manner, this diverging shall always
take place in the direction of the just discussed right line d,2+-d,y=0,
which after that represents in first approximation for each of the
two separated umbilics one of the directions of departure of lines
of curvature, 2°¢. that these separated umbilies are always of a dif-
ferent kind, namely one of ihe first kind, the other of the second.
Moreover d,v + d,y =O indicates for that of the second kind the
middle direction of departure, whilst also the remaining directions
of departure of the diverged umbilics nearly correspond to the direc-
tions of departure of the original double umbilic discussed in § 4,

( 390 )

All this being stated it is not difficult to guess how in general the
form of the lines of curvature must be, shortly after the breaking
up of the double umbilic *).

Fig. 4. Fig. 5. Fig. 6.

That form is represented in fig. 4, where O, indicates the umbilic
of the first kind, O, that of the second. At O, the angle of the two
other lines of curvature, starting from the umbilic, which contains
O, O, is a little larger than a right angle, at O, on the contrary it
is a little smaller.

If after that we allow the umbilics to coincide again, they meet
at about half the distance and the figure now formed where the lines
of curvature situated at some distance to the right and left of O,
and (, must have retained in general the same direction, can hardly
be otherwise but such as has been indicated in fig. 5°), apart from
the symmetry which in general does not exist of course, no more
than in any of the other figures.

1) After the publication of the Dutch version of this paper we found that Mr.
A. Gvtisrranp already in 1900, in his memoir “Allgemeine Theorie der mono-
chomatischen Aberrationen und ihre niichsten Ergebnisse fiir die Ophtalmologie”
(see Nova Acta Regiae Societatis Scientiarum Upsaliensis, ser. 3, vol. 20, pp. 90
and 114) arrived also, starting from other considerations, at the investigation of
the double umbilic and its breaking up and that we obtained the same results.

2) However, a closer investigation of this subject by another method would not be
unwished for, It would have to be a systematic study of the lines, if possible in
their entire length, satisfying the differential equation :

(Gi) fleas + 3e,27 + 4(e, — 2¢,*)ay + 3e,y°] +

di
iy. = [6d,y -+ 2(¢, —Ge, + 4e,*)u? + 6(e,—e,)ay + 2(6e,—e,—4e,*)y?] = 0.

For this is the form which the differential equation of the lines of curvature
assumes in the neighbourhood of a double umbilic at second approximation,
when we place the A-axis in the direction in which the two single umbilies diverge
by a slight deformation of the surface. We then have d;=0 and ds = 3d); the
latter on account of (5).

( 391 )

If we then continue the deformation in the same manner so that
now the two umbilics diverge imaginarily, a figure seems to be-
formed as is represented in fig. 6.

In no case there occurs a transition proper from the first kind
to the second on a point-general surface continuously deforming itself.
When the relation A,=0O meets its fulfilment then we find that
two umbilies of different kinds approach each other to disappear from
the surface after the coincidence.

b. The nodes of a point-general surface as
twelvefold umbilies.

6. When there is a node, the equation of the surface in its neigh-
bourhood cannot be given in the form indicated in equation (1). After
a fit choice of the axes we can however start from:

Gia Uys 1-6 22 OS. 2. (9)
or after a slight deformation, from:
ast B.e+ By + Bz + aa? + by? + cz? +....=0. . (10)

It is soon evident that to determine at first approximation the
umbilics which appear in the neighbourhood of the place where
formerly the node existed, the terms of order two are sufficient.
So the surface may be treated there as a quadric, which immedia-
tely makes the behaviour of the umbilies clear. If namely we have
to do with an isolated point, made to appear after the gradual
disappearance of a sheet, then at the very instant four real*) umbilics
disappear, which were situated on that sheet, whilst eight others were
imaginary and become so again after the disappearance of the sheet.
If the node is a conical point then, when the two sheets are disunited,
four real *) umbilics make their appearance, becoming imaginary at the
union, whilst eight others again meet likewise for a moment in the
node, but are previously and afterwards imaginary. For an imaginary
node of course all the twelve umbilics coinciding there for a moment
remain imaginary.

The umbilics at infinity. General considerations.

7. The umbilies are distinguished from the plaitpoints and many
other singular points by the fact, that they cannot stand a projective

1) These are at first of the third kind. They can, however, gradually pass during
a continued deformation into those of the second kind without giving rise to the
_ appearance of a double umbilic.

2) Also for those holds good what was remarked in the preceding note.

-

( 392 )

transformation. The cause of this is that they are in a definite
relation to the plane at infinity and in particular to the spherical
points in that plane. This obliges us to give a separate consideration
of the cases of the first order of exception, where umbilics reach
infinity. It was a priori not improbable that this would be accom-
panied by the occurrence of multiplicity in all or in some of those
cases, as really it proved to be for some.

The method of investigation with respect to this was as follows :
first the umbilics were exchanged for a more general kind of
singular points which are capable of projective transformation. To
this end it is sufficient to observe that an umbilic can be defined

as such a point of a given surface which — when regarded as
a node of its section of the tangent plane — has the property

that both nodal tangents pass through the circular points of the
tangent plane.

After applying the general projective transformation the problem of
the umbilics of the original surface is in this way reduced to the
following :

Given a surface w, a plane a, and in that plane a conic ¢; to dejine
on the surface w the points 2 which have the property that the two
nodal tangents of the section of the tangent plane @ in 2 pass
respectively through the points A, and A, where c is cut by 9.

For this more general problem the plane at infinity has been
replaced by the plane « and we have but to study the points 2
which as singularities of the first order may appear in the section d of 2
and « which can be performed by choosing an appropriate system
of axes with such a point for origin, by calculating for this system
of axes the approximate equation of the surface, and by then applying
a slight deformation. The results obtained in this way can be imme-
diately applied to umbilies.

In this manner it became evident that umbilics can appear in four
different ways at infinity as singular points of the first order of
exception, which we shall successively describe in short.

ec. The point of contact of a point-general surface with the
plane at injinity as a fourfold umbilic.

8. It is clear that whenever the surface @ touches the plane a,
such a point of contact must be regarded as an 2-point; for its
tangents in the section of the tangent plane will certainly meet the
conic c in the plane «. By regarding the surface as a quadrie we
can then by returning to the problem of the umbilics decide without
‘calculation that the point under observation is a fourfold 2-point.

( 393 )

At the same time ensues from the behaviour of the quadries that
when there is a real contact with the plane at infinity, the point
of contact, if it appears in the section of the tangent plane as an
isolated point, breaks up at the deformation into two real and two
imaginary umbilics in whatever direction the deformation may take
place. In the opposite case we have to do with four imaginary
umbilies. So transition from real umbilics to imaginary ones never
takes place in this way.

d. The point of contact of a point-qeneral surface with the curve
of the spherical points at infinity as a double umbilic.

9. It goes without saying that when w touches ¢ the point of
contact must be an 2-point, for the points A, and A, coincide with
this point of contact and so they are situated on the nodal tangents
in this same point.

By analysis it proves to be a double @-point. As the spherical
points at infinity are all imaginary, these umbilics and the single
ones into which they break up, are also always imaginary.

e. The points of mfjinity of the spinodal line as single umbilics,
when the tangent of the spinode hes in the plane at infinity.

10. If we consider a point in which the spinodal line of & cuts
the plane «, it is easy to see that this point must be regarded as an
2-point as often as the cuspidal tangent of the section of the tangent
plane lies in plane e, which isa single condition. It appears, however,
that this point cannot be driven asunder by deformation, so it must
be regarded as a single @-point and the umbilie corresponding to it
likewise as a single umbilic. This umbilie can be real or imaginary.
The manner indicated here is the only one in which real umbilies
can reach infinity without passing into a multiple umbilic, i.e.
without meeting other umbilies there.

f. The points of intersection of the surface with the curve of
the spherical points at injinity as single umbilics, when
one of the nodal tangents in the section of the
tangent plane lies in the plane at injinity.

11. It is immediately evident that the corresponding points on w are
Q-points and after investigation they prove to be single ones. As
umbilies they are of course always imaginary.

Application to quadrics.

12. The equation of a quadric can be brought with an appropriate

( 394 )

choice of axes when the origin is placed in one of its umbilies, into
the finite form:
2=c, (27+ 9’) + hee Rye et ET)
Bringing the value of z into the second member this furnishes
the development in series
z= 6, (@ + y?) + kye,a* + k,c,27y + kyc,vy*? + kc,y® + . (12)
Comparing this to (1) it is immediately evident that for the
umbilies on a quadrie we always find d,=d,, d,=d,, so K, <0.
Furthermore the cubic (7) passes into (dyn — d,)(n? +1) =0; so
K,<. 0. From this it is evident, as indeed is known, that on a
quadric never other umbilics than those of the third kind can appear.
From this ensues again immediately that on a quadric no common
double umbilics can appear. Indeed beside the nodes the only
possible multiple umbilies at finite distance on a quadric are the
vertices of a surface of revolution; but these are fourfold umbilies
whose occurrence on surfaces of higher order would demand more
than one relation between the coefficients of the equation. So it is
not astonishing that for such vertices the lines of curvature bear
themselves in an entirely deviating way.

13. Passing now to the umbilics of quadrics at infinity we observe
that the case given sub ¢ appears for paraboloids. If, however, we
regard more closely the section with the plane at infinity, then this
is evidently degenerated into two right lines. Each of these right
lines meets the curve of the spherical points in two points. If we
make tangent planes to appear in those points, then also there the
section of the tangent plane degenerates, namely, into one of the
recently considered right lines and into another. These two must at
the same time be regarded as the tangents of the section of the
tangent plane. One of these tangents therefore always happens to lie
in the plane at infinity and we are in case //.

To the fourfold umbilic at infinity four single umbilics are in this
way added for the paraboloid. For finite distances four such points
only are thus left, which furnishes here the proof to the sum.

Inversely case d requires as is easy to see, at least for quadrics
with real equation, that these should pass into surfaces of revolution.
There is then double contact of the surface and the curve of the
spherical points. Indeed in this case four umbilies pass into infinity;
the eight remaining ones coinciding four by four in both vertices.

The remaining case e cannot make its appearance for quadrics.
The case / has just been discussed. It can as is easy to see make
its appearance for quadrics only in the manner indicated there,

( 395 )

Chemistry. — “On the preparation of pure o-toluidine and a

method for ascertaining its purity.’ By Prof. A. F. Hotieman.
(Communicated in the meeting of November 26, 1904).

Whilst p-toluidine being a solid, well crystallised substance may
be very readily obtained in a perfectly pure state from the commer-
cial product by recrystallisation and distillation, this is by no means
the case. with the liquid ortho-toluidine. The latter stands a good
chance of containing its para-isomer as it is prepared from o-nitro-
toluene, which is rather difficult to completely separate from the
p-nitrotoluene simultaneously formed in the nitration of toluene,
particularly because the ortho-nitrocompound is liquid. It is further
stated that o-toluidine sometimes contains aniline.

Of the various ways mentioned in the literature on the subject
for the purification of o-toluidine, the conversion into oxalate seemed
to me the most appropriate. According to Briisteiss Handbuch, the
solubility of ortho-toluidine oxalate amounts to 2.38 parts by weight
in 100 parts of water at 21°; that of the acid oxalate of p-toluidine
(the neutral compound does not exist) 0.87 parts in 100 parts of
water at 10°. If, therefore, the o-toluidine contains a few per cent
of para, the oxalate thereof must remain in the aqueous mother-
liquor when the mixture is submitted to recrystallisation, and the
use of ether, which is given as an accurate method of separating
the oxalates, becomes superfluous. Even any aniline which happens
to be present, may be removed in this manner.

In order to see whether a complete purification might indeed be
attained in this way, it was necessary to first obtain a characteristic
test for asceriaining the purity; for the processes found in the
literature for ascertaining the purity of o-toluidine, of H&vussERMANN
(Fr. 26,750), Remuart (Fr. 38,90) and Lunex (Fr. 24,459) appeared
but little suitable for the detection of very small amounts of impurities.

For this purpose the determination of the solidifying point of the
acetyl compound proved serviceable. By determining a portion of the
solidifying point curve of o- and p-acetotoluidide the amount of the
impurity could then be ascertained quantitatively at the same time.

The following solidifying point figures were found :

Percentage Solidifying
of para. point.
0 109.°15
1.12 108.45
2.42 107. 75
9.58 103.°2
13.6 100.°8

( 396 )

That 109°.15 is the solidifying point of pure aceto-o-toluidide was
proved by recrystallising the oxalate prepared from a “chemically
pure” o-toluidine and then recovering the toluidine, which was then
treated once more in the same way.

After each crystallisation of the oxalate a small quantity of o-tolui-
dine was converted into the acetocompound; the observed solidifying
points were both the above figure, which moreover did not suffer
any change when the acetocompound was again recrystallised.

In order to ascertain how far small quantities of para-toluidine
and aniline-may de detected by means of the solidifying point tigures,
the above purified o-toluidine was mixed with 2°/, of aniline and
another portion with 2°/, of p-toluidine and tested as follows:

25.2 grams of oxalic acid (‘/, mol.) are dissolved in a litre of
boiling water and to this are slowly added 42.8 gram of toluidine
*/, mol.). On cooling, the oxalate crystallises out; after placing the
flask in ice the liquid is thoroughly removed by suction and the
crystals washed once with a little water; the toluidine is then
recovered from the crystals as well as from the motherliquor by
adding alkali and distilling in a current of steam. In order to avoid
loss it is necessary to extract the water, which has also distilled
over, twice with ether. The toluidine so obtained is converted into
the acetocompound by adding per gram a mixture of 2 cc. of glacial
acetic acid and 1 ec. of acetic anhydride. The mass is now evaporated
on the waterbath and the dry residue once distilled in vacuo when
everything passes over leaving but a small black residue. The solidi-
fying point of both products is then determined. We found :

Added
2"/ p-toluidine 2°/) aniline
Solidifying point of the acetotoluidide from the crystals : 109.°15; 109.°15

x = tae 3 » » motherliquors: 103.°2 ; 103.°O

This shows that while the oxalate crystallised out, the added
impurities remained completely in the motherliguor and that the
acetocompound prepared from the latter shows the serious depression
of about 6° If now we consider that the determination of the
solidifying point is accurate to 0.°2 and with practice even to 0.°1
it follows that we may detect in this way */,, part of the impurities
now present, viz. 7/,, or 0.03 °/,.

Using this method I have examined two samples of o-toluidine
from different makers and both marked “chemisch rein” as to their
purity with the following result.

I. Converted into oxalate in exactly the same manner as described. Flask cooled
in ice water.

( 397 )

From the erystals were obtained 31 grams, from the motherliquor 10.2 grams,
total 41.2 grams, 42.8 grams having been started with.
Solidifying point of the acetocompound from the erystals 109.915. Therefore pure.

, motherliquor 107. 15, corresponding

nn nn ” ” n
with 3.69/, of p-toluidine or 0.37 gram. The sample therefore contained
0.37 X< 100 2 ,
a 0.9 95 impurity.

II. 42.8 grams of toluidine converted as before into oxalate. From the crystals
are taken 30.5 grams, from the motherliquor 11.2 grams, total 41.7 grams,

Solidifying point of the acetocompound from the crystals 108.°45 so this still
contained 1.1 %/, or 0.34 gram of byproduct. After having been converted once
more into oxalate, the newly prepared acetocompound now solidified at 109.°15.
Solidifying point of the acetocompound from the motherliquor 101.°9 corresponding
with 12.1% or 1.36 gram, Total impurity present, therefore, 1.36-- 0.34 = 1.70
corresponding with 4.1 °/o.

Assuming the impurity to be either aniline or p-toluidine the
following plan was followed to ascertain which of these two was
present. Of a mixture of acetanilide (6 grams.) and acetoorthoto-
luidide (4 grams) the eutectic point was determined. For this was
found 64.°6 and 65.°1, mean 64.°8,. On adding to this mixture 0.1
eram of p-acetotoluidide, the said point was found to be 63.°1 and
63.°6, mean 68.°3,; the latter, therefore, seemed rather sensitive to
small additions of para.

5.64 grams of acetanilide were now mixed with 4.386 grams
of the acetocompound prepared from the motherliquor (1) which,
according the above examination, contain 4.20 gram of acetoorthoto-
luidide and 0.16 gram of an impurity, which might be p-acetotoluidide.

The point of initial solidification of this mixture was found to be
72.0 and 71°9, the point of complete solidification 62.°6 and 62.°8.
A mixture prepared from 5.64 gram of acetanilide, 4.20 grams of
acetoorthotoluidide exhibited these same points at 72.°1 and 62.°8, so
that the impurity seems to be indeed p-acetotoluide; acetanilide is
out of the question as then the point of complete solidification ought
to have coincided with the eutectic point of the pure mixture of
acetanilide and aceto-o-toluidide.

The above method will no doubt be found applicable in a number
of other cases as it is based on a general principle. By its means,
it is possible to ascertain the purity of organic preparations with a
greater degree of quantitative precision than has been the case up
to the present, particularly when dealing with liquid substances.

Mr. F. H. van per Taan has ably assisted me in the experimental
part of this research.

Groningen, Chem. Lab. Univers. November 1904,

.

( 398 )

Botany. — “On the influence of nutrition on the jluctuating varia-
bility of some plants.” By Miss Tixe Tames. (Communicated
by Prof. J. W. Mott).

(Communicated in the meeting of October 29, 1904).

That nutrition has an influence on the development of plants has
long been known. Also that some parts are much more sensitive in
this respect than others and that, for example, the size of the stem
and leaf is much more affected by good or bad nutrition than the
number of stamens. As yet our knowledge on this point, especially
our quantitative knowledge, is very superficial. The introduction of
the statistical method, however, into botany has enabled us to for-
mulate more sharply the formerly vague and insufficiently defined
question of the influence of nutrition and also to interpret the results
obtained easily and accurately.

Although the number of statistical investigations on plant charac-
teristics, carried out in recent years, is fairly numerous, yet the
influence of nutrition on the value of these characteristics has not
often been studied.

Dr Vries') carried out an extensive investigation in this direction
with Othonna crassifolia. He compared plants that had been grown
in a greenhouse in pots with very dry ground with garden-cultures
and found that with the plants from the greenhouse the median of
the length of the leaves was only about half that of the plants that
had grown in full ground, the average number of ray-flowers per
head being 12 with the former, 13 with the latter. In his work
“die Mutationstheorie’” Dr Vrirs*) describes experiments and obser-
vations, the chief object of which has been the comparison of the
influence of nutrition with that of selection, but which at the same
time increase our knowledge about the influence of nutritive con-
ditions as such. He investigated the influence of these two factors
on the length of the fruit of Oenothera Lamarckiana and Oenothera
yubrinervis, on the number of umbel-rays of Anethwn graveolens and
Coriandrum sativum and on the number of ray-flowers of Chrysan-
themum segetum, Coreopsis tinctoria, Bidens grandiflora and Madia
elegans. From his observations pr Vers concludes that nutrition and
selection act in the same direction and that by stronger nutrition as
well as by positive selection the median value of a character is
increased. Moreover he generally observes that the variability of the

1) Huco pe Vries, Othonna crassifolia, Bot. Jaarb. Dodonaea, 1900, p. 20,

2) Hueco pr Vries, Die Mutationstheorie. Bd. I, p. 368.

(399 )

characters is increased when nutrition and selection act in opposite
directions, i.e. when, as in his experiments, strong nutrition goes
together with negative selection.

Also the experiments by Ruinént *) on the variability of the number
of stamens of Stellaria media show that with good nutrition the
median of this character possesses a higher value than with bad
nutrition. Besides Ruméut finds that the index of variability, which
is a measure for the variability, becomes smaller under unfavourable
nutritive conditions.

Werssr *) investigated the influence of nutrition on various charac-
ters of Helianthus annuus and found that the arithmetical mean for
all the characters studied is smaller with plants cultivated on a sandy
soil than with well-fed plants. His numbers, (for each culture about
forty) are too small, however, to allow us to calculate the constants
for median and variability from them and to draw conclusions from
these.

Mac Lrop*) made experiments in order to determine the influence
of nutrition on the number of ray- and disk-flowers of Centaurea
Cyanus and found that this number is the smaller the more the
nutritive conditions are unfavourable. Besides he investigated the
influence of good and bad nutrition on the number of stigmatic-rays
of Papaver Rhoeas coccineum aureum. He arrived at the result that
with the badly-fed plants the median is considerably smaller, but
that the variability of the character is increased by the bad nutrition.

From this short summary it will appear that in very few cases
only the quantitative change, caused in the median by varying nutri-
tion, has been determined. It is desirable to extend the number of
observations on this point, but it is especially important to learn the
influence of nutrition on the variability for several characters and
plants. Two questions here arise, in the first place whether this
influence is different for different parts of the same plant, in agree-

,

ment with Verscnarrent’s ‘) result that the variability itself of diffe-

1) FrrepricH Reméut, Die Variation im Andréceum der Stellaria media Cyr. Bot.
Zeit. 1903, p. 159.

2) Antoun Wersse, Die Zahl der Randbliithen an Compositenképfehen in ihrer
Beziehung zur Blattstellang und Erniihrung. Jahrb. f. wiss. Bot. Bd. 30, 1897, p. 453.

8) J. Mac Leop, On the variability of the disk- and ray-flowers in the cornflower
(Centaurea Cyanus): Hand. y. h. 3de Viaamsch Nat. en Geneesk. Congres, Sept.
1899, p. 61 (in Dutch) and On the variability of the number of sligmatic-rays in
Papaver. Hand. v. h. 4de Viaamsch Nat. en Geneesk, Congres, Sept. 1900, p. 1!
(in Dutch).

1) Ep. Verscuarrett, Ueber graduelle Variabilitiit von pflanzlichen Wigenschaften.
Ber. d. d. bot. Gesellsch. Bd. XII, 1894, p. 350,

( 400 )

rent parts differs considerably, and secondly whether bad nutrition
causes either an increase or a decrease of the variability for all
characters, or an increase for some and a decrease for others.

With the object of answering these questions, I made some culture
experiments in the botanical garden at Groningen in the summer of
1903. The description and results of these experiments will be found
in what follows.

For the cultures four beds of 2 metres breadth and 6 metres length
were prepared in April. Two of them were manured with hornmeal,
about half a kilogram per square metre. The other two beds were
dug out to a depth of about half a metre and filled with a very
meagre loamy sandsoil, originating from Harendermolen, a sandy
region in the neighbourhood of Groningen. In the middle of April
on one of the manured beds and on one of sandy soi] equal quan-
tities of seed were sown of Jheris amara Linn., obtained from
Haace and Scumipt at Erfurt, Ranunculus arvensis Linn., obtained
from various botanical gardens and mixed, and of Malva vulgaris Fr.
(Mala rotundifolia Linn.), obtained from the botanical garden at
Leiden. The seeds of three other species, which were sown at the same
time on the remaining two beds, did not germinate in sufficient
numbers, so that about the middle of June we resolved to weed
them all out and to sow afresh. This time Anethum graveolens
Linn., from the trade, Scandix Pecten- Veneris Linn. and Cardamine
hirsuta Linn., both obtained from various botanical gardens were
chosen, three species of which it might be expected that, although
sown so late in the summer, they might still fully develop. This
seed was sown in germinating dishes, each species partly in meagre
and partly in fertile earth taken from the beds in the garden. In the
course of the following days part of the germplants were placed
into small pots with meagre as well as with manured earth, special
care being taken that no selection from the germplants should be
made. At the middle of July the young plants were placed in the
beds at such distances from each other that each could freely develop.

Already at the beginning a considerable difference between the two
cultures could be observed in all three species sown in the garden.
The seed in the bed that had been manured with hornmeal came
up sooner and the plantlets developed much more vigorously. With
Malva vulgaris the difference between the plants of the two beds was
at first very great. Those on the fertile soil showed already abundant
leaves and flowers when the plants on the sandy soil had only
formed few and small leaves. This difference remained till the begin-
ning of July, when suddenly also the plants on the meagre soil

—

( 401 )

began to develop vigorously, so that in the autumn searcely any
difference could be observed. The reason of this late, very rapid
development appeared when the plants were dug out. It turned out,
namely, that some of the strongest roots had reached the underlying
earth through the layer of sand. As long as the plants only obtained
their food from the sand, they remained tiny and backward, but
when the roots had penetrated into the fertile earth they still deve-
loped vigorously and with great rapidity. Also with /beris amara
the roots appeared to have reached the earth underneath but in a
much less degree. It was difficult here to trace the fine terminals
of the principal roots as far as the underlying earth, whereas the
roots of MJalea vulgaris, where they passed from the sand into the
earth below, were strong and penetrated at least a few decimetres.
Of Ranunculus arvensis only few roots had reached the underground
with their tips, the same being the case with Scandix Pecten- Veneris
and Anethum yraveolens ; the roots of Cardamine hirsuta were restricted
to the sand, as far as I could see.

Although with most of the species studied the nutrient material
was not entirely derived from the sandy soil, vet all these plants
were in less favourable nutritive conditions than the plants on the
manured soil. So the experiments will show us the consequences
of the difference in nutrition.

For the investigation I chose some characters that are easily
expressed quantitatively and numerically and took cave that the
determination was made at the same time for both cultures and that
the same parts of both were always taken.

In this way I determined in the first place the length of the leaf
of Lberis anara. In July the length of the five oldest leaves, which
were already adult then, was measured. Besides in the autumn, after
the plants had been dug out, the length of the plant was determined
from the base to the top of the inflorescence of the principal stem ;
at the same time were counted the number of branches of the second
order, the number of branches of the third order and the number
of fruits on the inflorescence of the principal stem.

Of Malva vulgaris the number of akenes of the schizocarp, the
length of the leaf-blade and the leneth of the leaf-stalk were deter-
mined. These countings and measurements were made in the beginning
of July, when a very distinet difference in the development between
the two cultures was visible, hence probably before the roots of the
plants on the meagre soil had penetrated the layer of sand, and in
any case before a better nutrition had any perceptible effect.

In the case of Anethum graveolens and Scandix Pecten- Veneris the

28

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 402 )

number of lobes of the first leaf was counted in the plants that
had survived in ‘ germinating dishes. Besides I determined in adult
plants of Scandix Pecten-Veneris the number of umbel-rays and
with Anethum yraveolens also the number of umbel-rays and at the
same time the number of flowers of the umbellet. For the determi-
nation of this latter character only the umbellets of the oldest umbel
of each plant were taken. Of Ranunculus arvensis the number of
fruits per flower was determined and of Cardamine hirsuta the length
of the silique, of each plant the siliques of the principal stem being
measured.

For each of the characters mentioned I took of each of the cultures
on fertile soil and on sandy soil 300 measurements or countings, a
number which, according to the calculations of Prof. Kaprryy, gives
in investigations of this kind a sufficient guarantee of accuracy. For
certain characters 1 had to be contented with a smailer number
since the material in these cases was deficient. For those cases in
which the variability concerns the number, the numbers were noted
increasing by unity; for those characters that vary in length, the
length was determined in’ fractions of a millimetre, in millimetres
or in centimetres, depending on the absolute size of the parts. By
means of the numbers obtained, curves were plotted in order to have
a general survey of the observations and to facilitate a comparison
of the observations of the culture on fertile soil with that on sandy
soil. In most cases the observations were combined into groups, so
that from seven to seventeen intervals were obtained. In this way
curves are obtained that admit of easy inspection and in which the
smaller irregularities have disappeared. Only for the number of
branches of the third order of /heris’ amara, tig. V, the observations
of the plants on the fertile soil had to be combined to 25 groups,
since only then a comparison with the plants from the sandy soil
was possible.

The curves for the various characters are reproduced on the
accompanying plate. Since for all cases the frequencies have been
ealeulated, all the curves have the same area and can be mutually
compared. For each character the curve of the well-fed plants has
been drawn as a continuous line and that of the badly-fed plants as
a dotted one, both having the same absciss. Of both the observations
have been combined to groups with the same interval. In all the
figures the size or the number of the part in question increases from
left to right.

These curves now show us the way in which the studied characters
vary and the limits of this variation.

( 403 )

Looking at the various figures we notice that the studied charae-
ters generally give fairly symmetrical curves, disregarding smaller
irregularities. Only in a few cases, as with Anethuwin graveolens for
the number of umbel-rays of plants on the sandy soil, fig. VI, for
the number of lobes of the leaves of the well-fed plants, fig. VIII,
and besides for the number of lobes of the leaves of Scanudie Pecten-
Veneris of the fertile soil, fig. IX, the curve is markedly oblique.
Only for the ntmber of branches of the third order of /beris amare
from the sandy soil, fig. V, a semi-curve has been obtained.

Examining in the various figures the position of the two curves
with respect to each other, it appears that they partly coincide.
This means that in the two corresponding cultures plants are found
in which the organ under consideration is as large or occurs in
equal number in the well-fed and in the badly-fed plants. But at
the same time they show that in one culture individuals occur, in
which a definite part is so strongly or feebly developed, as are not
to be found in the other cultures. The figures further show that in
all cases except of the number of akenes of JZa/va vulyaris, fig. X11,
the curve of the plants on sandy soil has been shifted to the left
with respect to that of the well-fed plants.

The observations now enable us to determine how great the
influence of the nutritive conditions is in the various cases and
whether this difference in development between the two cultures is
the same for various parts of the same plant.

Examining the figs. I—V, relating to the characters of Jheris
amara; figs. VI—VIIL of Anethum graveolens and XI—XIL of
Malea vulgaris it appears that, whereas with the two former plants
the shifting of the curve is very different in the various cases, it is
about the same for the three characters of Jalea vulgaris and for
aul three of them relatively small. So the curves enable us to form
an approximate idea of the influence of various nutritive conditions,
but a clear insight is only obtained when the curves are defined by
definite constants and these are mutually compared. In this way it
is possible to determine what influence feeding has not only on the
median value of the character, but also on its variability. In order
to obtain these values, the median value J/ and the quartile Q were
deduced from the observations. From these the coefficient of varia-
SoG gat
bility —, which is a measure of the variability and enables us to

mutually compare the variability of different characters, was caleu-
lated by the method introduced by Verscnarre.t'). Also for the

1) Ep. Verscuarrecr, |. c.

28*

( 404 )

somewhat skew eurves these values have been determined, since
these curves do not considerably deviate from the symmetrical ones
and besides, in all cases the average of both quartiles has been taken.
Only from the semi-curve for the branches of the third order of
[beris amara, fig. V, no constants were calculated. This curve will
be dealt with later on.

I give here the values found for the various characters in the plants
studied in the same order as that of the curves of the plate. In the
table, G means the constants of the well-fed, 4 those of the badly-
fed plants. For each character are given: the median value, the
quartile, the variability-coefficient and the minimum and maximum
value. Besides the differences of these values in the well-fed and the
badly-fed plants have been calculated as well for the median as for
the variability-coeflicient. This difference, divided by the value for
the well-fed plants and consequently expressed as a fraction of this
value, [I will call the sensibility-coerjicient of the median or the
variability.. This coeflicient is given in the table under the two values.
A + sign for the sensibility-coefficient means that the value is
greatest with the well-fed plants, a — sign that with these the value
is smallest.

It appears from this table as well as from the curves that in
general the median value of the characters of the badly-fed plants
is smaller than of the well-fed ones. Only with MJalea vulgaris the
median value of the number of akenes of the plants from the sandy
soil is slightly larger, the difference being very small, however. The
sensibility-coefficient is only — 0.015. With the remaining characters
the sensibility-coefficient of the median is positive and differs very
much; on the whole it varies between 0.015 and + 0.54.

Let us now see from the table whether nutrition has the same
influence on the median value of the different characters of the same
species. We shall leave JMalva vulgaris out of account here since,
as Was mentioned above, its roots had in the bed of unfertile earth
penetrated into the fertile underground and possibly on this account
the differences were very. slight for all the characters considered.
Comparing the sensibility-coetlicients of the median of the various
characters of one species, we find that they diverge largely.

While the sensibilityv-coefficient of the median of the number of
branches of the second order of J/beris amara is +- 0,54, it is
+0,15 for the number of silicles of the principal stem; the sensi-
hility-coeflicients of 7 for the length of the plant and the length of
the leaf lie between these values and amount to + 0,24 and + 0,28.
With Anethum graveolens the sensibility-coefticient of the median of

M Q = “Minimum. Maximum.
i
——_—_—_—$<—$— ee —_—
Iberis amara.
G 44.4 cM. | 4.65 cM. | 0.1444! 26 cM. | 56 cM.
I. Leneth of the plant. |
> i | | B Seg weg os op | 0-403] 12. » | 51.8»
| sensibility | |
| coefficient... —- 0.24 + 6.09 |
}
| G | 7.9 cM. | 1.085 cM.| 0.137 4.5 cM.| 14.2 cM.
Ul. Length of the leaf
. | B SEAR Pee RORSOoe om OoOn e2h3! pe" 8:5)»
sensibility
coefficient... —- 0.28 | — 0.47
| | |
G 55.4 7.5 0-43) 29 91
Il. Number of silicles:
B 47 | 6.8 O14 11 M8
|
sensibility |
| coefficient...) + 0.15 | — 0.08
|
| 2 5 | 0.45 5 5
IV. Number of bran- G | 22.4 3.35 24 » 35
ches of the 24 order B | 40.3 | 3.78 | 0.36 0 29
sensibility Sa | |
coefficient... 4+ 0.54 | —— 4] 740) |
| |
Anethum graveolens. Sie al UR ane , a
VI. Number of umbel-, G mat he P 2 Yu
re eres a ee B WORE ty 1685! 0s8o Te rT
sensibility Le
coefficient... + 0.44 — 0. ee
i | oo | J =
; VII. Number of flowers G | 33.3 6.55 | 0.19 ! 67
in the umbellet. B Weroees ae nl. ost 0.21 L i
sensibility | | | i
coefficient... + 0.20 | — 0.105 |
|
| |
2 }
VIII. Numb. of lobes ae hharelt eee Co Eee:
pitts pins se lew B. | 16.8 24 Cem. 7 | 38
|
sensibility | r | |
coefficient... + 0.08 + 0.29 |

Het:

( 406)

M Q - Minimum. Maximum.

Scandix Pecten-

Veneris. : a 4 or a ee.
1X. Number BF lakes G 97 2 3.85 O44 16 56
of the first leaf. . B 25.4 2.6 0.106 i 43
sensibility
coefficient... + 0.08 4 0.26
X. Number of umbel- 2 pa ao yee : 40
FAYS.-<-+--- Sok { B 5.03 0.55 0.400 4 7
sensibility
coefficient...| -- 0.17 + 0.01
|
Malva vulgaris. ane
G 53.8 mM. 3.85 mM. 0.071 40 mM. 65 mM.
XI. Length of the blade}

B 51.9 » 3.95 » 0.075 30 » 70 »
sensibility | |
ceellicient...| +- 0.03 OOD |

Xi Leneth of the G 172.4 mM.15.4 mM. 0.089 128 mM. 289 mM.
ioe B \467 » (13.65 » | 0.081/ 115 » | 2 »
sensibility
coellicient.... -+- 0.03 + 0.09.
|
|
XI. Number of | 6 13.38 oe fale Y iy
akenes ......- | B 13.6 0.6 0.044 rT 17
| sensibility
coefficient...) — 0.015 + 0.12
Ranunculus arvensis.
G 8.5 Op75is 0 09 5 12
XIV. Number of
MKENCS ces. es" 1| B 6.9 0.775 0.11 4 11
| sensibility
coeflicient.... + 049 — 0.22
|
| |
Cardamine hirsuta. ean ae ‘ c
XV. Length of the | \ G 17.5 mM.) 2.75 mM.) 0.15 | 4 mM. 24.1 mM.
silique.....+++-.| (B. |4h7 > | 2.75 3 | 049" [9.9 2s" ee

| sensibility 1
coefficient... + 0.16 — 0.27

{ | |

( 407 )

the number of umbel-rays is + 0,44, that of the number of lobes
of the first leaf only -+-- 0,08. To some extent this may be explained
by the circumstance that the influence of nutrition on the first leaf
is not so great as on characters which appear later, since the food,
stored in the seed, is the same for both cultures and possibly has
not been entirely used when the first leaf develops. In agreement
with this the sensibility-coefficient of the median of the number of
lobes of the first leaf of Scandia DPecten- Veneris is +- 0,08, whereas
it is + 0,17 for the number of umbel-rays of the same plant.

From what precedes it will be seen that the influence of nutrition
of the median value of different characters of the same plant varies
greatly, some organs being very sensitive for differences in nutrition,
others experiencing little difference in their development on this account.

Concerning the value of the quartile the table shows that we do
not obtain in all the eases studied, a variation in the same sense
by bad nutrition, as was the case with the median value. In some
‘ases Q is greater in the plants from the fertile soil, in other cases
it is smaller, as great or nearly as great as with the plants from
meagre soil. In order to be able to compare the variability of the
characters in both cultures, however, and to draw conclusions from
this comparison about the influence of nutrition on the degree of
variability, we must not take the quartile but the variability-coefficient —

If, to begin with, we consider the value of this variability-coefticient
in the various cases, we see from the table that it varies between
wide limits 0,044 and 0,386. Also Vurscnarren? ') found equally

' Q
divergent values of = for the characters of different plants studied

by him. The smallest variability is found with the different cha-
racters of Malva vulgaris, as well in the well-fed as in the
badly-fed plants. Hence this plant appears to be little variable.
Comparing the variability of the different characters of the same
species with each other, we see that they diverge relatively little
with the well-fed plants, as well with /ber/s amara, as with Anethum
graveolens and Malva vulgaris. For the different characters of /beris
amara =) is respectively O,114, 0,137, 0,13, 0,15; for Anethum
graveolens O09, OAI and 0,18 and for Malva vulgaris 0,071, 0,089
and 0,05.

It will be seen that for the same species these values are nearly

1) Verscuarrett, |. c, p. 353,

( 408 )

the same, while they differ considerably among the three species.
Doing the same with the badly-fed plants we find a much greater
difference between the variability-coefficients of the various characters
Q

of the same plant. For this culture = varies between 0,10 and
0,36, for the characters of J/heris amara and between 0,127 and
0,35 for those of Anethum ygraveolens. Hence it follows that the
influence of nutrition on the variability of the different properties of
a plant is not the same; how much this influence varies will be
seen from what follows.

Comparing for each character separately the variability of the
well-fed with that of the badly-fed plants, we find that the difference
between the variability-coefficients for the two cultures varies greatly
in different cases; for some characters it is very considerable, for
others small. In order to compare these differences, they were divided

ey
by the value of a of the well-fed plants, as stated. The resulting

number is the sensibility-coeflicient of the variability. This sensibility-

+

‘ed . a
coefficient of 77 BPpears to vary between — 0,140 and + 0,29. In
a comparison of various characters of the same species the fact that
the roots of the bad culture had more or less penetrated into the
subsoil, obviously is of no consequence, so that the results obtained

with Malva vulgaris ave also available here.

The sensibility-coefficient of = of /heris amara is for the four

characters respectively 1,40, —O17, —0,08 and + 0,09; for
the characters of Anethum graveolens — 0,74, —OA05 and + 0,29;
and for those of Malea vulgaris — 0,055, — 0,09 and + 0,12. Especi-
ally with the first two plants these sensibility-coeflicients diverge
considerably, which proves how very different the intluence of
nutrition is on the variability of the different characters of a plant.
3y the same change in nutrition the variability of one character is
hardly modified at all and that of another character of the same
plant very considerably increased or diminished.

It is very important to know in what direction the nutrition reacts
on the variability, whether under unfavourable nutritive conditions
the variability is either always greater, or generally smaller or whether
the two cases are equally frequent. In this respect the table shows

us that for 6 out of 14 characters the sensibility-coeflicient of u'

4

ce i

( 409 )

positive and the variability-coeflicient of the well-fed plants greater
than of the badly-fed ones, whereas in the other characters the
sensibility-coeflicient is negative and the variability-coeflicient greatest
in the badly-fed plants.

Even with the same species one character shows a greater, another
a smaller variability when the cultures grown under favourable and
unfavourable nutritive conditions are compared. With /beres ainara
the length of the plants from the fertile earth is more variable than
that of the plants from the sandy soil, other characters, on the other
hand, show greater variability in the badly-fed culture. In the same
way in Anethum graveolens the variability is greatest with the num-
ber of lobes of the well-fed plants and with the number of flowers
and umbel-rays of the badly-fed ones, while with Jala vulgaris
the length of the leaf-stalk and the number of akenes of the well-fed
plants, but, on the other hand, the length of the blade of the plants
from the sand, show the greatest variability.

Summarising the results obtained, we see that nutrition influences
the median value and the variability of the characters. Besides it
appears that the sensibility-coefficient of the median is very different:

1. for different species compared among each other.

2. for different characters of the same species.

And about the variability we saw :

J. that with good nutrition the variability-coefficient

=|

is fairly
constant for different characters of the same species, but very diver-
gent for the different species.

2. that with bad nutrition two of the species studied show great
differences between the variability-coefticients of the different charac-
ters of the same species, while with one species the variability-
coefficients of the various characters diverge relatively little.

3. that the sensibility-coefficient of — diverges greatly for different

species and characters and varies between —1,40 and + 0,29.
Q
4. that for some characters the sensibility-coefficient of a is

positive and good nutrition results in an increase of the variability ;
while for other characters, even of the same species, this coefficient
is negative.

In what precedes, there has only been question of those charac-
ters which show symmetrical or sensibly symmetrical curves and
which, when expressed in constants, yielded the results mentioned.

From these the curve of the number of branches of the third

( 410 )

order of J/heris amara, grown on the sand, deviates entirely, being
a semi-curve. For the culture on fertile earth, however, this same
character gives a symmetrical curve. In fig. V this latter is very
flat and extended in length, as the observations were divided over
a great number of intervals in order to allow a comparison of the
two curves. If, however, the observations ave arranged to a number
of groups equal to that of the other figures, the curve thus obtained
is not different from those of the other characters. For this culture
the median is 58, the quartile 17.25 and the variability-coefficient

7 0.32, the minimum number of side-branches being 1, the maxi-

mum 162.
With this character now, bad nutrition does not result ina simple
shifting of the curve to the left, accompanied by greater or smaller

: QO
changes in the values of J/, Q and a as in the other cases, but

here the symmetrical curve changes into a semi-curve of which the
apex lies at zero.

We can explain the origin of this semi-curve in the following way.
The lower limit for the number of branches of the third order
of /beris amara is 0. Since the plant also blooms on the principal
stem and on the branches of the second order, it may exist without
branches of the third order. Under favourable nutritive conditions
the development of the plant is so vigorous that in all individuals
branches of the third order are formed, but in greatly diverging
numbers, as is shown by the curve of fig. V for this culture. With
unfavourable nutrition, however, also individuals arise in which no
branches of the third order are originated and as nutrition becomes
worse the number of these individuals will become ereater. Hence
we see that with the very bad nutrition of the sandy soil, a great
number of plants has no branches of the third order and so has
reached the lower limit, the other specimens bearing a greater or
smaller number of these side-branches, as is shown by fig. V_ for
this culture. This leads us to the conviction that the semi-curve for
this character is a necessary consequence of the fact that by the
unfavourable nutritive conditions the variation-curve is shifted in
such a way that it strikes against the lower limit of the whole
range of variation of this character, a great many of the individuals
showing this lower minimum value.

Also with Anethum graveolens a great difference is noticed in the
shape of the curves of the number of umbel-rays in the two cultures,
fig. VI. The curve of the well-fed plants is nearly symmetrical, while

‘

T. TAMMES. “On the influence of nutrition on the fluctuating variability of
some plants.”

SREREDSEyeEcOm BRVGREEREE.
aN ANE =ERRRR
BABY ahs Seis

Z n 15 18 24 24 27 30 33 36 39 42 45 48 SI 54 57

25945 6789 OW 12 13 1eute

ON 6 aie, 16. ome 30 = m LA am 60 66 2 78 84 90 96 12 108 114 720 126 132 135 14 150 156 162 168

OF 9) WIG ele oreo dae OTT arse, a7, 57, Gf te 60 If Tele) 2b SIaso. S/O STO MGT 66 7

6 ae Je 56 38 40 42

ao tok = P55 4=5=:
I Ve aaa ed ee 2. 30. BI¢ DO IIe, NOI) Ola SLOT,

30 45 60 WS 90 105 120 145 150 165 180 1% 20 235 24 25

Proceedings Royal Acad. Amsterdam. Vol. VII.

~( 411 )

that of the plants from the sandy soil is asymmetrieal in sueh a
way that the top of the curve lies nearer the minimum. It can not
be stated with certainty whether in this case we have the same
phenomenon as with /heris amara, i.e. whether the lack of symmetry
of the curve indicates that it has been shifted to the proximity of
the lower limit. But the fact that the minimum now obtained, viz. 7,
is already very small compared with the maximum 41 and that this
lower limit cannot be zero, renders this view probable. Yet we must
bear in mind in cases like the present, that the appearance of an
asymmetrical curve need not in general be a proof that the curve
is located near one of the limits of the range of variation, but that
the asymmetry of the curve may also be the consequence of entirely
different causes,

Botanical laboratory at Groningen. July 30, 1904.

EXPLANATION OF THE FIGURES.

The figures are all reproduced at about half size. In the original figures the
distances of the intervals, placed along the absciss, are 1 em., each mm. of the
ordinates having a value of 1°/). So we can find from the length of the ordinates
the percentage number for each interval. In most figures the ordinates are drawn
between the two numbers indicating the interval, only in figs. X, XUL-and XIV,
where the observations are not arranged in groups, the ordinates stand above the
number, The curves of the well-fed plants are drawn in continuous lines, those
of the badly-fed plants are dotted.

Fig. I. Iberis amara. Length of the plant from the base of the principal
stem to the top of the inflorescence of this latter, in em.

eal Iberis amara, length of the leaf, in em.

» Il Lberis amara. Number of silicles of the inflorescence of the principal stem.
» LV. Iberis amara. Number of branches of the second order,

aah Iberis amara. Number of branches of the third order.

» VL <Anethwm graveolens. Number of umbel-rays.

» YIL Anethwm graveolens. Number of flowers in the umbellet.
» VILL. Anethwmn graveolens. Number of lobes of the first leaf.

» IX. Scandix Pecten-Veneris. Number of lobes of the first leaf.
a Os Scandia Pecten-Veneris. Number of umbel-rays.

» AL Malva vulgaris. Length of the leaf-blade, in mm.

» All. Malva vulgaris. Length of the leaf-stalk, in mm.

» All. Malva vulgaris. Number of akenes of the schizocarp.

» AIV. Ranunculus arvensis. Number of fruits per flower.

» AY. Cardamine hirsuta. Length of the silique, in mm,

( 412 )

Botany. Prof. J. W. Moni, presents the thesis for the doctorate
of Mr. B. SYPKENS: “On the nuclear division of Fvitillaria

imperialis LL”, and gives a summary of the results.
(Communicated in the meeting of October 29, 1904.)

The subject of this investigation is especially the nuclear division
in the embryo-sac of /riti/iaria, formerly a favourite material for inves-
tigations on the subject of nuclear division.

Mr. Sypkens studied the free nuclear divisions in the parietal layer
of protoplasm as well as the nuclear divisions in the first layer of
endosperm-cells which are directly followed by tangential cellular
divisions. besides some observations were made on the nuclei in the
ovules of Tulipa and in the growing-point of the root of Viera
Faha.

All the material was fixed by means of the strong chromo-aceto-
osmic acid of Freminc. It was for the greater part imbedded in
paraffin in various ways and was examined in series of sections
of 2 to 4 w thickness, stained with gentian violet. Some observations
were also made by means of the method introduced by vax WIssELINGH,
in which the nuclei are dissolved in chromic acid of about 50°/,. These
iwo methods supplement each other; the chromic acid method is to
be preferred for observations about the chromatic parts, sections
give more information about the nuclear spindle. But in this investi-
gation the excellence of both methods was again proved as compared
with the observation of the nucleus as a whole, which in many
cases renders it impossible to form an accurate idea about its
internal structure.

I will briefly mention the chief results obtained by Mr. SypKENs
for the various stages of nuclear division.

The resting nucleus was studied by means of sections and of
chromic acid and the results so obtained were in the main a complete
confirmation of the results published by van Wissrnincn and by
Grecorre and his co-workers WyGarrts and Brureus. The framework
of the resting nucleus consists of numerous larger and smaller lumps
of chromatin, connected by fine threads so that an anastomosing
network is formed. There is no reason for assuming in this network
the existence of two constituents, Chromatin and linin; the chromic
acid method as well as coloured nuclear sections show the contrary,
if only partial washing out of the stain is prevented, as Mr. SYPKENS
did. Those who wish to maintain the assertion about the existence
of linin-connections will have to bring forth new and valid proofs,

( 413 )

Also for the nuelei of the integuments and nucellus of /yii///aria and
of the ovules of Tudipa the same results were obtained.

Concerning the individuality of the chromosomes vax WisseLincu
has shown that it exists in the spirema, since at that stage a
continuous thread is never found. but his further observations as well
as those of Gricore and Wyearrts indicate that probably, even in the
resting stage, this individuality never entirely disappears. Mr. SypKeNs
was led to the same conviction by his observations about the
formation of the spireme and of daughter-nuclei from the daughter-
spiremes. He speaks of a ‘centralisation and decentralisation of a
number of chromatine masses, which in certain stages form as
many chromosomes.”

About the behaviour of the chromosomes during the process of
division little that was new could be found in this investigation for
the reason mentioned. The number of chromosomes was fixed at
about 60, but in certain nuclei it decidedly is much smaller. Neither
is the shape of the chromosomes constant; in the same nucleus (7
shaped, as well as V- and J-shaped ones could be found.

The study of the nuclear spindle on the other hand gave important
results, not so much about the formation of the spindle as about its
further history and the part played by it in cellular division.

The formation of the spindle could be followed in details. Round
the free nuclei in the parietal layer of protoplasm of the embryo-sac
granular protoplasm occurs with many very small adventitious vacuoles ;
round the nuclei of the first endosperm-cells also protoplasm with
several small vacuoles. Now, when the nuclei begin to divide and the
nuclear membranes are dissolved, the surrounding protoplasm pene-
trates into the nuclear space, at first without many vacuoles, and forms
at the interior the spindle-threads, which at first consist of coherent
granules and later become smoother. They gradually assume parallel
directions and are connected to a bundle without strongly converging
towards its poles. The nuclei ave then in the spireme-stage. Later, in
the aster-stage, besides the threads already mentioned, others are
formed in exactly the same way, which grow thicker and only
proceed from the poles to the equator, where they are attached to
the chromosomes, which have been formed in the mean time. They
are found not only at the circumference of the spindle, but also in
the interior part of all the longitudinal sections of a nucleus. SrRas-
BURGER has called the former sort of threads, running from pole to
pole, “Stiitzfasern”, the shorter and thicker ones “Zugfasern”’.

Now metakinesis follows and in the dyaster-stage a separation
of the two sorts of spindle-threads has taken place. The shorter

( 414 )

and thicker ones have much contracted and form at both poles,
adjacent to the daughter-nuclei, two small caps which soon disappear
in the protoplasm. The long threads on the other hand remain between
the daughter-nuclei, extending from one to the other and hence are
often called connecting threads. They occur in numbers from 200 to
300 and cross-sections show that they form a massive bundle lying
free in the surrounding protoplasm, which can freely penetrate
between them.

Hence Mr. Sypkens arrives at the conclusion that the nuclear
spindle is entirely formed from the cytoplasm within the nuclear
space and so agrees with what has been found by most other inves-
tigators and on main points also with the results obtained by STRASBURGER
and Hevser for the nuclei of /ritillaria.

Now with regard to the part played by the nuclear spindle in
cell-division zoologists and botanists lave divergent views. Con-
cerning animal cells the general opinion is that the nuclear spindle
is dissolved in the cytoplasm after the nuclear division has been
completed and takes no active part in cell-division, the cell subse-
quently dividing by constriction. Botanists on the other hand, attach
ereat importance to the spindle in cell-division and especially in
the formation of the wall. Their generally accepted representation
is that the above-mentioned connecting threads of the spindle grow
thieker in the equatorial plane and form so-called dermatosomes.
By fusion of the dermatosomes the so-called cell-plate is then formed,
which subsequently participates in some way or other in the for-
mation of the new cell-wall between the nuclei. STRASBURGER is one
of the chief representatives of this much spread conception.

It is a consequence of the fact that the study of this phenomenon
has for the greater part taken place with nuclei that were seen
from the outside. By means of his sections Mr. SypKeNs was enabled
to prove that, for the objects studied by him, the opinion now
prevailing in botany is incorrect and that, at any rate as far as the
behaviour of the nuclear spindle is concerned, the phenomena have
ereat resemblance with those of animal cells.

In describing the later phases of the nuclear spindle it is desirable
to distinguish three different cases of nuclear division. In the first place
we have the free nuclear divisions in the parietal layer of protoplasm of
the embryo-sae of Fritdlaria, which will be followed by still other
nuclear divisions before there is question of cell-divisions. Here in the
beginning a system of connecting threads between the daughter-nuclei
appears, as in all other cases, but this soon becomes narrower at the
equator and so assumes the shape of an hour-glass and is then

( 415)

absorbed in the protoplasm and disappears. So this case needs no
further consideration.

The second case regards the parietal layer of protoplasm of the
embryo-sae, which has already become partly divided into cells. Now
when here also free nuclear divisions take place, the nuclear spindle,
consisting of connecting threads, behaves at first in exactly the same
manner as in tissue-cells in which the cell-division follows immediately :
the system of connecting threads swells laterally and forms a so-called
nuclear barrel. After this, however, the spindle here is also lost
in the protoplasm and not until later one sees successive divisions
take place between these nuclei, progressing regularly from that part
of the parietal layer of protoplasm that is already divided into cells, so
that finally a complete pavement of endosperm-cells is formed from
the protoplasm. This description renders the existence of a connec-
tion between the nuclear spindle and cell-division not very probable.

The most important case is the third, in which the just-mentioned
endosperm-layer divides into two layers of cells by tangentially
directed walls. Here the nuclear divisions are immediately followed
by cell-divisions, in the same way as in the formation of various
sorts of tissues.

Hence this case, as was proved by comparative observations, must
be considered as completely analogous with what happens in the
cells of the growing-point of the roots of Vicia Faba.

From Mr. Sypkens’ sections it appears that in the two latter cases
the connecting threads soon cease to deserve that name, as their
extremities are not attached to the daughter-nucle: but end freely in
the protoplasm. In Vicia aba moreover, the equatorial parts are
soon dissolved so that the system of connecting threads falls asunder
into two halves.

Meanwhile the protoplasm round the nuclei of the parietal layer of
protoplasm penetrates with its small adventitious vacuoles into the
space between the daughter-nuclei where the massive complex of
connecting threads is found. These threads are consequently forced
asunder towards the circumference and thereby united to spindle-
shaped bundles, which lie free in the protoplasm; they form what is
usually called the nuclear barrel. The result is that the two daughter-
nuclei are at last separated from each other by the same granular
protoplasm, which also surrounds them and in which also the remains
of the connecting threads are found. The spindle-shaped complexes,
formed from these, are united to a barrel-shaped, equatorially swol-
len, cylindrical mantle, which, if the nuclei are only observed from
the outside, still seems to join them, although in reality this is no

( 416 )

longer the case by any means. On the contrary, the remains of the
connecting threads gradually disappear as if they were dissolved in
the protoplasm and this process has long been completed when the
cell-walls successively appear between these nuclei also.

Also in the divisions of the endosperm-cells of Friti//aria and in
the root-tip of Vieca Faba mainly the same occurrences take place
although there are some points of difference to which I shall refer
presently, and although the formation of the cell-wall follows sooner here.

How this wall-formation takes place has for the present not been
investigated by Mr. Sypkens, but that it stands in no relation to the
nuclear spindle or to a cell-plate formed by it, is pretty clear from
what precedes. A cell-plate in the sense of botanical authors does not
even occur.

Although the opinion, so generally spread in botany, that in many
cases the formation of cell-walls is dependent on nuclear spindles,
may have a certain probability when we only think of the cross-
divisions of the cells of growing-points and suchlike, it lacks,
generally speaking, every foundation. For any one knows that the
formation of cell-walls can in many eases have nothing to do with
a nuclear spindle. Not to mention all possible cases of thickening of
the cell-wall which do not correspond to the formation of a primary
membrane, | will only mention zoospores which, after having come to
rest, form a wall; plasmolvsed protoplasts of Spirogyra and other
Algae which cover themselves with a new cell-wall; Cawlerpa and
other Coeloblasts, the protoplasm of which after a lesion produces a
new wall-piece.

But also in other cases, which resemble more the cell-divisions in
erowing-points, it is often easy to show how newly-formed cell-walls
cannot possibly have been formed in the nuclear spindle. I mention
the antipodal cells, which so frequently are formed projecting inwardly
in the embryo-sac connected only for a small part of their surface
with the cell-wall of the embryo-sac; in any case no more than a
small part of the free wall-surface can have been formed here in a
nuclear spindle. A corresponding case is that of the U-shaped walls
in the epidermal cells of the leaves of ferns, by which the mother-
cells of stomata are formed. More clearly still one sees the same
thing in the formation of the stomata of Aneimia fravinifolia: the
stomata lie in the middle of an epidermal cell of the leaf and the
nucleus of this cell is still pressed against the stoma. A nuclear divi-
sion has taken place here before the stoma-mother-cell was formed
in the epidermal cell, and between the two cells so formed there
certainly was a spindle at first. But in the subsequent cell-division a

(417 )

evlindrical wall was formed at a certain distance round one of the
nuclei, which consequently could for a small part only have been
formed in the spindle. Finally we have the formation of the first
pavement of endosperm-cells from the parietal layer of protoplasm of
the embryo-sac as well in /riti/laria as in many other plants. When
the number of nuclei of this laver of protoplasm has very greatly
increased, separations between the nuclei arise, so that a layer of flat,
pentagonal or hexagonal cells is formed, which at last are separated
from each other by cell-walls. These cell-walls are formed at a period
when of the originally present nuclear spindles no trace is left.

In relation with these facts the result of Mr. Sypkens about the
negative part played by the nuclear spindle in cell-division cannot
surprise us and it even gains in inner probability by them. This
result also shows the way to a more profound study of the pheno-
mena of cell-division and wall-formation in the vegetable kingdom.
The cell-divisions in growing-points, in the above-mentioned epider-
mal cells of ferns, also in the parietal layer of protoplasm of the
embryo-sac, must now be more closely investigated, preferably by
the method applied by Mr. Sypkens, and important results may be
expected of this investigation. Also the study of living, dividing cells,
in the same sense as was formerly done by Trrus') deserves again
our attention in this respect.

It is by no means impossible that by such investigations the con-
ception of cell-division in’ plants will come still nearer to that of
the same phenomenon in animals than is the case at present.

From all that precedes it appears that the nuclear spindle is formed
entirely from the cytoplasm and returns to it. Besides, all investi-
gators agree that in nuclear division the nuclear membrane and the
nucleoli are dissolved and later are formed anew in the daughter-
nuclei. An uninterrupted individual position with regard to the eyto-
plasm is consequently, among all the parts of the nucleus, occupied
by the chromosomes alone, ouly here there is question of a hereditary
organisation.

The opinion of some authors that the nucleus during the whole
process of division would form an isolated whole with respect to
the cytoplasm and that at first there would be a sort of vesicle,
joming the daughter-cells and separating the spindle from the cyto-
plasm, must consequently be abandoned.

In relation with this T may briefly point out the complete agree-

1) M. Trevus, Quelques recherches sur le réle du noyau dans la division des cellules
végétales. Publié par Académie Roy. Néerl. des Sciences. 1878.

29

Proceedings Royal Acad. Amsterdam. Vol. VIL.

——-——

( 418 )

ment between the results of Mr. Sypkens and the theory of pr Vries
and Went, which looks upon the vacuoles as hereditary organs of
the protoplast. If the nucleus were, during division, an isolated whole,
the question about the origin of the vacuoles, present inside the
spindle, would perhaps give some difficulty. But we saw, how the
observations of Mr. SypKeNs prove that we have here ordinary
vacuoles, already present in the granular protoplasm and which are
shoved in between the spindle-threads from the outside with the
protoplasm.

Yet it will be desirable to give some nearer information about
this process, since two somewhat divergent cases occur and here
again a distinction must be made between the nuclear divisions in
the parietal layer of protoplasm of the embryo-sac and those in the
first endosperm-layer or in the meristem of the roots of Vicia.

In the latter cases, in which ordinary division of tissue-cells takes
place, Mr. Sypkens observed what follows. In these cells there are
a number of vacuoles, which are about equivalent and lie round
the nucleus in the granular protoplasm. After nuclear division this
protoplasm with its relatively large vacuoles, penetrates into the spindle
between the connecting-threads, as we saw above. This penetration
here occurs as well in the equator as more in the neighbourhood of
the daughter-nuclei. Hence it is the ordinary vacuoles of the mother-
cell, which shove in between the daughter-nuclei with the protoplasm
in which they le. Later, when the connecting-threads have been
dissolved and cell-division takes place, these vacuoles, as well as
those which did not penetrate into the spindle, are divided equally
between the two daughter-cells. So the question is here very simple
and in complete accordance with what van WisseLincu found in
Spirogyra. Only in this latter case the mother-cell has not several
equivalent vacuoles but a single large one which penetrates laterally
into the nuclear spindle.

Somewhat different are the circumstances in the divisions of the
parietal layer of protoplasm of the embryo-sac. This cell not only
contains many nuclei but has also a somewhat different structure
with regard to its vacuoles. It has namely one single large vacuole,
filling the middle part of the cell, but besides in the parietal layer
of protoplasm a great number of very small adventitious vacuoles,
which were very conspicuous in the preparations of Mr. SyPKENs,
stained without washing out of the stain. Now, after nuclear
division, the granular protoplasm with its many adventitious vacuoles
penetrates between the daughter-nuclei and the free extremities of
the connecting threads. From there it penetrates further towards

( 419 )

the equator between the connecting threads. Hence the daughter-
nuclei are finally separated from each other by granular protoplasm
with adventitious vacuoles of the embryo-sac. Now, when later the
parietal layer of protoplasm divides into cells, the large embryo-sac
vacuole does not partake in this process, but each newly formed
endosperm-cell is provided with a certain number of adventitious
vacuoles.

So there is a certain antithesis here with what happens in ordinary
cell-divisions in young cells, but with the vacnole theory of br Vries
and Wexv this process also is in complete harmony, for Wrvxt has
shown that small adventitious vacuoles can oceur in large numbers
in all sorts of ordinary cells and can in all respects be compared
with the large vacuole, from which they can also be produced
by division. I should not be surprised if further investigation
showed that their occurrence is much more general still than is now
supposed.

The case met with in the embryo-sae of /riti//aria and many
other plants stands by no means isolated, and is also met with in
the division of other multinuclear cells. Wrxr mentions some cases
of this kind in his investigation about the vacuoles of Algae.
(Chaetomorpha aerea, Acetabularia mediterranea, Codium tomentosum).

I had an opportunity personally to observe a similar case of
division in the formation of asexual zoospores in the cells of Hydro-
dictyon utriculatum. While the zoospores, which had been formed by
division of the parietal layer of protoplasm, were partly in motion and
partly had already arranged themselves to a network, all this mside
the wall of the large mother-cell, IL saw the middle part of this
cell occupied by three great tonoplast vesicles, having their origin
in the great central vacuole of the cell and which, upon being
heated under the microscope, first shrank and then burst. Hence
here, no more than in the embryo-sac of /7ritti/laria, the great
central vacuole took part in the formation of new cells. That the
zoospores were provided with very small vacuoles, present in the
granular protoplasm, cannot be doubted according to the above-
mentioned investigations of Wernvr. Lalso observed them very distinctly
in the cells of the young nets very soon after their formation.

Finally it requires to be mentioned that the doctoral dissertation
of Mr. Sypkens will soon appear in a German translation in the
second Part of Volume I of the Recueil des travaux botaniques
Neerlandais.

29%

( 420 )

Botany. — “An investigation on polarity and organ-formation with
Caulerpa prolifera.” By Prof. J. M. Janse. (Communicated by
Prof. Hugo pre Vrigs).

(Communicated in the meeting of October 29, 1904).

Polarity is a property of very many of the lowest organisms as
well as of a great part of the cells in the body of the higher plants
and animals.

The regular exterior shape and internal structure of organs must
be partly attributed to the agency of polar influences during their
development, while the definite vital phenomena of organs must
also, among other causes, be ascribed to polar actions of the con-
stituent cells.

The cause of this polarity, i.e. the property of acting or reacting
in a certain direction otherwise than in the opposite direction, is
unknown, and the great difficulty of finding suitable material for
investigation is perhaps the principal cause of this.

Former observations made with Caulerpa prolifera had convinced
me‘) that this unicellular, relatively gigantic and morphologically
highly differentiated alga must be suitable for this purpose.

Having had the opportunity during last summer, of submitting this
plant to a renewed investigation at the Zoological Station at Naples,
I wish to relate briefly the primecipal results obtained.

For a description of the structure of Caulerpa prolifera, as well
as of its protoplast and the very intense currents that take place
in it, I refer to my quoted paper.

For the new experiments the “leaves” were exclusively used, namely
the outgrowths of the “rhizome” measuring in extreme as much as 22
centimetres in length and 20 millimetres or a little more in breadth.
Their littke thickness allows us to examine them also microscopically
in a living condition, while their considerable length and breadth
make them particularly fit for experiments.” Moreover cut leaves or
parts of leaves can form new rhizomes and rootlets and so can
regenerate to complete plants by neo-formation.

Formerly already | used these leaves for experiments concerning the
course of the protoplasm-currents, in which it was often required to
make large incisions in the leaves.

These plants, to be true, offen sustain serious lesions which heal

1) Die Bewegungen des Protoplasma von Caulerpu prolifera, Pringsheim’s
Jahrb. f. wiss. Bot. 1889, Bd. XXI, pag. 163—284, with 3 plates.

( 421 )

in one day, but this is always accompanied by a great loss of proto-
plasm by which the cell is much enfeebled.

This time | sueceeeded in finding a new method in order to get at the
same result, based on the observation that every laceration of a
part of the numberless protoplasm-threads, which run through the
whole plant as an extremely fine network, is immediately followed
by the local secretion of a white, tough, wiry substance, whichtvery
soon becomes stiffer, assumes a bright yellow colour and then forms
a perfect partition. If at the same time the cell-wall had been injured,
the external wound is closed in this way. But the same laceration
of the plasm-threads can be brought about by pressure and without
external lesion; the partition is then restricted to the place where
sufficient pressure was exerted. In this manner one can at any
arbitrary point of the leaf produce, as it were, a cross-wall, to which
any desired direction and length can be given. If one proceeds with
care this partition is no broader than '/, millimetre.

In this way one can also physiologically, namely without external
lesion, divide a leaf into two or more parts.

This treatment, which in all respects has the same consequences
as are observed with a wound, is not accompanied by weakening of
the cell, since no protoplasm is lost, and besides the plant is already
after a minute fit for further manipulations or for examination.

Caulerpa prolifera derives its specific name from the circumstance
that the “leaves” which spring forth from the “rhizome” very often
produce new leaves, prolifications. Especially by this circumstance
I succeeded formerly in showing that, in accordance with pr Vries’
views, there exists also in this plant a direct relation between the
intensity of the motion of the protoplasm in various places and that
of the nutrition in these places. The bundles of protoplasm bands,
coloured dark green by chlorophyl grains and very often visible to
the naked eye, which pass from the stalk of the prolification into
the primary leaf and then tend to the leaf-stalk of this latter were
a very important aid in this investigation.

These bundles are lacking where very young prolifications are
found and only gradually develop in the leaf, and in doing so always
begin at the stalk of the prolification and extend towards the base
of the leaf. These stream-bundles are never seen developing in the
opposite direction, i.e. beginning at the leaf-stalk and extending
towards the stalk of the prolification, neither do they proceed from
the prolification to the top of the leaf. So they originate from above
and tend downward.

Moreover if an existing prolification is cut off, one sees the bundles

gradually disappear: this disappearance also proceeds from above
downward.

Both phenomena point to the existence of a polarity in the regulation
of the protoplasmic currents, of which the impulse proceeds in the
direction from the organic top to the base.

If this stream-bundle is interrupted by a large cross-wound the
communication is restored round the end of the wound. Now, however,
the currents above and below the wound behave quite differently :
the bands which proceed from the stalk of the prolification remain
on the whole unchanged until they have arrived near the wound; they
then partly deviate transversely and bend round the end of the wound,
after which they go in a straight line to the leaf-stalk. Another part
often turns back with a bend, namely if the currents are strong. So
above the wound there occurs as it were a thrust and often a
reflection which are entirely absent below the wound.

Also this difference in the course of the currents above and below
the wound points to a polarity in the regulation of the protoplasmic
currents, the impulse evidently here also proceeding from the top
and being directed to the organic base.

We must add here in the first place that the currents, running
in a non-proliferous leaf, which assemble like a fan from the top
and the edge of the leaf and all pass into the leaf-stalk, behave in
exactly the same manner, when interrupted, as the stream-bundle
which proceeds from a prolification downward ; only this latter is
generally more powerful and so more suitable for experimenting.

Secondly we must remember that everywhere in the leaf there
exists a very complicated network of currents, stretching between
the numerous (+ 800 per sq.mm.) cross-beams which join the two
sides of the leaf; so there exists an almost straight, but little intensive
connection between any arbitrary pair of points on the leaf; so when
we speak here for simplicity’s sake of the generation of currents, we
mean the strengthening or thickening of the currents in such away
that they become visible with the naked eve or with the eye-glass.

Thirdly the protoplasm in all currents moves continually or alter-
nately in both directions and this applies also to those which develop
from above and to those which disappear from above.

Thus far my previous investigations had led me.

The renewed investigation was begun again with these experiments ;
they gave entirely concordant results.

As the experiments with cross-wounded leaves had shown that
it is possible to deviate the large nutritive currents from their way
and to cause them to assume a lateral direction, the question was

whether it would be possible fo go farther still and
to lead the current in an opposite direction.

Formerly already [| had made similar experiments,
which had given a favourable result, but there was
reason to repeat them now on a more extensive scale.

The arrangement of the experiments was such that
two internal partitions were produced forming two
hooks, embracing each other, and the short arms of
which extended as far as the edge of the leaf (fig. 4

Hence the connection between top and = base lay
through the whole middle piece between the two
longitudinal partitions and in this piece the develop-
ment of the current would have to take place in a
direction opposite to that in the normal leaf.

Now the experiments proved that indeed such a
development, and thus so to say the “reversion”, in this
middle piece is possible, and that the typical direction
of the currents is then, as indicated in the figure by

the continuous line. The experiments proved besides :

Fig. 1. 1. that it takes a long time before by this route

a powerful connection between top and base is formed, some weeks
being required ;

2. that the attempt is suecessful only when the distance between

50 mm. being the

the cross-wounds is not too considerable, 25
extreme limit;

3, that for success it is desirable that the impulse from above
be powerful, which is the case, for example, if above the highest
cross-wound one or more vigorous prolifications occur ;

4. that the leaf always strongly opposes the reversion.

Concerning this latter point I must add what follows:

When a leaf of Caulerpa is eut off, either at the leafstalk or at
a higher level, rootlets are formed at the cut piece and this nearly
always exactly at the sectional plane i.e. at the organic lower side ;
a middle piece from the leaf does exactly the same.

The plant thus makes an attempt at beginning an independent life
by neo-formation. (In nature this is the most powerful, if not the only
means of propagation for Cun/lerpa, since it seems to have no sexual
organs).

Now one sees the same happen with the double-hooked wounds :
along the whole breadth above the lower cross-wound rootlets often
grow, proving that communication has become so much impeded,
that the first piece of the leaf (1, which is in communication with

( 424 )

the top) and the middle piece, IH, evidently meet their want of con-
nection sooner and perhaps better by means of the neo-formation of
rootlets, than by strengthening the existing but feeble communica-
tion with the old ones. *)

The third part of the leaf, II1, (which consequently is in direet
communication with the base) never shows any inclination to the
formation of rootlets, obviously because the communication has
remained unimpaired here.

If we must assume that the stream-bundle in the uninjured leat
is regulated by a basally directed impulse, then, when the ‘‘reversion”
has succeeded, the newly formed current in the middle piece must be
directed by an opposite impulse, or, to speak more correctly, by the
same impulse, after it has, so to say, been reflected by the cross-wound.

That this current in fact behaves in this manner, follows at once
from the fact that the new current is first visible below in the middle
piece and is gradually prolonged upward.

A still more convincing proof of this can be given by a further
experimental operation: if namely these new currents are interrupted
in I and II by a small ecross-wound (as in fig. 1) one sees the thrust
in I oceur above, in II on the other hand below the wound, and
the currents take their way as is indicated by the dotted line in the
figure. This is a proof that these two adjacent pieces behave oppositely.

Though we finally often succeeded in bringing about the “reversion”
in the middle piece, yet this reversion is very incomplete, as I infer
from the following observation. In one of the leaves with a double-
hooked wound a prolification had been formed above in the middle piece,
while the complete reversion was being brought about; the new
leaflet lay a litthe sideways of the current. Proceeding from this
leaflet a little bundle of three currents had developed. One of them
proceeded along the lower side of the upper cross-wound into the
third part of the leaf, afier having joined the main current coming
from below. The other two, however, took their way straight down-
ward as if the connection with the base of the leaf were still exactly
as before the lesion. Hence one of the currents, when coming forth
from the leaflet, obeyed the action of the reflected impulse, whereas
the other two experienced no influence. In that place of the middle
piece the old basipetal impulse must consequently have been preserved.

A similar case, occurring in) another’ experiment, will be .men-
tioned later.

1) Above the upper cross wound also rootlets are sometimes formed, although
only when the top-part is large and so powerful enough, or when prolifications
occur on it.

Also one of the leaves, mentioned above, in which a small cross-
wound was made in the middle piece and at the same time in the
first piece (as in fig. 1), showed a phenomenon which [ can only
explain in this manner. It was pointed out already that above wounds
new rootlets are regularly formed; we shall hereafter describe the
phenomena preceding the formation of the rootlets, phenomena which
always make themselves felt in a basipetal direction. These preli-
minary phenomena now appeared in that leaf in the first part (1)
above the small wound, as usual, and in the middle piece also
above the small wound. If the polarity of this whole piece had been
reversed, these changes should have appeared there be/ow the small
wound. Now this indicates, in my opinion, that the reflected impulse
was localised and had no influence on the lateral part of the middle
piece, after it had been withdrawn from its direet action, and
this piece, having retained the old impulse, reacted therefore as
normally.

After I had succeeded in ‘reversing’ a current, it was probable
that it would also be possible to cause a whole plant to develop
inversely.

When, however, this experiment was made in such a way that a
whole plant, with rhizome and rootlets, was reversed and the leaf-
tops were buried in mud, it gave no result; for seven weeks such
a plant remained absolutely unaltered; only the top of the leaf
became white on account of the loss of chlorophyl-grains, caused
by the darkness, while the rhizome grew a little and made some
new rootlets.

Cut leaves, freely suspended upside down or planted with their
top in mud, gave quite different results, however.

Nearly all the leaves, and especially and most quickly the youngest,
first formed new rootlets, which also in this position of the leaf always
arose at the end of the stalk ; very many appeared already after two days.

After that several prolifications appeared generally in various
places, and then a first consequence of the reversion could be observed
in the course taken by the stream-bundles coming out of the proli-
fications and continuing their way through the old leaf.

In a cut but erectly planted leaf these go always, without exception,
to the base of the leaf; here in nearly all prolifications the greater
part of these currents went to the base also, but some of them took
their course towards the top of the leaf, without reaching it however.
Gravity, acting in the opposite direction during their formation and
development, had evidently deviated them.

Still more clearly the existence of an antagonism between gravity

( 426 )

and the basipetal impulse was visible in some of these leaves from
the fact that stream-loops were formed. From the young prolifications,
namely, some currents were seen going to the top of the leaf, which
however later suddenly returned with a very sharp bend and then
went back straight to the leaf-stalk. So gravity had first deviated them,
but the continuous counteracting influence of the basipetal impulse,
which was evidently felt in every point of the leaf, had at last
overcome gravity and got the upper hand. These loops had in large
leaves a length of 5 to 10 mm.

All these changes took place’ before at the apical side rootlets
developed. This occurred finally with very many leaves; the earliest
appeared after 9 days, the majority came later, but after almost four
weeks they had not yet developed in all of them. That most of them
had formed rootlets at the extremity of the leaf-stalk much earlier,
proves that they possessed to the full the power of forming them.
That the presence of rootlets at the leaf-stalk was no impediment
for the development of new rootlets elsewhere, appeared also from
the fact that, with respect to these latter, no difference could be noticed
between leaves with and without rootlets at the stalk.

If now these leaves were planted in mud with their apical rootlets
(which, however, were hardly ever placed exactly at the top, but
at a smaller or greater distance from it) the prolifications grew on
or, if they had not been present beforehand, they always appeared
after this. A connection was generally formed between the stream-
bundle issuing from them and the rootlets and so a plant was obtained
in which, under the impulse proceeding from the prolification, the
nutritive current had developed in a direction opposed to the impulse
existing in the leat.

Here also it could be proved in the same way,
A as before by means of a cross-wound (as in fig. 2,
in which the basal half of a leaf was planted:
upside down), that this bundle in the leaf obeyed

indeed the impulse of the prolification A, since
from this side the thrust occurred.

Yet here also the reversion appeared to be
only local. A small prolifieation 4 had namely been
formed below the ecross-wound, after this had
been made (consequently at the side of the apical
rootlets). This new leaflet in its turn formed a

Fig. 2. small stream-bundle of which some thinner currents
went in the direction of these rootlets; one thicker current however
took his course athwart alongside the wound, turned at the end with

a sharp bend and went to the old base of the leaf. In doing so this
current crossed ') the bundle going from the other prolification to the
apical rootlets, buf even this did not cause it to change its direction.

So here also a reversion was obtained, this time by the influence
of gravity, but it also was proved to have a very local character.

The inverted leaves gave me material for still another experiment.

These leaves, as has been remarked, had at last for the greater
part formed prolifications and rootlets towards the apical side. Now
in some of them a prolification and a rootlet were found at about
the same height, but the one on the left side of the leaf, the other
on the right.

What would happen now if this piece of the leaf, isolated from
the other basal rootlets and prolitications by a cross-wound, were
planted separately > Since a manifold direct, but feeble communication
actually existed between the two organs by means of the numerous
fine protoplasmic currents, if was possible that the direct communication
would be strengthened and so a cross-current would arise, in the
same way as above a large cross-wound. But it was also possible
that the basipetal impulse of the prolification and of the piece ot
the leaf would not admit a communication or not one in that
direction.

The three experiments for which suitable material was obtained,
were not entirely at an end at my departure. Yet it then appeared
already with perfect distinctness that nowhere a strengthening of
the cross-communication had taken place. On the contrary, the currents
everywhere went from the prolifications straight to the basal wound ;
currents communicating with the rootlet showed the same. Even an
indirect communication between the two, via the basel wound, was
not established.

In one of the leaves a young rhizome was formed beside the
prolification at a distance of 1'’, millimetre from it; a communication
between the two was established, but by a very roundabout way,
viz. via the basal wound, which lay at a distance of 6'/, mm.

In another leaf a rhizome was developed near the rootlet and
another a little above the prolification. In both cases the communi-
cation between each of the two groups was again established via
the basal wound, but none between prolification and rootlet. The
complete physiological separation between prolifieation and rootlet

1) The current seemed to intersect the bundle, but as the currents proceed
from beam to beam and often two currents are attached to one beam at different
heights, these currents must have crossed each other. The same often happens with
stream-loops.

( 428 )

finally appeared from the fact that in a small spot at the wound,
exactly in the place where the currents from the prolitication reached
if, a number of small rootlets were formed.

Hence the basipetal impulse was so strong that it entirely prevented
a cross-communication, as a consequence of which each of the parts
of the leaf formed two individuals, cohering in a morphological
sense but scarcely in a physiological sense.

We spoke above of currents that were reflected by the wound;
this expression was chosen because the direction of the wound evidently
influences the direction which the current assumes afterwards and
this in a similar way in which a solid wall affects an impinging
wave-front.

This influence is most clearly seen when of three leaves the top
part is cut off, (this latter being taken as large as possible) following
in the first leaf a transverse line, in the second a V-shaped one, the
point of the V being downward, and in the last leaf an inverted
V-shaped line. After a few days the currents are seen to bend near
the wound in such a way that the lines bisecting these current arches,
are in the first case parallel to the longitudinal axis of the leaf, in
the second converge and in the third diverge. These currents are
often so strong that one can follow them over long distances with
the naked eye.

However, only those paris of the currents that lie near the wound
must be taken into account, firstly because the reflection is not
sudden but gradual, so that the currents assume a more or less sharp
bend with a radius of ‘/, to 2 mm, secondly because the leaves are
rather narrow and so the reflected currents cannot, for a long
distance, freely continue their new course.

That in the formation of wound-cork in higher plants the new
cross-walls in) the phellogen are always parallel to the direction of
the wound in the nearest place, suggests a similar influence of the
wound in these plants.

The basipetal impulse, indicated by the experiments mentioned,
shows itself no less distinctly in the formation of new organs in cut
leaves. ')

1) | never saw rootlets or rhizomes arise on intact leaves, attached to the rhi-
zome; cut rootlets die off at once, while loose rhizomes, when they are strong
enough, form new organs, but always in an entirely normal way.

.

( 429 )

Wakker') already pointed out that in these the young rhizomes
and rootlets always arise above the basal wound.

Investigation has shown that immediately after the lesion the
formation of these organs is prepared, namely by a division in the
protoplasm, This I could only observe in leaves which were in very
eood condition of life; in these, however, the changes were well
visible with the naked eye or else with the hand-magnifier.

Above the basal wound a clear white spot is gradually seen to
arise, often several millimetres in size. In these places only the
rootlets are later formed, while in the immediate vicinity of them
the rhizomes appear.

Where it was mentioned above (pag. 425) that the wounded leaves
showed an inclination for forming rootlets, the arising of such a
white spot was meant.

The first question now was: what causes this white spot?

In vigorous cut leaves one sees often already one day after the
cutting whitish streaks occur, of which no trace can be detected in
the intact plant. As far as they are rendered visible by a strong hand-
magnifier they begin at some distance from the top as well as from
the edges of the leaf, but become soon thicker and proceed in a
feeble curve (which is concave towards the edge) towards the middle
and there assemble and proceed together to the leafstalk ; here and
there they are connected among each other. So their mode of proceeding
is exactly the same as that of the green currents.

But also in other respects they behave like these latter: if the
cut leaf bears a prolification, from the stalk of this latter a bundle
of these currents passes into the leaf; when the currents meet a
cross-wound they proceed as far as this, move sideways and when
they have arrived at the corner of the wound, continue their way
straight to the leaf-stalk.

These currents, which sometimes appear light greenish because they
are seen through the peripheral layer of chlorophyl, consist of a
very fine-granulated and therefore milky white protoplasm, very dif-
ferent from the much clearer protoplasm of the green currents, in
which the chlorophyl-grains are moved along. The white currents
partly originate from the green ones: these latter are namely seen
to become feebler when the latter arise, while at the disappearance
of the white currents the green ones gradually become more distinct
again.

1) Die Neubildungen an abgeschrittenen Bliattern von Cawlerpa prolifera; Versi.
en Meded. der Kon. Akad. van Wetenschappen te Amsterdam, 1886, 3d_ series,
part 2, p. 252.

( 430 )

From all points of the wounded leaf consequently fine-granulated
protoplasm flows together; it gathers immediately above the wound,
replaces the chlorophyl-containing peripheral layer and currents and
so causes the leaf locally to assume a white colour.

When the white currents are observed microscopically, also in them
a distinct streaming is observed, mostly in the two opposite directions
at the same time or otherwise alternately, while a number of unco-
loured granules are dragged along. But chlorophyl grains are entirely
lacking. Yet after the lesion the quantity of plasm above the wound,
“white” as well as ‘green’, increases, while in the top it diminishes,
occasionally to such an extent even that the top becomes empty
and dies. From this follows that the mass of plasm, conveyed down-
ward by the currents is greater than the mass which is taken back
upward, so that the resultant of the two motions is equivalent to a
current going to the organic base.

So the white currents behave exactly like the green ones; yet
there is a difference between them, although only a quantitative one:
while both groups of currents obey the same basipetal impulse, this
latter appears to exert a somewhat greater influence on the white
currents than on the green, for the green protoplasm is always
observed to be pushed aside by the white.

Now, when it had appeared that the white currents move towards
the basal wound, the question arose whether they only strove to
reach this wound or, perhaps not contented with this, would also
try to occupy the very lowest (most basal) place near this wound.

Experiments showed this latter to be indeed the case.

If a wound be made in a slanting direction with regard to the
diameter of the leaf, the white plasm flows down along the wound
and assembles in the sharp point; if the wound be V-shaped all
gathers in the middle, while with a ,-shaped impediment the white
currents flow off to the two points near the edges of the leaf. With
these lesions the green currents behave exactly as the white ones,
but again their terminal point is left a few millimetres behind that
of the uncoloured currents. From this it appears more clearly: still
that these latter feel the basipetal impulse more strongly.

Especially the current of uncoloured: protoplasm which flows off
along the wound is seen to follow a wavy course, since it consists
of very short pieces of current, which go longitudinally downward,
are then retlected by the wound and soon afterwards bend down
again. Not unfrequently two currents run close to each other and
in doing so cross each other repeatedly. The height of these waves

is small, no more than */, to */, mm.

( 431 )

The basal part of the so wounded leaves underwent no change
when if remained in connection with the rhizome. If, however,
shortly before, it had been separated from it, the lower piece behaved
like a cut entire leaf. White currents here also appear; they however
only begin at some distance below the lesion as fine lines and,
growing thicker, pass all into the leaf-stalk. At the lower end of
this latter the accumulation of white protoplasm then takes place.

The very sharp antithesis between the phenomena below and
above the wound, again furnishes a striking proof of the existence
of the basipetal impulse and of its influence on the white plasm.

With regard to the origin of this plasm it must be remembered
that all organs during their growth always contain a large quantity
of such plasm at their top. Behind it, when growth has been com-
pleted, it is clear and contains in leaves and rhizomes a very great
number of chlorophylgrains. When an organ stops growing, the
white top soon disappears.

For this reason and on account of its appearance, | compared
already formerly ') this fine-granulated, turbid protoplasm to that which
fills the meristem-cells in higher plants.

Hence Caulerpa possesses, notwithstanding its being unicellular, a
“meristem-plasm’” which, however, is only to be found during growth
and in the growing tops. After the growth has ceased it disappears,
which disappeareance must be regarded as a mixing up with the
remaining protoplasm *).

The experiments now showed that after serious lesion this meristem-
plasm is secreted again (which can take place evidently in all points
of the leaf), after which it unites to currents of increasing thickness
and flows together at the organic base.

On the thus formed white spots the rootlets are produced, while
the rhizomes take their origin in the immediate vicinity of them,
mostly on the transition of the white spot to the dark green part,
but still within reach of the white currents. So both arise in conse-
quence of the resulting descending current after the lesion.

Hence the rootlets and rhizomes derive their meristem-plasm from
the same confluent, turbid protoplasm and therefore this latter may
in itself be regarded as meristem-plasm.

Although the source of the meristem-plasm of rootlets and rhizomes

1) 1. c.-p. 203.

*) Such a secretion of meristem-plasm from the protoplasm of the cell and its
resolution in it, has recently been described by Nott for a closely related plant
(Bryopsis); cf: Beobachtungen und Betrachtungen tiber embryonale Substanz;
Biologisches Centralblatt, 1903, No 8,

( 432 )

is the same, vet there exists a sort of antagonism between the two.

So, for example, it is not unfrequently seen that when somewhere
on a leaf a rootlet has been formed, immediately behind it a rhizome
arises, or the reverse.

The most striking case in this respect I observed with a leaf which
had formed two rhizomes laterally of the leafstalk (which is a rare
occurrence) one close above the other: at the other side of the leafstalk,
exactly behind each of the rhizomes, a well-developed rootlet was
found.

Properly speaking this antagonism is already observed when a
rootlet is formed on a rhizome in the ordinary way; it namely does
not arise at some distance from the top, but quite close to it, so that
sometimes the impression is given as if the top, of the rhizome would
divide dichotomically, i.e. into two equivalent branches, whereas
later one point develops into a rhizome, the other into a rootlet.

So the two meristem-plasms are very nearly equal, until at a
certain moment a division takes place. The principal cause of this
division is, in my opinion, light.

The rootlets can very well form and develop in light, but, if
possible, they seek the shaded side or turn away from the light.
Rhizomes, on the other hand, as well as prolifications, generally
are formed at the bright side.

Now taking into account that Noun?) has shown that a rhizome of
Cailerpa forms rootlets at the upper side if this is shaded above
and only illuminated from below, | think we have every reason to
look upon the difference in the intensity of the light on both sides
of the rhizome as the principal cause of the separation, which takes
place in the at first homogeneous meristem-plasm, and hence also of
the antagonism between rhizome and rootlet.

That also internal causes play a part here, follows already from
the fact that ‘the rootlets as well as the leaves, are formed on the
rhizome at distances which for each of them are pretty regular.

A rhizome-top is even occasionally seen to dissolve entirely into
rootlets, which proves that there can be no considerable difference
of origin between them.

So we are naturally led to the question: how do the leaves arise?

In this respect I must restrict myself to a few hints, since the
investigation of this point has not been completed yet.

In the intact plant they arise either on the rhizome or as_prolifi-

!) Einfluss der Lage, u.s. w.; Arbeiten aus dem botanischen Institut in Wiirzburg,
Ba. Ill, 1888, 8. 470.

( 433 )

cations on the leaves. On the rhizome they arise on the upper side
but, in opposition with the rootlets, always at a great distance (a few
centimetres) from the top and consequently quite out of reach of
the meristem-plasm there. | presume that their formation on the upper
side is also determined by light, although this has not been proved yet.

In unwounded leaves, and hence in the riormal life of the plant,
they are formed on full-grown leaves and then generally near the
top, in cut leaves they very rarely occur near the top ; in these they
as a rule arise on the lower two thirds of the leaf, preferably even
on the lowest third part, but hardly ever immediately above the
wound. So here also they arise out of reach of the meristem-plasm.

The formation of the leaf begins with the appearance of a very small
white spot on the dark green organ. This rapidly grows out into
a cylindrical, soon broadening appendix, which often remains entirely
white until it has reached a length of one centimetre, after which
it becomes green from below during further growth. The top remains
white as long as the leaf increases in length, but turns green when
erowth is arrested, either by the leaf having reached maturity or by
unfavourable external circumstances.

In no case the formation of a leaf was preceded by the appearance
of a large white spot with affluent streams of meristem-plasm. This
leads to the conclusion that the young leaf derives its meristem-plasm
evidently from the protoplasm of the whole neighbourhood ; so a
preferred direction of motion, as a consequence of a basipetal or
acropetal impulse, cannot be detected. As a consequence of this each
of these currents is so feeble, that it could’ not be observed with the
hand-magnifier. So the formation of leaves appears to be independent
of the descending current of meristem-plasm.

In one case only I have seen white currents in connection with
a young leaf; in a cut leaf I observed that a strong white bundle
had differentiated itself, running close to the base of a young leaflet
(prolification) that had arisen after the cutting. This was a little over
a centimetre long and still as white as ivory. From its stalk six
white streams passed into the leaf; they all ran in a basal direction
and soon became absorbed in the white principal bundle.

59 these also obeyed the basipetal impulse; they had gradually
formed during the development of the leaflet and so had not appeared
as preliminaries to the formation of it, as is the case with the rootlets.
Also the large white spot was absent here.

Since the white currents in this cut leaf also flowed together
at the base in order io prepare there the formation of a rhizome
and of rootlets, we may infer from this that there is no essen-

Bi

Proceedings Royal Acad, Amsterdam. Vol. VIL.

( 434 )

tial difference between the meristem-plasm of leaves on one hand
and that of rhizomes and rootlets on the other.

Hence there must be causes which in the cut leaf or in the
rhizome bring about a division in the plasm for the formation of new
prolifications and leaves, and since in cultures it is regularly observed
that new prolifications arise on the lighted side of the leaf, light
certainly plays a part here. Undoubtedly, however, there are still
other, internal, at present unknown, factors which cooperate in deter-
mining the origin and place of origin of the prolifications.

From the here briefly described observations we may infer that
in the leaves of Cuulerpa a basipetal impulse is active, proceeding
from every point of the leaf and revealing itself in two ways :

1. in leaves, connected with rhizomes and rootlets, in the course
of the “green” currents of protoplasm in the unwounded leaf as
well as in the severely wounded leaves.

2. in cut, vigorous leaves in the occurrence and course of the
“white” currents of meristem-plasm which partly assemble at the
most basally situated place. It is this descending current which
prepares the formation in that place of new rhizomes and rootlets.

None of the observations, on the other hand, gave reason for
assuming also the existence of a contrary, acropetal impulse, thus
even not with the formation of leaves and_prolifications.

It proved possible, by certain lesions and by planting invertediy,
to cause the formation of currents which developed contrary to the
currents in the normal leaf. It could be proved however that this
was not an inversion of the impulse itself, but that gravity or reflection
by a wound caused a change in the direction of the curreut, whereas
the basipetal impulse underwent no or hardly any change, and even
then in any case only a very local one.

The chief phenomena observed with Caulerpa remind us of obser-
vations which have been known for a long time in higher plants.

The consequences of annular wounds or of large transverse wounds
in the bark-tissue of trees, the formation of much plasm-containing
tissue (callus) above the wound, the frequent mortification of the
tissue below the wound, the formation or sprouting of adventitious
roots exclusively above the wound and in the very lowest place of
the living bark-tissue, are, mutatis mutandis, evidently analogous.
The same may be said of the formation of adventitious roots exelu-
‘sively at the base of cut leaves of very many plants, of the regene-
rative phenomena of the fruit-stalks of Marchantia, ete.

The former phenomena were explained by the older physiologists

( 435 )

by means of the hypothetical descending ‘“sap-current’’, by the newer
physiologists they were, as a rule, not explained at all. Now that the
existence of a resulting descending current could be proved with
Cauulerpa, which shows so many analogous phenomena, it seems to
me to be probable that on closer investigation it will also be found
with higher plants, although perhaps in an entirely different form
than was originally thought.

Physics. — “Double refraction near the components of absorption
lines magnetically split into several components’’, according to
experiments made by Mr. J. Grust. By Prof. P. Zepmay.

It has already appeared from experiments which | had the honour
to communicate to the Academy on a former occasion that the
magneto-optic theory of Voter’), who established a simple and
rational connexion between the magnetic splitting up of the spectral
lines and dispersion, accounts extremely well for all the phenomena
observed in the region of the absorption lines.

If light traverses parallel to the lines of force very attenuated
sodium vapour placed in the magnetic field, the plane of polarization
is rotated in the positive direction for all periods lying outside the
components of the doublet, but in the negative direction, and very
strongly *), for periods intermediate between those of the components.

If light traverses the vapour normally to the field, there is double
refraction as predicted by Vorer from theory. When placed in a
magnetic field, all isotropic bodies should show double refraction,
but to a measurable degree only in the neighbourhood of the absorption
lines. Vorer in collaboration with Wiscnrrt experimentally verified
this result, using a small grating and a flame with relatively much
sodium vapour.

I have extended these results*) by working with sodium vapour
so dilute that, in a strong magnetic field, there were seen the four
absorption lines corresponding to the components of the quartet into
which the line D, is split by the magnetic field. The mode of depen-
dency of double refraction on the period could, in this special case
with some reserve, be predicted from Voier’s theory. Observations,
in which Mr. Grrst took part, confirmed the theoretical result. Mr.

1) Voier, Wrepemany’s Annalen. Bd 67, p. 359, 1899.
*) Zeeman, Proc. Acad. Amsterdam, May 1902, see also Hatxo, Thesis for the
doctorate, Amsterdam, 1902.
%) Zeeman en Geust, Proc. Acad. Amsterdam, May. 1903.
30*

( 436 )

Grest has now extended these observations and will give a more
detailed exposition of his results elsewhere’); I intend to give here
a short explanation of them.

The arrangement of the apparatus was for the most part the
same as in our former experiments. Plane polarized light, under
azimuth 45° to the vertical, falls on a Basiner’s Compensator with
horizontal edges. The light then traverses a second Nicol with its
plane of polarization perpendicular to that of the first. An image of
the system of parallel interference bands in the compensator, is
thrown on the slit of the spectroscope. The light is then analysed
by means of a large Row Lanp grating mounted for parallel light.
The greater part of the experiments were made with a compensator
of which the prisms had angles of about 50’, but for the study of
some details compensators were used with angles of 10! or of 3°. In
the spectroscope a few dark horizontal interference bands are observed
as long as the magnetic field is off. The fine absorption lines of the
vapour are then coincident with the reversed sodium lines due to
the are light. As soon as the field is on, the bands become distorted.
Their vertical displacements are, with the method used, proportional
to the difference of phase between vibrations respectively parallel
and normal to the field.

For the simplest case of a line split by the field into a triplet,
Voret deduced a formula giving the difference of phase as a function
of the wave length*). The sodium lines YD, and D, being split, —
however, by the magnetic field into a quartet and a sextet, it was,
in order to compare theory with observation, necessary to deduce
the formulae for these cases. Mr. Grest has made these calculations
according to the method already indicated by Voier*) on another
oceasion. According to his calculation, the difference of phase between
vibrations normal and parallel to the field, the light having traversed
a layer 7 of the absorbing vapour, is given by:

x Wel 4d?—d90? 1
~ V2 | (40?9—d? 93)? 4 R20? ~ 4d? —d? 90?

In this formula V indicates the velocity of light in the aether,
R the strength of the field, «,d,d' and ¢ being constants characte-
ristic of the medium. Moreover 227%, —r, is the period of vibration
and d=0%. The formula given applies to the case of the sextet; for
the quartet, d@’ =o and for the triplet, moreover d=o. Figs. 1—3

1) Geest, Thesis for the doctorate, Amsterdam, 1904.

2)eVoreralenc:

3) Voret Wied. Ann. 68 p. 352. 1899.

give the graphical representation of 4 as a function of Jd for each
of these three cases.

The result of the observations is represented in figs. 4—8.
These drawings are made with the aid of photographic negatives.
We have not yet sueceeded in getting negatives that showed all
details simultaneously and equally well. Hence ocular observations
had to supply the imperfections of the photographie records.

Figs. 1, 4, 5 refer to the triplet (type line D, in feeble fields) ;

figs. 2, 6, 7 to the quartet (type line D,); figs. 3, 8 to the sextet
pe line D,).
When comparing the results of observation with theory, it should
be taken into account that the theoretical curve indicates the distortion
which one single interference band would undergo. With the method
of observation used, the central part of the field of view contained
also parts originating from bands lying higher and lower than the
one considered. The theoretical figure must therefore be completed
with parts of theoretical curves lying above and below the one
represented.

We will first of all consider the quartet. We indicate the bands
by a, 6, c, a being the superior one, and by 1, 2, 3, 4, we indicate
the positions in the spectrum which would be occupied by the com-
ponents. The double curved line between 2 and 3 shows entirely
the same character in both figures. This sinuous line (figs. 6 and 7)
thickens out at the extremities into more intense parts (where the
double refraction is at a maximum or at a minimum) turning their
concave side towards band 6. These intense parts correspond to the
loop of the theoretical curve, the loop between 1 and 2 belonging
to band c, and the one between 3 and + to band a. It was not to
be expected that the two branches which asymptotically approach
the components, would be seen separated from the loops. The distance
is too small by far to allow that. The vertical central line in the
figure is the reversed sodium line due to the are. With increased
vapour density the loops increase their distance from their band.
Fig. 7 relates to this case, which is also in accordance with theory.
As the vapour density increases, fewer details become visible, but
we will not go further into this point now.

The observations concerning the sextet are very difficult on account
of the extremely small distance of the components. It is already
difficult to observe the inverse sextet, and hence so much the more to
observe phenomena occurring between its components. Only under
very favourable circumstances -could ihe phenomenon be observed
as it is represented in fig. 8. The other phenomena observed with

( 438 )

D, are most readily interpreted by considering them as originating
from a triplet and not from a sextet.

It seems rather superfluous to give any further explanation of
figs. 8, 4, 5; in the case relating to fig. 5, the vapour density is
again greater than in fig. 4. All the phenomena we have considered
are qualitatively in excellent accordance with Vorer’s theory.

The phenomena described for D, and D, again demonstrate the
existence of very characteristic differences between different spectral
lines, differences no less striking here than in the case of the related
phenomena of the magnetic separation of the spectral lines and of
the rotation of the plane of polarization in the interior‘) of, and
close to, the absorption line. It is certainly very interesting that the
theory explains the entirely different behaviour of D, and D, in the
case now considered by differences between the velocities of propa-
gation of vibrations normal and parallel to the field, assuming, of
course, the magnetic division of the lines.

Physics. — “The motion of electrons in metallic bodies”. 1. By
Prof. H. A. Lorentz.

It has been shown by Riecke*), Drupr*) and J. J. Tomson *)
that the conductivity of metals for electricity and heat, the thermo-
electric currents, the THomson-effect, the Haui-effect and phenomena
connected with these may be explained on the hypothesis that a
metal contains a very large number of free electrons and that these
particles, taking part in the heat-motion of the body, move to and
fro with a speed depending on the temperature. In this paper the
problems to which we are led in theories on these subjects will be
treated in a way somewhat different from the methods that have
been used by the above physicists.

§ 1. I shall begin by assuming that the metal contains but one

1) Zeeman, Proc. Acad. Amsterdam May 1902, see also the description of another
phenomenon in Voter, Géttinger Nachrichten, Heft 5, 1902.

2) E. Rrecxe, Zur Theorie des Galvanismus und der Wiirme, Ann. Phys. Chem.
66 (1898), p. 353, 545, 1199; Ueber das Verhiiltnis der Leitfiihigkeiten der Metalle
fiir Wairme und fiir Elektrizitiit, Ann. Phys. 2 (1900), p. 835.

5) P. Drupr, Zur Elektronentheorie der Metalle, Ann. Phys. 1 (1900), p. 566;
3 (1900), p. 369. :

4) J. J. Tuomson, Indications relatives 4 la constitution de la matiérve fournies
par les recherches récentes sur le passage de |’électricilé & travers les gaz, Rapports
du Congrés de physique de 1900, Paris, 3, p. 138

“
{

( 439 )

kind of free electrons, having all the same charge ¢ and the same
mass m; the number of these particles per unit volume will be
represented by NN, and I shall suppose their heat-motion to have
such velocities that, at a definite temperature, the mean kinetic
energy of an electron is equal to that of a molecule of a gas. Deno-
ting by 7’ the absolute temperature, I shall write for this mean
kinetic energy «7’, where @ is a constant.

We shall further consider a cylindrical bar, unequally heated in
its different parts, so that, if v is reckoned along its length, 7’ is
a function of this coordinate. We shall also suppose each electron to
be acted on, in the direction of OX, by a force mX, whose intensity
is a function of x. Such a force may be due either to an electric
field or, in the case of a non-homogeneous metal, to a molecular
attraction exerted by the atoms of the metal. Our first purpose will
be to calculate the number of electrons » and the amount of energy
W crossing an element of surface perpendicular to the axis of x in
the positive direction, or rather the difference between the numbers
of particles in one case and the quantities of energy in the other
that travel towards the positive and towards the negative side. Both
quantities » and JI’ will be referred to unit area and unit time.

This problem is very similar to those which occur in the kinetic
theory of gases and, just like these, can only be solved in a rigourous
way by the statistical method of Maxweti and Boutzmann.

In forming our fundamental equation, we shall not confine ourselves
to the cylindric bar, but take a somewhat wider view of the subject.
At the same time, we shall introduce a simplification, by which it
becomes possible to go further in this theory of a swarm of electrons
than in that of a system of molecules. It relates to the encounters
experienced by the particles and limiting the lengths of their free
paths. Of course, in the theory of gases we have to do with the
mutual encounters between the molecules. In the present case, on
the contrary, we shall suppose the collisions with the metallic atoms
to preponderate; the number of these encounters will be taken so
far to exceed that of the collisions between electrons mutually, that
these latter may be altogether neglected. Moreover, in calculating the
effect of an impact, we shall treat both the atoms and the electrons
as perfectly rigid elastic spheres, and we shall suppose the atoms
to be immovable. Of course, these assumptions depart more or less
from reality; I believe however that we may safely assume the
general character of the phenomena not to be affected by them.

§ 2. Let dS be an element of volume at the point (wv, y, 2). At

a0.)

the time 7¢, this element will contain a certain number (in fact, a
very large number) of electrons moving in different ways.

Now, we can always imagine a piece of metal of finite dimensions,
say of unit volume, in which the ‘concentration’, as we may
call it, of the electrons and the distribution of the different velocities
among them are exactly the same as in the element d/S. In studying
the said distribution for the .V electrons, with which we are then
concerned, we shall find a diagram representing their velocities to be
very useful. This is got by drawing, from a fixed point O, .V vectors,
agreeing in direction and magnitude with the velocities of the electrons.
The ends of these vectors may be called the velocity-points of the
electrons and if, through the point OV of the diagram, we draw axes
parallel to those used in the metal itself, the coordinates of a velocity-
point will be equal to the components &, 7, § of the velocity of the
corresponding electron.

Writing now

A (S, 1, $) da
for the number of velocity-points within the element dé at the
point (§, 7,6), we make the exact solution of all problems relating
to the system of electrons depend on the determination of the
function 7 (S, 1, $).

We may also say that

FAS: My EVES ADS = Ws. es a, ee

is the number of electrons in the element dS, whose velocity-points
lie in da; in particular

FAS, 7, 6) Ss dyds— eh 2 ee
is the number of electrons for which the values of the components
of velocity are included between § and §-+ ds, 4 and 4+ dy,
§$ and $+ d. The expression (2) is got from (1) by a proper choice
of the element da.

If the function in (1) were known, we could deduce from it the
total number of electrons and the quantities » and JI” mentioned
in § 1. Integrating over the full extent of the diagram of velocities,
we have

N= [7G 5) dh TOO eee. (2)
v= fesGnga . MB teeta cs KS

and if, in treating of the ‘flux of energy, we confine ourselves to
the kinetic energy of the particles,

( 441 )

W= tm {sr 2G nl Al Mee ies See tc a) |

In. the latter formula, 7 denotes the magnitude of the velocity.

It ought to be observed that, in general, the state of the metal
will change from point to point and from one instant to another. If
such be the case, the function /(S, 7,6) will depend on «, y, 2 and
t, so that. the symbol may be replaced by /(&, 1, 5, x, y, z, t). We
shall, however, often abbreviate it to //.

As to the integrations in (8), (4) and (5), in performing these, we
must treat v,y,2 and ¢ as constants.

§ 3. We shall now seek an equation proper for the determina-
tion of the function 7. For this purpose we fix our attention on the
electrons present, at the time 7, in the element dS at the point (7, y, 2),
and having their velocity-points within the element 2; we shall
follow these particles, the number of which is

TAU Le SUE? GSC) a a ee en (6)
in their course during the infinitely short time dt. At the end of
this interval those particles of the group which have escaped a
collision with an atom will be found in an element dS’, which
we may get by shifting @S in the directions of the axes over
the distances &d¢, yd, Sdt. At the same time, if there are external
forces, the velocities will have changed. I shall suppose each elec-
tron to be acted on by the same force (mX,mY,mZ). Then, for
each of them, the components of the velocity will have increased
by Adt, Ydt, Zdt and, at the end of the interval dt, the velocity-
points will be found in the element d2’, which may be considered
as the original element 4, displaced over those distances.

_ We must further keep in mind that, while travelling from dS to
dS’, the group (6) loses a certain number of electrons and gains
others. Indeed, all particles of the group that strike against an atom
have their velocities changed, so that they do not any longer belong
to the group, and, on the other hand, there are a certain number of
encounters by which electrons having initially different velocities,
are made to move in such a way, that their velocity-points lie
within d2. Writing

adSdidt
for the number of electrons leaving the group and

bdSdidt
for the number entering it, we may say :

If, to the number (6), we add (4—a)dSdadt, we shall get

( 442 )

the number of electrons which, at the time ¢-+ dé, satisfy the condi-
tions that they themselves shall be found in the element dS’ at
the point (e+ S$dt,.y+ydt,2+d2) and their velocity-points in
the element 2’ at the point (§-+ VYdt,7-+ Ydt,$+ Zdt). Hence,
since dS’ = dS and dd’ = da,
a ve 5 x,y, 2,0) + (6—a d=

=f (s +. Xdt,y + Ydt,$ + Zadt, x au Sdt, y + dt, 2 + Sdt,t + dt),

or
af ar

ay ;
pon Expt ritzy Het fate 4 (7)

This is the equation we wanted to ett 25

It is easily seen that, in calculating the numbers of collisions
adSdidt and bdSdidt, we need not trouble ourselves about
the state of the metal varying from one point to another; we may
therefore understand by ad2dt the decrease, and by bdadt the
increase which the group of electrons characterized by da would
undergo, if we had to do with a piece of metal occupying a unit
of volume and being, in all its parts, in the state that exists in the
element dS.

§ 4. We are now prepared to calculate the values of a and 3d.
Let R be the sum of the radii of an atom and an electron, n the
number of atoms in unit space, and let us in the first place con-
fine ourselves to encounters of a definite kind. I shall suppose that
in these the line joining the centres falls within a cone of the infinitely
small solid angle dw.

Taking as axis of this cone one of the straight lines that may be
drawn in it, and denoting by & the acute angle between the axis
and the direction of motion of the group (6), I find for the number
of electrons in this group undergoing an encounter of the kind chosen,

Meal (Ss N35) 7 1COR CD: CA) sea ne ee)
per unit time, a result which leads to the value
an eR KS, 5S) Fs = Sone

if we take into account a// encounters, whatever be the direction of
the line joining the centres.

Now, if we ascribe to a metallic atom so large a mass, that it
is not sensibly put in motion by an electron flying against it, the
velocity of the latter after the encounter is given by a very simple
rule. We have only to decompose the initial velocity into one

1) See Lorentz, Les équations du mouvement des gaz et la propagation du
son suivant la théorie cinétique des gaz, Arch. néerl. 16, p. 9.

( 443 )

component along the line of the centres and another perpendicular
io it; the latter of these components will remain unchanged and the
former will have ifs direction reversed.

In applying this to the encounters of the particular kind specitied
at the beginning of this §, we may take for all of them the line of
centres to coincide with the axis of the cone dw. Our conclusion
may therefore be expressed as follows: Let J’ be a plane through
the origin in the diagram of velocities, perpendicular to the axis of
the cone. Then, the velocity-point of the electron after impact will
be the geometrical image of the original point with respect to this
plane. It is thus seen that all electrons whose velocity-points before
the encounters are found in the element @2 will afterwards have
their representative points in d4,, the image of @4 with respeet to
the plane V.

By this it becomes also clear, in what way the number / can
be calculated; indeed, in encounters taking place under the cireum-
stances considered, velocity-points may as well jump from d, to
dd as from d2 to d2,. The number of cases in which the first takes

- =

place is found from (8), if in this expression we replace §, 7, by
the coordinates §', 7,6 of the image of the point (, 4, $) with respect
to the plane V. It is to be remarked that the factor 7 cos 9 d 4 may
be left unchanged, because the lines drawn from the origin of the
diagram to the points (6, 7,5) and (§', 2,6’) have equal lengths and
are equally inclined to the axis of the cone. Also d4,—=da. The
increase per unit volume of the number of electrons in the group
(6), insofar as it is due to encounters in which the line of centres
lies within the cone dw, is thus found to be
n RR? TS 7',$)rcsPdidw

and, in order to find 6, it remains only to divide this by da and to
integrate with respect to all cones that have to be taken into account.

Using the formula (8) we may as well calculate directly the
difference b—a. By this the equation (7) becomes

n R* fis (§', 4', 51) —f(S, 9, $)} cos 8d w@ =
Kee ei nA ro nT Ue 11 OF
==) + Z y+ —¢ —.
0S ay 0s a Oy saa Ot
We must now express §',7/,$ in §, 2,6. Let fg, be the angles
between the axes of coordinates and the axis of the cone dw, this

last line being taken in such a direction that it makes the acute
angle # with the velocity (§, 7, §). Then

of of of oF
MP J j J (10)

ie rs

=)

$

= § — 2r cos Heos f, x’ = 7 — Arcos 9 cosg, $' =F — Arcos cosh, (11)

( 44 )

These formulae show that, as we know already, the magnitude of
the velocity ($', 7, $), which I shall call 7’, is equal to the magnitude
ry of the velocity (&, 4, §).

As to the integration in (10), it may be understood to extend to
the half of a sphere. Indeed, if in the diagram of velocities, we
describe a sphere with centre VU and radius 1, and if P and Q are
the points of this surface, corresponding to the directions (§, 4, $) and
(f,9,, we must give to the point Q all positions in which its
spherical distance from P is less than $2. For dw we may take a
surface-element situated at the point Q.

§ 5. At the time ¢ and the point (7, y,2) the metal will have a
certain temperature 7’ and the number V, the concentration of the
swarm of electrons, a definite value.

Now the assumption naturally presents itself, that, if 7’ and N
had these values continually and in all points, the different velocities
would be distributed according to Maxwe.i’s law

F691, 5) Ae eS OP ee

Here, the constants A and / are related to the number JN and the

mean square of velocity 7* in the following way

aye
A=N Sai ee

= 3
* = —.,
2h
Since 4 mr? = eT, the latter relation may also be put in the form
om
= : Pea tae (1!
deh =

Tt appears from this that the way in which the phenomena depend
on the temperature will be known as soon as we have learned in
what way they depend on the value of h.

§ 6. The function / takes a less simple form if the state of the
metal changes from point to point, so that A and / are functions of
z,y, 2. In this case we shall put

SF (Ss M$) = Ae” + pS.) - » » + + (15)
where gp is a function that has yet to be determined by means
of the equation (10). We shall take for granted, and it will be con-
firmed by our result, that the value of @ (S, 7, §) is very small in
comparison with that of Ae~'’*. In virtue of this, we may neglect
the terms depending on ¢ (&, 7,6) in the second member of (10), this

ont

( 445 )

having already a value different from 0, if we put f= Ae—'*. For
a stationary state and for the case of the bar mentioned in § 1, the
member in question becomes

Aa wid BIN ase: |
—2hAX + ey ay | ) Sem ae fae BAL)
da: da

As to the left hand side of the equation (10), it would become 0,
if we were to substitute f— Ae—”*. Here, we must therefore use
the complete value (15), the deviation from Maxwen’s law being
precisely the means by which this member may be made to become
equal to (16).

The occurrence of the factor § in this last expression makes it
probable that the same factor will also appear in the function g. We
shall therefore try to satisfy our equation by putting

PASM Si— SROs esse « - = (17)

This leads to

F(S, 1, §) = Ae? + Sy (r)
and
FS) 1,5) =A eh + Sry (r');
consequently, since 7’ =v, if we use (isD).
F(S,7',$') —f (6, 4, $) = — 2 rcos 9 cos f ¥ (7),
so that the first member of (10) becomes

—2Ank ry (>) feos CGS OO ios Sapiet ws a2) i(LS)

Denoting by uw the angle between the velocity (§, 7,5), i.e. the
line OP, and the axis of w, and by w the angle between the planes
QOP and XOP, I find for (18)

Li a *

er

— 2nk*r? (7) | | cos* B (cos H cos te +- sin H sin w cos Wp) sin Fd Hd y=
De
=— an Rr’ x (7) cosp = — rank §ryx(r).
If this is equated to (16), the factor § disappears, so that x (7) may
really be determined as a function of v. Finally, putting
Be 4S ee I
an
we draw from (15) and (17)
Saas =A ee i( 2 AX — = +rA =)
du da:
I must add that, as is easily deduced from (9), the quantity /
defined by (19) may be called the mean length of the free paths of
dA

gahia (20)

= | Sus

the electrons, and that, in the equation (20), the terms in and

( 446 )

dh
— are very small in comparison wifh A e—””, provided only the state
ae

of the metal differ very little in two points whose mutual distance
is /. This is seen by remarking that the ratios of the terms in
question to Ae—”* are of the order of magnitude

dA

==

da —dh

and J 7? —,
£ dx

_ 1
and that, in the second of these expressions, 7? is of the same order as A

If the term in (20) which contains X, is likewise divided by
Ae-*?, we get
2hl X.

Now, 2/X is the square of the velocity an electron would acquire
if, without having an initial motion, it were acted on by the external
foree m X over a distance /. If this veiocity is very small as com-
pared with that of the heat-motion, the term in X in our equation
may also be taken to be much smaller than the term A e—*”.

It appears in this way that there are many cases in which, as
we have done, the function ~ (5, 7,5) may be neglected in the second
member of the equation (7).

The above reasoning would not hold however, if, in the case of
two metals in contact with one another, there were a real discon-
tinuity at the surface of separation. In order to avoid this difficulty,
I shall suppose the bodies to be separated by a layer in which the
properties gradually change. I shall further assume that the thick-
ness of this layer is many times larger than the length /, and that
the forces existing in the layer can give to an electron that is initially
at rest, a velocity comparable with that of the heat-motion, only
if they aet over a distance of the same order of magnitude as the
thickness. Then, the last terms in (20) are again very small in com-
parison with the first.

As yet, a theory of the kind here developed cannot show that
the values we shall find for certain quantities relating to the contact
of two metals (difference of potential and Prttrer-effect) would still
hold in the limit, if the thickness of the layer of transition were
indefinitely diminished. This may, however, be inferred from thermo-
dynamical considerations.

§ 7. Having found in (20) the law of distribution of the veloci-

( 447 )

ties"), we are in a position to calculate the quantities v and W(§ 1)
with which we are principally concerned. If the value (20) is sub-
stituted in (4) and (5), the term A e~/” leads to an integral containing
the factor §; this integral vanishes, if taken over the full extent of
the diagram of velocities. In the remaining integrals the factor &

, “yi : Dy,
occurs ; these are easily found, if we replace §* by gr the element

di by 4a7r°dvr, and if then we integrate from 7=0 to r =o.
Taking 7? =s as a new variable, we are led to the integrals

3) ioe) nD
fs es ds, fe e—hsdsand [s* e's ds,

0 0 0
whose values are

1 2 6
h2? he and rte

Finally, the “stream of electrons” and the flux of heat are given by

2 ; 1 Bh) ALK dA of dh
==, — al sae
; 3 ae h* : daz a h® du

W— 2 i 1 OhAX dA go dh AG
— 3 wm 7 ah 2 = “ais — TER . . (22)

These are the equations that will be used in all that follows.
For the sake of generality, I shall suppose (though, of course, this
is not strictly true) that, if only a proper value be assigned to /,
the formulae may still be applied even if we make other assumptions
concerning the metallic atoms and their action on the electrons. From
this point of view, we may also admit the possibility of different
kinds of electrons, if such there are, having unequal mean lengths
of free paths, and of, for each kind, / varying with the temperature.

Provisionally, we shall have to do with only one kind of electrons,
reserving the discussion of the more general case for a future com-
munication.

§ 8 From the equation (21) we may in the first place deduce a
formula for the electric conductivity o of the metal.

Let a homogeneous bar, which is kept in all its parts at the same
temperature, be acted on by an electric foree ZL in the direction of
its length. Then, the force on each electron being ¢ /, we have to put

1) It may he observed that, as must be the case, the value (20) gives NV for

’ : 3 eee
the number of electrons per unit volume and on for the mean square of velocity:
v

( 448 )
nu eE
ia m ;
Also,
dA dh
= (1) and —=0,
av ae
so that (21) becomes
4al Ae
D— ———— FF
3hm

Multiplying this by e, we find an expression for the electric current
per unit area, and in order to find the coefficient of conductivity,
we must finally divide by “4. The result is

__ 4alAe’ 93
Scapa? Rie Soc y= oe = ((22))

or, taking into account the relations (18) and (14) and denoting by

: : 3 .
wu a velocity whose square is the mean square oh of the velocity of
U

2 Ne?
o=|% ee de ee
3a as

Drupr gives the valie

heat-motion,

1 lNe?u

Sa a Pe

§ 9. The determination of the coefficient of conductivity for heat,
which we shall call % (expressing quantities of heat in mechanical
units) is rather more difficult. This is due to the circumstance that,
if initially Y=O, the equation (21) implies the existence of an
electric current in a bar whose parts are unequally heated. This
current will produce a certain distribution of electric charges and
will ultimately cease if the. metal is surrounded on all sides by non-
conductors. The final state will be reached when the difference of
potential and the electric force arising from the charges have inereased
to such a degree that everywhere v= 0.

Since it is this final state, with which one has to do in experiments
on the conduction of heat, we shall calculate the flux of heat in the
assumption that it has been established.

In the first place we have then by (21), putting vr = 0,

Shy a Se oe ee eos

da h dex

and next, substituting this in (22) and again using the formula (14),

( 449 )

82lAa dT
ee ames
Oh* da
Consequently, the coefficient of conductivity has the value
8alAa

k = wana , . . . . . . . . (26)

8 2
b=?) Zan, oa. (a7)
9 32a

Drupr’s result for this case is

or

1
a ida

The ratio of the two conductivities is by my formulae

==5(<)r. eet ec: (28)
oO 9\e

k 4 =
eS =(<) T.
oO 3 \e

Here again, the difference between the two formulae consists
‘merely in the numerical coefficients.

and by those of Drupg

k
Just like Drupr we may therefore conclude that the value of a

does not depend on the nature of the metal and that it varies pro-
portionately to the absolute temperature, consequences that have been
verified with a certain approximation in the case of many metals.
It need hardly be observed that these conclusions could only be
arrived at because we have neglected the mutual encounters between
electrons '). In fact, these would tend to diminish the conductivity
for heat, but not that for electricity, since they cannot have an
influence in a phenomenon in which all electrons move in the same

= ; } , k: :
_ way. It is clear that, under these circumstances, a value of = inde-

pendent of the nature of the metal could hardly be expected.
Let us next consider the absolute values.

: If
The value of — that can be deduced from those of / and o and
for which, using (28), I find

= Sh ast A ee)
1) See Tuomson, |. c., p. 146.

ol
_ Proceedings Royal Acad. Amsterdam. Vol. VIL.

(450 )
may be compared, as has been observed by Drupr and Remweanum'),
to a value of the same expression that is obtained from other data.
I shall suppose that the charge e of an electron is equal to that of
an ion of hydrogen in an electrolytic solution and I shall represent
by p the pressure that would be exerted, at the temperature 7’, by
gaseous hydrogen, if a unit of volume contained one electrochemical
equivalent. Then

aT

— =dp.

é

The proof of this formula is as follows. We may write for the

: ; : 1
number of atoms in unit volume of the gas considered —| for the
e

1
number of molecules =, and, since the mean kinetic energy of a
PAG

py

’, . . . a .
molecule amounts to ¢7’, for the total kinetic energy ——. As is well
ae

known, the numerical value of the pressure per unit area is two
thirds of this.

Using the C.G.S. system and electromagnetic units, we have for
the electrochemical equivalent of hydrogen 0,000104 and, putting,
T = 273° + 18’,

3p = 38.

On the other hand, the measurements of JAnGuR and DimsseLnorst
have given for silver at 18° C.

k
Ss Sa (OS AOE
6

whence, by (29),

a it ale
é
The agreement between the results of the two caleulations, for
which the data have been furnished by widely different phenomena,
though not quite satisfactory, is close enough to make us feel con-
fident that Drupr’s theory rests on a sound basis”).

§ 10. We might now return to the formula (25) and, denoting
by @ the electric potential, so that

1) Retcanum, Theoretische Bestimmung des Verhiiltnisses von Wiirme- und
Elektrizitatsleitung der Metalle aus der Drupe’schen Elektronentheorie, Ann. Phiys.,
2 (1900), p. 398.

3) A better agreement is found if, instead of (28), we use Drupe’s formula.

we might deduce from it expressions for the fall of potential in each
point and for the difference of potential between the ends of the bar.

It is more interesting, however, to make a calculation of this kind
for a more general case. Before doing so, we may observe that the
equations (21) and (22) may be applied to a thin curved wire or
bar and that we may as well suppose the normal section slowly to
change from one point to another. The line passing through the
centres of gravity of the normal sections may be called the axis of
the conductor and we shall understand by « the distance from a
fixed point, measured along this axis. We shall also assume that in
all points of one and the same normal section the properties of the
bar and the temperature are the same, but that, generally speaking,
both depend on 2, changing from one section to the next. By making
different assumptions in this respect, we come to consider circuits
of different kinds, composed of one or more metals and with any
distribution of temperature we like.

For the sake of generality we shall introduce the notion of
“molecular” forces of one kind or another exerted by the atoms of
the metal on the electrons and producing for each electron a resulting
force along the circuit in all points where the metal is not homo-
geneous. Actions of this nature have been imagined long ago by
Hrnmnorz for the purpose of explaining the phenomena of contact-
electricity. We may judge of their effect in the simplest way by
introducing the corresponding potential energy J of an electron
relatively to the metallic atoms. This quantity, variable with «
wherever the metal is not homogeneous, will be a constant in any
homogeneous part of the circuit; we shall suppose this even to be
so im case such a part is not uniformly heated. If, as before, we
write @ for the electric potential, the foree Y divides into two parts

AUX p,: =| XG,
: eaViiss e dg Nee ee pe c1i(30)
Bn = ge = — ae | oe
m da m dx

We shall now consider an open circuit, calling the ends Pand Q,
and reckoning wv from the former end towards the latter. Putting in
(21) » =O and attending to (80), we obtain for the stationary state

dg Vm cs fal m dlog A
de Ss se dw += ee da \h) eh de

whence by integration

Rae Bl)

31*

m (1 dlog A

2e nr de
P

a formula which may now be applied to some particular cases.

dz, . . (82)

a. Let all parts of the circuit be kept at the same temperature.
Then, 2 is a constant, and

1 m
— =—|V_,—V, )+-—— A,-— :
Ame ee (7 > r,) 5 “(tg 4» log 4g) (33)

The potential-difference will now have a positive or negative value,
if the ends of the circuit are made of different metals. It appears
in this way that the differences that have been observed in this case
may be attributed either to an inequality of Vp and Va, i.e. to
“molecular” forces acting at the places of junction (HELMHOLTZ), or
to an inequality of Ap and Ag, i.e. to a difference in the “con-
centrations” proper to the metals (Drupe).

It need hardly be added that (83) becomes 0 whenever the ends
are made of the same metal and that the law expressed in VoLta’s
tension-series is implied by the equation.

6. Let the metal be the same everywhere. Then A is a function
of h and (32) will always be O, if the ends P and Q are kept at
the same temperature, whatever be the distribution of temperature
in the intermediate parts.

c. Let us next examine the potential-difference between the ends
of an open thermo-electric circuit, a difference that may be regarded
as the measure for the electromotive force / existing in it. Starting
from P. and proceeding towards Q, the state of things I shall
consider is as follows: 1st Between P and a section fF’, the metal /
maintained at a temperature varying from 7’p to 7" in R!. 24 Between
R' and S', a gradual transition (§ 6) from the metal / to the metal //,
at the uniform temperature 7”. 3'¢ From S' to S", the metal // with
temperatures varying from 7” to 7”. 4 Between S” and R", a
gradual transition from the metal // to the metal /, the temperature
being 7” in every point of this part of the circuit. 5 Finally, between
R" and Q, the metal / with a temperature changing from 7” to
Ta Tp. It being here implied that the ends of the circuit consist
of the same metal and have the same temperature, the equation (32)
reduces to the last term, and we find, after integration by parts,

( 453 )

Q
pe iF pike 1 ; 34
=a mas(zle- ss. - OH
ie

This integral may be divided into five parts, corresponding to the
above parts of the circuit.

Distinguishing by appropriate indices the different values of 4 and A
that have to be considered and keeping in mind that / is a constant
both in the second and the fourth part, we have

, R"

S

ee Nea ian! We = 0
; og re in a#z= VU, Te i ’
BE SY

, U

Q

pep eet (al Pee Wa ee Va

A (alias ——(— lida === ;

f of Samper COA Ta re IGT es aa
P . Tee h"

h' and h' being the values corresponding to 7" and 7", the tempe-
ratures in 2’ and Rk". Similarly

” "

fi A iG dz = {log A ied dh
Og As 7 ve) — og er) Fe l.
K

,

If we combine these results, the formula (34) for the electromotive
force becomes

A
ed), Sees Ser
2e, lane
h’
or, if we use (13) and (14),
Y fds
2a Niu
Se log ds is ene Oe te (88
3e "a NT Ce)

Geodesy. “The connection between the primary triangulation of
South-Sumatra and that of the West Coast of Sumatra.’ By
Mr. 8. Brox. (Communicated by Prof. J. A. C. Ouprmans).

I. Short description of the triangulations of South-Sumatra
and the West Coast of Sumatra *).

Towards the end of 1896 the measurements for the primary
triangulation, which will serve as a basis for the topographical sur-

1) For a more detailed description I refer to the papers of Dr. J. J. A. Mutter,
occurring in the proceedings of the International Geodetic Association of 1892,
1896 and 1903.

vey of South-Sumatra, were begun at the station Langeiland P68.
These measurements were carried from the West Coast of Java over
the Strait of Sunda and are lately completed at the station B' Gadang
P39, situated in the Government of the West Coast of Sumatra.

The triangulation consists of one continuous chain of triangles,
which, beginning at the side Langeiland ? 68 — Gs Radja Basa P 67,
is connected with the side Gs Talang P38 — Bt Gadang P 39 of
the triangulation of the West Coast of Sumatra.

It is true that this side does not exceed the length of 17120 meters,
but a connection with the longer side Bt Poenggoeng Parang P 45 —-
Ge Talang P38 had to be abandoned after it was found that the pillar,
erected at Bt Poenggoeng Parang during the triangulation of the West
Coast, was so damaged that it no longer could be used for this purpose.

The experience made during the measurements of the base at
Padang for the triangulation of the West Coast of Sumatra, executed
by means of a 20 meters steel tape, did not tempt us to measure
also the base line for South Sumatra with this comparatively unreli-
able apparatus; and as an instrument admitting of a high degree of
accuracy was not available, no special base was measured, but the
length of the first side of the chain was based upon the two sides
Batoe Hidenng P 15 — Gs Karang P35 and Gs Karang P35 —
x2 Gede P36 of the Java triangulation. For the Java triangulation
3 base lines had been measured with an apparatus of Repsoip, which
had been sent back to Europe in 1882.

For the orientation of the South-Sumatra chain, determinations of
latitude and azimuth were made at the station Gt Dempoe P 71 in
the Lampong Districts in 1897. The geographical longitudes were
reckoned from the meridian of 38° 15’ West of Batavia. This meridian,
which nearly passes over the middle of South-Sumatra, is deter-
mined by the geographical longitude of the Java station Gs Karang
P35, as given in Abtheilung V der Triangulation von Java, p. 207.

To obtain a zero mark for the determinations of altitude, tidal obser-
vations were made during a year at Telok Betong in 1897 and 1898.
From these the mean height of the sea level in Lampong Bay, the
Lampong-zero, was derived. This was transferred to the pillar 771559
at Telok Betong by levelling, and thence by reciprocal but not
simultaneous zenith distances measurements to the primary point
Gs Betoeng P 70°).

1) In 1902 and 1903 tidal observations were also made at Benkoelen and from
them the Benkoelen-zero (the mean height of the sea level at that place) was
derived, which wili be used afterwards, when the secondary measurements will
be so far advanced.

With respect to the triangulation of the West coast, I have remar-
ked above that the steel tape, with which a base line near Padang
of 4860 M. was measured in 1888, did not admit of a high degree
of accuracy. The length of the steel tape was determined before
and after the operation by measuring with it under the necessary
precautions a line of 200 M., of which the true length was accurately
known from measurements with the base apparatus of Repsoip.

Determinations of latitude and azimuth for the orientation of the
chain were made at the West end of the base in 1883 °).

The geographical longitudes were reckoned from the meridian of
Padang, which passes through the West end of the base, for which
meridian 6° 26’ 42’ West of Batavia has been preliminarily accepted,
a difference in longitude formerly determined by chronometers.

As zero mark for the altitudes was taken the Padang-zero, the
mean sea level at Padang, formerly determined by observations
during some months of 1874 °).

For the astronomical determinations, the measurements of the
horizontal angles and those of the altitudes, the 10-inch Universal
instruments of Piston and Martins and of WreEner were used in
both triangulations.

The telescopes of these instruments are placed excentrically ; each
circle is read with two micrometer microscopes.

With the exception of the Padang base-net, where directions
were measured, the triangulation was made according to SCHREIBER’S
method; the measurements of all combinations of angles were repeated
so often that the weight of a direction adjusted at the station was
about 24, the weight of one observation of a direction being adopted
as unit.

For the trigonometrical determinations of altitude, reciprocal but
not simultaneous measurements were made; at each station, whenever
possible, 6 zenith distances were measured for each point, under
conditions as favourable as possible. With the exception of the first
measurements on the West Coast, where signals were employed,
all observations were taken on heliotropes.

As to the adjustments and computations | remark that, for the
South-Sumatra chain, exclusive of the connecting pentagon with Java,
which was adjusted according to the method of least squares, the

1) In 1896 determinations of azimuth and latitude were also made at the station
Tor Batoe na Goelang, P 62.

2) In 1889 the mean sea level at Siboga, about 350 kilometers off Padang,
was determined by tidal observations; the connection of the two marks showed a
difference of 0,85 M.

( 456 )

adjustment was effected by equally distributing the error of closure of
each triangle over the 3 angles. The computation was made in a
plane by transference by means of a Mercator’s projection according
to the method of Scnots.

The adjustments of the triangulation of the West Coast of Sumatra
were made in portions; only for the base-net and for the Northern
part the method of least squares was applied; in most cases an
approximation method was used. The computations were made on
the ellipsoid.

The following remarks may be useful for a judgment of the
accuracy which may be expected in the different connections.

(1). The distance between the base of Simplak, on which the
triangulation of South-Sumatra rests, and that of Padang is about
850 kilometers; the least number of triangles, necessary for the
transference of the length of the side Poetri-Dago of the Simplak base-
net to the first side Gs Gadoet P1 -— Poelau Satoe 2 of the
triangulation of the West Coast of Sumatra, is 49.

(2). The distance between the stations G+ Dempoe and the West
extremity of the base at Padang, used for the orientation of the net,
is about 700 kilometers; the least number of triangles, by means of
which the azimuth of the line Gf Dempoe — Gs Tenggamoes can
be transferred to the first side of the triangulation of the West Coast, is 40.

(3). The distance between Telok Betong and Padang, where the
tidal observations were made, is over 700 kilometers; the least number
of steps, necessary for the transference of the altitude of the pillar
at Telok Betong to the zero mark of Padang, is 24.

The difference between the two values of the logarithm of the
length of the connecting side is expressed in units of the 7%
decimal and corresponds to about =a of the length of the side
or to 43.2 mms. per kilometer.

The differences found are comparatively so small that their origin
may be easily explained by the accumulation of errors of observation
and by the irregularities of refraction. The difference between the
values found for the latitudes does not indicate a local deviation
at GS Dempoe with respect to Padang.

For the length of the connecting side a better result might have
been expected, if for the base measurement at Padang a more
suitable apparatus had been available.

For the rest, the differences are such that for the purpose of the
triangulation, namely, to afford a basis for the topographical work,
they do not come into consideration.

( 457 )

Il. Mean Errors.

ennehe West |
Nature of the errors. lg adn, | Coast of Remarks,
umatra STnaih
umatra
In the determination of the}
geographical latitude fF ) O24 0".35 | (1) Por the orientation of
In the determination of the | the nets,
azimuth of a night signal | Ov.25 | 0.46
| (2) Determination — derived
In the azimuth of the Ist side, in from 37 closing errors
so far as they arise from errors | (see appendix).
of the base-net. — O".85
| 3) Determination derived

In the angles adjusted at the from 73 closing errors.
Station (weight 12) ;

; | (4) If we consider only the

a from the results of the ad- | 10 triangles’), which

justments at the station; O".34 | O52. | in the shortest way
(a) | connect the first side
4 from the errors of closure of | with the connecting side,
the triangles according tothe | we find: @0."59, 1 0" .86,
formula : ;
= (3)
m= Viel | ov.6e | 0.96
oh | 2) | )

Ill. Differences found in the adjustments.

7 ce aPec South West Goast Diffe-
Nature of the determination. Sumatra of Sumatra | rence.
Logarithm of the connecting 4,2335135.7 | 4,2334948 0 !187.7

side
Length » » » 17120.39 M -| 171419.65 M. |0.74M.
Azimuth » » » 247°26'18 07 | 247°2613"13 |!" 94
Geogr. latitude Gs. Talang 2°6'9". 312 S | 2°6'8".699 S. |0".613
» » Bt. Gadang
229143", 165S | 29942555 S. |0" 610
Altitude above the sea level 1375.5 M | 4376.7 M. {1.2 M.
Gs- Talang
» » » Bt. Gadang 981.8 M. | 9844 M. (|2.3M.

From the triangulation of South Sumatra we derive:

the geographical longitude of Gt Talang 5° 32’ 48/7525 ) West of
4 ,, Bt Gadang 5° 41’ 20’’,236 | Batavia

» 2?

and from that of the West Coast of Sumatra:

*) These triangles occur under the numbers 1, 2, 16, 17, 35, 36, 43, 50, 51
and 52 on pp. 603 and 604 of Comptes Rendus des séances de la dixiéme conférence
générale de |’Association Géodésique Internationale,

( 458 )

Appendix.

Errors of closure of the triangles in the South Sumatra chain.

2 oe i) angie | a?
2* la 4
S{2 eed se
| !
4 | Pistor and Martins, 27 em. a" |2 | 2% | 0.44] — | 0.4936
2 1" 12 |94.95! 4.82] — | 3.319%
3 » 1" |2 |94.95) 0.44] — | 0.4936
4 » (S| (5 | 24 25, — | 0.63 | 0.3969
5 » AS 2s 1 24ae25 0234 — | 0.0964
6 | Wegener, Pistor and Martins, 27 cm. 2".4") 2 | 24.25) — | 0.65 | 0.4295
7 > > on4| 2 194.95) 0.58] — | 0.336%
> > on.4"| 2 | 94.95 0.93 | 0.8649
9 » » gna) 2 | 24.25 0.21 — | 0.044
410 » » ,Qr.4"1 9 | 9% — |4.74 | 3.0276
441 » » DF AM so 24 — | 0.29 | 0.084
412 » » PUR ME ie) 24 | 0.04) — | 0.0016
13 » » PUR Uy 9% | 1.20) — | 4.4400
14 | Pistor and Martins, 27 cm. Le) DY — | 0.73 | 0.5329
45 | Pistor and Martins, Wegener, 27 em. 1”.2") 2 24 | — | 1.33 | 4.7689
46 | » » AH eo, 2% |}1.93) — | 3.7249
47 | ‘ > w.orle | a8 | — | 4.44 | 4.9394
48 | > > 4".on} 2 | 9% | — | 4.76 | 3.0976
49 | Pistor and Martins, 27 cm. Au 2 2% | 2.36] — | 5.5696
20 » qn oO | Of |} — | 470) | 28900)
24 » Wegener, 27 cm. | 41”.2" 24 — | 1.06 | 1.1236
92 > > 4".o"|2 | 9% | 0.38] — | 0.4444
23 | Wegener, 27 cm. 2" 12 | 24 | — | 0.03 | 0.0009
24 » QP 02, 24 — | 1.41 | 4.9881
25 » 2M | 2 24 — | 1.36 | 4.8496
26 > Di ez 24 — | 0.66 | 0.43856
27 » Pe a 24 — | 0.55 } 073025
28 » Q" | 2 94.95) 4.83 | — | 3.3489
29 » gu QD 194.95) — 41.05 | 4.1095
30 » 2! 2 }24.95] — | 0.42 | 0.0144
34 » Qu | 2 |24 25) 0.75 | — | 0.5625
32 > OH 2. 24 — | 0.56 | 0.3136
33 > a |2 | % [449] — | 4.9586
34 » DW HD 24 | 0.81 — | 0.6561
35 > or |o | 94 14.56) — | 9.4936
36 » Qt 24 |/0.40; — | 0.1600
37 » gr) 2 % |0.43) — | 0.1849

Pit -

( 459 )

the geographical longitude of Ge Talang 0° 53/37/7833 ) East of

: 3 = » Bt Gadang 0? 45’ 6/7151 |) Padang,
whence for the longitude of the West end of the base of Padang
respectively ; 6° 26’ 26’’,358 and 6° 26’ 26’',387, of which the mean
value is 6° 26’ 26/7373 West of Batavia, corresponding to
100° 22’ 10/’,68 East of Greenwich.

Hence follows that the difference in longitude between Padang and
Batavia, as determined by geodetic measures, is less by 16’’ or over
1s than that found by chronometers.

Probably the difference is due for a small part only to the aceu-
mulation of errors of observation in the triangulation and almost
exclusively to the method of determination by means of chronometers.

Physics. — “On the melting of floating ice’, by Dr. Cu. M. van
Deventer. (Communicated by Prof. VAN pur WAAts).

In what follows I shall diseuss a physical fact, which though one
of the simplest and most important of phenomena, seems to have
escaped the attention of physicists up to now. The author asked at
least some twenty men versed in physics after it, and not one of
them had heard about it: many of them and specially the most
experienced in this branch of science were not a little astonished
at it. It is therefore not devoid of interest to discuss the fact in
question, though the explanation can be followed even by beginners
in physics.

§ 1. In order to show how surprising the fact is, we put the
following

Problem.

Given a tray of a certain dimension, in which water is up to a
certain Jevel, and in which floats a piece of ice of a certain weight,
everything at O°. Required to find: in what way will the level of
the water be changed, when the ice melts?

Solution : the level of the water does not change.

§ 2. This answer may be derived as a simple application of
the law of ARcHimEpEs. ')

If the piece of ice weighs A kg., the upward pressure is also
A kg., and so the weight of the displaced water also .A kg. As now
the melted ice weighs also A kg., the melted mass will occupy

1) The weight of the air is neglected in this discussion.

( 460 )

exactly the place of the immerged part, and accordingly leave the
position of the surrounding water intact.

In short: the ice when melting contracts into the volume of the
immerged piece.

§ 3. A more elaborate, but perhaps more graphical demonstration
is the following.

Let the piece of ice have the volume of A liter. Then the volume
of the free part is 83 A em. and of the immerged volume 917
A em. The immerged part gives, when melting, 841 A em. of water,
the free piece 76 A cm. of water. The water of the free piece supplies
therefore what the immerged piece had lost in volume when melting,
and there is no reason for change in the level of the surrounding
water.

In this the specific gravity of ice is put at 0,917.

§ 4. It is obvious, that the same reasoning applies to other sub-
stances, so that the following general rule may be drawn up : when
a substance, floating in its own melting-liquid, melts, the level of
the liquid will not change.

§ 5. An application of everyday interest is this: if a glass is
filled to the brim with water, in which ice floats, the water will
not flow over when the ice melts.

We should, however, take care, when making the experiment with
a full glass, not to mistake water that is condensed on the cold
outside wall, and runs down, for water flowing over. A better proof
is furnished by a glass which is not quite full of water, and on
which the first level is indicated: after melting we must find the
same level.

§ 6. Attention may be called to the fact that not only after, but
also during the melting the level is the same as before.

For if one gramme of ice (or a given part of it) melts and gives
one em. (or an equally large part of it) of water, the weight of the
floating piece and so also the upward pressure will be diminished
by one gramme (or an equally large part of it), and consequently
the volume of the immerged piece will be decreased by one em. (or
an equally large part of it). For the additional water room has been
made by the decrease in displacement.

§ 7. The law of the permanent level holds also when the floating
ice has empty cavities.

( 461 )

This is obvious for cavities which are in the piece rising above the
surface, as these cavities have no influence on the upward pressure.

If the immerged piece has a cavity of A em., the upward pressure
is equally large as for a solid piece of ice of the same weight, but
there are A grammes of ice more above the water. When melting,
these A grammes of ice form A grammes of water, just sufficient
for filling up this cavity.

The law of the permanent level holds also when the ice contains
air bubbles, at least by the same degree of approximation, with
which we may neglect the weight of the air.

§ 8. When fresh water ice floats and melts in salt water, the
level does rise, though slightly, the immerged part now being smaller
than before, and so the melted ice cannot be contained in the volume
of the immerged part.

Here and in what follows the change of volume, caused by the
mixing of salt water and fresh water, is neglected, which is certainly
permissible when the proportion of the salt is slight.

For A liters of ice, which weigh 917 A grammes and float in
salt water of 1,03 specific gravity, the volume of the immerged piece
is 890 A em.; the available space can therefore hold 890 A em. of
the melted water, but the remaining 27 Acm. raise the level.

This remaining part is about one fourth of the piece which rises
above the surface of the water (110 A em.).

§ 9. If in salt water a piece of one liter of ice floats, which has
a cavity under water of Lem., then there are (1000-4) em. of solid
ice of a weight of 0,917(1000-4) grammes.

The upward pressure is therefore 0,917 (1000-2) gramme, and
with a specifie gravity’ of the salt water of 1.03, the immerged
volume is 0,89(1000-6) cm. When melting, we get 0,917(1000-B) em.
and so there is a surplus of 0,027(1000-2) em. of water to the
volume yielded by the immerged piece, which raises the level.

The piece of ice rising above the surface was 1000-0,89(1000-B) em.
or (110 + 0,894) em., and the ratio of the remaining piece mentioned

7—0,027 B
The smaller £6 is, the more this relation approaches to about
a fourth.

B
to this volume is as one to (41 ee )

§ 10. History. A remark made two years ago by a pupil of the
third year of the “Gymnasium Willem III” at Batavia to the writer

( 462 )

suggested this paper. This pupil, called van Erprecum, said that he
had observed that a glass filled to the brim with water and floating
ice, does not flow over, when the ice melts.

This fact leading easily to the law of the permanent level and
this law — as the writer is bound to believe — having up to
now escaped the attention of physicists, physical science owes the
discovery of a remarkable fact and the addition of a paragraph to
this pupil.

Amsterdam, Dec. 1904.

Chemistry. — “On trinitroveratrol’. By Dr. J. J. BUANKsMA.
(Communicated by Prof. H. W. Baxkuuis RoozeBoom).

It has been previously stated *) that the dimethylether of trinitro-
pyrocatechin is formed by the nitration of the dimethylether of 3.5
dinitropyrocatechin. As the nitro-group might have been introduced
either in the position 4 or 6, it was still necessary to ascertain the
constitution of this compound. The substance which melts at 146°—
147° is identical with trinitroveratrol, which has already been des-
cribed by Tiemann and Marsmoro*) and is obtained by nitration of
veratrol (the dimethylether of pyrocatechin) or of veratric acid.
Tiemann and Martsmoto have shown that veratric acid on nitration
yields nitroveratrol and nitroveratric acid. Afterwards, Zrinckr and
Franckr*) have proved that nitroveratric acid formed by nitration
of veratric acid has the following constitution:

OCH,
/ \OCH
ae
COOH.

Now, on further nitration with fuming nitric acid this nitroveratrie
acid yields trinitroveratrol so that the constitution of trinitrovera-
trol is

OCH,
/ NOCH,

|
NO. AN0,
NO,.
1) Recueil 23, 114.

2) Ber. 9, 937.
5) Ann. der. Chem. 293, 175.

( 463 )

Dinitroveratrol prepared by nitration of veratrol') and of meta-
hemipinie acid?) and which is consequently formed as follows:

OCH, OCH,
AX OCH; AN OCH,
fare 3)
HOOCK. if, NO.\4
COOH NO,

also gives on subsequent nitration the same trinitroveratrol, again
showing that the constitution of that substance may be expressed
by C, H(OCH,), (NO,), 1, 2,3, 4,5.

Now, trinitroveratrol obtained from veratrol is identical with
that from the dimethylether of 3.5 dinitropyrocatechin ; the melting
points of both substances are the same; a mixture of the two sub-
stances shows no lowering of the melting point, whilst the same
reaction products are obtained from both substances by the action
of alcoholic ammonia or methyl-aleoholic sodium methoxide. We
therefore see that in the nitration of 3.5 dinitroveratrol, the nitro-
group is introduced between the two existing nitro-groups.

OCH, OCH,
Y\ OCH, ny /N OCH,
ee aes

NO,\ NO, NO,\ ANO,

NO

2

Tiemann and Marsmorto*) have already demonstrated that trini-
troveratrol reacts readily with alcoholic ammonia. As they thought
that the two OCH, groups were replaced by NH,, they have not
been able to identify the product formed in this reaction.

On repeating the experiment, I noticed that ammonium nitrite is
formed so that also one of the NO, groups is replaced by NH,. The
substance formed melts at 247° and is identical with the compound
afterwards obtained by Nimrzxkr and KurrenBacuer *) which is formed
by the action of alcoholic ammonia on trinitrohydroquinonedimethyl-
ether.

OCH, OCH, OCH,
/\OCH, AXNH; 7XSNO,
e | = ie a amas

NO,\ “NO, NO,\ “NO, NO,\ NO,

NO, NH, OCH,

1) Briigcemann, Journ. f. prakt. Chem. (2). 58, 252.

2) Rossin, Monatsh. f. Chem. 12, 491. Hemiscu, ibid. 15, 229,
c)) Lien OS BY al be aleile

4) Ber. 25, 282.

( 464 )

This also shows that the NO, groups in trinitroveratrol are situated
in the positions 3, 4 and 5

If this dinitrodiamidoanisol is treated with KOH we obtain the
monomethylether of dinitrotrioxybenzene, a substance already obtained
by Nierzki and Kurrensacuer from the said reaction-product of
trinitrohydroquinonedimethylether and ammonia.

In quite an analogous manner the same result was obtained for
the oxyethyl compound :

OC.H, OCH, OC,H, OC,H,
oN /NNXO, /NOCH, /~ OCH,
fo Ss SOvi- mes: eee? Sa
wo\ “xo, NO.\/NO, NO,\/NO, NO,\/ANO,

NO,
|
OC.H, OC,H, OC,H,
OCH, /\ OCH, /N\NH,
| => (199°) —> (245°|

Ky NO,\ “NO, NO,\ “NO?
NO, NH,
This latter substance has been formerly obtained by Nierzki*) by
treating trinitrohydroquinonediethylether with alcoholic ammonia.
Although now the constitution of trinitroveratrol and of trinitro-
pyrocatechindiethylether seemed to be satisfactorily’ determined, I
have tried to furnish additional evidence by treating these substances
with sodium ethoxide or methoxide; then it was to be expected that
the following changes might occur:

OCH, OCH, OCH,
Ave: /~ OCH, /\ OCH
as ens) ee ( 1 |
NO, no, ) NO.\ AN0, NO,\ “NO,
OCH, OCH, NO,

If now trinitrohydroquinonedimethylether (1) is treated with a
solution of sodium methoxide in absolute methylaleohol the addition
of each drop causes a brownish coloration which nearly instantly
disappears. After a partial evaporation of the solvent, crystals are
formed which melt at 92°; according to an analysis this is the
trimethylether of dinitro-oxyhydroquinone.
OCH,

ye OSE
NOX AO,
OCH,

1) Ann. der Chem. 215, 153.

( 465 )

When we treat this substance with alcoholic ammonia two OCH,’s
are readily replaced by NH, and we obtain the same dinitro-
diamidoanisol as that obtained from trinitrohydroquinonedimethylether,

Trinitroveratrol (II) however behaves quite differently from Na OCH,.
If to the methyl-aleoholic solution is added sodium methoxide a
purple-red coloration is obtained, which only disappears after heating for
a few minutes on the waterbath, after which the liquid turns yellow.
On cooling, fine yellow crystals are deposited (m.p. 152°) which are
not affected by alcoholic ammonia or by potassium hydroxide.

The motherliquor contains besides Na NO, a small quantity of a
substance which is perhaps identical with that from trinitrohydro-
quinonedimethylether.

Fine crystalline compounds are also obtained by the action of
potassium cyanide on trinitroveratrol in alcoholic or methyl-alcoholic
solutions; in either case two different substances are produced.

It is probable that trinitroveratrol (in common with other nitro-
compounds) first forms an additive product with Na OCH, or KCN‘),
which then suffers decomposition and causes the formation of the
said products.

The fact that the course of the reaction is a somewhat unusual
one is most likely to be attributed to the presence of three adjacent
nitro-groups in the benzenecore. I hope a further study will throw
some more light on the subject.

AMSTERDAM, Dee. 1904.

Chemistry. — “On W. Marckwanb’s asymmetric synthesis of optically
active valeric acid.” By Dr. 5. Tustra Bz. (Communicated
by Prof. Baknuis RoozEBoom).

Some time ago, MarckwaLp’*) prepared active valeric acid in a
manner which according to his opinion entitled him to look upon
this synthesis as the first purely asymmetric one. Shortly afterwards
this opinion was challenged in an article from Messrs. Conen and
Parrurson *), who denied that the synthesis could be an asymmetric
one as being opposed to the theory of electrolytic dissociation. A fter-

1) Lorine Jackson. Amer. Chem. Journ. 29, 89, (1903).

Losry pe Bruyn. Rec. 23. 47.

2) Ber. 37, 349.

3) Ber. 37, 1012.

32

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 466 )

wards, Marckwaup ') defended his standpoint in such a manner that
no further controversyhas taken place.

Although the theoretical aspect of the question might be considered
as solved, it still oceurred to me that from an experimental point
of view, the synthesis might be capable of some improvement.
MarckwWaLp starts from methylethylmalonic acid; of this the acid
brucine salt is made in which now occurs an asymmetric carbon atom in

CH,
the residue of the methylethylmalonic acid: CO,H .C———CO,H . Br.

Nowy,

Owing to the influence of the active brucine, the two possible
forms will not be produced in equal quantities and as a transfor-
mation between the two forms is possibly owing to the ionisation,
the solution, on evaporation, will only deposit one salt, as during the
crystallisation the equilibrium between the two forms is constantly
being restored. The active brucine salt is now heated at 170° when
carbon dioxide is eliminated and the brucine salt of methylethyl-
acetic acid is formed. As this elimination of carbon dioxide will
take place exclusively, or nearly so, at the free carboxyl group, the
result will be a brucine salt of active methylethyl acetic or in other
words /-valeric acid. By acidification with dilute sulphuric acid, distil-
lation in steam and rectification, MarckwWaLp obtained a product which
in a 10 cm. tube showed a rotation of [elo = — 1°.7 which corre-
sponds with not quite 10°/, of /-valeric acid. Marckwatp attributes
this low yield of active material to the high temperature employed
(170°), which may have caused atomic shiftings.

The problem appeared to me of too great importance not to try
and obtain a better yield of active valerie acid by altering the
modus operandi. The idea struck me that it ought to be possible to
considerably lower the temperature at which carbon dioxide is
eliminated and thus remove one cause of atomic shifting.

In my preliminary experiments I used the methylethyimalonie awd
itself which melts at 118° and of which it is stated in the literature
that it rapidly loses its carbon dioxide at 180°. As a rule the acids,
which possess two carboxyl groups attached to one carbon atom,
lose carbon dioxide when heated above their melting point; we rarely
find, however, in the literature cases where this temperature is correctly
indicated and very often, at least in the case of substances melting at
low temperature, the uniform temperature of 170°—180_ is accepted.

The methylethylmalonic acid was now heated in vacuo at 130° in

1) Ber. 37, 1368.

ie Ieee

ee
{

( 467 )

a tube connected with a mercury barometer and also with a mer-
cury airpump, which caused a fairly rapid decomposition ; the pressure
rose beyond 1 atmosphere. Even at 100°, decomposition takes place
if we only take care to continually evacuate; the mass first becomes
partially liquid owing to the valeric acid formed and now we can
plainly see the evolution of carbon dioxide from the solid particles
of methylethylmalonic acid still suspended in the liquid. We cannot,
therefore, speak of a definite decomposition point of acids with two
carboxyl groups attached to one carbon atom. The statement that
these substances lose carbon dioxide by heating above their melting
point is consequently not only very vague but to some extent also
incorrect as meihylaethylmalonic acid already loses CO, when still
in the solid condition.

Whilst, however, it takes days before the methylethylmalonic acid
is decomposed at 100° at the ordinary pressure, this process is
finished in a few hours if we continually evacuate. This would be
most readily explained by assuming that this decomposition is a
dissociation phenomenon. At each temperature, there would then
exist a definite dissociation tension and if now by a continual
evacuation care is taken that one of the decomposition products is
always being removed, it is obvious that finally all must be decom-
posed. The only difference between this phenomenon and the classical
example of CaCO, is this that one has never succeeded in obtaining
an acid with two carboxyl groups by heating an acid, containing
one carboxyl group, in carbon dioxide. This may be explained either
by assuming false equilibria, or by supposing that the velocity of
reunion or the decomposition products is exceedingly small. I intend
to further investigate this point.

_ As it had now been proved that the temperature of decomposition
of acids with two carboxylgroups to one carbon atom could be greatly
decreased by diminution of pressure, it was obvious that the synthesis
of l-valeric acid might also be improved by allowing the CO,-elimination
to take place in vacuo at Jeast if Marckwap’s idea was correct
that the bad yield of active material was due to atomic shiftings’).

I have now heated the acid brucine salt of methylethylmalonic

acid with continual evacuation at 120°, therefore far below its melting

1) It is easy to understand that a decrease of the temperature at which carbon
dioxide is expelled is in itself not capable of improving the synthesis. The velocity
of the atomic shiftings would no doubt have much diminished but then also the
velocily of the carbon dioxide elimination, and the complete decomposition of the
substance would take a much longer time. The evacuation, therefore, merely serves
to accelerate the decomposition process.

29%

( 468 )

point (155°), and after the whole mass had turned to a thick liquid
and no more carbon dioxide was evolved the product was dissolved
in boiling water. The solution was acidified with sulphurie acid and
distilled in a current steam. The distillate was shaken with ether,
the etheral solution was dried and after the ether was evaporated, the
residual valerie acid was rectified and its boiling point found to be
174°—176°. The rotation of this was determined at [a]p = —4°.3
which corresponds with 25,8°/, -valerie acid. It made no difference
whether the first or last fraction of the distillate was taken.

The synthesis of /-valeric acid has, therefore, been much improved
and it is possible to still further increase the yield of active acid by
operating at still lower temperature as I have observed that the acid
salt of methylethylmalonic acid possesses even at 100° a fairly large
decomposition tension.

Amsterdam, Org. Chem. Lab:

Chemistry. — “On the system pyridine and methyl todide.” By
Dr. A. H. W. Aven. (Communicated by Prof. Bakauts RoozEBoom).

Among the binary systems which have been studied up to the
present in the gaseous, liquid and solid condition there are many in
which occur chemical compounds formed from the two components.
In most of those cases, those compounds possessed but little stability
so that the conditions of formation and decomposition were situated
within an easily attainable range of temperatures.

In the case of the more stable chemical compounds, however,
those conditions of gradual formation and decomposition are less
sasy to attain. Still, their study promises a clearer insight into the
changes which a system undergoes when a chemical compound is
formed therein, and in the systems which form very stable com-
pounds; such a comparison can be made all the more readily at a
lower temperature because the reaction velocities are then generally
so reduced that the system can be studied at will in the presence
or absence of the compound so that these two cases may be compared.

A first example in which this could be at least partially attamed
is given by the system pyridine and methyl iodide. These two sub-
stances are capable of forming a quaternary ammonium compound
C,H,N. CH,I which possesses a fairly great stability. At 60° and
higher temperatures this compound is rapidly formed in the mixtures
of the two liquids; at the ordinary temperature this formation takes
place rather slowly and exceedingly slowly on cooling. On cooling

( 469 )

rapidly, we should therefore undoubtedly get from the liquid mixtures
solid pyridine (m.p. —50 ) and solid methyl] iodide (m.p. below —80’).
Moreover, all those liquid mixtures in which no compound has
formed as yet are homogeneous.

If, however, the liquid mixtures are kept for some time, the
compound is formed with a considerable evolution of heat and if
separates at the ordinary temperature in the solid condition, the
amount depending on the temperature and the proportion of the
mixture. At higher temperatures, however, it may cause the formation
of two liquid layers. The peculiar behaviour shown is elucidated in
the annexed figure in which the composition of the mixtures is
expressed in molecule-percents of pyridine.

Let us first glance at the
right side of the figure. In this
. 1 is the melting- or solubility-
line of the compound, commen-
cing with the melting point of
the compound (117°) and ex-
i tending to a_ eutectic point
very close to the melting point
of pyridine, because at lower
temperatures the compound is
but little soluble in pyridine.
ae At 81°, from 77.5—85.5 mol. °/,
of pyridine, the line 1 is however
interrupted as no homogeneous
liquid can exist between the
20" two concentrations. The line 2
“30° ineloses with its two branches,
-49 which meet in M, an immisci-
5° bility-region whieh becomes

enlarged at higher temperatures.

The fused compound is therefore
miscible with pyridine toa limited extent only. The point M, however,
can only be reached when there is no separation of solid compound
which may be easily prevented for some time.

At the left side of the figure we meet with nearly the same
series of phenomena: 3 is here the solubility line, 4 the two branches
of the immiscibility line. The immiscibility region is here very great,
at 88° from about 0.5 to 41 mol.°/, of pyridine, whilst no change
could be observed at higher temperatures and consequently no
critical mixing point is known.

Q 4 20 30 4° §0 60 ae Seb

( 470 )

The line 5 is the solubility line of a metastable form of the solid
compound ; this line, however, can only be partially determined in the
presence of an excess of pyridine. With a large excess the stable
form was formed too readily. If the line could have been continued,
it would have been continuous, in distinction from 1, as it is situated
entirely below the mixing point M.

It could not be determined at the side of the mixtures which are
richer in CH,I as these erystallise very slowly and then we always
obtain the stable form.

The most noteworthy result of this research is, however, that two
liquids which are miscible in all proportions, may yield two sets of
coexisting liquids owing to the formation of a chemical compound.
In the formation of less stable compounds such has never as yet
been observed and the better known stable compounds have not as
yet been studied from this point of view chiefly because the com-
parison of combined and uncombined liquids is so often rendered
difficult by the great differences in the melting points of the components.

The sharp intersection of the melting point lines 1 and 3 at 117°
and the strong elevation of the boiling point after the combination
(pyridine 116°, CH,I 42°, combined liquid 270°) show that even in
the liquid state, the compound is certainly for the greater part undis-
sociated.

Probably the partial miscibility of this combined liquid with its
components is connected with the fact that the chemical nature of
the compound differs so greatly from those of the components. On
this point also we possess but very little knowledge at present.

Chemistry, — “The reaction of Friepen and Crarts”. By Dr. J.
36ESEKEN. (Communicated by Prof. A. F. Honiemay).

As is well known, the reaction of FriebeL and Crarrs does not
always proceed uniformly. Sometimes traces only of the catalyzer
seem to suffice for the preparation of large quantities of the desired
product; in other cases equimolecular quantities of the products to
be condensed require from '/,, to ',, mol. of the reagent. In a
great many condensations it has been shown that at least 1 mol.
of AICI, is required in order to obtain the highest possible yield.

The réaction is also dependent on a number of circumstances
which are either connected with a secondary action of aluminium
chloride (Ree. XXII p. 302) on one of the substances present during

(471 )

the reaction, or else depend on the nature of these substances
themselves.

It strikes me that the number of different syntheses made since
the discovery of the catalytic action of aluminium chloride is large
enough to enable us to explain the cause of this different conduct
by a somewhat systematic consideration.

It must be well remembered that aluminium chloride can only
then exert its power when it is capable of rendering the chloride
(or anhydride) active; that is to say it must be present in the mass
either in a free or loosely-combined state.

This mass contains besides the catalyzer (and eventually some
diluent, such as CS,): A the chloride (or anhydride), B the benzene
derivative, C the formed product. If now we disregard the above-
mentioned secondary decomposition phenomena the following cases
may occur.:

I. The aluminium chloride combines with none of these substances
or the compounds are completely dissociated at the reaction-tempe-
rature.

We are then dealing with the catalytic action in the truest form.
A trace of aluminium chloride will suffice to convert unlimited quan-
tities of A and B into C. This is for instance the case in the chlo-
rination (bromination) of benzene at the ordinary temperature. If the
substances used have been carefully dried more than 1 kilo of chloro-
(bromo) benzene may be prepared with the aid of 0.5 gram of
aluminium chloride without a visible diminution of the quantity of
the catalyzer. When preparing diphenylmethane from benzyl chloride
and benzene we can also work with very small quantities of the-
catalyzer if the strongly diluted benzyl chloride is poured into a
large excess of benzene and the reaction-mass is from time to time
replaced by new benzene; yet the decomposition of the benzy!
chloride by the catalyzer cannot be entirely prevented (Recueil
XXIII p. 98).

II. The aluminium chloride combines with the chloride (A) to a more
or less strong additive product. When these compounds are very
stable, the reaction may not take place at all: the phosphorus oxy-
chloride combines with strong evolution of heat with a mol. of
aluminium chloride (Cassenmann, Ann. 98 p. 220), and this product
is not attacked by benzene or toluene. In the other cases the reaction
proceeds, however, very favourably. As aluminium chloride also
combines with the formed product (C) only one mol. of the catalyzer
is required for equimolecular quantities of the components.

Here we must still distinguish between the following categories;

( 472 )

a. the catalyzer is situated closely to the place where the eonden-
sation takes place, which is the case in all syntheses of ketones,
sulphones ete from the corresponding acid chlorides, where it is
linked to the carbonyl {sulphuryl) group, for instance:

CH,COCIAICI, + C,H, = CH,COC,H, AICI, + HCl
(Recueil XIX p. 20).

Presumably, this ought to include the syntheses of GaTTERMANN,
B 1897 p. 1622, where the aromatic aldehydes are constructed from
CO and HCl and the amides of the avomatie carboxylic acids are
obtained from carbamine chloride (Cl CO NH,) B. 1899 p. 1117.

4. the aluminium chloride is combined to the chloride but not
near the place where the condensation occurs, for instance :

AICI,p.NO,C,H,CH,Cl + C,H, = AIC],p.NO,C,H,CH,C,H, + HCl

(Recueil XXII p. 103),
the catalyzer is here combined to the nitro-group.

HI. The aluminium chloride combines with the benzene derivative
(B) and not or with great difficulty, with the chloride (A).

In this case, the benzene-group which has combined with the
catalyzer may increase (a) or diminish (4) the activity of the other
H-atoms.

a. In the first case although the reaction may take place it will
be much retarded.

Anisole, for instance, which yields a well-defined additive product
with aluminium chloride hardly reacts at the ordinary temperature
with carbon tetrachloride; the chlorine atoms of this chloride do
not, apparently, get under the influence of the combined catalyzer.

Benzyl chloride, which acts very violently on benzene, attacks
anisole so slowly that the velocity of the reaction could be measured
at the ordinary temperature. (H. Gonpscumipt, Central-Blatt 1903
II p. 820).

b. In the second case, the reaction does not take place. Nitro-
benzene, aceto- and benzophenone, sulphobenzide ete. do not suffer
condensation with carbon tetrachloride, chloroform, methylene chlo-
ride, sulphur chloride ete. by means of aluminium chloride.

IV. The aluminium chloride unites both with the chloride (A)
and the benzene derivative (B). In this case it will depend chiefly
on the influence of the groups present in the benzene whether the
condensation takes place or not.

Whilst nitrobenzene cannot be acetylised or benzoylised, the nitro-
anisoles may be converted into the corresponding acetyl compounds.

One does not succeed in introducing a second acetyl group into
acetophenone, but on the other hand m-xylene, mesytilene, sym.

( 473)

triethylbenzene, and sym. durene appear to be dacetylised ; from the
experiments it appears that at least two mols. of the catalyzer ave
wanted (V. Mryrr, B. 1895 p. 3212; B. 1896 p. 846; B. 1896 p.
2564; H. Writ, B 1897 p. 1285).

V. The catalyzer combines but little or not at all with the chloride
(A) or the benzene derivative (B) combines only with the formed product.
When this is a molecular compound (as in the cases known up to the
present) at least one mol. will be required for one mol. of the chloride.

I have found that one mol. of carbon tetrachloride exactly requires
the molecular quantity of aluminium chloride for the formation of
triphenylmethane chloride

CCl, + 8C,H, + AICI, =.3HCl + (C,H,),CCI.AICI,.
5,Cl, and SCI, also require one mol. of the catalyzer when being
condensing with benzene.

The behaviour of sulphur itself towards benzene is very interesting
in this respect; from an investigation, the details of which will be
published elsewhere, it appears that this condensation must be repre-
sented by the following scheme :

S, ++ 6C,H, + 3AICl, = 2(C,H,),S. AICI, ++ (C,H,)S,. AlCl], + 4H,S
diphenylsulphide thianthrene

For one mol. of sulphur, three mols. of the catalyzer are absorbed;
the element itself does not combine with aluminium chloride.

As stated above we have only mentioned the cases where no
secondary actions occur or where these may be greatly prevented.
In a number of syntheses this is very diflicult to realise particularly
where we start from chlorides where the carbon atom which carries
the chlorine atom is also combined with hydrogen atoms (Recueil
XXII p. 306), or where hydrogen and chlorine occur near adjacent
carbon atoms. ((Moungyrat, Bull. Soe. chim [8] 17 p. 797; [3] 19
p. 179, p. 407 and p. 554).

To this belong all the syntheses of the homologues of benzene
where we also have the circumstance that the more alkyl groups
enter into the benzene, the more readily it will be decomposed by
aluminium chloride ; the quantities of aluminium chloride required there-
fore become larger and vary in each individual case. In order to get
a better insight in the actual catalytic action of aluminium chloride
these last reactions will furnish in my opinion, a less suitable ma-
terial than the first five categories which I have mentioned. These
will have to be submitted to a systematic and, if possible, also
quantitative research.

I have been engaged for some time in experiments in this
direction, which will be published from time to time.

Assen, Dec. 1904. Chem. Lab. H. B.S.

( 474 )

Physics. — “The influence of admixtures on the critical phenomena
of simple substances and the explanation of Tricuner’s expe-
riments.” By J. E. Verscuarrett. Supplement N°. 10 to the
Communications from the Physical Laboratory at Leiden by
Prof. KAMERLINGH ONNEs.

§ 1. Zntroduction. That small proportions of any admixture cannot but
have a great influence on the critical phenomena of a simple substance
has repeatedly been demonstrated by KaMERLINGH Onnes and his pupils.
This conviction led them to look for an explanation of the abnormal
phenomena at the critical point — on which some observers base their
doubts of the validity of the theories of ANDREWs and VAN DER
Waats — by preference in small quantities of admixture, and gave rise
in the Leiden laboratory to several researches in which the greatest
care was bestowed upon the cleaning of the substances investigated.

As early as Oct. °93, in Comm. N°. 8, p. 15, Kugnen has demon-
strated the importance of phenomena of retardation, due to the
irregular distribution of admixtures. In Comm. N°. 11 (Proc. May ’94)
he proved experimentally that, when pure substances were used, the
deviations found by GauirziInge were not observed. The subject of
Comm. N°. 68, p. 4 (Proc. April ‘01, p. 629) was a difference in
opinion between DE Hern and KameruiNGcH Onnes about the significance
of the former’s well-known experiments, of which the results were
ascribed by the latter to admixtures. I have taken part in some
preliminary experiments undertaken in consequence of this difference
of opinion. They gave us the conviction that pr Hren’s observations
required systematic, corrections and that, if these were applied, the
observations would agree with the theories of ANDREWs and VAN DER
W als’).

Indeed, according to KaAMERLINGH ONNES’ opinion, maintained by
him in Comm. N°. 68, p. 13 (Proceedings, April ‘O01, p. 637), the
deviations found should be ascribed for a good deal to impurities, and
should be explained by means of yAN per WAALS’ theory of mixtures *),
l.c. p. 6 (Proc. p. 631). Moreover, if attention were paid to the
variation of the molecular pressure the deviations to be expected in
consequence of yadmixtures would show exactly the same nature as
those observed by pe Hern, while the variation of the molecular pressure
owing to impurities, however small it may be for a small quantity of
admixture, would yet cause considerable differences of density owing

1) A more careful repetition of those preliminary researches is begun at Leiden
soon after the controversy with pe Heen.
2)Cf, also Hartman, Suppl. N°.3 to the Comms. from the Phys. Lab. at Leiden, p. 47.

( 475 )

to the high degree of compressibility of the substance in the critical
state; le. p. 13 (Proc. p. 637).

At the time it was not possible to form a true judgment about
the influence of admixtures and the required corrections. While other
corrections, which had probably to be applied and which might
have the same influence, were fully discussed, about the correction
for admixtures, nothing could be said but that (l.¢. p. 6, Proc. p.
631) measurements were being made at the laboratory, which would
spread the desired light on the influence of small admixtures.

Since that time have been published those measurements by myself
on mixtures of carbon dioxide and hydrogen (Comms. N°. 45, Jan.
"99 and N°. 47, Febr. °99) and those by Krrsom on carbon dioxide
and oxygen (Comm. N°. 88, Jan. ’04). In the series of “Contributions
to the knowledge of van pur WAALS’ y-surface”’ occur several calculations
of Kresom (Comms. N°. 75, Dee. “OL and N°. 79, April ’02) and of
myself (Comm. N°. 81, Oct. ‘02 and Suppl. N°. 6, May and June ’03).
These calculations in which the law of corresponcing states has been
applied according to KAmeruiNGH Onnrs for substances with admix-
tures, reduce all the deviations from the properties of the pure

; mi I alin:
substances to the knowledge of the two quantities « = a =
k\ dx /r—0

1 (dprr : Hk -
and § = 5 ) and of the empirical equation of state.
Pk\ de y=

I have availed+ myself of the obtained results to investigate what
differences of density will be observed in a tube of CaGniarD DE LA
Tour, containing carbon dioxide mixed with a small molecular
composition of oxygen, if in the manner indicated by Twicuner ‘)
floats are placed into it to determine the density. I found it confirmed
that the nature of the deviations which would be observed in Trrcuner’s
experiments in consequence of small admixtures (if pressure and
temperature are in equilibrium), corresponds entirely to that of the
deviations observed.

It seems to me an important result that, on the strength of the
knowledge of the behaviour of the mixtures of carbon dioxide and
oxygen, we can calculate that even very small quantities of oxygen
in carbon dioxide (a few 0.001 mol.) are sufficient to produce the

1) Drupe’s Ann., 13, 595, 1904. The explanation of TetcHner’s experiments
covers that of Gatirzine’s experiments, where the density was determined at dif-
ferent heights by an optical method. In tubes filled with carbon dioxide, Govy
(C. R. 116 p. 1289, June 1893) has observed a slow displacement of the meniscus
a little below the critical temperature, and has ascribed this phenomenon, and
rightly I hold, to impurities.

( 476 )

differences of density which pr Herren observed in carbon dioxide.
Small admixtures of the same kind as those by which pe HEEn’s
experiments can be explained, may, until we have a proof to the
contrary, also be assumed in the carbon tetrachloride with which
TEICHNER experimented. I therefore hold that TrrcHner’s researches,
which from an experimental point of view leave less to be desired
than those of DE HereEn’s, must be explained in the same way.

They are now being repeated at the Leiden laboratory with carbon
dioxide of the greatest possible purity, while in order to omit all
doubts of temperature equilibrium *), thermoelements are sealed in
the tube.

§ 2. Difference in density between two phases with slightly differing
proportions of admixture, when equilibrium of pressure and of tem-
perature exists. We imagine that in a tube, at a temperature yf he
which differs only little from the critical 7) of the pure substance,
there are two layers of which the one contains per gramme molecule
x, mol. of the admixture, the other z, mol.; the pressure is supposed
to be the same’), i.e. equal to p, and also to differ little from the
critical pressure jj), of the pure substance. In order to determine the
density of a mixture with an (infinitely small) composition a, we
may proceed as follows. The quantities a, 3, and y = «— determine
the critical elements 7%, prxr, Vie Of the point which for the mixture
corresponds to the critical point of the pure substance, in first
approximation (Comm. N°. 81 equation (14)) by the equations :

T.4= T,1 +42), pr=pd+ 62), v..=%0+ 72).

Hence to the temperature of observation 7’, i.e. the temperature

of the mixture, a temperature 7” of the pure substance corresponds

ab) y

Tk
in such a way that w= and we may therefore write in first
xk

approximation: 7” = 7(1— ew). In the same way the pressure
p =p(i1— Be) of the pure substance corresponds to the observed
pressure p (pressure of the mixture). Suppose that at the temperature
T’ and the pressure p’ the pure substance occupies the molecular
volume v’, a volume which may be derived from the empirical equation
of state or which may be read on a diagram of isothermals, then,
under the circumstances observed (7p), we have for the molecular
volume of the mixture considered »=w (4 + yx).

1) Cf. Vittarp, C. R. 118 and Comm. N”. 68 (April ’01).
2) Doing so, we neglect the influence of gravitation, which is much smaller than
at of the admixtures, and moreover increases the differences of density.

Determining the value of v, either by calculation or by means of

ae Po
a diagram, we find that, if the proportion ~ differs much from
«a

7, ) 0 ;
Pr OD -= (5*) =7.3") (cf. Suppl. N°’. 6, p. 14; Proc. June ’03,
pe \OT Jeg Ot/k:

p. 121) owing to the particular shape of the isothermals near the
eritical point, the difference v’

ve is much larger (of lower order, viz.
i : ’ ny)
3) than the correction term v y#. For that reason and also because

of the uncertainty about the volumes which belong to a definite
pressure, again owing to the shape of the isothermals, we need not
distinguish between v and vo, in other words, we may neglect
the correction term v’ y.2?). As, however, we intend to determine
the density of the mixture, we must bear in mind that v is the
volume oceupied by JZ, (1—a) + M, «a gr., JZ, and WM, representing
the molecular weights of the pure substance and the admixture. Thence
M,(1—«) + M,«

follows the density ——, for which, for the same reason
i

Uae. S Mar
as above, we may put —,, i.e. the density of the pure substance itselt
7"

at the temperature 7” and the pressure p’.

On the strength of this consideration we may conclude that the
densities of the two mixtures v, and 2,, at a temperature which is
about the critical temperature of the pure substance 7), may be read
approximately on a p, @ diagram of the isothermals of the pure
substance; on the isothermals of the temperatures 7’? = 7;,(1—e.r,)
and 7’, = 7;,.(1—ewx,) we seek two points for which the pressures
are p, = px(i—=,) and p,’ = px (1—? 2,) respectively.

Besides these two layers, however, the tube really contains still
several others of different composition, because the composition varies
eradually *). If for different compositions we determine the densities

1) Further on we shall see that, in the cases known thus far, this condition is
satisfied.

*) The circumstance that we must determine the difference between the «’s for
two mixtures, does not alter this conclusion in the least. For also the difference

between @,’ and v3’ is found to be of a lower order & thar the first.

8) What has been said here about accidental impurities, holds also for the
experiments of Camerer and Corarpeau (C. R. 108, 1280, 1889) where jodine,
which had been dissolved in liquid carbon dioxide, was not diffused equally through
the tube at the temperature at which the meniscus disappeared; it also holds for
similar experiments of Hacexsaca (Drude’s Ann., 5, 276, 1901), who dissolved

( 478 )

at the same temperature and pressure, we obtain points which
all lie in one curve, this curve therefore represents the variation of
the density in the tube; from the shape of this curve, which very
much resembles that of an isothermal in the neighbourhood of the
critical point, it is evident that the substance in the tube cannot but
show considerable differences of density.

I assume that between the two ends of the tube there is a certain
difference in composition; then the greatest difference in density
depends not only on the « and the ? of the substances considered,
and on the difference in composition, but also on the temperature, on
the mean density and on the mean composition. For those mixtures
of which the « and the 8 are known, I now shall give the difference
in density which corresponds to «,—a, = 0.001, if the temperature
is about the critical temperature of the pure substance and the mean
density is also the critical density :

CO, with 0.001 mol. CHsCl , #= 0.378, B= 0.088 , A=34"/)) or the
CHG se oy. kO, - oe — 0.29 A=) Oe Sa ae

CO, Sees: ela ls ,a=—117 , B=—162 , 7p =42% ;
COME Sc.) Soy Ry Be ee =O B65 1h kOe ee

The following differences in density would be observed in carbon
dioxide with small quantities of oxygen, with different temperatures
and differences of concentration, the mean density being still the

critical

critical one:

t @ 0, = 0.001 2232, = 00005. 7a, — 0-000
Si0 A = 36/5 A= 308/e Np bY
alee 24 17 6
S25 17 10 2.5
Son 12 5 = =a
34° 6 3 Eas

How the difference in density depends on the mean density of
the substance may be seen from the following table, which relate to
carbon dioxide with oxygen at a temperature of about 31° C. and
for 7,—2v, = 0.001.

Mean density 1.30, ay ee
1.2 8
3 Koa 24
1.0 36
0.9 24
0.8 6
0.70: 1.5°/,
salts in liquid sulphur dioxide. These experiments, therefore, where an admixture
had intentionally been added, have been erroneously adduced as arguments against

the theory of ANprRews and van per Waats; for the rest Hagensacn himself has
understood the cause of the deviation he had found.

Dr. J. E VERSCHAFFELT, “The influence of admixtures on the critical phenomena

of simple substances and the explanation of Teichner’s experiments.”

ae
2620 22 29 33 39 42 46 50 58 66 S39 60 2 4g 25 22 29 2020

& «
3 .@)
3 a
— Ss
“ —E
iS) a
= 3
Ss 8
°
i")

’

a]

is) a
S S
ie
E =
oS =
bo} ‘"
5 a
3 is)
S 8
[oy

Fig. 2.

Proceedings Royal Acad. Amsterdam. Vol. VIL.

al

( 479 )

The next table shows how for carbon dioxide with oxygen, at a tem-
perature of about 51° C., the mean density being the critical density
and a,—wv, — 0.001, the difference in density depends on the mean

composition.
4 (7, + v,) = 0.0005 A = 36°/..
0.005 47
0.01 12
0.015 6

All these numbers relate to carbon dioxide with oxygen as admixture ;
it is probable that these results will also be more or less applicable
to carbon dioxide with nitrogen, hence also with air, and as in
‘arbon dioxide, which had been purified with great care, KuEsom
detected about 0.00025 mol. of air, the possibility is not excluded of
explaining the anomalies observed with carbon dioxide, by impurities
of air.

The variation of the difference in density with the mean density
reminds of a diagram concerning DE Hrey’s experiments, formerly
made by me (cf. Comm. N°. 68, Appendix p. 26; Proc. April 1901,
p. 695); in Comm. N°. 68, Appendix p. 22 (Proc. April 1901 p. 691)
KamprtincH Onnes has derived the same diagram for the course of
the differences in density that would result from differences of tempe-
rature; therefore part of the deviations observed by pe HEEN are
perhaps due to differences of temperature.

§ 3. Survey of the experiments of Tuicuner. In the influence of
impurities we have a complete qualitative explanation of TrIcHNeEr’s
observations. The results of his second series of observations, of
which I have used only those above the critical temperature, are
represented in fig. 1. The positions of the floats are indicated on
vertical lines and the points occupied by the same bulb at different
temperatures are combined by lines. In this manner curves of equal
densities are obtained; for each curve | have given the corresponding
density. In this series of experiments TrIcHNER has first made obser-
vations at gradually increasing, and then at decreasing temperatures;
after each variation of temperature the observer waited till the
temperature had become the same throughout. As abscissae I have
not taken the temperatures themselves, but I have placed the different
observations at equal distances, that is to say, I have taken time
as abscissa, thus assuming that between two observations there is
always the same interval of time, which will not probably be tar
wrong. The temperature 282°.0 C. (uncorrected) is that at which

( 480 )

the meniscus with increase of temperature was seen last and reap-
peared when the temperature was lowered; hence very nearly the
critical temperature. It will be seen that most of the curves of
equal density, when the temperature is raised, leave the point where
the meniscus was seen last, bend away from that point more and more
rapidly, turn round at about the highest temperature observed and
return to the same point, which only few, however, reach when the
temperature is fallen to the critical temperature.

From this last circumstance we conclude that the course of
the curves of equal density is not only governed by the variation
of the temperature but also by diffusion. both through increase of tem-
perature and also through diffusion, the distribution of the substance
becomes more regular, and hence the curves of equal density ascend
and would finally project beyond the drawing, if not the decrease
of temperature in the second part of the experiment caused the
withdrawing curves, partly at least, to return. But the very fact that
the curves of equal density in the second part lie higher than those
at equal temperatures in the first, is a proof that the progressing
diffusion opposes the influence of the temperature; the following
numbers may show which is about the course of the greatest difference
of density in the tube throughout the series of experiment :

t = 282° 283° 284° 285° 286° 288° 286° 284° 283° 289°
A = 50°/, 40°/, 30°/, 25°/, 20°/, 15°/, 15°/, 20°/, 25°/, 30°/,

It will be seen that the difference in density first decreases, then
increases, but the values at equal temperatures are lower in the
second part than in the first and the deviation mereases; from this
appears the influence of diffusion.

The value of 4 is not even smallest at the highest temperatures ;
the smallest value is not reached until the temperature is falling, in
harmony with whieh is the fact that the bulbs 0.555 and 0.578 have
reached their highest position not at 281°.1 C. but at 286°.0 C.,
hence during the period of decreasing temperature. This proves that,
at least at the beginning of the decrease of temperature, the diffusion
has a preponderating influence. That the heaviest bulbs did not show
this peculiarity must probably be ascribed to the circumstance that
in the lower part of the tube, where the substance is much denser,
the diffusion takes place much more slowly; in those lower curves of
density, however, we can clearly distinguish a point of inflection,
which also, though less striking, points at the progressing diffusion.

That these circumstances can actually be explained by the diffusion
of impurities I have tied to demonstrate by caleulating and by

( 481 )

representing graphically in the same way as in fig. 1 how the
density of a substance is distributed in a tube which is filled with
carbon dioxide, mixed with a small proportion of oxygen, if that
admixture increases in concentration from the bottom upwards. I
also suppose that the temperature first rises from the critical tempe-
rature of 31° C. to 33° C., and then falls again to 31° C. Further
IT assume that the concentration of the oxygen which at first decreased
regularly from the top downwards, so that the greatest difference
of concentration was 0.001 mol., at last, owing to a more rapid
diffusion in the upper space, varies there less rapidly with the height
than in the lower space’). Fig. 2 thus obtained, may really be
looked upon as a diagrammatical reproduction of fig. 1; in the
falling period the density curves, as in fig. 1, show a point of in-
flection; in the upper half no maximum has yet been reached by
the curve 0.450, but by adopting a more rapid diffusion in that
space I might have brought about also this circumstance.

§ 4. Conelusio,,. On the strength of what precedes we can there-
fore firmly deny that Tricuner’s observations*), at least with respect
to the nature of the phenomena, are incompatible with the theory of
Anprews and vAN DER Waats. Down to details these phenomena can
be explained by the presence of admixtures, which are slowly diffu-
sing through the substance; and calculations based on existent data
have shown that in order to reach a quantitative agreement, we
must assume a proportion of the admixture of the same order as that
which actually was present in other experiments with so-called pure
substances. Whether in the carbon tetrachloride, used by Txrcuner, the
required proportion of any admixture, of which neither the nature
nor the « and @ are known with certainty, has occurred, is a ques-
tion that cannot be answered. It does not seem impossible, however,
because carbon tetrachloride is a substance which, owing to the manner
in which it is prepared, might contain many foreign components,
and the constancy of the boiling point (to within 0°.1 C?) is not
deemed by us a guarantee for sufficient chemical purity. We are
even inclined to consider the existence of the deviations as a proof
to the contrary, and the non-existence of the deviations (other

1) Starting from a given condition, I might evidently have worked out this problem
in perfect harmony with reality; it appeared to me, however, that this would have
been useless trouble, and that the scheme, I have given of it, does at any rate
represent the phenomena qualitatively.

2) The same conclusion holds for similar observations (pe Hern, Gatirzine, ete.)
about the so-called abnormal phenomena near the critical point.

33

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 482 )

causes taken into account) as the only certain physical criterium
of purity.

As long as it has not been proved that existing impurities cannot
account for the phenomena quantitatively, 1 see no reason to aban-
don the thesis that each substance shows a critical point at which
ihe two coexisting phases become identical, so that one single critical
density belongs to the critical temperature and the critical pressure.

Geodesy. — “Determinations of latitude and azimuth, made in
1896—99 by Dr. A. Pannekork and Mr. R. Posraumus Mryses
at Oirschot, Utrecht, Sambeek, Wolberg, Harikerberg, Sleen,
Schoorl, Zierikzee, Terschelling (the lighthouse Brandaris),
Ameland, Leeuwarden, Urk and Groningen.” Short account
of the report published under this title by Prof. J. A. C.
OvUDEMANS.

drawn by the Dutch Geodetic Committee, contained also the stations
Leyden and Ubagsberg, where the observations were made under
superintendence of Prof. H. G. VAN DE SanpE Bakuvuyzen, who himself
will publish them.

The observations of Messrs. PANNEKORK and Postaumus Meyszgs at the
above named thirteen stations, have been made under my super-
intendence, and in an introduction I have given an account and a
criticism of them. Here the following details may suffice:

The mean latitude of the four northernmost stations, Terschelling,
Ameland, Leeuwarden and Groningen is 53°18'39", that of Schoorl,
Urk and Sleen 52°42'45", that of Leyden, Utrecht, Wolberg and Hari-
kerberg 52°10'40", that of Zierikzee, Oirschot and Sambeek 51°35'51",
while the latitude of the southernmost station Ubagsberg is 50°50'53".

The entire arc of meridian, of which the length will be computed
as soon as the results of the entire triangulation will be known,
amounts therefore to 2°27'46" and may be considered to consist of
four parts of 35/54", 32/5", 34/49" and 44'58”" respectively. Thus
it will appear afterwards whether the curvature of the meridian, as
found here, agrees with the form adopted.

The Universal instruments used for the observations were of
Repsoup; they were provided with a horizontal circle of 315 mms.,
and a vertical circle of 245 mms. in diameter, and belonged to the obser-
vatories of Leyden and Utrecht respectively. The circles were gradu-
ated to 4, whereas the microscopes of the Utrecht instrument are
read direetly to 2", those of the Leyden instrument to single seconds,

Besides the stations mentioned in, the title, the programme, as

( 483 )

The micrometer screws, the levels and the differences in diameter
of the pivots were accurately investigated and all irregularities were
accounted for. For the illumination, electric lamps were always used,
for which the current was supplied by accumulators.

The latitudes were determined by zenith distances of northern and
of southern stars. For the northern stars only the two pole stars,
« and d Ursae Minoris were used; the southern stars were chosen
so that they had a northern declination from 6 to 14°, and conse-
quently culminated at zenith distances almost equal to that of the
pole, i.e. equal to the co-latitude.

As a rule, for each determination 16 zenith distances of the pole
stars were observed, without regard to the point on the parallel
they occupied; of the southern stars, four in number, 8 zenith dis-
tances were observed, four before and four after culmination ; so
that each complete determination of latitude rests on 32 zenith
distances north and 32 south of the zenith.

At each station four such determinations were made in four
positions of the circle which differed by 45 degrees. If we bear
in mind that the reading was always made by two opposite
microscopes, the zenith distance of each star may be said to be
determined by eight different arcs of the circle, hence the periodic
error of the graduation may be considered as almost entirely elimi-
nated,

The declinations of the stars used were taken from the Berliner
Jahrbuch, while due account was taken of the latest corrections,
published by Auwers in nos. 3927 -29 of the Astronomische Nach-
richten. Finally the latitudes found were corrected for the polar
motion, according to the latest data furnished by ALBRECHT.

For the azimuth determinations only the Polar star was used at

different points of its parallel. The horizontal distance between the Polar
star and the object was measured four times in 12 positions of the
cirele, differing 15 degrees; this was done according to the follow-
ing scheme :
Object, Polar star, Polar star, Object, reverse the instrument 180°;
Object, Polar star, Polar star, Object, while for each pointing at
the Polar star the level was read in two positions. Accordingly
each determination of azimuth consisted generally of 12 series of 8
observations i.e. 2 complete determinations each ; hence of 24 complete
determinations.

As object was used either a lamp, or a heliotrope, in most cases
a heliotrope. Its position with relation to the adopted centrum of
the station was determined by the Triangulation Service.

( 484 )

The following may be remarked about the accuracy attained :

For the mean error of one result from two zenith distances = 0"455*)
was found as mean value; the mean error of each final result,
derived from say 128 double observations, was then caleulated in
different manners to be + 0'065.

For the determinations of azimuth the mean error

of a single determination was found to be + 1"22,
hence that of the mean of 12 determinations = 0,355.

The amount of all these mean errors can very well stand a com-
parison with the determinations of other observers.

To this criticism of the determinations executed for geodetic pur-
poses two appendices are added, namely :

I. “A comparison between the latitude, determined at the station
Utrecht, Cathedral tower (Domtoren), by Mr. Postaumus Mryvses, and
the determinations made at the Observatory.”

The final result of this investigation was the following: Latitude
of the Universal instrument at the Observatory :

derived from observations of cirecummeridian

venith distances, <2 62 c. = a > See ia, ee nomen eres

derived from the observations in the prime vertical 52 5 10,29,
48 » > result of Mr. Postaumus Merysrs, reduced

from the “Domtoren” to the Observatory . . . 52 5 9,84.

This agreement is quite satisfactory, especially if we consider that
the observations of the cireummeridian zenith distances at the Obser-
vatory, which had been made for exercise, were executed in only one
position of the vertical circle, which was also a motive for neglecting
the polar motion.

Il. “A comparison between the azimuth of Amersfoort, determined
by the author in 1879 and °80, and the same azimuth determined
by Mr. Posrnumus Meysrs in 1896.”

The final result of this comparison, after due regard was paid to
all reductions, was: Azimuth Utrecht (Centre) — Amersfoort (Centre):
Determination of 1879,80: 68° 22’ 44"71 + 0"31,

. > 1996: 45,59 == 0,29:

Between these two determinations there is a difference of 0’’88
+ 0’’42 (mean error), which partly may be explained by the acci-
dental errors of the observation and the graduation, and partly by
the uncertainty in the different reductions which occur in this com-
parison. We should also bear in mind that in the results of Mr. Postaumus

1) For Mr. Pannexoek + O'49, for Mr. P. Meyses + O"42, two numbers that
are nearly reciprocal to the magnifying powers of the telescopes of the two
instruments (60 and 68 times).

( 485 )

Mrysus three out of twelve differences from the arithmetic mean exceed
the negative quantity —0’’88, whereas in the author’s results five
out of nineteen differences exceed the positive quantity -- 0'.88.
Accordingly the difference between the two results may be considered
as purely accidental.

(The last sentence does not occur in the original. It should be
remarked that in the publication of 1880, the last difference from the
arithmetic mean for 1879, must be ++ 0",74 instead of + 1°,74).

ERRATA.

Page 238, line 5 from bottom, for “increases” read ‘decreases.”’
pe SENOS es eae 2 een, MROACh. ae.

Pre ar LO, js » I" read T"” (twice).

» 241, in the formula for Xjp7, X,, Xipr read rip7, £,, ZpT-

(January 25, 1905).

eh, Pee ee ee

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Saturday January 28, 1905.

a

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 28 Januari 1905, DI. XIII).

(SO) ANP ae aa ING else

P. H. Scnoure: “The formulae of Gutpry in polydimensional space”, p. 487.

W. Karreyn: “On a series of Bessel functions”, p. 494.

H. G. Jonker: “Contributions to the knowledge of the sedimentary boulders in the Nether—
lands. I. The Hondsrug in the province of Groningen. 2. Upper Silurian boulders. — First
communication: Boulders of the age of the Eastern Baltic zone G”. (Communicated by Prof.
KX. Martin), p. 500.

J. J. van Laar: “On some phenomena, which can occur in the case of partial miscibility of two
liquids, one of them being anomalous, specially water’’. (Communicated by Prof. H. A. Lorenrz),
p. 517. (With one plate).

J. CarpinaaL: “The equations by which the locus of the principal axes of a pencil of
quadratic surfaces is determined”, p. 532.

The following papers were read:

Mathematics. — “The formulae of Guivix in polydimensional
space.’ By Prof. P. H. Scouts.

(Communicated in the meeting of December 24, 1904).

: , ° . ci 6 o(a)
We suppose in space S, with » dimensions an axial space ‘Sp

; ic se
and in a space S,41 through this ee a limited part with p—+ 1

0 : . ale
dimensions rotating round |S,

limited space, which may be called a polytope independent of the
shape of its limitation and may be represented by the symbol (/0),41,
describes a spherical space of n—p dimensions lying in the space

Then an arbitrary point 7 of this

S,—» through P perpendicular to Sins having the projection @Q of P
3
Proceedings Royal Acad. Amsterdam, Vol, VII.

( 488 )

on SS; as centre, PQ as radius; so it can be represented by the symbol
SPn—p (Q, PO):
The question with which we shall occupy ourselves is as follows:
“How do we determine volume and surface of
the figure of revolution generated by (Po),41 ro-

tating round Sp if we assume that (/),4) and
though lying in the same space Sy; have no points
in common?

This theorem is solved with the aid of a simple extension of the
well known formulae of GunpIx, which serve in our space to deter-
mine the volume and the surface of a figure of revolution. To prove
these generalized formulae we have but to know that the surface of
the above-mentioned spherical space Sp,» (@, PQ) is found by multi-
plying PQ-e—! by a coeflicient s,, only dependent on n—p; for
its application however it is desirable to know not only this coefti-
cient of surface s,—, but also the coefficient of volume Un-—p by which
PQ? must be multiplied to arrive at the volume of the same

spherical space. To this end we mention beforehand — as is learned
by direct integration — that between these coefficients the recurrent
relations
2ar 2ar
Un al Un—2 ? a Sai Fa) as as). ce ennGS)

exist, whilst the well known relation between volume and surface
leads in a simpler way still to the equation
1

—s§

by ee eee
Wt

vn =

In this way we find as far as and inclusive of = 12 out of the
well known values of v,, v, and s,, s,

nol 3 kN 5. ales

) 9 elas) 10 44 >|
| | | | | |
| : |. | Bke
’ a 4 1 <2 8 a2 ! 5 16— a | 4d aif || 32 af = 1 as
Spa ImSE 22." || Teo M Geen OS (2S Babies | 120 * 10396 we
8. | ; 16 1 Bde nl ed ee
Snl2x| 4a 9-2 | 2 3 x3 : <4 =4 | 19 =o ey: | 2

1. Determination of volume. If « indicates the length
of the radius PQ and the differential de the p+ 1-dimensional
volume-element, lying immediately round P, of the rotating polytope
(Po),+-> then the demanded volume is

( 489 )

V = sp fo -l dv,

X.
if the integral is extended to all the elements of volume of (70),4. If
now Voi is the volume of (Po)p41, we can imagine a quantity 2,
satisfying the equation

for dv = un—p-l

.

{ dv = a P— Vi

and we ean insert this quantity in the above formula. By this it passes into

a

v= V1 . Sn=p pe
If we call x the ‘radius of inertia of order n—p—1” of the
volume V,4: of the rotating figure (0),4; with relation to the

axial space SS lying in its space S,41, we find this theorem :
We find the volume of the figure of revolution
generated by the polytope (Po),41 rotating round

an axial space Ae not cutting. this polytope ofits
Space Spi, if we multiply the volume V,1; of (Po)p41
by the surface of a spherical space Sp,», having
Due Eradigs of inertia of order n—p—t of Vps1
with relation to se as radiws.”

2. Determination of surface. If in the above we
substitute the p-dimensional element of surface for the p + 1-dimen-
sional element of volume and in accordance with this for the volume
Vp41 and its radius of inertia the surface Sw,4, and its radius of
inertia, we arrive in similar way at the theorem:

We find the surface of the figure of revolution gener-
ated as above if we multiply the surface Su,4, of (Po),41
by the surface of aspherical space Sp,_,, having for
radius the radius of inertia of order n—p—1 of Sup4i

with relation to See

3. The segment of revolution. The opinions will differ
greatly about the use of the #-dimensional extension of the Gurprin
formulae proved above. Those regarding only their generality and
their short enunciation may rate them too high, though reasonably
they cannot go so- far as to believe that these formulae allow the
volume and the surface of a figure of revolution to be found when
the common principles of the caleulus leave us in the lureh, as
the quadratures can be indicated but not effected in finite form.

34*

( 490 )

Others, whose attention is drawn to the fact that these formulae
displace the difficulties of the quadratwres but apparently — in this
case displace them from definition of volume and of surface to the
definition of radii of inertia — will on the other hand perhaps fall
into another extreme and will deny any practical use to the formulae
in question. Here of course the truth les in the mean. Though it
remains true that the GuLpin formulae help us but apparently out
of the difficulty in the case where the direct integration falls short,
yet by the use of those formulae many an integration is avoided
because the radii of inertia appearing in those formulae of volume
and surface of the figure of revolution are known from another
source, which latter circumstance appears in the first place when
p=n—2, thus each point P of the rotating figure describes the
circumference of a circle and the radii of inertia relate therefore
to the centre of gravity of volume and surface of that figure, whilst
for p=n—3 the knowledge of the common radius of inertia of
mechanics gives rise to simplification.

As simplest example of the case p=n—2 we think that a
segment Spr (7,9) of a spherical space Spn—. with r and @ as
radii of spherical and base boundary generates a segment of revo-

lution Sp (7, 0, @, by rotation round a diametral space Ss situated
in its space S,—1, having no point in common with it and forming
an angle «@ with the space S,—» of the base boundary. For this we
find the following theorems:

“We find the volume of the segment-of revolution
Sp(7,9,@,. by multiplying the volume of a spherical
space Sp, with @ for radius by cos a.”

“We find the surface of the segment of revolution
Sp(t,0,@n Which is described by the spherical boundary
of Spzi(7,e) when rotating, by multiplying the cireum-
ference of a circle with 7 for radius by the volume of
the projection of the base boundary of Sp,—-i(7,9) onthe

>

axial space Sys.

These theorems are simple polydimensional extensions of well
known theorems of stereometry. They can be found by direct inte-
gration where the case «= 0 is considerably simpler than that of an
arbitrary angle « And now the formulae of GuLpIN teach us exactly
to avoid the integration in the general case, showing us immediately
that the theorems are true for the case of an arbitrary angle @, as
soon as they are proved for «=O. If namely «, and x, are the
distances from the centres of gravity of volume J7,—-; and surface

( 491 )

’ oO (a ' ae ‘ ‘
Su,—) of Sp,—1 (7, 0) to So where Sv,—1 now again indicates exclusive-
a—I1 Pn | v n— 2, >
ly the spherical boundary, then the formulae of GuLpIN furnish us with

V., — 250 Vy COS. Vi, a Ste == aor @y, CORE . SUpn—

Vi = Phere ee ate SU, = 250 vs el
and from this ensues immediately
Vin => Vi, COS ; Si, = Silo COS

and therefore what was assumed above, so that only for @ =O the
proofs have yet to be given. We commence with the volume. If «

C : (a) (x) x y
is the distance from S,—~s to a parallel space S,—~2 cutting Sp,—i(7, 9)

in a spherical space Spe. with y =V 7? — 2” for radius, then the
demanded volume is

ct
Vi= er vp 2fu— ada
— V 732

and this passes, as a? + y?=7° and adx + ydy = 0, into

V=22 Paap dy = a Un—9 OT == Un 0",
- nr
with which the special case of the theorem for the volume has
been proved.

In the special case of the theorem for the surface we regard the
superficial element generated by the rotation of the surface Su,—, (7, 9)
situated between the parallel spaces S2o- and eae? If ds is the
apothema of this frustum the demanded surface is

to

Su = 2a a ads,

x=) He

With the help of the relations yds =rdzx and «dx + ydy=0 this
passes into

°
.

2z
Su = 2ar af EAI] 5 7 §n—2 O™ SEAT s Une Ola

u—ae

0
i.e. the desired result.
Of course we can represent to ourselves the more general segment
of revolution Sp, 9, @n,x of order & generated by the rotation ofa

. ‘al . el a) *
spherical segment Sp,—x(7,9) round a diametral space SO 4 of its

space S,—¢: Of the various possible cases
RAL, atieie susie Bie » n—2

the first is the one treated above extensively. As any point generates
at the rotation the surface of a spherical space Sprpi, we find —
if along the indicated way by means of the formulae of GuLprn the
general case of an arbitrary angle @ is reduced to the special
case @=0 — for volume V,, and the surface Su,, of Sp (7,9, @n,t
the formulae

@rt—r \

Vine na Sk cosk af y- k-1 xk dz

a
z=V

i
Sunk = "Sn—k—1 Sk1 cost yr—k—3 vk dx

r=V PZ
and from this ensues the general relation
Stn ~ == 220 7 cos*.a Vy—v4

by which all cases of determination of surface except Sw,,,—2 and
Stn 3 ave deduced to simpler cases of the determination of volume.

When determining the volume the integral gives a rational result,
an irrational one or a transcendental one according to & being odd,
n odd and & even, or n even and / even. And this is evidently
likewise the case for the determination of surface.

4. The torusgroup. By rotation of aspherical space Sp,—z (7)

~( se , .
around a space so zp—-1 of its space S,-~% at a distance a>r from
the centre a ring is generated in S,, the ring or “torus” 7’(7,a),,-.
For volume V(r,@nxz and surface Su(r,@, of this figure of revo-

lution of order / we find
\

a
z —— nk 1
V(r, @)nke = 8h Mrn—k—i | V7? — 2 (a+ea)kda
—a

a
= n--k—3
Su (r, @)n,b = 7 Sk S—k—1 | V0? — 2" (ata)kde
—a
from which ensues again the formula of reduction
Suny == 2 ot 7 Vansak esi, soos) As meee

For the case k=1 and /=2 the results are calculated more
easily by means of the formulae of GuLpIn, if one makes use of

thy

\
\
\

( 493 )

the centre of gravity and of the oscillation centre of the rotating
spherical space.

Case k=. The centre of gravity of volume and surface of
the spherical space Spp—1() lying in the centre, we find

= PIG BUR ISSA ne ee
Case k=2. The radii of inertia of volume and surface of a

=
. Y . Iu" G
spherical space Sp,z—2 (7) with respect to the centre are VA

vt
: . . Z 1
and 7, those with respect to a diametral space S, 3 are thus =

vt

“a
and 7 mo So we find
n—2

1 1
V= 4x (« + — 7? Jono 2, Su = 4r{ a? + =a r? | . sn—9 73,
n—

If instead of a whole spherical space Sp,—-(7) we allow only

half of it to rotate around a space (Che in its space S, x parallel
to its base at a distance «a, then the limits (— 7,7) of the two
integrals (1) change into (0,7) or (—vr,0) according to the half
spherical space Sp,—~ (7) turning its base or its spherical boundary

to the axial space ar We shall occupy ourselves another moment
with the former of these cases, namely for £=1 and /: = 2.
Case (0,7), 4=1. We find immediately

= 2 Un 2 Y 2 §n—2
V=a( at+— —— >). v,-,7r"— , Su=a[ a +-——— {7} ,5,_) n—2,
m Un—t1 n—2 Sy}
Jase (0,7), k= 2. We determine the moments of inertia of
; (b)
volume and surface first with respect to the base aa and then
successively with respect to the parallel space S,’s through the centre

of gravity and with respect to the axial space S35. Thus we
finally find the formulae

i 2 Un—2 = 2 Un—2 :
— —( = =r) t( = Seta)! . ems,
n iis Ory 9 i. rah

aoe

+ 6y—2 17S,

S 2 aa 4° bn—2 7"
Su=2arla ee ea x
2 ae

Which pass for @ =O appropriately into volume and surface of the
spherical space Sp, (7’).

( 494 )

Mathematics. — “On a series of Bessel functions.” By Prof. W.
KAPTEYN.

(Communicated in the meeting of December 24, 1904).
In the following we shall try to determine the sum of the series
1 (QL) + 3L (QL) $51, OL0) +++ = Sn hil In).
To this end we begin to determine the sum of the simpler series
S=—= = Ly (@) cos ng.
If we introduce, m being Si odd number, for the Bessel function

the form
= (: fa was

Lf, («) = — ae e

then
= 1
= (Cas)
— 3 -
s=—&€, ; (t cos p + t* cos By -} ...),
and
t (1—#?) cos @
tcos gy + t° cos 3m + athe Ty = Ta ae Danae ee
hence
= 1
ee
a al ser
Ss On = 2 tts 2p ee

If we put

R= cos ~
~~ 1—2 #? cos 2g-+t”’
then
x 1
=| -——— 5
soph Beate
or

; 3 (“3)
S hrosnp=— Lye (1 —#) Rk
Differentiating this equation, we get

x 1
2 » a Uae _ dR
Yn I, (x) sin ng = oe eu ( Bi — t) oo

1.3

( 495 )

bith: :
If now we multiply this equation by — sin (@ sing) dp and if
Tv

we integrate between the limits 0 and 2 we find
ea aru F "dAR ; :
= m J, (a) ZL, (a) = Oy —t i Bie sin (asin yp) dy

0
r 1 x
al i) ae
(1 — t*) | 2 cos (a sin @) cos g dy.

a
—— Sy e
a 0

Putting for the further reduction

™

u = | PR cos (a sin ~) cos y dp

cos* @ cos (a sin @) a
1—2 #? cos 2g + t* tee

du (cos? Pp sin (a sin g)
< 1—2 t? cos 2g + t*

we arrive at

sin p dg,

au cos* Pcos(asing) . , 5
Che 1—2 #? cos 2p + #4 ee

and because
1—2 # cos 2p + #4 ah -—t)

sin? g = a

40° 4¢?
we find
ny
au 1 :
ae a mu — vie cos* @ cos (a sin p) dy,
0
—72
where m= Fon
1 + cos 2
If we replace cos? p by aa we can easily reduce this
differential equation to
Pu See < ie ie
AGE ie 7 Bye = (a) + J, (@))

xz I, (a)
a

( 496 )

Let us now determine the integral of this equation satisfying the
conditions that for «= 0

x cos? yp d ~ ad
v,— -— a 2s F
(pee cos p+ tt 2 (1 —t*)
0

and
du

da
We then find

a maz —mz < oD VV m(a«- §) —m(az—s

ae a ee eee
4 (1—?) 8t?m B

0

and by this

r= Ae 1 wt ea 1
Suto ti=—$ Efe’ | Dyer ie
7 x—2-++8 1 r-2—f 1
T, (8) 1 3 (‘—— 9 (<=)
“a ; 3 dp oe E ae e t | 5
0

Remembering now that

=| =|
—— t——
2 t
é

=/,@) +¢46 +446 +-.-
1 1
Tee a (2)
we see that the residues are easily determined. We have

Jr h@h@= Tho + he +a +

1.)

7 BLL e+) —Le+e—A)l- (1)

a
eG
0

From this result another important relation may be deduced. To —
show this, we shall again develop

I, (w#—a)+ TL, (# + a)
into a series.
From

pi
i. (e— a) = = fe ¢ sin (a sin p — asin g) dg
Eg
0

and

(497 )

I, («+ a)= = fon yp sin (w sin p +- asin gp) dy

0
follows

one
I,(e—a4+h(¢#+a)= =f Zp sin (wv sin @) cos (a sin py) dy.
0
If we write
sin (@ sing) = 21, sng+27,sn3p+...

we obtain

4 ~
L(—adt+ihw«+a=—f (0) f sw —p cos (a sin ~p) dp

Ed
‘ 0

4 am ise |

+— TZ, (o) {sn gp sin 3p cos (a sin g) dy
x
0

4 T
4+ — 1, (0) f sn yp sin 5g cos (a sin p) dp
x
0

or as
2 [si @p sin (2n + 1) — cos (a sin g) dp =
?
=f 2np — cos (2n + 2) 4] cos (a sin 7) dip
= a [Lo (a) — see (a)]
d Is,
= 29 pace ate rtrl MOY (@)
da
we get

Pe aye Fite aoe ‘+ h@)S'+1@)o eS)
Substituting here

ds E ib
ir ie n(a)—ea n+1 (@)

we arrive finally at

PG ace Ta) = = San 1-48 Ln 4-1 (a) Tn (2).

( 498 )
With this equation the result (1) may be written

“r 3
SO" Oa oe ap i, (« —a+§)—J, (a t+-a - p)] (2)

If here we develop
I, (e—a+@) = I, (x) J, (a—B)4+27, (w) Z, (ag—B)+27, (x) L, (a—8)+ ...
I, (e7+a—p) = L, (x) £, (a—p)—2T, (x) T, (a—B)+ 22, (7) ZT, (e—8A)— .-..
we find

2, ee ee 7, (8)
S In41(@) Ln(e) = = Iq | Tala 4p) ae
1.3 ls B

0

and consequently by comparing the coefficients of L, (x)

Ike
et er ee sae

By means of this formula we can give equation (1) another form.
For, according to (3),

I, (« — a) = fue-«-9 Pap
0
=—fie—a4 Pap
0
a 1)
L@@+a= [here ~ pa
0

hence the second member of (1) takes the form

a—zr

a 1, (8) 7“
ae fz (oop) Pars fateten 8) = 13]

0

ef NC Wraae srrag ot sa
+$[ fue-a+a2 Pa free 8) 3]
0

or

1.0) = 9 Ba]
ei mee et Os us fi tt @ 3 Bl.

( 499 )

If we now put in the first integral § =a — y and in the se-
cond one ate + y¥ this becomes

c , ae (4+ ¥)
T u— y) ” ay Y, yon XN -
aif A + five ary |
0

with which equation (1) assumes ne final form

of

Cs) Ii, = I, (a+ 7
Fn tn nO) =F fL(e— le BI | ar.

a—y a+y

A closer investigation of formula (3) teaches us, that it holds
good for even values of 7 too, also that many analogous relations
exist. So we find inter alia, /; being any integer,

[es ue uae =
a—Pp n

. Ly, Ly a
fro ga
0

je eee

0

Sea —8) , TG) za)
ee
(a— B)? 4, (8) 48 abe 1 aig aes Ol ie

a

{| I, (a— B) I, (8) d8 = sin a.

0
We shall not dwell upon this at present; we only remark, that
when a very great positive value is assigned in (1) to z, so that

we find

Oy on
I, (e—a)+ 1, (¢«—a)=2 ae cos (« — ) COs @t,
UL. 4
oe I
I,(¢«—a+t s)—T1,(¢# + a—8)=2 yee sin (« -- =) sin (a — ).
Es ds

This changes (1) into

ra nm a Cn (el (3)

SS =f — nae z 1

= n I, (a) sin ey cos @ + = | aaa sin (a — B) dg
3 4 i

( 500 )

or, noticing that

S L ( ) , nt a rT ( )
ad a) sn ——- — — a
ie ieee peels ad

we have

8

If we differentiate this equation, we find

“I
I, (a) = cosa +f : @) sin (a — B) dp
0

a

fi
I, (a) = sina —f os cos (a—f) dp
e t
from which we conclude that
AC ee
3 sin B dg = 1 — cosa Tl, (a) — sina J, (a),
0

7,
f ae cos 8 d3 = sina I, (a) — cos aT, (a).
f

u

Geology. — “Contributions to the knowledge of the sedimentary
boulders in the Netherlands. 1. The Hondsrug in the province
of Groningen. 2. Upper Silurian boulders. — First commu-
nication: Boulders of the age of the Eastern Baltic zone G.”
By Dr. H. G. Jonker. (Communicated by Prof. K. Marty).

This communication introduces the description of the Upper Silurian
boulders of Groningen and its surroundings, in which my contribution
that treats of the Cambrian and Lower Silurian erraties and appeared
in 1904, is continued (86). The circumstance that in the summer
of last year I had-an opportunity of getting more intimately acquainted
with the Scandinavian-Baltic strata by investigations of my own
has aided me considerably in the study of these younger rocks.
Owing to nearly a month’s stay in Gothland I managed to collect
a great number of different species of rocks together with fossils
characteristic to them in order to compare them with erraties that
are found here. Much I owe to the kindness and assistance of
Drs. O. W. Wennerstex, who accompanied me on some excursions
and whom I had very often reason to admire for his extensive
knowledge of his native country, the classical ground for the study
of the Upper Silurian formation. But I have as yet not been able

( 501 )

to pay a visit to Seania and Oesel; the material for comparison
from those regions (present in the Min.-Geol. Institution in this
town, for the greater part collected by Mr. J. H. Bonnema), however,
will make up for it to a large extent, though not all questions can
be solved.

Some days’ stay at Upsala enabled me, thanks to the kindly
assistance of Dr. C. Wiman, to examine the collections present in
the Geological Institution from the different Lower Silurian regions
of Sweden. This examination, which of course had to be made in
haste, obliges me to introduce some alterations into my former
description which however are not very important. By this time the
material has been increased by new finds, and as more recent
publications always make some alterations or completions necessary,
I have made up my mind not to introduce them now but to collect
all these corrigenda and addenda in an appendix at the end of the
treatise of the Groningen erratics.

The real description of the Upper Silurian species of boulders of
which two have been dealt with in this communication, is preceded
by some pages which, from an_ historical point of view, are not
unimportant. After the appearance of my first contribution Dr. L.
Hoitmsrrém at Akarp was so kind as to draw my attention to some
parts of his lately published biography of Orro Torrin. From this
I learned that, in 1866, the latter had written a prize-essay on a
subject suggested by the Dutch Society of Sciences at Harlem, and
treating of the origin of the stones and fossils of the Groningen
Hondsrug. His essay was rewarded, but was never published and
not given up to the Dutch Society tll after the author’s death.
Thanks to the kindness of its secretary, Prof. dr. J. Bosscua, I have
been able to study Tore.1’s essay, and now comprehend his relation
to the Groningen boulders which formerly really puzzled me. His
ideas about our subject are a necessary completion of the historical
outline. :

Finally, it pleases me to state that this year as well as last year
the support of the Groningen University Fund fell to my share,
while the expenses of my investigations in Sweden have for the
greater part been defrayed by a subsidy granted to me by the
“Central Bureau for the promotion of the knowledge of the province
of Groningen” after reeeiving the approval of “the Board of the
Physical Society at Groningen.” This obvious interest taken in the
subject of my study has been a source of much delight to me.

( 502 )
Supplement to the Historical Outline.

O. Torell.

The prize-subject of the Dutch Society of Sciences at Harlem
(1865) ran as follows:

“On sait, surtout par le travail de M. Roemer a Breslau, que
plusieurs des fossiles, que lon trouve pres de Groningue appartien-
nent aux mémes especes que ceux que l’on trouve dans les terrains
siluriens de Tile de Gothland. Ce fait a conduit M. Roemer a la con-
clusion, que le diluvium de Groningue a été transporté de cette ile
de Gothland ; mais cette origine parait peu conciliable avec la direc-
tion dans laquelle ce diluvium est déposé, direction qui indiquerait
plutét un transport de la partie meéridionale de la Norvége. La
Société désire voir décidée cette question par une comparaison exacte
des fossiles de Groningue avec les minéraux et les fossiles des terrains
siluriens et autres de cette partie de la Norvege, en ayant égard
aussi aux modifications que le transport d’un pays éloigné et ses
suites ont fait subir a ces minéraux et a ces fossiles.”

TorELL’s answer to this question consists of two parts. The first
part deals with the essential question and is entitled: ‘‘Essai sur la
question proposée de la Société Hollandaise des Sciences a Harlem.”
Here the author enumerates the Groningen fossils known to him with
their geological occurrence and the literature on this subject. Hardly
any new fossils are mentioned, so that this description is little
more than a development of Rormer’s treatise of the Groningen
fossils. Nor is this wonderful, because he, too, had received the
greater part of this material from Conen, whom he had paid a visit
in 1865. No doubt there were among the collection sent to him by
the museum of Natural History at Groningen, about which I have
spoken in my first essay (86, p. XXXIII), various fossils unknown
to Rormer, but Torri seems not to have paid much attention to
the determination of new fossils. From his enumeration he arrives
at the conclusion that the sedimentary boulders mzgAt originate in
Norway, but that there is not the least proof for it and that most
likely the origin from Oesel-Gothland is much more probable.

In the second part of this first essay, however, he deals with the
rocks themselves. By a comparison with limestones from Norway
and Gothland he is led to exelude the first region altogether and
this result is further on supported by what the crystalline boulders
teach, which are described next. The dispersion of the different
erratics being examined, his conclusion with regard to the question
which had been put runs:

( 503 )

“Le résultat de ces recherches tend ainsi a confirmer
Vopinion déja émise par M. Frerp. Roemer, que les
bloes siluriens de Groningue proviennent de |’Esthonie
et de l’ile de Gothland, mais nullement dela Norveége”.

This first essay was inserted by him in 1866; the next year
followed a second, entitled :

“Recherches sur les phénomenes glaciaires de | Kurope du Nord”,
which for more than one reason is a remarkable treatise. As, how-
ever, its contents do not refer directly to the question we are discussing
it be only said that this treatise is, in short, a pleading in all its
details for the glacial theory, which is here for the first time con-
sistently adopted and asserted, under a motto borrowed from L. von
Bucu (Ueber die Ursachen der Verbreitung grosser Alpengeschiebe,
1811, Abhandl. d. Berlin. Akad., p. 185, 186), too interesting not
to be cited here:

“Wer sich etwas mit den blécken beschaftigt hat, welche in so
zahlloser Menge die Ebenen des nérdlichen Europa bedecken, wird
nicht einen Augenblick zweifeln, dasz nicht auch in dieser Zerstreu-
ung dasselbe Phanomen wiederholt ist, was in der Schweiz so auf-
fallend wird. Ware die Granitzone des Wallisausbruchs nicht von
den Jurabergen zuriickgehalten worden, so wiirde sie an den
Ufern des Doux und der Saone eben so zerstreut iiber die Flachen
gelagert sein, eben so dicht wie in soviel Gegenden der Mark
Brandenburg, von Pommern, Meklenburg, Holstein. ..... Das
nordische Phanomen ist daher wohl bei weitem grdsser als das
schweizerische, allein von derselben Natur; und walhrscheinlich liegt
thi deswegen auch eme dhntiche Ursache zum Grunde.

Torri. gives here a compendium of his opinions, founded on
insights acquired by many travels about the origin of diluvial
deposits and the grounds which in his opinion argue a glacial
covering. The older theories are amply criticised, and after describing
the formation of ice in Greenland, where the inland-ice covers an
extent of country of °/, of the North-European erratic zone, he says:

“Serait-il done absurde de supposer, qu'une couverture glaciaire
semblable, mais plus grande des */,, a existé aussi dans I’ Europe
du Nord pendant une époque, ot la faune marine du Spitzberg
vivait entre les 50° et 60° de Lat., o& le Betula nana croissait dans
le Devonshire et ot le renne avait son domicile dans la France
méridionale |”

This quotation sufficiently illustrates the importance of this essay. It
has however never been printed. Torin did claim back his work
from the Dutch Society to revise it for the press and various emen-

y~

oo

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 504 )

dations and marginal notes have been introduced, but he did not get
farther than that. This is much to be regretted as it was now not
until 1875 that his insights and opinions found adherents among
the German geologists; it was the year when ToreLL on the me-
morable day of Nov. 3 by his lecture for the German geological
society in connection with the glacial scratches once more discovered
by him on the Muschelkalk of Riidersdorf convinced different colleagues
of the correctness of his theory. For, if the above-mentioned essay
had been printed as early as 1867 it would have contributed in a
high measure to propagate the novel ideas more rapidly.

The bulky manuscript written in French and provided with French,
Swedish and Dutch annotations (the Dutch annotations are by STARING,
who was a members of the jury, as well as Bosqur? and vay Brepa)
is at present again in possession of the Dutch Society of Sciences.
The maps (2) and plate mentioned in the text are not wanting.
For further details about the contents the reader is referred to
ToreLL’s biography by Hotmsrrém (385, p. 18—25).

UPPER SILURIAN BOULDERS.

In the description of the Upper Silurian boulders various difficulties
present themselves, which all may be reduced to the fact that the
exact succession of strata in the Scandinavian-baltic zone is not known
for certain. Especially with regard to the eastern balticum the struc-
ture has long ago been made out by Semmupr and never refuted by
anybody that I know of. His division of the strata in Gothland, on
the contrary, corresponding with Murcuison’s conceptions has found
but few adherents, and is especially called in question by LinpsTrom,
who has a quite different opinion. This discrepancy as to the strue-
ture of Gothland, which has already existed many years, has not
yet been satisfactorily removed. It must be said, however, that well-
nigh all other investigators who have pronounced their opinion about
this question, have taken Linpsrrém’s side; a.o. Dames, who has
made a division which differs but a littke from Lixpsrrom’s; then
Srouiey, Wimax, Barner, Kayser and others. | myself, owing to
my short stay in Gothland, am not so fortunate as to be able to
pronounce a decided opinion, though it does seem to me that, on
the whole, Scuipr’s arguments are stronger than LixpsrrOM’s, so
that it appears scarcely possible to me that new investigations will
confirm the opinions of the latter in every respeet. In collecting fos-
sils in Gothland, 1 frequently doubted of the correctness of LinpsTROM’s

( 505 )

division, and in some cases noticed certain contradictions. Anyhow
for the present it is impossible to parallel the Upper Silurian strata
of Gothland with those of Oesel, a question, indeed, which for a
determination of boulders of that age can hardly be dispensed with.
We may sincerely hope that the researches by Hoim, who has been
engaged in this question, may not be long in coming, and that this
solution may finally settle the question !

(The chief literature about this controverted question follows here :
AST lowe oilewe2 2A 20. p. kor ete.).

Nevertheless in enumerating the species of boulders we must
adopt a certain succession of strata to arrive at the determination
of their age. I select for this purpose Scumipt’s division of the
Eastern Baltic Upper Silurian (8, p. 41—54), corresponding to the
method hitherto followed in the museum :

G.1. Jorden Beds.
2. Borealis bank.
3. Raikill Beds.
ET. Pentamerus-esthonus zone.
Ee Lower Oesel zone.
K. Upper Oesel zone.

Northern yellow zone.
2. SOuthern grey zone.

Dames (22), Stotimy (380), SteeerT (32), and others have founded the
determination of their erratics on the division of Gothland by Linp-
sTROM; as it seems to me, however, that our boulders approach the
Eastern baltic rocks much more, I did not follow this example,
the more so, as I have said before, the above-mentioned opinion,
which is quoted below with the alteration introduced by Dames,
does not appear to me to be the right one in every respect.

a. Oldest red shale beds with Arachnophyllum.

6. Stricklandinia-shale.

c, Shale beds and sandstone.

d. Bands of limestone and shale, in some parts oolite.

e. Pterygotus-beds.

Crinoid- and Cora/-limestones with intermediate Slromatopord-
rifls, Gastropoda- and Ascoceras-limestones, together with Mega-
lomus-banks.

gy. Upper Cephalopodan strata.

The material may best be subdivided into four groups: | Boulders
of the age G,—G,; Il... H; Wl... 7; IV... K. The last division
will appear to be by far the most important. Besides there ave some

35*

( 506 )

characteristic boulders, which cannot be placed in the Eastern baltic
scheme; these, together with others whose age lies between limits
too far apart to reduce them to one of these divisions, will be
described at the end.

After these introductory remarks we may proceed to the
description of the boulders of the first-mentioned group.

G,— G;.

The boulders belonging to the oldest zone G,, those of the Jérden
beds in Esthonia with Leptocoelia Duhoisii De VeRN., which are
occasionally mentioned by German geologists, are not found near
Groningen. The two younger zones G, and (,, however, have
been met with.

29. Borealis-limestone.

These well-known and characteristic boulders consist of limestone
or dolomite, and usually contain in large quantities remains of

Pentamerus borealis Eicaw.,

while other fossils are absolutely wanting. As regards the kind of
rock my material from Groningen may be divided into two varieties :

a. Limestone, as a rule distinctly crystalline but somewhat marly,
as may be easily observed on its weathered surface. The slightly
variegated colour of the ground-mass shades from gray to brownish-
yellow at the fresh fracture; if weathered, however, it has mostly
a sallow-yellowish-gray tinge. In this ground-mass the valves of the
above-mentioned species of Pentamerus always occur in great numbers;
they are invariably changed into crystalline calcite and this is very
often of a bright white colour, so that the always very thick shells
sometimes stand out very distinctly against the surface of the boulders,
which is sometimes polished. Besides the ground-mass weathers more |
readily than this calcite, so that the fossils appear in relief. The
number of these petrifactions has influenced the exterior appearance
of the boulder. Though always numerous, the ground-mass may yet
occur in sufficient quantities to give a compact character to the stone.
These limestones which are rather hard when not partially weathered
make up the majority of the stones found. The dimensions of some
of them amount to about 17 ¢.M. In other pieces the ground-mass
recedes much to the background and the stone consists almost
exclusively of fragments of the valves of this species of brachiopoda

and thus forms areal shell-breccia. The ground-mass then is commonly
weathered to a more or less earthy yellow mass, which also covers
the surface of the shells, by which the whole assumes a yellow
colour. In other cases, however, the weathered ground-mass is almost
white, sometimes also brown-ochre-yellow. Though they differ so
much in’ exterior appearance, all specimens have in common that
this Pentamerus oceurs almost always only in single valves which
themselves are, for the greater part broken into more fragments.
I have never been able to produce a wholly preserved specimen,
though some fragments actually show that parts of both valves occur
in natural position. So this confirms in the main the results of
Rormer’s examination (18, p. 74), though I doubt of the truth of
his opinion, according to which these boulders should contain only
ventral valves of this species. This conception was supported by
KicHWALD’s Communication that also in the parent rocks both valves
were never seen in connection. Nevertheless Eicuwaup did know the
smaller dorsal valve and describes it as having half the length of
the larger one, being much broader and much less vaulted. This
can hardly be right, for afterwards Scumipt found complete specimens
at Weissenfeld in the neighbourhood of Hapsal in Esthonia. Among
my material for comparison there are three such specimens from
the above-mentioned place, collected by Boxnrma. These, however,
show a dorsal valve, but little smaller than the ventral one, but
much flatter and so comparatively wider. This causes the great
difference between the two shells to disappear, and so there is no
reason left for the inexplicable fact that in boulders only the ventral
valve should occur. Meanwhile the interior structure of the small
shell has to be examined still to confirm this. I have not been
occupied with this work.

b. Dolomite, very fine-grained, sometimes even impalpable, of a
light-gray or light-brownish-yellow colour. This dolomitic ground-
mass also contains great numbers of nuclei of Pentamerus berealis
Eicuw., which are covered all over with little, graceful, dolomite-
rhombohedra, which, however, are easily perceptible by the naked eye.

Of these boulders, which in literature is usually called ‘Penta-
merenkalk”’ are found here :

Limestone : Boteringesingel, Groningen 2
Behind the “Sterrebosch’, — ,, 1
Helpman 1
the “Huis de Wolf’, near Haren 1
“Old Collection” 9
Dolomite : Boteringesingel, Groningen 2

(508 )

So in all 16 pieces. From this list appears that at an early time
already these bonlders have attracted the attention. Quite in corre-
spondence with this is the fact that as early as 1878 Martin men-
tioned 11 pieces from Groningen (6, p. 21, @ and c), and even
earlier still Rormer observed such boulders from here (1, p. 387,
n°. 16; 3, p. 269, n°. 27). Afterwards van Caker also pointed out
their occurrence in the Hondsrug (19, p. 357; 25, p. 363).

As regards the further spreading of this species of boulders, I refer
to Rormer’s excellent treatise about everything known at the time
about this subject (13, p. 75), and only wish to state here, that in
Germany these boulders are found in a great many places, but nowhere
in large quantities. So everywhere in East- and West-Prussia (20,
p. 538), in Posen, Silesia and Brandenburg, near Sorau in the district
of Frankfurt on the Oder, in South-Holstein in various localities
(18, p. 45). Further north they seem not to occur, more westward,
on the other hand, Liineburg in Hannover and Jever in Oldenburg
are still to be called as places where they are found. Afterwards
WanHNSCHAFFE has made mention of a specimen found near Havelberg
(14), and various observations attached to it as to the value of these
boulders for the determination of the direction of the ice-flow and the age
of the diluvial deposits, in which they are found. I hope afterwards
to recur to this question. In Pomerania the Borealis-limestone is not
(yet) known (31, p. 83), no more, it seems, in Mecklenburg. STOLLEY
afterwards states that he has found it again in Sleswick-Holstein,
but differs in this respect from all other notations Known to me that
he has come upon greater numbers of dolomites than of limestones
(30, p. 98). Lastly, these boulders are neither rare in the regions
south of the Russian Baltic provinces.

While, as we see, an enormous tract is taken up by the erratics,
the Borealis-limestone occupies but a very small part as solid rock.
In the eastern baltie (8, p. 43) it forms Scumipt’s zone G,, the
Borealisbank, which stretches in E.-W.-direction throughout Esthonia,
in the shape of a zone narrowing to the west, which also appears
in the island of Dagé. The rock consists of limestone or dolomite,
just like the boulders, and for a long time only single valves of
Pentamerus borealis Eicuw. have been found in it. Afterwards Scumipt
has discovered also complete specimens of this species, in a marly
variety of the rock from the neighbourhood of Hapsal, as already
stated (27, p. 180).

Of this eastern-baltic oecurrence I possess limestones for comparison
from Risti in the extreme west of this zone on the mainland and
dolomite from Pantifer in East-Esthonia, Our boulders correspond

( 509 )

very well with those limestones as regards the principal features,
though they are not interchangeable with the latter. More perfect
still is the correspondence of our dolomites with the sample from
Pantifer. From this it appears sufficiently that we have to lool for
the origin of our boulders in’ the eastern-baltic zone. Besides all
authors agree about this question. Of course we do not mean to say
that these Groningen erratics must of necessity originate in the zone
now known. As the Borealis-bank is also found in Dago, it may be
surmised that it stretches, or formerly stretched, still farther westward
under the sea, and the very uniform petrographical character of the
rock throughout Esthonia leads us to adopt the opinion that this sub-
marine continuation may also be considered as the possible place of
origin of our boulders. Of course it is impossible to indicate a definite
point in this zone,

30. Elegans-limestone.
With this term, referring to one of the most important fossils of
this species of boulders, I denote a crystalline-limestone, generally
fine-grained, sometimes almost impalpable, but still oftener rather
coarse-grained. Calcite, bright as water often occurs rather regularly
spread through the stone, but not in large quantities. The limestone
is not perceptibly dolomitie nor marly, as in the solution in nitric
acid only a small part is left and this solution produces no or hardly
any reaction with magnesia. Its colour is bright-gray, sometimes
rather yellowish-gray, rarely blewish-gray ; when weathered, however,
the stone shades from white to yellowish white. Its surface is very
often marked by distinctly visible glacial scratches. Layers are but
seldom perceptible and moreover not very distinet. Fossils are by
no means rare, but belong to a relatively small number of species,
which are mentioned here :
Phacops elegans Sars and Borck sp.
Leperditia, Hisingerd Scumipe.
Strophomena pecten a.
Vincularia nodulosa Ercuw.
Vincularia megastoma Ercuw.

Enerinurus punctatus WAnLB.

Calymene sp.

Orthoceras sp.

Proetus sp.

Ptilodictya sp.

Beyrichia sp.

Murchisonia sp.

(510 )

The first five species almost occur in every piece. Head-shields of
the said Phacops-species are very common, pygidia oceur as well,
and an almost complete thorax (which has not been figured by Scrap?)
has also been fonud. I have named these boulders after this charac-
teristic species. Equally important is further the presence of the
Leperditia-species, whose valves, both right and left, are occasionally
present in large numbers in a single stone; in the unweathered rock
they are bright brown, weathered nearly white. The mentioned
Strophomena-species is very plentiful, while especially the two bryozoa-
species, mentioned next, sometimes give the stone a peculiar appea-
rance. Though occurring in each of these boulders they are hardly
perceptible in the unweathered rock; they are together in great
numbers at the fractured surface, split along the foliaceous ‘‘Mittel-
schicht”, like graceful little white feathers. But the structure of this
fossil may be more distinctly perceived at the weathered surface of
the boulders. The other fossils mentioned are found but rarely and
do not contribute in a great measure to the diagnosis of the rock.

Besides these fossils, however, remains of brachiopoda are very
frequently met with, which no doubt are characteristic, though I
have failed to determine them satisfactorily. Some Rhynchonella- and
Orthis-species are undeniably present among them. One piece also
contains white, globular and angular crinoid-stems. Also the presence
of graptolite-remains is most interesting; these, however, have been
preserved too incompletely to be specifically determined.

Of this species of boulders, thus petrographically and palaeontologi-
cally characterized there are among my material 33 pieces from the
following places :

“Noorderbegraafplaats”’, Groningen 2
“Boteringesingel”’, : )
‘“Noorderbinnensingel ’, 3 af
“Nieuwe Boteringestraat’’, 5 1
Between “Parklaan” and ‘‘Heerebrug”, ,, 1
“Nieuwe Veelading”’, : x 3
Behind “het Sterrebosch”, 3 1
“Schietbaan”’, » 1
Café “de Passage’, Helpman 2
‘“Hilghestede”’, » 4
$3 i

Between Helpman and Haren 2
Villa “‘Edzes” near aa 1
< q

Groningen. J

“Old Collection” 2

(511 )

To determine the age of these boulders, which, as the above list
shows, are by no means rare near Groningen, all that is known
about the oceurrence of the characteristic fossils is Communicated

below as completely as possible.
Phacops elegans Sars and Borck sp.

NOMMIDI MO ano, lente et, LO 2:1) XT f. 17,

is said by Scumipt to occur in the Raikiill strata and the Estonus-
zone in Ksthonia. It was first found in the oldest of the two zones
G,, near Wahhokill in the centre of East-Esthonia, together with
Strophomena pecten Li. and Diplograpsus estonus Scumupt ; its locality
in the H-zone is almost straight to’ the south of it near Té6rwe in
the neighbourhood of Talkhof, on the border of Livonia. Complete
specimens, however, have not been found; the thorax found here is
there unknown.

Most probably P. quadrilineata ANG. Lixpsrrém, 12, p. 48; 17, p. 2;
is identical with this species; it has been described by the latter
from the oldest strata of the Upper Silurian formation @ and 4, near
Wisby. Moreover Scumipr mentions Faré and Lau there, places which
according to him belong to his middle and youngest zone in Gothland
(8, p. 74); this notation borrowed from Lrypsrrém seems to me to
want confirmation. In Sweden this fossil is also found in Dalarne
7, p. 27) and if P. elliptifrons Esmarck must be identified with
this species (which I cannot state with perfect certainty), in Jemtland
(29, p. 269) as well. The stage there argues a conclusion in the
affirmative. This fossil is not known from Scania. On the other hand
it is found together with Leperditia Hisingert Scumipr in Malmé
in the bay of Christiania, it seems in a corresponding stratum
(8, p. 74).

This species is not known in the literature of German boulders,
though Wicanp makes mention of Phacops Stokes’ Mitxn Epwarps,
the English fossil, which is most like our species (16, p. 40). The
illustrations of this fossil found near Rostock in Phacites-sandstone
prove, however, that this species certainly does not correspond in
all respects with our specimens. Phacops prussica Pomprcks, may
also be taken into consideration but neither the latter is completely
corresponding with those from Groningen; the rock in which this
species occurs in East-Prussia, ‘krystalliner, gelblich-grauer ober-
silurischer Kalk” would not argue against it (23, p. 19). Roemer
does not mention our species,

( 512 )
Leperditia Hisingeri Senor.
Scour, 10,p. 14—16, 7. 1, £ 5—*7.

identical with Leperditia Schmidt Woumopty, has already been known
for a long time from the neighbourhood of Wisby, where it frequently
occurs in the Strick/andinia-shale; esp. near Snickeiirdet I found
beautiful loose specimens. But it also occurs south of Wisby in
LinpstroOM’s stage ¢,: according to Ko LMopIN moreover also in the
shale of Westergarn (¢,) and Capellshamn (7, p. 133). In Esthonia
this fossil belongs to the zones G, and G,, and is found there in
many places, also in Dagé. Our specimens are on an average much
smaller than those of Gothland, but for the rest correspond very
well in their relative dimensions with the description of the true
form. As already stated, this species is also found near Christiania.
LInpDsTROM states moreover, that it is found in Seania (17, p. 25);
I failed to find out on what grounds this notation is based, and
have reasons to doubt of the truth of it.

Kirsow writes that he has found it in German boulders from
Spengawsken in’ West-Prussia and in a limestone (not corresponding
with ours) which curious enough also contains Leperditia baltiea
lis. (11, p. 274). Camrenewskt on the other hand has not come
upon the true species in Kast-Prussia and Kowno (34). Krause,
again, has found it in Neubrandenburg (24, p. 7) and Sronury in
a bright yellow, crypto-Golitic limestone from Sleswick-Holstein (80,
p. 109).

Strophomena pecten L.

is a fossil generally occurring in the Jérden and Raikiill beds in
Esthonia; in Gothland it is frequently found near Wisby and our
specimens correspond most with this occurrence. LinpstréM, however,
mentions it from e¢-h; hence | should not be surprised if different
varieties of this species were to be distinguished. Wian also states
to have found it in Jemtland in the quartzite with Phacops ellipti-
rons Es. (29, p. 270).

Gace, has described it from boulders of Beyrichia-limestone
(20, p. 47) from East- and West-Prussia; various authors moreover
mention it in boulders of different age, which strengthens my opinion
to draw no important conclusions from this species.

Vincularia nodulosa Eicuw. and V. megastoma Eicnw,
Ercawap, 5, T. XXIV, f. 8 and 9,

are very characteristic of the Raikill stratum in Esthonia and are

( 513 )

found there everywhere, though they oecur in the Estonus-zone as

well (8, p. 43).

Hnerinurus punctatus Wann.
is only present in a single piece and is a fossil found in all Upper
Silurian regions throughout all zones so that this species is of no

value for the determination of the age.

If we take these results together we get:

Esthonia. Gothland.
Phacops elegans Saxrs and Borck. = G,— H a— b.
Leperditia Hisingeri: Scum. GHG b—e.
Strophomena pecten Ia. Ge gra; c—h.
Vineularia nodilosa Ereuw. G,—A
Vincularia megastoma Facnw. G,—H

It appears from this distinctly, that these boulders are remains of
an equivalent of the Raikill zone G, in Esthonia. As to Gothland,
the comparison with Lixpstréw’s zone 4, if a comparison is desired,
is the most probable one.

Moreover this result is especially interesting, because boulders of
this age are not known in literature that I know of. In the Groningen
collection on the other hand some pieces have been brought to this
zone long since. But once Norrnine mentions a stone belonging to
this stratum which, however, contained no determinable fossils and
was only under reservation by reason of the great correspondence
to a piece of limestone from Raikiill, counted as a representative of
this zone (9, p. 291). Rormer doubts of this (13, p. 77).

As regards the origin of these boulders, it may first be stated,
that none of the regions where only one or a few of the fossils
characteristic of this oecur, viz. Norway, (Scania), Dalarne and Jemt-
land, can be taken into account. Besides the petrographic nature of
these deposits precludes this supposition altogether. In Gothland on
the contrary these fossils, with the exception of the two bryozoa-
species, are all found. But the rock occurring there (almost always
shales) does not show petrographically the least correspondence with
our limestone. In fact these boulders must not be considered to origi-
nate in Gothland.

Lastly as regards Esthonia: The Raikiill zone, G,, (8, p. 48)
extends from Laisholm in Livonia and Wahhokiill in East-Esthonia
westward as far as Dagd; in the eastern part the zone is wider and

( 514 )

narrows westward. It almost always consists of two systems, now
limestone, then dolomite. In the above-mentioned passage Scumipt
gives no further petrographical description of the rock; but afterwards
he speaks once more (33, p. 308) of a ‘“‘dichten, festen, etwas kiesel-
haltigen hellgelben Kalkstein, der demjenigen unsrer Raikiill’schen
Schicht am meisten gleicht”. For want of material for comparison I
dare not conclude from this a great correspondence with our lime-
stone. Further it is striking that Sciumipr says that petrifactions are
comparatively rare in the Raikiill stratum, except corals. Now our
boulders contain a comparatively great number of fossils, whereas
corals are altogether wanting. Just the reverse argues the fact that
graptolites occur in both, which though shortly described as Deplo-
graptus estonus Scum. (2, p. 226), are not yet figured. Perhaps the
same species may be found in our pieces.

By reason of the differences adduced above, I deem it little probable,
that the Raikiill stratum in Esthonia itself can be considered as the place
of origin. If is not impossible that the submarine continuation of this
zone consists of a rock more corresponding with our boulders. For
the present this question cannot be solved more completely though
material for comparison esp. from G, in Dagd, may render valuable

services.
LITHRA TURE.
1. Roemer, F. — » Ueber holldndische Diluvialgeschiebe”’.
Neues Jahrbuch fiir Mineralogie, ete., 1857, p. 385—592.
2. Scuaipr, F. — » Untersuchungen iiber die Silur-Formation von Ehstland,

Nord-Livland und Oesel’’.
Sep.-Abdruck a. d. Archiv f. d. Naturkunde Liy-, Ehst-und Kurlands, le Ser.,
Bd. Il, Lief. 1, p. 1—248. Dorpat, 1858.

3. Roper, F. —- » Versteinerungen der silurischen Diluvialgeschiebe von

Groningen in Holland”.
Neues Jahrbuch, ete., 1858, p. 257—272.

4. Scumipr, F. — »Beitrag zur Geologie der Insel Gotland, nebst einigen
Bemerkungen ueber die untersilurische Formation des Fest-
landes von Schweden und die Heimath der norddeutschen
silurischen Geschiebe’’.

Archiv f. d. Naturk, Liv-, Ehst- und Kurlands, le Ser., Bd. II, Lief. 2, no. 6,
p. 408—464; 1859.
5. Ercuwaup, E. pv’ — »Lethaea rossica ou Palaeontologie de la Russie’.
Atlas. Ancienne Période.
Stuttgart, 1859.

( 545 )

6. Martin, K. — »Niederlindische und nordwestdeutsche Sedimentdrgeschiebe,
thre Uebereinstimmung, gemeinschaftliche Herkunft und
Petrefacten”’.
Leiden, 1878.
7. Kotmopix, L. — » Ostracoda Silurica Gotlandiae’’.
Ofvers. af Kong]. Svensk. Vet.-Akad. Férhandl., 1879, no. 9, p. 1833—189;
Stockholm, 1880.
8. Scumipt, F. — »Revision der ostbaltischen silurischen Trilobiten, nebst
geognostischer Uebersicht des ostbaltischen Silurgebiets’’.
Abtheilung I.
Mém. de l’Acad. Imp. d. Se. de St. Pétersbourg, 7e Sér., T. XXX, no. 1; 1881.
9. Norriine, Ff. — »Die Cambrischen und Silurischen Geschiebe der Pro-
vinzen Ost-und Westpreussen’’.
Jahrbuch d. k. pr. geol. Landesanstalt ete, fiir 1882, p. 261—324 ; Berlin, 1883.

10. Scumipr, F. — »Miseellanea Silurica I11,

1. Nachtrag zur Monographie der russischen silurischen
Leperditien. ;
2. Die Crustaceenfauna der Eurypterenschichten von
Rootzikill auf Oesel”.
Mém. de l’Ac. Imp. d. Se. de St. Pétersbourg, Je Sér., 1. XXXL, no. 5; 1883.

11. Kirsow, J. — »Ueber silurische und devonische Geschiebe Westpreussens’’.

Schriften d. naturf. Ges. in Danzig, N. F., VI, 1, p. 203—300; 1884

12. Lixpsrrém, G. — »Forteckning pa Gotlands Siluriska Crustacéer”’.

Ofvers. af Kongl. Vet.-Akad. Forhand]., 1885, no. 6. p. 37—100.

13. Rormer, F. — »Lethaea erratica oder Aufzihlung und Beschreibung der
in der norddeutschen Ebene vorkommenden Diluvialgeschiebe
nordischer Sedimentirgesteine’’.

Palaeont. Abhandl., herausg. v. W. Dames und E. Kayser, I[, 5, 1885.

14. Waunscuarre, F. — »Bemerkungen zu dem Funde eines Geschiebes mit
Pentamerus borealis bei Havelberg”.

Jahrbuch d. k. pr. geol. Landesanstalt ete. f. 1887, p. 140—149; Berlin, 1888.

15. Lixpsrrém, G. — » Ueber die Schichtenfolge des Silur auf der Insel Got-

land”.
Neues Jahrbuch, 1888, I, p. 147—164.

16. Wicanp, G. — »Ueber die Trilobiten der silurischen Geschiebe in Mecklen-
burg”. 1.

Inang.-Dissert , Rostock; Berlin, 1888
17. Lanpstrém, G. — » List of the fossil faunas of Sweden. IT. Upper Silurian”.
Stockholm, L888.

18. Zetsz, 0. — » Beitrag zur Kenntniss der Ausbreitung, sowie besonders der
Bewegungsrichtungen des nordeuropdischen Tnlandeises in
diluvialer Zeit”.

Inaug.-Dissert., Konigsberg, 1$89.
19. Van Canker, F. J. P. — »Die zerquetschten Geschiebe und die nihere

Bestimmung der Groninger Mordnen-Abla-
gerung”.
Zeitschr, d. deutsch. geol. Ges., XLI, p. 343—358, 1889.

(516 )

20. Gace, C. — »Die Brachiopoden der cambrischen und silurischen Ge-

schiebe im Diluvium der Provinzen Ost- und Westpreussen”.
Beitr. z. Naturk. Preussens, herausg. v. d. phys.-oekon. Ges. zu Konigsberg,
6; Konigsberg, 1899.

21. Scumipr, F. — »Bemerkungen wiber die Schichtenfolye des Silur aut
Gotland”.

Neues Jahrbuch, 1890, If, p. 249—266.

22. Dames, W. — » Ueber die Schichtenfolge der Silurbildungen Gotlands und
thre Beziehungen zu obersilurischen Geschieben Nord-
deutschlands”’.

Sitz.-Ber. d. k. pr. Akad. d. Wiss. zu Berlin, 30 Oct. 1890, Bd. XLII,
p- Li L1—1129.
23. Pomprcxy, J. F. — »Die Trilobitenfauna der ost- und westpreussischen
Diluvialgeschiebe”.
Beitr. zur Naturk. Preussens, herausg. v. d. phys.-oekon. Ges. zu Kénigsberg,
7; Kénigsberg, 1890.

24. Krausr, A. — »Die Ostrakoden der silurischen Diluvialgeschiebe”.

Wiss. Beilage z. Programm der Luisenstidtischen Oberrealschule zu Berlin;
Ostern, 1891.

25. Van Carxer, F. J. P. — »De studie der erratica’’.

Hand. y. h. 8e Natuur- en Geneesk. Congres te Utrecht, p. 360—370; 1891.

26. Sreustorr, A. — »Sedimentdrgeschiebe von Neubrandenburg’, p. 166?

Archiv d. Ver. d. Fr. d. Naturgesch. in Mecklenburg, Bd. XLV, p. 161—
179; 1891.

27. Scumipr, F. — »Kinige Bemerkungen tiber das baltische Obersilur in
Veranlassung der Arbeit des Prof. W. Dames iiber
die Schichtenfolge der Silurbildungen (Gotlands”.

Bull. de V’Ac. Imp. d. Se. de St. Pétersbourg, N. S. [IL (XXXIV), 1892,

p. 381—400; also: Mél. géol. et palcont., tirés du Bull. ete., T. I, p. 119—138.
28. Barner, F. A. — »TVhe Crinoidea of Gotland’. 1.
Kongl. Svenska Vet.-Ak. Handl., XXV, no. 2, 1893.

29. Wiwan, C. —. » Ueber die Silurformation in Jemtland”.

Bull. of the geol. Lust. of the Univ. of Upsala f. 1893, I, p. 256—276;
Upsala, 1894.

30. Srottey, E. —- »Die cambrischen und silurischen Geschiebe Schleswig-
Holsteins und ihre Brachiopodenfauna’. 1. Geologischer
Theil. ;

Archiv f. Anthrop. u. Geol. Schleswig-Holsteins u. d. benachb. Gebiete,
1, 1, p. 85—136; 1895.
31. Conen, E. and Dreckr, W. — » Ueber Geschiebe aus Neu-Vorpommern
und Rigen”. Erste Fortsetzung.
Sep.-Abdr. a. d. Mitth. d. naturw. Ver. f Neuvorpommern und Rigen,
Jg. XXVILI, 1896.
32. Stecerr, L. — » Die versteinerungssihrenden Sedimentgeschiebe im Glacial-

diluctum des nordiestlichen Sachsens”.
Zeitschr, f. Naturwiss., Bd, 71, p. 37-138; 1898,

( 517 )

33. Scumipt, F. — »UVeber eine neue grosze Leperditia aus lithauischen Ge-
schieben”.
Verhandl, d. k. russ. Min, Ges. zu St. Petersburg, 2e Ser., Bd. XXXVIII,
Lief. 1, VI, p. 307—811; 1900.
34. CumeLewski1, C. — » Die Leperditicn der obersilurischen Gleschiehe des
Gouvernement Kowno und dev Provinzen Ost- und
Westpreussen”.

Schrift. d. phys.-oekon. Ges. zu Kénigsberg, Jg. 41, 1900, p. 1—38.

5
35. Hotmsrrém, L. — »Otto Torell’’. Minnesteckning.
Geol. Moren. i Stockholm f6rhandl., XXIII, H. 5; 1901. (Separat-Abdruck).
36. Jonker, H. G. — »Bijdragen tot de kennis der sedimentaire zwerfsteenen

in Nederland.

I. De Hondsrug in de provincie Groningen.

1. Lnleiding. Cambrische en ondersilurische zwerjsteenen”.
Acad. Proefschrift, Groningen, 1904.

Groxtwaun, Min.-Geol. Instit., 31 December 1904.

Chemistry. — “On some phenomena, which can occur tn the case of
partial miscibility of two liquids, one of them being anomalous,
specially water.” By J. 3. vax Laan. (Communicated by Prof.
H. A. Lorentz).

1. In the second part of his Continuitdt (1900)') Prof. vax per
Waars has given the theory of the so called longitudinal plait on
the y-surface, and in the last Chapter (§ 12, p. 175 sequ.) he gives
moreover a special, ample discussion of this plait, in particular with
regard to anomalous components. It is shown there, that for the
appearance of certain complieations, which can present themselves
at this plait, one of the two components must be anomalous *).

In the following pages I shall try to explain the appearance of the
different particular forms, which can present themselves, when one
of the components is associative, specially when this anomalous
component is water.

2. We begin to remember brietly the theory of the phenomenon
of partial miscibility for binary mixtures of normal substances.

It is well known, that the total thermodynamic potential is repre-
sented by

1) p. 41—45.
2) Also compare These Proceedings of Noy. 5, 1902.

( 518 )
Z = — D(n,k,) T (log T— 1) + Z(n,(e,,.) — T S(n,(,).) —
| pde + pe + RT Z(n, log n,),
or

Z= J(n,C,) — | foe — RT Yn, . log Ln, = ye | + RT X(n, log =).
=n,

Differentiating subsequently at constant 7’ and p with respect to
n, and 7,, we get:
OZ dw n
= — lo
On, jy Ons 5 > A |
__ 0%
AD Sai
where C, and C, are pure functions of the temperature, represented by
C, = > k, T (log Js (e:)o i T(%,),
C, — > kT (log sears 1) Sir (¢s)o a T(%3)o

whereas the quantity @ is given by

dw n %
Se Og |
ae wen. a 4 in, |

?

@ = | pdv — RT Sn, .loeg Sn, — pv. . . - . )

The meaning of the different quantities 7,, (@,),, (%,).. ete. ete. is
supposed to be known.

We will substitute now the variables n, and n, by wz, so that
n, =1—a2, n,=wz and =n,=1. As @ is, just as Z, a homogeneous
function of the jirst degree with respect to m, and n,, we may write:

dw as
pu, =C,— (« —u =) + RT log (A — x) |
: Wore (2)
u. == C, — ( + (1 — 2) i) + RT log x |
be !

Now, when there is a plait on the Zsurface, the spinodal-curve,
that is to say its projection on the 7’ 7-plane, will be given

ae
by the condition - == 0. or also: p, being =Z—-2x tel) and —
Ou? Ow
et) a by as == ute ae ae
We therefore find for the equation of this curve in the 7,c-plane:
ie) Tage be
‘ Oz? 18 Trae

or
Cw

eat 2 (Se ae Peres

( 519 )

If we use the equation of VAN DER WaAAIs:
=n; al ail hi a

2

P=

’
v—b v

then we obtain:
w = Sn, . RT log (v — b) + . — RT En, . log En, — pv’).
Supposing now, that in the case of liquids the external pressure p
(or the vapour-tension) can be neglected with respect to the mole-
cular pression - the equation of VAN per WAALS may be written:

a Pie dadls

— b)

v v—b

and the expression for @, when in the same manner pv is omitted

a
by the side of — , passes into
v

Sn, . RT

iq a 2 |
w= =n, . RT log ee ase =n, . log Sn, ,

) y2
or
Ld al a
o= =n, . RT log +—,
yp? v
that is to say into
raw a
w = RT log a EES
[v2 v
: y i *w :
when 2n,=1. For —— we find consequently :
av

rw 0? a RT 0? 1 a
Ow? men v . Ow? “9 ye

by which the equation (8) of the projection of the locus of the points
of inflection on the 7’, .v-plane passes into

RI =20— | 5 is S) RE ay. =
0? fa

2 (1—z2) ale)
Figg Ee

= (> Aiea eee (5)

a

L + w (law) log =

or into

Nd eer CFT eS aaa

.

5 yi b=), then pae still gives a term ne | db . But this term may be
; v—b
regarded as independent of w, and so can be added to the temperature function C;.
ae
ob

e Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 520 )

. a . .
The term with Jog — was introduced some time ago by VAN DER
v i
Waats *); in the original theory this term was neglected, and so the
‘ é F he Arr
equation (4) was simply R7’= wx (l—2) “\
Le

- A a ‘ as a
In consequence of the relations — = [pis die SSS ps where ac-
v . v

cording to the variability of the liquid-volume v, the coefficients f
and y will still yary slowly with the temperature (7 is the well
known factor of the vapour-tension, which may be put circa 7), we
can-also write for (4):

7 pea 0°77,
r Serena (4a)
—— a ee . . . . . . +0
1+. (1—z2) Dias
xv

We see, that only in the case, that the critical pressures of the two
components differ little, the term with log pe can be omitted. This
will be also the case, when x is in the neighbourhood of 1 and 0.
But in all other cases if would be inaccurate to omit a priori the
designed term.

Further we write:

a = (l—za)* a, 4- 24 (1—2) a,, + a? a,

v = (l—2z) »v, + av,
since for liquids at low temperatures v can be supposed dependent on
wv in entirely the same manner as 6 = (1—.v) b, + vb,. The molecular
volumes 7, and 7, must then be regarded, just as 6, and 6,, as constant
or as slowly varying with the temperature *). We then find after some
reductions :

1) These Proceedings, in Ternary Systems, specially IV, p. 96—100. (June 12,
1902); see also July 18, 1904, p. 145 sequ.

2) If we substitute in the case of liquids v by #, and then write b = (1—.r) 6; + wba,
the difficulty arises, that in that way quantities of order v—6 are neglected
against those of order v, and the question would present itself, if this is only
upon yery definile conditions of in contradiction with omitting p by the side of

ih Toe : :
(This observation was kindly made to me by Prof. Loreyrz).

per

[ hope to escape this diflicully by not substituting » by 6, but by simply
supposing the velume vc /inear/y variable with «°in the case of liquids at
low temperatures; by writing therefore for 7, analogous to the expression for 0,
p= (l—w) ve, + ry. As I remarked, 7, and 7, still vary slowly with the tempe-
rature, whereas 4, and 6, of course would be perfectly constant. Now it

0? fa 2 ;
eT ene (a,0," + 4,¥,* — 24,40 304)s

or — when we suppose for normal components the relation of

BervuELor, viz. a,, = a,a,, as approximately exact:
OF “fa 2 rahe
Ow? aa hat (ray ds) sw, (8)
aL v )
ee

As the second member will be always positive, even if a,, might
be <Va,a, *), the curve 7 = /(v) will always turn its convex side
to the w-axis.

We will now determine = log =. With a,, =V a,a, ”) the expres-
sion for a becomes:
a= [(l—2) Ya, + « Ya,}’,
so that
(l—a) Ya, + « Vi
(l—2) v, + av,

a ,
log i 2 log
Consequently we have:

: : a t
will be better justified to subsitute pared fRT,, than 7 (and afterwards “1 by
1

(lg
V
temperature. For it is easy to show, that the expression for the vapour-tension
v

. 5 a) y2 a), db
for a single substance at low temperatures is Jog —-= —_— | —— (vy is in
Pp RT v—b

FRT, and by fRT3), where f will vary in the same manner as 7 with

the first two terms the liquid volume), whence we can deduce, in connexion with
a : a foe a
the empirical relation log" = AG — 1), where f is circa 7, that a yard ye

The error made by supposing v linearly variable with x, will certainly be
much smaller than by putting vd. In that way errors of,at least 16°/, would

; Oats Be Pisa bir. ;
be made, since a will be nearly */, for liquids in the neighbourhood of the

melting-point.
The quantities 7, and v, can now also immediately be substituted by the expe-
rimentally determined values in the liquid state.

1) See van per Waats, These Proceedings of Oct. 8, 1902, p. 294.

) Although there is no sufficient reason for this relation, | have supposed it
approximately exact, also because only in this case a simple expression could

: o? a
be obtained for aga 109 ce

36*

(522 )

ky Sane D eee ’
Oz ” v? Va v
and therefore

0? we a “ [a eo) ay,

v? a

This expression can be reduced to a different form, and then
we find:

0? a 2
5A log Sea (vy, Va,—v, Va,) [(v, V¥a,—v, Va.) + 2 v(Ya,—Va,)],

whence it appears, that the factor »,Va,—v, Va, occurs in the

0° aa
expression for 503 9 = as well as in that for ani “).
& 7 v

a a
+ / 1 : +e
Now when v,/a,=v,V/a, or —; = —., when in other words the critical
vy Us
, . 0? a
pressures of the two components are equa/, then > log — becomes = 0.
wv v

Ow? \ x
tudinal plait will disappear, (at the same time the curve 7c = /(«)
will then pass into a straight line).

We see therefore, that for occurrence of the phenomenon of partial
miscibility at attainable, that is too say at not to low temperatures, the
critical pressures of the two components must dijjer as much as possible.

Now this is not the case for the greater part of nurmal substances,
and that is the explanation of the well known fact, that for mixtures
of normal substances the phenomenon of limitated miscibility has
been so very rarely found at the common temperatures.

When we substitute (5) and (6) in the equation (4), then we
find finally :

; Orn/aG :
But then simultaneously —|(| — ) will be = 0, and the whole longi-
Af oD

(v, Va,— vy Vv a a)”

RT = 2 «(1—2) EEA |

where te

No «(ima v,)’ vv

a

This would be a pure parabola, if » and 1-+ 4 were indepen-
dent of w.

3. We will now determine the values of « and 7’ for the “critical

point of miscibility.” For this the conditions 42> and —t =

combined must be satisfied, or —- what is the same the conditions
Ou, dT
= — == (0
Ou da

as is obvious. Now from (7) follows, when 1 -+ 4 is supposed
independent of , which will be certainly permitted, in consequence
of the small values of 4 in the case of normal substances :

dP _2(%, Vee, Va,)(1-2e Se(l—a
t.—_ = — = sa Vs— 2) {5
Ale pele me | Ta ay! 2

as vv, + a(v,—r,). This expression becomes = 0, when
(L— 2) (1+ra) — 3 rae (1L—a) = 0

where *» = -——. This yields:

re? — 2(r + 1)a+1=—0,

whence
SN Ge a ps a een.

When += 0, that is to say when v,=>,, then a =0,5. At all
events this will be approximately the case, if A should still be in
any way dependent on «.

We will reduce now the equation (7) somewhat. With a, = / R7,r,
and a, =/RT,v,, where T, and T,-are the critical temperatures of
the components, these equations pass, after substituting 7, for «, into

[v, V fF RL», — », VERT»)

Dee (: + 2 st)
“il

a een (vy—v,)? VT, Us 2 YER i)"
fe. Cc. 2 IV Tr, tat Tv, v,-V To) \"

st"

RE. 2 a, (1 — a); :(1 + A,)

or with 7, = 6T; and v, = g?, into

— 6)? |
(1 + (» — 1) 2)’
hes ra | Ce (V6g—1)
. “U4 (@=D ar)? LE VW Op—D) x)? |
since (p — V4g) = 9 (Vy — VA)’.
We shall illustrate these equations by an example. In order to’
find the critical point as high as possible, we will choose two

DSO f ac (he San)
(8)

( 524 )

normal substances, of which the critical pressures differ as much as
possible. We take therefore ether and carbon disulphide. The eritical
data are the following:

CS, | T,=548° ; p,=76 atm.

ether | T,=467° ; p. = Bb5iatm.

; v Tried k
In order to determine g=—-, we remark, that ee hea
vy : VaR
Ofek ie way a, Fe. a,
v, = —.—, as for instance —=/fRY, and —=yp,. We have there-
Y Ps Ui On
fore:
Dp IP le Pr
aS SS eS
v, Ps oP. * BAO PY

that is to say g = 6a, where the proportion Pr is represented by a.
ae Ps ;

Now for the designed substances 6 = 0,852, a2 = 2,17, so that
we find g= 1,85. Since » = p~—1, the equation for 2, passes inte

Fat Ot Gee a Mee es |)

and hence we find for «, the value 0,29. Further /@=0,923,
Vy = 1,36, f=7, and so (8) becomes:
7 LEX 0,206 x 1,85 X 0,191
“ie (1,247)?

te =

548:(1 + Ad),

1,019
em 1,94

We have further:

_ (0,723 0,0650
A,.= 0,412 1,335 1,153.
so that we find for 1 + A. the value 1,17.

Hence 7. becomes 288: 1,17 = 246 = — 27° C.

The eritical point of the chosen substances lies therefore still a
thirty degrees beneath the common zero of Celsius. And for the
greater majority of other normal substances we will find for 7, still
much smaller values — because the critical pressures will differ
there in most of the cases less than in the case of ether and CS,.

548::(1 + A.) = 288: (1 + AQ).

= 0,412 << 05409) = 08695

4. All that precedes now undergoes important modifications, when
one of the two components is anomalous, specially water. For in
the first place the critical pressure of the water is very high, not

ene ot

as

SSC we wey

~ ee

( 525 )

less than 198 atm., so that it will differ much from the critical

pressures of most of the other substances. And in the second place the
value of 7, is here so extraordinarily variable with the temperature.
Water is in this respect exceptional in Nature, and gives therefore
rise to very peculiar phenomena, which are not found with other
substances, or not in that degree. Alcohol e.g. is also an anomalous
substance, but neither is the variability of the molecular volume
there particularly great, nor the critical pressure particularly high.
We know, that the variation of the molecular volume finds its
canse in the decomposition of the double molecules with the temperature.

Because vr, gradually grows smaller and smaller, the quantity

(v,Va,—2,V (r-.) Pe

which principally determines the value of 7, will become greater

1

and greater. And the initial value of that quantity is in the case
of water as one of the components already higher than for mixtures
of normal substances. This is connected with the high critical
pressure of water, being 198 atm., whence can be calculated, that
the critical pressure

if water continued to consist of only double
molecules — would yet still amount to circa 66 atin., Le. higher than
that of most of the normal substances. [Of course the designed express-
- , é : ; Va, ly
ion will increase with decreasing values of v, only et ;
1 2

that is to say, when the critical pressure of the first component is
greater than that of the second. This condition will nearly always
be satisfied, when we assume water as the /i7sf component}.

As said, the decrease of v, is very considerable in the case of
water. | remember, that I found some years ago‘), that for 18 Gr.
water v, = 19,78 cem., when all the molecules are double ; and only

= 11,34 cem. for 18 Gr., when all the molecules are single. When

therefore the temperature increases from nearly — 90° C., where all
the molecules are double (supposing, that the water had not congealed
long before), to circa 230° C., where all the molecules have become
single, then +, will diminish down to nearly */, of its original
value.

[In the same Memoir I showed, that in this fact lies also the
explication — qualitative as well as quantitative — of the well-
known phenomenon of maximum density at 4° C.]

Now the consequence of this variability of v, will be, that the
second member of (7) — we will represent it (livided by R) in

Z. f. Ph. Ghemie 31 (Jubelband van ‘2 Horr), p. 1—16, specially p. 13.

the following by A — will be no longer a constant for a definite
value of v, but a funetion of temperature.

If we draw therefore (see fig. 1) the straight line OM, which
divides into halves the angle of coordinates (O7' is the axis of
temperature, OA’ that of the values of A’) — then for mixtures of
normal substances the point of intersection of the straight line K =
const., which runs consequently parallel with the Z-axis, with the
line OM will represent the temperature, corresponding in the 7'-
projection of the spinodal curve with the chosen value of w. If this
were «,, then we should find in this manner 7’. That temperature will
be — as we have shown on the preceding pages — extremely low.

On the other hand, in the case of anomalous mixtures, that is
here: where one of the components is an associative substance, the
straight line AA’ will transform itself into fo straight lines, joined
by a curve (see fig. 2). The first straight line corresponds then with
the temperatures, where all the molecules are double, that is therefore
in the case of water below — 90° C.; the second straight line will
correspond with the temperatures, where all the molecules have become
single — so for water above 230°C. The joining curve will cor-
respond with the temperatures between — 90°C. and 230° C., where
the process of dissociation of the double molecules is going on.

Several cases can occur here, which presently we will briefly discuss,

5. We should now have to deduce an expression for RZ’ and
4, analogous to (7), but this time for the case that one of the
substances is anomalous. The required considerations and calculations
will not be reproduced here, however, because I shall do so in the
more ample Memoir, which will soon be published in the Archives
Tyler. We therefore will limit ourself to the communication of the
final result, viz.

RT = 22 (1—2) (2 te lott Siete

ma = : (1+ A)

A =22(1—2) 1+ ee) | S»
xt tae a ae | ae
— (1-8) ae x is Nes | ,

a
These expressions come in the place of the former expressions
(7). Of course they are somewhat more complicated, but they
have essentially the same form, as will be discussed amply in the
ip ie

5 tee

(v,—v,)* a (Va,—Va,)?

2

— (10)

v a

designed Memoir. It will only be remarked, that 22, =

a

(597 )

where ? is the degree of dissociation of the double molecules ; that
Uy, =*'/, 1—B)v, + Bv,, where v, is the molecular volume of the

double molecules, and v, represents that of the single molecules ;

’
OL

and then once more the relation @,, = WV«a,e, has been used, by
which again the calculation of 4 was practically possible.
The expression for @ reduced, in consequence of ¢,=4a,, @,,=2a,,

Gu A0;., 10

a=(1 —~2)?a, + a2’ a, + 2a (1 — a) a,,.

That for 6 or »v to v=(1—2)»v,, + a@v,, where »v,, has the
meaning as is indicated above. (The index 0 relates to the double
molecules, the index 1 to the single molecules of the associative
substance ; the index 2 relates to the second, normal substance).

As is already briefly indicated above, it will be principally the
factor (v, ¥a,—%,, Va,)*, on which the phenomenon, studied by us,
depends. The great variability of the quantity v,, with the temperature
is the only cause of all these peculiar phenomena of partial miscibility,
occurring in the case of mixtures, when one of the components is
anomalous, specially water.

That factor will increase more and more with the temperature,
because 7,, decreases in consequence of the continual formation of
new single molecules from the dissociating double molecules — a
single molecule being much smaller than half a double molecule.
(compare § 4).

It is evident, that the denominator * (by v,,) will equally diminish
with the temperature, so that the value of the second member of
(10) will increase still more. The variations of the other terms
have comparatively but little influence.

6. What will now be the different forms of the plait — i.e. in
the 7\~ representation — when the course of the curve A = /(7’)
(see §4) is continually modified with the different components added
to the water? (We call attention to the fact, that A’ represents the
second member of (10), divided by F, and that the following figures
indicate therefore the graphical solution of the equation 7 =
with respect to 7’).

a. The case of normal substances has already been considered by
us. It is represented by fig. 1. The spinodal curve will have the
same form as in fig. 2.

b. In fig. 2 the straight part of the curve A=/(7'), where K
has the initial value A, (all molecules are still double), intersects
the line OM in the point A; whereas the curved part, and the

(528 )

second straight part, where A’ assumes the final value Ay (all
molecules have become single), lie wholly on the right of OM. The
plait will consequently be identical with that of the preceding case
— only with this difference, that the point A lies below — 90° C.,
where the dissociation of the double molecules begins, so that this
point lies wholly beyond the region of attainable temperatures.

c. As soon as the value of A’ increases a little, we get the case
of transition of fig. 3. The curve A= /(7’) touches now the line
OM in B,C, and trom this moment the ¢so/ated plait will begin to
appear, extending itself above the just regarded normal plait, which
lies in unattainable depth. Here it is only two coinciding critical
points in the one point LC.

d. When the value of A, is still a little greater, the case of fig.4
will present itself, where the line OJ is intersected, besides in A,
in still two other points 6 and C. The isolated plait above the
normal one is formed now, with two critical points, a /orer one
in B and an upper one in C. Everywhere between & and C
K is > 7, just as below A, so that we are, in consequence of
eZ : stat : ‘ é
wet in the unstable region, i.e. within the spinodal line of
the plait.

This case — or the case of fig. 6 — is realised by a great
number of substances, also in the case of vo anomalous substances ').

a. In some cases the wpper critical point is found, as in the

case of water and COC (Rorumunp), and of H,O and isobutyl-

alcohol (AuExnsew); propably also in the case of water and ether
(Kropsre and Arexesew), of H,O and CO (C,H;), (Rornmunn), of
H,O and ethyl-acetate (Avexemew), and of H,O and amyl-alcohol
(Atexesew), in which latter cases, however, the point C was not
reached. As to water and ether e.g., Kiosppie has already found,
that the values of 2 of the two coexisting liquid phases reapproach
each other, when the temperature is lowered. That is an indication
for the existence either of a lower critical point, lying still more
down or of a contraction as in fig. 6.

8. In other cases it is only the /ower critical point, that is observed,
as in the case of water and triethylamine (RorHMUND), water and
diethylamine (Gurnee), and of water and -collidine (RoTHMunD).

1) Many anomalous substances namely can be regarded as normal ones, because
the variation of ¢ is so small; only in the case of water this variation is excep-
lionally great.

‘fg

According to these observations the first mixture has its eritieal
point (B) at nearly 18° C., the third at 6° C.*).

In the case of water and nicotine*) Hupson (Z. f. Ph. Ch. 47,
p- 113) has observed the complete isolated plait. But here a hydrate
is formed, being decomposed continually, when the temperature rises.
The theory of the phenomenon remains however formally the same:
everywhere, where a pretty considerable variaton in the value of
v presents itself — whatever should be the cause of it the
existence of such a plait may be expected — as soon as the required
conditions are satisfied.

Still another example is found in mixtures of carbonic-acid and
nitrobenzol (Bicuner), which makes it probable, that CO, in liquid
state is an associative liquid. Indeed, there exist important reasons
in the thermal behaviour of that substance which would confirm
that supposition.

Aten has observed, that CH,Cl and pyridine mix in every pro-
portion, but that the combination, which is soon formed. is nearly
unmixable with both components. In this case again there is founda
lower critical point, for both plaits — i.e. for that, formed by CH,Cl
and the combination, and for that, formed by pyridine and the
combination.

It is a matter of course that the existence of a /ower critical
point necessarily determines that of an wpper one. With rise of
temperature the /iquid mixture approaches more and more to a
gaseous one, where of course miscibility in every proportion takes
place. (How the plait can transform itself there, and pass into the
transversal plait, lies entirely without the plan of this inquiry).

Inversely we can not always conclude from the existence of an
upper critical point to that of a lower one, because — even, when
the connodal curve begins to contract downward — the case of
fig. 6 can occur.

But this is certain, that when an upper critical point is found
at ordinary temperatures, we have always to deal with the point C,
and not with A, the latter always lying (see fig. 2) in the ease of
mixtures of water and a normal (or anomalous) substance below
— 90° C., and in the case of mixtures of two normal substances
(compare § 3) at most some thirty degrees below 0° C,

Nearly always there may therefore be expected the case of fig. 4,

1) Kuenen (Phil. Mag. [6] 6, p. 637—653 (1903)) could however not confirm
the existence of a lower critical point for diethylamine. In an earlier Memoir
Kurven has found also a lower critical point for mixtures of C.Hy and ethyl:
isopropyl- and butyl-alcohol.

( 530.)

or that of fig. 6, when partial miscibility presents itself. The normal
plait with the critical point in A will appear only in a great minority
of cases, and can be regarded as highly exceptional. So the mixtures
of water with phenol (ALEXEJEW;, with suceinitrile (SCHREINEMAKERS),

with aniline (ALEXEIEW), with tsobutylic-acid (id.), ete., ete. — which
all present an upper critical point — will offer with great certainty

examples of the very general case of fig. 6 or of that of fig. 4.

e. Fig. 5 again represents a transitory case, where the value of

K, is still a little greater than in fig. 4. The two plaits — the
normal one and the isolated one — will coincide from this moment

into one continual plait.

fv. This will be the case in fig. 6. It is observed for mixtures
of water and secondary butylalcohol (ALBXKIEW). But, as we already
remarked above, many observations with an upper critical point
may belong just as well — whether the compositions of the two
coexisting phases approach each other at lower temperatures or
not — to this case as to that of fig. 4. The example mentioned
belongs with certainty to the class of fig. 6, because it is observed,
that the values of x after beginning to approach each other diverge
again at still lower temperatures.

Fig. 7 shows, that the contraction at ), where the curve A = 7 (7)
comes into the neighbourhood of the line O/, gradually vanishes, so
that the plait at last again will assume the xormal form — only
with this difference, however, that the critical point C of our quasi-
normal plait will appear at higher temperatures than the critical
point A of the real normal plait. ;

Remark. It will be superfluous to remark, that the m2merical
calculations by means of the formula (10) can be executed only
then, when the conditions are satisfied, on which that expression is
_ deduced. That will accordingly only be the case, when really p is to

a :
be omitted against — (see § 2), that is to say at temperatures, which
2 0 Sa)

are not higher than circa half the critical temperature (in the ordi-
nary meaning) of the mixture.

7. The question rises now, what will be the conditions to be
satisfied, that the transitory cases of the figs. 3 and 5 may present
themselves. Here too we only communicate the results of the caleu-
lations, that we have made on this subject. We found namely, that
the ¢solated plait (fig. 4) is only possible, when the second (normal)
substance has a critical pressure between circa 35 and 70 atm., and

( 531 )

this nearly independent of the critical temperature of these substances
(provided that the latter is between *
water).

/

, and 1-time of that of the

All normal!) substances, which possess a critical pressure above
+ 70 at., mix in every proportion with water; all such sub-
stances, having a critical pressure below + 35 atm., will form a
continual plait (tig. 6).

ry ea. . A -

lo the first group of substances belong those with relatively sia//
molecular volume (many anorganie substances and salts); to the
second group those with relatively great molecular volume (many
organic substances).

z 0? log Pe ;
As to the factor 1+ A=1-+.#(1 — ez) ae , the calculations

Lv

have taught, that this factor at higher temperatures, where 8 comes
into the neighbourhood of 1, can become very great, and also will
be pretty strongly variable with v. So I found for that factor for
B= 1 (7’= 280°) the values 2,57, 2,54, 2,25, 1,94 and 1,70, resp.
for v= 0,1, 0,2, 0,8, 0,4 and 0,5. But at such high temperatures
the deduced formulae are not longer exact, p being in that case
no longer to be neglected against °/,».

However, for /ower temperatures, where 3 approaches 0, 1 + A
will not differ much from 1, and will be little dependent on «. At

these temperatures — and for these temperatures the formulae are
deduced — 1+ 4 can, when not neglectable, yet be regarded as

a constant factor. So I found for 1-- 4 the values 1,08, 1,10, 1,10,
1:09 and 1,08, resp. for «0,1 unto 0,5.

Finally, L have applied the formula (10) for the case of triethy/-
amine and water, and found that, whenever the critical pressure,
viz. 30 atm., lies below the above designed limiting pressure of
35 atm., the appearance of a lower critical point at circa 18° C.
is not in contradiction with the given. theory. It must not be
forgotten here, that when the temperature, where 2 is practically
=0, lies above — 90°C., the limit in question also will lie below
35 atm.

1) And as we have already seen above, also many wnomalous substances, where
the variability of v is small.

( 532 )

Mathematics. — “Zhe equations by which the locus of the principal
aves of a pencil of quadratic surfaces ts determined” by
Mr. CarpINaat.

1. The communication following here can be regarded as a con-
tinuation of the preceding one included in the Proceedings of Nov. 26
1904. It contains the analytical treatment of the problem, of which
a geometrical treatment is given there. It ought to have been con-
ducive to the finding of a surface of order nine; this has not been
effected on account of the calculations becoming too extensive;
however, the form of the final equation has been found.

2. In the first place the equation must be found of the cone of
axes of the concentric pencil of quadratic cones, at the same time
director cone of the locus of the axes of the pencil of surfaces. To
this end we regard the intersection of the two cones, determining
the pencil of cones, with the plane at infinity P?,, and besides the
isotropic circle situated in this plane; then we have the three
equations in rectangular cartesian coordinates :

A=a,, 0? + a,, 7’ + a,, 2° + 2a,, ay + 2a,, az + 2a,, ye = 0,

B=b,, 27+ b,, y? + ,, 27 + 20,, ey + 2b,, xz + 2b,, yz = 0,

CHH4+y7-2=0. ;

Out of these equations we find that of the cone of axes in the
same way as we determine the Jacobian curve of a net of conics:

where A,, A,, A,, etc. are the derivatives of A with respect to a, y, 2.
So the equation of the cone becomes

| av Q,,%+ 4, y + 4,2 b,e7+b.y == ba;
| | a,,% + 4,,y + 4,2 b,,¢%+6,,y +4,,
|

a ©
i=)

a

z a,,0 + 4, y + 4,32 b.,@ + bas dee bs3

Without harming the generality we can always assume that the
principal axes of one of the cones coincide with the axes of coordinates ;
from this ensues that we may put 6,, = 6,,=6,,—=0, by which
the equation of the cone is simplified.

3. After having found the equation of this cone we can pass to
the formation of the set of equations, by means of which is found
the equation of the locus of the axes,

( 533 )
The equation of the pencil of quadratic surfaces now becomes
eB Ot hs ay

where however A and & have a wider meaning than before, A being
a,, v7 + a,, y* + a,, 2° + 2a,, vy + 2a,, wz + 2a,, ye +

+ 2a,,0 + 2a,,y + 2a,,2+ 4,,,
and & being the same expression with the coefficients /.

Let us now put the coordinates of the centre of the surface (4
p,q? and let us regard this centre as origin O’ of a new system
of coordinates with axes parallel to the original ones. We then
arrive for surface (1) at an equation in w’, y’, 2’, in which the terms
of order one are missing and those of order two possess the same
coefficients. The principal axes of this surface are given by the three
equations :

(a,, 2} +a, y +4, 2) +4(),, 2’ + b,, y' + 5,,2') + ke’ =0,

(41. a = Coa y' a 3 “) =F A (5,, a == OS y' == bas 2’) = hy! = 0,
! > ! All =} Uy ut: ape —

(4,2 + 4,9 + 4,,2)+4(0,,2 +4,,y' + 4,, 2) ke! == 0.

As could be foreseen the elimination of 2 and / furnishes the
same equation as was already found for the cone of axes.

If we wish to form the equation with respect to the original
system of axes, we must put 2’ —2—p, y’=>y—q, 2’ =2- r
and make use of the equations of condition for p,q, 7:

(a,, + 40,,)p + (4. + 4,,)¢ + (G4, + 40,,)7 + 4,, + 20,,= sy
(a,, + 46,,)Pp + (Gee + 4b,3) 9 + (@43 + Abi)? -- G,, — 20,, — 05)

(@5 + 20,5) P + (a, + 2045) 9 + (ag + 28,5)" + a,, + 2d, = 0-}

By this substitution the equations assume the following form:
(a,,0 + ayy + aye + 444) FA + By + Oye + d,,) + Hap) =0,
(ita $ aigay + yg2 + ty) + 2(Or@ + yay + Base + by) + H(y-g) = 0, (3)
eens yeas he) Aas bay 10,22, +40, |
or written shorter

A, +.B,4+ k(e¢ — p) = 0,
A, + BA t+ky—q fel ee eee (4)
ee eae (a4) = 0

The surface S, is obtained by eliminating p,q,7, 4,4 out of the
equations (2) and (4).

4. This elimination leads to extensive calculations as the variables
appear also as products two by two. We shall here point out the

( 534 )

general course by which at the same time the application in special
cases is rendered possible.
The equations (4) can be written as follows:

kp = A, + B,2-+ ke,
kg = A, + Bat ky,
ky = A, + B,a + ke.
Let us multiply each of the equations (2) by / and replace the
values kp, kg, kr; we then obtain:

(a, + 2b,,) (A, + BA + he) + (a,, +20,,) (4, + Bathy) + |

(a,, + ab,,) (A, + By 2+ kz) + ka,, + kb,,4=0,
or:

(Ay + Bak + (@,, +b, 4)(A, + B, a) + (a, + 5,,4) (4, +.B) +

(a,, +0,,4)(A, + B,2)=0.

We likewise find: j®)

(A, + B, a) b+ (a, +0,4)(A, + B, a) + (a, +8448) (A, + By) +

(a4: +5,:4) (A, + B, a) =0,
and finally :

(A, + B, a)k + (43 + b,,4)(A, + 4B,) + (a,, + b,,4)(A, + Ba) +

(a,, + 5, 4)(A, + B, a) =0. /

If we reduce these equations and if we regard / and 2 as variables,
we shall get as result three quadratic equations, out of which / and 2 can
be eliminated. As however these equations are linear in /, the elimi-
nation of # can take place without any difficulty. By putting the

values of & in the first and second equations equal to those in the
third and the fourth we deduce from (5):

(a,, +, 4) (A, +B, 4) (4, 4B, A)+(G.4+4,, 4 (A, 4B, a)?+ \

(ay3-+4s 2) (A, +B, 2) (A, +B, =(4.+D), 4) (4,-+B, 2)*+

(453-44, 4) (Ay-+-B, 4) (A, +B, 4) + (Ga O05 4) (As +B; 4) (A, +B, 4)
and (5)

(a5 +4434) (A, +B, a) (A, + B34) + (Gas t+ O94) (A, +B) (A, +B) +

(a,, 10,4) (A, +B,4)?=(a,,+0,,4) (4, + B,4) (4,+8,4)+

(5-+0452) (Ag+ B,a)? +(a,5 +0552) (A, +B,2) (A, +B, A).

When reduced these equations prove to be of order three in 2; we
can write them in an abridged form:

_ =

—- el ee

( 535 )

Ma? + Ni? 4+ PR4+Q=0, }
M+ N+ PA+Q—0, |

which give, according to the method of Bezour, the following resultant:

(7)

(MN') (MP’) (MQ)
(Py) (rd) ae) «6(Na) | =—0. . . (8)
(Q) (VQ) (PQ)

5. From this is evident that the form of the final equation is
found, but if is a very intricate one, as is proved from the values
of the coefficients, given here:

i) epehe h Be he BB. Bo? b,, B, B, —b,, BB,
Ge Bab A Ree, ASB Ob. A, Boke. Bo? +
ESN AN, Age AB eam. Be ol
ly, _B, Beare A. BN pe Ae g, B. Bae iit B=
Anes
P=6,, A, A, +4,, B, A, + 4, A, B, + 2a,, A, B, + 6,, A? +
eee He Bet AB bl A Oye Ak
be. A. A, —a,, A, BS — a,, A, B, — b,, A, A; — a,, A, B, —
Q=—«a,, A, A, + a;, A,* + @,, A, A, —4,, A? —a4,, A, A, —2,, A, A,
M'=»,, B, B, + b,, B, B, + 0,, B? — b,, B, B, — b,, B,? — b,, B, By;
Wea, 2 Bo -b A Bb A, B ta, BB, + b,. A,B, +
AB ge BORA. Bi ag BOBS — be AaB. =
ee Bo) Bg eB Bb AR
bs; A, B, ;
aera Ae ge Ae tea AB ab. A Ag AB

a,, A, B, + 2a,, A, B, + b,, A,? — 6,, A, A, — 4,, A, B, —
a,, B, A, — 2a,, A, B, —b,, A — ey A, A, —4,, 4, B, —
a, A, By;

Q =a,, A, A, + 4,, A, A, + a,, A,? — a,, A, A, —a,, A,” — 4,, A, 4,.

6. With the aid of these expressions the equation of the locus
of the axes can be determined for each separate case, which was
the purpose of this paper; we shall conclude by giving a few
observations.

a. Even in the general case abridgement is possible in the operation.
If we assume that the axes of coordinates coincide with the principal
axes of one of the surfaces, e.g. of B—0O, then 5,,—0, 6,,=—0,

( 536 )

4, =O) 6, =0,;. 6,,—0) 6,0; arhilst-alse BB. fh eacoome
simple forms and all coefficients except Q and Q’, are simplified.

At the same time this substitution shows that in equation (8) a
factor may be omitted; if namely we make use of the above named
values for by, we shall find :

M = (6;, — b;,) B, B Di (b,— 05.) Bebe:

from this ensues that the first column of the determinant (8) is
divisible by 6,. This divisibility is connected with the fact that the
equation of the locus of the axes must become of order nine, whilst
when developed the determinant (8) becomes of order twelve. So
when a complete operation is executed factors must disappear out of (8).

4. Out of the former geometric treatment it is evident, that in
some cases the locus of the axes S, breaks up. As one of the special
cases appearing there the case of a circular base curve of the pencil
was treated where S, broke up into a cubic surface and into a
surface of order six. The equations of the algebraic treatment of this
case become, when one chooses the plane YO Y as the plane that
is intersected according to a pencil of circles:

A=a,, 27 +4,, y? + 2a,, vz + 2c,, ye + 2a,,2 + 4,,=0,
B=b,, =? + 2b,,@2 + 26,, ye + 26,, 2 + 2b,,2=0.
From these equations the simplified values for WW, V.... can be
deduced.

(February 23, 1905).

ue

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING

of Saturday February 25, 1905.

—DECo—

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 25 Februari 1905, Dl. XIII).

© OPNGE HIENg AES:

M. C. Dexuuyzen: “On the osmotic pressure of the blood and urine of fishes’. (Communicated
by Prof. C. A, PekELHARING), p. 537.

H. Zwaarpemaker Cz.: “On the relative sensitiveness of the human ear for tones of different
pitch, measured by means of organ pipes”, p. 549.

H. W. Baxkuurs Roozesoom and E. H. Bucuner: “Critical terminating points in three-phase
lines with solid phases in binary systems which present two liquid layers”, p. 556.

P. H. Scuoure: “On non-linear systems of spherical spaces touching one another”, p. 562.

JAN pe Vriss: “On a special tetraedal complex”, p. 572.

Jan DE Vries: “On a group of complexes with rational cones of the complex”, p. 577.

M. W. Beiermck: “An obligative anaerobic fermentation sarcina”, p. 580.

H. A. Lorentz: “The motion of electrons in metallic bodies”, II., p. 588.

The following papers were read:

Physiology. — “On the osmotic pressure of the blood and urine of
fishes.’ By Dr. M. C. Drexnvyzen at Utrecht. (Communicated
by Prof. C. A. PExnLHarinG.)

(Communicated in the meeting of November 26, 1904).

Migratory fishes (eel, salmon, shad) move in a relatively short
time from sea-water, having an osmotic pressure of about 24 atmo-
spheres, into fresh water, in which this pressure is !/, of an atmo-
sphere or less, without experiencing any harm. And this same transition
also takes place in the opposite direction. It is very mysterious how
such an emancipation from the laws of osmotic pressure is possible.
It is known in general outlines that bony fishes, as well in salt as
in fresh water, keep up an osmotic pressure in their blood which is
relatively independent of that of the surrounding medium. But it is

o7
Proceedings Royal Acad. Amsterdam. Vol. VII.

(538 )

unknown between what limits the organism regulates thepercentage
of salt (for this is the principal factor) in the different fluids of the
body, or by what means it keeps up this percentage. It is certain
that bony fishes are in general stenohaline, i.e. that each species is
bound to an osmotic pressure of the water in which it lives, and
which must not vary too much and especially not too quickly. A
behaviour like that of the migratory fishes is exceptional. The Baltic
Sea which contains about all gradations between salt and fresh water
and the fauna of which has often been studied, furnishes a proof of
the statements made. Of every species of fish, found in the Baltic
Sea, I have traced the range of distribution, and the lists, for the
publication of which we have no room here, show that most fresh
water fish go some length into the brackish water and most sea-fish
sustain a certain diminution of the percentage of salt, but that cer-
tain limits are not exceeded.

If we want to penetrate into the mechanism of these physiological
phenomena it is of primary necessity, to know the osmotic pressure
of the blood of the various species of fish. Some determinations were
made by Borrazzt and Ropirer. About five years ago I began to take
part in these measurements under very unfavourable circumstances
with sea-fish that had been transported alive from Katwyk to Leyden.
These animals were mostly alive, in any case entirely fresh. The
results were not published until in the summer of 1904 they appeared
to be quite concordant (and sometimes fully to agree) with results
that had been obtained under more favourable conditions.

The excellent opportunity of obtaining live sea-fish in great variety,
afforded by the fishmarket at Bergen in Norway, induced me last
summer to take up the investigation again and to extend it. After
that determinations were made on fresh-water fish from the environs
of Utrecht and finally I was enabled through the kindness of Dr.
Kersert, director of the zoological garden and aquarium at*Amster-
dam, to study sea and freshwater fish, among these species that are
difficult to procure. I wish to express here my indebtedness to
Dr. KErBERT.

At Bergen the fish are offered for sale alive in a large num-
ber of open wooden troughs through which a vigorous current of
sea-water is passed, which, as I was assured, is pumped np from the
fjord at a great distance from the town. The fish are not entirely
normal, however; the catching, the lack of food, the transport, their
being handled by sellers and buyers, all harm the animals. Before
they come into the market tanks, namely, they swim in caufs, closely
packed together in the surface water of the harbour, which sometimes

( 539 )

is considerably diluted by the rains. There are reasons for assuming

that these influences make themselves felt in the osmotic pressure of

the blood. An investigator who should stay for a long time at Ber-
gen, choose his material carefully, keep it for some time in aquaria

and note for each specimen everything that could have any influence,
| would without doubt obtain more constant results than can be published
here. Still it would be more recommendable to accompany the fisher-
men and to collect blood and urine immediately after the catching.
The figures here given must be judged as one of the first attempts
in this almost unexplored region. Only during and by my investigation
I have become aware of the necessity of taking the condition of
health of the animals very much into account.

The specimens bought were conveyed to the biological station
either in pails of seawater or without this precaution, a distance of
twenty minutes. and placed there again in aquaria in which sea-
water circulated containing about 32°,°, salt, corresponding to a
freezing point of —1.731° to —1.742°. The fishes that showed
signs of debility were examined first, the others remained for some
hours and even for two or three days in a spacious aquarium
without special food. Many specimens proved to have still filled
stomachs and to lodge few parasites, others were in a less satis-
factory condition, but all these details were not recorded. The
quantity of blood furnished by each fish is relatively small and
varies as well with different species as with different individuals.
As a rule, for a determination of the freezing point the blood of
several specimens is required, since ten to fourteen cubic centimetres
must be put into the freezing tube. The fishes were washed in
tapwater, well wiped and their tails cut off with a pair of bone-
scissors. Sometimes it appeared to be necessary to make an incision

be
4
j
y
4
i
,
b

in the heart; in this case the gills were once more cleaned from
seawater with a dry towel.

Would it not be better to use serum? This does not appear to
be necessary to me, since the same sample of blood generally gives
the same freezing point in repeated measurements and later a serum
separates which as a rule is not coloured red, even with Raja clavata
and Trygon pastinaca which are supercooled to — 2.7°. Also Hepix
and Hampurerr') have found that it is not necessary to separate
the serum.

In order to diminish the quantity of blood necessary for a deter-
mination, which is desirable especially with small or rare species, I

1) Hampurcer, Osmotischer Druck und lonenlehre in den medicinischen Wis-
senschaften. Wiesbaden 1902. I. p. 453.

37%

( 540 )

have tried whether it is admissible to add some soft organs, spleen
or liver to the blood. With the blood of three specimens of a fresh-
water fish from Surinam, /rythrinus unitaeniatus, Spix, the same
freezing point was found twice with the bulb of the thermometer
not quite immersed, — 0.577°; then the livers of two specimens
were added and —0.60° was found. So improvement must not be
sought in this direction’). With the rabbit still more considerable
differences were obtained, probably on account of the conversion of
glycogen into glycose.

The determinations were made with an in many respects modified
BECKMANN apparatus, of which the description will be given later.
Here it may suffice to remark that if a supercooling of 0.5° is used
as well for the determination of the zero point as of the freezing
point, the figures obtained for pure salt solutions are very accurate.
For 1 °/, NaCl (1 gram NaCl dissolved in 100 grams of water, the
weights reduced to a vacuum) the apparatus gives — 0°.589, the
result of the “Prazisionskryoskopie” (Hampurcer |. ec. pag. 96).
BECKMANN’s correction for increase of concentration with supercooling
1 il
160" 3=05
calories being the latent heat of melting ice). The temperature of
the cooling bath was — 2°.5, when necessary it was lowered to
— 2°.9 by strong stirring.

We will first deal with the results obtained with freshwater
animals. Let 4 be the freezing point in degrees centigrade, omitting

was always applied, amounting for 07.5 to

the — sign.
Freshwater bony fishes. A
Perch, Perea fluviatilis L. me 2 and5 specimens from Utrecht.

Remark. The perch occurs in the whole Baltic sea as far as the Sound, Le.
in water containing to 12/o, salt and with ~ =0.64.
0.527 1 spec. from Bergen.

Carp, Cyprinus carpio L. x 1 , , the Amsterdam Aqua-
sole 4 0.540 ‘rium, lived in water of 4 =0,039.

Remark. The carp seldom penetrates into the Baltic sea, oftener into the Asow
and Caspian seas.

caught healthy in October and

Tench, Tinea vulgaris Cuv. thy i
kept some time in the caufs.

0.514
Remark. The tench goes from the Haffs and bays as far as Gothland, where
A is about 0.42.

0.466 | 1 resp. 3 spec. from Utrecht,

1) By reducing the size of the freezing tube and by using a smaller BrckMAny’s
thermometer of an old pattern; the quantity required can be reduced to 5 or 6 ce.
By means of salt-solutions it will be controlled how far the results need correction,

.

( 541 )

0.519 | Resp. 4, 5 and 3 freshly caught

: 6 aca spec. fr Jirecht, Sept. andOct
hae 4 594 spec. from Utrecht, Sept. andOct.,
Pike, Esox lucius L. 0.526 some of them examined in a half

0.530 ] dead condition.

Remark. The pike in the S.E. part of the Baltic Sea leaves the coast to fairly
great distances and is occasionally caught near Bohusliin, where 4 is at least
0.69. Goes even some distance into the Arctic sea }).

18 fine spec. from the Amsterdam

Aquarium, Noy, occurs in all

Rudd, Leuciscus erythrophthalmus L. 0.495
brackish bays of the Baltic Sea.

Bleak, Abramis blicca, Bloch. 0.497 12 fine spec. Aq. Amst. Noy.
Remark. In the Baltic sea the bleak occurs in the brackisn bays.
Trout, Salmo fario L. 0.567 ‘1 fine spee. Aq. A/dam. Noy.

Remark. The trout is a freshwater fish which seldom occurs in the brackish
haffs, but belongs to the Salmonidae, a family of migratory fishes, fishes of the
sea-coast and freshwater. Nisstin looks upon them as original seafish.

Waranga, Erythrinus unitaeniatus Spix. 0.577 3 fine spec. Aq. A/dam. Noy.
Remark. Surinamian freshwater fish, living in water of 20° C. By evaporation
during the collection of the blood the number is probably slightly too high. Also
the quantity of blood available (8 cc.) was somewhat too small. They belong to
the Characinidae, old genuine inhabitants of freshwater (Ostariophysi).

Average of 13 observations on freshwater bony fishes 0°.521.
Excluding the trout and waranga, the first as being a Salmonida,
the other because the observation is less reliable than the others and
because we have here a fish of which the somatic temperature is
higher than that of the others, we obtain an average value of 0°.512,
round which the 11 observations are pretty regularly grouped.

To these I can add still six measurements on other cold-blooded
freshwater vertebrates :

Lamprey, Petromyzon fluviatilis L.

0.473 ) Observations at Leyden, on six, resp.
five specimens in tolerably good con-

0.500 ) dition, made in 1899.
vy >
Frog, Rana esculenta L. Gane | id. a fine spec, caught in the autumn.
37 spec., Sept. 1904, animals sent from
Salamandra maculosa. Laur. 9.479 | Berlin, kept a day ima terrarium with
a dish of water.

Observation of Borrazzt 1897, quoted

Fr.water turtle, Emys europaea. Gray. 0.474 | from R. Quinton 2)

Average of these: O0°.476 and of the above mentioned eleven
together with these six: 0°.499.

The freezing point of freshwater is about 0°.02. Borrazzi’s Emys
lived in water of this freezing point. In the tanks of the Amsterdam

1) Patacky, Die Verbreitung d. Fische. Prag. 2e Aufl. 1895. p. 54.
®) R. Quinton. L’eau de la Mer, milieu organique. Paris 1904, p. 441,

( 542)

aquarium a somewhat brackish water circulates of 4 = 0.039 (ori-
ginally water from the river Vecht). The percentage of salt of the
lake of Geneva is given in Cart Voer’s Lehrbuch der Geologie 1.
p- 538 as 0.1574°/, which would point to a freezing point of only
0°.01. According to the figures collected by Dvusots (see Verslagen
1900 p. 12 and 30) 4 in Lakes Wener and Wetter is still lower.

The osmotic pressure in atmospheres at 0° which we shall hence-
forth denote by /P,, is obtained by multiplying A by the factor
12.08 according to Srenius*) or 12.03 according to Jorissen *) and
hence is in freshwater of the order of */, to */, atmosphere. In
such a medium the cold-blooded vertebrate animals, breathing mostly
through gills, maintain in their blood an osmotic pressure of six
atmospheres! With birds and mammals (see the table of HampurGEr,
lc. I. p. 456) also a pretty constant freezing point of the blood
has been found. I have proposed to call the power of keeping ?,
at a certain level, albeit within certain limits “‘7deotony” *), a property
comparable with the homoiothermic power. That also the freshwater
bony fishes possess this ideotony can hardly be doubted from the
results communicated. The limits between which the figures of the
same species lie, are narrow, only in the tench the differences are
fairly considerable. The ideotony is mest conspicuous when the
agreement between the cold-blooded freshwater veriebrates among
each other and the great difference with P, of the surrounding medium
are remembered. One is led to the supposition that for these animals
which indeed are not closely related: Cyclostomes, Teleosteans,
Amphibians and a reptile, the P, of about 6 atmospheres is an
optimum. For warm-blooded animals there seems to be a tendency
to maintain P, at 6°/, to 7'/, atmospheres; A’s of 0.570 with man
and of 0.6 to 0.625 with mammals and birds are namely kept up
with great constancy.

The kidneys are the regulators. For the 4 of the urine of man
varies between 0.12 and 3 (Hampcreer |.c. Il. p. 317) when the
separately discharged portions are examined, whereas A for the whole
quantity of 24 hours varies from about 1.8 to 2.4. For normal man
Scnovte*) found that A of the blood, provided digestion were eli-

1) Srentus. Ofversigt af Finska Vetenskaps-Societetens Férhandlingar 46. No 6.
1903—4.

2) W. P. Jorissen. Physisch-chemisch onderzoek van zeewater. Chem. Weekbl.
le jaarg. No 49, p. 731. Sept. 1904.

3) M. C. Dexauyzen. Ergebnisse von osmolischen Studien, namentlich bei Knochen-
fischen, an der Biol. Stat. d. Berg. Museums. Bergens Museums Aarbog. 1904. No 8.

4) D. Scuovre. Het physisch-chemisch onderzoek van menschelijk bloed in de
kliniek. Diss. Groningen. 1903.

minated, by taking the blood in the morning before breakfast, only
varies between 0.56 and 0.58,

We can only to a limited extent imagine why the percentage of
salt (for this is the chief point) of blood and lymph may only vary
between narrow limits. The globulines require a certain concentration
of “medium salts” in order to remain in solution. If horse serum is
diluted with 1'/, volume of distilled water, a precipitate is already
formed, i.e. with a percentage of salt corresponding to 4 — 0.24.
Why an increased percentage of salt should be injurious is less
clear. Danger for precipitation of albumens would only occur with
much higher concentrations, at any rate with horse serum. Yet the
fact, found by Ropmr’) that the blood and the somatie fluids
(pericardial and peritoneal) of rays and sharks are isotonic with
seawater but contain less salt, the deficiency being compensated by
the retention of 2 to 2.7°/, of urea, points to a strong need of the
organism of the Vertebrates to keep the percentage of salt below a
certain value. Grins *) has found that blood-cells are permeable for
urea so that this substance helps to bear the osmotic pressure against
the seawater but discharges the cells of a third of 23 to 24 atmos-
pheres. I have proposed l.ec. to call this power of being isotonic
with respect to seawater but of taking away from the cells them-
selves part of the osmotic pressure ‘“metisotony’’.

The blood of Teleosteans has a freezing point which differs considerably
from that of the seawater, in which they live. They possess ideotony
but the individual differences are greater than have been remarked
with the remaining vertebrates, so that it appears that they only
imperfectly possess the faculty of rendering their P, independent
of the surrounding medium. Before the figures are given, a summary
of A and P, of different seawaters may be inserted. The numbers
have been taken from M. Kyupsen’s Hydrographische Tabellen, from
Perrrrrson’s Review of Swedish hydrographical research in the Baltic
and North seas and from Morsivus und Hnrckn, Die Fische der Ostsee *).

1) Ropier. Sur la pression osmotique du sang et des liquides internes des
poissons sélaciens. Comptes rendus. Dec. 1900. p. 1008.

*) G. Gruns. Ueb. d. Einfluss gelister Stoffe auf die rothen Blutzellen. Pfliiger’s
Archiv. 63. 1896. p. 86.

8) M. Knupsen. Hydrogr. Tab. Kopenhagen 1901, Perrersson in Scottish geogra-
phical Magazine 1894. X.; Morsivs u. Heicxe. Fische d. Ostsee. Berlin 1883.

( 544 )

Ha was
25 &e
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=$3 BED a i.
ee eh ys ee in atmo-
aor ash
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Bea eee
ba <5
a 3S
6 1.00478 0.318 3.8 Gulf of Bothnia in summer.
Surface water (till 60 M.) of the Baltie
7 1.00559 (0.371 4.45 | sea proper between Riigen and Gothland

in summer. With 7.5°/o) salt, M. and
8 1.00640 0.424 5.09 | H. assume the limit between brackish
and salt water.

9 1.00721 0.478 5.74
10 1.00802 0.531 6.37 Water of the shallow part of the Baltic
sea south of the Danish isles in
12 1.00963 0°639 7.67 summer.
20 1.01607 1.074 12.89 The same in winter (Moesius and
HEINCcKE).
32 TROZ5 7 sg 20.9 Seawater in the fjord before Bergen,

8 M. below the surface in summer.
35 1.02813 1.908 22.9 Northern Atlantic.
38 1.03055 2.078 94.9 a Seawater, Gulf of Naples, in Nov.
1903, 2.105.
Seawater as it circulates in the tanks
of the Amsterdam aquarium 2.085.
Bony fishes living in the sea.

A
Gadus morrhua L. — Cod, 0.644 2 apparently normal spec. Aq. Amsterdam,

Noy. 1904. a
. é a 0.673 1 spec. bought on Monday at Bergen,

remnant of the fish supply of the end of

the preceding week.
A A 5 0.708 8 spec. from Katw. immediately conveyed
j to Leyden.

5 4 5 0.721 8 spec. id.
” ” ” 0.724 ” ” ”
. 4 : 0:729) 13>" | -

(April 4 1900, an ample quantity of blood
had been taken from 13 fine spec.).
0.729 Bergen 1904, summer, 3 large spec.

bl *
‘ . = 0.742 Leyden, as above.
4 5 5 0.744 Bergen,, 3 2 spec
5 J ” oJ 0.753 * ” ” 3 s
u 4 7 0.808 Leyden ,_ , On
a n > 0.811 oJ b ” 8 ”

1

(545 )

Remark. The cod penetrates very far into the Ballic Sea. We do not mean
to assert that individuals caught in the Bothnian gulf and not larger than 45—50
centimetres have swum in from the Atlantic. They may very well belong to local
races, propagating in the brackish water and which do not reach 4 greater length.
Near Stockholm the cod only reaches a length of 60 centimetres, in the Sound
80 to 90 em. at the utmost, on the coast of Bohuslin 90 em., but near the
Lofoden Islands even 140 to 150 cm, Also the common mussel is much smaller
in the brackish water of the eastern parts of the Baltic sea than in the more salt
containing sea.

In the Gulf of Bothnia the percentage is in summer north of the Quarks
3—4°%o9, 40.159 to 0.212, Py 1.9 to 2.54 atm., north of Stockholm resp. 5/9,
0°.265, 3.18 atm., at Stockholm 6°/o), 0°.318, 4 atm., on the north coast of
Gothland 7°/o9, 0°.37, 4.45 atm., and till Riigen—Schonen 7 to 8°/o), 0°.424, 5 atm.
At Bohusliin the salinity of the surface is in summer 13"/), 4 =0°.69, but in
the depth North sea water occurs of 32 to 33°/), and 1.8°.

A
Gadus aeglefinus L. Haddock. 0.767 Leyden, summer, spec. were dead but fresh,

Remark. The haddock does not penetrate further than the Mecklenburg coast.

A
Gadus virens L., Coalfish, Green Cod 0.760 Bergen.
3 ; 0.761 - 3 spec.
’ i 0.837 aan ES
5 _ 0.838 -

Remark. Gadus virens does not penetrate further than the bay of Kiel and is
rare there. In the fishmarkel at Bergen it is always supplied in large quantities
but generally in a bad condition, showing wounds and traces of having bled,
many specimens lie on their backs at the surface and breathe little. When a
purchase was made good specimens were selected, but I think it very probable
that the animals whose blood froze at — 0°.837 and — 0°.838 were abnormal.

4
Gadus merlangus, L. Whiting 0.760 Bergen. 14 spec.
Remark. The whiting enters the Baltic Sea with difficulty, about as far as
Bornholm. Only once it has been caught near Gothland.
A
Molva vulgaris, Flemm. Ling, 0.716 Bergen. 3 fine spec.
Remark. The ling no more than G. virens penetrates into the Baltic Sea.
A

Moiva byrkelange (Walb.), Trade Ling, 0.730 Bergen, 4 fine spec. were dead
but fresh.

Remark. Deep sea fish, not to be had alive. Had been caught at a depth of
400 metres. Does not come further than the Cattegat.

i
Motella tricirrata (Bloch), Whistler, 0,605 Bergen.
Remark. Motella tricirrata has only once been caught near Géteborg

( 546 )

A
Hippoglossus vulgaris, Flemm., Halibut, 0,671 Bergen. The specimen suffered
from a disease of the skin, had
lived long in the aquarium and

threatened to die.
Remark. The halibut does not wander into the Baltic sea further than Mecklenburg,

A
Pleuronectes platessa L., Plaice. 0.672 Bergen.
=, . 0675 Leyden.

Remark. The plaice goes as far as Stockholm. The Pleuronectides everywhere
show a tendency of penetrating into brackish or even fresh water. The flounder
has been found in the Moselle near Metz. Wicumany found species of flounder in
small mountain lakes of New Guinea.

A
Pleuronectes microcephalus Donovan, Lemon Dab 0.681 Bergen.

Remark. P. microcephalus very rarely comes as far as Eckernforde.

A
Labrus bergylta Ascan., Ballan Wrasse 0.694 Bergen. 3 spec.
= 5 0.704 : 6, of which one very
ill, liver and intestine full of nematodes.
. 5 0.708 :

Remark. This Labrus is only seldom found in the western part of the Baltic
sea, where the freezing point is in summer about —0°.6, —1° in winter.

A
Labrus mixtus L. Blue lipfish, Striped Wrasse 0.681 Bergen. 4 spec. all 2
oy

? . OA

Remark. L. mixtus (¢ red, 2 blue) seldom comes as far as the Sound.

A
Conger vulgaris Guy. Conger-eel. 0.696 3 spec. Aquar. Amsterdam.
~ : 0.786 1 ., Bergen.

Remark. Only seldom caught in the Baltic sea, repeatedly in the lower course
of the Weser.

A :
Salmo trutta L. Sea or Bull trout. 0.785 6 spec. Bergen, caught with the rod in
the fjord, in bad condition and partly dead.
Remark. The Sea trout is an anadromous migratory fish.

A :
Labrax lupus Cuy. Bass, 0.720 1 spec. Aquar. Amsterdam.

Remark. Rare in the western part of the Baltic sea.

A :
Trigla hirundo Bloch. The gurnard, 0.669 2 spec. Aquar. Amsterdam
Remark. Not often occurring in the western part of the Baltic sea.

A

Anarrhichas lupus L., Sea-wolf, 0.665 2 spec. Bergen. The fishermen use to beat
5 * 0.681 1 =, (- out the teeth of these
> - O69) 93", somewhat dangerous ami-

mals; in any case the
sea-wolyes arrive at the
market alive but not in
a normal condition. They
react slowly and die when
they are too much
handled.

Remark. The sea-wolf penetrates at the utmost as far as the coast of Pomerania.

The average of these 38 observations is A = 0 .7245 or P, = 8.7
atmospheres. The figures are grouped pretty regularly round this:
13 between 0.600 and 0.700, 13 between 0.700 and 0.750, 12
between 0.750 and 0.850. The average lies fairly well at the same
distance from the two extreme values. By omitting the extreme values
0.605 for Motella, 0.808, 0.811, 837 and 0.888 for codfish and
G. virens, which latter pretty certainly are based on pathological
deviations, the average is only little shifted and becomes 0°.716.

The differences between the extreme values and the average are
relatively large, 0.120 and 0.118, about *, of the probable normal
value. If the 5 extreme figures are rejected, the deviations from the
new average, 0.716, are only 0.072 and 0.070. We found a similar
result with the freshwater fishes; only after rejecting the values for
waranga and trout we obtained an average of 0.499, differing only
0.041 and 0.035 from the extremes.

If we bear in mind that these fishes live in a medium of which
the osmotic pressure is 21 to 23 atmospheres or even more, no one
will object to ascribing ideotony to these animals. But the considerable
oscillations in P,, which we noticed e. g. in the cod, give the impression
that the power of maintaining P, at a certain level, is limited. And
one is involuntarily reminded of the oscillations in somatic temperature
which homoiothermic organisms show with many disturbances in
the general well-being.

There exist in literature still a few data concerning the freezing
point of the blood of bony fishes living in the sea. borrazzi*) found
with Charar Puntazzo Gm. — 1°.04 and —1°.035, with Serranus
gigas L. —1°.035 and —1°.034, but these figures do not deserve

1) F. Borraza. La pression osmotique du sang des animaux marins. Arch. ital.
de biologie 28, p. 67, 1897.

( 548 )

too much confidence as he was wont to use a cooling bath of —12°').
Neither could he apply the later published correction of Brockmann.
With Chelone imbricata L. (sea-turtle) he found —0°.61 and —0°.62.

Ropier*) found in a Ganoid (sturgeon?) & = 0.76, in Lophius
piscatorius Li. 0°.68 and 0°.80, in Orthagoriscus mola I. 0°.80, in
the sea-turtle Thalassochelys corticata Rondelet 0°.602 and in a
mammal, living in the sea, the grampus Phocaena communis Less 0°.74.

The numbers obtained for the blood of the eel Anguilla vulgaris
Flemm. are very remarkable. With vigorous specimens I found
formerly at Leyden — 0.773°, at Bergen — 0.653°, at Utrecht
— 0.587°. Now the eel belongs to a family of tropical sea-fish ;
most species occur in the Dutch Indian Archipelago, they often go
into brackish water, others are deep-sea animals. Our common eel
excellently bears quick variations of the percentage of salt. Born in
the sea, it enters the river mouths as a young animal and remains
in fresh water until the time of propagation approaches. The eel
which is caught in the fresh or somewhat brackish waters of Frisia,
is put at Workum into caufs into which the seawater has free
entrance, goes to London and is sold on the Thames from these
caufs. The layer of slime, with which their skin is covered facilitates
this transition. Pavn Berr*) noticed that all the eels which he put
from the fresh water into the seawater himself, supported the sudden
change, whereas those which his assistant handled, all died. He used
a little net, whereas the assistant took them with his hand, held
them in a rough towel and in this way removed the layer of slime.

The eel shows in its osmotic pressure sometimes the type of a
seafish, sometimes it approaches that of a freshwater fish. The high
P, with the trout, as an original migratory fish, now also becomes
to some extent explainable.

Here a field of study lies open which may be urgently commended
to the Committee for the international investigation of the sea.

How do the marine bony fishes maintain in their blood a so
much lower osmotic pressure than exists: in the seawater? Some
observations on the urine of the cod, sea-wolf and G. virens ean
perhaps throw some light on this question. The A of the urine was
always lower, the osmotic pressure less than that of the blood.

1) G. Fano et F. Borrazzt, Sur la pression osmotique du serum en différentes
conditions de l’organisme. Arch. ital. de biol. 26, p. 46, 1896. See especially p. 47.

2) Hampurcer |. c. I. p. 466. The original article of Roper in Travaux des
laboratoires d. 1. soc. se. et station zoolog. d’Arcachon. 1899. p. 103, I have not
at my disposal.

5) P. Reeyarp, La vie dans les eaux. p. 438. Paris, 1891.

(549 )

With a large specimen of the sea-wolf, whose blood had given A
0.681°, the urine gave 0.6381°. With other individuals I found 0.555°.
The urine taken from some twenty specimens of G. virens gave
4 0.630°. With the cod 0.652 and 0.619 have been stated.

It is very simple to take the urine. A sea-wolf, e.g. is taken
behind the gills and suddenly lifted from the seawater, the skin of
the belly is dried, while the assistant stands ready for collecting the
urine which often is ejected in a vigorous jet. By some pressure
on the belly a little more is obtained, but often the ‘bladder’ (the
extended part of the ureters) is empty. Most animals gave little or
nothing and were given back to the seller so that a comparison of
4 of the blood and urine was only possible in exceptional cases.
At Bergen I had for the three species that were studied, found not
a single figure for A that was lower for the blood than for the
urine. At Amsterdam, however, it has appeared that there also
occur specimens, the blood of which shows a still somewhat smaller
osmotic pressure than any of the urines (cod).

The remarkably low /P, of the secreted product of the kidneys
with marine Teleosteans certainly points to this: that these animals
do not keep the osmotic pressure in their blood 23—8.6—14.4 atmo-
spheres lower because the kidneys so quickly eliminate the surplus
of salts taken in. The relative richness in water of the urine rather
points to these fishes resorbing from the sea-water in opposition to the

osmotic pressure, hence by using chemical energy, water or if one
prefers, a diluted solution of salt. But Rrenarp has stated (Lc. p. 391)
that certain freshwater fishes secrete from their gills soluble car-
bonates! About the mechanism of ideotony we are still in the dark.

Physiology. — “On the relative sensitiveness of the human ear for
tones of different pitch, measured by means of organ pipes.”

By Prof. H. Zwaarpemaker Cz.

(Communicated in the meeting of January 28, 1905.)

Almost simultaneously, but by different methods, the relative
sensitiveness of the human ear as depending on pitch, was investigated
by Max Wien’) and by F. H. Quix and myself*). The result of

1) Max Wren. Physik. Ztschr. IV p. 69. Pfliiger’s Archiv Bd. 97. p. 1. 1903,
2) ZWAARDEMAKER and Quix. Ned. Tijdschr. v. Geneesk. 1901 II p. 1374: 1902
Il p. 417. and Engelmann’s Archiv. 1902 suppl. p. 367.

( 550 )

these parallel investigations were concordant in some respects, different
in others. They agree in this that:

1s*. there is only one maximum of sensitiveness ;

2°4. that this maximum lies at g';

3, that the zone of fair sensitiveness extends from c' to g'.

4) that outside this region toward the limits of the scale the
sensitiveness diminishes very strongly.

They differ in this that:

dst. with Max Wien the sensitiveness still diverges very much
within the zone of fair sensitiveness, whereas with us it is of the
same order.

2.4. that the perceptible minimum for the most sensitive point is
with him 100.000.000 times smaller than with us.

In this state of affairs it seemed desirable once more to determine
the perceptible minima throughout the whole scale by an entirely
different method. Telephone as well as tuning-forks ought thereby
to be avoided. So we had recourse to wide roofed organ pipes of
which a wooden set of uniform pattern, extending from C' to g*
was at our disposal which partly coincided with the well-known
EprtmMann whistles and could be continued by the Galton whistle.

Some series of such experiments were made, partly on the heath
at Milligen, partly in the gallery of the university library at Utrecht,
partly in the sound-tight room of the physiological-laboratory. Since
the results, generally speaking, agree fairly well and a full account
of them will be published later, for the present only two series
taken under the simplest conditions, will be dealt with. These are:
a, the concluding series on the heath, 6, in the gallery. The arran-
gement, which was the same for both, will first be described.

The organ pipe which serves as the source of sound, is mounted
vertically on a stand, near the floor, with as little contact as possible.
It is connected with a HurcmiNson spirometer. Close under the air-
room of the organ pipe and connected with this latter by a wide
opening, is a ligroine manometer. The> manometer being bent
into an obtuse angle as little as ‘/, mm. of waterpressure can be
read. The spirometer is now loaded with a little box containing
sand, so that it forces out the air very regularly and causes the
organ pipe to emit a soft sound without an audible frictional noise
and without partial tones. The air used is read off on the scale of
the spirometer and calculated per second by at the same time starting
a timing watch. The product of the volume of air, pressure and
acceleration of gravity (all in em.) then give the energy supplied
per second in ergs.

What part of this energy is converted into sound is unknown.
Wessrer ') values the “efficiency” at 0,0013 to 0,0038 ; Rayieren ’)
on the other hand supposed in 1877 as a preliminary estimate, that
all was converted into sound (“supposing the whole energy of the
escaping air converted into sound and no dissipation on the way’’).
The truth will probably lie between these two, since we have always
paid attention to clear and easy sounding. For such a case Max
Wien remarked in 1888: A_ loss of energy certainly takes place,
first on account of the fact that part of the air-current is not
converted into sound-waves at all, but is lost by the formation
of vortices, partly inside, partly outside the pipe. We shall see
later that this part is small only for a definite position of the
lip of the pipe and for a definite pressure. A second loss of
energy takes place by friction on the walls of the pipe and by
tremors imparted to them; a third on the way between source and
observer by friction on the floor, motion of the air (wind) and
viscosity of the air. This latter part especially is relatively large
with RayLeien, since by viscosity a loss of energy of + 22°/, took
place *).

If 22°/, is considered relatively much, we may assume that Max
Wien at that time supposed for the losses by other causes a similar
or smaller amount. But whatever the ‘efficiency’ of the supplied
energy may have been, there is no reason for assuming that it has
been appreciably different for the different pipes. The wooden pipes
at any rate belonged to the same set of uniform pattern. So the
method suffices for comparative measurements.

While one observer read the scales of spirometer and manometer,
the other moved to the greatest distance at which the tone was just
heard and recognised (‘“Erkennungsschwelle”’). This distance was
then later taken as the radius of a hemisphere through which the
energy of the sound spread.

A. Experiments on the heath at Milligen.

Perfectly level ground, trees only at 600 metres. Quiet, fine evening,
October 19, 1904. Acoustical observer F. H. Quix, optical observer
H. F. Minxema (See Table I).

B. Experiments in the gallery of the university hbrary.

Afternoon of January 3, 1905. Acoustical observer H. Zwaarpe-
MAKER, Optical observer H. F. Minkema. (See Table II).

1) A. G. Wessrer Boltzmann’s Festschrift 1904 p. 870

*) RayLeicH Proc. Roy. Soc. vol 26 p. 248 1877.

3) M. Wien, Die Messung der Tonstirke, Inauguraldissertation. Berlin 1888 p. 45,

( 552 )

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Minima perceptibilia in the course of the scale; the minimum for g* =
(absolute value of the chosen minimum: in 1902 0,79.10—* Erg, in 1904 0.32.10—* Erg).

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—— ee o> :
le ee elie ee ee ee ee” ’ 4

Now, if for the present we only take into account the energy
supplied and neglect the necessary loss of energy in the organ pipe
and in the air; if we further assume the validity of the theoretical
law of distances (extension over a hemisphere), we obtain the following
results :

1. that the sensitiveness of our ear has only one maximum,
lving in the four times marked octave.

2. that there is a zone of fair sensitiveness, extending from q' to g*.
3. that outside this zone the sensitiveness diminishes very rapidly.

4. that in the zone of fair sensitiveness the perceptible minima
are of the same order.

5. that, for the most sensitive part of the scale the perceptible
minimum is 0,32 13-8 ergs for Mr. Quix, 1,9 < 10-8 ergs for myself.

The true perceptible minimum for the most sensitive point of the
scale will of course lie lower. How much lower cannot be determined
for the present, but at any rate the perceptible minimum found with
organ pipes certainly remains a million times greater than that which
was calculated by Max Wien from his telephone experiments. The
minima, found on the heath and in the library, are in satisfactory
agreement, however, with the minimum which we formerly caleu-
lated for tuning-forks, using the data of T6pLer and BourzMann ‘).

Taking into account the “efficiency” of an organ pipe, found by
Weester (0,0013 and 0,0088), the perceptible minimum for the
most sensitive point of the seale becomes lower, namely 0,45 to
1,8. 10-' ergs, but it does not reach the amazingly small value of
Max Wievy’s telephone experiments by a long way. Even if we
assume that one hears better at night in the profound silence of a
laboratory, than on the heath, not to mention an afternoon hour in
the library, yet this difference is by no means accounted for. But I
see no reason why the results of experiments made on perfectly
level ground, far from woods or buildings, which, according to
Max Wien’s former valuable investigations, fall perfectly under the
theoretical law of the distribution of sound, should deserve less
confidence than experiments with a telephone, which require very
complicated calculations.

1) TépLer u. Botrazmann. Ann. d. Physik u. Chemie Bd. 141 p. 321.

(555 )

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Chemistry. — “Critical terminating points m three-phase lines with
solid phases in binary systems which present two liquid layers.”
By Prof. H. W. Baksvuits Roozesoom and Mr. E. H. Bicuner.

(Communicated in the meeting of January 28, 1905).

Up to the present only one critical terminating point has been
found in systems of one component, namely in the equilibrium
liquid-vapour. If this is represented by a p,¢-line this suddenly
terminates in the point where liquid and vapour have become
identical. There exist at the moment no well-founded reasons to
assume critical terminating points also in the equilibria solid-liquid
and solid-vapour. We will not further consider these here.

In systems of two components we get instead of a critical point
liquid-vapour, acritical
line A,A, which con-
nects the critical points
of the components. In
a p, t, a-diagram this
line runs in- space,
here its p, ¢projection
is only indicated. (AK,
and BA, are then the
equilibria-lines liquid-
vapour for the com-

Fig. 1. ponents A and A). If
there is a homogeneous mixing for all concentrations in the liquid
phase, this will then be the on/y critical line.

The recent researches of Smits")

have shown how in some binary

nap mixtures the equilibrium line

x for solid B with liquid and

A 8 vapour may meet this critical

line (it is necessary that the

e melting point of 4 should be

Fig. 2. situated higher than A,). There

are then two such meeting-points p and g with the two parts op

and qg& of the three-phase line. This line therefore acquires two

critical terminating points owing to its meeting the critical line liquid-
vapour. Between p and q both lines cease to exist.

P

1) These Proceedings 1904.

Other cases of similar critical terminating points through the
meeting of three-phase lines in binary mixtures might be conceived
when no homogeneous mixing in all concentrations is possible in
the liquid phase and when, therefore, two liquid layers are possible.
It was of great importance for the knowledge of the conditions of
existence of solid phases at high temperatures and pressions to also trace
the critical terminating points in similar cases. If we indicate the two
liquid layers by 4, and /
component in the solid state by JS, then besides the three-phase line
SLG, two other three-phase lines SL,/, and L,4,G will be possible.
In order to trace the critical terminating points of these lines we

the vapour by G and the one or other

49)

first consider the line L, 4, G. Here
we first take exclusively the cases
where the p,#-diagram has the form
of Fig. 3, in which G, Z, and ZL,
indicate pressure and concentration of
coexisting vapour with two liquid
layers. The three-phase pressure for
this system lies, therefore, between the
vapour pressure of liquid A and 6

A ze S (Pe-and: Py):

Fig. 3. The p, ¢t-line CM for the three-phase
equilibrium L, L,G is situated accordingly in Fig. 1 between AX,
and Bk,.

On elevation of temperature Z, and ZL, may approach each other,
or recede. If the first happens and if they still coincide below the
line AA, for instance in JZ, then the two layers become identical,
a case of which many instances have been found by ALEXEsEW and
Rorumund. This point has been quite properly called a critical point.
This critical terminating point of the line for L,14,G must of course
necessarily be a point of intersection with a critical line. The said
critical line is the line DAV in Fig. 1.

If the hquids 1, and LL, can be made to mix more completely
by an increase of pressure a mixing point J will be found to exist
in Fig. 3 at a sufficiently high pressure. Ifnow L, /, approach each
other at a higher ¢, D will then be situated at a lower pressure
whilst the concentration in the mixing point may differ. In this way
is obtained the critical line DJ/ for the phases 1,/,. Its one ter-
minating point lies at the meeting point J/ where G occurs together
with £,4, and where consequently the lowest possible pressure is

1) The other case where the three-phase pressure is higher than those two does
not lead to materially different results.

attained. In the upper direction a terminating point would only be
conceivable in the case of the occurrence of a solid phase. If an
increase of pressure promotes the separation of the two layers, DM
would then run from the point J/ to the right instead of to the left.

In the case, therefore, in which the two liquid layers possess an
upper mixing point J/ which lies below A,A,, the critical lines
A.A, for GL and DA for L,L, are quite independent of each other.

If, however, the upper mixing point is not yet reached below
K,kK,, CM will continue up to a meeting point with A,A,. As the
phases occur in the order GL,L,, G will then become identical
with £, in O, Fig 4.

The critical line A,O is here at the same time broken off. From
a consideration of a series of p, x-diagrams for successive temperatures
we may, according to Mr. Bicuner, easily demonstrate graphically
that the other end of the critical line GZ, which commences in K,
now amalgamates with the upper part of the critical line Z,/, the
lower continuation of which is not realisable owing to the absence
of J/. In this way is formed the critical line A,PD whose upper
part may eventually also run to the right.

A junction of the three-phase line GL,L, with the critical line
GL in O will, secondly, always take place when ZL, and L, diverge
by an elevation of temperature. This may frequently occur with two
liquid layers whieh have a lower mixing point J/'. The three-phase
line GL,L, then terminates in O by intersection with the critical
line GL and in J by intersection with the critical line 12P for
L,=L,. From the p,2-diagrams we may now again deduce that
the upper continuation of this line is not now realisable because the
lower part coincides with the second part of the critical line GL
which proceeds from A’. In this way the critical line J PA, originates.

Lately, KurNEN has
found instances of the
cases represented by
Figs. 4 and 5.

The figures 1, 4 and
5 exhibit the three
main types of the
manner in which the
three-phase line meets
either the critical line
GL. or the critical line
L, ZL, and then finds
Fig. 4. 1 or 2 terminating

points, also of the fusion of
parts of the one critical line
with those of the other.

In the second place we
will now consider the occur-
rence of critical terminating

points on those three-phase

lines where one of the phases

Fig. 5. is solid. We may then limit
ourselves to the case where B occurs as a solid phase as no critical
phenomena can occur in presence of solid A.

In many cases where the line GL,L, proceeds to lower tempe-
‘atures the solid phase will occur ina point Q. According to previous
research by myself, 4 three-phase lines then meet in the quadruple
point Q. If we take the case of Fig. 1 we obtain in Fig. 6 QM
for GL,L,, QB for GL,S, EQ for GL,S, whilst QN indicates the
equilibrium of the solid phase S with the liquid layers L, /,. Similar
lines have been formerly
studied by me in cases
where hydrates of SO,
HBr, HCl oceurred as
solid phases. For the com-
ponent 6B as solid phase
their courses will be ana-
logous, and like all ordi-
nary melting lines — the
direction will diverge only

Fig. 6. a little from the vertical
either to the right or the left according to the volume differences
of the phases.

If the line runs to the

N

right and the critical
line MD to the left there
might be a possibility of
their meeting in a critical
terminating point iV as
the two liquid layers
*,; might here become ident-

‘ ical in presence of solid 2.
th t4,¢ The chances that this will
Fig. 7. ocew with an attainable

pressure only exist when Q and J/ do not differ too much in temperature.

( 560 )

Fig. 7 represents a similar meeting point .V for the case corre-
sponding with Fig. 4. Far greater chance of attaining a critical ter-
minating point of the line L,L,S is offered by the case of Fig 5
which would lead to Fig. 8. The line VQ is here supposed not. to
proceed as far as the lower mixing point ./’ because the solid phase
occurs previously at Q. For this reason the downward continuation
of the critical line ZL, = L, is wanting.

If, however, the mixing
point J/' should he but very
little below Q (metastable)
the point of intersection V
might be found at a compa-
ratively low pressure. Mr.
Bicaner has frequently no-
ticed a similar proximity of
M' and Q in systems of all

Fig. 8. kinds of organic substances
which on being dissolved in liquid CO, give rise to two layers.

Besides the three
cases fig. 6, 7, 8 in
which there exists a
critical —_ terminating
point of the line QN
a fourth type is pos-
sible. This occurs
when the liquid on the
line EQ already be-
comes critical with the
vapour before Q has
been reached, that is
beforethe second liquid

Fig. 9. occurs in presence of
the solid phase. “RA then intersects the critical line liquid-vapour ina
critical point R which quite corresponds with the point p of Sirs
(fig. 2). The line QO is now wanting, namely, below & we notice
nothing of a second liquid. Mr. Bicuner has here again graphically
deduced that in this case the line QA, for the second series of liquids
in presence of solid B and vapour, fuses with QV to a single line
BQN where the one fluid phase has the character of vapour on the
lower part and of liquid on the upper part, whilst these however,
gradually pass into each other.

In this case, like in fig. 4 the lower part of the critical line GL

( 561 )

fuses with the upper part of the critical line L,4,, A,7?.V, and we
have here again the possibility that the line L@.V also possesses a
critical terminating point V. This point bears some resemblance to
the second critical terminating point found by Sits in ¢ fig. 2. That
we are dealing in fig. 9 with a region of two liquid layers could
only be made plainly visible, if, owing to the non-appearance of the
solid phase, two liquids, in presence of vapour (metastable), occurred
below &. Otherwise it is only the peculiar course of LON, which
shows that we have this type.

Mr. Bicuner has succeeded in finding a case where this course
could be indicated (although NV remained unattainable). Fig. 10 gives

Fig. 10.

a correct representation of the p,élines HR and LQ for solid
diphenylamine in presence of solution in liquid CQO, and in vapour
very rich in CO,, on HR, and in presence of a much more dipheny|-
anine-containing second liquid phase on BQ. The point P is situated
only 0°.6 above the critical point of pure CO,, consequently dipheny|-
amine is but very sparingly soluble in CO, at that temperature.
Between 31°6 and 38°8 two liquid phases are not capable of
existing in presence of solid diphenylamine; above these we again
find the second three-phase line with two fluid phases now much

( 562 )

richer in diphenylamine. This line was determined up to 120 atm.
pressure. The significance of all the regions in which three-phase
lines are absent can only be expressed by a series of p, z-diagrams.

The above considerations foreshadow the possibility of enunciating
in general terms the conditions for the existence of a solid phase in
presence of one or two fluid ones, when traversing the region of
the critical phenomena of those latter ones, also for those binary
mixtures which in the liquid state are not miscible in all proportions.

Mathematics. — “Qn non-linear systems of spherical spaces touching
one another.” By Prof. P. H. Scnourer.

1. Before passing to our real investigation it is necessary to find
how many spherical spaces touch 7-1 spherical spaces given arbi-
trarily in the n-dimensional space S,. And in its turn the answer to
this question demands a knowledge of the situation of the centres
of similitude of those given spherical spaces. So we start with a
study of these centres of similitude. To this end we represent the
spherical space, which is in S, again the locus of the points situated
at a distance 7 from the centre J, by the symbol Sp, (MW, 7).

2. Just as is the case with two circles lying in the same plane,
two spherical spaces Sp (J/,,7,) and Sp (J,,7,) lying in S, admit
of two centres of similitude on the line J/, 17, connecting the centres,
an external one l’,, and an internal one /,,; through UU, pass the
lines P, P, connecting the extremities P?,, P, of direct parallel rays,
through /,, pass the lines ?, P', connecting the extremities P,, P’,
of opposite parallel rays.

Supposing that in S, a number of v-+1 spherical spaces Sp (Mz, 7x),
(k= 1,2,...,n +1) is given arbitrarily, we shall now investigate
the situation of the (n-++ 1), pairs of centres of similitude (Up 9, Jp9)
with respect to each other. To this end we first notice that the
three pairs of centres of similitude of the three spherical spaces
Spa (Mi, vi); (¢= 1, 2,3) form the three pairs of opposite vertices
of a complete quadrilateral, each of the four triplets of points

(U,, U,; Uys), (U,, Ii, L,5)s (Z,. Uy, 153)5 (i Ly U;,)
consisting of three points of a right line; we indicate these lines in
the given order by

{) (2) iS

Lis ; 12 j Qs :

If we now further regard the n—1 pairs of lines (dj), IY through

( 563 )

U,, and the n—1 pairs of lines (/j,, Ly) through /,, where
successively p assumes the 7 1 values 3,4,...,0-+1 , we
see immediately that each space S,—) through 7» | lines / through
U,, (or /,,) — with all indices p differing mutually — contains one of
the two centres of similitude of each of the (2 -+- 1), pairs (l,,, L,,).
Thus a space S,—; through »—1 lines / through U,, will contain

the point or the point /,,, according to the two lines / with

~ ps wD
p and q as third index being of the same kind or not; just the

reverse is found for a space S,—; through m—1 lines / through
/,,. As the choice of the lines / corresponds in both cases to n— 1

bifurcations 2"~! of those spaces S,-; pass through each of the two
points U,,, /,,. So the theorem holds good :

“We can indicate 2” spaces S,—,, each of which contains (n-+-1),
centres of similitude of a system of 2-1 spherical spaces Sp, given
arbitrarily in .S,, and namely one of each of the (+4), pairs
(Oy. Lpg)-” \

We need not enter into further details about the situation of the
centres of similitude for the purpose we have here in view.

3. From the well known properties of the figure consisting of two
circles and their centres of similitude we read (fig. 1):

u u

UP," = UC," . UC,|

rp. 76". "re"

1 2 1

UP“: UP,=r,:+7,| UP,
np: 2 ==, ee) IME

1 2 1

With the aid of these relations we can easily find the following

Fig. 1.
theorems, where for Sp, (./,,7,) and Spr(W,,7,) we shall write the

: S44 (2
abridged form Sp,’ and Sp.

( 564 )

)
}

5 y (1 el 3 .
“The spherical spaces Sp,’ and Sp,’ are homothetic and directly

similar with €7, homothetic and inversely similar with / as centre

r ° . . *)- nr °
of similitude and + —~ as quotient of similitude. The points corre-

"sy

; L : , i
sponding to each other, P,“ and P, in the first case and P, and P,
in the second, are called homologous.”

= . Gl ~ (2
“The spherical spaces Spy? and Spr correspond to each other

in an inversion with l/ as centre and UC,". UC, as positive power

A : : i ;
and in an inversion with / as centre and /C, . /C, as negative

power. The points P,“ and P," corresponding to each other in the

first case and P,’ and P,' in the second are called antihomologous.

And the two inversions appearing in these theorems shall furtheron

be indicated for shortness’ sake by the symbols (1, 2) and J (1, 2).”
‘

“Each spherical space Sp, through a pair of antihomologous points

P, and! P, ‘of Sp? and Sp cuts these spherical spaces at equal

angles. If the spherical spaces Sp, through P, and P, touches the

5 ie ; P ; 9
spherical space Spi? in P,, it will touch the spherical space Sp?
in P,. And these contacts will be of the same kind or not, according
to l or J being the centre of the antihomologous correspondence.”

In connection with the general theorem concerning the situation
of the centres of similitude the second and the third of these three
simple theorems form the foundation of a method of solving the
problem to construe a spherical space Sp, touching 1-1 spherical

spaces Sp, Sp, ea: Spry given arbitrarily in S,. As will

immediately be evident, to each of the 2” spaces S"—! through (7+1),
centres of similitude answers a pair of tangent spherical spaces Sp,

and the contact of one of these spherical spaces with Sp and Sp?

is of the same kind or not, according to the chosen space S,—; con-

taining of the centres of similitude U,,,, J), of Sp\” ) and Sp? either

the first or the second. So 2"+! is the number of the theoretic solutions.
And if we indicate external contact by ++ and internal contact by
—, then the 2” pairs of solutions are indicated by the pairs of
completely opposite combinations of signs of the series consisting of

n+ 1 terms

( 565

ee ks ob. ey

where the two solutions of a selfsame pair correspond in all signs
or differ in all signs.

The construction of the tangent spherical spaces proves the above
assumed concerning the number of the solutions and their connection
with the 2" spaces S,-;. We give it here to avoid prolixity
for the case n = 4 in a form, in which it is immediately transferable
to the case of an arbitrary n. It is:

6)

Seas Y . = {Q)) y (2) 5 :
a. “If in S, the spherical spaces Spy’, Spy’,... Spy’ are given

EP Rial aA ;
arbitrarily, ifd,,, is the space through the points

U;;, U,;, U,3; Oe Dy Liye Ty 43 Hee Less fy,

and if P, is an entirely arbitrary point of Sp”, then the antihomo-
logous points P,, P,, P,, P, of P, in the inversions U (1, 2), U (4, 3),
Z(1,4), 71,5) are to be determined and the spherical space Sp, (7)
through the five points P,, P,, P,, P,, P;.”

= . ats : » «=—f5 ° s
b. If &, is the plane of intersection of ¢,,, with the radical space
2 eG : : ;
of Spy? and Sp,(P), let us bring through «, two spaces touching

Spy and let Q, and Q,’ indicate the points of contact.”

c. Finally must be determined the pairs of points (Q,, Q,’),
(Q;, Q;’), (QQ), (@;, Q;") which are antihomologous to (Q,, Q,’)
in the inversions U7 (1, 2), (1,3), 71,4), 21, 5) and the spherical
spaces Sp,(Q) and Sp,(Q’) passing through the quintuples of points
Q,,Q,,..-Q, and Q,’, Q,’,...Q,’. These spherical spaces Sp, (Q)
and Sp,(Q’) form one of the 2” pairs of solutions of the problem.”

The proof of this construction is plain. When P, moves over

y (1 F . ay tomers 5 4 45
Spy the power of each of the ten centres of similitude lying in d_,,
with respect to the spherical space Sp,(P?) remains unchanged; con-
sequently the spherical spaces Sp,(2?) which are possible form a

* . AS . . .
pencil with d,,, as common radical space and «, is a common radical

plane of Spi? with each of the spherical spaces Sp, (2) of that
pencil. If now we choose for P one of the two points of contact

. a : -
Q or Q of Spy’ with a space through ¢,, then this tangent space

( 566 )

must likewise touch the spherical space Sp,(Q) or Sp, (Q’) passing
through this point in the same point, ete.

We have now arrived at the first part of our investigation propet
concerning the system of the spherical spaces Sp, touching 7 spherical
spaces Sp, (My, 7x), (k= 1, 2,...n) given arbitrarily in S, and we

reduce the general case — following the way indicated by Reyer for
our space — to a simpler one, in which the centres J/; of the x

spherical spaces which must be touched lie in a space S,~2.

The centres VW; of the m given spherical spaces Sp, (Mz, 7) deter-
mine a space .S,—;, intersecting these spherical spaces according to
“central spherical spaces” Sp,—) (Af; 7%), thus intersecting them at
right angles. Let OU be the radical centre of these n spherical spaces
Spry and 7? the power of this point with respect to the spherical
spaces Sp,—i, provisionally supposed to be positive. Then the spherical
space Sp,—1(0,7) lying in S,—: intersects at right angles the n
spherical spaces Sp,—1 (My, 7%), thus also the 2 spherical spaces
Spr (Vi, 7). So an inversion with an arbitrary point O”’ of the
surface of the spherical space Sp,-1(0,7) as centre makes the n
given spherical spaces Sju (.V/;, 7.) and the spherical space Sp,—1 (O, 7)
cutting them at right angles to pass into m new spherical spaces Sp'n
and a space S,-2 cutting them at right angles. This special case
where the centres W, of the m2 spherical spaces which must be
touched lie in a space S,—2 shall be treated first.

5. If Sp", is a spherical space touching the new spherical
spaces Sp',, then this spherical space Sp", rotating round the space
S,-9 through the 2 centres /', will touch in any position the n
spherical spaces Sp’, and will thus form a singular infinite series of
tangent spherical spaces. In an arbitrary space S,—; through the
axial space S,-2 we find according to the results obtained above
2"—! pairs of spherical spaces Sp",—1, touching the central spherical
spaces Sp’, 1 — lying in S,—1 — of the 2 spherical spaces Sp’, and as
a matter of course each of these pairs consists of two spherical spaces
Sp"n—1 lying symmetrically with respect to S,—o. As each of those
pairs by rotation leads up to a singular infinite series there are 2"—!
of such series. The spherical spaces of each of those series are
enveloped — compare my preceding communication on page 492 —
by an w-dimensional torus 7',,; their centres lie on a circle. And if
we confine ourselves to one of the 2"— series, we can extend the
system of the 7» touched spherical spaces Sp’, to a n—2-fold infinite
series by representing to ourselves all the spherical spaces described

out of the points of S,-», in such a way that they touch one of
the spherical spaces of the singular infinite series, thus all the
spherical spaces of that series too.

6. If we now confine ourselves to a single series of the 2"—!
singular infinite series we have found two systems of spherical spaces
possessing the remarkable property that each spherical space of
one touches all the spherical spaces of the other. Of these two
systems one is a singular infinite series of equally large spherical
spaces with a circle C(M,, 7,) having WW, and 7, as centre and as
radius and lying in the plane ¢, as locus of centres, whilst the other
is an n—2-fold series with the space S,—2 perpendicular in .J/, to
€, as locus of centres. How do these two systems transform them-

selves if we apply to both — in order to return to our n given
spherical spaces Sp, — the inversion with 0’ as centre and the

formerly used power?

To answer this question is made easy by the observation that the
n-dimensional figure consisting of the two systems Sy',, Sz';,—2 and
their inverse systems Sy,, Syj,—2 have a plane of symmetry, the
plane o through J/,, O' and the projection O" of O' on S,—~». This
plane o forming the plane of fig. 2. has with ¢, in common the
diameter m' parallel to O' 0" of the circle C(MV,,7,) and is according
to that line im’ perpendicular to €,; so it is a plane of symmetry for
Sy',. It has moreover with S,—2 the line J/,0" in common and is
according to that line @ perpendicular to Sy,—2; so it is also a plane
of symmetry for Sv7/',-2. And if it is a plane of symmetry for Sy’,
and Sy’',—2, then it is so too for Sy, and Sy,—s, because it contains
the centre (' of the inversion.

We prove to begin with that the centres of the spherical spaces
of Sy, lie in a conic. To this end we regard in the plane of sym-
metry o (fig. 2) the points of intersection J/', J/" with the circle
C(M,,7r,), the circle of section C(J/,,7,) with the spherical space
Sp"n(M,,7,) of Sy’, and the point O of the line J/'0', for which
M'O'.M'O=r,?. Then point A of a, which is at an equal distance
from O' and QO, is the centre of a sphere Sp,(A, 40) with AO’ as
‘adius, intersecting Sp", (J/',7') and so all spherical spaces Sp", of
the singular infinite series at right angles. This sphere is transformed
by the inversion with O' as centre into a plane € perpendicular to
O'A, intersecting 6 according to a line m normal to O'A; this plane
€ must contain the centres of the spherical spaces of Sy, as it cuts
all those spherical spaces at right angles. And farther, when inverting,
the centre of a spherical space remains on the line connecting this

Fig. 2.

point with the centre O' of the inversion; so the oblique cone with
QO as vertex and circle C(J/,,7,) as base must contain the centres
of the spherical spaces of the series Sy,, and the locus of those
centres is the conic of intersection of this cone with the plane «.
Of this conic m is an axis of symmetry and the points 1/' and 11",
becoming the centres of the inverted spherical spaces Sp", (JZ, 7’),
Sp"n(M',r') are vertices. This conic is an ellipse #, a parabola
P or an hyperbola H, according to none, one or two of the spherical
spaces Sp", of Sy', passing through O', i.e. according to O' lying
outside the two circles C(M',7r') and C(.M",7'), on one of those
circles or inside one of them. Of these three cases fig. 2 represents
the first and this will be furtheron exclusively under consideration.

If we suppose that the conic obtained is an ellipse /# the inverse
spherical spaces Sp, (4/', 7) and Spr (JZ", r") of the spherical spaces
Sp", (MW, 77) and Sp", (M", 7’) will touch every spherical space Sp, (Mf, r)
of the system \Sy,—2 in the same kind. From the triangle 1 M' i"

(569 )

then ensues, if we represent the radii vectores M' and MW" M of M
with respect to the fixed points J/’ and J/" by w and », that
u—v=+(r'—r"). So the locus of the centres of the spherical
spaces Spi (MV, r) of the system jSy,—2 is the figure of revolution,

which is generated when the hyperbola /7 with MW’ and J/" as foci
and + (r'—r") as half real axis rotates round m in the space S,—1
through S,-2 and O'. And beeause each spherical space Sp, of Sy,
touches the spherical spaces of Sy,—; having the vertices of the
hyperbola #7 as centres, those vertices of the hyperbola // are reversely

the foci of the ellipse /“. Thus the theorem holds good:

“The spherical spaces Spx’ touching » spherical spaces Spr given
arbitrarily in JS, form 2”-! singular infinite series. The spherical
spaces of any of those series are connected by this that they intersect
a definite spherical space Sp‘, at right angles and that their centres
lie on a definite conic (A’); the determining figures, the spherical
space Sp’, and the conie (4'), change from seriés to series. To each
series corresponds as envelope of its spherical spaces a definite curved
space of order four, the n-dimensional cyeclid of Dupm. And if we
confine ourselves to a single series, the system of -given spherical
spaces Sp, can be extended to an m—2-fold infinite series of spherical
spaces Sp,, connected by the fact that they cut another spherical
space Sp,\° at right angles and that their centres are situated on the
surface of a figure of revolution generated by the rotation of a conic
(KX). These two conies (A’) and (X’') lie in mutually perpendicular
planes in such a way that the foci of one are vertices of the other
and reversely.”’

7. The inversion applied becomes impossible within the region of
reality when the common power of the radical centre O of the n
given spherical spaces Sp» with respect to those spherical spaces is
negative. In this case before inverting we can diminish the radii of
the n given spherical spaces by a common quantity in such a way
that the radius of one of those spherical spaces disappears. Then the
power of the radical centre O of the new spheres is certainly positive.
By operating now with the new system and after that, when the
system Sv, has been found, by adding the assumed quantity to the
‘adii of the spherical spaces of Sy,, we arrive at the desired aim.
As is evident we can even augment the radii of some of the
given spherical spaces by the radius of the spherical space that is
to become a spherical space reduced to a point if only the series of
the tangent spherical spaces is chosen so as to correspond to this.
39
Proceedings Royal Acad. Amsterdam. Vol. VIL.

ae

(570 )

8. Are there not also non-linear systems Sy, and Sy,—~—-1 of
spherical spaces Sp, respectively 4-fold and n—k—1-fold infinite
situated in S, in such a way that each spherical space of one system
touches all the spherical spaces of the other?

This question must be answered in the affirmative as we shall
prove here analytically.

If in a space S,-; of S, the spaces S; and S, ~~~, which have but
the point O in common are perpendicular to each other in this point,
if OP is the normal in O on S,-1, OQ an arbitrary line through
O in S, OR an arbitrary line through O in S,—,—; and if we assume
(fig. 3) in the planes OPQ and OPR an ellipse (/) with the half
axes OA=a, OB=b and an hyperbola (//) with the half axes
OC =c=Va'—b?, OD=b, then by rotation of (2) round OP
in the space Sy41—= (OP, S,) — when every point describes a spherical

space Spx a quadratic space of revolution Qt is generated, by
rotation of (H) round OP in the space S,_,.=(OP,S, rt) —
when every point describes a spherical space Sp,—.—; — a quadratic

. 2 :
space of revolution Q, is generated.

@

Fig. 3.
If now # and // are arbitrarily chosen points of those figures of
revolution the distance HH can easily be calculated. If namely we
use a rectangular system of coordinates with O as_origin, OP as

axis OX,, the plane OPE as plane OX,X,, the plane OPH as
plane O.X,X,, then the coordinates of the points / and HT are

13.6 opin 2, CHE OD, TO SO, CRAY Re Oy a re taree
pea
and we find

ceecntys aU) AOR ALS i =) Ona

EH = a sec — ¢cos p
From this ensues that the spherical spaces Sp,(2,ccosg--e) and
Spr (Hf, a sec w-|-9), where 9 represents an arbitrary constant, touch
each other and that this contact is an external one or an internal
one, according to ¢ cos g +o and asecy +e having the same sign
or not. Thus the theorem holds good: j

“If we describe out of each point / of Qr4 with acos@ as x,
a spherical space Sp, (/,¢ cos -+ 0) and out of each point H of

Qe, with csecyp as «w, a spherical space Sp, (H, a secw- @)
where y represents an arbitrary constant and w and y assume all
possible values, then two systems Syx, Sij,—z—1 of spherical spaces
Spx are generated with the property that each spherical space of

one system touches all spherical spaces of the other.”

Both systems of spherical spaces are enveloped by the same curved
space of order four. If namely of a rectangular system of coordinates
with O as origin and OP as axis OX, the axes OX,, OX,,
OX,, .. . OXzto are situated in Sp, the axes OX,, OXp45.
OXp44,--- OXn in S,-~, then the coordinates of two points / and

oi : 2) 2 :
HT lying arbitrarily on Qe and Q, can be written in the form

E EX
£, =acosp wv, =csecw
xz, =Dbsingoosg, Cant —10
ce =O z, —btg cosy,
L, =bsingpsing, cosy, i)
a, —bsingsing, sing, cos py, ae == 0
Lei = Dsin P sing, sn P, --. e411 = 0
SUN Pk —2 COS Pr—1
12 Dsin PsinP, sin P,.+-- r4+2— 0
SUN Pj-—Q SM PJ; -1
re+3—= 0 LL43 = btg wsin wy, cos w,
aep4=0 Lh44 — btg wesin yp, sin, cos w,
Gee 0 &p—1 = btg wsiny, sinw,....
SiN Wn—L—3 COS Wn—fL—9
wt, 0 Ly =btywsiny, siny,....

SIN Wyn K—3 SIN Wr—K—2

39*

(572 )

and
(x,— acosp)* + («,—b sing cosp,)* + «,? + (a,—b sing sing, cosg,)? +...
+ (th 2 — bsing sing, sing, ... sinpr—g singy. 1)? + @p3- -.. + ry? =
=(ccosp + 9)"
is the equation of the spherical space Sp,(E,ccosp+e). If we
write this equation in the form.
= «7? + b—o?=
=)
= 2 fax, cos p+hsing [x, cosy, +x, sing, cosp,+...+ap49sing,...sin Pri}
and underneath the / equations formed out of it by differentiation
according to 9, 9,,...@—1, then addition of the 4+ 1 equations, after
having squared them, furnishes us with

n k+2
(2 a? + 8 — ')* = Alar, + og +h + Za). .
i= —
And this same equation is obtained in the form

(S a; — B — 9°) = Alex, + a9)? — 8 (@,? + = 2r)),
i=E3

a

if we consider the spherical space of system Sy,—x—1-

9. For a variable parameter 9 equation (1) represents a system
of parallel n-dimensional cyclids of Dupin. Here we can then ask
after the n numbers indicating successively how many of those eyelids
pass through a point or touch a line, a plane, a space, etc. In this
investigation the £-+-(n——1)-, i.e. the n—1-fold congruence of
the right lines is in prominence, connecting an arbitrary point £ of

Qe with an arbitrary point 7 of QO: the case of 2 = 3 has been
treated before in a small paper (‘Prace matematyczno-fizyezno”’,
vol. 15, pages 83—85, 1904). And the more general case we do
not touch here.

Mathematics. — “On a special tetraedal complex.” By Prof. Jan
DE VRIES.

1. By the equation
y? =

=to+5=" ‘a se,
a c

a system of similar ellipsoids is indicated.
The normal in a point P, on the ellipsoid containing this point is

determined by

( 573 )

a* (e—2,) = b° (y—y,) 2 e (z—2z,)
a, Wy oat
or also by

atte bv? + u e+tu 4
aa Pu, Ayia Wi POT + + + (2)

For its orthogonal coordinates of rays, i.e. the quantities

pi a A yp, = e—e
0M, =ye — ay, pp, 2a — ez, pp, = ay'—ye' ,
we find
u—u (b?—c?) (u'—u)

= - CC a — Pe Y, 7, ete,

From this ensues that the «* normals of the system of ellipsoids
form a quadratic complex with the equation

a pip, pep, tC pepe 05 2. . « - 2 (8)

2. For the traces of the normal with YOZ and YOZ, we have
successively uw” = -— 6? and u' = —a’, so
?—b? e—a

ee and 2'=
¢ c

BI)
~

Now follows from
2": 2! = (c?—6") : (?—a’)
that the complex can be built up out of o* linear congruences, of
which the directrices form two projective pencils of parallel rays
situated in NOZ and YOZ having the direction OX and OY. So
the complex is ¢etraedral and has as principal points O and the
points X,, Y,,Z,, lying at infinity on the axes.

The trace of the ray of the complex with XOY is determined
by w" = — c*. If we notice that the parameter w is proportional to

the distance of the point P indicated by it to the point P,, we see
that out of

r

I
uw —y' ai—c?

A at! b2—e?

the characteristic anharmonic relation of the complex is obtained,

namely
@ yp pli iP) a (a°—c’*) : (b°—c’).

3. The footpoints P, of the normals let down out of P, lie |

1
evidently on the cubic curve

era, a Lave Gren
atte’ y pees eds oi! oh ictal ine, etn (4)

( 574 )

which passes through the points P,,0,X,,,V,,.2,,.(v=0, ©,—a*,—b*,—c’),
and which is thus an orthogonal cubic hyperbola w*. Each of its
points P, determines an ellipsoid, for which P,P, is the normal
in.

Through a given point P, pass o' curves w*; their ‘foci’ (Null-
punkte) P, are indicated by

Vr, =(@ 4+ uYe,, Py =U + u)y,, ba, =(F+uz,; - (5)

so they lie on the normal having P, as footpoint.

These curves are all situated on the surface determined by the
equations

a’ + u)e, (b° + u)y, (c? + u)z, 6
— , y = SO —__, = = —

awe Bee mee (9)
or also by the equation obtained from these by elimination of u
and v

vi

(ae) ieee

BF (9) ye Ws ee ee)

oe (z —z,) eres

This same equation we obtain out of (3) if we express the
coordinates of rays in the coordinates of points. So the locus of
the curves w* passing through P, is the complex cone of P,.

Corresponding to this we find out of (6) for v= const. a right
line through P,, whilst «const. indicates a curve w* through P,.

4. Out of the preceding ensues that all bisecants of a curve w*
are rays of the complex. This is further contirmed by the calculation
of the coordinates of rays of the bisecant (v, v'). We find out of (4)

2a esas
eS ea Wo , ete
(@ + r)(@ +e)
b*c*y 2 (b° —c’) (v' =)
PF VE+)C Lye +0)
from which ensues readily
a'P,P, + O°psPs + O'PaPs = O-

Pi

ete.

5. The planes of coordinates and the plane at infinity are the
principal planes of the complex. The complex cone of a point lying
in a principal plane must degenerate.

We truly find out of (7) for z,=0 the planes 2=0O and

(7 —c*)y,¢—(0?—e)axy=(e—B)ay,. . . . (8)

In connection with this the curve w* consists now of the hyperbola

z=0, (C—O )ey+byae—cary=—0 ... . (YY)

_—
cr
=I
OX
—

and the right line cut by it
(=) a — are 8) 8 (07 = oy b*y,, >. - . (20)
lying in the plane (8).
If in (7) we substitute 4e,we,ve for v,, y,, 2, and reduce it to
the form

a? ( _ i) (vy — uz) + 0? (: — «) (Az — va) + ¢? (é — » ie —Ay)=0,
2 2 2

then @ =o furnishes the equation
(a? — 6?) Auz + (67 — c*) pve + (c? —a*)vpay—0, . -. (11)
which represents the plane containing the normals with the direction
(4, u,v). The footpoints of these normals lie evidently on the right line
MELUP SS 7 8 (Ne ear Se Go so o 9 6 Alyy)

6. We determine with respect to the ellipsoid (/,) the polar line
of the normal n, having P, as footpoint.
For an arbitrary point P' of that normal we find the polar plane
s et
3 a

2 == Io

For all values of w’ this plane passes through the line of inter-
section of the planes
Wy
b?

LA
a0

+2 =k, . it dyad eed ew SIS)

a
Wy
te

This line of intersection is the required polar line. When /, changes
it displaces itself evidently parallel to itself.
Out of (18) and (14) we obtain the equation
¢@—B)yy C-—e)z2
b* r c* aa

which becomes identical with the equation

uu 2,2
ASS er OI ee IS Por icgien sae eta (UZ 9)
a &

(Op Gs

Ya 2s

of a projecting plane of the normal n, with footpoint P,,
conditions are satisfied

(a*@—b’) y, b° (a? —c’?) z, —?¢
k,*b4 Ss GSa\in. hneCe a (b?—c?) z,

From this we can deduce that the polar line of the normal »,

with respect to the surface (/,) is again a normal n,; the footpoints

P, and P, are connected with each other by the involutory relations

= 6? — ¢?

if the

( 576 )

«, 2, =a' k,* : (a?—0?’) (a?—c’),
Yr Ys = O° K,*: (b’—a*) (b°—c’),
Z, Z, = c° k,? : (c?—a’) (c?—3’).
By polarisation with respect to each of the ellipsoids the complex
is thus transformed in itself. This agrees with a well known property
of the tetraedal complex.

7. The footpoints of the normals are then arranged in an involutory
quadratic correspondence, which transforms a right line into a twisted
cubic, thus the tetraedal complex into a ae of twisted cubics
which all pass through the points O, X,, Y,, 7. Let us now regard
in general the transformation

ea a" yy Bie ee ay oe ee

It substitutes for the ray of the complex indicated by (2) the

twisted curve

. a? «e F 5? B? : ce y?
a“ > f= S| OS == SS
© @+wa,’ 7 @+ay, + yz,
If we still put
a Be? 7°
——=—f,, ~ =I —-—s 2 ee
ae yy ea

then this curve is indicated by
p= aa, eae bry on Cz
jac a7 Te Tie aS ae nee

So it is the curve w’* belonging to the “focus” P,, which corre-
sponds in the transformation to the footpoint ?, of the normal.

The complex of normals is thus transformed into the complex of
the curves o°.

In connection with this the cone of the complex of /, passes into
the locus of the curves w* containing the point /,, thus (§ 3) into
the cone of the complex of P,. Indeed the equation (7) does not
change in form if we apply the relations (15) and (16).

8. If the vertex of the cone of the complex moves along the
right line / represented by
pa en es , Pic nee : pate ts
: aay een a
then the cones form a system with index two represented by
4,7 U, + 24,4, U,, +4,? U,=9,
where

0, = 2a (e¢ —2,)(,y9—9,2) » U,= Za? (w—2a,) (2, 4 — Ye 2)s
3

— > 7 , > ~ . . ~ ~\i
2 U i 2 a {(w Th #,) (2, U aie 45 (ae v;) (2, Y — Ya 2)5-
The envelope of this system, at the same time the locus of the
conics of the complex having / as chord, has for equation
U =U,U,— Uv, =0.
The eight nodes which this biquadratic surface of the complex

1

must possess are the points of intersection of the surfaces
eR ae Oe UE 10
For we have

0U OU, 4u oU, ay 0U;,
Op Corp com, Ne Oa
so that — disappears for each of those eight points of intersection.

Ou

To these nodes evidently belong the points 0, Y,, Y,,
four other ones change their places with the right line /.

That 7 is double right line of the surface of the complex is
immediately proved by the substitution «= #, + 20, y= y, + ue,
2=2,-+ 0; on account of 2,y — 7,2 = 0 (uz, — vy,) we see that
U then obtains the factor 0°.

Z.; the

Mathematics. — “On a group of complexes with rational cones
of the complex.’ By Prof. JAN pe Vrirs.

§ 1. In a communication included in the Proceedings of May
19038 *), I have treated a group of complexes of rays possessing the
property that the cone of the complex of an arbitrary point is
rational. In the following a second group will be indicated with the
same particularity.

We consider a pencil (s) with vertex S in the plane o, and in a
second plane t a system of rays [{f],, with index » (thus the system
of the tangents of a rational curve t,) and we suppose the rays ¢
to be projectively conjugate to the rays s. The transversals of homo-
logous rays form a complex, which will be investigated here.

Out of an arbitrary point P the pencil (S, 0) is projected on the
plane t in a pencil (S', 1), projective to [f]n. Together these systems
of rays generate a curve of order (7-++1) having in S' an n-fold
point; for on an arbitrary ray s' through S' lies outside S' the point

1) “On complexes of rays in relation to a rational skew curve.” VI, p. 12—17.
Pp Pp

(578 )

of intersection of s' with the corresponding ray ¢; on the rays s’
conjugate to the m rays ¢ passing through S’ this point of intersection
falls in S', so that the locus of the point (s’,¢) must pass m times
through |S’; the curve is therefore of order (n+ 1).

The cone (P) of the complex is of order (n-+-1) and of class 2n
and has an n-fold edge PS.

§ 2. If the point S’ lies on the envelope rt, two of the n rays ¢
passing through |S’ coincide, so also two of the tangent planes through
PS to the cone (P).

The locus of the points P for which two tangent planes through
the n-fold edge of the cone of the complex coincide is the cone S of
order 2(n—1) projecting the envelope t out of S.

The 3(z— 2) cuspidal edges of S contain the points ?, for which
three of the tangent planes of (/?) coincide along the n-fold edge.
The 2 (n— 2) (n—3) double edges of S form the locus of the points
P, for which two pairs of tangent planes of P coincide along PS.

The cone = is a part of the singular surface of the complex;
the remaining parts are planes.

To these belongs in the first place the plane 6. Each right line of
o is cut by m rays ¢, can thus be regarded nm times as ray of the
complex. Consequently o is an n-fold principal plane. In connection
with this the cone of the complex of a point P? assumed in 6
degenerates into n planes coinciding with o and into the plane
through P and the right line ¢ corresponding to the ray s deter-
mined by P.

On the contrary t is single principal plane, for each of its right
lines rests on but one ray s. The cone of the complex of a point P
lying in rt degenerates into t and into the » planes through P and
the m rays s corresponding to the right lines ¢ through P.

Finally there are still (7 +1) principal planes y,, (k=1 ton-F4),
each connecting two homologous rays s,¢. For the points of the
line of intersection of o and 7 are arranged by the projective systems
(s) and [¢]}, in a (1,7) correspondence; in each of the (n-+1) points
of coincidence C, two homologous rays meet. In connection with
this the cone of the complex of a point ? assumed in one of these
principal planes degenerates into the combination of this principal
plane with a cone of order x, for of the projective systems (s’) and
{t|, lying in t two homologous rays coincide.

§ 3. The curve of the complex (x) in the arbitrary plane = is of
class (7 +1) and has the line of intersection (6 2) as n-fold tangent;

so it is of order 2n. Its points of contact with (7a) are determined
on (62) by the 7 rays s corresponding to the 7 rays ¢ through the
point (Or a).

If a passes through one of the 2 (—41) points of intersection
of o with the envelope r, two of the points of contact of (6 2) coincide.
Regarded as locus of points (a) then consists of a curve of order
(2n —1) and the right line (62).

The planes containing curves of the complea for which two points
of contact of the multiple tangent coincide form 2(n—1) sheaves
having their vertices on the line of intersection of 6 and xt.

If a passes through a ray s,, then (2) as envelope consists of a
pencil having its vertex in the trace of the homologous ray ¢, and
of the pencil (S, 2) of which each ray belongs 7 times to the complex,
because it is intersected by nm rays ¢. As locus of points (x) is here
the line connecting the vertices of the pencils counted 27 times.

If a contains a ray ¢, the envelope (a) consists of a pencil having
the trace S, of the homologous ray s, as vertex and of a curve of
class n for which (62) is an (n—1)-fold tangent. As figure of order
2n the curve (a) breaks up into a curve of order 2(7—1), its (2 —1)
fold tangent and the tangent which can moreover be drawn to it
out of S,.

If one brings a through one of the coincidences C;, then (a) breaks
up in the same way into a pencil with vertex C, and a curve of
class 7.

The complex possesses an n-fold principal point S and (n+1)
single principal points Ch.

§ 4. Let us now consider the surface of the complex 4 of an
arbitrary right line /, thus the envelope of the rays of the complex
resting on /. The rays in a plane a brought through / envelop a
curve (x) of order 2n (§ 3). If a is one of the 2n tangent planes
through 7 to the cone of the complex of the point P lying on J,
then two of the tangents drawn out of P to (x) coincide, so that
P is a point of (a). So each point of 7 belongs to 2n curves of the
complex; consequently / is a 2n-fold right line of 4.

The surface of the complex is of order An.

In the planes connecting / with the principal points C, the curve
(t) breaks up into a curve of order 2 (”—1) and two right lines.
This also takes place when a passes through one of the n rays ¢
resting on /. In the plane through / and S the curve (2) degenerates
into a right line to be counted 27 times.

In each of the planes connecting / with the points of intersection

( 580 )

of t, and 6 the curve (2) consists of a curve of order (2 —1) and
a right line (§ 3).

On A lie besides the 2n-fold right line and a 2n-fold torsal right
line 6n single right lines more.

The plane 6 contains 2(m— 1) right lines of A and touches 4 in
the points of a curve of order (n+-1), which is the locus of the
points where the curve of the complex (2) touches its n-fold tangent
(oz). For, if the ray s, resting on / corresponds to the ray ¢, cutting
6 in 7’, then one of the points of contact of the curve of the complex
of the plane (/7',) with o lies in the trace Z, of 7; consequently
the indicated points of contact lie on a curve of order (n-++1). This
curve is generated by the pencils (Z,) and (S) arranged in a (1, n)
correspondence ; so it has in S an n-fold point.

The plane rt touches 4 according to a curve of order (nm + 1)
which is the locus of the points of contact of the curves (7), in
planes a through /, with the traces (ar). This curve has an n-fold
point in the trace L, of / on +; the tangents in this multiple point
are the traces of the planes a cutting (or) on the n rays s conjugate
to the rays ¢ drawn out of L,.

The plane rt has farthermore the envelope t, in common with A.
For, while a point P of the right line (ar) bears in general n
tangents of the curve of the complex (a) determined by the rays s
corresponding to the m rays ¢ drawn through P, two of those
tangents coincide as soon as P lies on the envelope r,,; then however
P belongs to the curve (x), thus to the surface of the complex <A.

Microbiology. — “An obligative anaerobic fermentation Sarcina.”
By Prof. M. W. Brtertnck.

The following simple but yet delicate experiment gives rise to a
vigorous fermentation, caused by a sarcine, wherein microscopically
no other microbes are perceptible and which, when rightly performed,
can produce a veal pure culture of this fermentation organism. The
simplicity of the experiment is the result of many previous invest-
igations, partly made conjointly with Dr. N. Gosiines, which have
gradually rendered clear the conditions of life of the examined
microbe.

3ouillon with 3 to 10°/, glucose, or malt wort, is acidified with
phosphoric acid to an acidity of 8 cc. normal per 100 cc. of culture
liquid and introduced into a bottle, which is quite filled with it
and fitted with a tube to remove the gas. The infection is done

————ee—e——————Ke-™

( 581 )

with an ample quantity ') of garden soil, from which the heaviest
and roughest portion has been removed, but in which so much
solid substance is left behind that in the nutrient liquid it forms
a muddy deposit from 5 to 7 or more millimeters thick. The
culture is effected in a thermostat at 37° C. After 12 hours already
the liquid is in a strong fermentation, which lasts from 24 to 36
hours, and whereby the surface is covered with a rough seum,
produced by gas bubbles mounting up from the depth. Whilst
the liquid itself remains wholly free from microbes, the micros-
copical image of the deposit shows a luxuriant, pure or almost
pure culture of a sareine, of which the elementary cells measure
for the greater part about 3.5 u, so that the species belongs to
the largest forms known, and the multicellular sarcine-packages are
easily visible to the naked eye. The cells are colorless and
transparent and the packages present irregular sides. Here and there,
but much less generally, a brownish intransparent form is seen,
with more regularly cubical packages of which the cells measure
2 to 2,5 w.

The scum floating on the fermenting fluid consists of slime in
which the evolved gas remains for a time imprisoned. This slime
is produced by the outer side of the sarcine cells, whose walls
for the rest consist of cellulose, which becomes violet-blue by zine-
chloride and jodine. This reaction was discovered in 1865 in the
stomacal sarcine by Surincar*), who on this account argued the
vegetal nature of this organism, which fully corresponds to the
small-celled fermentation sarcine. The large-celled form more resembles
the figures which Linpner*) gives of his Sarcina maxima, found, as
he expresses it, in “Buttersiuremaischen”’, hence, in wort wherein
a spontaneous butyric fermentation. I am not, however, convinced
that both these forms do really belong to two different species of
sareine, as it is well known that in this genus of microbes great
morphological differences may occur in the same species.

The gas is a mixture of about 75°/, carbonic acid and 25 °/,
hydrogen; methan is not present. Besides, a moderate quantity of acid
is formed, which for example, in a nutrient liquid with an acidity
of 6 ce. per 100, may mount to 12 ec., a percentage only found
back in the technical lactic fermentations. Furthermore a peculiar
odor originates, reminding of the ordinary lactic-acid fermentation, by

1) With Jitile soil fcr infection, the experiment becomes doubtful.

») W. F. R. Surincar, De sarcine (Sarcina ventriculi Goovstr), pag. 7
Leeuwarden 1865. Here very good figures are to be found.

5) Mikroskopische Betriebscontrolle in den Girungsgewerben, 3e Aufl. p. 432, 1901.

?

(582 )

Lactobacillus. Vt, as is probable, this acid will prove to consist
entirely, or for the greater portion, of lactic acid, the fermentation
sarcine may be considered as the most differentiated lactic-acid ferment
hitherto known.

When using a suflicient quantity of soil for the infection, that is
a relatively great number of sarcines, which thereby, in the given
circumstances, may compete with advantage with, and conquer
all other microbes, the experiment described succeeds within very
wide limits. Thus the sarcine fermentation may in this case be
obtained as well in an open flask as in a closed bottle, whence it
follows that the sarcine can suffer a moderate quantity of oxygen;
and it will appear below, that a slight quantity is even wanted
under all circumstances. Notwithstanding this, the name of obligative
anaerobic remains applicable as the cultivation at full atmospheric
pressure is impossible. The acid may further be varied between 3
and 11 ee. normal phosphoric acid per 100 ec.. The phosphoric acid
may be replaced by lactic and even by hydrochloric acid, if the
acidity of the latter is not taken higher than 6 to 7 cc. per 100 ce.,
but not by nitric acid.

Instead of glucose cane sugar may be used, but with milk sugar
and mannite the experiment does not succeed. As source of
nitrogen only peptone can be used, such as found in malt-wort or
bouillon; simpler nitrogen sources, like asparagin, ureum, ammonia
and saltpeter, are unfit for the nitrogen nutrition of the sarcine. The
limits of the temperature are wide and may vary between 28° C.
and 41° C.

Although the experiment may thus be modified in many respects,
the first described arrangement is recommendable, as it is best adapted
to the optimum of the different conditions of life of the organism.

A property peculiarly important for this research is the readiness
with which the function of fermenting, that is the power of evolving
gas, gets lost under the influence of a secretion product, probably
the acid, and through which all transports with old material become
perfectly useless. Hence it is necessary to transport cultures still in
fermentation to insure the success of further experiments.

That some aeration enhances the life-functions of this obligative
anaerobic and that access of a little air is even necessary in the long
run, is evident from the fact that the most vigorous fermentations are
obtained in a closed bottle, with the deposit got in an open flask,
whereas renewing of the nutrient liquid formed above the deposit
in a closed bottle will after few repetitions give rise to diminuation
or cessation of the fermentation.

errs"

( 583 )

For the continuation of the culture by inoculating s/ight quan-
tities of material of a rough fermentation into the same nutrient
liquid, two precautions should be taken. First, the inoculation should
be done into the medium, freed from air by boiling, the bottle
being entirely filled with the hot liquid, so that on cooling no air
can dissolve. Second, an acidity of less than 7 proves not suffi-
cient, hence this should be 8 or 10 c¢e., as otherwise the lactic acid
ferments might prevail and supplant the sarcine.

From the necessity of expelling the air we see that the fermen-
tation sarcine undoubtedly belongs to the ordinary anaerobics, which,
considering the success of the rough accumulation experiment 2th
aeration, might perhaps not have been expected ; but the fact holds good
in the same way for the butyric acid ferment, generally accepted as
an obligative anaerobic, so that, also with respect to the fermentation
sarcine, there should be spoken of ‘“microaerophily.” Further exami-
nation shows that in deep test-tubes with maltwort-agar, very easily
pure cultures may be obtained, whereby the sarcine is recog-
nisable by the obvious size and the remarkably rapid development
of its colonies. On the other hand, on maltwort, or broth-bouillon-
glucose-agar-plates with or without acid at 37° C., with access
of air, no growth at all of the sarcine takes place, as might
be expected. Of course the packages can also be seen on the
plates without growing and be removed in a pure condition. When
we make use of little acid for the rough accumulation, colonies of
lactic acid ferments, belonging to the physiological genus Lacto-
bacillus, will develop on the plates at the air, which can grow
as well with as without air, but whose other life conditions corre-
spond to those of the sarcine. In this case the experiment shows at
the same time that everywhere in garden soil real lactic acid fer-
ments are present, whereof the proof had not been given until now.

When using much acid, for example 10 cc. or more normal
acid per 100 ce. of culture fluid, through which the vital functions
of the sarcine, such as rapidity of growth and the faculty of assimilating
oxygen, are lessened, certain alcohol ferments, proper to garden soil,
come to development, but they can, together with some of the other
impurifications of the rough accumulations, as moulds, JZucor and
Oidium, be checked and expelled by exclusion of air, hence, by
culture in closed bottles. To this end however, it is necessary to render
the conditions for the sarcine as favorable as possible and not allow
a temperature below 37° C.

The staying out of the butyric acid fermentation (caused by Granu-
lobacter saccharobutyricum), which so readily originates with exclusion

( 584 )

of air in glucose-bouillon and maltwort, is due to the acidity of
about 8 ce. or more, whereby this fermentation becomes impossible.

Although it is evident from the foregoing, that the growth of the
sarcine is less inhibited by the acid than that of the lactobacilli and of the
butyric ferment, it may still be easily proved that already 7 ce. acid per
100 ce., are less favorable than 3 or 5 ec., also for the development of the
sarcine itself, so that the higher amount of acid in the accumulation
only serves to render competition with the said ferments possible.
If by timely transports into maltwort with more than 8 ec. phosphoric
acid, or by separation in solids, real pure cultures are at disposal, the
further transfers, with entire omission of the acid, show that then
also vigorous growth and fermentation may occur. We thus see how
wide the limits are of the life conditions of the sarcine, as soon as
competition with all other microbes is quite out of question.

The discovery of this certainly unexpected fermentation has sprung
from the working out of the general question which organisms of
the soil can develop in a sugar-containing culture fluid in presence
of an acid and with imperfect aeration. At temperatures of about
30° C. and lower, alcoholferments, J/ucor racemosus and Oidium
prove to be the strongest, but then already a few sarcines are
observed. At about 40° C. most alcoholferments of garden soil,
besides Mucor and Oidium can no more compete with the sarcine
and the lactobacilli, which then become predominant. This being
fixed the last steps which led to the culture of the fermentation
sarcine alone, were the recognition of the obligative anaerobiosis,
and of the superiority of the resistence of the sarcine with respect
to anorganic acids compared with that of Lactobacillus and the
butyric ferments.

Above, already, I pointed to the perfect correspondence of the
small-celled form of the fermentation sarcine to the description which
SurinGar gives of the stomacal sarcine, and I suppose that in the
cases of non-cultivable Sarcina ventriculi, of which, for instance,
DE Bary speaks’), there should really be thought of the ferment-
ation sareine. This view is supported by different observations in
the older literature, cited by Surmear. But still more convincing is
my accumulation experiment, which proves that the conditions for
the existence of this sarcine are just of a nature to render its life in
the stomach possible.

It will be easy to obtain certainty thereabout by a repetition of

") Vorlesungen fiber Bacterien, Je Aufl. pg. 96, 1887.

this experiment, not with garden soil for infection material, but by
using the stomacal contents of such a case of stomacal sarcine. The
“not cultivability” of pe Bary may mean the. same as anaerabiosis,
for it is well known how difficult it is, even at the present time,
to cultivate anaerobies if the particulars of their, life conditions are
not exactly known.

For the rest I do not doubt of the precision of FaLkENieim’s ') and
Miaua’s *) observations, wno have seen aerobic colonies of micrococei
originate from stomacal sarcine. It is true that I for my part have
not succeeded in confirming this observation with regard to the
fermentation sareine, but for other species of Sarcina I have, with
certainty, stated the transition into micrococci, and with various
anaerobics, although not belonging to the genus Sarcina, | have seen
now and then colonies originate of facultative anaerobics, which in all
other respects, corresponded to the obligative anaerobies used for the
cultures. Therefore this modification also seems possible for some
individuals of the fermentation sarcine. Accumulation or transfer
experiments with stomacal contents will however only then give
positive results, if these are used when still in fermentation; with
long kept material nothing can be expected.

Already the older observers*) as ScHLosspercer (1847), Simon
(1849) and Cramer (1858) have tried, although in vain, by a kind of
accumulation experiments, to cultivate the stomacal sarcine, wherefore
they prepared, as nutrient liquid, artificial gastric juice with different
additions. Remarkable, and illustrating the biological views of
those days, is the fact, that for the infection they did not use
the stomacal contents themselves, but beer yeast, supposing, that the
sarcine might originate from the yeast cells, which somewhat resemble
it, and are always found in the stomach together with the sarcine
itself.

Physics. — “Vhe motion of electrons in metallic bodies.” UW. By
Prof. H. A. Lorunrz.

(Communicated in the meeting of January 28, 1905).

§ 11. By a mode of reasoning similar to that used in the last §,
we may deduce a formula for the intensity ¢ of the current in a
closed thermo-electric circuit. For this purpose we have only to
suppose the ends P and Q, which consist, as has been said, of the

1) Archiv f. experiment. Pathologie und Pharmacologie. Bd. 10, pg. 339, ISS:
2) System der Bacterién. Bd. 2, pg. 259, 1900.
3) Cited from Surinaar (I. c.).

2

1

40
Proceedings Royal Acad. Amsterdam.. Vol. VIL

( 586 )

same metal and are kept at the same temperature, to be brought in
contact with each other. The potentials gp and gq will then become
equal, but the stream of electrons » will no longer be 0. We shall
have on the contrary, denoting by > the normal section, which may
slowly change from point to point, as has already been observed,

P= Ps... 2c ee a ve oe
Taking this into account and using (23), we get from (21) and (30)
dg es! 1 dV md (1 mad log A i
p=-7E-22l)-m ~ oe

We shall integrate this along the circuit from P to Q. Since ¢
has the same value everywhere and
Pp =Fa> us = Vo ; hp =ho
we find
Q
m ld log A dx

at — 2 SS =
QeJh dz o=
P

Here, the first term is reduced to the form (34), if we integrate
by parts. Hence, if we put

da ‘
o= —s R ’

F

Rp

as was to be expected. Indeed, 6 being the coefficient of conductivity,
R is the resistance of the circuit.

the result is

Bile. |

§ 12. We shall now proceed to calculate the heat developed ina
circuit in which there is an electric current 7, or rather, supposing
each element of the wire to be kept at a constant temperature by
means of an external reservoir of heat, the amount of heat that is
transferred to such a reservoir per unit time. Let us consider to this
effect the part of the circuit lying between thie sections whose positions
are determined by w and w+ du and let wdt be the work done,
during the time dt, by the forces acting on the electrons in this
element. IVY being the quantity of heat traversing a section per
unit time, we may write

“ weya
fis aie

for the difference between the quantities of heat leaving the element
at one end and entering it at the other, and the production of heat
is given by

( oor )

d
g=uw — (CHESS Gr We, re aw ee ON)
dar

In order to determine iw, we observe in the first place that the
work done, during the time (ft, by the force acting on a single
electron is

m X & dt

and that, by the formula (J), the element v/v contains

7 (5, 1, $) = da da
electrons having their velocity-points within the element (/2 of the
diagram of velocities. Taking together the forces acting on all these
particles, we find for their work

mX > da dt .§ f (8, 7, $) da ,
an expression that has yet to be integrated over the whole extent
of the diagram. On account of (4), the result becomes
m Xv + da dt,

so that, by (36) a
miX
w=

da.

é
Now, the value of XY may be taken from (21). Substituting
z
5
and using at the same time (23), we find

E 1 dlog A aol et :
X=— : 4. (7) + Dict ak aoe (oS)

2h dz da\h mo

1)

so that

ww, + Ww, ,

__mifldlgA df1\ ; a6
UO air rr GET i Th 5 an 5 (GW)

2
i dx.
o>

if we put

and

The expression (22) may likewise be transformed by introducing
into it the value (88), or, what amounts to the same thing, the
value of

eth aa
2hAX— 5

da

that may be drawn from (21). One finds in this way
WWE Wd

( 588 )

if
* 2amlA dh
=a ee
and
We a
, h eh

§ 13. The expression (37) for the amount of heat produced in
the element de may now be divided into three parts.
The first of these

aoe 5 :
corresponds to Jour’s law. Indeed Ss is the resistance of the part

of the cireuit extending from (x) to (a + d2).
The second part

d , >I
—— (W, 5) de

is entirely independent of the current, as appears from (40). It may
therefore be considered to be due to ordinary conduction of heat.
This is confirmed by comparing it with what has been said in § 9.
It remains to consider the quantity of heat
d

q = w, ——(W, =) de,
da

or, if (89) and (41) are taken into account,
,i mi dlogA
q —a =
2eh dea
This expression, proportional to the current and changing its sign
if the latter is reversed, will lead us to formulae for the Prtimr-
effect and the THomson-effect. Reduced to unit current, it becomes

at.

,. _. m dlogA ie a

7, = Deh Fag tid “ eres, hes (42)

a. 1 shall suppose in the first place that, between two sections

of the circuit, there is a gradual transition from the metal I to the

metal Il, the temperature and consequently / being the same through-

out this part of the cireuit. Then, reckoning « from the metal I

towards II, and integrating (42), I find for the heat produced at the

“place of contact” by a current of unit strength flowing from I
towards II,

( 589 )

Hence, if we characterize the Puntir-effect by the absorption of
heat JT, ;, taking place in this case,

2aT Aj PACH hi N
Ti 3H by ( 1 ) = 3 - loy ( ‘ ) ay (43)
é ALY] ve LN J]

b. In the second place, substituting again for 4 the value (14),
we shall apply (42) to a homogeneous part of the circuit. We have
then to consider /oy A as a function of the temperature 7’, so that
we may write
2aT dlog A
q i=) a Je: “aT
for the heat developed between two points kept at the temperatures
Tand 7+d7, if a current of unit strength flows from the first
point towards the latter. What Krtvin has called the “specific heat
of electricity” (THomson-effect) is thus seen to be represented by
2aT dlog A

3e . aT

“= — (44)

§ 14. An important feature of the above results is their agreement
with those of the well known thermodynamic theory of thermo-
electric currents. This theory leads to the relations

d IT, Il
See pg ee ee men id he SES a 3 ee
2 LE ala al T ) (49)
and
FUG

i ~eE
b= (Par, seid Gewese forth (46)

1

f

in which my and wy are the specific heats of electricity in the
metals I and II, at the temperature 7, whereas /’ denotes what we
have calculated in § 10, viz. the electromotive force in a circuit
composed of these metals and whose junctions are kept at the
temperatures 7” and 7”, the force being reckoned positive if it
tends to produce a current which flows from 1 towards I through
the first junction.

The values (44), (43) and (85) are easily seen to satisfy the
equations (45) and (46).

Instead of verifying this, we may as well infer directly from (42)
that our results agree with what is required by the laws of thermo-
dynamics. On account of the first of these we must have

PE fs ery ie

and by the second

(590 )

> is £2.
7
the sums in these formulae relating to all elements of the closed
circuit we have examined in § 11. Now, by (42), these formulae

become
Q
yh? kes
2e h da
P
and
Q
f 1 d log A Ms
= —— dx — 0.
AT dz«
P

The first of these equations is identical with (84) and the second
holds because 47’ has everywhere the same value.

It must also be noticed that the formula (35) implies the existence
of a thermo-electric series and the well known law relating to it.
This follows at once from the fact that the value (35) may be
written as the difference of two integrals depending, for given
temperatures of the junctions, the one on the properties of the first
and the other on those of the second metal. Denoting by I] a third
metal, we may represent by Fy, 77, Lv, 11, Fir, 7 the electromotive
forces existing in circuits composed of the metals indicated by the
indices, the junctions having in all these cases the temperatures 7”
and 7" and the positive direction being such that it leads through
the junction at the first temperature from the metal indicated in
the first towards that indicated in the second place. Then it is seen
at once that

Froir+ Fuom + Fir,r= 9. Pee ey te (47)

Strictly speaking there was no need to prove this, as it is a con-
sequence of the thermodynamic equations and our results agree
with these.

§ 15. In what precedes we have assumed a single kind of free
electrons. Indeed, many observations on other classes of phenomena
have shown the negative electrons to have a greater mobility than
the positive ones, so that one feels inclined to ask in the first place
to what extent the facts may be explained by a theory working
with only negative free electrons.

Now, in examining this point, we have first of all to consider the
absolute value of the electromotive force /. If we suppose the tem-
peratures 7" and 7” to differ by one degree and if we neglect the

— lo i et i ee

( 591 )

variability of M7 and N7 in so small an interval, we may write
for (35)

The value of the first factor on the righthand side may be taken
from what, in § 9, we have deduced from the electrochemical
aequivalent of hydrogen '). We found for 7’= 291

aT A
SS Bis MNF
e

so that

Ni Ass ,
log —— = 0,00011 F >.
Nr

In the case of bismuth and antimony, /° amounts to 12000,
corresponding to

N
log — = 1,82

. Nu =
Ny Nr

= 97

I see no difficulty in admitting this ratio between the number of
free electrons in two metals wide apart from each other in the
thermo-electric series *).

1) The numbers of that § contain an error which, however, has no influence
on the agreement that should be established by them. The value of 3 p and that of

7

a
— deduced from the measurements of JAkGerR and Diessetnorst are not 38 and
e

47, but
3 p = 38 < 10°
and
.
a 2 10,
é

—

N
2) Let 2 be the mean value of log ie between the temperatures 7" and 7".

Then the equation (35) may be put in the form

2
Le qe (7" — 7’).

This may be expressed as follows: The work done by the electromotive force

9
in case one electron travels around the circuit is found if we multiply by 3”

the increase of the mean kinetic energy of a gaseous molecule, due to an elevation
of temperature from 7" to 7",

( 592 )

The question now arises whether it will be possible to explain
all observations in the domain of thermo-electricity by means of
suitable assumptions concerning the number of free electrons. In order
to form an opinion on this point, I shall suppose the Prrier-effect
to be known, at one definite temperature 7’, for all combinations
of some standard metal with other metals and the THomson-effect to
have been measured in all metals at all temperatures. Then, after
having chosen arbitrarily the number .V, of free electrons in the
standard metal at 7’, we may deduce from (43) the corresponding
values for the other conductors, and the equation (44) combined with
(13) and (14), will serve to determine, for all metals, the value of
N at any temperature we like. Now, the numbers obtained in this
way, all of which contain N, as an indeterminate factor, will suffice
to account for all other thermo-electric phenomena, at least if we
take for granted that these phenomena obey the laws deduced from
thermodynamics. Indeed, these laws leading to the relation

Wy, + My, 11 + Min,1 = 9,

similar to (47), the values of N we have assumed will account not
only for the Pritier-effect at the temperature 7’, for all metals
combined with the standard metal, but also for the effect, at the
same temperature, for any combination. Finally, we see from (45)
that the value of W777 at any temperature may be found from that
corresponding to 7, if we know the THomson-effect for all inter-
mediate temperatures and from (46) that the values of the electro-
motive foree are determined by those of J.

There is but one difficulty that might arise in this comparison of
theory with experimental results; it might be that the assumptions
we should have to make concerning the numbers N would prove
incompatible with theoretical considerations of one kind or another
about the causes which determine the number of free electrons.

As to the conductivities for heat and electricity, it would always
be possible to obtain the right values from (24) and (27), provided
only we make appropriate assumptions concerning the length / of
the free path between two encounters *).

It must be noticed, however, that, whatever be the value of this
length, the foregoing theory requires that the ratio = shall be the

1) If the electric conductivity were inversely proportional to the absolute tempe-
rature, as il is approximately for some metals, and if we might neglect the varia-
tions of N, the formula (24) would require that 7 is versely proportional to
v7. 1 am unable to explain why V should vary in this way.

(593 )

same for all metals. The rather large deviations from this law have
led Drupe to assume more than one kind of free electrons, an hypo-
thesis we shall have to discuss in a sequel to this paper. For the
moment I shall only observe that one reason for admitting the
existence not only of negative but also of positive free electrons
lies in the fact that the Hani-effect has not in all metals the same
direction.

(March 22, 1905).

-~ 4

i a

hs ieee 4

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday March 25, 1905.

OC -

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 25 Maart 1905, DI. XIII).

(GS) any ae aan aes

Rh. Menmike: “On moments of inertia and moments of an arbitrary order in spaces of arbi-
trary high rank”. (Communicated by Prof. P. H. Scmoure), p. 596.

S. J. pe Lance: “On the branchings of the nerve—cells in repose and after fatigue”. (Com-
municated by Prof. C. WINkLEs), p. 599. (With one plate).

J. A. C. Ouprmans: “A short account of the determination of the longitude of St. Denis (Island
of Réunion) executed in 1874”, p. 602.

W. Huiskame: “On the presence of fibringlobulin in fibrinogensolutions”. (Communicated
by Prof. C. A. PEKELHARING), p. 610.

J. D. van pen Waatrs: “The transformation of a branch plait into a main plait and vice
versa”, p. 621. (With one plate).

JAN DE Vrigs: “A group of algebraic complexes of rays”, p. 627.

Jan DE Vrigs: “On nets of algebraic plane curves”, p. 631.

Errata, p. 633.

The following papers were read:

Mathematics. — “On moments of inertia and moments of an
arbitrary order in spaces of arbitrary high rank.’ By Prof.
Dr. R. MreuMke at Stuttgart. (Communicated by Prof. P. H.
SCHOUTE).

In the ‘“Mathematische Annalen” Vol. 23 (1884) pages 143—151
I have pointed out a manner of calculating the moments of inertia,
leading easily and quickly to the purpose and being independent
of the number of dimensions. As an instance I chose the case of
a figure filled with a homogeneous matter in the space of (n—1)
dimensions, analogous to the tetrahedron, thus according to the well-
known expression of Mr. Scnoure a simplex S,. Without being
acquainted with this Mr. Scrourr has lately treated this case in

41
Proceedings Royal Acad. Amsterdam. Vol. VII.

(596 )

another way in the ‘“Rendiconti del Cirecolo Matematico di Palermo”,
Vol. XIX (1905) and has arrived at the same result. Instead of
contenting myself with the reference to these facts | wish to com-
municate how in the same way moments of any higher order than
the second can be found. It is true this problem has been prepared
in the above mentioned place, pages 146—147, for a simplex so far,
that but a slight step would have been necessary to bring about its
solution.

Let us presuppose a flat space of (7 — 1) dimensions, a space of
“rank”? (“Stufe’) 2 as GRASSMANN expressed it as early as 1844, or of
“point-value” 2 as Prof. Scnoutr has said in his excellent textbook
on polydimensional geometry. The moment J/, of order v of an
arbitrary material figure belonging to this space with respect to a
space / of rank (n—1) (thus n—2 dimensions) contained in the

same space is
Ve uf r’ dm,

where r indicates the distance of a central point p in an element
of that figure from £, dm the mass of the element. According to
GRASSMANN however

r= [Ep],
i.e. equal to the “outer” product of /£ and p, when we assign
both to # and p the numerical value 1, consequently

M, = { Eph a . emer

[ assume that pv is a positive integer. If » is an even number
and if the moment is to be calculated with respect to a space A of
a rank smaller than (mn — 1), if thus it is e.g. a case of a moment
of inertia with respect to an axis (»=2), then according to GRASSMANN

7 = [Ap| Ap],
where the symbol | denotes the “inner” multiplication, and we

arrive at

v

2
M, = { [Ap | Ap] dn. aes) 5) he nD

The integrals appearing in a) and 6) can be evaluated by one
and the same integration, if we make use of the very useful notion
of the ‘gap-expressions’” introduced by GrassMaNnn. If namely we

place the point p appearing in [Zp|’ or in [Ap | Ap|? symbolically

( 597 )
outside the brackets and if with PEANo we indicate every gap thus
formed by +, we arrive at

y
m— [help resp. 7° = [As|Ae]? . p’

or

ah
where the expression / (furnished with » gaps) is equal to [ H«|’

y

in the first case and to [As|A*|? in the second. The expression ZL
remaining constant in the integration it can be placed before the

symbol fof the integral, so that we get

M, = 1. fp dm = Lp? + see wae ice 1a! * CL)

This has reduced our problem to the determination of the ‘“point-
quantity of order v”

file | CR) OR gs Bene? BN Be Ne)

belonging to the given material figure. (The v power of a point
p we have to imagine as the v-fold point p. The algebraic product
of » different points is the total of these points, where on account
of the interchangeability of the factors of an algebraic product the
order of succession of the points is arbitrary. The sum of an arbitrary
number of such like quantities has primarily but a formal meaning,
but then it may be represented geonietrically by a figure of order
y, the analogon of the ellipsoid of inertia). The integral 2) is depen-
dent only on the form and the distribution of the mass of the given
material figure, and whilst when treating our problem in the usual
way with the aid of cartesian coordinates the space / or A may
have a very disturbing influence upon the integration this influence
is here entirely done away with. Various other problems lead to a
similar integral as 2). If inter alia we wish to calculate the kinetic
energy 7’ of an (invariable or affinitely variable) continuously moving
system of masses for an arbitrary epoch, then

2 T= { v%dm,
where v denotes the velocity of a central point p in the element
7 s ”» . d,
dm. Bat v* is equal to the “inner” square of the vector repre-

senting the velocity of p according to length and direction, ice.
41*

(598 )

1 &

7 ata :

and when the symbol 4 denotes a certain affinity the momentaneous
system of velocities of the system of masses is indicated by

dp

— = Yip

dt Ps
hence we have

T x Lp),

L= [Use | Ae], pO | ptdm.

The evaluation of the static sum of the forces of inertia of an
arbitrary order called forth during the motion of the system of mass
at any epoch and the evaluation of the energies of higher species
inter alia considered by J. Somorr also lead to the integral p®.

It does not raise the slightest difficulty to find the integral p™ for
a simplex of constant denseness with the vertices a,, a,, .
We can put

ean

pasA,a,t+4a,+... +’aq,
where all points inside the simplex are obtained, when to the
numerical quantities 2,, 4,, --, 4, are given all positive values
compatible with the condition

> iy aes er ae Mee
If we think the simplex broken up into elements of the shape of
the parallelotop, i. e. of the (2—1) dimensional analogon to the
parallelepiped of our space, and with edges parallel to the edges of
the simplex starting from a,, then a slight calculation to be found
(l.c.) on page 147 gives us
dm = (n—1)! M da, da,...dhy,

where .J/ indicates the mass of the entire simplex. Hence we find
p”) = (n—1)! M | (a, a, + 4,0, +... + Anan) dd, day «ss dy.
The polynomial theorem gives

(Q).4)-deaeee eat ana,’ = SS

with

v!
: % ¥, AQ % ¥,
Ss Aya, ae eA” A, 1G," An®
1/0, !.--Pni

Bay Dare > Datel Oy oe My Dot Dee. ae lee
On the other hand we find according to a wellknown theorem
of LiovuvitLe under the above conditions for 4,, 4,,...an:

aya Ann da, a 12 v/v)... Dy!
TieAn SL = Ar! Aa Aus ovilAy = —— =
ae are atc meee ves. ee, one

v,!v,! v,!
(» + n—1)!
hence
n
v! (n—1)!
pr) — eR OTE, M A, A,% «2» Ayn eS a(S)
f (» + n—1)! wana
4,4 i
with

rp,t+ry+...+rm=?.

The sum to the right could evidently be arrived at out of

(a, +a,+...-+a,)’ by developing it according to the polynomial
theorem and by suppressing all the polynomial coefficients. The factor
v! _v! (n—1)!

(v + n- n—1l)!

is nothing else but the reciprocal of the number of terms. By introducing
in 1) the obtained value of p”, we find, when the distance of the
vertex a; of the simplex from the space /# or A is indicated by ¥;,
eee Set ET yarns clap
ri (v + n—1)! wy: bee to
(»y, +r, +...-+- m= ?).

For »=2 I have deduced (I. c.) the sum in 3) to asum of (7+1)
respectively m squares, in other words I have substituted for the
simplex a system of (z+ 1) resp. 7 single material points, which
is equivalent to it with respect to all questions connected with the
moments of inertia. For » > 2 a similar reduction seems to be less
easily effectible.

Stuttgart, March 1905.

Physiology. — “On the Hee of the nerve-cells in repose and
after fatigue.” By Dr. 8. J. pe Lanen. (Communicated by
Prof. C. Winker.)

In the laboratory of Marrntas DuvaL some experiments have been
made by MANovukLIAN in order to ascertain whether it is possible
to demonstrate modifications in the dendrites of the ganglion-cells in
cases of sleep through fatigue. His results have been published ji
the “Comptes Rendus de la Société de Biologie, 28 Févr. 1898” and
subsequently.

The animals he made use of for his experiments were mice, and
he proceeded in the following manner: For the space of an hour

( 600 )

together a mouse was driven to and fro in a cage, without granting
it any rest; after that the exhausted animal fell asleep or at any
rate remained perfectly quiet. The control-animal was kept in perfect
repose. Both animals were then killed, and small pieces of the
brain were immediately fixed after the method of Goer He
obtained manifest results already when only feebly magnifying: the
collaterals of the dendrites have vanished, instead of these the
dendrites have globular tumefactions, retracted branchings which
seem to have loosened themselves from the neighbouring end-arbor-
isations.

MANOUELIAN writes :

“On pense, en présence de ces images, a celle d’une sangsue vue
comparativement dans l'état d’elongation et dans l’état de rétraction
en boule.”

Previous to these experiments, Rasi-RickHarpt had published a
theory on the amoeboid motion in the cells of the central nerve-
system, a theory not founded however on microscopical data. (Neurolog.
Centralblatt 1890, p. 199). The investigations of Wirprrsnem who
experimented on a living Crustacea, Leptodora hyalina, and those
of Prrerens and others on the retina of Leuciscus rutilus, seemed to
confirm the conjectures of Rasi-Ricknarp.

Wiepersuem has been able to follow the motion of the processes
of the nerve-cells with the microscope and arrives equally at the
conclusion: ‘“dasz die centrale Nervensubstanz nicht in starre For-
men gebannt, sondern dasz sie activer Bewegungen fahig ist.”

J. Demoor injected dogs with lethal dozes of morphia, and studied
a small piece of the cortex cerebri, which he extirpated before the
death of the animal. He too, and likewise Srrranowska, after injecting
mice with ether, found similar changes as observed by MANovxgLIAN:
the branchings having become smaller and shaped like a string
of beads.

Two american authors however, Frank and Wein, did not obtain
these results on animals under narcosis.

In order to obtain some certitude whether any differences might
in reality be observed, | tried a few experiments in the laboratory
of Professor WINKLER.

Firstly I did repeat the experiments of Sreranowska and Dermoor,
albeit the methods employed were not in every respect the same
as theirs.

The mice were brought under narcosis by means of chloroform
instead of ether: immediately after death they were decapitated, the
head was caught into a liquid, prepared after the method of Gouer

: |
r ra
Vie: Pay

S. J. DE LANGE. “On the branchings of the nerve-cells in repose
and after fatigue.”

Nerve-cells of the cornu Ammonis from a mouse, exhausted by incessantly
running in a turning cage for four hours together.

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 601 )

modified by Cox, whilst the brain was prepared directly in the liquid.
For the control another mouse, not having been put under narcosis,
was treated in the same manner. No differences whatever were to be
observed in the microscopical preparations, obtained by means of the
freezing-microtome.

Neither did I observe any differences in the case of mice, injected
in the manner used by Demoor with repeated doses of morphia until
death ensued.

- Thinking these results might have been impaired by the fact that
the animals were decapitated only after death, | next tried with the
utmost accuracy a repetition of the experiments of MANOUELIAN.

A mouse was put into a turning cage, being therefore constrained
to run incessantly, whilst the cage was kept in continual motion by
means of a small motor driven by water. The motion was continued
for four hours together, the animal experimented upon being therefore
perfectly exhausted. Meanwhile the control animal had been kept in
darkness, enveloped in wadding. The four hours having elapsed,
both animals were very quickly decapitated, the heads being caught
into the fixation-liquid, and the brain being further prepared in it.

After ten weeks the preparations were impregnated with celloidine
and section-series in frontal direction were made of both brains. In
this way it became possible to obtain a comparable material.

For further control another pair of mice was sacrificed, for the
purpose of demonstrating by means of the method of Nissi the
presence of the well-known modifications in the easily tinctured parts
of the protoplasma of the nerve-cells.

For whilst under normal conditions the elective tincturing part of
the protoplasma of the ganglion-cells is divided into small granula,
in case of fatigue these granula tend to dissolving more and more,
the tincturing of the cellular body thus becoming homogeneous.

These modifications are clearly to be observed in the ganglion-cells
of the exhausted animal experimented upon: the fatigue therefore
must have been exquisite.

The preparations, made after the method of Gonet modified by
Cox, offer however beautiful arborisations as well in the case of the
non-fatigued animal, as in that of the exhausted one used for the
experiment, the annexed photographical reproduction of the exhausted
animal presenting no trace of retracted branchings, or of globular
tumefactions, neither of being shaped like a string of beads.

I have therefore not succeeded in demonstrating after this method
modifications in the branching system of the nerve-cells of the cortex
cerebi, caused by intense fatigue.

( 602 )

Astronomy. — Prof. J. A. C. Oupemans presents as a first communi-
cation on his journey to Réunion for observing the transit of
Venus: “A short account of the determination of the longitude

of St. Denis, Island of Réunion), executed in 1874.”

In our ordinary meeting of October 30, 1875 I communicated
a few details on the state of the computation of the observations
at St. Denis on the transit of Venus of December 9, 1874. The
purport substantially was, that the computations had been carried
out as far as was possible at that moment.

Several circumstances, independent of my will, were the cause that
this state of things remained the same till the middle of last year,
and that the computations could not earlier be taken in hand again.

What I communicated then has been inserted in the Proceedings
of that meeting. Passing by all that refers to the heliometer
measures, which I hope to take up at some later time, I will only
mention the fact that the necessity was pointed out of determining
with precision the longitude of the place of observation.

For this purpose we, viz. Mr. Ernst vaN DE SANDE BAKHUYZEN,
Mr. Sorrers and myself, have observed a number of oceultations,
not so much of the brighter stars, announced in the Nautical Alma-
nac, as rather of fainter stars, of the 8" or 9" magnitude, the posi-
tions of which were not yet known with precision at that time.
These had to be determined therefore by meridian observations ; our
honoured president readily undertook the task of having these deter-
minations made at the observatory under his direction.

As a rule at least four determinations have been made of each star.
Though the added epochs show that this was done between the
years 1879 and 1884, it lasted a considerable time, till November
1901, before the reductions of those determinations had proceeded so
far that the results could be communicated to me.

At the same time my attention was cailed to the fact that most
of these stars had been since also observed at other observatories.

It thus beeame necessary, in my opinion, to look for all these
determinations in the several Annals and to reduce them to the same
epoch, (of course 1874), in order to make allowance for proper
motion, wherever necessary. In many cases it proved sufficient to
retain the Leiden determination unchanged.

But besides, the errors of the lunar tables, that is to say of
the positions published in the Nautical Almanac, had to be derived
from observations. For this purpose the observations at the meridian-

( 603 )

circles of Paris, Greenwich and Washington and those at the Alta-
zimuth of Greenwich have been used.

It is true that, in a remarkable paper, /nvestigation of corrections
fo Hansnn’s Tables of the moon, with tables for their application,
Newcoms brought together the corrections to be applied to the for-
mulae by means of which Hansen caleulated his tables of the moon,
The paper contains the terms which had to be added according to
the state of science in that year, and also an empirical correction
determined by the most recent observations.

Moreover a table of corrections for 1874 was given, founded on
these data. But after having made a diagram representing, both the
corrections found by direct observation and those furnished by
Nrwcomp’s table, I came to the conclusion that the former was to
be preferred *).

As for the longitude of St. Denis, I will remark, that it has been
determined by the French naval officer Gurmain in 1867 and 1868
by means of 13 culminations of the first and 12 of the second limb.
In the Connaissance des Temps of 1871 a short report of that deter-
mination is to be found. Though the 25 results there given, agree
tolerably well, this kind of determinations is always liable to the
drawback that the difference in the constant error, made in observing
the culmination of the moon’s limb and of the comparison stars,
enters into the result, about thirty times magnified. There is no fear
of such an injurious influence in a determination of longitude by
occultations *). If the voyage to reach the isle of Réunion did not
last so long, and if the Indian Ocean were not so wild and _bois-

1) The present state of science requires a correction of one of the tables of
Newcoms. He points out (page 9) that the parallactic equation of Hansen is founded
on the value 8”.916 of the solar parallax, whereas the value which he derived in
1867 from all the available materials is but 8".848, which is less by 0".068.
Further that later determinations require rather a diminution than an increase of
that number. At present 8.800 is generally adopted as being the most probable
value of the solar parallax, which is less than Newcoms’s value by 0.648. The
parallactie correction of Newcomp must therefore be increased 1,7 fold; in other
words: three terms have to be added, viz. :

+ 0".67 sin D + 0".05 sin (D—gq) — 0".09 sin (D + 9'),

where D represents the mean elongation of the moon from the sun, g the mean
anomaly of the moon and g’ that of the sun.

*) Newcomp says at the beginning of his paper above mentioned: *Determi-
“nations of longitude from moon occultations are found by experience to be
“subject to constant errors which it is difficult to determine and allow for. It
“was therefore a part of the policy of the American Commission to depend on

“occultations rather than upon culminations for the determinations of longitudes, ete.”

( 604 )

terous, these voyage would also present an occasion of determining
the longitude by transport of chronometers. Unfortunately the results
given by the different chronometers were so diverging as to be of no
value whatsoever.

The report above mentioned of GrrMain’s determination is accom-
panied by a plan showing his place of observation. We see from
this plan that west of the town the river St. Denis runs nearly in
a north-north-westerly direction towards the sea and that the place
of observation of Germain was still on the west of the river.

A brick pillar, on which stood his transit instrument in 1867
and 1868, was still extant during our stay in 1874.

The result of Gerrmar’s determination of longitude and latitude
was given by him as follows:

Longitude of the place of observation east of Paris 332™258,7

Reduction to the flag-staff, east of the Barachois (7. e.

of the little creek which protects the sloops in landing) + 1,07
Longitude of the flag-staff east of Paris, (sic.:) . . 3°32™268,8
Southern latitude of the place of observation deter-

mined by 4 northern and 3 southern stars . . . . 20°52! 2",0
Reduction to thé flag-stafl eo 2) 2 Ge 2) — 23 ,7
Southern latitude of the flag-staff. . . . . . . 20°51'38",3

Our observations of occultations took place at different points, the
relative position of which was accurately determined by Mr. Sorrers.
Taking the difference of longitude of Paris and Greenwich
= 920863 from the Nautical Almanac of 1874, (as given at that
time both in the C. d. T. and in the N. A.), we got from the num-
bers just mentioned, for the flag-staff 3'41™47,43 east of Greenwich.
Corresponding therewith :

Place of observation : Long. E.of Gr. —_ Latitude
1st on the ground of the harbour office 3'41™47s,382 — 20°51'40",6
Dade ane Es ,, our dwelling house,

NGS Olen COnsely aan e 48 11 46 1
3'¢ Near or in the pavilion of the helio-
meter on the battery . . . .. . 47 ,81 35 ,3

The calculation of the longitude from the oceultations has been
carried out on printed forms, arranged according to the method which
I developed in the Astronomische Nachrichten N°. 1763.

In this method the declination of the moon is taken from the

4

( 605 )

astronomical almanac, using an adopted longitude; the parallax is
then computed for that point of the moon’s limb, where the star
has disappeared and which therefore has the same right ascension
and declination as the star. We then have to add or to subtract two
terms to or from the right ascension of the star, to get that of the
moon’s centre, and finally we find from the almanac the Greenwich
time corresponding with that right ascension.

The longitude of the place of observation, then found, is the right
one, if it agrees with the adopted longitude. If it does not agree,
we have only to repeat a small part of the calculation with a
modified longitude of the place, to derive the true longitude from
the two differences.

This method corresponds with the method, which was customary in
the 18 century (which we find inter alia explained in the well known
treatise of BonnenBerGER : Anleitung zur geografischen Ortsbestimmung)
with this distinction that then the whole computation was carried
out in longitude and latitude, whereas we use the right ascension
and declination. Further, that for BoHNENBERGER ec. s. there is no ques-
tion of any second hypothesis.

I will readily grant that Brsset’s method of computing ecliptic
phenomena and thus also for the prediction of occultations and for
the calculation of the longitude from an observed occultation, is
justly considered to be the classic method. It is also the only
one explained in most of the textbooks. But it seemed to me that
the method indicated by myself is more expeditive and only in a
few cases inferior to that of Brssen in point of accuracy. The
drawback of this last method consists in the troublesome preparatory
calculations, which it requires. Any one may convince himself of
the truth of this statement by consulting the wellknown textbook of
Cuavuvenet: A manual of spherical and practical Astronomy, Phila-
delphia 1874, vol I, p. 5507).

The horizontal equatorial parallax of the moon could be derived
from the Nautical Almanac, without any correction. As for the appa-
rent semidiameter of the moon, I myself made a determination of this
quantity, based on an elaborate investigation in 1859, (vid. Vers/agen
en Mededeelingen der Natuurkundige Afdeeling, Vol. V1, p. 25 seqq.)

1) I have calculated a single example by this method ; the result differed only by
05,1 from that obtained by the other method; in the first however 57 logarithms
had to be taken out, against 37 in the latter. Thinking the matter over, however,
I believe that the method of Besset will probably admit of a modification by which
this difference will be materially diminished. I hope shortly to investigate this more
thoroughly.

( 606 )

which furnished 0.27264 for the proportion of the mean moon’s semidia-
meter and that of the earth’s equator (at least this is the result of the
occultations discussed). After mature consideration, however, I now
adopted the value 0.2725 hor. equ. parallax + 0"04. This leads
approximately to the same value as when we take the sine of the
moon’s apparent semidiameter =0.272525 of the sine of the equatorial
horizontal parallax.

This factor is the mean of those which were derived from occul-
tations during total eclipses of the moon by Lupwie Srruve in 1888
and by J. Prrers in 1895 (0.272535 and 0.272518). The Nautical
Almanac, which used both the semidiameter and the parallax as given
in the Tables of Hansen, gave a value larger by 1”4 to 1"6. This
difference has remained the same up to the present time.

About the observed occultations we may communicate the following
particulars. They were mostly observed by myself, partly with the
Fraunhofer telescope, (aperture 11 cm.) mounted on a stand, which
Mr. Stoop of Amsterdam had kindly lent to the commission for the
observation of the transit of Venus, partly with the telescope of the
heliometer (aperture 7'/, em.). At a later epoch, when the assistance
of Mr. Ernst BaknvuyzEN was not so constantly required, as in the
beginning, for the experiments of Dr. Kaiser with the photoheliograph,
he also took part in the observation of the occultations, as also
did Mr. Sorrers in one case.

Altogether 35 disappearances and 4 reappearances were observed ;
but 12 disappearances and 1 reappearance had to be rejected. There
thus remained 23 disappearances and 3 reappearances, that is al-
together 26 observations, which furnished useful results.

The reason of the rejection lay partly in the fact that, already in
recording the observation, the remark “uncertain” had been added,
an addition due to the faintness of the star as it approached the
moon’s limb, or to passing clouds.

For another part the correction of the longitude determined by
GerMAIN and adopted by myself, came: out so extravagantly large
that some mistake or other seemed probable. There seemed to be
reason to suspect that a wrong star had been taken for the occulted
one. In five of the cases I sueceeded to find out the right star by
means of star catalogues, but in four other cases all my endeavours
proved in vain. Ultimately there remained five cases in which
the correction to the adopted longitude was found so considerable
(— 215, 208, —283, 24s and + 335), that there was no escape
from the conclusion that either a mistake, however improbable in
itself, had been committed in writing down the time, or that the

ee ee ee

a es

( 607 )

Results for the longitude of St. Denis-Réunion, (flag-staff), obtained by
occultations, without making a difference between disappearances
and reappearances.

5 : cal | Az
yom | §| Sar Momest FEE\—cm.| ¢ [eas] « [ow
S | s|~ |Germain

Sept. 19 | 0. | Arg. Z. 223, No. 75 | D |p 42°96 | 0 70 | 44.58 | +3.18 | 7.08
» » | 0. | Cordoba 111.1589 | » |p| 46.64 | 0.74 | 44.91 | 47.56 | 37.29
» DO: » XVITI.124. D \D +8.21 | 0.60 | 44.93 | +9.12 | 49.90
» 22 | O. 33 Capricorni D |D) +4.00 | 0.29 | +0.29 | 41.92 | 1.07
» » | O. | Arg. Z. 255, No. 27 | D |D} —6.10 | 0.50 | —3.15 | —5.48 | 413.42
» 5 oy COP Dy nD? » 32! D |D| —1.54 | 0.03 | —0.97 | —0.62 | 0.24
» pe LO: >» » » » 34] D ||P} —41.54 | 0.89 | —1.34 | —0.59 |] 0.31
» De ROE » » » » 35 | D |D) —5.75 | 0.97 | —5.57 | —4.83 | 92.63
» 26 | O. 73 Piscium R |D| +3.41 | 0.91 | 42.83 | +4.03 | 14.78
October 2 | O. 53 Geminorum R |P| 44.97 | 0.28 | +0.36 | 49.19] 1.34
me L0: = 4. 99°5738"7 R |p| 44.39 | 4.00 | +4.39 | 45.31 | 98.90
» 46] B. | Arg. Z. 223,No. 47 | D |D| —3.91 | 1.00 | —3.91 | —2.99 | 8.94
ed) |e bs » D |D| +9.67 | 0.40 | 13.87 |4410.59 | 44.86
De 2 OSB.) > » » 49} D |D} —5.99 | 0.95 | —5.69 | —5.07 | 24.42
S. v/0.Bi x >» » 52] D |D| 43.84 | 0.515) 44.98 | 14.76 | 411.67
D>) 038: > > » 51] D |D| +4.65 | 0.49 | 42.28 | 45.57 | 15.20
my ¥, SB: a igeee D |p| —4.96 | 0.99 | —4.92 | —3.34 | 14.08
Disne dese Gould 24851 D |D\ +5.84 | 0.87 | 45.08 | 46.76 | 39.76
» 47/0. = ae p |p| +5.39 | 0.49 | 44.03 | 46.31 | 7.57
» » | O. | Arg. Z. 244,No.9 | D |D| —5.140 | 0.58 | —2.96 | —4.48 | 10.43
» » | O.} » » 234, » 12 | D |D} 44.45 | 0.35 | 40.40 | 49.07] 41.50
Ty a ea NO) DDD » 41 D '\D) —4.73 | 0.62 | —2.93 | —3.84 9.00
» 18) B. | » » 239, 103 | D |D| —5.40 | 0.95 | —5:07 | —4.18 | 16.60
» 19) B. | >» » 247, » 99 | D |D| —2.62 | 0.98 | —2.57 | —1.70 | 2.83
» » |B. | x Capricorni D |D| —4,22 | 0.97 | —4.09 | —3.30 | 10.56
>» » |B. (5 _Sresgnaa } | p |p| —8.95 | 0.9% | —8.44 | —8.03 | 60.61
18.30° |+-33.93 | 25 m? = 450.96

— 548) < ati, 48,04
—16.85 m= 4095

16.85 _ __ 05,99-+ 05.99,

at =— 79305

( 608 )

Results for the longitude of St. Denis-Réunion, reappearances
and disappearances separately.
The 3 reappearances give: ©G= 2.19 £G4L =-+ 7.58 therefore A L= + 3s.462

The total sum was: 18.30° —16.85
Therefore the disappearances —__
separately give : 16.115 —9iS , AL=—it oe
Mean : + 08.97
£ z= | G | G
Disapp, |
a | al N.B. As there is no reason to suppose that
‘ 14.28 dF 10.00 3
eg a fiassties ae a reappearance at the dark limb should be so
+8.16 66.59 0.74 49.28 much more accurate than a disappearance at
| the dark limb, I have combined them.
-++9.73 | 94.67 0.60 | 56.80
49.52) 6.35 | 0.99 | 4.86
—4.58 | 20.98 06.50 | 10.49
—0.02 0 0.63 0
-+-0.01 0 0.89 0
—4.23 | 17.89 0.97 | 17.35

—2.39 | 5.71 1.00 5.71

4-11.19 |125.22 | 0.40 | 50.09
—4.47 | 19.98 | 0.95 | 18 98

929 m?= 406.16
+5.36 | 98.73 | 0.515 | 14 80
+6.17 | 38.07 0.49 | 18.65 ne= 18.06 |

8s

A) AG 0.99 | 7.48 n= +4.34
47.36 | 54.17 | 0.87 | 47.43 (not used)
16/94 | 47.75 | 0.19 | 9.07

—3.21 | 40.30 | 0.62 | 6.39 Together:
24m? = 408 .47

Ar
= 4100) Ano 0.98 1.419 m pale
—2.70 | 7.29 | 0.97 7.07 m =+ 4.13
—7.43 | 55.20 | 0.94 | 51.89 ! uS t
——- =1.086 v=+1.03,

—0.35| 0.412 | 0.91 | 0.44

m
—2.19| 4.80 | 0.27 | 1.34 | 2m?=2.31 spp VS Ee
ee
0.93 | 0.86 1.00 0.86 m= 1.155 | 209 v=+1.49.

set
m= +1.08
(not used)

( 609 )

occultation had taken place at a point considerably elevated above
the rest of the limb. In the following lines we will only communicate
the results of those observations which have been retained.

We remark that the weights G, which have been added, were
taken equal to sin? yw, 2y being the arc, of which the star would
describe the chord behind the disk of the moon, were this disk at rest;
(according to the notation of Cuauvenrr this would become cos* yp).
This quantity could be easily derived from the numbers occurring
in the computation.

The calculations have been all made in duplicate; the first by
myself, the other by Mr. Kress, amanuensis at the observatory of
Utrecht.

We thus find:

Taking disappearances and reappearances together :

Correction to GurMain’s longitude : — 03,92 + 0s,99 (m. err.)

Treating them separately : = ORO 2149 5, 5; )

We thus come to the conclusion that the occultations observed by
us leave undecided whether the longitude of St. Denis, according
to the determination of Germain in 1867 and 1868, must be increased
or diminished; in other words they confirm his result.

Only one of these days I noticed, that since 1886 the Connaissance
des Temps gives a longitude for that place, which is larger by 152
or 18"; in the last column of the table of the geographical positions

Gece ens tgs Newcomb. N. — Merid.
= 7 : 7 = 7
Sept. 19, | —0.52 | —4.3 0.300 | 0S 40.13 | +44.6
9917, | —0.51 | —1.9 =049)) ||) =974 +0.02 | —0.5
ae. —0.54 |) —9.4 —0.49 | —2.6 40.02 | —0.5
Seer c=0:73° } — 89 = 0210) 458 5P0 03" |), “HOM
Oca May 0.79" | 47 —0.95 | 40.6 0416 phot
Ml, | —0.75 0.0 007K il) cEg.9. —0.02 | +42.9
46, | —0.35 | —44 —0.47 | 40.4 —0.12 | +445
17, | —0.48 | —2.0 —0.46 | —0.4 G05) Ere
at | = Os850 |) 41.9 045 | —1.3 =(40), | 0.4
460) =o; 34) 1 — 95 —0.44 | —2.0 =00 4) 220.8

s
Mean: —0.03 +1.3

( 610 )

we find: GerMarn corr. 86; the reason for the correction is however
not stated. | have therefore written to Paris asking for information.

Moreover I will observe that the difference of longitude Paris—
Greenwich above used, must be increased according to the determination
executed by French and English observers in 1902. The result
obtained by the English observers was 9™20°,932 + 0°006; by the
French observers 9"20s,974 + 08008. Mean 9™205953. (Monthly
Notices of the R. A. 5. Jan. 1905).

Finally we subjoin a comparison of the corrections to the moon’s
ephemeris of the Naut. Alm. of 1874, furnished by the meridian
observations on the one hand, and by Newcoms’s formulae on the other.

It might be worth while to ascertain, whether the agreement of
the results is improved, if we adopt the corrections according to
NEWCOMB.

As for the meridian observations, some have been made at other
observatories (Leiden, Pulkowa ete.). I hope to investigate this more
closely ; it is not probable however that the result will be greatly altered.

A last remark in conclusion. According to the “Post en Telegraaf-
gids” the isle of Mauritius is already connected telegraphically with
Europe. There is reason therefore to expect that the same will
shortly be the case for Réunion also. In that case the “Bureau des
Longitudes’ will no doubt endeavour to obtain a telegraphically
determined longitude of St. Denis.

Utrecht: 1905 Mareh 24.

Physiology. — “On the presence of jibringlobulin in jibrinogen
solutions.’ By Dr. W. Huiskamp. (Communicated by Prof.
C. A. PEKELHARING).

After Hammarsten had proved that in fibrinogensolutions, which
had been coagulated either by heat to 55° or by means of fibrin-
ferment, a proteid, afterwards called fibringlobulin, appears which
coagulates at 64°, there existed several possibilities with regard to
the formation or appearance of this proteid.

Firstly the origimal fibrmogensolution might already have contained
the fibringlobulin as an admixture; in the second place it was possible
that at the heat-coagulation or by means of fibrinferment, the fibrin-
molecule was disintegrated, and that in such a way that an insoluble
substance, fibrin, is formed, along with a soluble one, fibringlobulin;
and lastly the fibringlobulin might perhaps be an altered fibrinogen,
which has remained in solution, a sort of soluble fibrin.

(611 )

Against the first of these possibilities HAMMARSTEN |

has raised
serious objections, and by his later researches he came more and
more to the conviction that fibringlobulin must be a somewhat
changed soluble fibrin.

A research of CanuGAREANU *) was the occasion for experiments to
be made in this direction. The author inter alia demonstrates that
natriumfluoride, in strong concentration, greatly increases the effect
quantities of fibrinferment. Canucarnanu prepared horseoxalateplasma,
which contained a quantity of fibrinferment so small, that the plasma
remained fluid for a considerable time.

If this plasma was mixed with natrium fluoride to a quantity of
about 8°/,, either by addition of a saturated solution of NaF or also
of finely powdered NaF, then there ensued an almost immediate
coagulation. That the formed precipitate really is fibrin, CALUGARBANU
derives from the fact, that it is like fibrin insoluble in diluted salt
solutions. Further Canucarnanu discovered that horseoxalateplasma,
if it was only perfectly free from ferment, did not coagulate by
addition even of several volumina 38°/, Na Fl. When therefore no
ferment is present the natriumfluoride remains inactive, from which
CaLuGAREANU concludes that the Na FI exercises its influence on the
fibrinferment but not on the fibrinogen.

When the experiments of CaLuGarnanu were repeated I obtained
results which partly differed from. his.

It namely appeared that perfectly fermentfree solutions containing
fibrinogen gave a precipitate with natriumfluoride; this precipitate
is in case horsefibrinogen is used gelatinous and in consequence
reminds one more or less of coagulation; if however oxenfibrinogen
or oxenbloodplasma is used, the precipitate is floeculent and does
therefore not, outwardly at least, resemble coagulation.

In the second place it appeared that the precipitate formed by
Na Fl could be easily dissolved, when treated properly, and that
these solutions coagulated with fibrinferment.

Some experements I will describe here in detail.

A rabbit was injected in the vena jugularis with 65 cem. leech-
extract, next the blood out of the Carotis was received in a centri-
fugalglass covered with paraffine and the corpuscles where centri-
fugalised off. Plasma in this way prepared contains no ferment as
PEKELHARING *) has demonstrated; the plasma, meant here, remained

1) Pfliigers Archiv., Bd. 22, p. 431.
*) Arch. internat. de Physiol. Vol. Il, p. 12.
5) Untersuchungen tiber das Fibrinferment. Verhand. Kon. Akad. van Wet.
Amsterdam 1892.
42

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 612 )

fluid for a number of days, as long as it was kept, yet by addition
of three times the volume of saturated natriumfluoridesolution a
flocculent precipitate was slowly formed; a fibrinogensolution prepared
from the plasma could also be precipitated by the addition of
saturated natrium fluoride solution; by saturating with solid natriam
fluoride a precipitate ensued immediately.

Other experiments were taken with horsefibrinogen. The fibrinogen
solutions used, which were prepared by three times precipitating
with salt from oxalateplasma showed even after being preserved
for several days, no trace of clotting; by addition of Ca Cl, no clotting
was caused either at 37° or at the temperature of the room. In such
a fibrinogen solution a thick precipitate is then immediately formed*) by
addition of the double volume of saturated natrium fluoride solution ;
this gelatinous precipitate can be easily wound round a glass rod and
in this state be taken out of the liquid for further research. The
precipitate washed by water showed the following properties. It did
not dissolve perceptibly at the temperature of the room in 3——5°/, salt,
more easily the solving succeeded in this way, at the temperature
of the body, or still better at 40—45°. On cooling, the precipitate
does not return. The surest way to obtain a complete solution is to
make use of *‘/,,°/, ammonia as solvent; if the precipitate is divided
with a glass rod, rather concentrated solutions can easily be prepared
in this way. Such a solution can after addition of salt, to a quantity
of 3—5°/, be neutralised without a precipitate forming anew (only
when the concentration of the solution was very great, a part of the
dissolved substance precipitated often again after some time; this preci-
pitate was solved however at 37°). Such a solution may be preci-
pitated again in the same way, with the double volume saturated
natrium fluoride solution and may be dissolved in */,,°,,, ammonia. Such
neutral solutions containing 38—5‘/, salt and prepared by being once
or twice precipitated with Na Fl possessed all the properties of fibri-
nogen; by addition of an equal volume saturated salt solution, a
ereat precipitate was formed; acetic acid caused a precipitate soluble
in excess; the coagulation temperature was at 54°; the solutions
coagulated quickly and completely with fibrinferment for which I
mention the following experiments as example.

5 cem. fibrinogen solution of 0.842°/,-+-1 cem. fibrinferment solution ;
the coagulation begins at 37° after half an hour; the tube further
coagulates completely.

5 cem. of the same fibrinogen solution ++ 5 drops of oxenblood-

1) The mixture contains then not much more than 3 °/) Na Fl.

( 613 )

serum, the coagulation begins (at 37°) after ten minutes; after an
hour a solid clot. was formed.

Placed at 37° a tube with 5 ecem. of the same solution, without
ferment for control, remained perfectly fluid.

The above mentioned experiments were now repeated with horse-
oxalateplasma, which was not perfectly free from ferment, as was
obvious from the partial clotting of the received blood; the results
were in general the same; the precipitate obtained with Na Fl dis-
solved only with somewhat more difficulty; the solution of this
precipitate meanwhile possessed the properties ofa tibrinogen solution
and coagulated with fibrinferment.

Further experiments were taken with fermentfree oxenfibrinogen
prepared after the method of Hammarstren. It was stated that to
precipitate this fibrinogen with Nall more natrium fluoride solution
was needed than for horsefibrinogen. The flocculent precipitate obtained
with NaFl dissolved at 37° more easily in a diluted salt solution
than the horsefibrinogen precipitated with NakFl; on the contrary less
easily in ‘/,,°/, ammonia; rather great quantities dissolved already at the
temperature of the room in 38——5°/, NaCl. The coagulation temperature
of the neutral solution, containing about 3°/, salt was at 538—54°
addition of acetic acid caused a precipitate which dissolved in excess ;
by half saturating with NaCl the fibrinogen could be precipitated.
That the solution coagulates with fibrinferment appears from the
following experiment.

5 eem. of the solution in 3°/, NaCl + 5 drops of oxenbloodserum.
Complete clotting after two hours.

Although it might seem after the above mentioned experiments
that the fibrinogen remains unaltered on being precipitated with
Na Fl, a closer inquiry brings to light a remarkable alteration. If
namely a solution of fibrinogen precipitated with Na F] is heated to
55—58", very little fibringlobulin is found in the liquid filtered off
from the coagulum; if the fibrinogen is precipitated twice with
uatrium fluoride, no or only few traces of fibringlobulin can be
obtained from the solution as appears from the following experiments.

I. A solution of fibrinogen prepared after the method of Ham-
MARSTEN was partly precipitated twice with Na Fl; the last precipitate
was dissolved in '/,,°/, ammonia and the solution was neutralised
after addition of salt; 8 cem. of this solution, which contained
0.445°/, fibrinogen were heated for five minutes to 55—60°, then
it was filtered; the clear filtrate was heated to 72°, by which only
a small opalescence ensued, which did not increase perceptibly after
the liquid had been made slightly acid and afterwards boiled.

42°

( 614 )

For comparison the fibrinogen from 8 ccm. of that part of the
fibrinogen solution which had not been prepared with NaFl, was in
the same way first coagulated and afterwards the fibringlobulin in
the filtrate; although the fibrinogen solution used for this experiment
contained ©0.565°/, fibrinogen and so had been but little more con-
centrated than the solution prepared with Na Fl the quantity fibrin-
globulin found was remarkably larger, as there was formed abundance
of floceulent precipitate by heating to 70°.

Another experiment gave the following results.

II. The solution of the fibrinogen not precipitated with Na Fl con-
tained 0,634°/, fibrinogen, that of the fibrinogen precipitated with
Na Fl 0,452°/,. after this the fibrinogen being removed from the two
solutions by heating to 55—58° and by filtering off of the coagulum,
5 cem. of each of the filtrates were mixed with 1'/, cem. of a
saturated solution of picric acid. In the filtrate of the fibrinogen treated
with Na Fl there was formed only an opalescence which after some
time passed into a very slight precipitate; in the filtrate of the fibri-
nogen not precipitated with NaFl there was immediately a con-
siderable flocculent precipitate.

Ill. A solution of oxenfibrinogen was precipitated by four times
the volume of a saturated natrium fluoride solution; afterwards the
precipitate was centrifugalised off, washed with water and dissolved in
4°/, salt; the solution contained 0,232°/, fibrinogen; after its having
been removed by heating to 55—58° and filtering off of the coagulum
the filtrate remained perfectly clear on being boiled, and so contained
no fibringlobulin, although the original fibrinogensolution had been
precipitated with Na Fl only once.

So it appears that by means of natrium fluoride fibrinogensolutions
may be obtained, which by heating produce no fibringlobulin or

only traces of it.

This confirms the opinion that the fibringlobulin was present
already in the original, not heated fibrinogensolution either combined
with fibrinogen or simply as admixture, and that consequently fibrin-
elobulin is not formed by alteration of the fibrinogen during the
heating ; in the last case it could not be explained why the fibrinogen,
prepared with Na FI should not be altered as well in the same degree
by heating in fibringlobulin. If however the fibringlobulin was present
already in the fibrinogensolution, every thing may be explained in
this way that on being treated with Na Fl the fibringlobulin passes
into the filtrate at any rate for the greater part, while the fibrinogen
proper precipitates. The possibility that the fibringlobulin does not
precipitate also appears from the following experiment. In a solution

of horsefibrinogen prepared after the method of HamMarsren the
fibrinogen was coagulated by heating to 60° and filtered off; to the
filtrate was added a double volume of saturated natrium fluoride
solution; the liquid remained perfectly clear.

The question whether the fibringlobulin passes into the filtrate
when the fibrinogen is precipitated with Na Fl cannot be answered
immediately by examining the filtrate, while the fibrinogen with
Na Fl does not precipitate completely, so a certain quantity of fibri-
nogen exists still in the filtrate, and when, after heating, fibringlobulin
is still found, the possibility exists, that all this fibringlobulin proceeds
from the quantity of fibrinogen present in the filtrate; only the quan-
titative research can decide here; if on precipitating with Na Fl the
fibringlobulin passes into the filtrate it must be possible to prepare
from this filtrate nearly as much fibringlobulin as from the original
fibrinogensolution. As the fibrinogen precipitated with Na FI is not
perfectly free from fibringlobulin, an accurate agreement is not to be
expected. In the first place I subjoin the results of such an experiment.

a) 100 cem. of a pure horsefibrinogensolution, prepared after
HAMMaksTEN’s method were precipitated with 200 ccm. saturated
natrium fiuoride solution. The precipitate was taken with a glass rod
out of the liquid, pressed out firmly, dried to constant weight and
weighed, the substance was burnt carefully, the weight of the ashfree
substance proved to be 0,2485 gram. After the precipitate obtained
with NaFl had been removed a clear liquid remained, which was
neutralised with some drops of diluted acetic acid, as the reaction
of the solution of NaF] used was faintly alkaline, which mostly is the
case. The liquid (285 cem.) was heated afterwards for a quarter of
an hour in a waterbath to 55—60°; the coagulated fibrinogen was
filtered off on a weighed, ashfree filter, with a diluted saltsolution
and after that washed with water, dried to constant weight and
weighed together with the filter; the filter and the substance was
carefully burnt. It proved, that 0.2262 gram ashfree fibrinogen had
been present on the filter; this quantity was obtained from 285 cem.;
so in the original 800 cem. there would have been found 0.2381 gram.

In order to determine the quantity of fibringlobulin 250 cem. liquid
filtered off of the coagulated fibrinogen were heated during a quarter
of an hour to 67—69° in a waterbath. The liquid remained perfectly
clear till 64°; to obtain a coagulation as perfect as possible 5 cem.
1°/, of a sulphas cupri solution were added as soon as the liquid
became turbid; by this the coagulum became roughly flocculent and
could easily be filtered off. The weight of the filtered fibringlobulin
was afterwards determined in the same way as was done with the

( 616 )

coagulated fibrinogen and amounted to 0.1141 gram ashfree sub-
stance; so from 3800 ccm. filtrate would have been gained 0.1369
gram. In the liquid filtered off from the fibringlobulin no proteid
could be demonstrated.

b). For comparison it was determined how much fibringlobulin
the fibrinogen solution used gave without treatment with Na FI.
Therefore 100 cem. of this solution was again mixed with 200 cem.
+°/,°/, salt through which volume and salt quantity in this experiment
was made equal with that of experiment a). By heating for a quarter
of an hour to 55— 60° the fibrinogen was coagulated and was
treated further as mentioned above; the weight of the fibrinogen
amounted to 0,4548 gram ashfree substance. 250 cem. of the liquid
filtered off from the coagulated fibrinogen were heated for a quarter
of an hour to 67-—69 ; the liquid remained perfectly clear till 64°,
just as in experiment a); here also 5 cem. 1°/, CuSO, were added
as soon as the first turbidness became visible.

The coagulated fibringlobulin was filtered off and treated as men-
tioned above; the weight of the fibringlobulin amounted to 0,1354
gram; in the liquid filtered off from the coagulated fibringlobulin no
proteid could be shown.

Taking together the results of these experiments we find, that in
experiment a) after the removal of the precipitate obtained with
Na Fl, 0,2381 gram fibrinogen and 0,1369 gram fibringlobulin were
present; and in experiment 4) 0,4548 gram fibrinogen and 0,1625

gram fibringlobulin. So the quantity of fibrinogen was in experiment
a) reduced to 100 through precipitation with Na Fl while the quantity

~

ae Consequently there
must have passed a considerable quantity of fibringlobulin into
the filtrate after precipitation with Na Fl. The difference of 0,0256
gram between the quantities of fibringlobulin, found in both experi-
ments, must be attributed, apart from any errors of determination
to the fact that the fibrinogen which was not precipitated with Na Fl
is not perfectly free from fibringlobulin; the weight of this precipitate
amounted to 0,24385 gram; if we abstract from this 0,0256 gram as
being fibringlobulin, this precipitate contained to 100 mg. at 55°
coagulable fibrinogen 11,7 mg. fibringlobulin; in experiment 6) 35,7
mg. fibringlobulin was found to 100 mg. at 55° coagulable fibrinogen
and in experiment a) after removal of the precipitate obtained with
Na FI] 57,5 mg. fibringlobulin to 100 mg. at 55° coagulable fibrinogen.
By precipitating with Na Fl the fibrinogensolution was consequently

of fibringlobulin only showed reduction to

( 617 )

divided into a precipitate, which contained relatively little and a
filtrate which contained relatively much fibringlobulin.

In experiment a) about half of the fibrinogen was precipitated
with NaFI; the liquid poured off from this precipitate was clear;
if however such a liquid is left standing for some time it becomes
turbid and a new precipitate has formed itself after 24 hours, in
the filtrate of this precipitate a new turbidness forms again ete.,
till at last after some days all the fibrinogen has precipitated. — It
may me expected after the above mentioned experiments, that, as
more fibrinogen precipitates, relatively (that is to say with regard
to the quantity of fibrinogen which was not precipitated) more
fibringlobulin will be present in the filtrate; this supposition is
confirmed by the two following experiments.

1. 100 cem. horsefibrinogensolution of 0.642 °/, were precipitated
with 200 cem. saturated natrium fluoride solution ; the precipitate was
removed with a glass rod and the liquid remained standing after
that twice 24 hours. When the fibrinogen, precipitated after that
time, also was removed by filtering, the quantity of fibrinogen and
fibringlobulin in 250 cem. filtrate was determined in the same way
as in the above mentioned experiment @). It appeared that in these
250 cem. 0.0742 gram fibrinogen and 0.1113 gram fibringlobulin
were present.

2. 100 ccm. of the same fibrinogensolution were precipitated with
200 cem. saturated natrium fluoride solution; after removal of the pre-
cipitate the liquid remained standing for eight days; putridity did
not occur from this on account of the quantity of Na Fl, the new
formed precipitate was filtered off, the filtrate became again turbid
and after 24 hours a slight precipitate had again formed, that was
filtered off. The filtrate was neutralised with a few drops of diluted
acetic acid; by heating of the neutral liquid to 55—60° there followed
only an exceedingly slight opalescence; the fibrinogen was therefore
precipitated almost completely by the Na Fl; when the opaline liquid
was filtered a considerable flocculent precipitate was formed by
heating the filtrate to 67—69°.

While in experiment a) after the removal of the precipitate obtained
with NaFl still 1°/, times more fibrinogen than fibringlobulin was
present in the filtrate, the analogous filtrate in experiment 1) con-
tained only *, times as much fibrinogen as fibringlobulin, while in
experiment 2) with a considerable quantity of fibringlobulin only
a small quantity of fibrinogen was present.

The results of the above described experiments lead to the con-
clusion that at the coagulation of the fibrinogen, the fibringlobulin

( 618 )

does not proceed from the fibrinogen, but that this proteid was
already present in the fibrinogensolution, for it could not be explained,
that on one hand, the fibrinogen precipitated with Na Fi produces
no or but little fibringlobulin, and that on the other hand the liquid
filtered off from this precipitate contains fibringlobulin in such greater
quantity.

It here is necessary to discuss still a few objections that might be
raised against this conclusion.

Firstly —— on account of the fact that the fibrinogen precipitated
with Nak l, dissolves with more difficulty in diluted saltsolution than
the usual fibrinogen, and that the solution does not produce any
fibringlobulin by heating — it might be asked whether the substance
precipitated with Nakl might not be a kind of soluble fibrin, as for
instance the “‘fibrine concrete pure” described by Denis. The latter
also principally dissolved in’ diluted saltsolution at 40°; while the
dissolution went very slowly at the temperature of the room. Against
the opinion that the substance precipitated with NaFl is a soluble
fibrin speaks first the coagulation temperature which was found by
Denis for the dissolved ‘“fibrine concrete pure’ at 60—65°, while in
every case if is not higher than 55° for the substance precipitated
with Nal}. The strongest argument against the opinion that this last
is fibrin, namely the power of this substance to clot with fibrinferment,
I have already stated several times; if we further take into consider-
ation that the fibrinogen prepared with Nall behaves with respect
to acetic acid, half saturation with salt ete. quite as common fibri-
nogen, the opinion that this substance is fibrin may be considered
as having been refuted.

As to the slight solubility of the fibrinogen precipitated with Nakl
and in diluted salt solution, this peculiarity may be explained in
this way, that on being heated with NaFl it forms a slight soluble
fluorine-compound of the fibrinogen, which dissolves only very slowly
in saltsolution; by the great abundance of chlorine-ions then present,
this dissolving will probably be accompanied by an exchange of the
fluorine by chlorine. It is still rendered more probable that a fluorine-
compound is formed, when we consider that the slight quantity of the
natrium fluoride solution cannot be put on one line with the precipitating
of proteid by the saturating of the solution with a neutral salt.

It might be imagined that the tibrinogen, it is true, is precipitated
as such by natrium fluoride, but that also (especially as natrium fluoride
solutions usually react slightly alkaline) part of the fibrinogen is
changed into fibringlobulin; by which the presence of fibringlobulin
in the filtrate would be explained.

( 619 )

Apart from this that then it would not be explained why the
fibrinogen precipitated with Nall does not produce any fibringlo-
bulin by heating, it would have to be expected according to this
view that, if the fibrinogen were precipitated with Nall for the
second or third time also a part of it would be changed into fibrin-
globulin, which ought to be found in the filtrate. This however is
not the case; under these circumstances only very little or no fibrin-
globulin is found in the filtrate.

So, when it should be assumed, that the fibringlobulin is present
in the fibrinogen solutions beforehand already, the question remains,
whether this proteid is combined with the fibrinogen or must be
considered as a simple admixture.

For a compound plead some experiments of HAMMARSTEN '), in
which is demonstrated that from concentrated fibrinogen solutions
after heating to 56—60° and filtering off of the coagulum, relatively
less fibringlobulin is obtained than from the same solutions after
their having been diluted. If the fibringlobulin were only an admixture
it would be expected that the relation between the quantities of
fibrinogen and fibringlobulin would always be the same; on the
other hand, if the fibringlobulin is combined with the fibrinogen the
results of HamMarsteNn could be explained thus, that in diluted fibri-
nogen solutions the fibringlobulin is more easily disintegrated. To a
compound also points the fact, that when a fibrinogen solution is
precipitated for the first time with NaF] a not inconsiderable quantity
of fibringlobulin is precipitated also.

Against a compound speaks however that by precipitating with
Nak the fibringlobulin passes into the filtrate, at least for the greater
part, for it is difficult to believe, that addition of alkali salt, as
NaFl up to a quantity of about 3°/, would have for its result a
splitting off of fibringlobulin. The following observation may perhaps
give some light.

100cem. of horsefibrinogensolution were precipitated with the double
volume of saturated natrium fluoride solution; the solution of NaFl
used reacted almost neutral by exception ; with litmuspaper the alkaline
reaction was hardly perceptible. Part of this natrium fluoride solution
was now made weakly alkaline by addition of 0.8 cem. normal
sodium hydrat with 200 cem. of the natrium fluoride solution ; with
this 100 cem. of the same fibrinogen solution were precipitated twice
in the same way.

From the precipitates obtained with neutral and with alkaline
NaF! two fibrinogen solutions of equal concentration were prepared.

1) Loe. cit.

( 620 )

The fibrinogen was in both cases coagulated and filtered off by
heating to 55—60°. The filtrate of the fibrinogen prepared with slight
alkaline natrium fluoride solution gave a slight precipitate by heating
to 70° or by addition of picric acid, while the precipitate of fibrin-
globulin in the other filtrate was clearly greater, perhaps twice or
three times.

From this it would follow that the supposed splitting off of
fibringlobulin is not brought about by NaFl but by the alkaline
reaction of the natrium fluoride solutions ; for this disintegration however
exceedingly small quantities of alkali are already sufficient, for also
that fibrinogen solution which was prepared with almost neutral
Na Fl produced much less fibringlobulin than a fibrinogen solution
of the same concentration, not prepared with Na FJ. The supposition
that water also, in particular at a rising temperature could bring
about the splitting off of fibringlobulin is obvious; if this is the
‘ase there would be present in a fibrinogen solution a compound of
fibringlobulin with fibrinogen, which is disintegrated more or less
by hydrolysis and this idea is, as appears to me, most easily recon-
ciled with the facts. The disintegration will in this case with raised
temperature e.g. at 55—60’ be rather complete; from diluted solutions
relatively more fibringlobulin may however be obtained than from
concentrated solutions, because in the first case the disintegration
will be more complete owing to the greater excess of water. That
not all the fibringlobulin passes into the filtrate by the precipitation
with Na Fl, becomes clear if only a partly hydrolytic disintegration
is accepted.

If the fibringlobulin is mixed simply with the fibrinogen in conse-
quence of hydrolysis, be it then for a part only, it cannot be expected,
— with a view to this, that by half saturation with salt as is usual
with the preparation of fibrinogen, no complete precipitation of the
fibringlobulin takes place, — that in every fibrinogensolution the
relation between the quantities of fibrinogen and fibringlobulin will
be the same; this may perhaps lead to the explanation of some
observations of HamMarsteN ') from which it appeared that fibrinogen-
solutions prepared from different plasma produce, it is true, relatively
different quantities of fibringlobulin, that however a diluted solution
does not always produce relatively more fibringlobulin than a con-
centrated solution.

In conclusion [ will discuss some facts here, relating to clotting
by means of ferment.

1) loc. cit. p. 456,

( 621 )

Of the indentity of the fibringlobulin which is obtained by the
coagulation by ferment and that which is found in filtrate after the
heat-coagulation of the fibrinogen, there is no doubt, on account of
the conformity in composition, coagulation temperature ete. When
however it must be assumed that the fibringlobulin is already before-
hand present in the fibrinogensolutions, then for the present falls
away every ground to assume that by the clotting by ferment the
fibringlobulin should be formed still in another way e.g. by trans-
formation of fibrinogen, the more so, as the quantity of fibringlo-
bulin which is obtained by clotting with ferment certainly is com-
paratively not larger than that which can be prepared by heating
from a fibrinogensolution. Hammarsren ') found, it is true, that in
weak alkaline solutions relatively little fibrin was formed by ferment
and so relatively much proteid remained dissolved; this may partly
be explained by the fact that the fibringlobulin was disintregrated
very completely by the alkaline reaction, partly also, as HAMMARSTEN
himself observes, by the fact, that under these circumstances part of
the fibrin remained dissolved as “soluble fibrin”.

From the fact that a solution of fibrinogen, from which the fibrin-
globulin is removed by means of NaF, clots with fibrinferment, must
be deduced that by removal of the fibringlobulin the fibrinogen
proper is not, as might be expected from the formula given by
ScCHMIEDEBERG and defended a short time ago by Heusner *) changed
into fibrin, and that in general the fibringlobulin does not play a
considerable part in the clotting. So the clottingprocess must consist
in an alteration of the fibrinogenmolecule itself. That fibringlobulin
is present in the serum of coagulated fibrinogensolutions can be
easily explained from this, that fibringlobulin was found already in
free condition in greater or smaller quantities in the fibrinogensolu-
tion, so the supposition, that the ferment causes a splitting off of
fibringlobulin is superfluous as may be deduced from this.

Physics. — “The transformation of a branch plait into a main
plait and vice versa.” By Prof. J. D. van per Waats.

If for a binary mixture the temperature is raised above the critical
temperature of one of the components, the y-surface has a plait,
which does not occupy the whole breadth from «=O to #=1,
but which is closed on the side of the component for which 7%
lies below the chosen value of 7. In normal cases such a plait

1) Pfliigers Archiv, Bd. 30, p. 479.
3) Arch. f. exp. Pathol. u. Pharmakol. Bd. 49, p. 229.

( 622 )

which is closed on one side, does not present any special particul-
arities, and starting from the open side a bitangent plane may be
rolled regularly over the binodal curve as far as the plaitpoint.
There are, however, also cases where we meet with complications,
and already in my ‘Théorie moléculaire” 1 have allowed in my
description of the w-surface, for the possibility of the existence of a
branch plait by the side of the main plait. If two plaits exist
simultaneously over a very great range of temperature, we may
properly speak of a transverse plait and a longitudinal plait, and
the non-miscibility in the liquid state may be ascribed to the long-
itudinal plait. But if these two plaits occur only over a small range
of temperature, it is better to speak of a main plait and a_ branch
plait; I have chosen these names, because really in such cases
one of the plaits may be considered as main plait, and the other
only as branch plait. But, what has not been observed as yet,
the circumstance may occur, that at a certain temperature these
iwo plaits reverse their parts. What was a branch plait, becomes
a main plait, and the main plait is reduced to a branch plait.

In saying this I have chiefly in view the description of the
modifications to which the y-surface is subjected with change of
the value of 7, to account for the observations of Kurnen on the
critical phenomena of mixtures of ethane and some alcohols.

These mixtures have, for a value of 7’ only little greater than
7). of ethane, a plait on the y-surface with a continuous course
without any complication. But with rise of 7, besides the plaitpoint
on the ethane side, a new plaitpoint appears lying more to the side of
the alcohol. So from this temperature 7’) we may speak of a three-
phase-pressure. With further rise of 7’ the new plait extends, and
at a certain higher value of 7’= 77, the first plaitpoint disappears.
Then the three-phase-pressure vanishes, and from that moment the
plait has resumed its simple form. Between 7’, and 7, we have,
therefore, a plait with two plaitpoints. If referring to a plait we
speak of a base and a top, we have between 7. and 7, a plait
with one base and two tops. Beyond the limits of 7’ equal to 7,
and 7’,, the plait has only one base and one top. But whereas just
above 7’, the top which has newly appeared, extends but little
beyond the binodal curve of the original plait, at a higher value of
T this top will extend further; the top on the ethane side contracts,
and disappears altogether at 7’,, and as we shall show, disappears
as a branch plait.

As therefore the plait appearing at 7’, is originally a branch
plait, a transformation must take place with increasing value of 7’

( 623 )

which converts this branch plait into a main one. On the other
hand that part of the plait, which at 7’, was situated in the neigh-
bourhood of the existing plaitpoint lying below 7’, and which was
then a main plait, must have been reduced to a branch plait for
values of 7’ slightly below 7).

That the distinction between a main plait and a branch plait
is not arbitrary, but essential, appears when we determine which
of the two tops which occur between 7, and 7, belongs to the
base of the plait, and when this is ascertained, examine in what
way the binodal curve of the other top must be completed.

So the question is, when the bi-tangent plane is rolled over the
binodal curve from the base part of the plait, which of the two
occurring tops will be reached by continued rolling.

If we consult fig. 1, it is easily seen that a rolling tangent plane
which comes from the right side, and which has reached the two
points of contact A’ and A", has obtained a new point of contact in
A, lying on the same isobar and in this way has become a plane
touching in three points. At the assumed temperature we have there-
fore a three-phase-pressure. In this case there are two tops of a
plait viz. P and Q. But there cannot be any doubt as to which
of these two tops belongs to the base part lying right of A’ A”.
If viz. we continue to roll the tangent plane when it has the
line AA’ as nodal line, the binodal line on the side of the small
volumes between the points A” and A is completed by the curve
A" BCA, the configuration <A'B'C'A' giving on the other hand the
completion on the side of the larger volumes. This harmonizes
with the diagram in my Theorie Moléculaire. (Cont. I p. 28).
So when continuing to roll we reach P as top of the plait. We
are therefore justified in considering the part of plait A’PA as
belonging to the main plait. There lies, however, on and by the
side of the main plait, a second configuration, of which AQA" is a
part. If a rolling tangent plane is moved over it, starting from Q,
the binodal curve described in this way does not end in the points
A and A", but if the plane has reached those points and has there-
fore again assumed the position of the three-phase-triangle, we may
roll it continuously further till it has reached a point of the spinodal
curve. This curve is denoted by PD in fig. 1. The binodal curve
under consideration has then obtained a minimum pressure; the
conjugate point D' is then a cusp‘).

1) For a proof of these and similar properties consult Cont. II, fig. 3. Further

the very important papers of Korrewee on the theory of plaits.

( 624 )

When the plane is rolled further the binodal curve passes the part
DE" on the left side and the part D'# on the right side, where the
spinodal curve is again met with. For this part there is a maximum
pressure, while there is now a cusp in /’. And finally this plait,
which has its top in Q, is closed by the portion /' RE of its binodal
curve. If we consider also unstable phases as realisable, states between
E' and R coexist with conjugate ones between / and F on this part.
The point R closes this branch plait as unrealisable plaitpoint.

There is not the slightest doubt that for the above mentioned
mixtures of ethane and alcohol just above 7’, the newly appearing
plaitpoint Q on the alcohol side leads to the diagram of fig. 1 and
that Q is then the top of a branch plait. If the points A and A” are
still very close together, then the distances from these points to points
of the spinodal line must, a fortiori, be extremely small, and we
have justly assumed that the tangent plane in A’ A” when rolled
further, passes through the spinodal curve on the side of A’.

That on the other hand at temperatures just below 7’, the plait
the top of which is P, must be considered as a branch plait, is
beyond doubt for the same reasons. Above 7’, namely, only the top
Q is found, and the whole plait does not present the slightest com-
plication. Only with decrease of temperature below 7’, an extremely
small bulging out appears in the beginning at P (i.e. in the position
which that point has at that temperature) and the same reasons
which led us to consider the point Q as top of a branch plait
just above 7, must lead us now to consider Pas top of a branch
plait. Fig. 8 represents the binodal lines in this case. Only we have
assumed there that the temperature has fallen already so much below
T,, that the branch plait has got such an extension, that at first
sight it is not to be distinguished from a part of a main plait.

Both in fig. 1 and in fig. 3 there is asymmetry between the two
binodal curves of the tops P and Q. But when 7’ is gradually
changed from 7’, to 7’, or vice versa, fig. 1 will gradually pass into
fig. 3 or vice versa. This transition requires a value of T, at which
the asymmetry between the two tops P and Q has vanished. What
the shape of the binodal curves must be at the transition temperature,
is represented in fig. 2. Then we have one plait with one base, but
with two heads.

If we compare fig. 1 and fig. 2, the only difference is that the
points B' and £' have coincided, which involves that the node
belonging to B' and that belonging to /’, so the points 6 and £,
also coincide. From fig. 2 we derive fig. 1 by separating again the
parts which have run together, at the points which have coincided,

ee A EEE eT ee

av

ot ¢ Malan

+

and which is denoted by B' and L’', and by doing the same with
the point BH. In the same way fig. 2 leads to fig. 3. But the way
in which this separation must take place is different for these two
transitions. What happens in one case in the left-hand point, takes
place in the other case in the right-hand point.

The coinciding of the points £' and £#' is represented in fig. 2
on the spinodal curve; also the coinciding of the points 4 and
kk. The spinodal line is namely the curve which is denoted thus
— — — —, and which runs through the points BL'PDRCQ BE.
That the coinciding must take place on the spinodal curve might be
anticipated from the characteristic whieh we have used to distinguish
between main plait and branch plait. We had to consider Q as
top of a branch plait, if the rolling tangent plane, arrived at the
position A’ A", reached the spinodal curve on the side of A" when
rolled further, so in the space lying within the top @. On the
contrary P was the top of a branch plait when this happened on
the other side. For the case that there is symmetry between the
two tops P and Q, the meeting of the spinodal curve must take
place on both sides simultaneously. But we might also have taken
as criterion for the main plait, that the main plait is such a plait
for which the points 6 and HL’ are separated '). The comparison of
these two criteria leads to the fact that the coincidence of the points
L' and E' must take place on the spinodal curve. But as long as
the two tops P and Q are present, whatever the character of these
tops may be, there is a third plaitpoint, viz. the pomt FP, belonging
to a composition of the binary mixture which lies between the com-
positions belonging to the points P and Q.

In the figs. 4, 5 and 6 the complete (p, 7) curves have been given
for the coexisting phases. Fig. 4 for a temperature which is little
higher than 7), and at which Q is still the top of the branch
plait, and fig. 6 for temperatures below 7, at which P is still the
top of the branch plait. Fig. 5 represents the transition temperature.
I may assume as known that the differential equation for this (p,.)
curve is:

Qs
v,, dp = (x, — Eas ae thoy oof. tte Nees cheer (CO) '
vy pr
Whenever that the (p,v) curve has a point in common with the
3
spinodal curve (= = 0), p is a@ maximum or a minimum. This
ve nT
P

1) Cf. Wiskundige opgaven enz. 1V4e deel, 54¢ stuk, Vraagstuk CXXXIX, where it
is also demonstrated, that the branches of the binodal curve which touch in B' Z ,
have the same curvature. Also the conjugate ones, which touch in BZ.

( 626 )

is the case in the plaitpoints, but also in the other points, in which
a phase coexisting with an other, passes through the spinodal curve.
In fig. 5 there must therefore be maxima or minima at P, Q, BL,
BE, D, C, R. Vt from the differential equation we calculate the

2
€ vr . .
value of 2; for the points B’'L' and BL, it appears, that for the
ae,
two branches which meet, this value is the same there. If we
differentiate equation (a), we get:
Gp dp dv,,) \ arg dv dp | 075 «d(x, - 2,)

Bi aes ak Tan > Oe Bh ee a

O27 aa;

dp arg ; AE Seep ee.
and being O, this equation is simplified to:
dix, 00,7,T
‘ dp ( ' as
v%,, — = («,—2,) | — =
ac dx? . : dr,* pT
ae
The quantities v,,, (@,—,) and (; 7) are the same for the
vy J pT

two branches, and so also — In fig. 5 this has not been fulfilled
ae |
in the tracing of the branches in the neighbourhood of the points
BE. Better in the neighbourhood of the points BL. Also in the
cusps an inaccuracy in the proper curvature of the branches may
be detected here and there. But the figs. should be considered as
only schematical. The properties that the two curves in fig. 2 which
touch have the same curvature, and that this is also the case with
the two curves which touch in fig. 5, are of course closely allied.

From
dp Op Op\ dv
pa Ea ee Cay
da Ou) oT Ov J pda

Pp 0p Op (dv Op (dv? dp dv
Fast +255 +-~ (—)+ 2 —
dz? Ov. Oa Ov \ da: Ov? 7\ da Ovr7 dx

follows for two curves, passing through the same point, and for
Op Op 02p Op

3 sel Fee eee
Ow? Oa dv Ov? Ov

and

which, therefore, is the same, and which

dv\ .
touch in that point, and for which also (=) is therefore the same, that

av
dee : dp
the equality of 72% involves also the equality of 5 and vice versa.
Ak

ae
Korrewra’s thesis, which has also been proved by Kivyver, might
therefore also be proved by the method followed here.

Mathematics. — “A group of algebraic complexes of rays. By
Prof. JAN pik Vrins.

§ 1. Supposing the rays @ of a pencil (A, @) to be projective to
the curves 4" of order 7, passing through 7? fixed points, 5;, of the
plane 8, we shall regard the complex of the rays resting on homolo-
gous lines. For n= 1 we evidently find the tetrahedral complex.

Out of any point P we project (A, @) on B in a pencil (A’, 8),
generating with the pencil (4") a curve c'+!. So we have a complea
of order (n+ 1).

Evidently the curve c"+! does not change when the point P is
moved along the right line AA’; so the intersections of the «* cones
of the complex (7) with the plane ® belong to a system o’. It is
easy to see that they form a net.

For, if such a curve c"t! is to contain the point NY and if Oe
is the curve through A, and X, and ax the ray conjugate to it
through A, the point A’ must be situated on the right line connecting
X with the trace of ax on the plane 8. In like manner a
second point through which c’+! must pass, gives a second right
line containing A’. The curve c’t+! being determined as soon as A’
is found, one curve c™t! can be brought through two arbitrary
points of ~.

On the right line @ the given pencils determine a (1, 7)-corre-
spondence; its (2-1) coincidences C;, are situated on each c+!
So the net has (n° + 2-1) fixed base-points *).

§ 2. When A’ moves along a right line a’ situated in 8 and
cutting the plane @ in S, the curve c*+! will always have to pass
through the m points D; which a has in common with the curve
bn conjugate to the ray AS. It then passes through (x + 1) fixed
points, so it describes a pencil comprised in the net.

To the 8n° nodes of curves belonging to that pencil must be
counted the 2 points of intersection of @¢ with that c” passing through
the points 5, and D;. Hence a’ contains, besides ,S, (3 n? — n) points
A' for which the corresponding curve ct! possesses a node.

If A’ coincides with one of the base-points B; then the projective
pencils (A’) and (/") generate a c"+! possessing in that point B a
node. According to a well known property #B is equivalent to
two of the nodes appearing in the pencil (e*+!) which is formed

1) To determine this particular net one can choose arbitrarily but $ 2(2-+-3)— 1
points B and three points C.
43
Proceedings Royal Acad. Amsterdam. Vol. VII.

( 628 )

when A’ is made to move along a right line a’ drawn through B.
From this ensues in connection with the preceding:
The locus of the vertices of cones of complex possessing a nodal
edge is a cone & of order n (8n—1) having A as vertex and
passing twice through each edge AB,.

§ 3. If ? moves along the plane e« then the cone of the complex
(P) consists of the plane @ and a cone of order n cut by @ along
the right lines ACy. So @ is a principal plane and at the same time
part of the singular surface.

The plane ? belongs to this too. For, if P lies in 8 then the rays
connecting 2? with the points of the ray a corresponding to the
curve 4" drawn through P belong to the complex. All the remaining
rays of the complex through P lie in 8. So 2 is an n-fold principal
plane and the singular surface consists of a simple plane, an n-fold
plane and a cone 4 of order n(3n— 1).

The complex possesses (7? ++ n + 2) single principal points, namely
the point A, the n* points B, and the (7 -+ 1) points Cy.

§ 4. The nodes of curves c? belonging to a net lie as is known
on a curve #7 of order 3(p—41) the Hessian of the net, passing
twice through each base-point of the net. This property can be
demonstrated in the following way.

We assume arbitrarily a right line /and a point J/. The ce touching
lin L, euts AZZ in (p —1) points Q more. As the curves passing
through J/ form a pencil, so that 2(j—1) of them touch /, the
locus of Q passes 2(j)—1) times through J/; so it is of order
3(p—1). Through each of its points of intersection S with / one
ce passes having with each of the right lines 7 and JZS two points
in common coinciding in SS; so S is a node of this ep.

Consequently the locus of the nodes is a curve of order 3 (p — 1).

If / passes through a base point 4 of the net then the pencil
determined by J/ cuts in on / an involution of order (p — 1). This
furnishing 2 (j — 2) coincidences L, the locus of Q is now of order
(8p —5) only. So & represents for each right line drawn through
that point two points of intersection with the locus of the nodes,
consequently it is a node of that curve.

If / touches in B, the curve cy? having a node in £B, and if one
chooses .V/ arbitrarily on this curve, then the curves of the pencil
determined by .W have in B, a fixed tangent and /, is one of the
coincidences of the involution of order (p—1). The locus of the
nodes has now in £, three coinciding points in common with /;
consequently it has in B, the same tangents as ¢,.

( 629 )

For the net .V’+! of the curves c"t! lying in the plane ? the
locus of the nodes 7 breaks up into the right line «@ and a curve
of order (87 1). For, «8 forms with each curve 6” a degenerated
curve c?t!,

The locus of the nodal edges of the cones of the complex is a
cone with verter A of order (8n—1) having the nv?
AB, as nodal edges.

§ 5. The tangents in the nodes of a net Ne envelop a curve Z
of class 3(p—1)(2p—3) '), the curve of Zeurnun. It breaks up

right lines

for the net N+! indicated above; for, the tangents to the curves
6" in their points of intersection with « envelop a curve, which
must be a part of the curve Z The pencil (4”) is projective to the
pencil of its polar curves p"~!' with respect to a point O; the points
of intersection of homologous curves form a curve of order (2n—1);
in each of its points of intersection S with «8 a curve b” is touched
by OS; so these tangents envelop a curve Z' of class (2n — 1).

So for V+! the curve of ZevuruEN consists of the envelope 7
and a curve Z" of class 3 (2n — 1)— (2n — 1) =(8n —1) (2n —1).

The pairs of tangents in the nodes of the genuine curves of
N+! determine on a right line / a symmetric correspondence with
characteristic number (2n — 1) (8n—1). To the coincidences belong
the points of intersection S of / with the curve H/; to such a point
S are conjugated (2n — 1) (8n — 1)— 2 points distinct from S; so
S is a double coincidence. The remaining 4 (2 — 1) (8n —1) co-
incidences evidently originate from cuspidal tangents.

The locus of the vertices of cones of the complex, possessing a
cuspidal edge consists of 4(n— 1) (8n—1) edges of the cone A.

A general net of order (n+ 1) contains 12 (mn —1)n cuspidal
curves, thus 4(2—1) more; therefore each of the 2(n—1)
figures consisting of the right line @ and a curve 4" touching it
is equivalent to two curves c’t! with cusp. Evidently the nodes of
these figures form with the point (C, the section of «8 with the
curve H.

§ 6. On the traces of a plane a with the planes @ and @ the
pencils (a) and (b”) determine two series of points in (7, 1)-corre-
spondence; the envelope of the right lines connecting homologous
points is evidently a curve of class (2-1) touching @= in its point
of intersection with the ray @ conjugate to the curve 6" through

') This has been indicated in a remarkable way by Dr. W. Bouwmay (Ueber
den Ort der Bertihrungspunkte von Strahlenbiischeln und Curvenbiischeln, N.
Archief voor Wiskunde, 2nd series, vol. IV, p. 264). 7

( 630 )

the point @f7, whilst it touches 82 in its points of intersection with
the curve 4,” for which the corresponding ray passes through «pz.

The curve of the complex of the plane x has the right line Ba
for n-fold tangent, so. it is rational.

If the curve 4," touches the intersection @2, then the multiple
tangent is at the same time inflectional tangent.

We now pay attention to the tangents 7 out of the point S= a8
to the curve 6" corresponding to a. The envelope of these tangents
has the right line « as multiple tangent; its points of contact are
the 2(n—41) coincidences of the involution, determined by the
pencil (6") on a8. As S evidently sends out n (7 —1) right lines r
the indicated envelope is of class (2 — 1) (n + 2).

The planes containing a curve of the complex of which the n-fold
tangent is at the same time injlectional tangent envelop a plane

curve of class (n—1) (n + 2).

§ 7. The curve (2) can break up in three different ways.

First the point «8a may correspond to itself, so that (2) breaks
up into a pencil and into a curve of class n. This evidently takes
place when z passes through one of the principal points C;.

Secondly the involution on Ba may break up, so that all its groups
contain a fixed point; then also a pencil of rays of the complex
separates itself. This will take place, when a passes through one
of the principal points by.

Thirdly the curve a may contain the principal point A. Then the
curve 4” corresponding to the ray @= aa determines on pz the
vertices of n pencils, whilst also A is the vertex of a pencil. The
curve a is then replaced by (7 + 1) pencils.

In a plane through @, thus through all principal points Cy, the
curve (x) consists of course also of (+ 1) peneils.

A break up into to pencils with a curve of class (n — 1) takes
place when the plane a contains two principal points 4, or a point
B; and a point Cy.

§ 8. To obtain an analytical representation of the complex we
can start from the equations
Al) ; f, Age 0);
coi) - a + ab” == (0
Here a: and &" are homogeneous functions of 2, 2, #3, of

order 7.
For the points of intersection Y and Y of a ray of the complex

( 631 )
with @ and ~B we find
Uy) + Pys = Uy * Pay = %y* Pass
Yr * Pris = Ya * Pas = Ys * Prag:

After substitution, and elimination of 4, we find an equation of
the form

Pas (41 Pas + % Pas + %s Poa)” = Prs (8, Pin + 5s Par + 9s Poa)”
by which the exponent between brackets reminds us that we must
think here of a symbolical raising to a power.
If in pra ee yy— 2X, Ye We put the coordinate «, equal to zero,
we find for the intersection of the cone of the complex of Y on?
the equation

(v5 Us— Ya vs) (a, vy ++ a, vy =a as vs) "= (y,2,— 4, vs) (b, vy == b @a+ b, x)”,
or shorter
y, @s bn —¥.t,+ yy (v7, a% — ay bn) = 0.

This proves anew, that the intersections of the cones of the complex
form a net.

Mathematics. — “On nets of algebraic plane curves”. By Prof.
JAN DE VRIES.

If a net of curves of order n is represented by an equation in
homogeneous coordinates

Yr tz + ys be + Ys cr = 0
to the curve indicated by a system of values y,:¥,:y, is conjugated
the point ) having y¥,, y,, y; as coordinates and reversely.

A homogeneous linear relation between the parameters ye then
indicates a right line as locus of J, corresponding to a pencil com-
prised in the net.

To the Hessian, H, passing through the nodes of the curves belonging
to the net, a curve (J) corresponds of which the order is easy to
determine. For, the pencil represented by an arbitrary right line /y
has 3(n—1)’ nodes. So for the order n" of (VY) we find n"=3(n—1)’.

If one of the curves of a pencil has a node in one of the base-
points, it is equivalent to two of the 3(2—1)? curves with node
belonging to the pencil. Then the image /y touches the curve ())
and reversely.

Let us suppose that the net has 4 fixed points, then H passes

( 682 )
twice through each of those base-points; so it has with the neteurve

cy indicated by a definite point Y yet (nn'—4) single points in
common; here »'=3(n—1) represents the order of H. The curve
ce” having a node in D, determines with cy a pencil represented by
a tangent of the curve ()). From this ensues that the class of (¥)
is indicated by k" = 3n (n—1) — 20.

The genus g" of this curve is also easy to find. As the points of
(Y) are conjugated one to one to the points of H these curves have
the same genus. So we have

g' = } (n'—1) (n'— 2) — b = $ (8n—A) (8n—5S) — B.

We shall now seek the number of nodes and the number of cusps
of (Y). These numbers 6” and x" satisfy the relations

2d" +32" =n" (n'—1)— Fk’,
o" 4 x" = 4 (n!—1)(n"—2) —9".
From this ensues after some reduction
d” = 3(n—1) (n—2) (8n*—3n—11) + 4,
x" = 12 (n—- 1) (n—2).
The curve ()’) has nodes in the points Vg which are images of

the curves cp possessing a node in a base-point of the net. For, to
each right line through a point ’, a pencil corresponds, in which

cp must be counted for two curves with node.

Each of the remaining nodes of (}’) is the image of a curve ¢”,
possessing two nodes.

So a net N® contains *(n—1) (n—2) (38n?—3n—11) curves with
two nodes.

To a cusp of (}") will correspond a curve replacing in each pencil
to which it belongs two curves with node. According to a well-
known property that curve itself must have a cusp. For a definite
pencil its cusp is one of the base-points;- this pencil has for image
the tangent in the corresponding cusp of (}’).

So a net N” contains 12 (n—1) (n—2) curves with a cusp.

The two properties proved here are generally indicated only fora
net consisting of polar curves of a c'+!. We have now found that
they hold good for every net, independent of the appearance of fixed
points B.

We can now easily determine the class z of the envelope Z of
the nodal tangents of the net.

Through an arbitrary point P of a right line / pass z of these

( 633 )

tangents. If we add the second tangent in the corresponding node to
each of these tangents, these new set of 2 tangents intersects the
right line / in z points P'. The coincidences of the correspondence
(P,P') are of two kinds. They may originate in the first place
from cuspidal tangents, in the second place from the points of inter-
section of / with the curve //; each of these latter points of inter-
section however is to be regarded as a double coincidence. Thus
22== 12 (n—1) (n—2) + 6 (n—1) = 6 (n—1) (Q2n—3).
The curve of Zevtuwn is of class 3(n—1) (2n—8).

ERRATA.

Page 504, line 13, for members read member.
» 904, ,, 15, ,, not wanting read wanting.
» 009, ,, 24, ,, blewish read bluish.

(April 19, 1905).

YSCHAPPEN

KONINKLUKE AKADEMIE VAN WETE
TH AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday April 22, 1905.

2 — SOC —

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige
Afdeeling van Zaterdag 22 April 1905, Dl. XID).

(SNS) ASEM AROS as Se

J.J. van Laan: “On the different forms and transformations of the boundary-curves in the
case of partial miscibility of two liquids”. (Communicated by Prof. H. W. Baxnurs Roozenoom),
p- 636. (With one plate).

J. J. van Laan: “An exact expression for the course of the spinodal curves and of their
plaitpoints for all temperatures, in the case of mixtures of normal substances”. (Communicated
by Prof. H. A. Lorentz), p. 646.

FE. M. Jarcer: “On miscibility in the solid aggregate condition and isomorphy with carbon
compounds”. (Communicated by Prof. H. W. Baxuvuis Roozrrnoom), p. 658. (With one plate).

F. M. Jarcrer: “On Orthonitrobenzyitoluidine”. (Communicated by Prof. A. F. Hotiemay), p. 666.

F. M. Jarcrr: “On position-isomerie Dichloronitrobenzenes”. (Communicated by Prof. A. F.
Horireman), p. 668.

H. Kamerriscu Onnes and W. Heuse: “On the measurement of very low temperatures.
V. The expunsioncvefticient of Jena— and Thiiringer glass between + 16° and — 182° C.” p. 674.
(With one plate).

H. A. Lorenrz: “The motion of electrons in metallic bodies”, ILI, p. 684.

H. G. Jonker: “Contributions to the knowledge of the sedimentary boulders in the Netherlands.
I. The Hondsrug in the province of Groningen. 2. Uppersilurian boulders. 2nd Communication :
Boulders of the age of the Eastern Baltic Zones H and I’. (Communicated by Prof. K. Marriy),
p: 692.

Ernst pe Vries: “Note on the Ganglion vomeronasale’. (Communicated by Prof. T. PLace),
p- 704. (With one plate).

J. W. van Bissexicx: “Note on the Innervation of the Trunkmyotome”. (Communicated by
Prof. T. Puacr), p. 708. (With one plate).

Jan DE Vrirs: “On linear systems of algebraic plane curves’’, p. 711.

Jan DE Vries: “Some characteristic numbers of an algebraic surface”, p. 716.

K. Bes: “The equation of order nine representing the locus of the principal axes of a pencil
of quadratic surfaces”. (Communicatec by Prof. J. Carpinaar), p. 721.

Pu. Kouystamm: “A formula for the osmotic pressure in concentrated solutions whose yapour
follows the gas laws”. (Communicated by Prof. J. D. van per WAars), p. 723.

Pr..Kounstamm: “Kinetic derivation of van ’r Horr’s law for the osmotie pressure in a
dilute solution”. (Commuuicated by Prof. J. D. van per Waats), p. 729.

Pu. Komxstamm: “Osmotic pressuie and thermodynamic potential”. (Communicated by Prof.
J. D. van per Waats), p. 741.

J. Wreper: “Approximate formulae of a high degree of accuracy for the relations of the
triangles in the determination of an elliptic orbit from three observations’. (Communicated by
Prof. it. G. van pe Sanne Bakuuyzen, p. 752.

A. W. Visser: “A few observations on autocatalysis and the transformation of j-hydroxyacids,
with and without addition of other acids conceived as an ion-reaction”. (Communicated by
Prof. H. J.;HampBurcGer), p. 760.

Artuur W. Gray: “Application of the baroscope of the determination of the densities of gases
and vapours’. (Preliminary Notice). (Communicated by Prof. H. Kameriincu Onnus), p. 770.

The following papers were read :
44
Proceedings Royal Acad. Amsterdam, Vol. VII.

Sale a

( 636 )

Chemistry. -- “On the dijferent forms and transformations of
the boundary-curves in the case of partial miscibility of two
liquids.” By J. J. vax Laar. (Communicated by Prof. H. W.
Bakunuis Roozesoom.

(Communicated in the meeting of March 25, 1905).

1. In a preceding communication’) I showed, that when one of
the two components of a binary mixture is anomalous, the 7’, 2-
representation of the spinodal curve, and consequently also that of
the connodal curve, the so-called saturation- or boundary-curve
w=f(T), can assume different forms, which are indicated there. It
depends principally only on the value of the critical pressure of the
normal component, with regard to that of the anomalous component,
which of the different forms may occur with a definite system of
substances.

An allirmation of the theory, developed by me, that is to say
of the cases and transformations deduced by me from the general
equations, is found in the circumstance, that these cases and trans-
formations may be realised im quite the same succession with one and
the same substance, and this by eaternal pressure. In the same way
as with different normal substances as second component the form
drawn in fig. 7 Le., presents itself at relatively /ow critical pressures
(with regard to that of the anomalous component), and that of fig. 2
le. at relatively Aigh critical pressures — the form of fig. 7 may be
realised at relatively /ow external pressure, and that of fig. 2 at
relatively /igh external pressure, whereas at intermediate pressures
all the transitional cases again will return in just the same succession.

2. For that purpose we but have to look at the p,7-diagram
of the critical curve for ethane and methylaleohol, as projected
by Kupnrn *) in consequence of his experimental determinations (com-
pare fig. 1). We see, namely, immediately from the form of the curve,
departing from C, (the higher critical temperature, that of CH,OH),
which indicates the pressures, at which for different temperatures
the two coexistent phases coincide, and above which we have
consequently perfect homogeneity, that according to the value of the
pressure one critical point a@ may occur (at the pressures 1 and 2),
two viz. a and b,¢ (at 3), three, viz. a, b and ¢ (at 4), again two, that
is to say (a, 6) and ¢ (at 5), and finally again one, viz. c (at 6). (also
compare fig. 2).

1) These Proevedings of 28 Jan. 1905.
2) Phil. Mag. (6) 6, 637—653, specially p. 641 (1903).

( 637 )

All this is rendered still more conspicuous, when we project a
space-representation, in connexion with fig. 1 and of some successive
p,a-sections. In fig. 1 D, and YP, represent the vapourpressure-
curves of the two components; AJ is the threephasepressure-curve,
which terminates abruptly in J/, where the gaseous phase 3 coin-
cides. with the liquid phase 1 (which consists for the greater part
of ethane), because it meets there the critical curve CLC), that is
to say the curve of the plaitpoints ?. Beyond W there is coexi-
stence only between the j/lwid phase 3,1 and the phase 2, which
consists principally of alcohol. It is the equilibrium between these
latter phases, of which in fig. 2 the 7. 2-representation is projected
at different pressures. (The dotted houndary-curve 0 corresponds with a
pressure inferior to the critical pressure of the second component, and
superior to that of the first one). The T, x-representation of fig. 8
corresponds, at the (variable) threephase-pressure, with the threephase-
equilibrium wnto J/. In fig. 4 the indicated space-representation is
drawn, which will be clear now without the least difficulty '). For
the different higher pressures the corresponding 7’, v-sections are
drawn in that representation.

Remark. From C, (see fig. 1) to the maximum at 126°, where
a and $ coincide, and also from the minimum at 26°, where 4 and
e coincide, to the lowest temperatures, increase of pressure will
lower the critical temperature (Q, and these critical points will be
upper critical points in the 7’, 2-sections at constant pressure (see
fig. 2). On the other hand, from the maximum at 126° to the
minimum at 26° increase of pressure will raise the eritical tempe-
rature, and the designed critical points will be /ower critical points.

That increase of pressure favours the mixing, as is clear from

fig. 1 and from the p, v-representations — as well in the case of
an upper critical point, as of that of a lower one — is also in

agreement with the 7’ x-representation of fig. 2. For in the case of
an wpper critical point (see also fig. 5) a point A, situated within
the boundary-curve will come — when the pressure is increased, by

. : : aT . ;
which 7, will be removed tothe lower point 7’, (for -— is negative, as

ie
we saw above) — without the new boundary-eurve. And the same
: : a , dT
will be obviously the case for a Jower critical point, where a
P

is positive.

1) This space-representation (without the 1’ a-sections) has been already pro-
jecled independently by Mr. Bicuner; but is not inserted in his communication.
(These Proceedings of 28 Jan. 1905).

44*

( 638 )

To the considered type also belongs SO, + H,O, C,H, + H,0,
and equally ether and water. This latter mixture only with this diffe-
rence, that the composition of the vapour-phase is here continually
between that of the two liquid phases (see fig. 3"). Kurnen*') found,
that at 201° the vapour-phase coincides with that liquid phase, which
consists for the greater part of ether. The threephase-pressure is then
52 atm. (At C, we have 7, = 195°, p, = 38 atm.).

The p, x-diagrams would now show a maximum-vapourpressure,
if the two liquid phases 1 and 2 could become identical. In
connexion with this the threephase-pressure will be higher (here only
some mM) than the vapourpressures of each of the components, and
it follows immediately from fig. 1, that the critical curve C,C,, or
rather C,J/, will at first run back from C,, that is to say will present
a minimum critical temperature. In the case of C,H, 4+- CH,OH, where
the composition of the vapour-phase is without that of the liquid
layers, the threephase-pressure will always be between the vapour-
pressures of the components.

3. Now, as to the representation of the so-called transversal- and
longitudinal plait on the y-surface at different temperatures (in its
projection on the v, v-surface) in the case of C,H, + CH,O9, it will
be obvious, that the critical point Q, considered above, of the longi-
tudinal plait always lies at the side of the smal/ volumes. For
increase of pressure finally favours (see above) the mixing.

The successive transformations of the transversal- and of the longi-
tudinal plaits are further represented schematically, in agreement with
the p,.a-sections, in fig.6. The longitudinal plait, occurring here,
is regarded by van DER WaAats *) and this equally in the case to

be considered presently — as a transformed transversal plait. Many
questions however, connected with these plaits, lose — as has been

remarked already by van per Waats*)— much of their weight, and
become of secondary interest, as soon as we succeed in connecting with
other properties of the components of the mixture the often so com-
plicated transformations, which may occur at the different plaits.
And to do this an attempt is made in my preceding communication.
There I showed, that the ordinary theory of the association is capable
of representing the different possible forms of the boundary-curves quali-
atively, and in many cases even quantitatively.

4. We will now consider the second of the three principal types,

1) Z. f. Ph. Ch. 28, 342—365, specially p. 352 (1899).
2) These Proceedings 7, p. 467 (1899).
8) Id. 25 Oct. 1902, p. 399.

( 639 )

indicated in a recent communication of Bakuvurs Roozesoom and
Bicuner !), the jirst of which is amply considered above.

Fig. 7 gives the p,7-representation of it; fig. 8 the 7’, v-diagram
of some sections at constant pressure (the dotted boundary-curves are
relative to pressures between that of JZ" and C,, and to that below 7’).

The series of p,v-diagrams, and also the space-representation (also
projected already by Bicuner) are omitted here. We find that case
with mixtures of ethane and ethyl-, propyl-, tsopropyl- and normal
butylalcohol, all examined by Kurnen ?). Also triethylamine +- water,
and some other mixtures *) with a /ower critical point (this lies at
18°,3 C in the last mentioned case) belong to that type. Indeed, it
is obvious from fig.8 and from the p,v-diagrams, that J/’ is at
present a lower eritical point, contrary to JM in fig.1, which was an
upper critical point. For, whereas in J/ (fig. 1) the threephase-pressure
ends, this pressure commences in M'. Farther however, in J/", the
gaseous phase coincides again with the first liquid phase (rich in ethane)
(because the threephasepressure-curve J7'J/" anew meets the critical
curve €,C)), after which the further course is the same as with the
preceding type. The 7’,x-representation with variable threephase-
pressure, that is tosay between J/' and J/" (fig. 9a), is in agreement
with it.

It appears that, as little as with the preceding type, there exists
here a properly-said upper critical point. For in both cases 1 and 3
coincide, when the threephase-pressure comes to an end, and not 1 and 2
(see fig. 8a and 9a). The vapour-phase becomes identical with the
upper liquid layer and vanishes, but then there remain still two
phases, the lower liquid layer 2 and the fluid phase 1,3. These
however always pass into each other with further increase of the
temperature, as is plainly indicated by the space-representations
(see e.g. fig. 4), at the vapourpressure-curve of the second component
at v,, =v, =1 (also compare the dotted boundary-curves in fig. 2
and 8). So, if we begin with a mixture of a definite composition,
then with increase of temperature we come jinally beyond the bourdary-
curve 1,3 at the moment, that the liquid layer 2 has entirely vanished.
So we have demonstrated more exactly at the same time what I have
said in my preceding communication, namely that the existence of
a lower critical point involves necessarily that of a higher upper one.
It is however not, as we have seen, a critical point proper.

1) Id. 28 Jan. 1905, p. 581—537.
2) Z. f. Ph. Ch. 28, p. 358—363 (1899).
8) Among others f-collidine, and the bases of the pyridine- and chinoline-series

with H,0.

( 640 )

From the series of p,a-diagrams we might still see, that the
boundary-curve of the two liquid phases with the plaitpoint Q below
the temperature, where 1 and 2 coincide (in JM’), does not come
within the boundary-curve vapour-liquid, as Kuenen *) thinks, but has
entirely vanished. In fact, there is no reason to suppose, that in J’
decrease of pressure should cause again permanent separation (if
that state were realisable), as apparently Kvugnen thought that he had
“undoubtedly” to expect with’ mixtures of triethylamine and water *).
What he has observed in another case with propane and methylalcohol*),
must be ascribed in my opinion to this, that the expansion just above
the (upper) critical point has caused the temperature to fall a little,
so that he came on the (metastable) part of the two liquid curves,
which lies below the threephase-pressure. But when the cusp was
reached, the metastable equilibrium became immediately stable, and
by further expansion homogeneous liquid and vapour reappeared.

Because increase of pressure in this case too favours the mixing, as
appears from fig. 7 and from the p, .a-diagrams, the plait on the y-
surface will have turned its plaitpoint Q also to the side of the small
volumes. Fig. 10 gives a schematical representation of the successive
transformations of the two plaits, or rather of the transversal plait,
for we can regard again with vaN per Waats the longitudinal plait
as a transformed transversal one.

In fig. 7 we see, that increase of pressure raises the critical points
Q, at least in the beginning, if the curve M'C, should present
a maximum; and from the p, a-diagrams, that these critical points
will be again in that case /ower critical points, just as in fig. 1 between
the minimum and the maximum in the curve of the critical points Q.

Equally in the case of the second general type the threephase-
pressure may be either between the vapourpressures of the two
components, as in the case of C,H, and the mentioned alcohols, in
which case the composition of the vapour-phase will be not between
that of the liquids — or may be /zgher than that of the components.
Then there is again a maximum-vapourpressure after the coinciding
of the two liquid phases below the lower critical point, and the
composition of the vapourphase is between that of the liquid layers *).

1) Phil. Mag. 1. c. p. 645.

2) Id. p. 652.

3) Id. p. 646.

4) That at the coinciding of the two liquid phases 1 and 2 in M’ the vapour-
phase does not necessarily coincide with 1 and 2, has been already remarked by
Kuenen, and still earlier has been deduced by me theoretically for an analogous case

( G41 )

(see fig. 94). It is a matter of course, that in fig. 7 the critical curve
GiGe or ratheraC.
presents a minimum; equally the critical curve M'C, will not

J", again turns back in that case, consequently
seemingly eut C,C, between C, and J/" in this case, but remain on
the left of C,. This case occurs e.g. with mixtures of water and
triethylamine, where in consequence of the almost complete uwnmis-
cibility of the two components above the lower critical point the
threephase-pressure will be but very little smaller than the sum of
the vapourpressures of the two components. So Kurnien found, that
at 93° C. the threephase-pressure was 142,6 em., whereas the vapour-
pressures of triethylamine and water were resp. nearly 86 ¢.m. and
58,6 em., consequently together 144,6 em.

5. It is worth remarking, that the region of the threephase-
pressure continually shrinks the more we ascend to higher alcohols
(Kurnen). In the case of C,H,-+C,H,OH the temperatures in
M' and M" were resp. 31°,9 (46 atm.) and 40°,7 (55 atm.);
in that of C,H, -+ C,H,OH these are 38°,7 and 41°,7; in that of
C,H, + isopropylaleohol ? and + 44°; in that of n-butylalcohol
38°,1 and 39°,8 (55 atm.). Finally with isoamylalcohol three phases
were no longer realisable, so that the critical points Q coincide
there with the critical points ?, one continuous critical curve being
formed from C, to C, (General type HI).

In agreement with this is the fact, that the anomaly of the
alcohols decreases, as these are higher. In fact, we approach then
more and more the case of mixtures of normal substances, where
only at very low temperatures (see my preceding communication)
a formation of two liquid layers can present itself. Kugnen found
indeed, that C,H, + ether mix in all proportions, whereas C,H,-+-H,O
again present a threephase-equilibrium.

The influence of the fact that the alcohol is higher is also sensible
in the case of mixtures of CS, and different alcohols. So we find for
the upper critical points of CS, with CH,OH, C,H,OH, C,H,OH and
C,H,OH successively -+-40°,5 (Roramunp), —10?,6 (KuENEN) —52° (K.)
and —80° (K.).

Equally the influence of the hydrocarbon was examined by Kurngn.
While, as we saw above, the separation between type I and II in
the case of C,H, + different alcohols was between CH,OH and

(equilibrium between two solid phases and one liquid phase). This will obviously
also be the case with an upper critical point, as appeared from the experiments

of ScHReINEMAKERS with water and phenol. We will return to this question
ih § 8.

( 642 )

C,H,OH, and that between II and IIL between n-butyl- and isoamyl-
alcohol, the separation between I and III in the case of CH,OH +
different hydrocarbons is lying now between C,H, and C,H,. Propane
and the following hydrocarbons -+-CH,OH belong consequently, just
as C,H, + isoamylalcohol, and the greater part of the mixtures with
an upper critical point, examined by Gurarim, ALEXEJEW and RorHMunD,
to the third general type, which we will briefly consider now.

6. The third general type is principally characterised by this,
that the threephasepressure-curve meets 7o longer the critical curve
C,C,, but has come to an end already before (fig. 11). This third
type may proceed either from the second type (see fig. 7), the
threephasepressure-curve J/' M/" shrinking more and more, and
finally vanishing, as is the case with the transition from C,H,+C,H,OH
to C,H, + C,H,,OH (see above) — or from the first type, when AW
ends already before C, C,. In the first case (e.g. with C,H,+-C,H,,OH,
C,H, + ether) there exists no threephase-equilibrium at all — or it
should be at very low temperatures, which even may be expected
according to the theory (see my preceding communication), so that
the question arises, whether also in the case of fig. 7 there exists
at low temperatures a new threephasepressure-curve, and we will
return to that question at the end of this communication — in the
second case there exists a threephase-pressure from the beginning,
which vanishes at a definite temperature (wpper critical point). To
this latter case belong the mixtures of CH,OH with C,H,, ete.,
H,O + CO,, Cs, + different alcohols (Kurnrn), and also the greater
part of the mixtures formerly examined (see above).

But in the case of this third type there exists still another diffe-
rence. Firstly the threephasepressure-curve again may lie either
between or without the vapourpressure-curves of the two components,
with all the consequences, connected with it in the p, 2-diagrams,
etc. (see above). As to the mixtures of CH,OH with C,H,, C,H,,,
C,H,,, with all these is found (Kuenen, I. c¢.), that the threephase-
pressure is higher than the vapourpressures of the hydrocarbons, con-
wary to C,H, + CH,OH, which belongs to type I, where the three-
phasepressure is lower than that the vapourpressure of C,H,. Equally
with phenol and water (ScHREINEMAKERS, Vv. D. Ler), H,O + CS,
(Reenavtt) — where, according to the exceedingly small miscibility,
the threephase-pressure is again a little smaller than the sum of the
vapourpressures of the components — H,O + Br, (Bakuuis RoozEBoom),
H,O + isobutylaleohol (KonowaLow), CS,-- CH,OH and C,H,OH
(Kurnen), H,O + aniline (Kurnen), ete. we find everywhere the three=

( 643 )

phase-pressure higher than the vapourpressures of each of the com-
ponents. Only of H,O-- SO, (Baxnuis Roozrsoom), and of some
systems more (S-++ Xylol and Toluol, CO, + H,O) we know with
certainty, that the threephase-pressure is between the vapourpressures
of the two components.

There exists, however, still another, important difference. Wheras in
the case of type I (fig. 1) the critical curve QC, presents alternately

re heres Qee: : : Soh ee
positive and negative me and in that of type II (fig. 7) a is of
course positive in the beginning in J/’ (indeed, the point (Q just
appears in J/') — in the case of type II the initial course of J/Q may
be as well to the left as to the right.

: pene

Is this course to the /eft, that is to say is ap negative, then —
just as in fig. 1 between C, and the maximum and between the
minimum and the lowest temperatures — increase of pressure will
lower again the critical temperature in the case of these upper critical
points, and the plait on the y-surface in its v, v-projection will again have
turned the plaitpoint @ to the side of the smal/ volumes. (This is
equally the case with C,H, + isoamylalcohol, where no threephase-
equilibrium could be stated, but where the plaitpoint @Q, which has
become here identical with P, has removed strongly to the «z-axis,
just as in the case of mixtures of C,H, and the /ower alcohols). We
find this e.g. with C,H, + CH,OH.

But when the initial course of J/Q is to the right, as in the case
of C,H,, -+- CH,OH, C,H,, + CH,OH, and of phenol and water, then
increase of pressure will raise the point Q, and the mentioned plait
will now have turned the plaitpoint @ for the first time to the side
of the /arge volumes.

The question, whether the longitudinal plait, as in the case of phenol
and water, will present still a second plaitpoint at very small volumes,
consequently at very high pressures -— in other terms, whether the
coexistent liquid phases, after diverging initially, will reapproach
afterwards in composition, has not yet been answered theoretically
with certainty. It however appears to me, that where in the
ease of C,H, -+ CH,OH the plait has turned the plaitpoint Q to
the side of the sma// volumes, whereas C,H,, and C,H,,, equally
with CH,OH, have turned this point to the side of the large
volumes, there must exist a continwous transition between the two
kinds of longitudinal plaits, and that also the latter (as long as it
has not yet detached itself from the liquid curve of the transversal
plait, that is to say below the upper critical point) must be regarded

( 644 )

as an appendix of the transversal plait. Only when the longitudinal
plait has detached itself entirely from the liquid curve of the trans-
versal plait above the critical temperature of mixing, it can be
regarded in my opinion as a separate plait by the side of the
transversal one. This is in full agreement which what we find e. g.
for the boundary-curves in the 7z-representation (see fig. 2).

As long as the two parts of the boundary-curve, for instance N°. 2,
are not yet separated, we can hardly speak of two boundary-curves :
it remains one continuous boundary-curve; only beyond the transitional
case N°.3, e. g. N°.4, we have a right to speak of two isolated
boundary-curves.

As to the values of the different critical temperatures, we still
mention, that with C,H,-+ CH,OH the upper critical point was
found at 21° C., with C,H,, + id. at 19°,5, and with C,H,, + id.
at circa 40°. With C,H,, + C,H,OH8 the latter temperature immediately
falls down to — 65°.

7. Resuming all that precedes, we have the following summary.
(p; designs the threephase-pressure, p, and p, the vapour-pressures
of the two components).

C,H, + CH,OH
Type I |, between p, and p, , C,H, + H,O

(fig. 1). 80, + H,0
| Ps > Pp, and p, Ether + H,O

C,H, + C,H,OH, C,H,OH,

| ps between p, and p, | oT enO.aOee

Type I
(fig. 7).
| p, > p, and p, Triethylamine ++ H,O

p, between p, and p, | ice ae Toluol
Type a | eerily OG Ca
(fig.41). © C,H,, and C,H,, + id. /ar +)
BA C,H,, + C,H,OH
[r > p, and p, \ H,O + Phenol (” /ar +)
H,O + Aniline; id + isobutylaleohol.
Br, + H,O ;CS, + H,O, CH,OH, C,H,OH,
\ C,H,OH, C,H,OH.

( 645 )

8. We saw above, that when the composition of the vapour-phase
is between that of the two liquids — which is the case, when the
threephase-pressure is /igher than the vapourpressures of each of the
components — there must be a maximum-vapourpressure after the
coinciding of the liquid phases 1 and 2. That maximum may however
still be present before the coinciding of these liquid phases, which
is connected with the fact, that this maximum, which appeared origi-
nally at lower temperatures as a minimum (see fig. 12) in the meta-
stable region, has become gradually a maximum, and has moved
outwards before the coinciding of 1 and 2. The vapour-phase 3,
which was lying at lower temperatures always between 1 and 2,
as to its composition, remains not necessarily between these till the
moment of coinciding of 1 and 2, as was thought formerly, but
may have come outwards long before (see also fig. 9). It would be
very accidental on the contrary, when 3 coincided in the same time
as 1 and 2 to one phase. In the case of phenol and water SCHREINEMAKERS
has in fact shown experimentally this moving outwards’).

In what manner the moving outwards takes place, has first been
clearly shown and considered quantitatively by me?) in a series
of figures, and this in the case of coexistence of two solid solutions
and one liquid phase, whereas we have here —- what of course is
quite the same*) — the coexistence of two liquid and one gaseous
phase. The figures 9—I4, drawn in the indicated communication
(which refer to meltingeurves, and consequently are 7\x-repre-
sentations) are to be turned upside down, and the figs. 12, reproduced
here, are obtained (fig. 12 of the mentioned communication is omitted).
It will be remembered, that the case, which is realised with respect to
liquid-vapour with phenol and water, is realised with respect to solid-
liquid with Ag NO,. + NaNO, (only the maximum of fig. 144 Le. at
D has been already vanished there).

Some months after the publication of my communication KuENEN *)
came independently of me also to entirely the same view. What
is described on the pages 471 and 472 of his communication,
is quite identical with that, which I have described and represented
on the pages 184—186 of the designed communication.

Z. f. Ph. Ch. 85, p. 462—470 (1900).

2) K. A. v. W. 27 June 1903.

5) The calculations were based on the equation of state of yAN DER Waats, so
that the results of it are a fortiori applicable in the case of two liquid phases
and one gaseous phase.

4) K. A. v. W. 31 Oct. 1903.

( 646 )

9. Now, that we have sufficiently characterised the three general types,
and have brought some harmony into the multiplicity of the phenomena,
the question arises, whether there is a still farther synthesis, a still higher
unity. More than once the occasion presented itself in the treatment of
the different general types to remark striking agreements and continuous
transitions, often accompanied with great differences. Equally the
fact, that with a higher alcohol or a higher hydrocarbon, suddenly a
quite different type often appears, must certainly draw attention in
a high degree. All this induces us to look for the one fundamental
type, of which the three types, treated above, are special cases.

Also the analytical consideration of the question suggests that idea
to us. Indeed, the coexistence of two liquid phases and one gaseous
phase, or of two liquid phases, or finally of one liquid phase and
one gaseous phase, is determined by one and the same equation of
state, and it must consequently always be possible to reduce all the
different cases, which may occur to ¢vo fundamental proportions :
that of the critical temperatures and that of the critical pressures of
the two components — entirely in the same way asI have formerly
deduced ad/ the different types in the case of mixed crystals, where
appear two solid phases by the side of one liquid phase, from tivo
fundamental proportions: that of the me/tingtemperatures and that
of the latent heats of melting of the two components.

In a following communication it will be shown theoretically, that
the three types may be deduced from the ordinary equation of state
of Prof. vax per Waats, even in the case of zorma/ components. In
connexion with this we must not forget, that in the neighbourhood
of the critical points of each of the components the influence of
anomaly vanishes nearly always. In the case of C,H, -+ H,O for
instance the water will be in the neighbourhood of 365° C. already
normal long before, and in the neighbourhood of 32° C. the liquid
phase, which consists nearly entirely of ethane, will contain the water
in such a dilute state, that this will be passed for the greater part
into the state of simple molecules.

Chemistry. — “An exact expression for the course of the spinodal
curves and of their plaitpoints for all temperatures, in the
case of mixtures of normal substances.” By J. J. van Laar.
(Communicated by Prof H. A. Lorenrz).

(Communicated in the meeting of March 25, 1905.)

‘4. It is well-known, that the points of the $-surface, corresponding

to points of the spinodal curve on the y-surface, are given by the

simple relation

( 647 )

0°75
— ——s
oh tks

Ow? Ov? Ouwdv

: ‘ —, 07y 07yp Op \? _ :
which corresponds with the condition — .—— — { —— }=0,*) when in-

stead of the thermodynamic potential the free energy is used, and

not w,p and 7, but v, v and 7’ are the independent variables.
As we have further in the case of normal components e. ¢.

0g : 0g
a => fi ==
: On, Ox?
On, 0s ; oe
we have also a2 =—w ay and the above-mentioned condition may
Ov aL os
be replaced by
Of, —0
Ox

Now

r dw ;
“u, = C, — ( — a =) + RT log (1—z2),
&

where C, is a pure function of the temperature, whereas w is
1 ’

given by
0) = { pee — pe.
=

Oye
The condition a= = 0 is therefore identical with
Uae

or

from which I also started in my preceding communication ’).
2

. . . . . @
Now the difficulty arises, that the exact calculation of —— leads

0x?

to rather complicated expressions, so that VAN DER Waats contented
himself most times with approximations. These consisted in this, that
in the liquid state at sufficiently low temperatures 1s* p was omitted

a
by the side of —, 2°¢ terms of order v—b were neglected against

?
v3

those of order v.

Starting namely from the equation of state of vAN DER WAALS

1) Compare van perk Waats, Cont. Il, p. 137.
1) These Proc. 28 Jan. 1905.

( 648 )

a
(» 2y = (v--b) = RT,

where 6 will be regarded as independent of v and 7, then we
find for w: j

@o = RT log (v—b) + = pp Py Mer he ete a As (2)
v 5
; RT ! ;

If we write now aa for v—+, and omit p, then we obtain:

PA Ay ut

dw Nee i) 0 (a

ae Rog Eh

Ow Ove a Oa (“)

in which vAN ppr Waats further wrote / for v, whereas for illustrating
Pan. ; : : Gs
several properties 5 was brought in connexion with 7’,, and 7 with pe.

This is consequently a complete -set of approximations, and
with good reason Prof. Lorentz remarked to me, that in such
cases we must be carefull, whether these approximations are
not in contradiction, and up to what temperatures the results,
dw
Ow
Van per Waats himself considered therefore the deduced expression
merely as a more or less rough approximation, but which is at all

; ~ 00 10 fe 0 fa
events better than the former expression <— = =e a)
v a

deduced with the above-mentioned expression for —,can be used.

dz Ox
0 a ‘
where the term with ae os was omitted.

Now, I showed in my preceding communication, that at /ow tem-
peratures, and in the case of normal substances, where the critical
pressures rarely differ much, this omitted term has in the greater
part of eases a very small value, and is of entirely the same order as
v—b

, which is constantly neglected.
Only at higher temperatures the term has a large value, but then
Ow
the deduced expression for aE is not exact enough by far, for then

a
; 2 a 4 v—b
neither p can be neglected against -, nor terms of order 7 ean
v .
be omitted in that case.

The matter is consequently this: at sufficiently low temperatures

( 649 )

Beda 40. fa
the former simple expression an On a. may be safely used, at
a“ wv )

least in the case of normal substances; but at higher temperatures

0 a
equally the new expression with the term a log —> will be insufficient.
Uv v

/ i 0 Vw

And we want a more accurate expression for aa and Try the
more, when we — specially with respect to the course of the plait-
point-curve — also wish to know anything about the course of the
spinodal curves from the lowest temperatures to the highest critical
temperature.

I therefore tried to solve that problem; I was the more encou-
raged to do so, as soon it appeared to me, that the entirely accurate
expressions are not so complicated as was expected. On the contrary,
the often occurring fact presented itself here, that the exact expres-
sion is relatively more simply than the approximated one.

2. If we write the equation (2) in the form
—— + RT log (v—b) — p (v—b) — pb,
Vv
then we obtain:

dw = 0 fa Fela 0(v—b) db
T=sl- 1 P Oc Lee

N RT a
Now a = ij) =

consequently we find further:

Tey

dm 1 da a Ov a a Ov a db db
Oz v de v0n vida vide P dx’

or
0M 1 da a db 2
ey Suis: Soe ere
Ow v du Po v? ) dx’ (3)
Ov
where 5, *ppears no more.
Ai

If we write now:
a=(l—2)? a, + 2x (1—2) a,, + vw a,,
and if we put a@,,—=Wa, a,, by which the calculations and the results
are simplified in some way, without affecting much the exactness of
these results *), then we have:

1) [ am convinced, that the expression @:=V a ,@ is exact in the case of nor-
mal substances. At all events the inaccuracy, which results from this supposition,
will certainly not be greater than that of the equation of state used.

( 650 )
a= ((l1—2x) Va, + a Ya,)*.

Further we admit for } the ordinary linear relation
b= (1—2) b, + ab, .

The suppositions, on which the following calculations are based,
are consequently the following.

1st. the equation of state of van pER Waats, with 4 independent
of v and 7.

2°4. the ordinary suppositions about a and 6.

34, the special supposition a,, =Wa,a,.

From the expressions for @ and & used results:

~ = 2 (a —x)Ya,+2 V%,) (Va, — Va.) =2 Va. (Va, — Va,)
da

dx? =2(Ya, — Va,)

db db

ae —— b, —- b, ; de =

If we did not put a,,—Wa,a,, then we should have found
Ma

> = =2(a,+a,—2a,,). so only somewhat less simple.
&

3

o
Ou?

3. We will now calculate
For (3) we can write :

0m 2a a

\ ere (Va, —Va,) — (» tr = (6,— ,);
so that we obtain, when for shortness’ sake @ is written for
Va,—Va,, and @ for b, —),:

Cw 2. eA dv
On? = - (Va, roe Va,) — Va, Ja, a
2 2a 0
= 6, = 0) |v, ee =
ES 20° =f mil 6 ae (- 2a Me)
v v v v

v

valet oh

Consequently we must calculate

Ov
an:

( 651

a
From the equation of state (» “+ ) (v — b) = RT we deduce:
7

a Ov db ( b) 1 da 2a 2a dv we 0,
¢ + v)\de da ee) vi da SE

yielding

or also

Ov _de RT wv dz

== 4
oe Be eB o
RT v3
. . . . . to) .
Substituting this in the last equation for sa WO obtain:
-
Oe 2 Va 2a/ feed
Spauaeill Cokie eae ean

Meola) Zev coo] (, tute
v Re) he v3 ee 3

F a : db - : :
since rr =2a//a and sea 6. Further treatment yields after im-
ax av

portant simplification :

= 5S oe eee
Ox? v 1 _ 24/(v—b)? (5)

ier RT v?

Comparing this entirely exact expression with that, deduced in my
: reece v-—b
former communication, where p and —— were neglected, we see
5
that the exact expression (5) is already simpler than the approx-
imated one, which may easily be written down by means of the

0? a 0? a
expressions for — and log —, deduced there.
dz? \v Ou? v3

4. Consequently equation (4) passes into
2a (1—2z) (av—8 Va)?
a 9a/ v—b)?’

a/v

Fel
Ry) 2

v v

Proceedings Royal Acad. Amsterdam. Vol. VIL.

T.-C | ao =

( 652 )

that is to say into
a (v—b)? 2a (1l—w
RT — 2 —- aot — a (av—p pa)’,

v v"

or into
RE = — E (1 — x) (av — BY a)? + a (ve — |

Now av—@Ya=a(v —b)+ ab — Bya
=a(v — b) + a(b, + xB) — B(Va, + wa)
= a(v—b)+ (ab, —BV a, )=a(v—b) + (6, Va, —b, Va,).
Therefore we obtain (compare also VAN per Waats, Cont. II,
p. 45):
9 2
Ro [sao (b,Va,—b,V/a,) + av — Hj ae a(o—tp | (|
Ee
being, with the above mentioned suppositions, the sought, quite
general expression for 7’=/(v,x), by which for each given tem-
perature the v, v-projection of the spinodal curve is entirely determined.
We may also construct a “spinodal surface” 7’= /(v, 2), and im-
mediately deduce from the subsequent sections 7’= const. the forms
of the spinodal curves of the transversal- and longitudinal plaits, and
this in just the same +, v-representation as is used by van per WAALS
for the projection of the spinodal curves of the surfaces y=/ (7,2)
for different values of 7.

5. The equation (6) gives rise to some results, which may be
deduced from it without further calculation.

1st. Is v=4, that is to say, is the limit of volume 6, reached at
any value of wv, then (6) reduces to the equation of the boundary-
curve, lying in the v,2-plane :

i)

Ri 3 v(l—a2)(6, Ya, —b, Ya,)?,. - .« « (6a)

viz. the same expression, which was formerly found for small values
of » by means of the approximating method.

It is obvious at present, that only for v= the expression (6)
holds rigorously good. In every other case terms with ¢+—4 must be
added. But it also results from the found expression (6), that as
long as terms with ¢——6 may be neglected, the formula (67) gives
approximately the projection of the spinodal curve on the 7’, -plane,
Without it being necessary to take into account the corrective-term with

3

— log —,
De manors

a . . . . .
~, indicated by van per Waans. In a former communication

( 653 )

I showed already, that this correction-term is small in the case of
normal substances, about of order v — db.

As the second member of the expression (6) is always positive,
even when a@,, should be (Va, a,, the longitudinal plait on the
w-surface (for if is obvious, that in the neighbourhood of »—=/
the spinodal curve belongs to the longitudinal plait, which can
be regarded as a prominence of the transversal plait) will always
close itself above a definite temperature’ at the side of the small
volumes.

This temperature 7’, is the plaitpoint-temperature, corresponding to

dT
(6%); it is given by (6%), in connexion with the expression for z= 0,
deduced from it, yielding for the plaitpoint after elimination of 7’
the value

1 re ee
on <|¢ +1)—Vrtrt |,

b —/j)
where 7 = << =: (compare VAN DER Waats, Cont. II, and also my
)
1
preceding communication, p. 579). Only when b, = 4, (r = 0), x, will
be ='/,. In each other case x. will be removed to the side of the

smallest molecular volume.

Just at 7, the closing will take place at the limit of volume
v=b(w=a,); for values of 7’?< 7, the longitudinal plait will
remain wrclosed up to the smallest volumes. For in that case (compare
the representation in space) a section 7=const. will cut the boundary-
curve (62), lying in the boundary-plane v = 4, in a straight line.

This temperature 7’, may consequently be regarded in any respect
as a turd critical temperature. For above that temperature a for-
mation of two liquid layers will never present itself at values of v in
the neighbourhood of 6, that is to say at very high pressures; just
in the same manner as above the ordinary critical temperatures of
the single substances can never appear a liquid phase in presence
of a gaseous one.

2nd, Is v=o, then for each value of wv, 7’ will be =O, that
is to say, the equation (6) cannot be satisfied in that case. The plait
will consequently never extend to v=o.

3 Is «=O or 1, then (6) passes into the two boundary-curves,
lying in the two limiting 7’ v-planes, viz.

a 2a.
RT = —(v—b,? and RT=—(v —3,).
rs

With » =3 4, (resp. 34,) these two curves yield duly:

45%

5 RT, Sa eee

which is again a good test of the exactness of our formula,
deduced above.
These two critical points are at the same time plaitpoints of the
(transversal) plait, for it can easily be shown, that (=) and also
%

a

or é
= will be there = 0.
dx),

Before deducing the equation of the plaitpoint-curve, I shall first
point out, that the second member of (6) is always positive, as
consisting of the sum of two essential positive terms, so that the
T,v, a-surtace possesses nowhere points beneath the v, x-plane, which
of course cannot occur, because 7’ cannot be negative. Further,
that from (67) and (6%) results, that as to the limiting-curve (6%),
there will be found 7=0O for «=O and «=14, and as to the
limiting-curves (6”), 7’ assumes again the value O, as well for v = d,
(resp. 6,), as for v=o. :

Since the values of 4/7, and %/;, can be very different, according
to different substances, the surface (6) will also present very different
forms. Generally a greater value of 4 corresponds with a greater
value of 7), and in that case the surface has the form, as is indicated
in the figure. It is manifest already at superficial consideration, that
this form will be pretty complicated. ;

6. We will now determine from (6) the locus of the plaitpoints.

( 655 )

This may be found by combination of the two conditions

On, —(0 : ad — 0),

Ow pv Oa? pv
of Of (Ov 0 ks
{ > | 5 ie == Uy aah tre oe ere 1004:

when / represents the second member of (6). Indeed, this second

leading to

member has in all points of the spinodal curve on the y-surface the
same value, so that we have, by passing along an element of that curve :

af af
5, da +- a Chi== 0).
But in the plaitpoint we may regard an element of the spinodal
curve also as an element of the connodal curve, that is to say as the
line which joims the two tangent-points of a double tangent-plane,
when the tangent-points have approached each other to an infinitely
small distance. And as in these two tangent-points the pressure has
the same value, the latter does not vary, when at the plaitpoint we
pass along the considered element of the spinodal curve. Consequently

] a /
v= | — x,
t e Sa D

which yields immediately equation (7).

we have:

For shortness, we will write in the following 6,/a, — b,/a, = 2,
by which the second member of (6) passes into

2 2 5
f= = Ec (1—2) | +a 0 +a "| é

Ov
The value of a will be found from (4), viz.
U)y,T

2aVa (v—b)?

And since the denominator of this expression cannot become a,
(7) passes into

24/,,(v—b)?\ OF 2a Va (v—b)?) OF
pea Lay (eee ete -)Z=0. . (qa
( RT a iu (1 ne ie ")

Now we have:

[=a ae -

( 656 )
of
ks aos 6? (1—2r)—2a(1 —.xr)Gap—2a(v—b) B+ 2ay/a(v—b)*
Fe ay — 2e(1—wx)Ga + 2a(v—b) _ bs
Ov 4

where @ is written for 2 + a(v—h), and @ for */, v*f.
The equation (7a) becomes consequently :

ae @ ai 2aVa as )3y

"RT wv? Sv

2aVa (v-b)? 24/y (v-b)?
x (1-2) ba +o (e-{| Ja eee ) \- 2| 1 — ( a | = 0:

eae

fats 2) + 24 Va(r-ty| - {a-="

2

The expression between [ | is obviously :

2M 4/y (v—b)? . a ea ee
a ear ane Ae aniaar Lipa roe:

29
as av —BYa=a+a(v—b)=—6. Further we have R7T= a in

v
consequence of (6), so that we obtain:

ala ve(v—b)? |"? os

a Glee ea (1—2) + 2a a oo |— 2 =
g 7

v

— eve |e 2 (1—a) Oa +a (—0)| =o.

And since g — a(v—b)? = « (1—2) 6?, we have, after multiplication
with g:

(t-ny'| 1-20 a Zaye —3

a(1-2)6* + a(v-b)? By-eV a.r(v-b,? =
<i

— 26 Ya (v-b)? | na-m Gata | ==)

In this expression the underlined terms vanish. And for
Bp —aYa.v(v —})* may be written:

pa (l—a) 6? — Ya (v—b)? (av—B Va) = Be (1—2) 6? — Ya (v—b)* O

so that we obtain, after dividing by 6, and multiplying by v:
x (1-«) 6° [29 v — 3a (1-2) a| + Va (v-b)? |- 2av(v-b) 4-32 (1-2) 6? —
— 3 Va Be (1-x) 6 + 3a oo | =i):

or finally:

( 657 )

Jota] 12090 -se1— a | | Ya(v | du (1l—«r) AA pYa)+

| -}- a (vu—b) (v—3b) | == | ene 1 oe (8)

\
rs

where 6 —BVa may be substituted by av — 28 Ya.

This is consequently the sought equation of the v, 7-projection of
the locus of all the plaitpoints, which can appear on the y-surfaces
at different values of 7. Combined with (6), we find the points of
the surface, represented by (6), which satisfy the plaitpoint-condition,
that is to say the equation of the plaitpoint-curve as space-curve.
Equation (6) may be written:

(9), \|7
{iQ = = E (l—w) @ + a(v 1} s 6 6 ((@)

D

where thus 6 =a + a(v— db), and 7=b, Va, — b, Yay.
For v = (8) passes into
(1 — 22) b — 3a (1 — «) B=—9O,

yielding 2, = — | +1) —Vr+t+r4+1 ; as we have deduced
” i

already above (in § 5) for that limiting-case.

To conclude, we remark, that the sections for constant volume of
the surface, given by (6), on/y extend down to 7’= 0 («=O and 1)
for v7 ==. For all volumes > 6, 7 will assume for « = O\and 1, as
x ‘ae _ 2a(v — 6)?
is Obvious from (6), a jinite value, viz. ; . The 7,2-boundary-

z
curve suddenly ends then at the Z-axis at the designed value of 7’
(also compare the space-representation).

The proper discussion of the equations (6) and (8) must be
reserved for a separate communication. It will appear then, that the
different forms of the spinodal- and plaitpoint-curves, which occur
specially in the case of anomalous substances, are already possible
in the case of norma’ substances, provided the proportion of the
two critical temperatures 7/7 be sufficiently large. The spinodal
curves, given by (6), will appear easily calculable, and as to the
course of the plaitpoint-curve (there are two, independent of each
other), some conclusions will be deduced in a simple way.

It will also appear, which indeed results already from (6), that
the longitudimal- and the transversal plaits — at least with respect to the
spinodal curves (compare also vAN pyr Waaus, Cont. Il, p. 175) —
are no separate plaits, but one single plait, of which the plaitpoint
is lying, according to the different circumstances, either on the side
of the small volumes, or somewhere else.

( 658 )

Chemistry. — “On miscibility in the solid aggregate condition and
isomorphy with carbon compounds.” {First communication}.
By Dr. F. M. Jarcger. (Communicated by Prof. H. W. Baknuts
RoozEBoom).
(Communicated in the meeting of March 25, 1905).

Since the discovery of isomorphy by Mirscnerticn the power of
isomorphous compounds to form, on being mixed, a homogeneous
solid phase of gradually-varying composition has been experimentally
demonstrated in numerous cases.

In recent years several investigators have started theories as to the
course of the melting curves likely to be exhibited by such mixing-
series, and in conjunction with the theory of the equilibrium of
phases and with the aid of thermodynamical developments, a fairly
clear idea has been formed of the special cases which may be
expected to occur with substances of the said kind.

On the other hand, it is not permissible to draw conclusions as
to existing isomorphy, so/e/y on account of the course of the melting
curve or the solubility lines. Since the introduction in chemistry
of the idea of “solid solutions’, many cases have already been
pointed out where amorphous or even erystallised solid solutions
exist of substances which bear either xo or an unknown crystallo-
nomic relation to each other. We have only to think for instance
of amorphous glasses and on the other hand of the cristalline mix-
tures of ferric-chloride and sal-ammoniaec. The difficulty is felt in a
particularly striking manner in the chemistry of the carbon compounds ;
not only do we know continuous series of crystalline mixtures
between morphotropously allied carbon-containing derivatives, as in
MvcrHMann’s terephthalic-acid derivatives, but such mixing even in
the crystalline condition, has also been observed in the case of
organic molecules which have little or nothing in common.

Brunt and his collaborators, who have made a long series of eryos-
copic determinations conclude that the most dissimilar organic sub-
stances may yield ‘mixed crystals” and ‘solid solutions” of whose
erystallonomic relation not only nothing as a rule is known, but of
which the erystallographer will think the chances of isomorphy but
very small.

In any case the relation existing between “crystallonomie form-
relation’: and ‘miscibility’ is as yet quite unknown. If substances
are isomorphous, that is if crystalline phases possess regulated mole-
cular structures, which may be assumed to be formed from each
other by a slight deformation, such phases may jointly yield a homo-
geneous mixing-phase of variable composition and their relations

( 659 )

based on the equilibrium of phases will take the course indicated by
theory. But the reverse is by no means the case and the question
as to the existence of “isomorphy” can only be satisfactorily solved
by a crystallonomic investigation.

The problem has a particular interest in so far as it relates to the
determination of the limits in which morphotropously-related kinds
of molecules may exhibit such a miscibility. For the word “isomorphy”’
relates to a number of special cases in a series of much more general
phenomena of crystallonomy, namely, to those which show the rela-
tion between the chemical constitution of the substances which have
substitution-relations, with their innate crystalline form, which pheno-
mena are expressed by the name of morphotropy. If the chemical
relation of such substitution derivatives is confined within certain
limits such morphotropous substances may become “isomorphotro-
pous” and will then be able to combine with each other in a
limited or may be unlimited proportion. And if the relation of such
substances has become so intense that a nearly identical property
must be attributed to their crystal-structures on account of a// their
physical properties, such isomorphotropous substances actually become
“isomorphous” and mixing is then always possible.

From the above it follows that the idea of “isomorphy’’ admits
of a certain gradation; only the crystallonomer can determine in
each case the degree of “isomorphy” by measuring the size indi-
cated by the parameters of the molecular structure in the cases to
be compared and particularly by studying the analogy in the cohesion-
phenomena of the crystal-phase. As the differences in the values of
the said molecular structure-parameters become smaller and a more
complete similarity in the directions of cleavage and nature of surface
of the similar limiting planes is found, a more complete isomorphy
exists and the probability of a complete miscibility in the crystallised
condition is at the same time enhanced.

It cannot be doubted that in the cases investigated by Bruni there
may be instances of such rea/ isomorphy and the following research
may even prove this fact. But it must also not be lost sight of that
many cases of miscible substances supposed to be instances of “‘iso-
morphy’’ are only cases of isomorphotropy or even only of morpho-
tropy within relatively large limits.

All this renders it highly desirable to undertake an exhaustive
investigation of organic substances as to their miscibility in the solid
condition, coupled with that of their crystalline form so as to elucidate
the matter. The following research is a first communication on this
subject.

( 660 )

I. Nitro- and Nitroso-Derivatives of the Benzene-Series.

In consequence of some crystallographic faets, 1 intended long ago
io make a special study of morphotrophic action of NO, and NO
substitution in organic molecules. The matter became still more
important to me by the observation of the transformation of 0-Vitro-
benzaldehyde into crystallised 0-Nitrobenzoic acid under the influence
of light, and by a recent treatise of Brunt and Carircart (Gazz.
Chim. It. (1904) 34. Il, 246) who determined this formation of solid
solutions according to cryoscopic determinations. These investigators
arrive at the following conclusions :

1. As a rule aromatic nifrosoderivatives may form solid solutions
with the corresponding ntroderivatives.

2. In those solid solutions, also in the liquid ones which havea
green colour the nitroso-compounds have the simple molecular size.

It was particularly the first conclusion, which I wanted to submit
to a further investigation.

a. Paranitrodiethylaniline and Paranitrosodiethylaniline.

First of all I have extended the investigations paranitrodiethyl-
aniline by Scurave and myself). As I could only get proper crystals
of the nitroso-derivative from ethyl-acetate +- ligroine it was necessary
to obtain the crystals of the n/tro-derivative from the same solvent
in order to get strictly comparable preparations.

1. p-Nitrodiethylaniline.

C,H,.(NO,).N(C,H,), ; meltingpoint : 73°,6.

Crystals from ethyl-acetate + ligroine. (Fig. 1).

p-Nitrodiethylaniline, from
ethyl-alcohol.

p-Nitvodiethylaniline, from
ethyl-acetate.

1) Jarcer, Z. f. Kryst. 40, 127. (1905); ef ibid. 11, 105, Ref.

F. M. JAEGER. “On miscibility in the solid aggregate condition and
isomorphy with carbon compounds.” (1st Communication).

{120}

Fig. 3.

Pleochroism of p-Nitro-diaethy]-aniline.

Fig. 5.

Pleochroism of p-Nitroso-diaethyl-aniline

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 661 )

Brownish-yellow crystals, short prismatic or somewhat extended
towards. the axis and flattened towards {100} with a beautiful
reddish-violet reflexion on {LOL}.

Monoclino-prismatic.

Me Oe == T0pYD pal aalefstep
B= 80°34’.

The measurements are identical with those formerly given by
me. The habitus of the crystals is, however, different from those
previously obtained from alcohol (Fig. 2): ¢ and m= give ideal
reflexes; a very good, but often somewhat curved; ¢ alone has a
violet reflexion ; the reflex is coloured light-lilac. Etch-figures on {100}
corresponded with the indicated symmetry.

Very completely cleavable parallel {001}; fairly so towards {010}
with conchoidal fracture plane.

Optical behaviour. The optical behaviour of the compound is very
interesting. (Fig. 3).

First of all the crystals are strongly pleochroic ; on {001}, sulphur
yellow and blood red; the inflexion indicates the direction of the
vibration of the polarised rays; on {O10} yellow and orange ; on
{100} yellow and orange-yellow. On {001} are seen in convergent
light two red absorption hyperboles like the opening arms of an
axial cross.

On {100} an eccentric axial image is visible, the axial angle is very
small so that there is apparently present the image of a monaxial crystal
with a crossing of the axial planes for red and green light. The axes for
the red lie in a plane | b-axis, for the green in one | c-axis. The angle
for the red rays is smaller than that of the green rays. The double
refraction is positive; the first diagonal inclines to the normal i
the plane of symmetry; on {100} there are, therefore, at the same
time an inclined and a horizontal dispersion.

The angle of inclination is somewhat larger than in the case of
the nitroso-derivative.

2. p-Nitrosodiethylaniline.

Some time ago*) I made some preliminary communications on
paranitrisodiethylaniline C,H,(NO) N(C,H,), melting point 82°,2. |
(4) (1)
have since succeeded in obtaining this compound in a form more
1) KF. M. Jarcer. Ueber morphotropische Beziehungen hei den in der Amino-
Gruppe substituirten Nitro-Anilinen. Z. f. Kryst. 40 (1904) 112—146.,

( 662 )

suitable for investigation, so as to be able to make a clear comparison
with the corresponding pavra-nitro-derivative.

The said crystals had the appearance of small emerald green
rectangular plates, which were most readily obtained from acetone.
They are very poor in combination forms and only exhibit @ = {100}
predominating, m = {110} and 4 = {O10;{ whilst @ = {100} is generally
present in a rudimentary condition. They were determined as mono-
clino-prismatic with @=about 85°53’ and a:+ = 10166:1; on
{100} an interfering image is visible with slight inclination to the
normal on that plane; the axial angle is small, the dispersion an
average of o <r round the sharp bissectrix, the double refraction
is positive.

More accurate investigations have, as will be shown, practically
confirmed these data. I obtained the improved material from a mixture
of ethyl-acetate and ligroine, in which the compound was dissolved
on warming. On slow evaporation small rectangularly bounded,
thicker plates or also emerald green prisms are formed, which on some
planes possess a splendid violet reflexion. They exhibit a particular
lustre and but rarely a curvature of {100}; therefore very accurate
determinations could be executed.

The symmetry is monoclino-prismatic : the axial relation :

QO = WOKS e tesa 9614e
B = 85°43’.

Forms observed: @ = {100}, broadest
developed of all and well reflecting, some-
times a little curved; m—={110} and 6 ={010},
about equally developed, but 4 generally a
little broader, although sometimes much
narrower than 7 and much sharper reflec-
ting with light-violet reflex; ¢ = {001}, with
ideal reflexion also with a violet reflex
image; 7 = §T02), lustrous, mostly narrower,
sometimes a trifle better developed and
sharply reflecting; the reflexes are often
violet, mostly, however, colourless or yel-
lowish; s = {101}, broad but very dull and
only approximately measurable; /={1 0.13%,
very narrow in vicinal form has been
observed in a few cases.

The habitus of the crystals is elongated
along the vertical axis with flattening towards {100}.

p-Nitroso-diethylaniline.

( 663 )

Measured : Calculated .
th — (100) : (OOL) —* 85° 43’
t:m = (100): (1410) =*45 25'/,
c:r = (001) : (102) =*45 591/, :

r:a = (102): (100) = 48 19 48°171/,'
Qs 10))) (LOL) —— 2659 26 29
Sac — (LOL). (001) = 59 4 59 14
Ge (O0L) (EtO == 87 <5 87 O

3 44 341/,

Heo — (EEO): (O10) == 443
m:r == (4110): (02) = 62 11 62 9?/,
mis — (110) (OL) = 52° Or (circa) Sol 5

The crystals cleave very completely towards {O01} with a lustrous
separation plane; incompletely towards {010} with a conchoidal fracture.
Etch figures were not obtained.

Optical behaviour. The compound exhibits interesting optical
properties.

First of all the splendid, violet reflexion of the planes {O01},
{010} and {102}, which is wanting on the other planes. This reflexion
is not due to a streaking of these planes. If we cleave a crystal
along c or / the plane of separation also has that reflexion and a
streak made with the crystals on porcelain also exhibits the same
phenomenon. The light reflexes of the goniometer lamp on these
planes are coloured a beautiful pale hlac; on the other planes white,
sometimes yellowish.

Further, the compound exhibits on {100}, {OLO} and {O10} the
pleochroic behaviour as seen in Fig. 5; the inflexion again indicates
the direction of the vibration of the two polarised rays, which arrive
along the normal on the respective plane. On {100} a difference is
only observable with thicker crystals, on {O01} the colours are light-
green and dark, somewhat bluish-green; on {O10} the difference is
most pronounced, namely lighi-green and dark violet; the latter colour
is indeed, as I noticed, no surface reflexion but the colour of the
phase in transmitted light. On {O01} feeble absorption bundles of
hyperbolic form are observable resembling an axial cross opening
when the table is turned.

The optical axial plane is nearly horizontal; the acute bissectrix
makes a small angle with the normal on {100}. In convergent light
avery fine axial image is visible with a small axial angle and an
average dispersion of rhombic character; the dispersion of the said
diagonal is 9 <7, the double refraction is positive.

Calling the erystallonomie axes respectively a,6,¢ the optical

( 664 )

orientation of the elasticity-ellipsoid is therefore in the same order:
y,8,@ in which «¢> > y. The double refraction is feeble.

On $100} and {O01} orientated extinction ; on {OLO} the very small
inclined angle of the elasticity axis with the vertical axis could not
be sharply defined on account of the strong absorption; it is not
distinguishable from 90°.

The specific gravity of the crystals was found to be 1.240 at 15°;
the equivalent volume is, therefore, 143.53 and the topical axes become:
421 @ = 4,2363 : 4,1623 : 8,1626.

The complete tsomorphy of p-Nitrodiethylaniline and p-Nitro-
sodiethylaniline is therefore firmly established. The properties of both
substances are given here by way of a comparison.

p-Nitrodiethylaniline : p-Nitrosodiethylaniline :

Monoclino-prismatic. Monoclino-prismatic.

a:b:e =1,03842: 1 sibsVictel, VaaEiierea al {0ulfis33 al 196195
yi: @ = 4.4276 :4,2807 8.4710. | 4: w:@ = 4,2363 :4,1623 : 81626.

AeA see)
v = 158,36 v = 143,53
Angular values : | Angular values :
(110) : (100) = 45° 34’ (110) : (100) = 45° 253’
(100) : (102) = 51° 13’ (100) : (102) = 48° 1734’
(100) : (001) = 80° 34’ (100) : (001) = 85° 43’
In ethyl-acetate + ligroine the In ethyl-acetate +- ligroine the

habitus is flattened towards {100} habitus is flattened towards {100}
elongated towards the c-axis; some- and elongated towards the c-axis.
times towards the /-axis.

Very completely cleavable towards | Very completely cleavable towards
S001, fairly so towards jOLO{, with {O01}, fairly so towards {O10} with

a conchoidal fracture. conchoidal fracture.
On {001} violet reflexion. | On {001}, and on {O10}, {102}
violet reflexion.
Optical orientation: 7. ~, @ Optical orientation : 7, p, «@&
Double refraction, positive. Double refraction, positive.

On $100) a but little-inclined On }100) a but little-inclined
axial image with small axial angle; axial image with small axial image;
axial plane parallel the /-axis for axial plane parallel the 4-axis;
the red, parallel the c-axis for the rhombic dispersion : @ < v.
green rays; dispersion: 9 < v. |

Colour: brownish-yellow. Colour: emerald green.

Strongly pleochroic: blood red- Strongly pleochroic: violet-pale
orange-yellow. ereen-bliuishgreen.

On ¢ absorptionbundles. | On c absorptionbundles.

( 665 )

There is not the least doubt that the two substances possess a
quite analogous structure ; cleavability, optical orientation whilst the
nature of the surface of the crystalplanes is quite in agreement.

Krom mixed solutions of the two components are formed smal
ereenish-black mixed crystals with a vivid steel-blue reflexion.

As generally happens in the case of most isomorphous mixtures, the
crystallisation power is considerably smaller than with each of the
components separately. Under the microscope such mixed crystals
consist of thin olive-green little plates, which on their predominating
plane show little or no pleochroism. In convergent light a splendid
interfering image may be observed: slight inclination to the normal
on the horizontal plane, elliptical rings, and small axial angle, larger
however than in the two components. The double refraction is
positive: the dispersion has a rhombic character and shows: 9 < vt.

From the last motherliquors are deposited mixed crystals of a
lighter shade representing silky needles as those above with less
surface reflexion. Otherwise they are optical continuations of the
above described mixed crystals. From mixtures of the two components
in a melted condition these mixed crystals depose on the sides of
the testtube im a fine steel-blue lustrous condition.

The behaviour of the two isomorphous substances in the liquid
condition is elucidated by the investigation of the melting curve of
binary mixtures. On account of the dark colow of the fusion, the
course of the solidification curve was traced by the graphical method;
the determinations were made as usual in the van Eyk apparatus.
It should be noticed that all these fusions solidify to solid phases,
which also exhibit a splendid violet or blue reflexion.

The nitroderivative has a ereater latent heat of fusion than the
nitrosoderivative; in both cases the caloric effect was, how ever, very
readily observable in the solidification. The lower solidifying line
can by no means be determined so sharply as the upper one.

It was found that:

A mixture of 100°/, of p-Nitro and 0°/, p-Nitroso-derivative
melted at 73°,6.

A mixture of 85,14°/, p-Nétro- and 14,86 "/, p-Nitroso-derivative
commences to solidify at 75°,2 and completely solidifies at 74°\9.

A mixture of 72,5°/) of p-Nitro- and 27,5 °/, p-Nitroso-derivative
commences to solidify at 76°,2 and completely solidifies at 75°,9.

A mixture of 54,4°/, of p-Nitro- and 45,6 °/, of p-Nitroso-derivative
commences to solidify at 77°,6 and completely solidifies at 77°,3.

A mixture of 38,64 "/, of p-Nitro- and 61,36 "/, p-Nitroso-derivative
commences to solidify at 78°,2 and completely solidifies at 77°,9.

( 666 )

A mixture of 10,0°/, p-Nitro- and 90,0 °/, of p-Nitroso-derivative
commences to solidify at 80°,8 and completely solidifies at 80°,6.

A mixture of 0°/, of p-Nitro- and 100°/, of p-Nitroso-derivative
melted at 82°,2.

The composition is given in molecule-percents.

In fig. 6 the course of the melting curve is represented graphically
and the double line for the initial and final solidifying points is
shown. It will be seen that the character of the line points to a
continuous series of mixed crystals; the average temperature-interval
between initial and final solidification amounts to about 0°,3.

400 90 8 #70 6 50 40 30 20 40%,
Fig. 6.
The result of the research reveals the complete isomorphy of
p-Nitro-diethylanaline and p-Nitroso-diethylaniline and also their
complete miscibility in the solid state.

Chemistry. — “On Orthonitrobenzyltoluidine’. By Dr. F. M. Janeen.
(Communicated by Prof. A. F. Houremay).
(Communicated in the meeiing of March 25, 1905).

Some time ago the o-Nitrobenzyl derivate of para-toluidine, (melting
point 72°C.) was investigated by NorprnskJOLD, who described it as
being fetrayonal with the parameter-relation a:c¢ =1:0,6230; the
compound exhibits only one combination-form, namely {111} and is
optically monaxial: positive. (Bull. Geol. Instit. Upsala, (1892), 84,
also Ref. Zeits. f. Kryst. 24, 147).

( 667 )

For comparison I have investigated the o-Nitrobenzy/-derivative
of orthotoluidine*).
Reerystallised from acetone in which the compound (m. p, 96°) is

very soluble, the substance forms

very large, transparent, pale yellow

or rather pale greenish-yellow crystals possessing a strong lustre and

assuming a more brownish tinge
Elongated, prismatic needles are al

on prolonged exposure to the air.
so occasionally obtained.

The first-named crystals are nearly isometrically developed and

possess many combination forms; they admitted very well of accurate

measurements.

Rhombic-bipyramidal.
The parameters are :
a:

b:

’

o-Nitro Benzyl-o-Toluidine.

ce = 0,8552 ;

1: 0,61388.

Forms observed: g = {021},
strongly predominating and lus-
trous; 0={211}, broadly developed
and yielding sharp _ reflexes;
7—=}101}, well formed and lustrous;
e¢ = {001}, narrower but
reflecting; 6 = {010}, dull some-

well

times present with only a single
plane; mostly a little broader than
c, but also somewhat smaller ;

s = }201}, narrow and unsuitable

for measurement; the symbol has been deduced from the zone-relation.

The habitus is mostly thick-prisi
the smaller crystals possess a very

natie along the a-axis; particularly
regular form.

Measured : Calculated:

@: = (001) 2 (024) =*50° 50 —

Car— (OOM MO —==3 5740 —

6. q¢=— (010): (021) ==39' 9 S97 10!
Osp == (Cb) 3 Goi) == Av 20 27 10
0:0 = (211):(211) = 65 8 65 18
O26 =(211)) O01) = 57926 57 21
Oa (ids (O01) 138) 46 38 40
Ge (OO Gid) == 53) 16%), 53 18
gir== (021): 01) = 59 1 59 «8
bo OLO i) == 7038 70 40

Completely cleavable parallel {021}, distinctly towards {211}.

1) Ber. d. d. Chem. Ges. 23. 3582.

Proceedings Royal Acad. Amsterdam. V

46
ol. VII,

( 668 )

In oil of cloves as immersion liquid the situation of the elasticity
directions on the planes of {O10}, {021} and {001} orientated normally
in regard of the a-axis. The optical axial plane is {100}; the first
diagonal stands perpendicularly on {010}. On the planes of {021} a
brightly coloured interferential image is visible in convergent polarised
light; extraordinarily strong dispersion of a rhombic character with
@>v around the first bissectrix. In oil of cloves the apparent axial
angle amounts to about 49° for the red and 46° for the green rays.

The oil caused on {021} little solution-figures, which had the form
of isosceles trapezia; they agree with the indicated symmetry of
the crystals.

The specific gravity is 1,278, at 15°, the equivalent-volume is
189,28, and the topical axes are:

YW: w = 60875 : 7,1175 ; 4,38688

Although differing from NorpENskJOLD’s para-derivative in symmetry,
the analogy of the two isomers is still distinctly recognisable in the
value of the relation }:c.

o- Nitrobenzyl-para-Toluidine: a:b: ce =1,000 : 1: 0,6230.
o- Nitrobenzyl-ortho- Toluidine : a:b: ¢ = 0.8552 : 1 : 0,6138.

The difference in position of the methyl- and amino-group with
regard to each other therefore causes chiefly only a variation of the
crystal parameters in one direction.

Chemistry. — “On position-isomeric Dichloronitrobenzenes.” By
Dr. F. M. Jancer, (Communicated by Prof. A. F. Hotieman).

(Communicated in the meeting of March 25, 1905).

Of the six theoretically-possible dichloronitrobenzenes, which I
received some time ago for investigation from Prof. Honiuman, |
succeeded in obtaining four in such a measurable form that their
erystallographical determination could be satisfactorily undertaken.

Notwithstanding the great power of crystallisation of most of them,
the preparation of properly developed crystals is a troublesome and
very tedious matter. This is partly due to the very great solubility
in most of the organic solvents, which in connection with the low
melting points of these compounds often causes a not inconsiderable
supersaturation. During the spontaneous crystallisation, which then
takes place, no well-formed individuals, but crystal-aggregates are
formed, which are difficult of investigation. In addition, the peculiar
softness of the crystals causes most of them to exhibit curved planes
and considerable geometrical deviations. Again, owing to the heat

( 669 )

of the source of light during the measurement the crystals soon
become a dull surface, so that the inaccuracy of the measurements
is still further increased by the less sharp limitation of the signal
reflexes.

Of the substances examined the ortho-dichloroderivatives are both

rhombie, the meta-derivatives probably all monoclinic and the para-
dichloroderivative triclinic; only the geometrically well-defined sub-
stances of this series are described here in detail. In crystalline
form they show comparatively little resemblance to each other,
chiefly in consequence of the considerable deformation of the molecule
owing to the mutual attraction of the Cl-atoms and of the (NO,)-group.

Fig. 1.

a. 1-2-Dichloro-3-Nitro-Benzene.

C, H,.Cl.Cl.(NO,) ; melting point : 61° a 62° C.
Q) @) (3)

This compound crystallises from a mixture of
ethyl-acetate and ether and also from glacial
acetic acid, on very slow evaporation of the solvent,
in colourless silky needles, which are limited by
small, lustrous pyramidal planes (Fig. 1).

Rhombic-bipyramidal.

a262 6016472 102780:

Forms observed : a= {100} and 6 = {O10} equally
strongly developed and both very lustrous;
p = {230}, m—={110}, n= {430}; the latter form
is the smallest of the three and reflects less sharply
than p and im; @ sometimes shows a delicate
streak parallel with o:a3; 0 {133} lustrous,
{-2-Dichloro-3-Nitro. Yielding good reflexes.

Benzene. The vertical zone is, geometrically very well
constructed. The angular values observed in different crystals differ
but inconsiderably from the average values.

Measured: Calculated :
a: p = (L100) : (230) = *44° 9’ —
Oe 0e-—= (ss) (los) — *30!511/. ==
(rd Ch (7-133 0) (G8 0) as Ga 11°14’
At 1. == (40) (430-7 © 74 (oa
i- b- ==(430) 3 (010) = 2793 27 35'/,
C0 — C00) sd) — 62016 82° 7
6b: 0==(O010)2 (103) == 74-36 74 34
6:0 = (133): 433)= 15 32 15 46

46%

( 670 )

Readily cleavable along 0. On m and p right-angled little etch-
figures are visible in cassia-oil, which correspond with the indicated
symmetry. In the vertical zone the direction of the optical elasticity
axis is orientated on all the planes. An axial image was not observed.

The specific gravity

of the needles as determined by means of a

solution of mercuric-potassium-iodide was 1,721 at 14°. The equi-
valent-volume is therefore 111.56 and the topical axes become;
X:W: ow = 5,5190 : 8.5272 : 2,3706..
b. 1-3-Dichloro-2-Nitro-Benzene.

C,H, .c1. Cl
Ql) (3

.(NO,) ; melting point: 71° C.
9)

Fig. 2a The compound cry-

Fig. 2b.

1-3-Dichloro-2-

stallises from carbon
disulphide in large,
colourless, thin plates
a" A Of parallelogram shape
or also in smaller thick
crystals as shown in
figs. 2a and 2b. The
crystals are often
Nitro-Benzene. opaque and difficult
to measure ; sometimes, how-
ever, they are more lustrous
and very clear.

Monoclino-prismatic.
a: 6 »c=0,6696: 1 10.4149

Biot ola.

Forms observed : a = {100}

generally strongly predomi-

1-3-Dichloro-2-Nitro-Benzene. nating and always sharply
reflecting; g = {O11}, lustrous and either quite as narrow as 0 or
else the broadest developed of all, so that the erystals appear short-
prismatic towards the clino-axis; 0 = {111}, generally small, mostly
streaked parallel with @:o and reflecting rather dullisly; 6 = {O10},
very small aud often only present in a rudimentary form.

@: 0 = (100):
: (011) = * 45 2%/, —

gig — (01d)

Gif — 00):
Obi fa—i el) es
Rt] —s GEG We
a+b = (100)

pia — (dale

Measured : Calculated:
(144) = * 58°44’ ae

(014) —* 88 1%, =

(011) = 127 55 128°25’
(014)—= 2917 29 17°/,
:(010)—= 8957 90 0

(1411) = 3888 (about) 3816

(671 )

\ distinct cleavability was not observed.

The crystals deposited from acetone, which were very large but
dull, show a predominance of @ over 4; they are much elongated
along the ie axis and further possess a form which is probably
{233} with (233) : (100) = 67°33’, calculated 67°24’. On a there is
diagonal Rae, the optical axial plane is {O10}. One optical axis
descends almost perpendicularly on a.

The specific gravity is 1,603, at 17°, the equivalent volume 119,77.

Topical axes: 7: yw: w = 5,0596 : 7,5561 : 3,1350.

Although the parameter-relation a@:6 and the angle 2 in this isomer

are comparable with those of the 1-2-3-derivative :
1-3- Dichloro-2-Nitro-Benzene: a:b = 0,6696 :1; B= 87 52’
1-2- Dichloro-3- Nitro-Benzene: a:b = 0,6472 :1; B= 90°.
their crystalline forms are still rather different; the relation of
the latter substance is about 1'/, that of the first derivative.

e. 1-38-Dichloro-5-Nitro- Benzene.

7s H;.Cl.Cl.(NO,) ; melting point: 65° C.
NG 5

In alcohol or glacial acetic acid, in which solvents the compound
exhibits a remarkably great crystallisation power, there are generally
formed very long, flat columns of considerable thickness, or also
right-angled or obtusely truncated pale-sherry coloured small thin
plates. Owing to the great softness of the substance and its great
plasticity, the crystals are in most cases so ill-formed and distorted
that measurements become impossible. With very slow evaporation we
sometimes get better formed erystals although they are very poor in
planes. They have a peculiar odour resembling nitrobenzene.

Monoclino-prisimatic.

a7 0 = 05940: 1 ; f= 58" 49’.

Forms observed: « = {100}, broad and very lustrous; 4 = {010},
narrower and less lustrous; it is often absent altogether ; m — {110},
narrow and c= {O01}, small but very reflecting; the habitus is
elongated along the c-axis and then flattened {100}.

Measured: Calculated:
OF: na) (010) + E10). = *:-63' 5
a:¢ —(100): (001) —* 58 43 =e

and (100) GO) == 26.58 26°55’
m:m = (110): (410) = 12615 126 10
mie == (10). O0li== == 62 25

b:¢ = (010) 7 001)— 89 57 90 O

( 672 )

The crystals are completely cleavable along {010}, readily so
along {O01}.

On 100) extinction occurs on orientation; on {010} under 28° with
regard to the vertical side. The optical axial plane is {010}; at the
border of the vision-sphere an optical axis is visible on {100}; the
axial angle is small. The direction of the vertical axis is here the
axis of the greatest elasticity.

On $100} etch excrescences were observable with a circumference
of isorceles trapezia, whose angular points appear to be connected by
straight lines with a point situated in the centre; this point lies
nearer to the smallest than to the largest of the two parallel sides
of the trapezium. They agree with the indicated symmetry.

Fig. 3. The specitic gravity is 1,692, at 14°C.;
the equivalent volume is, therefore, 113,4.

d. 1-4-Dichloro-2-Nitro- Benzene.
C,H,.Cl.Cl(NO,) ; Melting point: 54°,5 C.
4) (I 2
In most solvents this substance shows
a very great crystallisation power, but
measurable crystals are but rarely
obtainable, as most of the individuals
exhibit important geometrical deviations
on account of the great softness of the
material and often possess curved and
very dull planes.
Some time ago the crystal form was

incompletely determined by Boprwie; he

1-4-Dichloro-2-Nitro-Benzene. investigated crystals deposited from car-
bon disulphide but did not succeed in obtaining combinations admit-
ting of a complete determination of the crystal parameters (Zeits. f.
Kryst. 1. 589; Ann. Ch. Phys. (4). 15. 257).

From acetone I always obtained the largest crystals, sometimes
some centimetres in length; they are quite of the prismatic type of
the erystals investigated by Boprwie and possess in addition a lateral
prism; they exhibit, however, such considerable deviations and are
generally so opaque that an accurate measurement is out of the
question.

I succeeded best by erystallisation from ethyl-acetate mixed with
a little carbon-tetrachloride; the pale sherry coloured crystals flattened
towards {100} so obtained, are very well formed and admit of accurate
measurement.

( 673 )

Triclino-pinacoidal.
a@:b6:¢=0,8072 :1

Al = [7S)° ast
== 119169 ale

(= bis! Bho

: 0.8239

oa

Forms observed: @ = {100!, predominant,

and well developed ; 7 = {110}, narrow but well-reflecting ;

narrow very lustrous; 7 = {LO1!, somewhat
good reflexes.
The crystals are flattened along a and elongated in the direction

of the c-axis.

Measured: Calculated :
ace — C00))3(OO1)\ == *652-81/." —-
Ga 0e = (100) 2 (O10) S121 25 —
bse = (OlOys (CO) SS] 110) DY —

a:r = (100): (401) =* 50 127/, =e

Cage — OUR) = (0M) * 45.39 _

qb. = (011) 3010) = 54 44 54°48’

Ce — (COOL : (101) = 64 40'/, 64 40°).
eo — ClO) (O10) == fa) 23 (iy ks)

Nee — CellO) eel OO) 5 462 46 6

r:q = (101):(014)= 51 50 51 35

m:r =(110):(101)= 65 36 65 22

m:q =(410):(011)= 62 54 638. 3

The crysials are very completely cleavable towards {O01}; th

plane of cleavage is very lustrous.
On {100} obtuse-angular extinction ;

7°40' in regard to the vertical side;

hyperbole is noticed on this plane.

92° 48'
eel lene)
OWE:

broader

well-reflecting,
than 6 = {O10}, which form is also narrower; ¢ = {O01}, very lustrous

Dp?

its amount is small, only about
in convergent light a dark

The specific gravity of the crystals is 1,696 at 12

valent volume is, therefore, 118,20.
The topical axes are x: tp: w = 4,8484 :

6,0065 : 5.1422.

and yielding

( 674 )

Physics. — H. Kameruincnu Onnes and W. Heusr. “On the measu-

rement of very low temperatures. V. The expansion coefficient

of Jena and Thiiringer glass between 4+- 16° and — 182° C.”

Communieation N°. 85 from the Physical Laboratory at Leiden.
(Communicated in the Meeting of June 27, 1903)

§ 1. At Leiden the hydrogen thermometer (ef. Comm. N°. 27
May °96) is taken as the standard for very low temperatures.
To reach the degree of accuracy otherwise obtainable with this, it is
necessary to know the expansion coefficient of Jena glass 16™! to
about 1°/,. Hence we have determined the two coefficients in the
quadratic formula assumed for the linear expansion of glass below
O° C. At the same time we have, in precisely the same circumstances
made a similar determination for the Thiiringer glass, from which
the piezometers mentioned in Comm. N°. 50 (June 99), N°. 69 (April 01),
and N°. 70 (May ‘O1) were made, in order to be able to calculate
and apply the correction for expansion to the results attained with
these piezometers.

Some time previously we made measurements on expansion coefti-
cients, among others on platinum. The value for this metal was
required for the reduction, from the measurements mentioned in
Comm. N°. 77 (Febr. ’02), of the galvanic resistance at low temperatures.

But the results which we have lately obtained for the two above
mentioned kinds of glass appear to us to be the first that are worth
to be published; the final reduction of the measurements named
above was postponed till the required accuracy was reached. However
the measurements on platinum must be repeated.

Although the field of measurements at low temperatures is hardly
touched, still we consider that in this field preliminary and approx-
imate values are worth little. In the majority of cases approximate
values of this kind can be obtained by extrapolation, and thus only
those determinations which are accurate enough to allow a judgment
on the question whether such an extrapolation is allowed or not,
are really of use in advancing our knowledge. We have hence
arranged our observations on the expansion coefficient so as to reach

1
an accuracy of —.
200

For general the investigation of expansion at low temperatures it
will be required to determine on the one hand the linear coefficient
of solids and on the other the absolute coefficients for those substances,
which remain liquid to very low temperatures, e. g. pentane, in such
an hydrostatic manner as DuLone and Perrt’s (improved by Reenavr).
The determination of the relative expansion of the liquid chosen can

( 675 )

then serve as a control and as the starting point for further measure-
ments. The present investigation forms the first part of this general
program and gives the linear expansion of glass with an accuracy
which suffices for our present purpose. From the description of our
measurements it will be seen that with practically the same apparatus
and in nearly the same way it will be possible to determine the
absolute expansion of pentane.

§ 2. We have determined the two coefficients @ and é in the
formula for the linear expansion = L,(1-+ at-+ 6¢), for the
two varieties of glass from three observations for each. These

were made at ordinary temperature, at about — 90°, C. and at about
— 180° C., by measuring directly and at the same time the lengths
of the rods of the two substances.

The rods were drawn out at each end to a fine point which could
be accurately observed with a microscope. At the bottom and top, the
two rods project out of a vertically placed cylindrical vessel. The
bath is closed at the lower end and is filled with a liquefied gas giving
the required temperature. Care is taken that the points shall be kept
as nearly as possible at the temperature of the surrounding air, and
also that the air between the points and the objective of the microscope
shall be at the same temperature. The lengths are then read directly
against a scale by a cathetometer arranged as a vertical comparator.

Although this arrangement gives a convenient method for the deter-
mination of length it necessitates a considerable difference in temperature
between the middle and the ends of the rods. To correct for this, use is
made of the method employed in Comm. N° 83 (Febr. 03) for the deter-
mination of the corrections along a piezometer or thermometer stem.
This depends upon the use of a uniform platinum wire wound uniformly
round the rod. Its use depends upon the assumption, that the change of
resistance of a wire wound in this manner is nearly proportional to the
mean change of temperature of the rod. This will be further considered in§ 4.

After this general view we may consider certain details.

1st. The glass rods were about 1 m. long and had diameters of
5 mm.!). Round these 0.1 mm. thick platinum wires were wound
spirally and soldered to brass rings A, B, C, D (P11 fig. 1.) which
were tightened by screws.

Between 5 and C, the part which was immersed in liquified gases,
there were 140 turns with a pitch of about 0.5 em. Between A and B
or C and D where the temperature changes rapidly there were 25
and 40 turns respectively with a pitch of 0.25 em. Care is taken

1) A platinum tube provided with glass ends similar to those described above
was used for the determinations on platinum.

that the pitch remains constant in each section A to B, B to C, or
C to D. At A, B, C, and D platinum wires a, b, ¢, d, e, f,g, and
hk about 15 em. long and 0.5 mm. thick are soldered in pairs. At
the other ends they are connected to copper wires. In order to pre-
vent faults in insulation the spirally wound wires lay in shellac they
were also covered with a layer of tissue paper for purposes of pro-
tection. The portions A to B and C to D were enveloped in succes-
sive layers of fishglue and writing paper to about a thickness of 0.25 cm.,
in order that the distribution of temperature should be as even as
possible along the rod. This protection was found to be proof against the
action of either liquid nitrous oxide or oxygen. To allow of contraction
on cooling the paper layers were only pasted together at both ends.

20d, The cylindrical vacuum jacket. The bath for the liquid gases
has the form of a tubular vacuum glass. Usually vacuum glasses
are made so that there is but one edge connecting the cooled and
uncooled walls. When it is necessary to remove liquid at the bottom of a
vacuum glass the lower surfaces are connected by a spiral tube. However
we required something quite different i.e. a double-walled fue open at
both ends and capable of holding a rubber stopper in one. Ifsuch a vacuum
tube were made by blowing simply together inner and outer walls
it would certainly crack when cooled, owing to the different expan-
sion of the outer and inner walls. Also it did not appear to be pos-
sible to make the outer wall sufficiently elastic by blowing several
spherical portions in it (see fig. 1).

Fig. 1.

Hence the outer wall was divided by a thin brass case V,, PI. I,
which allows a compression or expansion of 2 mm. This copper
box was inserted by platinismg and coppering the two glass surfaces
and then soldering them to the copper box. The vacuum tube thus
produced was silvered and evacuated in the usual manner. In the
first arrangement the top was left clear in order to allow of the observation
of the surface of the liquid. In later arrangements we preferred a float,
Such tubes with compound elastic walls appeared to be suitable for our
purpose and will probably also be found to be useful for the solution of
various other problems. An example of how easily tensions arise which
cause such glass apparatus to crack, was found when the rubber
stopper at the bottom was pushed in too far. On admitting the liquid
oxygen the rubber became hard before it had reached the temperature of
the liquid, which temperature the glass immediately above had reached

low temperatures. V. The expansion coefficient of Jena and Thiringer

H. KAMERLINGH ONNES and W. HEUSE. On the measurement of very
glass between + 16° and — 182° C.

Proceedings Royal Acad. Amsterdam. Vol. VII.

( 677 )

already, and the lower rim cracked off. Later we made the connee-
tion tube and the stopper more elastic (cf. 7 PL. I fig. 2) by inserting
between them a collar formed of several layers of paper glued together
at the borders. In this way a closure was obtained which was
perfectly tight, a quite necessary item, for otherwise the escaping
liquid streams past the reading points as a cold vapour, which
disturbs the uniform distribution of temperature supposed to exist
in the ends of the rods and obtained by continually blowing air
on to the points which is necessary also for keeping them dry.
At the top, the rods are supported sideways so that no strain
is caused in them. They are protected from the cold vapours which
arise from the bath. From the front and side elevation of the
upper end, Fig. 1, the arrangement of paper used for this protection
can be clearly understood, and the course of the vapour can be
followed as it streams over the wall of the bath through channels
of cardboard. This arrangement has moreover the advantage, that
the outer surface of the vacuum vessel is also cooled. This is of
great importance in the beginning. The cold gas and cooled air are
so conveyed away by various paper screens, that they do not come
into the neighbourhood of the cathetometer or the standard scale, and
also that air at the ordinary temperature remains between these and
the points. At the commencement the liquefied gas is introduced in drops
through an opening in the cork at the upper end, and afterwards
carefully in small quantities. When the bath is once full, fresh liquid
is continually added in small quantities to keep the level at the
same height. The liquids used were nitrous oxide and oxygen obtained
in the manner described in Comm. No. 14 (Dec. °94) and No. 514
(Sept. 99). In both cases considerable purity was aimed at, in consequence
the temperature of the bath did not change during the measurements.
There is no doubt that the temperatures at the top and the bottom
of the bath were not the same but this introduced no difficulty
since in the calculation only the mean temperature as determined
by the platinum resistance was required.

3°, The comparator (cathetometer and scale). We used the
instruments which are described in Comm. No. 60 (Sept. 00). The
scale was very carefully enveloped in wool and paper to protect it
from changes of temperature. Its temperature was read by two
thermometers divided into */,, and symmetrically placed above and
below, while the room temperature was maintained as constant as
possible. The telescopes were provided with the microscope objec-
tives which had been used for the measurements on the viscosity
of liquid methyl chloride (Comm. No. 2, Febr. ’91) and which

( 678 )

can be used at a distance of 10 ems. In this case one revolu-
tion of the head (divided into 100 parts) of the micrometer screw
(cf. Comm. No. 60 § 15) was equivalent to 60 to 70 u. The levels
on the telescopes were carefully calibrated; at the distance used, one
division on the levels corresponded to from 4 to 6 « and the uncer-
tainty in reading was less than 0.2 division or about 1 uw. After each
setting, 80 seconds was allowed to elapse before reading and former
measurements have shown that this is sufficient for the attainment of
equilibrium.

The field of view of the microscopes was also investigated by
measuring at various points a */,mm. scale, but no irregularity could
be found.

4h Measurement of resistance. The doubled conducting wires
a, b, ¢ ete. at the ends of each measuring wire AB, ete. (cf. Pl. 1)
were lead to eight cups of mereury for each rod, which cups could
be connected in pairs to the wires from the Wueartstonr bridge.
By measuring

w,=atAB+d
w,=ce4t-ABtd
Ww, =a+e
w,=b+d

the resistance of the wire AB
w,+w,—w,—w,
2

can be determined’). The galvanometer with reading scale (see
Comm. N° 25, April °96) had a resistance of 6 @ and a sensitiveness
of 2.5 > 10-7. Thermoelectric forces in the circuit of copper leads, |
platinum leads and platinum resistances are unavoidable, they were,
however, onty small and could be eliminated.

§ 3. Survey of a determination. A complete determination com-
prises focussing the microscopes, referring to the standard scale,
and reading the thermometers, as well as the various determinations
of resistance between A and Bb, B and C, C and D.

In the following table all the readings for the determination of length
of the Jena rod in liquid oxygen are given. Column A contains the
readings of the micrometer heads, 2 the corresponding positions of
the levels, C the nearest division on the standard scale, D and #
the micrometer and level readings for this and / the temperatures.

i —

1) In our case the influence of the shunt between A and B, C and D was so
small that it could be neglected and then ws, wy could be determined at once,

( 679 )

TABLE I. JENA GLASS.

| |
| 95/5 ‘03 A Ban C dD | Z Pr
| | |
| LA3ul |
Point below | 97.82 | 6.4 16.4
| 446 | $4.93 | 5.9
| ahavi |
Millimeter | | 7 920.07 54
[Point above 19.44. Gel | 16.70
| | = ; ; -
ee ) abi haay, AB aoe | Bete:
Millimeter) |} 4498 | 17.44 | 5.8 |
Point below eve Soma On| | | 16.54
| | |
above 19.47 | 6.0 | | 16.80 |
| | | |
| Lass? | | | |

The readings on the micrometer head are now reduced to a
standard position of the level and the temperature readings are
corrected. This gives the following.

TABLE II. JENA GLASS

95/5 703 A B C D
4/30
Point below | 27.81 | 16.37
ee | HG || Bist
Millimeter 17 | 90.45
Point above | 49.43 | 16.63
aa ae 4127 33.35
Millimeter 1198 17.16
Point below 27.83 16.47
s above 19.47 16.73
AhAS : 7
Point below 116.458
| » above 1197.859
time 1h371.5 Length 1014.4C1

Nothing new was in the method used for the determination of resistance.
It is hence only necessary to give the final results, as the means of
the various measurements reduced to the same time.

To calculate the temperature we have used the following preliminary
formula, obtained in the measurements described in Comm. N°. 77

ae Lie pee. SD ee

( 680 )
TABLE III. JENA GLASS.
4B | sm | 6.90
1
BC | 8.77 33.95
wep | 437 10.17

TABLE IV. JENA GLASS.

[a] s oa te) Lise acre ae wale
ie

D
Date Mean | Tempe Lr | Lae) We | W,

time. scale. |
i

20 V.) 4410 15.58 |1012.594 1012.587 | top 6.66 6.29
.63 -599 .588 | middle 36.04 | 33.95 16.03
69 593 .587 | bottom 10.66 | 10.17
22 V. 17.74 1011.834 1011.865 | top 5.10 6.29
3Al5

middle 22.15 | 33.95 |- 87.87
bottom 6.98 | 10.47

17.82 836
18.00 844

868

880
92 V| 5420 18.32 |4011.827 |1011.868 | top 5.01 6. 29
18.4 815 858 | middle 22.43 | 33.95 |- 87.87
bottom 6.91 10.47

O38 Vi) AM5 16.68 }4012.567 }1012.579 | top 6 68 6.29
16.68 .b73 .585 | middle 36.09 33.95 16.41
bottom 10.72 | 410.17

25 V.) 12410 16.08 1011.408 1011.409 | top 482 | 6.29
13 AML 443 | middle .8.77 | 33.95 |-182.9)
ly) 406 -409 | bottom 4.37 | 10.17

25 V.) 14415 16.38 1011.407 |1011.414 | top 4.68 6.29
-5OD 401 -441 | middle 8.77 | 33.95 |-182.9%

bottom 4.34 | 10.47

26 V.| 3410 17.30 1012.565 |1012.588 | top 6.70 | .6.29
AD -567 -594 | middle 36.12 | 33.95 16.64
bottom 10.66 | 10.17

( 681 )

with platinum wire of the same kind as that used in the present
instance

w, = w, (1 + 0.003864 ¢ — 0.0103 ¢?)
thus tpg — — 182".99.

The calculation of the temperatures of the projecting portions from
the values wag and wep will be described in § 4.

In the following table the final results") for all the determinations
are given, the standard scale at 16° C. being taken as the reference
length. Column / thus contains the values for the rod lengths reduced
to this reference. We have used as the expansion coefficient of
brass between 16° and 17° the value 17.8 &* 10-6 Column / refers
to the ends, and its contents will be considered in § 4.

TABLE V. THURINGER GLASS.
Pee (cm lene | G Fe Nad

| |
20 V.| 245 | 15.42 1013.407 |1013.091 | top 6.47 | 6.42
AA | -108 .098 | middle 36.53 | 34.53 15.08

bottom 10.21 9.68

22 V. | 12/30 | 47.08 |1012.9%4 /1012.263 | top 459 | 6.42 =
| | | 26.6
33} 938 | —.262 | middle 92.55 | 34.53 |-87.71
sey 239 | .263 | bottom 6.51 9.68 ‘=
| | 85.4
| 35 240 264

23 V. | 11440 16.68 4013.086 1013.098 top 6.52 6.12

«68 088 .100 | middle 36.70 | 34.53 | 16.36 |

bottom 410.23 9.68

25) Vi 3h20 17.04 |1011.744 |1014.763 | top 3.81 Gxt2" |

a
; | 1.0
he ON = eres .768 | middle 8.95 | 34.53. |-182.79)
te AG 2720 .761 | bottom 5.29 | 9.68 | a=
25.9
| 25 .738 760 |
\ | a | | | |
26-V. 11450 | 16.56 1013.095 1013105 top 6.46 | 6.42
| 67 098 | .410| middle 36.60 | 34.53 | 45.61| |

bottom 10.18 9.68

1) The numerical values are slightly different from the values given in the original
Dutch paper according to a new and more exact calculation. The final results
for the dilatation given in the original are quoted § 6 footnote.

( 682 )

§ 4. Discussion of the measurements. In § 2 we have already
remarked that the mean temperature of the platinum wire, wound
round the portion BC of the rods, which is at the temperature

of the bath, may, with sufficient accuracy, be put as

equal to the mean temperature of that portion of the rod

apf itself. Throughout this length, the differences of temperature

or the length over which they are found, are on the whole

sinall, so that only the mean temperature comes into

account. Further consideration is however necessary in

respect to the relation of the temperatures of the ends
AB and CD and the resistances determined.

Fig. 2. Let us suppose that the level of the liquid reaches to a
position 4, fig. 2, and hence that the upper portion of AZ is outside
the liquid. We may suppose that, for the length 4, the rod has the
temperature of the bath. The resistance of the wire between B and 4
is then w, = w, (1 + pt-+ qi’) where ¢ is the temperature of the bath.

Also we may suppose that at A, which was damp but just free
from ice, the temperature was about 0° C. Further let us suppose that
between 2 and / the temperature gradient is linear, in other words
that the external conduction may be neglected in comparison with the
internal conduction of the glass. There is every reason to assume
that this was true to the first approximation, since the glass rods were
well enclosed in paper the conductivity of which is about ‘/,,, of that
of glass. Then, neglecting the conduction of the platinum wire, itself
the resistance of an element of the wire between 4 and LF is wda,

o~

oe
where w= w, (t + pte + qt.) and the whole resistance _fuede,
oa
Further for « between O and 4, ¢,-=t#,, between 4 and JZ,
t
i; =t — Z 5 and) for 7 — > i, — OF so that

2
Wap = WB), = (1 + pt + gt’) +
L

1 t 2 t .
y ae > a — ¢ [6 — 7) du.
+ [Wan z[+2 ((— 4-9) -a(¢- Ge ») | ;

cs

From this 2, ‘the only unknown, can be obtained. One of the
most unfavourable cases, that for the upper end of the Jena glass
rod in N,O, shows when calculated that the linear form for the
resistance can be employed in our measurements without difficulty,
in place of the quadratic form. We found 2= 8.4 em. with the

( 683 )

quadratic and 4= 9.0 cm. with the linear formula. The uncertainty
thus introduced into the determination of length, is less than Iu.

In order to determine the influence of various suppositions with
regard to the distribution of temperature in the rod, we have
‘alculated the change in length which would be produced, if the
temperature was — 87° C. from 0 to 4 and 0° from 2 to “4, in
place of the distribution assumed above. The change was hardly
O..u and thus lies within the degree of accuracy. However an
important control indispensable for more accurate determinations
would be obtained by measurements on a rod with similar ends
AB and CD, but where BC was only a few centimeters long ’).

To apply generally the method of this section for the determination
of mean temperature it may be necessary to subdivide the portion
of at variable temperature AZ into more parts while for each of
these separate portions the resistance would have to be found. In
our case this would have been an unnecessary complication.

§ 5. Influence of errors. These can be fully considered by the
aes vl

15; ee

1

aid of @=

The accuracy of the cathetometer reading can be put at 2u (the
whole contraction being 1200). This gives da = 2 >< 10—. For the
mean temperature of the portion SC the error is certainly less than
0.5 deg. C, whence di =1.5 « 10-8, and for that of the ends we
found 1g. Hence a greater uncertainty than d@ = 4 10~% is not to
be expected. Although the division of this error between @ and 4
cannot well be made, it is certain that an error in the temperature
determination has by far the greatest influence on 0.

§ 6. Final results. For the observed lengths Lyyo, at the tempe-
rature ¢yo, in nitrous oxide, Lo, in oxygen, and Lyjgo at ordinary
temperature we have the three equations

Lino, = (Lec, + 4: + 4s) (1 + atvo, + bo.) +
: : 1 1
—- Ly, “= Ls, sie (ts — A; + Lj, rag i) (: a S atn Ox = = Wexo,)
5

and two analogous ones for /io, and Lig, with Lec, = 840 mM.,
Lj, = 97, L., =59 for Jenaglass, and Lg¢, = 834, L;, — 96, iE, == G0!
for Thiiringerglass. For Lc, (the length of the part LC in the figure
at 0 C.), Lj, L,, (that of the parts CD and AB in the figure) are
assumed approximate values; the exact values Zy, and Z;, to be

1) For Jenaglass in oxygen we found a negative value of a, we made therefore
the calculation on another supposition viz. that from A in the direction of B the
rod has the temperature 0° over a length of A’ cm. (ef. Table LV).

47

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 684 )

ascribed accordingly to the lengths at O° C. of the points projecting
beyond A and PD follow from the equations. These equations give
Ly, + “Ls5,= 16.587 and Ly, + L;,—= 23.095 tor Jena and Thiiringer
glass respectively, and further ;

L=L, (1 + at + bt?)

V=V,(1+h,t+4,2)
Jena glass 16" |, — 7.74. 10~°,b = 0.00882. 10 §
a 23.21. 10-°,k.— 0.0265. 10-6
ees 9.15 10-*,b —0.0119 10-5
( k, =27.45 10-6, k, — 0.0857 10-6
The value found for Jenaglass 16"! differs much from that obtained

by Wrese and Borrcner*) and fyom those obtained afterwards by
Turesen and Scueet*) for temperatures between 0° and 100°.

Thuringer glass (n°. 50)

Physics. — “The motion of electrons in metallic bodies, Il.” By
Prof. H. A. Lorentz.

(Communicated in the meeting of March 25, 1908).

§ 16. We may now proceed to examine the consequences to
which we are led if we assume fro kinds of free electrons, positive
and negative ones. We shall distinguish the quantities relating to
these by the indices 1 and 2; e.g. .V, and .V, will be the numbers
of electrons per unit of volume, m, and m, their masses, = =
z oh, U
the mean squares of their velocities. For simplicity’s sake, all elec-
trons of the same sign will be supposed to be equal, even if con-
tained in different metals. As to the charges, these will be taken to
have the same absolute value for a// particles, so that

OS ek oct oy Poa: ae

Our new assumption makes only a slight difference in the formula
for the electrie conductivity ; we have only to apply to both kinds |
of electrons the considerations by which we have formerly found the
equation (21). Let a homogeneous metallic bar, having the same
temperature throughout, be acted on in the direction of its length
by an electric force /; then, just as in § 8, we have for each kind
of electrons

2) In the original was given
Jena 16"! “«=7.78 b=0.0090
Thiiringer n° 50 a=9.10 b=0.0120.
2) Wiese und Bérrcuer. Z. f. Inst. k. 10, pg. 234. 1890.
3) Turesen und Scheer, Wiss. Abt. der Ph. techn. Reichsanstalt. Bd, ILS, 129, 1895.

( 685 )

Aal Le
v — — <<
shm

The electric current per unit of area of the normal section, is the

Ue

sum of the currents due to the positive and the negative particles.
We may therefore represent it by

4, A,e,? 4atl, Aye,” E
3hym, 1 dhym, :

and we may write for the coefficient of conductivity

Ol Oe Ozemeci ere Tenu rs) We) os (49)
if
4a l, A, ON ee 4a Up A, fp
iS ¢ ’ 2-5 ets eT) ‘
3h, m 3h, m

or (cf. § 8)

5) AT se Oy Ne pu?
2 Le ING Oy 2 LINO ae
o, = a ease bt et es a ese? (50)
3av a 3m a

These latter quantities may be called the partial conductivities due
to the two kinds of electrons.

§ 17. In all the other problems that have been treated in the prece-
ding parts of this paper, we now encounter a serious difficulty. If
either the nature of the metal or the temperature changes from one
section of the circuit to the next, we can still easily conceive a state
of things in which there is nowhere a continual increase of positive
or negative electric charge ; this requires only that the ¢o¢a/ electric
current be 0 for every section of an open circuit and that it have the
same intensity for every section of a closed one. But, unless we
introduce rather artificial hypotheses, it will in general be found
impossible to make each partial current, i.e. the current due to each
kind of electrons considered by itself, have the same property. The
consequence will be that the number of positive as well as that of
negative electrons will inerease in some places and diminish in others,
the change being the same for the two kinds, so that we may speak
of an accumulation of “neutral electricity” in some points and of a
diminution of the quantity of neutral electricity in others. Now,
supposing all observable properties to remain stationary, as indeed
they may, we must of necessity suppose that a volume-element
of the metal contains at each instant the same number of really
free electrons. This may be brought about in two ways. We may
in the first place imagine that all electrons above the normal number
that are introduced into the element are immediately caught by the
metallic atoms and fixed to them, and that, on the other hand, in

7%

(*

( 686 )

those places from which electrons are carried away by the two cur-
rents, the loss is supplied by a new production of free electrons.
This hypothesis would imply a state of the circuit that is not, strictly
speaking, stationary and which I shall call ‘‘quasi-stationary”’. Moreover,
we should be obliged to suppose that the production of free electrons
or the accumulation of these particles in the metallic atoms could
go on for a considerable length of time without making itself any-
ways felt.

In the second place we may conceive each element of volume to
contain not only free positive and negative electrons, but, in addition
to these, a certain number of particles, consisting of a positive and
a negative electron combined. Then, the number of free electrons
might be kept constant by a decomposition or a building up of such
particles and we could arrive at a really stationary state by imagining
a diffusion of this “compound electricity” between different parts of
the circuit.

§ 18. The mathematical treatment of our problems is much sim-
plified by the introduction of two auxiliary quantities.

In general, in a non-homogeneous part of the circuit, the accele-
ration .Y will be composed of the part Y,,, represented by (30), and

é 5 : . Si . .
the part — /, corresponding to the electric force 4. The formula (21)
m

for the flow of a swarm of electrons may therefore be replaced by
2 1 2hAdV 2heA 1A A dh
pea ApS ee [ees phe) ey oe
3 h? m dx m dit

This will be 0, if the electric force / has a certain particular
value, which I shall denote by E and which is given by

eee m dlogA md /1
= = az} = ree Aas (52)

edz | 2he da e daX\h

For any other value of the electric force the flow of electrons
will be

and if, in order to obtain the corresponding electric current, we mul-
tiply this expression by e, we shall find the product of 4— E by
the coefficient of conductivity, in so far as it depends on the kind
of electrons considered.

Substituting in (52) the value (14) and applying the result to the
positive and the negative electrons separately, we find

( 687 )

K Welle 2 aT dlog A, 4adT
Cu meeGn, <a. ' 5 aaa |
: (53)
K ab 2 2 aT dlog A, Aa ar
camp Te) et ae ee 3 e, da

If the area of a normal section is again denoted by , the inten-
sities of the partial currents are given by
=O (2 — Ei), = 0,(2 —E) >, . =. . (a4)
and that of the total current, on account of (49), by
Nevin teh (OO eh On Be) sas fs ts + (DD)

Putting

we may also write

0,6.

A ; LA ? r 0,6.
oi oe); Jo, fla SE a

O (E, i E,) =.

It appears from these formulae that, whenever E, differs from K,,
the partial currents 7, and 7, will not be proportional to the conduc-
tivities 6, and o,.

§ 19. The above results lead immediately to an equation determining
the electromotive force /’ in an open circuit composed of different
metals, between which there is a gradual transition (§ 6) and which
is kept in all its parts at the same temperature. Let ? and Q be
the ends of the cirenit and let us reckon « along the circuit in the
direction from P towards Q.

The condition for a stationary or a quasi-stationary state is got
by putting 7= 0 in (55). Representing the potential by g, so that

dy
wi da 3
we get
dy Oa oO, -
ee SR es Sk . (56
da: aes Gia G (a0)
and finally, taking into account the values (53), in whieh we now
dT tole
have == 0, and integrating from P to Q,
av

Q
US (Gon ae leap Cie
ie == Tar 4 3 dz —
Pa Pp ~ {2 da se =| o dx ;

( 688 )

Q Q
2 aT (6, dlog A, ; 2aT (6, dlogA, ; 57
ar) 5 c Gece Sie, bo ieee: ae

P P
At the same time the intensities of the partial currents are given by

, = 22 GB) 5

’ U

y=

These values, which are equal with ee signs, will in general
vary along the circuit, so that, even in this simple case, we cannot
avoid the complications I have pointed out in § 17. Nor can the
difficulty be easily overcome. Indeed, we can hardly admit that the
state of two pieces of different metal, in contact with each other
and kept at a uniform temperature is not truly stationary. If, in
order to escape this hypothesis, we have recourse to the considera-
tions I presented at the end of § 17, we must suppose the neutral
electricity to be continually built up in some parts of the system
and to be decomposed in other parts. The first phenomenon will be
accompanied by a production and the second by a consumption of
heat. That these effects should take place in a system whose state
is stationary and in which there are no differences of temperature,
is however in contradiction with the second law of thermodynamics.

The only way out of the difficulty, if we do not wish to confine
ourselves to one kind of free electrons, seems to be the assumption
that there is no accumulation of neutral electricity at all, i. e. that
7, and 7, are simultaneously 0. This would require that E, = E,,
or in virtue of (53)

dV 2cel dilog A dV 2 ca) LdilogeA!

:
ee pe es he ae
e da 3 ey da e, da at oime, dx (3a
Since e, = —e,, we might further conelude that
2 pillog (Ar A) V+ VA) 9
3 da: da:

whieh means that
log (Ay A) + 57M, + Y= 9D)

ought to have the same mae in all parts of the circuit. We should
therefore have to regard this expression as a function of the tempe-
rature, independent of the nature of the metal‘).

If we suppose the contact of two metals to have no influence on
the number of free electrons in their interior, we must understand
by A, and A, in the above equation quantities characteristic for each

1) Cf. Drupe, Annalen der Physik, 1 (1900), p. 591.

( 689 )
metal and having, for a given temperature, determinate values, whether

the body be or not in contact with another metal.

~

By the assumption E, = E,, (56) simplifies into

and (57) becomes

ie es sf 2 aT A,p
Bea oe iP anlar 3 => a A =

v9 @, 1Q
tages 2 2aT A, Pp 3
y= (Vie iq) f= — log (ys: 9)
4 vo ¢, 444Q

a formula which is easily seen to imply the law of the tension-series.

§ 20. The question now arises, whether, with a view to simplifying
the theory of the thermo-electric current, we shall be allowed to con-
sider E, and E, as equal, not only in the junctions, but also in the
homogeneous parts of the circuit, in which the differences of tempe-
rature come into play. This seems very improbable. Indeed, supposing
for the sake of simplicity V7, and |’, to be, for a given metal, inde-

dV dV,

. . 1 _ P
pendent of 7’, so that in a homogeneous conductor —— = 0 and ==)
av av

we find from (53), putting E, = K,,

2 aT dlog A, 4 adT 2 aT dlog A, 4 adT

3 e, da 3 e@, dx a) Ge da 3 e, dx 1
or, since e, = e;
T eae) ee yeas
du de’

which can hardly be true. It would imply that the produet A, A, is
inversely proportional to the fourth power of the absolute temperature
and this would require in its turn, as may be seen by means of (13)
and (14), that the product V,.V, should be inversely proportional to
7’ itself.

We are therefore forced to admit inequality of E, and E,. Now,
it may be shown that, whatever be the difficulties which then arise
in other questions, the theory of the electromotive Jorce remains
nearly as simple as it was before. For an open circuit we have
again to put ¢=0; hence, the formula (56) will still hold, as may

4 : eas dp ,, ee
be inferred from (55), if we replace / by — —. The equation for
ae

the electromotive force becomes therefore

( 690 )

tt | ;

I =o ~G,=—| 56 EK, + 6,H,)de . . . (60)
P

In the case of a closed circuit, which we get by making the points

P and Q coincide, we shall integrate (55) along the circuit after

dz 5 dg ,
having multiplied that equation by = and replaced / by The

—_— wv

intensity 7 being everywhere the same, the result takes the form

i ae 61
i aS — . . . . . . . . . ( )

This is the mathematical expression of Oxnm’s law.

§ 21. It must further be noticed that the equation (60) agrees
with the law of the thermo-electric series. This may be shown as
follows. If we suppose the temperature to be the same throughout
a junction, we may easily infer from what has been said in § 19
that the part of the integral corresponding to such a part of the
circuit can be represented as the difference of two quantities, which
are both functions of the temperature, but of which one depends
solely on the nature of the first metal and the other on that of the
second. Considering next a homogeneous part of the circuit between
two junctions, we may remark that in this E, and E, have

’

dx

temperature. We may therefore write for the corresponding part
of (60)

the form 7 (7)

: 0 0. :
and that the rations = and = are functions of the

Yad

fucmar.

ui

This integral, which is to be taken between the temperatures 7” and
7” of the junctions, may be considered as the difference of the values,
for T= T7' and 7 = 7", of a certain quantity depending on the
nature of the metal.

Combining these results, we see that the electromotive force in a
given circuit is entirely determined by the temperatures of the
junctions, and that, if there are two of these between the metals
/ and //, the electromotive force /’);; we have examined in §10¢
may still be represented by an equation of the form

Friu=b (1) — 8 2") —Su (2) + Su (2"),
the function $,(7’) relating to the first, and the funetion $,(Z’) to

( 691 )

the second metal. The law of the thermo-electrie series may imme-
diately be inferred from this formula. However, in order to obtain
this result, if has been necessary to adopt the hypothesis expressed
by (58).

I shall terminate this discussion by indicating the way in which
our formulae have to be modified, if, in the direction of the eirenit,
the electrons are acted on not only by the electric foree caused by
the differences of potential, but also by some other force proportio-
nal to their charge and whose line-integral along the circuit is not 0.
Let us denote this force, per unit charge, by /, and let us write for

its line-integral
| Aa ei.

This latter quantity might be ealled the “external electromotive

foree” acting on the cireuit. Now, in the formulae (54), we must
replace by H+ .. Consequently, (55) becomes

i 10) (BE E.) oO; Ki, 0; K,},

and treating this equation in the same way as we have done (55),
we find instead of (61)

§ 22. I shall not enter on a discussion of the conduction of heat,
the Prirmr-effect and the THomson-effect.

In the theory which admits two kinds of free electrons, all ques-
tions relating to these phenomena become so complicated that |
believe we had better in the first place examine more closely the
Hatr-effect and allied phenomena. Perhaps it will be found advisable,
after all, to confine ourselves to one kind of free electrons, a course
in favour of which we may also adduce the results that have been
found concerning the masses of the electrons. These tend to show
that the positive charges are always fixed to the ponderable atoms,
the negative ones only being free in the spaces between the molecules.
If however a study of the Hatt-elfeet should prove the necessity
of operating with both positive and negative free electrons, we shall
be obliged to face all the difficulties attending this assumption.

( 692 )

Geology. — “Contributions to the knowledge of the sedimentary
boulders in the Netherlands. 1. The Hondsrug in the province
of Groningen. 2. Upper Silurian boulders. Second communi-
cation: Boulders of the age of the Eastern Baltic zones H and
I.’ By Dr. H. G. Jonker. (Communicated by Prof. K. Martry).

H.

Besides the Borealis-limestone, described in my preceding commn-
nication (33) and on which I am going to touch later on, boulders
with Pentamerus-remains near Groningen are rare. I can mention
but three pieces here, in two of which the species is not to be made
out, while in the third, found in the ‘“Noorderbegraafplaats” in Gro-
ningen, Pentamerus estonus E1rcuw. occurs. Nor is this determination
beyond doubt and especially the possibility of its being Pentamerus
oblongus Sow. can in my opinion not be excluded, as indeed in out-
ward appearance the latter corresponds almost perfectly with the
former (12, p. 81 and 3, T. XVIII, f. 4%). As however, the latter
form in Gothland has no doubt to be looked upon partly as the
real P. estonus Eicnw. (27, p. 98), nothing can be said for certain
about its origin, as the rock, a weathered, yellow limestone does
not give sufficient indications for it. I mention this boulder however
for completeness’ sake.

With regard to the Borealis-limestone I wish to add, that after
all I did find an almost complete specimen of Pentamerus borealis
Eicuw., in the Groningen museum, evidently from a Groningen
boulder. The correspondence with the specimens from Weissenfeld,
mentioned before, is however not very great, the top of the ventral
valve in our specimen being much more curved and thus agreeing
more with Ercuwa.p’s description.

A close investigation removing the existing confusion with regard
to the Upper Silurian Pentamerus-species is really most desirable.

31. Clathrodictyon-limestone.

White limestone, sometimes having a more or less light-yellowish-
gray tinge. At the surface and in cavities the colour is rather yellow.
It is always crystalline and the very irregular fractured surfaces
show a peculiar fatty silk-gloss, which is most characteristic of them.
If the colour becomes a little darker, as is sometimes the case, the

( 693 )

gloss remains preserved. The rock is a veal Stromatopora-limestone,
which may be distinetly perceived in some pieces, as they consist of
slightly curved, concentric layers the surface of which is covered
with small knobby mamelons (25, Pl. XVII, f. 14), which make it
more than probable that we have to do here with

Clathrodictyon variolare Rosxn sp.

Its structure, however, is not easily traced on account of the erystal-
line character of the stone.

This species of boulder further contains real fossils only in the
form of peculiar conical cavities, mostly slightly bent towards the
point. On the inside they are invariably set with annular edges,
which on an average are lying a little more than 1 mm. from each
other in specimens of an average size. The cavity is often completely
filled up with crystalline calcite bright as water. Its rather thick
wall presents on the outside small irregularly running lines of growth.
Frieprich Scumipt, Akademiker in St. Petersburg, whom I sent a
piece of this limestone, was kind enough to inform me that these
cavities originate from Cornulites sp. (1, T. 26, f. 5—8), a fossil of
the [-zone in Oesel, frequently occurring near St. Johannis.

These boulders are by no means rare near Groningen as appears
from the following list:

“Noorderbegraafplaats’’, Groningen 6
“Boteringesingel”’, 3 7
“Nieuwe Kijk in ‘t Jatstraat’’, ~ 1
“Nieuwe Veelading’’, i 3
“Sehietbaan’’, x 1
Behind the “Sterrebosch’’, 2 1

= 1
Café “the Passage’, Helpman 1
Villa “Edzes” near Haren 1
The “Huis de Wolf” near Haren J
“Klein-Zwitserland’” near Harendermolen 1

About the occurrence of the mentioned species of Stromatopora
Nicnoison records it from Borkholm and Worms in the Borkholm
stratum in Esthonia, but he has especially found them frequently in
the Estonus-zone there, chiefly near Kattentack. (25, p. 151). He
does not record it from Gothland, though this fact is not sufficient
altogether to exclude its occurrence there. Moreover Linpstrém

( 694 )

mentions three other species of this genus (16, p. 22). Among my
material for comparison is a specimen from Klein-Ruhde, to the
west of Kattentack in the H-zone in Esthonia. This rock is some-
what darker, more grayish; but yet examples are to be found among
our pieces which perfectly resemble it, so that the correspondence
may really be called striking. The described Cornulites do not occur
in it which it is trne cannot surprise us im a piece of so small
dimensions (7 & 6 >< 2 ¢.M.).

Finally I wish to state that in a boulder of stromatopora-limestone
in Gothland, I found analogous Cornu/ites-cavities, which petrogra-
phieally does not altogether agree with our pieces. The place where
it is found is immediately to the north of Hégklint, on the field
(not in the beach). But this fossil is of little importance for the
further determination of the age of the rock, as most likely various
species will be implied in the name of Cornu/ites serpularius SCHLOTH.
which is usually given.

Taking everything into consideration, it seems possible (perhaps
even probable) to me that this Clathrodictyon-limestone comes from
the H-zone in Esthonia or from its western continuation.

In connection with this must be said that among the very nume-
rous stromatoporae of the Hondsrug (of which specific determinations
are hardly ever possible) two occur which from their characteristic
astrorhizae may be called :

Stromatopora discoidea Loxsp. sp... . . 25, Pl. XXIV, f. 2.

Both pieces, found in the “Noorderbegraafplaats” and in the * Violen-
straat’” in Groningen, consist of fine-grained crystalline (stromatopora-)
limestone ; the former is all over white and therefore closely resem-
bles Clathrodictyon-limestone, the latter is rather grayish and also
partially weathered, which fact decreases the correspondence.

This species, very common in Wenlock limestone from England,
also ocenrs in the neighbourhood of Wisby in Gothland. Nicnonsox
‘alls those Gothland specimens however usually highly mineralised
(25, p. 191), which with my material from Gothland corresponds
but to this extent that this fossil occurs only as a not always very
thick crystalline crust in marl or marly limestone. LinpsTrR6M records
it only from 4 (16, p. 22), his youngest zone of the Upper Silurian
of Gothland (7, Dames). Contrary to this I allege to have found a
specimen (it is true somewhat differing in a smaller number of
astrorhizae) in the caleareous marl immediately to the north of
Hogklint, occurring there as firm rock; this fossil comes from a

( 695 )

stratum about 1 M. above the beach. This petrographical and strati-
graphical occurrence is, it seems to me, hardly to be referred to the
age fh; the other specimen in the Groningen museum supports my
observation only to the length of containing marly remains. still
distinctly to be seen. The place where if was found is, however, not
further indicated.

Our Groningen fossils have upon the whole but little in common
with these Gothland pieces; meanwhile this fossil also occurs in
Ksthonia near Klein-Ruhde in the Estonus-zone. That is why these
two pieces have been mentioned here though no further data can
be brought forward to prove their origin from these Eastern Baltic
regions through want of material for Comparison.

I.

Boulders which correspond in age with the Lower Oesel zone in
the Eastern Balticum, are not rare near Groningen. Bonnema already
pointed it out some years ago (31); this short essay, however, has
more of a palaeontological character, so that | wish to complement
these Communications and enter into further particulars.

32. Baltica-limestone.

In an unweathered state rather hard, tough, tine-grained-crystal-
line limestones of a bright-gray or light-brownish-gray colour. Some
pieces are almost impalpable; some parts are coloured bluish-gray
on the inside, so that the rock may originally have had that colour.
Through weathering the bright-gray tinge passes into light-yellowish-
gray; the uneven fractured surfaces then are very often covered with
sallow-yellowish and brown spots. Crystalline calcite rarely occurs.
The limestone is rather pure, but a littke marly and hardly ever
slightly dolomitic. Real dolomites are not among them. Stratification
is imperceptible. The dimensions of the pieces found amount to 25 em.

Fossils are not present in great numbers, chiefly Ostracoda, among
which Leperditia-shells are the most important. Whilst bright-brown
in the unweathered rock, the valves which sometimes occur frequently
in a single piece, have become nearly white by weathering. As is
often the case with the younger Leperditia-limestones, which are to
be described later on, this limestone is not unfrequently connected
with Stromatopora-limestone; the fossils to be mentioned below,
however, never occur in it. Besides these large Ostracoda-remains,
small Beyrichia- and Primitia-valves are also frequently found but

( 696 )

they become only distinctly visible through weathering. The fossil
fauna consists of the following species.

Leperditia baltica His. sp.

Strophomena rhomboidalis Wick. sp.

Strophomena sp.

Atrypa reticularis L.

Meristella sp.

Encrinurus punctatus WAH.

Zaphrenthis conulus LinpstR. ... 2... 28, p. 32, T. VI, f. 65—68.
Orthoceras sp.

Murchisonia sp.

Tentaculites sp.

Primitia seminulum Jones... ........ 14, p. 413, Pl. XIV, f. 14.
Primitia mundula Jones. . . . 23 T. XXX, f. 5—7; 18, p. 375, Pl. XVI.
Beyrichia Jonesti Bout... ....... .. 17, p. 13, T. Tl, £. 10a
Beyrichia spinigera Bou... 2... . 23, p. 501, T. XXXII, f. 19—20.

The first meutioned Leperditia-species is present in all pieces; all
other fossils, however, occur either few and far between or in a
single piece, excepting the small ostracoda. I have however not taken
much pains to increase the number of species of them (for the greater
put already mentioned by Bonnema), because their stratigraphical
value is still but tritling nowadays. Then to determine age and origin,
we can restrict ourselves to the communication where and in which
strata occurs the type-fossil of this group, Leperditia baltica His. sp.
(after which in accordance with the names of Phaseolus-limestone
and Grandis-limestone, generally in use, I have called these limestones).

First of all, however, the number of the pieces found here and
the special places where they were found, be given here:

“Noorderbegraafplaats’’, Groningen 7
“Boteringesingel’, - 4
‘“Noorderbinnensingel’, ce

“Violenstraat’’, i .
“Nieuwe Boteringestraat”’, “3
“Nieuwe Kijk-in ’t Jatstraat” _
“Nieuwe Veelading”’, n

“Old Collection”

Helpman

“Hilghestede”, Helpman
Between Helpman and Haren
Harendermolen

FPMmMRMwWwWRR NWR HB wx

So in all 81 pieces. The number found is presumably much larger,
because I have only mentioned here the boulders which beyond any
doubt belong to this group; among the numerous limestones with
Leperditia-remains which cannot be specifically determined there
will no doubt be a number of this age.

Leperditia baltica Wis. sp.

Literature: 1869. Kormopin, 2, p. 13, f. 2—3.
1873. Scampr, 4, p. 15—17, f.19—21.
1876. -Rormer, 5, T. 19, f. 7.
1878. Martin, 6, p. 45.
1880. Kotmopin, 8, p. 134.
1883. Scum, 10, p. 11—13, T. I, f. 1—3.
1884. Kuissow, 11, p. 275, T. IV, f. 4.
1885. Reneé, 13, p. 26, no. 226.
1888. Linpsrrém, 16, p. 5, no. 25.
1890. Kixsow, 19, p. 89—91, T. XXIII, f. 14—16.
1890. Scummpr, 20, p. 255.
41890. Dames, 21, p. 1125.
1891. Krause, 22, p. 5, 7.
1891. Krausr, 23, p. 488, T. XXIX, f. 1—3.
1891. Scumipt, 24, p. 123.
1895. Sroiiny, 27, p. 109.
1898. bonnema, 29, p. 452.
1900. CHmiEeLEewsk1, 30, p. 17—20, 33; T. I, f. 17—20.
1900. Bonnema, 31, p. 138—140.

From the literature about this fossil, cited above, which as regards

the later years is rather complete, it appears that for a long time
a certain confusion and uncertainty about the limits of the species
have existed, which have been removed but a few years ago.
Besides the real L. baltica His., characterized by the comb-shaped
striae on the inverted plate of the left valve (L. pectinata Scumipt) —
which characteristic may be distinctly perceived in twenty of the
boulders from here —, Scumimr had also described another species :
L. Kichwaldi Scum. Boxxuma has proved that both species have to
be united (81); at nearly the same time this has also been observed
by Cumietewski. The latter, however, distinguishes besides the typical
form two other varieties :

L. baltica, var. Hichwaldi Scumivr
23 5 formosa CHMIEL,

These two varieties are present among our boulders, var. Lichwaldi

( 698 )

not unfrequently, var. formosa less often. But the characteristics of
these varieties are by no means conspicuous, so that there are
specimens which partake of the nature both of these varieties and
the real species, as Cu irLewskt himself too has perceived. In
accordance with this is the fact that these varieties are practically
of no stratigraphical importance; it is on these grounds I have
thought it allowable to combine all these forms in+one species under
the name of Leperditia baltica His. sp.

It has been frequently found in boulders. Kirsow describes it from
“weisslich-grauen Mergelkalk” of Langenau, from ‘‘ziemlich verwit-
terter und in Folge dessen gelblich gefarbter Kalk mit zahlreichen
Schalen der Leperditia baltica Urs. (FP. Scummpr); daneben finden
sich Enerinurus punctatus, Alrypa reticularis, wid einige schlecht
erhaltene Beyrichien, u.s.w.’ from Zoppot-Olivaer Walde, also in
West-Prussia. The first stone corresponds perfectly with limestone
from Langers in the N.E. of Gothland, the second shows much
correspondence with the occurrence of Oesterby near Slite. Therefore
he vefers these pieces to Gothland. (Of the co-occurrence of L. baltica
His. sp. and L. Hisingert Scum., which question I treated of in my
previous communication (33, p. 560), he is afterwards not quite
sure — 19, p. 90). In his excellent, already frequently cited treatise
CronmeLewski briefly describes six boulders in which he has found
L. baltica in Kurland, Kowno, East and West-Prussia. Most cor-
responding with our boulders seems to be his: ‘hellbraunlich-grauer,
deutlich krystallinischer, wenig thoniger, fester, unebenbriichiger
Kalkstein mit Lnerinurus punctatus (80, p. 33),” from Kowno. He
does not give a decided opinion about the origin.

Farther to the west this species is still recorded from Brandenburg
by Rempté and Krauss, mostly together with fossils, which also
oceur in our boulders and from limestones which, so far as can be
gathered from ihe short descriptions, correspond in some respects
with ours. Srouuny describes also various of those limestones from
Sleswick-Holstein among which “ein gelber Kalk enhalt neben
L. baltica Uis., Atrypa reticularis L. und Encrinurus punctatus
WauLenperG” is again conspicuous. From Groningen our species
was already recorded in 1878 by Martin, from Kloosterholt after-
wards also by Bonnema (29, p. 452).

From these statements about the erratic occurrence of this species,
it appears sufficiently, that it has spread from Kurland and Kowno
to the Netherlands though nowhere, it is true, large numbers of sueh
boulders have been met with. In the Scandinavian-baltie area it is
found in different places in solid rock :

( 699 )

Is. In Malmo new Christiania which is not very important
to us;

2. In Gothland, where Scumipt describes its occurrence as follows :

“Das grosse Centralmergelgebiet von Follingbo bis Slite und Faré,
das bald aus reinen Mergeln, bald aus Mergeln mit Kalken wech-
selnd besteht, wird neben andern Fossilien besonders durch die
urspriingliche Leperditia baltica His. mit kammférmiger Zeichnung
aif dem Umsehlag der linken Sechale characterisirt, die einerseits
auch bis zu den Mergeln von Westergarn vordringt und andererseits
sich vielfach auch in den oberen Kalken der Wisby-Region findet,
so bei Heideby und Martebo. Auf F r6é bei Lansa kommt sie zusam-
men mit Zaphrentis conulus Lixpsrr., Strophomena imbrex Vurn. u. a.
im Kalk vor, weehselnd mit JMJegalomus-banken.” (20, p. 255). These
places belong to Scripr’s middle zone; besides Koumoprn records it
from Oestergarn and Hammarudd near Kriiklingbo (8, p. 184), where
no doubt younger strata are found. In these two places I have
been seeking for a long time, but failed to find it. According to
Linpstrom : b—e.

3°, In Oesel this species is a type-fossil of the Lower Oesel-
zone I. For a long time it was only known from dolomite from
Kiddemetz (var. /ichwaldi) but has later also been found in lime-
stones in the peninsula of Taggamois on the N. W. coast, thus
verifying Scumipr’s prediction. Only there this zone consists of erys-
talline limestone ; everywhere else of dolomite or marl (9, p. 46—49).

With regard to the origin of these boulders whose age has now
been determined, the following remarks may be stated. First of all
the fact that Skine cannot be thought of, as Leperditia baltica does
not occur there. In general the marly character of the rock found
in Gothland, argues against the possibility of its originating there ; no
doubt we have only to think of the north eastern part of the island.
Though indeed our boulders do uot make the least impression of
originating in marly strata, it does not say so very much, because
in Gothland the limestome with LZ. baltica cannot everywhere be
decidedly looked upon as being limestone from marl. The question
then about their origin is not to be solved without extensive material
for comparison, which I do not possess; only a single piece of lime-
stone from Slite does not correspond with our boulders. This lime-
stone from Slite is differently coloured and also much more crystalline
and betrays by marl-remains and a small concretion of little pyrite-
crystals its origin from marl. Now as regards Oesel, from this region,
too, I have but a single piece with Z. baltica for comparison. It is
from Kiro, immediately to the south of Taggamois, and corresponds

48

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 700 )

much more with our boulders. It is however but a badly preserved,
weathered piece, so that it is not very important.

Taking all this into consideration the origin of our boulders is
probably to be found between Oesel and Gothland, where there is
every reason to assume that along the line Faré—Taggamois lime-
stones of the age of the I-zone have been developed.

More or less closely allied to this Baltica-limestone are different
boulders which for their fossil contents may best be referred to the
Lower Oesel zone:

a. Yellowish-gray limestones with:

Proetus concinnus Datm., var. Osiliensis Scum. 26, T. IV, f. 1—9.

Calymmene tuberculata Brinn. 26,T.I, f.1—7.

Cyphaspis elegantula Lov. sp. 7; TX Vilgaeaet

Encrinurus punctatus W su.

Strophomena rhomboidalis Wi.ck. sp.

Orthis sp.

They closely resemble some pieces of Baltica-limestone and most
likely neither differ very much from the latter in age. Without
tracing their occurrence in particulars here the following list shows
sufficiently why they are mentioned here:

Gotland (16) Oesel (26)

P. concinnus Datm., (var. Osiliensis Scum.) (c—e) dh
C. tuberculala Brinn. c—f 5
C. elegantula Lov. sp. c vi
Eight pieces of this limestone are from the following places:

“Boteringesingel’’, Groningen 2

“Noorderbinnensingel”, __,, »

“Nieuwe Veelading”’, i 1

3 1

“Hilghestede”’, Helpman 1

i

”

Again the tract between Gothland and Oesel must be looked upon
as the place of origin by reason of perfectly similar considerations
as mentioned in dealing with the Baltica-limestone.

b. Perhaps two limestone-rocks also belong to this with
Bumastes barriensis Murcu.
found in the “Nieuwe Veelading” and the “Schietbaan” in Groningen,

( 701 )

while Horm records this fossil from the Eastern-balticum from /
(15, p. 37), Linpstrém from 6—A in Gothland (16, p. 4, N°. 64).

e. Thirdly various limestones with
Encrinurus punctatus WAune.
may be mentioned here. These ‘“Encrinurus-limestones’” are not
further to be determined in age on account of the want of other
adequate fossils. Some corals, Fuvosites and Halysites, together with
which they sometimes occur, can be of no use for that purpose.

d. Among the great number of corals from the Groningen
Hondsrug there are no doubt many of the age of the Lower Oesel
zone e.g, Thecia Swinderenana Goupr and others. However I do not
intend to occupy myself with this question, but later on I shall deal
with these together with the other corals, whose age is hardly ever
to be determined between narrow limits, under the heading “Coral-
limestone.”

e. Finally I wish just to make mention of a single piece of
dark-greenish-gray calcareous marl, which contains numerous pygidia
and head-shields of a Calymmene-species. This boulder found in
the “Boteringesingel” in Groningen suggests the marly stratum of
St. Johannis of the /-zone in Oesel, but also corresponds fully with
marls from different places in Gothland. About the origin, then,
nothing can be said. Probably we have to do here with Remes’s
“Griinlichgrauer Calymenekalk”. (13, p. 27).

Here ends the enumeration of the boulders of the age J. Be it
only added that this zone may possibly be well represented among
the very manifold dolomites of Groningen. These, however, but
seldom contain fossils and on account of this admit of no distinctly
separated groups. At the end of the description of the Upper-Silurian
boulders, I hope to be able to communicate some particulars about this.

LITERATURE.

1. Murcuison, R. J. -— »The silurian system’. -
London, 1839.
2. Kotmopiy, L. — »Bidrag till kainnedomen om Sverges siluriska Os-
tracoder’’.

Inaug.-Dissert., Upsala, 1869.
3. Davipsoy, Tu. »A monograph of the British fossil Brachiopoda.
ITI. Devonian and silurian species”’.
London, 1864—7).

48*

ve)

10.

13.

14.

15.

Whe

18.

19:

( 702 )

Scumipt, F. — »Miscellanea Silurica. — 1. Ueber die russischen siluri-
schen Leperditien mit Hinzuzichung einiger Arten aus
den Nachbarldandern’’.

Mém. de VAc. Imp. d. Se. de St. Pétersbourg, Vile Sér., T. XXL, no. 2; 1873.

. Romer, F. — »Lethaea geognostica. I. Lethaea palaeozoica”. Atlas.

Stuttgart, 1876.
Manrtiy, K. —— »Niederlindische und nordwestdeutsche Sedimentdrgeschiebe,
thre Uebereinstimmung, gemeinschaftliche Herkunjt und
Petrefacten”’.
Leiden, 1878.
Aneexin, N. P. — »Palaeontologia scandinavica’’.
Stockholm, 1878.
Kotmopin, L. — »Ostracoda Silurica Gothlandiae”.
Ofvers. af Kong]. Svensk. Vet.-Akad. Vérhandl. 1879, no. 9, p. 133—139; | S80.
Scummt, F. — » Revision der ostbaltischen silurischen Trilobiten nebst
geognostischer Uebersicht des ostbaltischen Silurgebiets”’.
Abtheilung I.
Mém. de Ac. Imp. a. Se. de St. Pétersbourg, Te Sér., T. XXX, no. 1; 18S.
Scumipr, F. — »Miscellanea Silurica. — LI. 1. Nachtrag zur Mono-
graphie der russischen Silurischen Leperditien™.
Mém. de VAc. Imp. d. Se. de St. Pétersbourg, Vile Sér., T. XNXAI, no. 5; 1$S3.

. Kirsow, J. — » Ueber silurische und devonische Geschiebe Westpreussens”’.

Schrift. d. naturf. Ges, in Danzig, N. F., VI, 1, p. 205—300; 1SS4.

. Roemer, F. — »Lethaea erratica’’.

Palaeont. Abhand!., herausg. v. W. Dames u. E. Kayser, U1, 5; 1885.
Rement, A, — »Katalog der beim internationalen Geologen-Congress zu
Berlin ausgestellten Geschiebesammlung’’.

Berlin, 1885.
Ruerert Jonus, T. and
Hout, H. B. — »Notes on the Palaeozoie Bivalved Entomostraca. —

XXJ. On some Silurian Genera and Species”.

Ann. and Magaz. of Nat. Ilist., 5 Ser., Vol. XVII, p. 403—414; 1586.

Houm, G. — »Revision der ostbaltischen silurischen Trilobiten’’.
Abtheilung III.
Mém. de PAe. Imp. d. Se. de St. Pétersbourg, Te Sér., 1. XXXULI, no. 8; 1886

. Linpstrém, G. — »List of the fossil faunas of Sweden.

IT. Upper Silurian”.
Stoekholin, 1888.
Ktesow, J. — »Ueber gotlindische Beyrichien’’.
Zeitsehr. d. deutsch. geol. Ges., XL, p. 1—16; 1888.
Rurerr Jones, T. — » Notes on the palaeozoic Bivalved Entomostraca.
XXVIII. On some North-American (Canadian)
Species”.
Ann. and Magaz. of Nat. Hist., 6 Ser, Vol. Iff, p. 373—3887; 1889.
Kirsow, J. — »Beitrag zur Kenntniss der in westpreussischen Silur-
geschieben gefundenen Ostracoden’’.
Jahrb. d. k. pr. geol, Landesanst. u,s.w. f. 1889, p. S0—103; 1890.

( 703 )

20. Scumirpr, F. — »Bemerkungen iiber die Schichtenfolge des Silur auf
Gotland”.
Neues Jahrbuch, 1890, [L, p. 249—266.

21. Dames, W. — » Ueber die Schichtenfolge der Silurbildungen Gotlands und
ihre Beziehungen zu obersilurischen Greschieben Nord-
deutschlands”’.

Sitz-Ber. d. k. pr. Ak. d. Wiss. zu Berlin, 30 Oct. 1890; Bd. XLIT, p. 11 11—1129.

22. Krause, A. — »Die Ostrakoden der silurischen Diluvialgeschiebe’’.

Wiss. eilagez Progr. d. Luisenstadtischen Oberrealschule zu Berlin, Ostern 1841.

23. Krause, A. — »Beitrag zur Kenntniss der Ostrakodenfauna in silurischen
Diluvialgeschieben.”

Zeitschr. d. deutsch. weol. Ges. NXLUL, p. 488~521; 189!.

24, Scnmpr, KF. — »Liniye Bemerkungen tiber das baltische Obersilur in Ver-
anlassung der Arbeit des Prof. W. Damus iiber die
Schichtenfolge der Silurbildungen Gotlands”’.

Bull. d. PAc Imp. d. Sc. de St. Pétersbourg, N.S. 11 (XXXIV), 1892, p, 381—
400; and: Mél. dol. et pa'éont., tirés du Bull. ete., T. 1, p. LLY —*3s.
25. Nicuorson, H. A. — »A monograph of the British Stromatoporoids’’.
‘The Palaeoutozraphieal Society, London, LSS6—s'2.
26. Scumipr, F. — » Revision der ostbaltischen silurischen Trilobiten.”’.
Abtheilung LV.
Mém. de l'Ac. Imp. d. Se. de St. Pétersbourg, 7e Sér., T. XLII, no. 5; 1994.

27. Srormey, E. — »Die cambrischen und silurischen Geschiebe Schleswig-

Holsteins und ihre Brachiopodenfauna. I. Geologischer
Theil’.
Arch. f. Anthrop. u. Geol. Schleswiz-Holsteins u.s.w., I, 1, p. 35—136 ; 1895

28. Lixpsrrém, G. -— »Beschreibung einiger obersilurischer Korallen aus der

Insel Gotland”.
Bihang till kh. Svenska Vet.-Akad. Hlandl., XXI, Afd. 1V, no. 7; 1896.
29. Bonnema, J. H. — »WDe sedimentaire zwerfblokken van Kloosterholt
(Heiligerlee)”.
Versl. v. d. gew. Verg. d. Wis— en Nut. Afd. d. Kon. Ak. v. Wet. v. 29 Jan.
188, dl. VI, p. 448 453.
30. Cuierewski, C. — »Die Leperditien der obersilurischen Geschiebe des
Gouvernement Kowno und der Provinzen Ost- und
Westpreussen”’.
Schrift. d. phys -oekon. Ges. zu héniysberg, Jy. 41, p. 1—38 : 1990.
31. Bonnema, J. H. — »Leperditia baltica His. sp., hare identiteit met Leper-
ditia Fichwaldi Fr. v. Schm. en haar voorkomen in
Groninger diluviale zwerf blokken’’.
Versl. v. d. gew. Verg. d. Wis- en Nat. Afd. d. Kon. Ak. v. Wet vy. 30 Juni
1900, dl. IX, p. 138—140.
32. Jonker, H. G. — »Bijdragen tot de kennis der sedimentaire zwer/steenen

in Nederland.
I. De Hondsrug in de provincie Groningen.
1. Inleiding. Cambrische en ondersilurische zwerf-
steenen”’.
Acad. Proefschrift, Groningen, 1904.

( 704 )

33. Jonker, H. G. — »Bijdragen tot de kennis der sedimentaire zwerfsteenen
in Nederland.
TI. De Hondsrug in de provincie Groningen.
2. Bovensilurische zwerfsteenen.
Eerste mededeeling: Zwerjsteenen van den ouderdom
der oostbaltische zone G”’.

Versl. v. d. gew. Verg. d. Wis- en Nat. Afd. d. Kon. Ak. v. W. v. 28 Jan.
1905, dl. XU, 2. p. 548—565.

Groningen, Min.-Geol. Institute, April 4, 1905.

Anatomy. — “Note on the Ganglion vomeronasale.” By E. pe Vries.
(Communicated by Prof. T. Puace.)

The description and drawings given in this note derive from a
wellpreserved human embryo. This embryo was fixed in a ten
percent solution of formaldehyde. After fixation the greatest diameter
was 55 mm. Precise information as to the probable age of this
embryo was not to be obtained, but the dimension of the embryo in
connexion with the fact, that the corpus callosum was not yet
formed, makes it probable that the age of the embryo may be
estimated between 2'/, and 3 months.

After the embryo being hardened in alcohol the head was cut off
along the base of the crane and imbedded in paraffin; a complete
series of frontal sections of 10 was made. A slight deviation from
the frontal plane existed, so that the top of the right hemisphere
first appeared in these sections. The greatest part was stained in
haematoxylin and eosine in the usual manner, the rest of the sections
with haematoxylin only, in slightly different ways.

A description is given of the right hemisphere, — which in the
microscopical sections corresponds with the left one —, concerning
only that part which has a closer relation to the rhinencephalon.
This description is illustrated by four drawings of successive sections
and by two semi-diagrammatic figures.

These figures (Fig. V, VI) are a projection of the olfactory lobe
on a sagittal plane and constructed from the series of sections.
Because the plane upon which the projection is performed is sagittal,
only these curvatures of the olfactory lobe are seen, which have a
component in that direction. The lines in these drawings denoted
from I to IV indicate the place of the four sections marked with
a corresponding roman number. ;

The olfactory lobe, as seen in this stage of development, forms

( 705 )

a hollow outgrowth from the base of the hemisphere vesicle. On
the external surface of the lateral wall of the hemisphere, the lobe
is limited by a shallow suleus, the fissura rhinica. This suleus runs
in a fronto-oecipital direction (fig. I, Il, III F. rh.). On the external
surface of the mesial wall of the hemisphere vesicle the olfactory
lobe is bordered by a very broad suleus which in the beginning
runs also in a fronto-occipital direction but bends afterwards more
vertically. This suleus is the fissura prima of His and only to be
seen in the first two figures (fig. I, II F. pr.).

Bordered by these two grooves the olfactory lobe shows a double
curvature from lateral to mesial and slightly from behind forwards.
The anterior cornu of the lateral ventricle forms a prolongation in
the olfactory lobe reaching into the top of the bulb. This cavity
shows the same curvatures as the lobe, which can partly be seen
from the diagrammatic figure V. In its general feature and apart
from its curvatures this cavity of the olfactory lobe has the shape
of a funnel, the mouth turned to the lateral ventricle the tube to
the top of the olfactory bulb.

A close relation between the form of the external and internal
surfaces of the hemisphere vesicle does not exist. The internal surface
of the lateral wall is thickened by the appearance of the corpus
striatum. This thickening of the wall begins wellmarked at some
distance (2mm) from the top of the hemisphere vesicle; a prolonga-
tion of this thickening, described by His as the “Crus epirhinicum’”’,
which, along the top of the hemisphere unites the striatum with the
rhinencephalon does not seem to exist. The ventral edge of the striatum
is also clearly marked by a_ prominent crest, the crista ventralis
corporis striati; (fig. I, Cr. v. str.) which is bordered by a deep sulcus
(fig. I, 5. v. str.). This suleus on the internal surface of the vesicle
does not agree in all respects with the fissura rhinica on the external
surface.

The ventral edge of the striatum first proceeds in a fronto-occi-
pital direction and then turns more ventrally over the posterior wall
of the funnellike outgrowth of the rhinencephalon. By its typical
configuration it is easy to follow this ventral edge of the striatum
till it goes over in an analogous formation belonging to the rhinen-
cephalon.

This formation of the rhinencephalon appears as a thickening of
the internal surface of the mesial wall of the hemisphere vesicle. It
begins pretty well marked a little more distant from the top of the
hemisphere than the striatum. Dorsally and ventrally this thickening
is limited by a deep groove, the sulcus rhinencephali dorsalis and

( 706 )

ventralis (fig. I and II, S. rh. d. and S: rh. v.). The ventral edge
of this thickened part of the mesial wall forms a prominence, which
goes over in a crest, the crista ventralis rhinencephali.

This crest first runs in fronto-occipital direction and then turns
more ventrally over the posterior wall of the funnellike outgrowth
of the rhinencephalon where it goes over continuously in the same
formation proceeding from the stratium. This is clearly seen in figure II
(Cr. v.) where the ventral crista is seen on the posterior wall of
the depression of the rhinencephalonu cut in a very oblique direction.

The line described by this ventral border of the corpus striatum
and thickened part of the rhinencephalon has, looked at asa whole,
the form of a horseshoe with its top directed to the occipital pole
of the brain and meantime turned ventrally, while its opening is
turned to the frontal pole of the hemisphere vesicle. The connection
of rhinencephalon and striatum, which les initially in the base of the
brain comes with the outgrowth of the rhinencephalon partly on the
posterior wall which borders the cavity, that proceeds in the olfaet-
ory lobe. This connection between striatum and rhinencephalon is
therefore a primary one.

The olfactory bulb in this stage of development of the rhinence-
phalon is limited by a circular groove, the suleus circularis bulbi
(Fig. I, Il, $.¢.b.), which deeply cuts in on the frontal pole of the
bulb, becomes more flat on both sides and is seen as a round
shallow groove at the posterior pole of the bulb (Fig. V 5. ec. b.).
The top of the bulb is turned to the mesial side and in a slightly
forward direction, while the form of the bulb can be seen in the
diagrammatic drawing figure V.

The nerves which belong to the formation of the rhinencephalon
are of two different kinds, and leave the brain at two different
places. The first kind of nerves proceed from the top of the olfactory
bulb. They are easily recognised by the fact, that their nuclei are
small and not very numerous, so that the fundamental substance in
which they are imbedded is distinctly seen

These nerves split up into very small tracts in the neighbourhood
of the mucous membrane of the nose, where they seem to end. These
nerves, which contain the olfactory nervelibres do not have any
connection with the ganglion olfactorium. They all pass along this
ganglion.

The second place from where the nervefibres proceed is given by
the mesial part of the sulcus cireularis bulbi. These nervefibres can
be differentiated from the olfactoryfibres by the fact, that their nuclei
are a little larger, and more numerous than the nuclei of the

"

( 707 )

olfactory nervefibres; the fundamental substance in which these nuclei
are imbedded deeply stains with eosine. Where these fibres leave
the brain, the superficial layer of the hemisphere vesicle becomes
richer in elongated nuclei (A vn. fig. I, II, Ill, 1V and VI). They
form four bundles (Rd. N. vn. tig. Hl, IV and VI), which all
converge into the ganglion olfactorium (G. vn. fig. HI, 1V, VD. The
nerves leaving this ganglion are arranged in five bundles (Fig. VJ
Nn. vn) which all went to the mesial side along the cartilagineous
septum nasi (Fig. 1 Sp.m.n.). Unfortunately the course of these nerves
could be no further traced out, the head being cut off too close along
the base of the brain. In a second human embryo however, of the
same age, which was not so well preserved, it was possible to find
baek the same relations and to see, that all these nervefibres run
exclusively to the organon vomeronasale (RuyscH, Jacosson). In the
whole course of these nerves ganglioncells are seen. The so called
ganglion olfactorium has therefore no connection with the olfactory
nervefibres but is the sensorial ganglion belonging to the organon
vomeronasale.

In figure VI is given a semidiagrammatic drawing constructed
from the sections where the whole apparatus belonging to the organon
vomeronasale is projected upon a sagittal plane. The ganglion vomero-
nasale (G. vn.) is seen in the niveau of the top of the olfactory bulb
from which proceed to the periphery the nervi vomeronasales (Nn.
Vn.) and to the centrum the so called internal olfactory roots (Rd. N. vn.).
which enter the brain in a large triangleshaped zone, the area
vomeronasalis (A. vn.)

In the guinea pig these relations are slightly different. From the
organon vomeronasale proceed two nervestrands, which at a short
distance and still in the submucosa of the nose have each a ganglion.
This paired ganglion vomeronasale has two roots which very soon
unite and intermingle with the nervi olfactorii, with which they
perforate the lamina eribrosa. Arrived at the base of the brain they
enter the olfactory bulb over a large area, reaching from the sulcus
cireularis bulbi at the mesial side to some distance from the same
suleus at the lateral side of the olfactory bulb.

Probably the same relations occur through the whole series of
vertebrate animals. Though the existence of an organon vomeronasale
can be doubted im anamnia, it seems very probable, that the
nerve described by Locy (Anat. Anz. 1905, Heft 2 and 3) in
Selachii is identical with the nerve of the organon vomeronasale
as described here.

From the preceding description it is obvious, that we have

( 708 )

to consider the organon vomeronasale as a special senseorgan of
which the function is unknown, while the duality seen in the
central tracts belonging to the rhinencephalon finds its source in the
anatomical independence between the system of the olfactory nerves
and the system of nerves belonging to the organon vomeronasale.

My thanks to Prof. J. W. LanceLtaan under whose direction these
researches were made.

Anatomy. — “Note on the Innervation of the Trunkmyotome”.
By J. W. vay Bisserick. (From the Anatomical Institute at
Leiden). (Communicated by Prof. T. Pacer).

These researches form a sequel to professor LaNnGELAAN’s first
communication “On the Form of the Trunkmyotome”*), and were
performed under his direction in the anatomical institute at Leiden.

The aim of this research was to know if one single spinal nerve
innervates only one single myotome.

The method followed, existed in dissecting a spinal nerve and to
see if the different territories to which the nervestrands can be
followed, belonged to one and the same myotome. To this purpose
an Acanthias or a Mustelus was cut through along the mid-sagittal
plane and treated with a one tenth percent solution of osmie acid.
The nerves stained black and were easy to follow with the naked
eye or with a magnifier.

As a first result it was found, that all nerves passed through the
connective tissue laying between the myotomes; therefore a minute
dissection of this tissue was necessary.

The myotome itself is covered by a very thin layer of a fibrous
tissue which constitutes a perimysium. This perimysium extends
between the muscular fibres of the myotome forming an endomysium.
It affords a continuous investment for every muscular fibre and
forms in this way a frame for the muscular tissue. Where this mus-
cular tissue is broken off the framework is continuous and enables
us to recognize paris of the myotome belonging together. The
myotomes covered by their perimysium are separated by a coarser
and denser fibrous tissue. This intermyotomal tissue forms lamellae
which have only a very loose connection with the perimysium, so that
it is possible to dissect these lamellae as discrete formations. These
intermyotomal septa pass over in the fibrous tissue of the skin and
form a continuous formation with the latter. Where the myotome

1) Proc. K. Akad. W. Amsterdam 28 May 1904.

(209°)

has a simple form, this line of insertion coincides with the border
of the myotome; where the myotome is elongated in a peak, this
line of insertion crosses this peak.

Figure I reproduces the external surface of the myotome extended
in a plane. The black line indicates the transition of the inter-
myotomal septum in the skin; where the myotome is elongated in
a peak, it has distended the septum, because the line of transition is
fixed upon the skin. The peak is covered by this distended part of
the septum, and as far as the peak is adjacent to the skin, this
part of the corium is doubled by this triangular sheath. Whereas on
the line of transition the passage of the intermyotomal septum into
the corium is a direct one, this is not the case with this adjacent
part of the septum, which is only loosely connected with the corium
by means of some fibres of connective tissue. This makes it possible
to dissect these triangular slips from the corium.

In the same way as the myotomes, the triangular slips of the
intermyotomal tissue overlap. In concordance with the direction of
the peaks it is seen, that slips belonging to peaks directed towards
the caudal end of the body cover each other, so, that the more
caudal slip covers the more cranial one. If the peak is directed
cranially the mode of overlapping is reversed, the more cranial slip
being uppermost. Figure II reproduces the intermyotomal tissue as
far as this formation is adjacent to the skin.

On the mesial side the intermyotomal septum goes over in the
connective tissue which covers the axial skeleton and beyond this
forms a lamella between the left and right half of the dorsal mus-
culature. Ventrally the same formation goes over in the fascia trans-
versa covering the abdominal cavity.

Figure III gives the line of passage of the intermyotomal septum.
As can be seen there are two places where the muscular tissue is
broken off, the myotome becoming thinner from outside to inside.
The lamellae, where the muscular tissue is interrupted, cover each
other and in this way two strong continuous septa are formed.
The distance over which the muscular tissue is discontinuous in the
neighbourhood of the sagittal plane amounts to four myotomes in
the first septum and to three in the second. In agreement with this,
the lamellae are built up resp. by four and by three sheaths of
intermyotomal tissue. The dotted fields in figure HI belong therefore
together, forming one myotome, as can easily be verified by dissecting
the myotome. .

Each spinal nerve springs from the cord with two roots, which
separately leave the spinal canal through two foramina (AR and

( 710 )

PR fig. IV). When they have quitted the canal each root separates
into two filaments, one of these filaments is ascending (Ase. f.) and
one is descending (Desc. f. fig. VI). Both ascending root filaments
unite to form a nerve, the internal branch of the posterior division
(fig. VI), the filaments of which pass over in the intermyotomal
septum at the places indicated by 3 D—5 D fig. IV, and leave the
septum to go over in the skin at the places indicated in the same
way in fig. V. Before these filaments go over into the corium
they each give off a small twig innervating the distended part of the
intermyotomal septum, which is adjacent to the skin.

Before the two ascending rootfilaments join, they each give off a
small branch, which also unite to form asmall nerve, the first external
branch of the posterior divisions (fig. VI) entering the septum at
2D fig. IV and leaving the septum to pass over in the skin at the
corresponding place of fig. V.

Both descending rootfilaments before joining each give off a
small branch, which form together a small nerve, the second
external branch of the posterior division (fig. VI), which enters and
leaves the intermyotomal septum at the places indicated by 1D in
fic. IV and V.

The nerves described, all together, innervate the dorsal part of the
myotome and the intermyotomal septum, and form the posterior
primary division of the spinal nerve.

The descending rootfilaments also unite to form a nerve which pretty
soon divides into two branches, one of these innervating the lateral
part of the myotome and the intermyotomal septum; the other is, the
continuation of the maintrunk, crosses the lateral part of the myotome
and innervates the ventral part of the myotome and the intermyotomal
septum. The branch innervating the lateral part of the myotome
divides into two branches, an external and internal branch of the
lateral division (fig. V1). The external branch splits up into two filaments
one of which is recurrent (recurrent br. fig, VI) and innervates the
top of the lateral part of the myotome. The external branch enters
the septum at 1.2. fig. IV and leaves the septum at 1/, 22
fig. V. The internal branch gives off several branches passing over
in the skin at 3 L—6 L fig. V.

The branch innervating the ventral part of the myotome and the
intermyotomal septum shows the same arrangement as the branch
for the lateral part of the myotome. It divides into two branches
one being the external branch of the anterior division, the other the
internal branch (fig. VI). The external branch passes over in the
septum at V 1.2. fig. IV, splits up into two smaller branches of

( 741 )

which one is recurrent (recurrent br. fig. VI) and leaves the septum
to go over into the skin at 1 V and 2 /. fig. V.

The internal branch can be followed up to the vena lateralis (V7
fig. 1V) and then goes over in a loose plexus. On its way to the vena
lateralis the internal branch gives off several filaments, whieh reach
the skin through the intermyotomal septum 3)—6J" fig. IV and VY.
Before passing over into the skin these filaments form a loose plexus
covering the most ventral part of the myotome.

The roots and mainbranches of the spinal nerve have a submyotomal
position and are not bound in their course by the form of the
myotome; these branches on the contrary, which go over into the
septum to reach the skin, are in their course fixed by the form of
the myotome. The final course of the branches in the corium was
not traced out with enough accuracy to give results here.

The descriptions given in this note only apply to that region of
the trunk which is situated between the thoracic and first dorsal fin.

Conclusions :

I. One single spinal nerve only innervates one single myotome
and the intermyotomal tissue through which the nerves pass.

II. The roots and mainbranches of the spinal nerve have a sub-
myotomal position; the branches never perforate a myotome, but
run always in the intermyotomal septum to the skin. In general
they are to be found between the perimysium and the intermyotomal
septum.

Ill. The spinal nerve shows a primary division into three parts, a
posterior, lateral and anterior division in agreement with the diffe-
rentation of the myotome in a dorsal, lateral and ventral part.

IV. All larger branches are mixed nerves containing elements of
the anterior and posterior roots.

Mathematics. — “On linear systems of algebraic plane curves”.
By Prof. JAN pE Vrins.

§ 1. The points of contact of the tangents out of a point O to
the curves c* of a pencil lie on a curve #"—! which I shall eall the
tangential curve of QO. It is a special case of a curve indicated by
Cremona’). By Eni. Weyr?), Guccta*) and W. 30UWMAN *). it has
been applied when proving the properties of pencils and nets.

1) Gremona—Currze, Einleitung in eine geometrische Theorie der ebenen Curven
(1865) p. 119.

2) Sitzungsberichte der Akademie in Wien, LXI, 82.

8) Rendiconti del Cireolo matematico di Palermo (1895), IX, 1.

4) Nieuw Archief voor Wiskunde (1900), IV, 258.

( 712 )

If a linear system (c")z of o* curves c” is given, we can consider
the locus of the points P.4:, where a curve of that system has a
(k-+ 1)-pointed contact with a right line, passing through the fixed
point O.

To determine the order g(4) of the locus (P*+') I consider the
curves (c"), having in the points P of the right line 7 a k-pointed
contact with the corresponding right line OP. Each ray OP cuts
the curve individualized by P moreover in (nx —4) points Q. Each point
of intersection of 7 with the locus of the points Q being evidently
a point P.41, the locus (Q) is a curve of order (A).

The curves of (c"), passing through O form a system (c"),-1. The
order of the locus of the points P; where a c" of this latter system
has a k-pointed contact with OP is evidently indicated by g(&%— 1).
So on / lie g(k—1) points P for which one of the corresponding
points Q coincides with O; in other words the locus (Q) passes
g(k —1) times through QO, so it is of order ¢(4 — 1) + (n — &).

To determine ¢() we have now the recurrent relation

g(k) = (fk — 1) + (a — 2b.
From this we deduce
g(t) = 9) +4&—1) Qn-- k—2).
Here ¢(1) represents the order of the tangential curve, thus (2 — 1).
So we find
g(k) = 3 (& +1) Qn — 2B).

The locus of the points where a curve °, belonging to a k-fold
injinite linear system has a (k + 1)-pointed contact with a right line
passing through a fixed point Os acurve of order (k +1) (2n — &,
on which O is a &k(&-+1)-fold point.

For (c"), determines on a right line 7 through QO an involution of
order n and rank &. The number of (4 -+ 1)-fold elements of this in-
volution amounts to (£-+ 1)(n— 4); that is at the same time the
number of points P.41, lying on 7. Consequently QO is an 44 (k-+ 1)-
fold point on (P41).

§ 2. Each ray 7 through a fixed point O is touched by 2 (2 — 1)
curves c" of a pencil (c"); the points of contact 7’ are the double points
of the involution determined by (c") on 7. The curves c” indicated by
these points 7’ intersect 7 moreover in 2(n—1)(n—2) points S.
When 7 rotates round O the points S will describe a curve which
I shall call the satellite curve of O.

This curve passes (n--1)(n—-2) times through QO; for if r
coincides with one of the tangents out of O to c” passing through

( 713 )

O one of the points S lies in O. So the curve (S) is of order
(nv ++ 1) (2 — 2) + 2 (n — 1) (n — 2) = (n — 2) (8n — LI).

If L is a base-point of (c”), then only 2 (x — 2) points 7’ (the double
points of an /”"—!) lie on OB outside O and B. So OF touches in
B the tangential curve of O whilst it is (~ — 2)-fold tangent of (S).

Each of the 2 (n — 2) curves c” touching OB projects a point Sin B.
So each base-point is a 2 (nm — 2)-fold point of the satellite curve.

The common points of the tangential curve “"~! and the satellite
curve so—U"—2) form four groups.

First there are (n + 1) (mn — 2) united in O.

Secondly 2 (2 — 2) lie in each base-point JB.

Thirdly the two curves touch each other at each inflectional point
sending its tangent through 0.

Fourthly they cut each other in the points of contact of each
double tangent passing through 0.

Now the inflectional tangents of a pencil envelop a curve of class
dn (n — 2). *)

So the number of points of contact of inflectional tangents through
O amounts to
(n — 2) (8n — 1) (Qn — 1) — (n — 2) (n + 1) — 2 (n—2) n? —6n (n—2) =

= 4n (n — 2) (n — 8).

The double tangents of the curves c™ belonging to a pencil envelop

a curve of class 2n(n — 2) (n — 8).

§ 3. Following Emm Weyer’) we consider the curve c’+! gene-
rated by the pencil (c") with the pencil projectively conjugate to it

- of the tangents in a base-point 4. As each c” cuts its tangent more-

over in (7 — 2) points, B is a threefold point of the c’+!. From this
ensues easily that through Lb can be drawn (n + 4) (n — 8) tangents
to c™+', As many double tangents of the pencil (c”) have one of
their points of contact in B.

We shall now consider the satellite curve of B. On each ray 7
through 6B lie 2(n—2) points of contact 7, so 2 (n — 2) (n —83)
points S. If 7» coincides with one of the double tangents just mentioned,
one of the points S lies in B. So B is an (n+ 4) (n—8)-fold point
on (S) and the order of (S) proves to be equal to (a + 4) (n — 3) +
2 (n -— 2) (n — 3) = 38n (n — 3).

The tangential curve of 4 has in 4 a threefold point; for a ray

1) For this is the number of tangents of 2”—!

of OB can be drawn through 0.
*) Sitzungsberichte der Akademie in Wien LXI, 82.

which besides the »? tangents

( M14)

through B bears but (2n—4) points 7 whilst the curve ¢ is of
order (2n — 1).

Of the common points of 2?—! and s*@—®*) there are 3 (n ++ 4) (n — 3)
lying in L, 2 (n— 3) in each of the remaining (7? —1) base-points and
two in each of the inflectional points sending their tangent through B.

The number of those inflectional tangents is 3n (n— 2) —9, as
each of the three inflectional tangents, having their inflectional point
in 4, must be counted three times. This is evident when we consider
a curve of (c*), where a base-point can lie only on inflectional
tangents for which it is inflectional point itself. This number amounts
to three, whilst the class of the envelope of the inflectional tangents
is nine.

So we find for the number of the points of contact, not lying in
B, of double tangents out of B
3n(n—3) (2n—1) —3 (n-++4) (n—3) —2 (n—3) (n?—1) —6 (n—38) (n4-1) =

=4(n— 3)(n—A (n+ 1).

So B lies on 2 (n — 4) (2 — 8) (n + 1) double tangents. This num-
ber is 2 (m— 3) (n + 4) less than the number of double tangents out
of an arbitrary point. The (7 — 3) (n-+ 4) double tangents having
one of its points of contact in 4 must thus be counted twice.

The envelope of the double tangents has in each base-point an
(n + 4) (n — 3)-fold point.

§ 4. The locus of the points of contact D of the double tangents
of (c*) evidently passes’ (” +- 4) (n — 3)-times through each base-point
(§ 3). An arbitrary c” having on its double tangents m (7 — 2) (n? — 9)
points of contact D, the curve JD and c” intersect each other in
n? (n + 4) (n — 3) + n(n — 2) (n? —-9) points. Consequently the locus
of the points of contact D is a curve of order (n—8)(2n?-+-5n—6). ')

We shall now consider the locus of the points HV” in which a
c” is intersected by its double tangents.

As each base-point 6 lies on 2 (m — 4) (n.— 3) (n + 1) double tan-
gents (§ 3) the curve JI’ passes with as many branches through B.
So it has with an arbitrary c? in common 2n? (n—4) (n—8) (n+1) +
-- 4 n(n — 2) (n® — 9) (n — 4) points. Krom this ensues that the curve
(W) is of order 4 (n -- 4) (n — 3) (5n? + 5n — 6).

The curves (PD) and (JI’) have outside the base-points a number of
points in common equal to

4 (n — 4) (n — 3)? (5n? + 5n — 6) (2n? + 5n — 6) —
— 2n? (n — 4) (n — 3)? (n + 1) (n -+ 4).

aa See P. H. Scnoure, Wiskundige opgaven, II, 307.

From this ensues :
In a pencil (c”)

4 (n — 4) (n — 3)? (10n* + 35n* — 21n? — 80n + 20)

curves have an injlectional point of which the tangent touches the
curve mm one other point more.

§ 5. The locus of the inflectional points / of (c”) has a threefold
point in each base-point and a node in each of the 3 (m— 1)’ nodes
of the pencil, out of which we immediately find that the curve (/)
is of order 6 (n— 1) and of class 6 (mn — 2) (4n -— 3) ’).

Let us now deduce the order of the locus of the points V deter-
mined by a c" on its inflectional tangents.

As a_ base-point / lies on 3 (2 — 3) (m7 -+ 1) inflectional tangents
the curve (JV) passes with as many branches through B. So with
an arbitrary c” it has 3? (2 — 3) (nm + 1) + 3n (n— 2) (n—3) points
in common.

Consequently (V7) is a curve of order 3(7— 3) (n? + 2n—2). Now
the curves (J) and (J’) have besides the base-points a number of
points in common represented by

18 (n — 1) (n — 3) (n? 4+ 2n — 2) — 9n? (n — 3) (n+ J).

These points can only have risen from the coincidence of inflectional
points with one of the points they have in common with the c” under
consideration, thus from tangents with fourpointed contact. Such an
undulation pot, being equivalent to two inflectional points, is point
of contact for (7) and (V’) from which ensues:

: wae 2o) ;
A pencil (c") contains a (n — 8) (n® + n? — 8n +4) curves with an

undulation point.

§ 6. Let a threefold infinite linear system of curves c" be given.

The c” osculating a right line / in the point P cuts the ray OP
drawn through the arbitrary point O moreover in (2—1) points Q.

The curves of (c"), passing through O form a net (c"), determining
on / the groups of an involution /,". The latter having 3 (n — 2)
threefold elements, the locus (Q) passes 3 (7 — 2)-times through O,
so it is of order (42 — 7).

Each of its points of intersection A’ with / is evidently a node on
a curve of (c”),, with 7 and OX for tangents.

Each right line ts nodal tangent for (4n — 7) curves of the system.

From this ensues that the locus of the nodes K sending one of

2) See Bosex, Casopis (Prague), XI, 283.

49
Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 716 )

their tangents through the point J chosen arbitrarily is a curve of
order (4n — 5); for J is a node of a c”, so it lies on two branches
of (K).

Each point A’ of the arbitrary right line / is a node of a curve
belonging to (c"),. The points of intersection / and J’ of the tangents
in A with the right line m chosen arbitrarily are pairs of a sym-
metric correspondence with characteristic number (4n — 5). To the
coincidences belongs the point of intersection J/, of 7 and m, and
twice even, because the ¢”, having in that point a node, furnishes two
points 7,’ coinciding with V,. The remaining coincidences originate
from tangents in cusps. From this ensues:

The locus of the cusps of a threefold injinite linear system of
curves of order nis a curve of order 4 (2n — 3).

Mathematics. — “Some characteristic numbers of an algebraic
surface.’ By Prof. Jax bE Vries.

In the following paper we shall show how by easy reasoning
we can find an amount of the characteristic numbers of a general
surface of order n'*). To this end we shall make use of scrolls
formed by principal tangents or double tangents.

§ 1. First I consider the scroll A of the principal tangents a of
which the points of contact A lie in a given plane «. The curve
« along which é@ cuts the surface 9" is evidently nodal curve of
A. The tangents in the 3x (2 — 2) inflectional points of e being
principal tangents of 9", the scroll A has 3 (7— 2) right lines and
the curve @ to be counted twice in common with 2”, so it is a
scroll of order nx (8n — 4).

The two principal tangents @ and a’ ina point of e” have each three
points in common with 9”; consequently @ belongs six times to the
section of A and ¢”. These surfaces have moreover a twisted curve
of order 7? (872—4)—6n in common containing the 3n(n— 2) (n— 8)
points where ¢” is cut by the principal tangents @ situated in @. In
each of the remaining » (lim — 24) points of intersection of this
curve with «@ the surface $" has fowr coinciding points of inter-
section in common with a. From this ensues:

The locus of the points in which 9" possesses a fourpointed tangent
(fleenodal line) ts a teisted curve of order n(Ain—24).

1) We find the indicated numbers in Satmon-Fiepter, ‘Analytische Geometrie
des Raumes”’, dritte Auflage, I, p. 622—644, and in Scuusert, *Kalkiil der abzihlenden

Geometrie’, p. 236.

(747 )

§ 2. I now determine the order of the scroll B formed by the
principal tangents cutting ¢* in points 6 of the plane 9.

Out of each point 6 of the section 8" start (m— 3) (n* + 2)
principal tangents; this number indicates at the same time the
number of sheets of B which cut each other alone ~B". The infleet-
ional tangents lying in ” evidently belong (mn — 3)-times to the
indicated seroll. So its order is equal to
n (n — 3) (n? + 2) + 3n (rn — 2) (2 — 3) = nn (rn — 1) (n — 3) (n+ 4).

According to § 1 2 (8? —4n—6) principal tangents have their
point of contact A on e” and one of their points of intersection
B on pr. So this number indicates the order of the curve along
which g” is osculated by B. Beside this curve of contact and the
manyfold curve p” the surfaces ¢”" and B have still in common the
locus of the points 5’ which determine the principal tangents Ab
moreover on og”. This curve (5') is of order n* (7—1) (w~—3) (n-+-4) —

3n(38n? —4n— 6) — n(n — 3) (nv? + 2) =n (n — 2) (n—A4) (rn? ++5n-+8).

§ 3. To tind how often the point A coincides with one of the
(1 — 4) points 4’, I shall project the pairs of points (A, 2’) out of
a right line 7. The planes through / are arranged in this way in a
correspondence with the characteristic numbers 2 (827—4 n—B6) (n—4)
and n (2 — 2) (n — 4) (n? + 5n +3). Each right line @ resting on /
evidently contains (—4) pairs (4, 5’), so it furnishes an (7 — 4)-fold
coincidence. The remaining coincidences originate from coincidences
A= B'. Now n (8n?—4 n—6) (n—4) + n (n—2) (n—A4) (n?+5 n-+-3)—

n (n—1) (n—8) (n-+-4) (n—4) = n (n—4) (6n?-+-2n—24). So this is
the number of fourpointed tangents which cut 9” in a point B of 3.

The points of intersection of $” with its fourpointed tangents form
a curve of order 2n(n — 4) (3n? + n — 12).

If f is the order of the scroll of the fourpointed tangents then it
is evident that we have the relation
nf = 4n (11n— 24) + 2n(n — 4) (8n? + n—12) = 2n? (n— 3) (8n—2).

The fourpointed tangents form a scroll of order 2n(n—8)(3n—2).

If we make the point of contact /” of a fourpointed tangent to
correspond to the (m—4) points G which that tangent has still in
common with g”, a system of pairs of points (/°, G) is formed, of

which the number of coincidences can be determined again with the
aid of the correspondence in which they arrange the planes through
an axis /. By the way indicated above we find for this number :
n(Lin—24) (n—4)-+ 2n(n—A) (3n? +n—12 )—2n(n—38) (8n—2) (n—4)=
n(n—A) (85n— 60).
The surface ~" possesses 5n(n —4) (Tn —12) fivepointed tangents.
49*

( 718 )

§ 4. Returning to the seroll B (§ 2) I consider the points of
intersection of the twisted curve (4') with the plane @. Each point
of intersection of ¢” with an inflectional tangent lying in 8 can be
regarded as the point £, each one of the remaining (2 — 4) as a
point 5’. Hence the curve (4’) meets 38x (nm — 2) ( 3) (7 — 4)-times
8" on the inflectional tangents of 3". In each of the remaining points
of intersection of (4') with 8 we find that 9" is touched by a right
line having elsewhere three coinciding points in common with g". Such
a right line is called by me a tangent 43, A being its point of
osculation, B its point of contact.

The points of contact of the tangents te3 form a curve of order

n(n — 2) (n — 4) (rn? + Qn + 12).

§ 5. In each point ( of the curve y" according to which 9” is
cut by the plane y I shall regard the (7—3)(n- 2) tangents e which
touch g" moreover in a point C’. On the scroll C of the double
tangents c the curve y" is a manyfold curve in which (7—38) (n+-2)
sheets meet. Each double tangent situated in y representing two right
lines of C the order of this scroll is equal to
n(n— 8) (n+ 2) +n(7 3) (w+ 3) or n(n —) (nv? + 2n—4

The surfaces 9g and C touch each other along the locus (C’) of
the two points of contact. Of this curve the plane y contains the
points of contact of the right lines ¢ lying in y besides the points
C=C’, where a right line ¢ is a fourpointed tangent. So the order
of (C’) is n(n—2) (n?—9) + vn A1n—24) or n(n? — 2n? + 2n — 6).

Besides the curve (€’) to be counted twice and the curve y”" to
be counted 2(2— 3) (n+ 2)-times C and g» have moreover in
common the locus of the points S determined by the double tangents
¢ on 9". The curve (S) is of order n? (a — 3) (nv? + Qn — 4) —
Qn(ni—2n? + 2n—6) — 2n(n—B) (n4-2) or n(m—A4) (n? + n?—+4n—6).

To the points of (S) lying in y belong the points of intersection
of y® with its double tangents c. As each of the two points of contact
of ¢ can be regarded as point C' these points of intersection S must
be counted twice. The remaining 7 (7 — 4) (> + 2? — 4n — 6) —
n(n —- 2) (n? — 9) un — 4) points S lying in y are apparently points
of osculation of the tangents f2;. So from this ensues:

The points of osculation of the principal tangents touching 9”
moreover elsewhere form a curve of order n(n — 4) (3n? + 5n— 24).

The curves (4) and (4) formed by the points of osculation and
ihe points of contact of the tangents 3 have the points of contact of
the fivepointed tangents in common. Taking this into account we
find (by again projecting out of an axis /) for the order of the

irre)

scroll of the right lines fs; the expression 17 (—2)(—4)(? 4-2-1 2) +
+ n (n—A4) (8n?+5n—24) — 5n (n—A4) (Tn—12).

The principal tangents of o" which moreover touch the surface
form a scroll of order n(n — 3) (n — 4) (n* + 6n — 4).

§ 6. The double tangents ¢ cutting ¢* in points ) of the plane J
form a scroll D, on which the section dé” of o” with dis a many fold
curve bearing 4 (7% — 8) (2 — 4) (n? + n + 12)-) sheets. As moreover
every double tangent of 6 belongs to (7 — 4) different points D the
order of D is {0

4 2 (n—8) (n (i? Sean ) + 4 2 (n-—2) (n—8) (n-++3) (n—4) =

n (n—1) (n-+-2) (n—8) (n—A4).

According to § 5 2 (m7 — 4) (n®? + n*? — 4n — 6) double tangents c
have one of their points of contact C in a given plane y and at the
same time one of their points of contact D in the plane d. So this
number indicates the order of the curve along which D and 9”
touch each other. If we take the manyfold curve d” into consideration,
it is evident that the points D’ which the right lines of D have in

common with g” besides the points of contact C and the points of
intersection J lying in d, form a twisted curve ()’) the order of
which in equal to
n* (n—1) (n+-2) (nv 7a n—4) — 2n (n—A4) (n*?-+-n?—4n—6) —

1 n (n—8) (n—A4) (0? -+-n-+2) = 4 n (n—2) (n—A) (n—5) (2n? +-5n-+8).

This curve ane cuts dS (7—A4) (7—5)-times on each double

tangent of dé”. In each of its remaining points of intersection with d
the surface o” is tonched by a right line, which is tangent to the
surface in two more points. From this ensues :

The points of contact C of the threefold tangents of o” form a
curve (C') of order $n (n—2) (n—4) (n—S) (n?+5n-+-12).

§ 7. On each right line ¢ of the scroll D lie (n—5) points D’
5) (n—6) pairs D', D". If these pairs

which can be arranged in 4 (7

of points are projected out of an axis / by pairs of planes 4’, 2", these
form asymmetric system, the characteristic number of which is $ 2(2—2)
(n—A) (n—5) (2n?+5n-+3) (n—6). Each right line ¢ cutting / deter-
mines a plane 4 evidently representing (7—5) (n—6) coincidences
A'== 4". The remaining coincidences of the system (4) originate from
coincidences D' = D", thus from threefold tangents d. As however
1) In Cremona—Curtze, Theorie der Oberfldchen, page 66 we find the expression
1 (n—3) (n—-4) (v?+n—2) hy mistake for the number of double tangents cutting
g” in one of its points,

(720 )

each of the three points of contact of a right line 7 can be formed
when J’ coincides with D" the number of threefold tangents cutting
o”" on the curve dé” is but the third part of the number of the indi-
cated coincidences of (2), thus equal to

‘in (n—4) (n-——5) (n—6) {(n—2) (2n?-+-5n-+3) — (n—L) (n-+-2) (n—3)j=
1 n (n—A) (n—5) (n—6) (n*+-3n?—2n—12).
This is at the same time the order of the curve (2) formed by
the points D which the threefold tangents d have still in common
with 9”.
Now we can also find the order 2 of the scroll (/). This scroll
being touched by 9” in the points of (C) and being cut in the points
(D) we have namely

na = n (n—2) (n —A4) (n

5) (? +5n+12) +
+n (n—A4) (n—5) (n—6) (n?+3n7—2n—12).

Out of this we find

The threefold tangents of a form a scroll the order of which is
+n (n—3) n—A) (n—5) (n?+-3n—2) *).

§ 8. To find the degree of the spinodal curve I consider the pairs
of principal tangents a, a’ of which the common point of contact A
lies in the plane « If two rays s and s' of a pencil (S, 6) are con-
jugate to each other, when they rest on two right lines @ and a’,
then in (S,6) a symmetric correspondence with characteristic number
n(3n — 4) is formed. The coincidences can be brought to three groups.

First @ and @ can cut the same ray s; their plane of connection
is then tangential plane, their point of intersection A lies on the
polar surface of S. Such a ray s coincides with two of the rays s'
conjugate to it. So the first group contains 7 (2 — 1) double coinei-
dences.

Secondly s can cut the curve @; then too it coincides with two
rays s’. So the second group consists of 7 double coincidences.

Finally a single coincidence is formed when @ coincides with a’.
The number of these coincidences evidently amounts to 27 (82 —4) —
2n (n — 1) — 2n = 4n (n — 2). From this ensues:

The parabolic points form a twisted curve (spinodal line) of order
An (n — 2).

1) In Satmon-Fiepter we find on page 638 by mistake ”?-+-3-- 2 instead of
n2+3n— 2.

On page 643 we find the derivation of the number of fourfold tangents and of
the numbers of tangents f,,0, f5,0,9 and /s,9.

Mathematics. “7 equation of order nine representing the locus
of the principal axes of a pencil of quadratic surfaces. By
Mr. Kk. Brs. (Communicated by Prof. J. Carpinaat).

1. In These Proceedings of Jan. 28 1905 appears a communication
by Prof. Carpinaan: “On the equations by which the locus of the
principal axes of a pencil of quadratic surfaces is determined.”

2. Prof. Carpinaat deduces three non-homogeneous equations of

order two between two variable parameters 4 and /:, and tries to
arrive at the equation of the demanded surface by elimination of
these parameters. The result obtained by him (8) seems to be an
equation of order 12. This is incongruent with the result arrived at
geometrically, which made an equation of order nine to be expected.
This ineongruency is attributed to factors, which the equation arrived
at may contain, but these factors are not indicated.
3. The method of elimination described in my paper “Théorie
generale de elimination” (Verhandelingen, Vol. VI, n°. 7) gives the
means to set aside this incongruency and to determine in reality the
equation sought for by Prof. Carpinaan.

To this end we can start from his equations (5) after having made
them homogeneous with respect to the variable parameters, which
may be done by assuming the equation (1) of the pencil of surfaces
in the form:

pA 1B = 0-

If now we develop the equations (5), they assume the following form :
(a,,4,+4,,4,14,,4,)u?-+(a,,B,+0,,B,40,,B,40,,A,+b,,A,+5,,4,)au+)\

(6, ,B,+0,,B,40, ,B,)22+A,uk+A,ak—0,
(a,,A,+0,,4,+a,,A,)u?+(a,,B,+a,,B Cr eneas +b,,A,+0,,A, ar |

+b ale +b, B, +b,,B, )22-+A, +B, Ak=0,

(a,,A,+0,,4,+0,,A,)u? Le ORE Sareea &
1(b,,By+0_,B,4+0,,B,)\?-+A,uk+B,ak—0.

The coefficients of these equations are linear functions of the
variable coordinates, y and z. To simplify we can introduce the
following notations :

P= An et= dy, At t= Gy, Ais's

P, = 4,,A, + @,,d, + a,,4, ,

DG At Gad Dea,

Q, = Gao. oh GB ote a,,P, 4 bi As a b,,A, + b,,A; ’
oe

v2 a,,B, st a,,B, + a,,B, at b,,A, “f b,,A, =f b,,A, ’
Q, = 4,8, + a,,B, + a,,B, + b,,A, + b,,4, + by
Rh, = 6,,B, + b,.B, + 6,,B; ,
R, = b,,B, + 6,,B, + b,,B35
t, = 6,,B, + b,,B, + 5,,B, ,

( 722 )

by which the equations (a) pass into the following :

Pw? + Qu + Ry? + Auk + Byak=0,
Py? + Q,au + Ra? + Ayuk + Byrak=0,{/ . . . (0).
Py? + Q,4u + Bj? + A,uk + Bak=—0, |

4. Which condition now must exist between the coefficients of
these equations if they are to allow of a mutual system of roots?
The answer is that no condition is demanded for this. These equations
are namely satistied independent of the value of the coeflicients by
the system of roots:

A=; ~w= 0. kearbitrary.

The result arrived at by applying the method indicated in § 118
of my paper. “Théorie générale de l’élimination” agrees with this.
According to this method we should have to find for the resultant
ihe quotient of two determinants successively of order 15 and of
order 3. In the case under consideration where we have

a, == 05 b= 0vand cn — 05
we always obtain, in whatener way we choose the determinants,
as quotient a quantity which is identically zero.

So the above-mentioned equation (8) can be nothing else but an
identity.

5. This result having been fixed it is no longer difficult to answer
the question how to obtain the equation of the demanded locus.
To this end we must express the condition that the equations (4) are
satisfied by a second system of roots.

The condition in demand is, that all determinants are equal to zero
contained in the assemblant (55) appearing in § 118 of the already
mentioned paper. Applied to the equations (4) it gives but one
equation, namely

B, B B,

i a !
this being the equation of the demanded locus. It is of order nine
agreeing to the geometrical researches of Prof. CARDINAAL.

( 723 )

Physics. “A formula for the osmotic pressure in concentrated
solutions whose vapour follows the gas-laws’. By Dr. PH.

KounstamM. (Communicated by Prof. J. D. van per Waats).

§ 1. The formula for the osmotic pressure may be derived in two
different ways: by a thermodynamic and by a kinetic method. When
putting these two in opposition | mean by no means an absolute contrast,
on the contrary [ believe — an opinion which | hope soon to treat
more fully elsewhere — that without an equation of state based on
kinetic considerations thermodynamics has nothing to start from
and that therefore we can only oppose ‘purely kinetic” and “thermo-
dynamic-kinetie”’” considerations.

Not numerous are those who have tried to find formulae for the
osmotic pressure of more concentrated solutions by a thermodynamic
method. Only Honpius bonpincu’) and after him Van Laar?) have
pointed out that it appears from the theory of the thermodynamic
potential that the concentration of the solution should not be taken
into account in the form «, but as log (1—) and that for further
approximation a correction term of the form az* must be applied,
and lately the latter has again come forward to advocate with great
zeal the validity of this result.

More numerous are the attempts to determine the osmotic pressure
in concentrated solutions by direct, molecular-theoretic methods; |
may mention those of BrepiG*), Noyes *), Barmwarer *), Winp *).

This fact is surprising because Van ’r Horr himself, though he
has a definite conception of the nature of the osmotic pressure, has
never dared to base his equations on it, but has clearly indicated
as basis of his theory of the osmotic pressure the thermodynamic
considerations, by means of which he derives the osmotic pressure
from the gas-laws. And; it is the more surprising because all
these attempts wish to follow the train of thought which led
Van prR Waats to his equation of state, though Van per Waats
himself has clearly shown, that in his opinion the osmotic pressure
must not be sought in this way, but by the thermodynamic method, in
connection with the equation of state given by him. That notwithstanding
this so often the other way has been followed, seems noteworthy to
1) Diss. Amsterdam 1893.

8) Zsch. phys. Ch. 15, 466 (1894).
8) Zsch. phys. Ch. 4, 444.

4) Zsch. phys. Ch. 5, 53.

5) Zsch. phys. Ch. 28, 115.

6) Arch, Néerl. (2) 6, 714.

( 724 )

me on account of the predilection which it shows for purely kinetic
considerations. The reasons why | do not share this predilection in
this case, will appear from another communication, occurring in
these Proceedings; here I shall confine myself to the thermodynamic
method, and specially to the form given by Van pER WaAAts.

§ 2. In § 18 of his Théorie Moléculaire Vax per Waats treats the
case, that of a binary mixture the first component can expand through
a given space, whereas the other is confined to a part of that space.
He demonstrates that for equilibrium a difference in pressure between
ihe parts of the space is required which for dilute solutions has the
value indicated by the law of Van ‘t Horr. In this a thesis is used,
which is very plausible (and which moreover may be proved in the
same way as the condition for equilibrium in the general case)
that namely equilibrium is established when the thermodynamic
potential of the first component is the same in the two parts of
ihe space. i shall here apply this condition to a binary mixture
of arbitrary components and arbitrary concentration, the vapour
of which follows the gas-laws, and which is in equilibrium with
one of the components in pure condition under the pressure of its
own vapour through a semipermeable wall. How such an equili-
brium might be reached in reality in a special case, and whether
this would be possible, need not be discussed.

§ 8. We assume that there are (1—v) molecules passing through
the membrane and w non passing molecules, then the thermodynamic
potential of the first substance in the mixture is

ow Op)
a ent Gl ee ( ote

7 if
“79
= yf pdv + pv + MRTI(l —2)— 2 : i dv + F(Z)
ey) 7

in which the integrations must be extended from a volume y so
large that all the laws of ideal gases apply there, to the volume in
question, /(7’) being a function of the temperature, which occurs
here only as an additive constant. In order to be able to carry out the
as mentioned above — an equation of

integrations, we require
state p =f (v, 7).

For this purpose I shall adopt Vax per Waats’ equation with
constant 4; though in this way we certainly do not get strictly
accurate results, yet we shall be able to decide about the quantities
which must occur in the formula.

§ 4. If in fig. 1 the isotherm of the mixture is indicated and

the horizontal line is drawn according to the well-known law. of
Maxweut'), then the pressure indicated by that line is what VAN DER
Wars calls the pressure of coincidence of the mixture and denotes
by the symbol p.. The volumes at ihe end of that line we call
ve, and »,, and p, and v% represent pressure and volume of the

.
C1

mixture in equilibrium in the above mentioned way.

Now the integral { ree may be split into three parts:

Ye Vig 7
fra +f pdv +f pdv
Ve, Vey

Vo

vo

For the middle quantity we may write:
Ye

.

Joe = Pe lve. — %, )

Ve,

As according to our suppositions the vapour follows the gaslaws,
we have:
Pe Yc, = MRT.

For the same reason we may replace p in the third integral by

1) Through a mistake in the plate this line is drawn much too high here. Also
the form of the isotherm is imperfectly represented. But the figure is merely given
as a schematic representation,

( 726 )
MRT ve. Carrying out the integration we get MRT log y/ec,, for
which we may also write MRT log pep, We get then:
Vey

[ paw + porto = | pdv + pores — pe ve, + MRT + MRT I p/p,

Vo 05

.

§ 5. Let us consider the first three terms. The first is represented
in the figure by the area C+ D, the second by A+ B, the third
by B+ D. The three terms together are therefore A + C. If now
as we assumed, the vapour is very dilute, and therefore the tempe-
rature far from the critical, hence also the isotherm very steep when
cutting the line of coexistence liquid-vapour (or strictly speaking: at
the pressure p., which however is very near the line of coexistence
on the liquid side), then we may neglect C by the side of A, and
we are the more justified in this as the pressure p, is higher, so the

Vo

mixture in question more concentrated. For C= {ple— D. It we

V
introduce
MRT ay
P v—by v?

and integrate, we get:

MRT
BRT oe =) MRE Gp) (S = 5) (v, — %)
v

. . 2
= Cy Vo a b Ye,

Yee lige ) = a
= =a Vo — —= aor Vo

Ve, i Vo,

If we arrive at very high pressures, 7, — 6 approaches zero and
numerator and denominator become both infinite, but the denominator
of a higher order than the numerator. It is already apparent from
the form of the isotherm which becomes steeper and steeper, that
when neglecting © by the side of A we make proportionally a smaller
mistake the higher p, is. And that the neglect is allowed for small
osmotic pressures is selfevident. We may therefore put for the three
terms discussed in this §:

A= (p.— p.)%.

7
fr)
§ 6. It remains to calculate the term {(2) dv. This integral

v

Vo, Ue, y

too we separate into three parts | a { ; iF The last integral

U U

vy ie a
is now zero according to the law of AvoGrapo. The middle one
we find from the equation already used above:

tes

{ va 705 | Oa,

Ve,
by differentiating, taking into consideration that the limits of the
integral are functions of wv. We get:

v

° Op Ove |? Ope Ove, ve,
| — dv 4+- | p =— (v%, — te) + px Pe :
0a) n7 ~ Oak Ow Ow Ow

Ve,
Now at the limits of the integral p is pe; we retain therefore on
the left and the right only the first members.
Ve,

‘(Op
Finally the first part i (sr) As we were allowed to neglect

ed

Vey hy

>

free, we might be inclined to think that this term too might be
Vo

omitted. But as follows form the equation of state :

Op MRT db 44/g,

dx (v—b)? dz v?

J

this integral appears to be of higher order than the other for small
values of v — 6. We therefore retain it. Carrying out the integration

we get:
E RT db da =| Ve,

v—b dz 0 |v,

Here we may substitute p+ a/v’, for MRT — 4, so that our
expression for the thermodynamic potential becomes:

Ope

Mu, = MRIU (1-2) + puvo~ pers + MRT MRI Ipo/p- (v2, -¥e,) +
xv

db

db vee da( 1
—(peo—Po) + te — ;
eo oe E Fe

§ 7. This value must now be equated to the thermodynamic

a a

1 es
{+A

Vo

potential of the same substance in pure condition. As we suppose

( 728 )

it to be under the pressure of its own vapour, the quantity to be
calculated is the same as the thermodynamic potential of its saturated
vapour, 1. e.
ie
fide + Peoer. Vcoer. + F (1)
Vcoen.
where we denote by the index coev. that the quantity must be taken
on the line of coexistence. Now is on account of the assumed
validity of the gaslaws
vt
| pd v + Peoer. Vcoex. = MRT 1 Peoer./Py + MRT
Vcoex.
If we equate the expression obtained here with that of the

preceding §, then #(7’), IZRT and MRT logp, neutralise each other
on both sides. What is left we may write in this way:

Ib c 1—wa 0 Cc
(Po — p(m _ of =~ uRE jog PERS) ys OE ee

du Pcvex. Ow

S| SS > =

+ va — {— — —
de \ ve, 5

dx (0,7 Vo"

da (1 ‘) db (1 1

Now v,, and v can never differ much. If the osmotic pressure
of an aqueous solution amounts e.g. to 1000 Atm., these volumes
differ only a few percents. In the two last terms, which themselves
can only be correction terms, we may therefore put 7, = vo in any
case, so that those terms vanish. Further we may neglect vc, by the
side of v,., and write MRT logp. for ve, Our equation becomes then:
MRT I pPc(L — 2) dlogpe
db a, nese dx
dix

The remaining v may, of course, not be replaced by ve,, first
because this expression occurs here in the principal term and then
because the substitution of 7, for +, would of course be more felt
in a term of the order 1/e—#/ than in 1t/v. But in any case, when
we have really to do with osmotic pressures, the pressure will never
be so large that we could not compute v, with the aid of the coefti-
cient of compressibility of the saturated liquid without any difficulty.

§ 8. The quantity po—p,, which we have found, is not identical
with the osmotic pressure; the latter is rather po—pcoer., but the
transition of one quantity to the other is without any difficulty. If

SS >

Vy — vx

( 729 )

we neglect in our formula the terms, which are multiplied by «by
the side of those in which this is not the case, if we put py=proer.
and if we take Veo,, instead of v,, which is permissible for very

dilute solutions we get:

MRT
fe = Po--Poex. a log L—wa
V% UL.

which gives the well-known formula of vant Horr when the /oy
is developed and the higher powers omitted.

I wish to point oui, that also a more accurate treatment yields
the logarithmic form which Bonpinau and van Laar have advocated

and there could not be any doubt but it must be so but that
it also shows that van LaAar’s statement') was too absolute when
he asserted that a correction term need never be applied in the
numerator Veoer, (OL Vy) in Connection with the size of the molecules.

In the second place | draw attention to the fact that we find the
osmotic pressure exclusively expressed in what van per Waats has
called thermic quantities (in opposition to caloric quantities). — It
appears to be unnecessary to take into consideration the heat of
dilution or other quantities of heat, which van ‘rr Horr *) seems to
deem necessary for concentrated solutions and which Ewan *) has
taken into consideration. Even if we had avoided all the introduced
neglections, so when we had not assumed, that the vapour follows
the gaslaws, nor that 7, = v,, may be put in some terms, nor that
the area C’ may be neglected compared to A, nor (the most important)
that 6 is constant, we should evidently not have had to deal with
any quantity of heat. This seems important to me, as both theore-
tically and experimentally the caloric quantities are much less accessible
than the thermic ones.

Physics. — ‘“Ainetic derivation of vax “vr Horr’s law for the
osmotic pressure in a dilute solution.” By Dr. Pu. Konnstam™.
(Communicated by Prof. vax Der’ WaAAts).

§ 1. When we leave out of account the more intricate theories
as that of Poynrine *), who tries to explain the osmotic pressure
from an association of solvent and dissolved substance, and that of

Hlenc:

2) K. Svenska Vet. Ak. Hand. 21. Quoted by Ewan Zsch. phys. Ch. 14
499 en 410.

3) Zsch. phys. Ch. 14, 409 en 31, 22.

4) Phil. Mag. 42, 289.

(730 )

BackLUnb ?), who seems?) to require even ether waves to explain it,
chiefly two theories have been developed about tne nature of the
osmotie pressure: the static and the kinetic theory. The first theory
finds warm advocates in Pupin *) and Barmwarer ‘); it seems however
doubtful to me whether they have closely realised the consequences of
their assertions. At least the latter brings forward as an objection
io the kinetie nature of the osmotic pressure: “Kin molekulares
Bombardement in einer Fliissigkeit ist mir immer etwas sonderbar
vorgekommen”; notwithstanding he considers the equation of state
of Van prr Waars by no means as a “sonderbar’” instance of false
ingenuity, but as an example to be followed. However this may be,
he who does not want to break with all our conceptions about
heterogeneous equilibrium, will not be able to explain such an equili-
brium in another way than statistically i.e. as a stationary condition
of a great number of moving particles. This does, of course, not
detract from the fact that the question may be put what forces are
required to bring about that state of equilibrium. This implies that
the adherents of the static theory need not be altogether mistaken
when they assert that the cause of the osmotic pressure is to be found
in forces of attraction. On this point | shall add a few remarks at
the end of this communication.

§ 2. Of much more importance than this static theory of the osmotic
pressure is the kinetic theory. The great majority of its advocates
(I shall speak presently about the few exceptions) take as their basis

the equality, which has been proved experimentally and by means
of thermodynamics, of the osmotic pressure and the gas pressure (the
pressure which the molecules of the dissolved substance in the same
space would exercise, when they were there alone and in rarefied
gas state) and derives from this that they have both the same cause
in this sense that the dissolved substance is present in the two cases
in the same state and so acts in the same way: this is then
expressed in about this way that the solvent converts the dissolved
substance into the rarefied gas state. This conception seems doubly
remarkable to me; first because it seems to be pretty well generally
prevailing *), secondly because it alone seems to me to be able to

1) Lunds Univ. Aarsskrit 40.

8) | know his paper only from an abstract in the Beibl. 29, 375.

3) Diss. Berlijn 1889.

4) Diss. Kopenhagen 1898 and Zsch. phys. Ch. 28, 115.

6) It is naturally difficult to give a proof of this opinion, therefore I shall only
adduce the following citations as a confirmation.

“If we look a little more closely into the matter, we find that in the case of
dilute solutions, at least, there is far more likelihood of the dissolyed substance

.

(731 )

explain, why the theory of the osmotic pressure has become so quickly
popular, whereas Gibps’ method for the solution of the same problems
was scarcely noticed. In fact the view mentioned possesses all qualities
required for great popularity: it seems to give a very simple, clearly
illustrating explanation for the striking law discovered by van ’? Horr;
it is allied to the universally known gaslaws; it seems to make us
acquainted in the osmotic pressure with a quantity, which is as
characteristic for the dissolved state as the well-known external pressure
for a gas. On the other hand it does not seem to carry weight
that this “explanation” is, properly speaking, no more than an
explanation of words, which leaves undecided exactly that which
had to be explained, viz. how it is, that the solvent acts on the
dissolved substance in this way. It is, however, worse that this
explanation clashes with everything we know of liquids and gases,
and therefore is to be rejected. We need only think of the well-
known experiment with a bell jar, closed at the bottom by a
membrane, filled with a solution of cane sugar and placed in a
vessel with pure water, which forces its way in till equilibrium
has been established. If now the pressure ?, exerted on the mem-
brane, was a consequence of the fact, that the dissolved substance
in the bell jar was in a state which more or less resembles the
gasstate, then those molecules of the dissolved substance would have
to exert the same pressure also on the glass wall of the bell jar, in
other words, the water molecules would exert the same pressure

being in a condition comparable with that of a gas.” (Watxer, Introduction to
Physical Chemistry, 148).

“Ich glaube dargethan zu haben —- im Gegensatz zu der zur Zeit allgemeinen
Aujffassung — dass es nicht notwendig ist eine freie Bewegung der geldsten

Molekiile wie fiir die Gase anzunehmen. Wenn ein fester Kérper in einer Fliissig-
keit gelost, oder eine Fltissigkeit mit emer anderen gemengt wird, so wird eine
neve Fliissigkeit erhallen, von deren Molekiilen es nicht gestattet ist, andere
Beweglichkeit anzunehmen, als diejenige, die Vliissigkeiten charakterisiert.” (BarM-
water |. c. pag. 143). “Aus den klassischen Arbeiten von vay ‘r Horr und ARRHENIUS
geht nun hervor, dass die Kérper bei Gegenwart von Lésungsmittel thatsiichlich
mehr oder minder dem Gaszustand niher geriickt werden,” and a little before:

“Andererseits konnle ich mir.... nicht verhehlen, dass gerade diese Gegenwart
und Einwirkung des Lésungsmittels doch die notwendige Vor-
bedingung fiir den Eintritt des gasiilinlichen Zustandes sei:.... daher ist

aber ein gasiihnlicher (also kinetischer) Zustand nur unter dieser Einwirkung
vorhanden und hort sofort auf, sobald diese Kinwirkung beseiligt ist. Es sei be-
tont, dass diese Auffassung durchaus nichts Neues bietet, duss sie vielmehr wohl
einem Jeden eigen ist, der den Begriff des osmotischen Druckes kennen gelernt
hat.” Brenig. |.c. p. 445 and 444). The italics are mine, the spacing the cited authors’.

Finally ef. Van Laar’s address in the “Bataafsch Genootschap”, p. 2 and 3 and
the example cited there.

50
Proceedings Royal Acad. Amsterdam, Vol. VII.

( 732 )

on that wall from the inside and from the outside (of 1 atm.). This
now is a perfectly unacceptable result, as immediately appears from
what follows. Let us imagine the same solution as in the bell jar
inclosed in a cylinder with a piston under the pressure of its satu-
rated vapour p— 4p, where 4p is the decrease of vapourpressure.
The cane sugar molecules contribute nothing to that pressure or hardly
anything *), as appears from the fact that they cannot pass into the
vapour (at least not in a measurable degree); all the pressure is
furnished by the water molecules. Now we compress the liquid, till
it has got a pressure P+ p, it is now in perfectly the same con-
dition as the liquid in the bell jar, when we except the immediate
neighbourhood of the membrane. On the supposition made just now
the water molecules would exert a pressure » against the piston, the
sugar molecules a pressure /?, i.e. the pressure of the latter would
have increased by an amount about 1000 times that of the former,
whereas their initial pressure was at least a hundred thousand times
smaller. And the result would be that the, let us say 2, sugar mole-
cules, which are found to every 1000 water molecules would exert a
pressure twice as great as the 1000 particles together. It is beyond
doubt that the pressure ?-+ p on the piston or the glass wall of
the bell jar is exclusively exerted by the watermolecules, and if he
meant this, LorHar Mnyer was certainly right when he asserted *),
that the Osmotic pressure was a result of the collisions of the solvent.

Also in this respect the theory of the gaslike character of the dissolved
substance falls short, as it leaves perfectly unexplained why in an
isolated solution, e.g. a cane sugar solution, which in a glass vessel
stands under atmospheric pressure, nothing is perceived of the gaslike
character of the dissolved substance. For that in this case solvent
and dissolved substance are less closely in contact than in the
osmotic experiment, cannot seriously be asserted.

§ 3. If therefore we must not seek the explanation of the
laws of the osmotic pressure in a particular condition of matter,
characteristic of dilute solutions, then the remarkable fact formulated
by Van’? Horr calls the more peremptorarily for an explanation.

Nobody less than Lorentz and BonrzMann have made attemps to
do this *), but even their endeavours do not seem to me to have solved
the problem entirely. In saying this | agree with Prof. Lorenrz’s own
opinion, at the beginning of his paper he terms it a “freilich nur zum
Teil gelungene Untersuchung’. As to the reasons of this partial failure,
however, I shall most likely differ in opinion with Prof. Lorenz.

1) Perhaps the pressure of these molecules would even prove to be negative,

*) Zsch. Phys. Ch. 5, 23.

#25.)

For what is the case? The behaviour of liquids is entirely dominated
by the oceurrence of the quantities @ and 6 in the equation of
state. Only matter in dilute solution seems to emancipate itself
from it, according to the law of Van ’r Horr, where neither
the @ nor the 4 occurs. This fact calls for an explanation.
Now it is not difficult to understand, why the « can disappear
here; the membrane is bounded on one side by the solution, on
the other side by the pure solvent. If we now think it thin com-
pared to the extent of the sphere of action, then it is clear that at

ly : : :
the membrane the force —— which works towards the solution, is

Vo”

in first approximation neutralized by the force a towards the other
vg”

side. It is more difficult to see why also the 6 vanishes, i.e. why

the molecules of the dissolved substance seem to move as through a

vacuum, instead of through a space, which is occupied for a very

great part by the molecules of the solvent.

Just on this most important point Prof. Lorrnrz’s paper leaves us
in the dark, for so far as I have been able to see. And it seems to
me beyond doubt, that in the first place this is due to an inaccurate
interpretation of the term “kinetic pressure”. According to Prof.
Lorentz it is always equal to */, of the kinetic energy of the centres
of gravity of the molecules which are found in the unity of volume.
It is therefore independent of the volume of those molecules. Now
this would only be a question of nomenclature, if not that kinetic
pressure was also defined as the quantity of motion, carried through
the unity of surface in the unity of time by the motion of the
molecules; and that this quantity is dependent on the number
of collisions and so on the volume of the molecules does not seem
open to doubt to me after Korrewne’s proof’). In agreement with
this the kinetic pressure is represented in the equation of state by
MRT/v—b. In consequence of his definition Lorentz replaces this

) Zsch. phys. Ch. 7, 37 and Arch. Néerl. 25, 107.

2) Zsch. phys. Ch. 6, 474 and 7, 88.

5) Verslagen Kon. Ak. Amst. (2) 10, 363 and Arch. Néerl. 12, 254. Compare
also the simpler, perhaps even more convincing proof for one dimension in Nature
44, 152. As the attentive reader will notice Prof. Lonentz’s proof (1. c. 39) does
not take into account the collisions and the fact ensuing from them, that a quantity
of motion skips a distance or moves with infinile velocity fora moment. And
the admission of the validity of KorrewrG’s reasoning appears, as it seems lo me,
already from the fact, that Prof. Lorentz has to assume for the solid bodies intro-
duced by him, that they are immovable (I. c. 40) or of infinite mass (I. c. 42)
which comes to the same thing in this case.

50*

( 734 )

quantity by WRT vr, and so his paper cannot give any elucidation on
the point which requires it most. But that notwithstanding we owe
to Lorenrz’s labour a considerable widening of our views, will as I
hope, appear from the continuation of this communication.

Also BourzMANn’s paper leaves us in the dark as to the question
why the quantity 4, which in other cases plays such an important
part for liquids, seems to have no influence on the value of the
osmotic pressure. In the equations, which he draws up, he never
takes the size of the molecules into account *) and it does not appear
why he does not do so. Further he stops at the result, that the
osmotic pressure is equal to the sum of the pressures exercised by
the two kinds of molecules, without discussing the part played by
the .different kinds. For these reasons I cannot see a satisfactory
solution of our problem in BourzMaNy’s paper either.

§ 4. To arrive at a solution it seems in the first place necessary
to give three definitions.

dst. Given a fluid. Placed in it a body of perfect elastic impermeable
substance, which does not exert any attraction on the molecules
of the fluid. The thickness of this body (or this surface) be infinitely 3
small; let us suppose it to have an area of 1 cm?. The “kinetic
pressure” in that fluid is then the quantity of motion in unity of
time transferred by the molecules of the fluid to this body (or obtained
in the elastic collisions from this body).

2nd. In the second place I imagine a body *), which is distinguished

1) See specially 1. c. 475 equation (4), which is evidently incorrect, when part of
the cylindre is not open to the centres of the molecules, because it is occupied by
distance spheres of other molecules.

2) That I assume that the body does not attract the molecules of the fluid, is
for simplicity’s sake, but it is not essential. If we imagine a wall, which doves
attract the fluid, more molecules will reach its surface (ef. the footnote p. 739)
and hence will impart a greater quantity of motion to the wall. But on the other
hand the particles of the surface will now be drawn into the fluid with an} equally
greater force. The elasic displacement of the particles of the surface of the solid
wall, and with it (with sufficient elasticity) that of the layers lying under it, in
other words the pressure which propagates in the solid body, and which would
be measured with a manometer of any kind, will be perfectly the same in the
two cases. If we wish to take also negative external pressures into account, we
shall even have to give the definition by means of an attracting body, because
in this case a non-altracting body would not even be reached by the molecules ~
of the fluid. (Cf the well-known fact that for the observation of the negative
pressure strongly adhering walls are required). In this case the impulse of the
attraction of the molecules is simply greater than the quantity of motion which
they impart to the wall (and which may still be very great), the elastic displacement
is therefore not from the fluid, but towards it.

Also in the ease that we wish to take capillary layers into account, our definition

from the just mentioned body only by its being very thick compared
fo the sphere of action of the molecules. The quantity of motion
transferred by this body per unity of time to the molecules, is called
the “external pressure” in that fluid.

34, In the third place 1 place in the fluid (whieh [now suppose
to be a mixture) a body, which is distinguished from that mentioned
under 2 only by the facet that the molecules of one component (solvent)
pass through it without any change in their velocity. | shall leave undis-
cussed here whether sueh a body can actually oceur. The pressure
to which this body is now subjected, and which might be measured
e.g. by the elastic displacement of the particles of its surface, I
eall the “osmotic” pressure in that solution.

From these definitions it is already clear that in dilute solutions
the osmotic pressure defined here must be of the order of the kinetic
pressure exerted by the dissolved substance, and not of that of the

. . . a .
external pressure. For these two differ, in that — has disappeared
;

for the kinetic pressure, and this will also be the case for the osmotic
pressure defined here, as appears from the reasoning given above
(§ 3). I shall further show, that in dilute solutions this osmotic
pressure has the value indicated by the law of Van ’r Horr, and that
in any case it is as great as the well known experimentally intro-
duced and measurable osmotic pressure, i.e. the difference in external
pressure of solution and pure solvent under the pressure of ifs own
vapour in equilibrium through a semipermeable wall.

ealls for fuller discussion. First of all this applies to what we have just now
said, for just as for negative pressures so also in the capillary layer, as
Van ver Waats has shown in his theory of capillarity, the attraction of the
surrounding layers is a necessary condition for stable equilibrium. But further,
as Hutsnorr has shown (These Proc. 8, 432 and Diss. Amsterdam 1900), the
above defined quantity does not obey the law of Pascan any more, because mea-
sured in the direction of the layer and perpendicular to it, it has a different value.
In this case we might perhaps speak of a total external pressure, which might
be split into an external fluid pressure and an external elastic pressure. The
consideration of capillary layers round a free floating sphere, teaches us further,
that the “external” in the name “external pressure’ must not be understood
in such a way, as might easily be done, viz. that the reactive force of this
pressure, as it prevails in a certain point, would act in points outside the system
in question, which would always be more or less arbitrary, as we may choose
the limits of our system arbitrarily. The assertion: the external pressure is in a
point of the fluid so great, comes simply to this, that when I should place a
strange body at that place, without altering the condition more than necessary
for this, this body would experience a pressure of such a value, and would
suffer an elastic modification in form which corresponds to it, so differing in the
capillary layer in different directions,

( 736 )

§ 5. For this proof I must refer to a formula of Cravstvs used
by me already before‘). Imagine a point which can freely move

in a space W. Crausivs*) shows — which is already plausible
beforehand — that the number of collisions of this point per second

against a wall of area S is proportional to S/W (the factor of
proportion depends only on the velocity of the point).

Let us now consider a wall as defined under 2, and draw a
plane parallel to that wall at a distance *), 6 (6 is the diameter of
the molecules, which we think spherical); this plane we call plane
of impact, because the centre of a molecule, which strikes against
ihe wall, lies in this plane. Now we apply Cravstus’ formula to
this wall. In this we must allow for the fact that the centre of a
molecule cannot move freely throughout the volume of the fluid;
for within the distance spheres (spheres drawn round the centre of
every molecule with a radius 6) it cannot come; instead of + we
have therefore to put 7—2h, when 2/*) is the volume of the distance
spheres. Now the whole plane of impact, however, is not accessible
to collisions either, part of it also falls
within the distance spheres. In order to
fix this part we draw two planes at
distances 4 and + dh parallel to the
plane of impact. We determine how
many centres of molecules are found
between them and what part of the
plane of impact is within their distance
sphere. In order to find what part of
the plane of impact falis at all within
distance spheres, we must integrate with respect to 4 between O
and ‘'/, 6. It appears then, that instead of S we must put S(1—6/?)
in the formula for the number of collisions against the wall, so
that the pressure becomes proportional to

or in first approximation

1) These Proc. VI. 791.

2) Kinetische Theorie der Gase, 60.

8) For simplicity | confine myself to the first term, even if we have to deal with
liquids; this is permissib'e here, because the other terms have no more influence
on our question (the derivation of the law of Van ‘r Horr) than the first.

1
a—b )

§ 6. Now we apply the reasoning of the preceding paragraph to
the collisions of the dissolved substance on a wall defined as under
3. We assume the solution to be so diluted, that the volume of the
molecules of the dissolved substance may be neglected compared

with the whole volume. For simplicity —- though it is not essential
to the proof — we assume now also that the molecules are spheres.

Then here too the available space must again be put equal to »—2é;
but the part of the plane of impact, accessible to collisions, is now
different. For as the molecules of the solvent pass through the
wall, their centres may now just as well be on the other side of
the plane of impact. We have therefore not to integrate with respect
to h from 0 to '/, 6, but from —?*/, 6to + '/, 6, which evidently
yields the double value. The pressure on the wall becomes therefore
proportional to
S(1—2b/x) 8 :

v—2b v

so that the influence of the molecules of the solvent vanishes and
van “Tt Horr’s formula is proved for the quantity defined by us.

§ 7. That this quantity has further always the same value as
the quantity whieh may be measured experimentally, is proved as
follows, Let us think the action of the membrane in such a way
that it suffers the molecules of the solvent to pass through freely,
but repels those of the dissolved substance perfectly elastically.
Something similar would take place when the membrane worked
as a ‘molecule sieve’, i.e. when the pores were such as to allow
the molecules of the solvent (thought smaller) to pass, the others
not. According to the definition the latter will then exert a
pressure on the membrane equal to our osmotic pressure. The other
molecules passing through the wall unmolested, there is no mutual
action with the wall, and so they co not exert any force on it.

1) If one should object to the train of reasoning followed here, one can find in
Bourazmann’s “Gastheorie’ a proof for this formula which intrinsically agrees per-
fectly with that given in this paper, but will appear stricter to some. There one
will also find the above given integration carried out.

2) [It is clear that we shall get the same result, when we do not take 2h,
but f (b/v.) for the voiume of the distance spheres. For as the place of the plane
of impact with respect to the molecules of the solvent is quite arbitrary in our
present case, the part of the plane of impact, which lies within the distance spheres
will stand to the whole area in the same proportion as the volume of the distance
spheres to the whole yolume,

( 738 )

The experimentally measurable difference in pressure on either side
of the membrane must therefore have the same value as the quantity
defined by us.

Lorentz’), however, has shown that the assumption made here
concerning the membrane is by no means necessary. On the contrary;
if we assume that the membrane is thick compared with the sphere
of action, that its substance fills a volume large compared with the
apertures present and that it feebly attracts the molecules of the
dissolved substance, whereas these are strongly attracted by the
solvent — none of which are improbable assumptions — we arrive
at the result, that none of the dissolved particles reaches the membrane,
much less exerts a pressure on it; the membrane is then quite
surrounded by the pure solvent. And that this case is really the
usual one in nature is made probable by the fact, that it is by no
means always the smaller molecules which pass the membrane, as
we assumed above. The membrane seems therefore not to work as
a molecule-sieve. We are then easily led to suppose that the mem-
brane does not exert a positive repulsion at all on the non-passing
substance, but that it only attracts those particles much less strongly
than the solvent, so that the dissolved particles do not pass through
the membrane, because they occur but extremely rarely in its neigh-
bourhood. This view is supported by the fact, that only those
substances seem to be non-passing which are not easily converted
to vapour, and so cannot reach the limits of the liquid in virtue of
their own thermal motion alone.

However this be, also in this case our conclusion holds good.
For when the molecules of the dissolved substance do not (or only
in an intinitely small number) reach the membrane, two planes will
be found not far from the membrane, A where the molecules of the
dissolved substance still have their normal density, B where this
density has diminished to zero. Between 6 and the membrane we
find then pure solvent. If we wished to discuss such a layer fully,
we should, of course, have to give a theory, as VAN DER WAALS
has given for the transition liquid vapour’), extented to a mixture
in the way van Enpik*) has done. But for our purpose this is
fortunately not necessary. We need only observe, that the layer
AB as a whole has now exactly the same influence on the condition
of motion of the dissolved molecules as the mathematical upper surface
of the membrane had just now. The layer AZ’ as a whole will now,

1) 1. c.

4) Verh. dezer Ak. (2) 1; Arch. Néerl, 28, 121 and Zsch. phys. Ch. 18, 657.
5) Diss. Leiden 1898.

just as the membrane just now, be pressed downward with a force
equal to the osmotic pressure defined by us, and transfer this force
to the underlying layer of the pure solvent, which is pressed outward
with this force. But this pressing force is evidently equal to the
difference in pressure which may be measured experimentally *).

§ 8. Thus it seems to me that van ’r Horr’s law for dilute solu-
tions is kinetically explained in the same way as the law of BoyLn-
Gay Lussac-Avocapro for dilute gases and that of vax per Waats
for liquids and gases, i.e. we have obtained an kinetic insight how
these laws result from the condition of motion in the homogeneous
mass, while we have left out of account what happens in the
eventually (probably always) present unhomogeneous bounding layers.

It appears from the explanation convincingly, that van LaAar goes
too far, when he states *), that we cannot speak of osmotic pressure
in an isolated solution. Here too this notion has a clear physical
signification, and the laws which govern it, are to be derived.

1) This hydrostatic proof may easily be replaced by a purely kinetic one, though
the latter is somewhat more elaborate. The layer AB, which (in consequence of
course of the neighbourhood of the membrane) behaves as a layer of water, through
which the dissolved substance cannot penetrate (Cf. Nernst’s well-known osmotic
experiment) imparts to the molecules of the dissolved substance per second a
quantity of motion equal to the osmotic pressure defined by us, and receives itself
an equally large quantity in opposite sense, which it transfers to the underlying
layers, as the kinetic theory teaches. (See e.g. Bonrrzmann, Zsch. phys. Ch. 6, 480).
Now the whole mass of water, which is in the neighbourhood of the membrane, (on
either side, reckoned on one side from B, on the other from a plane, so far from
the membrane that the latter does not act on it any more), does not moye downward,
so it must receive an equally strong but opposed impulse, which, of course, cannot
issue from anything but the membrane. Of what nature the forces acting here are
is quite unknown. It cannot be the ordinary molecular attraction, for then the
denser liquid found above the membrane would probably be drawn more strongly
downward than that found ander it upward, We might think of friction in the pores, but
it would then have to be different in one direction from that in the other ; in short I dare
not venture on any conjecture about this. This alone is certain, such forces must
exist, at least if the case put by us ever actually occurs. This appears already
from the fact that the pure solvent above the membrane is subjected to a higher
pressure, so has a greater density than under it. Such an equilibrium occurs for
all kinds of kinetic questions (liquid vapour, gas under the influence of gravity),
but the necessary condition is always a force, which at a cursory examination
seems to have the result, that the velocity of the molecules in one part (so the
temperature) would be higher than in the other, but in reality only proves to have
influence on the density. The membrane, which furnishes this impulse, receives an
equally strong one back from the reaction, and so here too, though indirectly, we
see a force equal to the osmotic pressure defined by us, exercised on the membrane
from the inside to the outside.

2) Chem. Weekblad 1905, N°, 9, § 3. Voordracht Bat. Gen. 3.

( 740 )

Whether this renders it desirable for us to give it a prominent place
in the theory of solutions and make all the rest proceed from it, is
a question to which | wish to revert in a separate paper.

First I must add this observation. The insight obtained in the
nature of the Osmotic pressure enables us to examine what quantities
must occur in the formula for more concentrated solutions. In the
first place it will no longer be true for concentrated solutions, that
the term @/,2 vanishes, both because on the two sides of the mem-
brane the density v differs, and because the concentration and so
the @ will differ. Further —- as appears from our proof — for
higher concentrations the volume of the molecules will assert its
influence, and not only that of the dissolved substance, but also of
the solvent. For as on the two sides of the membrane the density
differs, the part of the plane of impact that falls within the distance
spheres of these molecules, will no longer be represented by the
above given value. <As_ finally the molecules are of different size,
when the terms 4, and #/, occur, the term /,, is sure to appear.
The formula found in this way will certainly not agree with the one
found in the preceding communication by a thermodynamic method,
for the latter is derived from the equation of state with constant 4,
whereas the kinetic considerations exclude all doubt that 4 is a
function of the volume. If ihere should be a real diminishing of
the size of molecules when passing beyond the membrane, then this
fact is also to be taken into account.

Far be it from me to make an attempt to draw up such an
equation. To achieve this, if would be required, as appears from
What precedes, that one should be able to surmount at least all the
obstacles which stand in the way of an accurate equation of state.
And if this might be done — the preceding paper proves it — the
final formula could be found in a way, which would not expose
us again to the danger of making errors. T shall therefore not enter
into the question either, in what way the formula derived in a kinetic
way can satisfy the first requirement that may be put to every formula
for concentrated solutions: that it yields the value o for the case
that the substance passing the membrane has perfectly vanished from
the solution.

§ 9. IT shall just add a single remark on the question whether
our kinetic view implies that the so-called static theory of the osmotic
pressure, which ascribes the canse of the phenomenon to attractive
forees, is entirely wrong? It seems to me that from what Lorentz
has proved it appears that we must answer this question in the
negative. It is true that we have seen that the attraction of solvent

(aia a}

and dissolved substance begins to play a part only in sensibly con-
centrated solutions, and that we have to explain the osmotic pres-
sure by a ‘“moleculares Bombardement”. But the case treated by
Lorentz shows that the whole osmotic phenomenon might possibly
exclusively be the consequence, not so much of the presence of
attractive forces, but just of the reverse, of the want of attraction
between the molecules of most solid substances and certain other
solid substances which form membranes. If the adherents of the
static theory mean no more than this with their assertion: that the
osmotic pressure must be explained from forces of attraction, then
they seem to me for the present secured against every attack.

Physics. — “Osmotic pressure or thermodynamic potential’. By
Dr. Pu. KonnxstammM. (Communicated by Prof. J. D. van
DER WAALS).

§ 1. The theory of thermodynamic functions, through which
Gipss has enabled us to derive from the equation of state of a system
in homogeneous condition, what heterogeneous equilibria will oceur,
has attracted attention only in a very limited circle during a series
of years. However great the region opened for investigation by Grpps
was, the methods indicated by him seemed so abstract, that only
very few dared to grapple with them. At a stroke this was changed,
when in 1885 Van ’r Horr succeeded in replacing these methods
in appearance so abstract, by another, that of the osmotic pressure,
which strongly appeals to the imagination. The theory of solutions,
which up to that time had only existed for a few, rapidly became
one of the most frequently treated and diseussed subjects of physics
and chemistry; since then it haz continued to enjoy undivided
attention.

It stands to reason, that the attention, which now for twenty years
has been so lavishly granted to the questions of heterogeneous equili-
brium, have also been conducive to making Gipps’ methods for the
solution of such questions known in a wider circle. But though Gress’
name may be counted among the most famous and widely known
names in the sciences of physics and chemistry, yet even now his
methods cannot be said to have been universally accepted.

The adherent of a mechanical (or, if one prefers, statistical) natural
philosophy has by no means reason exclusively to regret this course
of affairs, for he sees in it a clear indieatien, that the views whose
truth he advocates, are by no means so antiquated, nay even dead,

( 742 )

as they are often declared to be. And if the current opinion —
which certainly greatly contributes to the greater popularity of the
osmotic pressure compared with that of the thermodynamic potential

were really correct, that we can form a clear idea of the
physical meaning of the first quantity and not of the second, then

there could not be any doubt for him which method to prefer, if

for the rest the circumstances were quite the same.

But this current opinion seems to me hardly tenable and on the
other hand I believe that in many respect the thermodynamical
potential is preferable to the osmotic pressure, and that therefore it
will be advisable to put the question whether it would not be
better to return to the older metbod both for scientifie investigation
and for instruction ?

§ 2. This question has lately again been put forward by Mr.
VAN Laar in an address for the “Bataafsch Genootschap” at Rotter-
dam’), which was followed by an article “Over tastbare en ontast-
bare grootheden’” (On palpable and impalpable quantities) *). Though
I readily admit, that these papers have induced me to consider the
problem of the osmotic pressure specially, there would not be any
reason for me to discuss Mr. van Laar’s views here, when only
his address had appeared. For I can fully subscribe to the general
tendency of this paper though of course I would not be responsible
for every statement, as moreover has already appeared from my
preceding communications in these Proceedings — and I should
therefore only have to consider what in my opinion would have to
be added to his address. His second paper, however — and in
this I have specially in view § § 6 and 7, pointed out as the gist
of his paper by the author himself — Mr. van Laar seems to me
to harm rather than to promote the good cause, which he has
espoused with so much ardour, and already for this reason I feel
it incumbent upon me to protest against this part of his reasoning.
I think that I accurately represent the gist of it as follows: It is
irue that we cannot form a clear idea*) of the nature of the ther-
modynamie potential, but we cannot do so for the osmotic pressure
either. Nor is this surprising, for the improved philosophical insight

of the last years gives us the conviction that our natural philosophy _

never works with any but fictitious (though sharply defined) ideas,

1) Also published in Chem. Weekblad, 1905, N°. 1.

2) Chem. Weekblad, 1905, N°, 9.

3) Mr. van Laan speaks of a “palpable conception” (lastbaar begrip). It would
lead me too far if I would account for the reason why I think that I may,
nay even ought to substitute the term chosen here for it.

( 743 )

which must not and cannot claim in the least to represent the real
nature of things. It is also owing to this insight, that several voices
have been raised of late in favour of the use of the thermodyna-
mical potential.

§ 3. Now I think that I have convincingly proved the incorrect-
ness of the second thesis in the preceding paper, and as I gladly
and with full conviction range myself with the “tastbaarheids-
menschen,” (those who want to form a clear idea of the physical
meaning of each term used), whose opinion Mr. van LaAar severely
condemns, his reasoning would lead me to take side against the
thermodynamie potential party when IT could subseribe to his. first
and his last thesis more than to his second. This however, is by no
means the case.

The last philosophical-historical thesis 1 can, naturally, not discuss
here and I confine myself therefore to that concerning the physical
meaning of the thermodynamical potential. It seems to me that we
can form an idea of this quantity which need not be inferior
to that of any other statistical quantity. That Mr. Vax Laar has
overlooked this fact seems chiefly owing to two circumstances of
which it may appear that one can have hardly any influence, for it is
sunply a question of nomenclature. Following a common way of
speaking, which does not seem to me the less reprehensible for the
fact that it is of frequent occurrence, Mr. Van Laar does not give
the name of “thermodynamical potential” to the quantity introduced
by Gipps into science by that name, but to one of the other functions
introduced by Gipps, the §-function. There are more reasons than
only a feeling of deference, which make this undesirable. The real
(Gisps’) potential zs really a potential, i. e. it is constant in a space
where equilibrium prevails, and its not being constant means, that
there is no equilibrium. At least when there act no capillary or
external forees; and in this case the resemblance of the thermody-
namic potential with the potentials of other energies stands out per-
haps the more clearly. For in this case we need only add to the
(Gipps’) thermodynamic potential the other potentials, which exist in
that space in order to get a quantity, the total potential, which now
also is constant throughout the space in case of equilibrium. The
S-function has neither the one property, nor the other, except when
we have to deal with a simple substance without capillary layers,
in which case it becomes identical with the thermodynamic potential.

If now also in §§6 and 7 Mr. Van Laar had directed his atten-
tion instead of to the $-function, fo the real potential, as he has done
in § 4, where he carries out his calculations by means of it, it would
probably not have escaped his notice that he wrongly represents the

( 744)

thermodynamic potential (whether it be in one sense or in the other)
as the last, most fundamental quantity, which determines the internal
condition of a body. As such we cannot take others than v and 7
(if necessary of course wv, y, etc.); that this is not only a subjective
“point of view” appears perhaps most clearly from the study of the
theory of capillarity, as van per Waats has given it.

§ 4. From this follows naturally, that we must try to form
an idea on the relation between the thermodynamic functions and
these fundamental quantities, and this does not seem so very difficult
to me just with regard to the thermodynamic potential. Let us only
consider the following. Thermodynamics teach, that however composite
the equilibrium may be, the total potential of every component must
be the same in two phases which are in equilibrium; the kinetic
theory, or in plain language, common sense that in all those cases
equilibrium is only possible when an equal number of particles of
each substance passes from the first phase into the second phase and
vice versa. Now Van per Waats has shown') that in the case of
equilibrium of vapour and liquid, whether in a simple substance or
a binary mixture, the two conditions are simply the same fact stated
in different terms. It does not seem hazardous to me nor jumping
to conclusions to conclude from this that these two conditions, which
are always at the same time fulfilled or not fulfilled, also in other
cases will agree in signification and that therefore the physical meaning
of the thermodynamic potential *) of an homogeneous phase, on which
no external forces act, is nothing but the number of particles which
per second reach a wall as defined in the preceding communication
§ 4 under 2, if this wall is thought in the midst of that homo-
geneous phase.

1) Verslagen Kon. Akad. Amsterdam (4) 3, 205 and Arch. Néerl. 30, 137.

2) I choose purposely the words “that the physical meaning of etc.” and not
“that the thermodynamic potential is equal to elec.” For the equality of the
two quantities would require an “absolute” scale of thermodynamic potential.
For from the equality of the conditions mentioned follows only:

Vf eee NE ON Cate weGlen oe oo (Il!)
where / is such a function, that My is a one-valued function of V and reversely
N of My. This however, is not of material influence, for formula (1) expresses
only, that we begin to count the thermodynamic potential from another point
than the number of particles (which agrees with the fact that our thermodynamic
potentials always include an undetermined constant) and that we make use of
another unily when measuring one quantity than when measuring the other.
There is therefore perfect concordance of our case ard that of the temperature
measured e.g. according to Celsius and certainly nobody will object to the statement,
also when he thinks of this temperature seale, that the physical meaning of the
temperature is the mean vis viva of the centres of gravily of the molecules.

( 745 )

Yet this definition requires some further elucidation, because the
number of molecules under consideration reaches a bounding plane
of the phase, which does not exercise any attraction on those par-
ticles, whereas on the particles discussed above and whose number has
heen calculated by Van per Wats, viz. those which pass from the one
phase into the other, a force docs work directed to the other phase. But
this difference is in my opinion, only apparent. Also in the equations
arrived at by vax per Waats, one member refers exclusively to one
phase, the other to the second phase; there are no terms in them
consisting of factors, one of which vefers to the first phase, another
to the second. That we had to arrive at that result, may be easily
understood, for the thermodynamic potentials themselves refer either
to the one or to the other phase and are quite determined by the
condition of that phase.

That at least in the definition of the thermodynamical potential
one number may be put instead of the other, appears as follows.
Let us consider a liquid in equilibrium with its vapour. The number
of particles that now passes, per unit of area, through the bounding
layer is that which Vax per Waats treats of; let us now place
on this liquid a layer of a substance which does not attract the
molecules; let this layer be thick with respect to the spheres of
action and provided with narrow channels. The number of particles
that penetrates into these channels on either side is the number,
which we used in our definition. Now I assert that the introduction
of this layer cannot disturb the equilibrium of the homogeneous
phases’), i.e. their pressure and concentrations will not change. For
if this had been the case we should have been able to construct
with the aid of such a layer a so-called perpetuum mobile of the
second kind, and should have come in conflict with the second law
of the theory of heat. From this follows that equality of the number

') The equilibrium in the non-homogeneous, capillary layer is disturbed by
introducing such a wall. For, as vay per Waats has shown (cf. the footnote p. 735)
the equilibrium in a plane of such a layer is only stable in consequence of the
attractive forces exercised by the surroundings. When introducing the solid layer
in question the condition in the transition layers will be considerably modified,

which might also be anticipated. This does not affect our reasoning,

for by the
word “homogeneous” we have positively excluded these transition layers in our
definition. That this was necessary in any case appears already from the fact, to
which we have already called attention above, that the thermodynamic potential
for such layers is no longer the quantity which determines the equilibrium, but
that it is replaced by the total potential. We must therefore certainly not have
recourse to such layers, in order to get acquainted with the thermodynamic poten-
tial in its simplest signification.

( 746 )

meant by Van perk Waals implies equality of that used by us in
the definition, and that we may therefore substitute the latter for
the former in the definition of the thermodynamic potential.

§ 5. In this way we have obtained a clear idea of the nature
of the thermodynamic potential, which so far as I can see is in no
respect second to that of temperature, external pressure, kinetic
pressure, number of collisions, mean length of path ete. That for
all this it is not always easy to derive in a special case the value
of the thermodynamic potential from this kinetic meaning is self-
evident, as well as that it will possibly always be more desirable
to derive the thermodynamic potential by means of thermodynamic
functions than from direct kinetic considerations. It is true that we
do not avoid the latter in this way either, but we make use of the
result of these considerations, as it is given in the equation of state.
In these two respects, however, the thermodynamic potential is in
no way inferior to the osmotic pressure, as appears from my two
preceding communications, specially from § 8 of the second.

§ 6. Mr. vay Laar informs us, that in connection with his address
he had been asked “to supply something as a substitute” for the
osmotic pressure and the kinetic conception of it, something that
“conveys some meaning”’.') This request seems by no means so
unreasonable to me as it seemed to Mr. van Laar and I think that
I have complied with it in the preceding pages. Now I may be
allowed to prove that this “something else” is at the same time
“something better”.

First of all it seems not very appropriate to me to give a quantity
of pressure such a prominent place in the theory of mixtures. As
soon as we deal with this theory in general, i.e., include also
external forces and capillary phenomena (which are very often of
great importance, I need only mention critical points) it appears,
that the pressure is a quantity we may only handle with great
‘aution and which may certainly not be treated as fundamental
variable.*) In a much higher degree this objection holds for the
osmotic pressure. For this is, as we have seen, not a quantity which
is characteristic for the state in which the solution is; the peculiar
laws of the osmotic pressure are not due to the fact that matter
in dilute solution is in a particular, peculiar condition, they originate
— in their generality — only from our arbitrariness, which by means
of fictitious ideas, calls peculiar conditions into existence on paper,
which never exist in reality. For let us not close our eyes to this
ai ‘Chem. Weekblad 1905 No. 9, § 3. The inverted commas are Mr. van Laar’s.

*) Cf. the footnote on p. 735.

( 747 )

undeniable fact and least of all should they do so who are so
averse to “hypotheses” that though all those semipermeable walls
may be realised in a few cases, yet we have on the other hand thou-
sands and thousands of cases, where we have not the slightest
foundation for belief in their existence. What reason can there be
for assuming, that there will ever be found a wall permeable to
toluol, but not to benzol, and another wall, permeable to benzol
and not to toluol, and what else is it but a fiction to speak of a
wall, permeable to cane-sugar and not to water. (For also this is
necessary, see Van ’r Horr, Vorlesungen II, 24). And let us even
put the most favourable case: that such walls existed really, does
it not remain perfect fiction then to try and treat the theory of
concentrated solutions with them? We need only bear in mind that
steel, our strongest material, however thick it is taken, can hardly
bear pressures above 5000 atms, what to think then of a semiper-
meable wall for which such a pressure is but a trifle. And now I
do not in the least object to such fictitious ideas when they are quite
unavoidable — this is sufficiently proved by what precedes — but
what is the use of using them, when we have another quantity of
simple signification, which zs characteristic of the condition in which
the mixture is, which can be defined solely from the properties of
the substance with which we have to deal?

To this comes another difficulty. He who works with the osmotic

pressure history teaches it — is but too apt to consider a mixture
not as an individual, which must be examined in itself and must be
known from itself, but as another substance (solvent), more or less
modified by the presence of the “dissolved substance’. In this way
we lose quite sight of the fact, that the two components in a mixture
are present in exactly the same condition (the singular theory of the
“oaslike nature” of the dissolved substance proves it); we begin to
overlook, that ‘dissolved substance” and “solvent” are perfectly arbitrary
names, which have only a right to existence when we confine
ourselves to one of the two extreme cases; we are led to try and
explain the properties of a substance from those of another, which
is often in quite different circumstances; we begin to apply all
kinds of hazardous approximations and compromises; we get to the
most extraordinary association and dissociation theories. How fruitful
on the contrary the opposite method is, the whole work of VAN prr
Waats, the experimental and theoretic material (inter alia on the
behaviour of mixtures with respect to the law of corresponding
states) gathered specially at Leiden may prove.

§ 7. Now one may object to this, that all these are theoretical

51
Proceedings Royal Acad. Amsterdam. Vol. VII.

( 748 )

objections of more or less value, but that they are outweighed by
the practical advantage that calculations with the osmotic pressure
are so much simpler than with the thermodynamic potential, but
ihis objection lacks all foundation. For kinetic calculation cannot be
meant in this, and for the thermodynamic calculation it holds on
the contrary, that when making use of the thermodynamic potential
we need not take one step, which we are not obliged to take in exactly
ihe same way when making use of the osmotic pressure. In order to
prove this, I should like to reprint and follow step by step the
proof given by Van “r Horr in his Vorlesungen, but as this proof —
carefully selected by Van *r Horr from considerations partly from
himself, partly from Lord Rayieien, partly from Dr. Donnan, so
undoubtedly the finest and simplest to be found — covers two pages
in print, I shall only indicate the principal operations and put in
juxtaposition the operations, which are required for the thermody-
namie potential with the same neglections.

1. Remove from a solution of 1. The thermodynamic poten-
osmotic pressure P a quantity of | tial is:
solvent, occupying a volume v. 1

The substance yields an amount | jfu—pv+ | pde + MRTU(A—«, +
of work — Pv.

-_—_—_-- vo
0
4 F(T) +{() dv
On) oP
Ori)
pe becomes here p,?,.
2. Neglect the change in vapour 2. Neglect the variability of p

tension and the contraction of the | with « and the compressibility of
solution. (This is not expressly | the liquid, then
stated, but is evidently necessary Yo,

U
for the proof). f(Qje= en fous 0.
wv

. . . % ro
3. Let the quantity of dissolved 3.
substance, dissolved in v, evaporate es
diosmotically ; let its volume be
x epee, : pdv. = pe (Ycs— Ve, )
I”, the work done is: —
Ve,

pV

(when we neglect v by the side

oL WA):

( 749 )

4+. Let the vapour expand to 4.
infinite volume, the work done is: 7
i Vv { rae = MRT I,
| pdv = MRT : sls
« v

Co

\ —

5. Now press the vapour again 5. The thermodynamic poten-
into the solution, then a work is | tial of pure water is:
done by the substance:

© Vy Mu = pe +{ pd -- F(7)
— | pte -— MRT 1 — 7

"

V

7

| pae — MRT 1 : ; pe — Pe Vex
Ve, ——
Vq

6. The total quantity of work | 6. The two potentials are the
must be zero, so: | same, so:
Va Piao) te — TT a)
which in spite of the different notation is the same, when log (1 — x)
is replaced by — 2.

So it is seen that to every integration on the right corresponds
an operation on the left of exactly the same nature, though it does
not always refer to the same substance. The only difference is that
on the right the integration is carried out directly and that on the
left pistons and membranes are worked with. Now I do not think
that any one can easily set greater store by a clear physical meaning
of operations than I do, but that we should not be able to carry
out an integration along an isotherm without bringing in two pistons
and three membranes, seems rather too much of a good thing.

§ 8. And now we have considered the most favourable case :
dilute solutions; how is it with more concentrated ones? It will
certainly be possible to devise also for them cycles so that the
calculations introduced in my first paper may be carried out without
mentioning the name: “thermodynamic potential’, but it will not be
found possible by a thermodynamic method to draw up a formula for the
osmotic pressure without determining the integrals occurring in it.

In this way it would seem as if the two methods were essentially
the same; it is not so, the osmotic pressure method has drawbacks,
of which the other is free. For what is it that we really wish to
learn by the «wo different methods > Not the osmotie pressure itself,
and the properties of the solutions under that pressure, that is
for concentrated solutions: in sensibly compressed state. What we

oL*

eS a

( 750 )

aim at are the properties of solutions under the pressure of their
saturated vapour: lowering of the freezing point and the vapour
tension, rise of the boiling point and coefficient of division or more
accurately (ef § 6 above) freezing point, boiling point, vapour tension
of the mixture and the concentration of coexisting phases. And this
does not only apply to physicists and chemists, who rarely if ever
work with membranes, but also to biologists, to whom they are of
the greatest importance. For differences in pressure of about ten
atmospheres will probably hardly ever oceur in biologie experiments
and a fortiori not in the living organism either. The equilibrium
between two solutions will therefore never be established by diffe-
rence in pressure, but by the difference in concentration required
to make the pressure equal. So we have not to deal here either
with compressed solutions. *)

For the calculation this implies that when making use of the
thermodynamic potential we need extend the integration along the
isotherm only to the pressure p- and the thermodynamic potential
may then be determined with sufficient approximation from the
well-kwown formula for the vapour pressure :

c Tr.
- log =1( ~ a ')
Pk =

thongh it be with the factor 7 determined experimentally instead
of the value 4 found theoretically. But if the osmotic pressure is
used we can naturally not do without the integration up to p,

“0 (ae :
(in the term | » ar) and it is exactly this part of the isotherm
a“

which is known the least, where e. g. the variability of 6 is felt
strongest, even the only term, on which it has influence when the
just mentioned formula for the vapour pressure is used. Quite
unnecessarily therefore the result is made less reliable by the intro-
duction of the osmotic pressure.

And supposed even that we had found the desired expression,
of what use could it be to us? It is true that the quantities, whieh
we really wish to know and which I mentioned before, are con-

1) For this reason hardly anything would be lost when in the discussion of
really osmotic questions it was made a rule to treat them without “osmotic pres-
sure’ and simply to introduce the concentrations on either side of the wall; whereas
in this way there would be a great gain in lucidily of expression, witness the
example cited by Mr. van Laar (Le. § 5). For the interpretation given there may
be correct or incorrect, it can hardly be denied that the cited phrases may be
easily misunderstood in the sense of the well-known question of Puriy, which has
so repeatedly been seriously discussed, how e.g. a CaCl, solution of no less than
53 alms. osmotic pressure could be kept in a thin glass vessel without making it
burst asunder!

G7pt )

nected in a simple way with the osmotic pressure in dilute solutions,
but we have not a single reason to assume this also for concentrated
solutions, or rather we may state with almost perfect certainty that
this will not be the case. How on the other hand those quantities
may be determined with the aid of the thermodynamic potential,
Van ppr Waats taught us already fifteen years ago.

§ 9. And let us finally not forget that though solutions of non-
volatile substances at low temperatures do play an important part
in nature, yet they are not the on/y substances which exist, nor the
only ones which deserve scientific consideration. And yet, the theory
of the osmotic pressure must necessarily be confined to them. One
is so customed to derive the laws of the rise of the boiling point and
the decrease of the vapour tension from the osmotic pressure, but
it is generally forgotten, that many mixtures have on the contrary lowe-
ring of the boiling point and rise of the vapour tension *), and that at
any rate if the dissolved subsiance is but in the least volatile, the
changes in boiling point and vapour tension cannot be derived
any more from the osmotic pressure. And it is obvious why. It is
inherent in every definition of the osmotic pressure, that it can only
be applied to those cases, in which one component may be separated
from the mixture in pure condition, as Nurnst has clearly stated
for the first time. Hence this does not only exclude the whole region
of higher temperatures, at which all substances become more or less
volatile, but also all cases of not perfect separation in the liquid
or solid state. Also the lowering of the freezing point is touched by
this objection. It is true that the lowering of the freezing point may
be computed from the osmotic pressure, but only when, as in water
and sugar, the solid substance, which deposits, is not of variable
composition. Solid solutions and mixed crystals, which attract at
present so much attention in chemistry, cannot be treated in this way.

Physical chemistry in its present state reminds us strongly with
regard to its quantitative part, of the navigation of a people, which
does not yet know the compass. The coasting-trade is carried on
with great vigour, the same limited region is traversed again and
again; but they do not dare to venture on the main sea far from
the coast, and with reason, for great is the danger of ruin in the
towering waves of random hypotheses. This can only be remedied
by a trustworthy compass. Physical chemistry may obtain it if it
will abandon the method of the osmotic pressure and adopt that of
the thermodynamic potential in connection with a well-grounded
equation of state.

1) Cf. Théorie Moléculaire §-17.

Astronomy. — “Approximate formulae of a high degree of ac-
curacy for the relations of the triangles in the determination
of an elliptic orbit from three observations.” By J. Waeper.
(Communicated by Prof. H. G. van DE SANDE BAKHUYZEN.)

The places in space occupied by the observed planet or comet at
the instants ¢,, ¢, and ¢, are indicated by P,, P, and P,, the posi-
iion of the sun is indicated by Z.

For the determination of an elliptic orbit we mainly proceed as
follows: first by means of successive approximation we derive the
distances P,Z=r,, P,Z=r,, P,Z=r, from the data of the obser-
vations, from which distances the elements of the orbit are directly
computed without using the intervals of time. From the obtained
ellipse we can again derive the intervals of time in order to test the
accuracy of the results and compare them with the real ones. In
case they perfectly agree, the ellipse found satisfies all the conditions
of the problem, but as a rule this is not so. The cause of it is
that, in order to calculate the distances 7,, 7,, and 7,, we use
triangle P,ZP,
triangle P,ZP,

approximate formulae to express the relations ny

triangle P,ZP,
triangle P.ZP,
three distances to be found, while neglecting the terms of the 2°¢, 34
or 4 order with respect to the intervals. Indeed, different expressions
have been proposed for 7, and ,, some recommending themselves by
greater simplicity, others by greater accuracy, but, so far as I know,
in the general case of unequal intervals none of them contain the
quantities of the fourth order with respect to the intervals.

The errors in the calculated distances 7,,7, and 7, and those in the
elements of the orbit derived from them are generally of the same
order as that of the terms omitted in the expressions for m, and 7,.

Accurate and at the same time simple expressions for n, and 7,
have been given by J. W. Grpps’).

The purpose of this paper is to develop, according to Gipps’ method,
expressions for n, and », which include the terms of the 4 order;
at the same time a new derivation of Gipps’ relations is given.

In the ellipse sought let P be the position of the heavenly body
at the time ¢, # and y its heliocentric rectangular coordinates in the

and =v, in terms of the intervals of time and of the

1) J. W. Gress: On the determination of elliptic orbits from three complete
observations. Memoirs of the national academy of sciences. Vol. IV, 2; p. 81.
Washington 1889.

( 753 )

plane of the orbit, and 7= ZP, then « and y satisfy the following
differential equations

dv a nf d*y y

— = — = Se —_ = ——, = J»

dt* ?" dt? ri

wherein we have put r= %(¢—1,) as independent variable instead
of the time ¢; 7 is therefore the time reckoned from the epoch of
the first observation and expressed in the unit for which, in the
solar system, the acceleration =1 at a distance from the sun
which is adopted as unit of length; / is the constant of Gauss
[log k = 8.235 581 4414 — 10).

While designating the rectangular coordinates of ?,,P?.,P, by corre-
SD oD ton) 1 3 3 .

‘ bee LY — Ye triangle P, ZP
sponding indices I remark that » = —————- => —————__
L5Y3—Y,4, triangle P,ZP,

satisfies a similar differential equation as v and y, namely :

dn n
—_ = — —_ =r,
dt? r
At the times i= O)ne eS) ee,
the values of 7 are 0 +n, +1 and
Re Ns 1
the values of 0 == ——
ins, rT;

Consequently in the development of » in a series of ascending
powers of t after Mac Laurin, the terms of the power zero and 2
will be wanting. If in this expansion we do not go farther than the
4th power of t, we require only 8 indefinite coefficients which may
be eliminated from the following 4 relations:

n,—Kor,-- K,v,°> 4+ Kyx,*+ f,
b= Ke, Ar Kee + OF,

Ny s nares :
aa at aa ae 6K,r, a 12K,r," sine

2

1 le ig 7 2 7
= a: = + Gkyr, + 12K,7,7 + F,.

The remaining relation yields an expression for , in r,, T,, 7%, 7°,
and the remainders /,, /,, 7, and F,.

The indices which I have used for the remainders, indicate the
order of these terms with respect to rt; F,, for instance, which
begins with A,7,° is of the 4 order of +, which is evident when
we express the coefficients A’ in terms of the derivatives of n for
<=0 and develop the latter by means of the differential equation
for m as products of #,. For clearness I shall here give this
development :

( 754 )
dn ies eee f 1
= = — ni — zi, where z is put for 3
d*
aa = 2) n— Be
Tt
din
Seem Dae irs

d
From the differential equation BoC sie: satisfied by 7, we
T

can derive the following differential equation for <= -—:
7

which may serve to eliminate 2 from the higher derivatives.

dn z ome Tal
a5 CS 5 a) ee — Sz)%.
For t= 0, n is equal to zero and #= K,, hence

i 1 22—32
oS ee : i Sa : tee
If we substitute the expressions for the coefficients A in the second

of the 4 relations, this becomes :

Ke

‘ i perme oleh 3, (eater
l= Ker era — apres warn see
and from this it clearly appears that A, and the other coefficients
ae : P 1
kK, in so far as they depend on the intervals, are of the order —.
T

3
From the 4 relations with the indefinite coefficients A,, X,, A, we

find by eliminating the latter :

T, Sik ere fe . ) > 5 Pile he hea ( 1 +

n,-f,- ———-—- | — +/ 1-F ,- —————~ | —_+ }
= 12 SP NELe ; 12 Ree

From this equation I solve :

—

Ts

2 2
Tv; "45?

—
ee T, 127,° LR
ee tT T, 1 T, +47, —Ts" ; ;
127,°

where

, , Pia tS Tet T; \ PF ide ete
Js site Ja = 7 12 = my ad sa | 4 am 2 nap OLN nL &
R a T, a)
a— — —--——-- > —________———.,
; 1 (pate IR T;
eae

This remainder is apparently of the 4° order with respect to the
intervals. If we neglect the terms of higher order than the fourth
we can replace in R,: 7, by K,r,', &, by K,z,', 7, by 20K,r,* and
I’, by 20K,7,°; and we obtain as supplementary term, accurate to the
fourth order

oe ;
-|- a Kt, (t, + t,) («, — t,) (20, — t,))(t. — 27),

which expression vanishes on account of the last factor, in the case
of equal intervals.

The corresponding approximate formula for 7, can be derived by
triangle PZP,
triangle P.ZP,?
ing powers of /(¢, — ?@) and further by proceeding in the same manner
as we have done for 7,. The result for n, is derived from the pre-
ceding result by interchanging the indices 1 and 8, in which case
t, stands for &(¢, — ¢,), hence :

developing the relation depending on the time, in ascend-

airat 2
+ EES
Des
T, 127, 5
n, =—- ——— = = lie,
Ty, 1 Ts, HTT, T,
anes

The remainder of 7, is not only of the same order as that of 7,,
but even in the 4' order it has the same absolute value, with a diffe-
rent sign however. This appears clearly when, using the relation
T, —=7T, +7;, We express the correction of the 4% order for n, in
terms of t, and r,; this correction takes the following form, which is
symmetrical with respect to rt, and 7, :

ae
3 Kye ye, (2c, --'%,) (2) (©, —*,)-

In the remainder of 7, the coefficient LZ, may be assumed equal
to A. Therefore these approximate formulae always give for
m,-+ 7, an accurate value (comp. p. 758), including the terms of
the 4" order of the interval.

The denominators of these expressions for n, and n,, although here
different in form, are indeed identical ; the expressions themselves
agree with those derived from the fundamental equation adopted by
Gisps between the 3 vectors ZP,, ZP, and ZP, which can_ be easily

reduced to the form:

oa (: - me) ZP. + es (: = ree see) Z7, ue
1 Ss
T

ca 127,° ‘ 129r,?
SS Pie t, +4,t,—?y ZP
17 129, >

This equation is satisfied by the real places of the object when
we neglect a residual of the 5 order with respect to the intervals
of time. This signifies little, however, when compared with the
accuracy of the places calculated after Gress’ method, which rigo-
rously satisfy them; for each set of vector corrections 4 7P,, 4 ZP,
and 4 ZP does not lessen the agreement below the 5 order with
respect to the intervals of time, provided they satisfy the condition

Tv, = T; ——
IY NY I RN NY ey A
2 , T,
and are not below the 3 order with respect to those intervals.

Because in Gipps’ method the relations n, and n, contain errors
of the 4th order, it would follow from this that the places computed
after this method are inaccurate in the 4% order also. But thanks to
the circumstance that Gipps’ method includes for n, -+- 7, the terms
of the 4 order in all cases, its results are yet correct in terms of
the 4% order.

This special feature of Gipps’ method has been pointed out by
BE. WHIss *).

In order to obtain for x, and nv, expressions including in all cases

the 4" order of the intervals of time and containing besides them

1 1

1
only ==—4,, — =, and —;=2,, I have used the relation derived
; Us vr, Ur

i
on p. 754 K,= mae K,.

Starting from the development
n= K,1+ K,2° + K,x* + K,t* + remainder of the 5 order
I can make use of the following relations between the coefficients
K, the quantities z,, z,, z, and ”,.
Ay Gig NG hy nts ai 1 Ot A a
beige iG PRE ee AC a api ie cS F,
— 2,2, = + 6K,+t, + 12K,'r,* + 20K, t,* +f,
a + 6K,7t, + 12K, 7,7 + 20K, t,? + F,
0=K, 2, + 6K,
1!) E. Weiss, Ueber die Bestimmung der Bahn eines Himmelskérpers aus drei
Beobachtungen. Denkschriften der Mathem. Naturw. Classe der Wiener Akademie.
Bd. LX (1893).

( 757 )

By eliminating A,, A,, AK, and K, we derive from them the
following equation:

t,* (27, — 5r,)
SY (9 tel epee eas St Leh | ae
3 Js) ( 60r, )

( 1h) 2r,°+- 27,71, 21,7,?— 3r,° t,*r, (4, —3r;)
— (n,2 - ———__ =.
AA 607, 720 i

Tt,’ (2t, — 5r,)
1 — F,){| 1 + ——————-z, ] —
iS eae .)

LP 2u, Tt, 2r,” t+ 27,2 T Braet T,°t,(47,—3r7,)
Gat Fs) 60, 720 ri

(ee
(=

For shortness I replace the expressions which only depend on the
intervals of time by single letters, putting

Ge * (21, —9t;) G2\(2e s—9T,)

SS ee As 9. ————_ —
60r, 60r,

—or,*_27,71,—217,1, baeoa —27,° —2r,’ le ~2r, Te *+8r,’

Bog == Se — Bo _
60r, 607,
ee Te 3 (4r, — 3r,) on ‘4, °T, (At, - DT.)
a 720 rns 720

then the equation, solved with respect to 7,, yields for this relation

the following expression :

a 1 + Ase 21 + Bas 23 + Cao 21 23
1+ Aogz + Bog zo + Co3 21 20

The Bae: R, contains the quantities F’,, 7;, /, and /,; for
these I set, in order to form the value of FR, in the 5 order
with respect to the. intervals. of time, -7, = <K,7,°, [; = K,*,',
a 30 KG x7 and: 2. — 30 K, z,*; 1 then find:

et Ke (C52 1, — 0 * 6,7 — t,t,” - 2,') v; (t, — *,)-

As the root of the 4° power equation 1 — « — «#? —2*°+2'*=0
lies between zero and 1, viz. «=0,5806, the terms of the 5 order
will vanish from the residual, if 7, = 0.5806 r,.

We obtain the corresponding approximation for n, when we derive
an expression from that for 2, by interchanging everywhere the
indices 1 and 3, hence

4. lt Aiozs + Bisa + Cis zs 21
a, “1 + Aoi zs + Bor 22 + Co 2 22

The meaning of the new letters agrees with the rules tor the
interchange of the indices 1 and 3.

ie

—

ie gee eee 0)

wt een CLERY

( 758 )

t,? (21, — 5r,) my x, 8,)

A\o = ————————_ Ao, —
i 607, = 607,
— 2r,°-2r,? r,-27, 7,7 31,’ —2r,°-27,? r,-2r, 7,7-+37,°
Bia = = : Bi — Sar
60 tr, 60 7,
t,? 1, (4t, — 3t,) 7 tT,” t, (4t, — 31,)
Crs —— = Co) =-
4 720 720

In the remainder which belongs to this expression for 7,, the term
of the 5™ order :
1, (ctr, t, —1,* 4 —7, t,t) 4 5)
will vanish if t,—=0.58067,, therefore the term can never vanish
at the same time for 7, and for 7,.

es ; , triangle PZP,
LL, oceurs as coefficient of +, in the development of ————.—_-
triangle P,ZP,

in ascending powers of t=/ (t,—d), while A, indicates the coefficient

triangle P,ZP
triangle P,ZP,’

6

of t° in the development of where the variable t

means /: (t—1,).

If the first of these developments were performed in powers of
k: (t—t,) = —t, there would exist between each pair of corresponding
coefficients a relation implying that its sum with regard to r, would
be of one order higher than the coefticients themselves. Therefore,
neglecting terms of higher order than the 5'", we may assume that
the coefficients A, and Z, are identical in absolute value, yet differ
in sign.

Of a similar relation I have made use on p. 755, where in the
remainders of the 4 order I assumed the coefficients identical. In
the new expressions for n, and n, we can now, by putting L,=—A,,
derive the following value for the remainder of the 5" order of n,-+-n,:

2 K, Cie as (t,—t;) (27,* = T,T;)-

Therefore when the intervals of time are equal, the error in
n, +n, is of the 6 order.

If according to the indicated method we include the terms of the
triangle P,ZP, __ Ms
triangle P,Z7P, ht

nN, Ts 1 4- As iz2 + Bsi23 + Cs.12228
nm, t, °°) 1+ Arezo + Biser + Ciseser
and with it as remainder of the 5 order

4th order, we find for the 3'¢ relation

. ay

Tv
+ 3K, = T3°(Ts! $F 75°T, — T3°T,* TT? + 7")
1
From one of the examples from Gauss’ Theorta Motus (Libr. HU,

Sect. 1 ce. 156—158) I have computed the 3 relations according to
the formulae I, I and Ill. The rigorously correct values of those

( 759 )

relations and the results of Gupps’ expressions for this example I
borrow from P. Harzer’s Bestimmung und Verbesserung der Bahnenvon
Himmelskivpern nach drei Beobachtungen p. 8."*)
The heliocentric motion of the planet Pallas was from the 1%* to
the 3° observation 22°33’.
log 1, = 9.8362703 log 1, = 0.0854631 log 7, = 9.7255594
log r, = 0 3630906 login; = 0.3507163 log r, = 0.8369508
These values for /og 7 are also taken from Harzer and differ a
little from those according to Gauss.

Results for log n, and for log n;
GIBBS 9.7572961 GupBs 9.6480108
formula I] 9.7572928 formula I 9.6480167
rigorous 9.7572923 — rigorous 9.6480201.

nl
Formula IIT yields: log ~ = 9.8907287.
ny
With the given logarithms agree the following values :

n, Ns Ms
rigorous — 0.5718634 0.4446518 0.7775491
ead 0.5718641 f.I 04446484 f. II 0.7775418

differences —§ — 0.0000007 + 0.0000084 + 0.0000073

From the expressions given for the remainders of the 5% order
I calculated that they are in the ratio of — 9, + 72 and + 140. If we
compare these numbers with the residuals, it appears that for our
example they would vanish to the 7 decimal if we succeeded
in including also the terms of the 5 order in the expressions.

As to the calculation of the quantities A and B dependent on
t,,7,, andr,, I remark that it may be performed quickly if we modify
these forms in the following way :

here sien eas?
Ajo —— ‘3 a ao z a By» — il
TOMI See 10

toa F(t) nye nse
tea (EE) mene Gl nant)
ee ae
toa (PEER) meme ft Haut
eee eae

1) Publication der Sternwarte in Kiel, XI.

( 760 )

Chemistry. — “A few observations on autocatalysis and the trans-
formation of y-hydroxy-acids, with and without addition of
other acids, conceived as an ion-reaction.” By Dr. A. W. Visser.

(Communicated by Prof. H. J. Hampurcer).

1. The general equation for catalytic processes as proposed by

OstWaLp *) is:

#2, HEE@iA—\aG—a

i TE Ree Ce ec:

in which Yf/ f(x) indicates the changeable catalytic influences. He
remarks here that he cannot give a general method for drawing
a conclusion from the observed progress of a definite reaction, as to
the form of the function 7 (7).

Whilst studying the transformations of cane-sugar by invertase and
of salicin by emulsin*), I have shown, that, by introducing a correct
measure for the intensity of a catalyzer, the changeable catalytic
influences which occur here, could be indicated during the whole of
the progress of the reaction and it appears to me, that the method
given in my dissertation for determining that change in intensity
during the progress of the reaction may be called a general method
for determining the above mentioned function.

The most simple case imaginable is this, that the change in intensity
of the catalyzer with the change in concentration of the substance
acted upon is constant. In that case :

di
NEC e
therefore :
I=hk,—k, C.

In this equation is J—%, when C=O; therefore &, is the
intensity of the catalyzer when all has been converted, whilst /,
is the increase in intensity, when the concentration diminishes with
the unity.

Schematically this may be represented as follows :

1) Osrwatp, Lehrbuch der Allgem. Chem. II, 2, p. 270.
2) Visser, Dissertatie, Amsterdam 1904. A translation will shortly appear in the

Zeitschr. f. phys. Chem.

C, C
Fig. 1.
The reaction-equation for an unimolecular reaction, where the inten-
sity of a eatalyzer plays a rdle, may be represented by :
dC

SSC I
dt
In the supposed case, therefore, by :
OS ye
ee el ee) hoe

This is the same equation as the one proposed by Ostwa.p') for
positive autocatalytic processes, but it seems to me that by introducing
the intensity-idea the constants, occurring in the formulae obtain a
more definite significance.

Then we have for the negative autocatalysis :

dl

hn

dC
RG

and here /, is again the intensity when all has been converted and
i, the decrease in intensity when the concentration diminishes with
the unity.

Schematically, this may be represented as follows :

J

'
'
‘
'
'
'
'
‘
'
i)
'
‘
'
‘
i)
1
1
'

(Ge C
Fig. 2.

1) Lehrb. der Allgem. Chem. II, 2 p. 263.

( 762 )

By positive autocatalysis must then be understood an increase in
the intensity of the catalyzer during the progress of the reaction ; by
negative autocatalysis a decrease in that intensity.

2. As examples of reactions in which autocatalysis occurs, may
be mentioned the spontaneous transformations of solutions of y-hydroxy-
acids into their lactones. The acid is split up into ions and according
to Paut Henry’), who studied the transformation of y-hydroxybutyrie and
hydroxyvaleric acids, these would convert the unaltered portion
of the hydroxyacid and therefore act only catalytically. I have put
to myself the question whether these transformations may be consi-
dered as being ion-reactions, as it seems to me that it is more rational
to view them in this way and as such to bring them within the
laws of the mass-action, as we are ignorant as to the true nature
of catalysis. Profiting by the researches of Pavunt Henry, I have
arrived at a very satisfactory result.

According to this view, the hydroxy-acid is at any moment in
eyuilibrium with its ions and these are converted into the lactone
according to the scheme:

y-hydroxy-acid = positive ions + negative ions |
positive ions + negative ions = lactone + water II

As has been stated, Paui Hrnry thinks that in this transformation
the non-split portion of the hydroxy-acid only changes into lactone
and that the ions exercise only a catalytic action, for he observes,
that on addition of HCl or H,SO, the dissociation-degree of the
hydroxy-acid is diminished, that is to say, the equilibrium is shifted
towards the side of the hydroxy-acid and the concentration of the
negative ions of the hydroxy-acid diminishes and if now the dissociated
portion of the acid were converted, the reaction-velocity would not
rise quite so much under these circumstances. It seems to me that
this argument is not correct; the concentration of the negative ions
of the hydroxy-acid diminishes by addition of H-ions in the form of
HCl, but on the other hand the concentration of the positive H-ions
rises enormously and in order to construct the reaction-equation we
must multiply these two concentrations by each other if we view
the reaction as an ion-reaction.

3. In the first place the spontaneous transformations of the y-hydroxy-
acids must be considered. In constructing reaction-equations it must
be borne in mind that the transformations of y-hydroxy-acids into
lactones is not complete.

a) Wh f, phys. Chem. X, p. 111.

( 763 )

Suppose the initial coneentration of the y-hydroxy-acid = C;,
Suppose the concentration after a time ¢ wien
Suppose that the portion p of the hydroxy-acid is then split into
ions, the concentration of the hydroxy-acid will then be (1 — p) C
and the concentration of the ions pC.
According to equation (1)
we = kh.
1—p
hk: is the dissociation-constant of the hydroxy-acid.
Suppose the concentration of the ions after a time ¢= y, then is:
y =pC=3{Y 4eC+P—RB

The scheme:
positive ions + negative ions = lactone + water,
gives as reaction-equation, when the concentration-change of the
water is neglected

dy c .
Beige ty) tal (Cx Gr)
In this
spit aed
(C, rar C) == : =

is the concentration of the formed lactone.
The above differential-equation gives, after introduction of this value
for C,— C and after integration
yt =l(y — a) — l(y — B) 4+ const.
For t=0 is y=y,, that is the initial concentration of the ions
and the equation becomes:

Pl aD) (¥,—B)
~ t (y—B)(y,—@)
This equation is the same as the one deduced in my dissertation
for all bimolecular reactions where an equilibrium is formed.

In this equilibrium :
Y — 3 {V4kC, + 2? — ft the initial concentration of the ions,
PKG ie RB the concentration of the ions after
: - a time f,

iV4kC, +k? — kt the end-concentration of the ions,

eh

s= i=
—BC, ;
c— ~ (see p. 14 of my dissertation).
Co
The C, and C were determined by titration with barium hydroxide.
fk is the dissociation-constant ot the hydroxy-acid and could be

obtained by conductivity-determinations.

Or
bo

Proceedings Royal Acad. Amsterdam. Vol. VIL.

( 764 )

4. In the spontaneous transformation of the hydroxy-acids the C,
has not been determined and this is necessary in order to calculate
3 and «. The question therefore arises how these may be determined.

se Ropes dy
In the condition of equilibrium aS therefore:
:

k, i= k, (C, caine C.)
ae {V4kC,, + i? — ki, so
. k, __ 2kC,, + # —k V4kC, 4-2
My C, a C.,
(’,, the initial concentration is known, also & the dissociation-con-

stant. If now we know ~ the reciprocal value of the equilibrium-
1
constant of the transformation :
positive ions + negative ions = lactone,
C,, may be calculated.

This equilibrium-constant may be found from a series of deter-
minations where the end-condition has been determined of a definite
quantity of hydroxybutyrie-acid when HCl was added, therefore having
H-ions as active constituents. On pg. 112 of his treatise the author
states that to 20 ec. of the hydroxy-acid was added 5 ce. of the acid
to be investigated (HCI or H,SO,); according to table 64 on pg. 116
5 ec. of N HCl are added, the solution therefore becomes N/, HCl
and in ease of complete splitting of the HCl in that concentration also
quadri-normal in H-ions.

In the condition of equilibrium the positive H-ions of the y-hydroxy-
acid and those of the HCl and the negative ions of the hydroxy-
acid are therefore in equilibrium with the lactone.

From the data of the above mentioned table 64 we may calculate that
Fy 1715 >< 10-2.

1
The transformation of hydroxyvaleric-acid also leads to an equi-

librium, but, whereas in the condition of equilibrium the hydroxy-
butyric-acid was converted to the extent of about 65°/,, the hydroxy-
valeric-acid had been converted to the extent of 95 °/,.

In this transformation was found for :

k, -
“2 — 15 x 10-7.
hy,

Pavt Henry disregards in this transformation the occurring equili-
brium as it is shifted so much towards the side of the decomposition-
products. This should not happen if this transformation is regarded

( 765 )

as an ion-reaction as shown from the data on pg. 766; then the end
concentration of the ions is in the one case O.00O701 and_ the
initial-concentration 0.002690 and in the second case the end-concen-
tration of the ions is 0.000501 and the initial-concentration 0.001885.

The following tables derived from a series of determinations by
Pau Henry show that the above reaction-equation is a correct one

For the spontaneous transformation of y-hydroxybutyric-acid we
find: (see table p. 766).

5. It has been shown by the writer, that, if so much of a foreign
acid (HCl, H,SO,) is added that the concentration of the H-ions during
the whole progress of the reaction may be taken as constant, the
usual reaction-equation applies; therefore in this case that of the
unimolecular reaction in which an equilibrium is formed.

If we~ consider these reactions as ion-reactions it may be easily
proved, that in the supposed case that equation appears.

If the concentration of the hydroxy-acid at a definite period is C, that
of the added H-ions C’ and p the part of the hydroxy-acid which has
split up into ions, then Chyaroxy-acia = (1 — p) C; Cirions = p C+ C';
Cueg. ions — P) C.

This acid is in equilibrium with its ions, consequently,

BSE: CN
1—p

If so many H-ions have been added that this concentration may
be regarded as constant, pC-+ C' is a constant and the above
equation becomes :

P

=fh-
1—-p

that is to say, whatever value the concentration of the hydroxy-acid (C)
attains, p remains constant and during the whole of the progress of
the reaction, the concentration of the negative ions will amount to the
same part of the hydroxy-acid present. In this case we therefore, have:

negative ions + positive ions = lactone + water.

Suppose the concentration of the negative ions = y, and that of
the positive ions = Cy, then if the concentration of the lactone
= Cy and that of the water = Cy we have:

——- =, Coy —*, Cw Cr=h,y —#, Cr=F,y — Kk, (C, — C);

then Cy and Cy may be taken as constant.
It has been shown above that during the progress of the action

( 766 )

Tab. 92 and 93. PAUL HENRY p. 128. | Tab. 94 and 95 PAUL HENRY p. 128

CQ, = 0.17166 0, = 0.3390
6 =0.000741 eg =0.001313
x =— 0.000703 a =—0.001318
¢in hours. | y } ¢in hours. y | yi
0 0.001810 -- 0 0.002549 | —
25 0.001742 0.00067 24 0.002476 | 0.00085
66 0.091634 71 48 0.002372 | 97
120 0.001520 71 72 0.002274 | 107
1481, | 0.001476 68 123 | 0.002118 110
172 0.001448 66 1511/, 0.092049 109
219 | 0.001377 66 174"), 0.002006 106
©0 | 0.000741 = 192 0.001925 119
ore 0.001313 —

The following tables may serve for the spontaneous transformation
of y-hydroxyvaleric-acid.

Tab. 66. PAUL HENRY. p. 118. Tab. 72 and 73, PAUL HENRY. p. 121.

Cy, = 0.3580 C0, = 0.1769

2 =e = 0.000701 ge =0.000501

« =—0.000703 a = — 0.000302
0 0 002690 — 0 0.001885 --
240 0.002607 | 0.000032 390 0.001830 | 0.0000193
450 0.002541 29 1170 0.081705 226
1170 0.002350 30 1860 ~) 0.001615 226
1500 0.002°66 3h 2640 | 0.001520 230
1890 0.002188 30 3300 0.001460 225
2810 0.002053 28 4080 0.001385 227
8530 0.001942 28 4710 0 001335 225
4310 0.001895 27 5550 0.001275 224
4940 0.001773 24] 6900 0.001185 225
6170 0.001580 26 re) 0.000501 —
7740 0.001557 25 |
x | 0.000701 — |

y= pe
dC 1
dy p
From this and the previous equation follows :
ee ata Cc — he = Gok C.— kG, — C)
lt Pp p ) 1 3 0

and this is the differential-equation for a unimolecular reaction in
which an equilibrium is formed.

6. If we add instead of HCl or H,50, an acid like acetic-acid
which is partly split into ions, then, although H-ions are being added,
the concentration of the H-ions during the whole progress of the
reaction must no longer be considered as constant.

Suppose the initial-concentration of the hydroxy-acid = C,, that
of the acetic-acid C’ and that of the hydroxy-acid after a time ¢
and let us suppose that a portion a of the hydroxy-acid is split up
into ions and a portion & of the acetic-acid then :

’ v Y "
( hydroxy-acid —= @l —, (1) (Cig Garstne =m —- be
; _ ( \ L t . . —_ '
Cacetic-acid = dd a b) C; Cree. ions hydroxy-acid —= al
> !
Chee. ions acetic acid = b6

At each period the hydroxy-acid is in) equilibrium with the
H-ions and its negative ions and the same applies to the acetic
acid, therefore the two following equilibrium-equations apply :

a(aC+6bC’) b(aC+bc’)
- — =f, and ———— => &,.
l—a 1—b $

The dissociation-constants 2, and 4, of the hydroxy-acid and the

acetic acid are nearly alike, consequently @=% and

2 (Oy (6x
eS) oy, ; fk = 0.000207.

ea

_ V4k(C+C) +R —k

a ee
2(C+ C’')
: , EW 4k(C-C) +H =e
C neg. ions hydroxy-avid —= CC awa ar SEL Goo
2(6+ 6!)
g r See TOBE i: MOTE, CH
Oreos = a (C= Cl) NV 4k (G22 OC) 226 — kh} = Re

From this follows :

C= Jig +) 4+ Vas © + EC}.

We again haye:
negative ions -+- positive ions = |

— lactone, therefore :

( 768 )
dy C+C
SS) | | ee =
dt 1 ( (e yy GS C)
kk, +h, oy tat A es,
= lv to +VOFH FEC ]y bay —8,0.
Suppose ytk+Vy +h +460 =z,
2? — 4k C'
; = ——— —ik
then y oe
dy 24+44kC
and aes SEO
d wl eee,
and when we ca bb Ek. ——s
A ies 2? + ARC! as
Nd z2\@— %z—4xC)(@ > Nek) — 2k, C, 2) ©
ate et4kC' a
~ 2? — (NEE, + 2k) 2? + 2 (Nk, — 2k C’ — Nk, C,)2 + 4NE RC
_ Ie qd wa 2° s ;
ae sees z—B 2—y¥
Therefore :

at+p+y=Nkk, +2k (a); af + ay + By = 2(NPk, —2kC'— NE,C,) (0);
= OB y= 2 INGKAC! eee. ee oe ae)
From (4) and (c) follows:
28 (Nik, — 2kC' — Nk?C,) + 4K, C'

= ees.
aia, 2 (

ANKE, C'
8
From (a) and (d) follows:
6? — (kk, + 2h)8" — 28(Nik, — 2kC' — N#C,) — 4NRR,C’ . (Ff)

ay = —

dz ;
If there is equilibrium — = becomes 0 and this happens when the
; R

denominator of the above differential-equation becomes nought. The
equation which we then obtain in z is the same as equation (/) in ,

consequently @—=<, (the value of 2 in case of an equilibrium being
established).
Again introducing the value for .V in equation (/) we obtain :
eA \ ais pine ae
) J.2 Dye h oO) kVau
= ; ys ee ; + Oe ae 23 —.(9)
eo | l: 23, be ee
a5 k, 5a ky ‘7 k, ie

For the constants p, g, 7 and s we find:

/ 440"

qe =
a(a—y)(a—P)

4kC"

ee apy

«

( 769 )

|

B?-+4kC'
(=,
B(a—py(p

and x
Y)

4k’
y(a—y) (8-7)

The following tables have been constructed from the observations

of Pau, Henry on the action of acetic-acid on hydroxy valeric-acid.

k, a
For this, 4 = 0.0000207 and a = 15 < 10-7 (p. 764). Given the

values of (,, (initial concentration of the hydroxy-acid) and C’ (con-
centration of the acetie-acid) @ may be calculated from equation (7)
«a and y may then be calculated from (d/) and (¢).

For this transformation the reaction-equation becomes,

Tab, 74 and 75. PAUL HENRY. p. 123.

Go Zo
Lees yt

—p

—Y¥

ao area en

Y

Tab 76 and 77. PAUL HENRY. p, 123.

C,= 0.1708 p= 26976,89 G= 0.1795 p= %5905,42
C'= 0.2058 7 232,46 C'= 0.01977 q=— 497,543
2= 0.004195 r= 933,84. 2= 0.001653 r=— 499,052
4 = —0.004153 s = —27005,45 4 = —(0.001597 $ = —25906,92
y= 0.000035 7= 0.000024
¢ in hours | Zz oe ¢ in hours Zz af as
0 0.003495 = 0 | o.oo40s3 | —
210 0.005439 0.014 390, | 0.003947 0.0126
390 | 0.005349 21 1170 | 0.003706 | 128
1170 0.005226 a 1860 | 0.003521 | 128
1860 0.004953" 21 2640 0.003330 127
2640 0.004806 21 3300 | 0.003206 128
3300 0.004692 29 4080 | (0).003082 126
4080 0.004594 23 4710 | 0.002587 125
4710 | 0.004529 93 5550. 0.002859 197
5550 0.004435 2

These tables also give satisfactory values for the reaction-constants.

( 770 )

Physics. — « Application of the Baroscope to the Determination of
the Densities of Gases and Vapors.” By Artaur W. Gray,
(Preliminary Notice.| (Communication No. 940 from the
Physical Laboratory at Leyden by Prof. H. Kamertincr Onxts).

For determining the densities of gases, especially while flowing
continuously, the principle of the baroscope has been variously
applied by FrrzcrraLp*), LomMEn *), SiGERT and Dirr*), Mrstans*),
Precut®), and others. In the apparatus here described the aim has
been great sensitiveness combined with simplicity, ease of operation
and small volume.

The accompanying figure illustrates the essential features. A capillary
glass tube carries at one end a closed bulb, and at the other a
hemispherical shell of the same diameter, weight, and kind of glass.
This is fastened to a horizontal quartz fiber stretched on a glass
frame, and carries a small mirror M, so that rotations about the
quartz fiber®) as axis can be measured with telescope and scale.
The whole is placed within a glass tube containing a sensitive ther-
mometer of some sort, and communicating with a manometer.

1) G. F. Fivzcerap. Fortschritte der Physik 41, 102, 1885.

2) E. Lomaer. Wied. Ann. 27, 144, 1886.

3) A. Siegert and W. Dirr. Zs. f. Instr.k. 8, 258, 1888.

4) M. Mestans. Comptes Rend. 117, 386, 1893.

5) H. Precur. Zs. f. Instr.k. 13, 36, 1893.

6) The use of the quartz fiber was suggested by the delicate chemicat balance
of Nernst and Rirseyrery, Beibl. 28, 380, 1904, to which Prof. Kamertinga Ones
had drawn my attention. Much more delicate instruments are, however, the quartz
thread gravity balance of Turetiratt and Poxtock R. 8. Trans. 193, A, 215, 1900,
and the magnetograph of Warsoy, Proc. Phys. Soc. London, 19, 102, 1904.

enigiy

If the instrument has once been calibrated, the scale reading gives
immediately the density of the gas within; while the thermometer
and the manometer permit the calculation of the density under
standard conditions, if the compressibility of the gas is known. The
calibration may be made either with a single gas whose density at
various pressures is known with sufficient accuracy for any one
temperature, or by employing in turn several different gases under
known conditions of pressure, temperature and density, or with a
rider. Counterpoising the closed sphere with the hemispherical shell
of equal surface tends to eliminate errors that would be introduced
if the apparatus contained a vapor which condensed on the glass.
The instruments should, of course, be protected from changes of
temperature by proper jacketing or by immersion in a liquid bath.
A fixed reference mirror (not shown in the figure) is desirable to
indicate any change in the leveling of the apparatus.

In order to get an idea of the sensitiveness that could be expected
from such an instrument, some rough preliminary measurements
were made.

The dimensions were as follows:

Diameter of bulb Os em.

” capillary beam Onis.
Length ,, % “3 ri) eee
Mass of entire suspended system 0.67 gms.
Length of quartz fiber 1.4 em.

The apparatus was filled with dry air, and the scale readings
noted for various pressures ranging from 0.3 cm. to nearly 90 em.
of mercury.. With a fiber about 0,005 cm. in diameter and the scale
255 em. from the mirror, 0.1 mm. change in the deflection was

em. fee cere
= ehange in the
liter 2

density; and this was the same for all densities tried; that is to say,
a change of 0.1 mm. in the scale reading indicated a change of
about one part in 6000 in the density of air under ordinary con-
ditions. The seale might easily have been placed much farther from
ihe mirror and the sensitiveness could have been greatly increased
by using a larger bulb, a longer beam, and a longer and thinner
fiber. And since the change in deflection is, in the first approximation
at least, directly proportional to the change in density, an accurate
knowledge of the deflections for a few densities is sufficient for the

found to indicate a change of about 0,0002 -

calibration of the instrument. Certain corrections, as, for instance
>

Cw73 )

for the effects of changes of temperature on the quartz fiber, must,
of course, be applied when the greatest accuracy is desired.

This instrument was devised in order to follow the course of a
separation of atmospheric gases by fractional distillation at low tem-
perature, which Prof. KamertincH Onnes wished to be made and to
be controlled by density measurements; but it is evident that its use
is not confined to this field. It might be used for determining the
densities of gases or vapors under various conditions, and therefore,
their compressibilities ; but it is especially useful as an indicator of
minute changes of density. Professor KAmERLINGH ONNurs has already
suggested its use to determine the composition of coexisting vapor
and liquid phases in cases where a chemical analysis would be
difficult or impossible, for example, in a mixture of two of the
inert gases of the atmosphere.

Constructional details and refinements, together with the results
of more careful and more varied tests will be communicated in a
later paper.

(May 25, 1905).

CONTENTS.

ABSORPTION LINES (Double refraction near the components of) magnetically split into
several components, 435.

ABSORPTION SPECTRA (Dispersion bands in). 184.

ACETONE (On Px-curves of mixtures of) and ethylether and of carbontetrachloride and
acetone at 0°C, 162.

ActID (Sulphoisobutyric) and some of its derivatives. 275.

ADMIxTURES (The influence of) on the critical phenomena of simple substances and
the explanation of TEICcHNER’s experiments. 474.

ALGEBRAIC PLANE CURVE (On an expression for the class of an) with higher singu-
larities. 42.

— (On an expression for the genus of an) with higher singularities. 107.

— (On the curves of a pencil touching an) with higher singularities. 112.
ALGEBRAIC PLANE CURVES (On nets of). 631.

— (On linear systems of). 711.

ALGEBRAIC SURFACE (Some characteristic numbers of an). 716.
Anatomy. J. W. LanceLaan: “On the Form of the Trunk-myotome”. 34.

— A. J. P. van pen Broek: “On the genital cords of Phalangista vulpina”. 87.

— E. ve Vries: “Note on the Ganglion yomeronasale”’. 704.

— J. W. van Bisseick: “Note on the innervation of the Trunkmyotome”. 708.
ANGLES (On the equation determining the) of two polydimensional spaces. 340.
ARBITRARY HIGH RANK (On moments of inertia and moments of an arbitrary order in

spaces of). 596.
Astronomy. C. Easton: “On the apparent distribution of the nebulae’. 117.
— C. Easton: “The nebulae considered in relation to the galactic system’. 125.
— J. WeepEr: “A new method of interpolation with compensation applied to the
reduction of the corrections and the rates of the standard clock of the Observatory
at Leyden, Hohwii 17, determined by the observations with the transitcircle in
1903”, 241.

— J. A. C. Oupemans: “A short account of the determination of the longitude
of St. Denis (Island of Réunion) executed in 1874”. 602.

— J. Weerper: “Approximate formulae of a high degree of accuracy for the

relations of the triangles in the determination of an elliptic orbit from three

observations”. 752.
Proceedings Royal Acad. Amsterdam, Vol, VIL. 53

iy CONTENTS.

ASYMMETRIC syNTHESIs (On W. Marckwatp’s) of optically active valerie acid. 465.

ATEN (a, m. W.). On the system pyridine and methyliodide. 468.

AUTOCATALYsIS (A few observations on) and the transformation of y-hydroxy-acids,
with and without addition of other acids, conceived as an ion-reuction. 760.

axes (The locus of the principal) of a pencil of quadratic surfaces. 341.

— (Ihe equations by which the locus of the principal) of a pencil of quadratic
surfaces is determined. 532.
— (The equation of order nine “representing the locus of the principal) of a pencil
of quadratic surfaces, 721.

azimutu (Determinations of latitude and) made in 1896—99. 482.

BAKHU1S ROOZEBOOM (H. W.) presents a paper of Prof. Eve. Dusors: “On the
origin of the fresh-water in the subsoil of a few shallow polders’. 53.

— presents a paper of J. J. van Laar: “On the latent heat of mixing for asso-
ciating solvents”, 174.

_ — presents a paper of Dr. A. Smits: “On the phenomena appearing when in a
binary system the plaitpointeurve meets the solubility. curve” (3rd communica-
tion). 177. ;

— presents a paper of Dr. J. J. Buanksma: “On trinitroveratrol”. 462.

— presents a paper of Dr..S. Tymsrra Bz.: “On W. Marckwaup’s asymmetric
synthesis of optically active valerie acid”. 465.

— presents a paper of Dr. A. H. W. Aten: “On the system ous and meth
iodide”. 468.

— presents a paper of J. J. van Laar: “On the different forms and trarisfovitintians
of the boundary curves in the case of partial miscibility of two liquids”. 636.

— presents a paper of Dr. F. M. Jarcer: “On miscibility in the solid aggregate
condition and isomorphy with carbon compounds”. 658.

— and f—, H. Bitcuner. Critical terminating points in. three-phase lines eit solid
phases in binary systems which present two liquid layers. 556. ae so

BAKHUIJZEN (H. G. VAN DE SANDE). v. SANDE Bakwulszen (H. G. VAN BE).

BARENDRECHT (H. P.). Enzyme-action. 2. =

BAROMETRIC HEIGHT (On a twenty-six-day period in dail means of the). 18.

BAROSCOPE (Application of the) to the determination of the densities of gases and
vapors. 770.

BEMMELEN (J. M. VAN) presents a paper of Dr. H. P. Barrnprecut: “Enzyme-
action”. 2.

— presents a communication of Prof. Euc. Dusots: “On the direction and the
starting point of the diluvial ice motion over the Netherlands”. 40.

— On the composition of the silicates in the soil which -have been coma from
the disintegration of the minerals in the rocks. 329.

BENZENE SERIES (On the preservation of the crystallographical eynitinagst in tlie sub-
stitution of position isomeric derivatives of the). 191.

BENZENES (The nitration of disubstituted). 266. : =

BENZOL (On the intramolecular oxydation of a SH-group bound to) ¥y an orthostanding
NO :-group. 63.

CONTENTS. i

BENZPINACONES (On intramolecular atomic rearrangements in), 271.

BENZYLPHTALIMIDE (On) and Benzylphtal-iso-imide. 77.

BES (K.). The equation of order nine representing the locus of the principal axes of a
pencil of quadratic surfaces. 721.

BESSEL FUNCTIONS (The values of some definite integrals connected with), 375.

— (On a series of), 494.

BEIJERINCK (Mm. w.). An obligative anaerobic fermentation sarcina. 580.

BINARY MIXTURES (The derivation of the formula which gives the relation between
the concentration of coexisting phases for). 156.

._— (The conditions of coexistence of) of normal substances according to the law
of corresponding states. 222.

BINARY SYSTEM (On the phenomena appearing when in a) the plaitpointcurve meets
the solubilityeurve. 177.

BINARY sYsSTEMS (Critical terminating points in three-phase lines with solid phases
in) which present two liquid layers. 556.

BISSELICK (J, w. VAN). Note on the innervation of the Trunkmyotome. 708.

BLANKSMA (J. J.). On the intramolecular oxydation of a SH-group bound to benzol
by an orthostanding NO.-group. 63.

— On trinitroveratrol. 462.

BLOK (s.). The connection between the primary triangulation of South-Sumatra and
that of the West-Coast of Sumatra. 453.

BLOOD (On the osmotic pressure of the) and urine of fishes. 537.

BOESEKEN (J.). The reaction of FrirepeL and Craris. 470.

BOULK (1.). presents a paper of A. J. P. van pEN Broek: “On the genital cords of
Phalangista yulpina”. 87.

Botany. H. P. Kuyper: “On the development of the perithecium of Monascus purpu-
reus Went and Monascus Barkeri Dang”. 83.

-— C. A. J. A. Oupemans: “On Leptostroma austriacum Oud., a hitherto unknown
Leptostromacea living on the needles of Pinus austriaca, and on Hymenopsis
Typhae (Fuck.) Sace., a hitherto insufliciently described Tuberculariacea, occurring
on the withered leafsheaths of Typha latifolia”. 206.

— ©, A. J. A. Oupemans: “On Sclerotiopsis pityophila (Corda) Oud., a Sphae-
ropsidea occurring on the needles of Pinus silvestris”. 211.

.— Miss Tine Tames: “On the influence of nutrition on the fluctuating varia-
bility of some plants”. 398.

— B. Sypxens: “On the nuclear division of Fritillaria imperialis L”. 412.

.— J. M. Janse: “An investigation on polarity and organ-formation with Caulerpa
prolifera”. 420.

BOULDERS. (Contributions to the knowledge of the sedimentary) in the Netherlands.
1. The Hondsrug in the province of Groningen. 2 Upper silurian boulders.
Ist Communication. Boulders of the age of the eastern Bultic Zone G. 500.
2nd Communication. Boulders of the age of the eastern Baltic Zones Wand J, 692.
BOUNDARY CURVES (On the different forms and transformations of the) in the case of
partial miscibility of two liquids, 636. _
53*

iv CONTENTS

BRAIN in Tarsius spectrum (On the development of the). 331.
BRANCH PLAIT (The transformation of 2) into a main plait and vice versa, 621.
BRANCHINGS (On the) of the nerve-cells in repose and after fatigue. 599.
BROEK (A. J. P. VAN DEN). On the genital cords of Phalangista vulpina. 87.
BRUYN (c. A. LOBRY DE). vy. Lopry DE Bruyn (C. A.).
BRUYN (H. E. DE). Some considerations on the conclusions arrived at in the com-
munication made by Prof. Euc. Dusots, entitled: “Some facts leading to trace
out the motion and the origin of the underground water of our sea-provinces”. 45,
BUCHNER (£. H.) and H. W. Bakuuis Roozenoom. Critical terminating points in
three-phase lines with solid phases in binary systems which present two liquid
layers. 556,
CARBON coMPoUNDS (On miscibility in the solid aggregate condition and isomorphy
with), 658.
CARBON DIOxIDE (The validity of the law of corresponding states for mixtures of
methylchloride and), 285. 377.
CARBONTETRACHLORIDE (On Px-curves of mixtures of acetone and ethylether and of)
and acetone at 0°C. 162.
CARDINAAL (J.). The locus of the principal axes of a pencil of quadratic sur-
faces. 341.
— The equations by which the locus of the principal axes of a pencil of quadratic
surfaces is determined. 532.
— presents a paper of K. Bes: “The equation of order nine representing the locus
of the principal axes of a pencil of quadratic surfaces”. 721.
carvacnoL (The inversion of carvyon and eucaryon in) and its velocity. 63.
cARVON (The inversion of) an eucaryon in carvacrol and its velocity. 63.
CAULERPA PROLIFERA (An i: -estigation on polarity and organ-formation with). 420.
Chemistry. H. P. Barenprec r: “Enzyme-action”. 2.
— C. A. Lospry pe Bruyn and 8. Tymsrra Bz.: “The mechanism of the salicylacid
synthese”. 63,
— J. J. Buanxsma : “On the intramolecular oxydation of a SH-group bound to benzol
by an orthostanding NO -group”. 63.
— J. M. M. Dormaar: “The inversion of carvon and eucaryon in carvacrol and
its velocity”. 63.
— J. J. van Laar: “On the latent heat of mixing for associating solvents”. 174.
— A. F. Hotneman: “The preparation of silicon and its chloride”. 189.
— A. F. Hotteman: “The nitration of disubstituted benzenes”. 266.
— A. P. N. Francumont and H. FriepMann: “On zz'-tetramethylpiperidine”. 270.
— J. Mont van Cuarante. “Sulphoisobutyrice acid and some of its derivatives”. 275,
— P.J. Monrtacne: “On intramolecular atomic rearrangements in benzpinacones’’. 271.
— J. Oxie Jr.: “Lhe transformation of the phenylpotassium sulphate into p-phenol-
sulphonate of potassium”. 328.
— J. F. Suyver: “The intramolecular transformation in the stereoisomeric g-and
@-trithioacet and z-and £-trithiobenzaldehydes”. 329.
— J. W. Dito: “The viscosity of the system hydrazine and water”. 329.

CONTENTS. v

Uhemistry. J. M. van Bemmenen: “On the composition of the silicates in the soil which
have been formed from the disintegration of the minerals in the rocks”. 829.

— A. F. Hoiteman: “On the preparation of pure o-toluidine and a method for
ascertaining its purity”. 395.

— J. J. Buaxksma: “On trinitroveratrol’’, 462.

— 8. Tymsrra Bz.: “On W. Marckwa.p’s asymmetric synthesis of optically active
valeric acid’, 465.

— A. H. W. Aven: “On the system pyridine and methyliodide”. 468,

— J. Boérsexen: “The reaction of Friepe, and Crarrs”. 470.

— J.J. van Laan: “On some phenomena, which can occur in the case of partial
miscibility of two liquids, one of them being anomalous, specially water”. 517.

— H. W. Baknuis Roozesoom and E. H. Bicuner: “Critical terminating points
in three-phase lines with solid phases in binary systems which present two
liquid layers”. 556.

— J.J, van Laar: “On the different forms and transformations of the boundary-
curves in the case of partial miscibility of two liquids’. 636.

— J. J. van Laar: “An exact expression for the course of the spinodal curves
and of their plaitpoints for all temperatures, in the case of mixtures of normal
substances”. 646,

— F. M. Jazcrr: “On miscibility in the solid aggregate condition and isomorphy
with carbon compounds”. 638.

— F. M. Jarcer: “On orthonitrobenzyltoluidine”. (66,

— F, M. Jarcer: “On position-isomeric Dichloronitrobenzenes”. 668.

— A. W. Visser: “A few observations on autocatalysis and the transformation of
y-hydroxy-acids, with and without addition of other acids, conceived as an ion-
reaction”. 760,

CLIMATE (Oscillations of the solar activity and the). 368.

COMPLEX (On a special tetraedal). 572.

COMPLEXES (On a group of) with rational cones of the complex. 577,

— of rays (A group of algebraic). 627.

components (Double refraction near the components of absorption lines magnetically
split into several). 435.

CONDITIONS of coexistence (The) of binary mixtures of normal substances according
to the law of corresponding states. 222.

— (The determination of the) of vapour and liquid phases of mixtures of gases at

low temperatures. 233,

CONGRUENCE (a) of order two and class two formed by conics, 311.

conics (The congruence of the) situated on the cubic surfaces of a pencil, 264,

— (A congruence of order two and class two formed by). 311.

CORRIGENDA et addenda, 382.

crarrTs (The reaction of FrrepEt and), 470.

CRITICAL PHENOMENA (The influence of admixtures on the) of simple substances and
the explanation of TErcHNEk’s experiments. 474,

CROMER FOREST-BED (On an equivalent of the) in the Netherlands, 214,

vI CONTENTS.

Crystallography. F. M. Jancer: “On Benzylphtalimide and Benzylphtal-iso-imide”. 77.
— F. M. Jarcer: “On the preservation of the erystallographical symmetry in the
substitution of position isomeric derivatives of the benzene series”. 191.
cusic surfaces of a pencil (The congruence of the conics situated on the). 264.
curve (On an expression for the class of an algebraic plane) with higher singu-
larities. 42,
— (On an expression for the genus of an algebraic plane) with higher singularities. 107.
— (On the curves of a pencil touching an algebraic plane) with higher singu-
larities. 112. _
— (The relation between the radius of curvature of a twisted) in a point P of the
curve and the radius of curvature in P of the section of its developable with
its oseulating plane in point P. 277.
curves (On Px-) of mixtures of acetone and ethylether and of carbontetrachlo-
ride and acetone at 0°C. 162.
— (On nets of algebraic plane). 631.
— (An exact expression for the course of the spinodal) and of their pligeame
for all temperatures, in the case of mixtures of normal substances. 646.
— (On linear systems of algebraic plane). 711.

a ‘
DALFSEN (B, M. VAN), On the function 7 for multiple mixtures. 94.

prpucTIoN (Simplified) of the field and the forces of an electron, moving in any given
way. 346.

DEKHUIJZEN (M. c.). On the osmotic pressure of the blood and urine of fishes. 537.

pENsITIES (Application of the Baroscope to the determination of the) of gases and
vapors. 770.

DEVENTER (CH. M. VAN). On the melting of floating ice. 469.

DICHLORONITROBENZENES (On position-isomeric). 668.

DILUTE sOLUTION (Kinetic derivation of van ’t Hoff’s law for the osmotic pressure
in a). 729.

DILUVIAL 1CE MOTION oyer the Netherlands (On the direction and the starting point
of the). 40.

DISPERSION (Spectrobheliographie results explained by anomalous). 140.

DISPERSION BANDS in absorption spectra. 134.

— in the spectra of d Orionis and Nova Persei. 323.

p1To (J. w.). The viscosity of the system hydrazine and water. 329.

pDORMAAR (gs. M. M.). The inversion of caryon and eucarvon in carvacrol and its
velocity. 63.

pUBOTs (EUG.). On the direction and the starting point of the diluvial ice motion
over the Netherlands. 40.

— (Some considerations on the conclusions arrived at in the communication made
by Prof.), entitled: “Some facts leading to trace out the motion and the origin
of the underground water of our sea-provinces”. 45. ; :

— On the origin of the fresh-water in the subsoil of a few shallow polders. 53.

— On an equivalent of the Cromer Forest-Bed in the Netherlands. 214,

CON fp N T's) WH

FAR (On the relative sensifiveness of the human) for tones of different pitch, measured
by means of organ. pipes, 549.
EASTON (c.). On the apparent distribution of the nebulae. 117,

— The nebulae considered in relatiori to the galactic system. 125.

— Oscillations of- the solar. activity and the climate, 368.

EINTHOVEN (W.). On a new method of damping oscillatory deflections of a gal-
vanometer. 315.

ELECTRON (Simplified deduction of the field und the forees of an), moving in any
given way. 346.

ELEctRONS (The motion of) in metallic bodies, I. 438. II. 585. IIT. 684,

ELLIPTIC ORBIT (Approximate formulae of a high degree of accuracy for the relations
of the triangles in the determination of an) from three observations. 752.

ENERGY (On artificial and natural nerve-stimulation and the TS of) involved. 147,

ENZYME-ACTION. 2,

EQUATION (On the) determining the angles of two polydimensional spaces, 340.

— of order nine (The) representing the locus of the principal axes of a pencil of
quadratic surfaces. 721.

EQuations (The) by which the locus of the principal axes of a pencil of quadratic
surfaces is determined. 532.

ERRATUM, 329, 485, 633.

ETHYLETHER (On Px-curves of mixtures of acetone and) and of carbontetrachloride
and acetone at 0°C. 162.

EUCARVON (The inversion of carvon and) in carvacro] and its velocity. 63,

EXPANSION COEFFICIENT (The) of Jena- and Thiiringer glass between + 16° and
182°C, 674.

“FERMENTATION SARCINA (An obligative anaerobic), 580.

FIBRINGLOBULIN (On the presence of) in fibrinogen solutions. 610.

FisuEs (On the osmotic pressure of the blood and urine of). 537.

PLOCCULUS CEREBELLI CESS Ros in the central nervous system after remoyal of
the). 282.

FoRMULA (The derivation of the) which gives the relation between the concentration
of coexisting phases for binary mixtures. 156.

FORMULAE of GULDIN (The) in polydimensional space. 487.

FRANCHIMONT (a. P, N.) presents a paper of Dr. F. M, HSS “On Benzyl-
phtalimide and Benzylphtal-iso-imide”. 77.

— presents a paper of Dr, F. M. Jarcer: “On the preservation of the crystallo-
graphical eyuely in the substitution of position isomeric derivatives of the
benzene series”, 191.

— presents a paper of P. J. Montagne: “On intramolecular atomic rearrangements
in benzpinacones”. 271. ;

— presents the dissertation of Dr. J. Mouu van Cuarante: “Sulphoisobutyric
acid and some of its derivatives”. 275.

— and H. FrrepMann. On gz'-tetramethylpiperidine. 270.

PRIEDEL and Craprs (The reaction of). 470,

Vill CONTENTS.

FRIEDMANN (H.) and A. P. N. Fraxcatmont. On z2'-tetramethylpiperidine. 270.

FRITILLARIA IMPERIALIS L. (On the nuclear division of), 412.

a
FuNCcTION >- (On the) for multiple mixtures. 94.

GALACTIC sysTEM (The nebulae considered in relation to the). 125.

GALVANOMETER (On a new method of damping oscillatory deflections of a). 315.

GANGLION VOMERONASALE (Note on the). 704.

Gas Laws (A formula for the osmotic pressure in concentrated solutions whose vapour
follows the). 723.

casks (The determination of the conditions of coexistence of vapour and liquid phases
of mixtures of) at low temperatures. 233.

— and Vapors (Application of the Baroscope to the determination of the den-
sities of). 770.

GEEST (J.) and P. Zeeman. Double refraction near the components of absorption
lines magnetically split into several components. 435.

GENITAL corps (On the) of Phalangista vulpina. $7.

Geodesy. S. Buox: “The connection between the primary triangulation of South-
Sumatra and that of the West Coast of Sumatra’. 453.

— J. A. C. Oupemans: “Determinations of latitude and azimuth made in 1896—
99”, 482,

Geology. Eve. Dusois: “On the direction and the starting point of the diluvial ice
motion over the Netherlands’, 40.

— H. E. pe Bruyn: “Some considerations on the conclusions arrived at in the
communication made by Prof. Eva. Dusors, entitled: “Some facts leading to
trace out the motion and the origin of the underground water of our sea-
provinces”. 45.

— Eve. Dusors: “On the origin of the fresh-water in the subsoil of a few shallow
polders”, 53,

— Eve. Dusors: “On an equivalent of the Cromer-Forest-Bed, in the Nether-
lands’’, 214,

—H. G. Jonxer: “Contributions to the knowledge of the sedimentary boulders
in the Netherlands. I, The Hondsrug in the province of Groningen. 2 Upper
Silurian boulders. 1st Communication: Boulders of the age of the eastern Baltic
Zone G@”. 500. 2nd Communication: Boulders of the age of the eastern Baltic
Zones H and J”. 692.

GERRITs (G. C.). On Px-curves of mixtures of acetone and ethylether and of car-
bontetrachloride and acetone at 0°C. 162.

atass (The “expansion-coefficient of Jena- and Thiiringer) between -+ 16° and —
182°C. 674.

GOLD WiRE (Comparison of the resistance of) with that of platinum wire. 300.

GRAY (ARTHUR W.). Application of the Baroscope to the determination of the
densities of gases and yapors. 770.

GULDIN (The formulae of) in polydimensional space, 487.

CONTENTS. 1x

HAMBURGER (iH. J.) presents a paper of Dr. A. W. Visser: “A few observations
on autocatalysis and the transformation of y-hydroxy-acids, with and without
addition of other acids, conceived as an ion-reaction”. 760.

HEAT OF MIXING (On the latent) for associating solvents. 174.

HEUSE (w.) and H. Kameruincu Onnes, On the measurement of very low fem-
peratures. V. The expansioncoéfficient of Jena- and Thiiringer glass between
+ 16° and — 182°C. 674.

HOEK (p, P. c.). An interesting case of reversion. 90.

HOFF’s LAW (Van ’t) (Kinetic derivation of) for the osmotic pressure in a dilute
solution, 729.

HOLLEMAN (A. F.). The preparation of silicon and its chloride. 189.

— The nitration of disubstituted benzenes. 266.

— On the preparation of pure o-toluidine and a method for ascertuining its
purity. 395.

— presents a paper of Dr. J. Borsrken: ‘The reaction of Frrepen and Crarrs”. 470.

— presents a paper of Dr. F. M. Jancer: “On Orthonitrobenzyltoluidine”. 666,

— presents a paper of Dr, I, M. Jancer: “On position-isomeric Dichloronitro-
benzenes”’. 668.

WoNDskuG (The) in the province of Groningen. 500. 692.

HUBRECHT (A. A. W.) presents a paper of Prof. Ta. ZiruEN: ‘On the development
of the brain in Tarsius spectrum’. 331.

MUISKAMP (w.). On the presence of fibringlobulin in fibrinogen solutions. 610.

HYDRAZINE and water (The viscosity of the system). 329.

HYDROXY-acIps (A few observations on autocatalysis and the transformation of y-),
with and without addition of other acids, conceiyed as an ion-reaction. 760.
HYMENOPSIS TYPHAE (Fuck.) Sacc. (On) a hitherto insufliciently described Tubereula-

riacea, occurring on the withered leafsheaths of Typha latifolia, 206.

Ice (On the melting of floating). 459.

ICE MOTION over the Netherlands (On the direction and the starting point of the
diluyial). 40.

INERTIA (On moments of) and moments of an arbitrary order in spaces of arbitrary
high rank. 596,

INNERVATION (Note on the) of the Trunkmyotome. 708.

INTEGRALS (Evaluation of two definite), 201,

— (The values of some definite) connected with Bessel functions. 375.

INTERPOLATION (A new method of) with compensation applied to the reduction of the
corrections and the rates of the standardclock of the Observatory at Leyden,
Hohwii 17, determined by the observations with the transitcircle in 1903. 241,

INTRAMOLECULAR atomic rearrangements in benzpinacones, 271.

INTRAMOLECULAR OXYDATION (On the) of a SH-group bound to benzol by an ortho»
standing NO,-group. 63.

INTRAMOLECULAR REARRANGEMENTS (On), 329,

INTRAMOLECULAR TRANSFORMATION (The) in the stereoisomeric z= and (-trithioacet and
a: and #-trithiobenzaldehydes, 329.

x CONTENTS.

INVERSION (The) of carvon and eucarvon in earvacrol and its velocity. 63.
1ON-REACTION (A few observations on autocatalysis and the transformation of y¢
hydroxy-acids, with and without addition of other acids, conceived as an): 760.
isomorPHY (On miscibility in the solid aggregate condition and) with carbon eom-
pounds. 658. i “nd of “) Segue
JAEGER (fF. M.). On Benzylphtalimide and Benzylphtal-iso-imide.- 77."

— On the preservation of the crystallographical symmetry in the substitution of
position isomeric derivatives of the benzene series. 191. a) am

— On miscibility in the solid aggregate condition and isomorphy with carbon
compounds, 658.

— On Orthonitrobenzyltoluidine. 666.

— On position-isomeric Dichloronitrobenzenes. 668.

JANSE(J. M.), An investigation on polarity and organ-formation with Caulerpa prolifera.420.
JONKER (uH. G.). Contributions to the knowledge of the sedimentary boulders in
the Netherlands. 1. The Hondsrug in the province of Groningen. 2. Upper
silurian boulders. 1st Communication: Boulders of the age of the eastern Baltic
zone G. 500, 2nd Communication: Boulders of the age of the eastern Baltic zones
H and J. 692.
JuLIUs (w. H.). Dispersion bands in absorption spectra. 134.
— Spectroheliographic results explained by anomalous dispersion. 140.
— Dispersion bands in the spectra of -3 Orionis and Nova Persei. 323.
KAMERLINGH ONNES (H.) presents a paper of B. Meitink: “On the measure-
ment of very low temperatures. VII. Comparison of the platinum thermometer
with the hydrogen thermometer. 290. VII[. Comparison of the resistance of gold
wire with that of platinum wire.” 300. ;

— presents a paper of Dr. J. E, Verscuarrett; “The influence of admixtures on
the critical phenomena of simple substances and the explanation of base
experiments”. 474.

— presents a paper of Artaur W. Gray: “Application of the Baroscope to” the
determination of the densities of gases and vapors”. 770.

— and W. Hevse. On the measurement of very low temperatures. V. The expan-
sion-coefficient of Jena- and Thiiringer glass between +- 16° and — 182°C, 674.

— and ©. Zakrzewskt. Contributions to the knowledge of van ver Waals surface,
IX. The conditions of coexistence of binary mixtures of normal substances
according to the law of corresponding states. 222. ;

— The determination of the conditions of coexistence of vapour and liquid phases
of mixtures of gases at low temperatures. 283.

. = The validity of the law of corresponding states for mixtures of methylehloride
and carbon dioxide, 285. 377.
«KA PTEYN (w.). The values of some definite integrals connected with Bessel functions. 375.

— On a series of Bessel functions, 494, ‘

KLUYVER (J. ©.) presents a paper of Prof, Eomunp Lanpav: “Remarks on the

(m)

> b : - Lid
paper of mr. Kiuyver: “Series derived from the series Sia 66.

CONTENTS. x1

KLUYVER (J. c.). Evaluation of two definite integrals. 201.
KOHNSTAMM (Pu). A formula for the osmotic pressure in concentrated solutions
whose vapour follows the gas laws. 723.
— Kinetic derivation of van ’r Horr’s law for the osmotic pressure in a dilute
solution. 729,
— Osmotic pressure and thermodynamic potential. 741.
KORTEWEG (D. J.) presents a paper of Mr. Frep. Scuum: “On an expression for
the class of an algebraic plane curve with higher singularities”. 42.
— presents a paper of Mr. Frep. Scuut: ‘On an expression for the genus of an
algebraic plane curve with higher singularities”. 107.
— presents a paper of Mr. Frep. Scutu: “On the curves of a pencil touching an
algebraic plane curve with higher singularities”, 112.
— and D, pe Lance. Multiple umbilics as singularities of the first order of except-
ion on point-general surfaces. 386.
KUYPER (u. P,). On the development of the perithecium of Monascus purpureus
Went and Monascus Barkeri Dang. 83.
LAAR (J, J. VAN). On the latent heat of mixing for associating solvents. 174.
— On some phenomena which can occur in the case of partial miscibility of two
liquids, one of them being anomalous. 517.
— On the different forms and transformations of the boundary-curves in the ease
of partial miscibility of two liquids, 636.
— An exact expression for the course of the spinodal curves and of their plait-
points for all temperatures, in the case of mixtures of normal substances. 646.
LANDAU (EDMUND). Remarks on the paper of Mr. Kiuyver: “Series derived

ne
from the series gt On 66.
7

LANGE (D. Dz) and D. J. Kortewec. Multiple umbilics as singularities of the first
order of exception on point-general surfaces. 386.

LANGE (s, J. DE). On the branchings of the nerve-cells in repose and after
fatigue. 599.

LANGELAAN (J. w.). On the Form of the Trunk-myotome. 34.

‘LATITUDE (Determinations of) and azimuth made in 1896—99. 482.

Law of corresponding states (The conditions of coexistence of binary mixtures of
normal substances according to the). 222.

— of corresponding states (The validity of the) for mixtures of methyl chloride

and carbon dioxide. 285, 377.

LEPTOSTROMA AUSTRIACUM oOUD., (On), a hitherto unknown Leptostromacea living on
the needles of Pinus austriaca. 206.

LiguID LayERS (Critical terminating points in three-phase lines with solid phases in
binary systems which present two). 556,

LIQUID PHASES (The determination of the conditions of coexistence of vapour and) of
mixtures of gases at low temperatures, 233.

Liquips (On some phenomena which can occur in the case of partial miscibility of

two), one of them being anomalous, specially water. 517.

xT CONTENTS,

tigurps (On the different forms and transformations of the boundary-curves in the
case of partial miscibility of two). 636,

LOBRY DE BRUYN (CG A.) presents a paper of Dr. J. J. Buanxsma: “On the
intramolecular oxydation of a SH-group bound to benzol by an orthostanding
NO,-group”. 63.

— presents a paper of J. M. M. Dormaar: “The inversion of carvon and eucarvon
in carvacrol and its velocity”. 63.

— presents a paper of J. O1ie Jr.: “The transformation of the phenylpotassium
sulphate into p. phenylsulphonate of potassium’: 328.

— presents a paper of J, F. Suyver: “The intramolecular transformation in the
stereoisomeric a- and 4-trithioacet and g- and #-trithiobenzaldehydes”. 329.

— presents a paper of J. W. Drro; “The viscosity of the system hydrazine and
water’. 329,

— and 8, Tymstra Bz. The mechanism of the salicylacid synthese. 63.

LONGITUDE of St. Denis (Island of Réunion) (A short account of the determination
of the) executed in 1874, 602.

LORENTZ (H. A.) presents a paper of Prof. A. SOMMERFELD: “Simplified deduction
of the field and the forces of an electron moving in any given way”. 346.

— The motion of electrons in metallic bodies. I. 438. If. 585. IIL. 684.

— presents a paper of J. J. van Laar: “On some phenomena, which can occur
in the case of partial miscibility of two liquids, one of them being anomalous
specially water”. 517.

— presents a paper of J. J. van Laan: “An exact expression for the course of
the spinodal curves and of their plaitpoints for all temperatures, in the case of
mixtures of normal substances”. 646.

MAIN PLAIT (The transformation of a branch plait into a) and vice versa. 621.

MARCKWALD’s (On W_) asymmetric synthesis of optically active valerie acid. 465.

MARTIN (K.) presents a paper of Prof. Eve, Dusors: “On an equivalent of the
Cromer-Forest-Bed in the Netherlands”. 214.

— presents a paper of Dr. H. G, Jonker: “Contributions to the knowledge of
the sedimentary boulders in the Netherlands. I. The Hondsrug in the province
of Groningen. 2. Upper Silurian boulders, 1st Communication: Boulders of the
age of, the eastern Baltic Zone G”. 500. 2nd Communication: “Boulders of the
age of the eastern Baltic Zones H and J”. 692.

Mathematics. Frep. Scuun. “On an expression for the class of an algebraic plane
curye with higher singularities”. 42.

— Epm. Lanpau: Remarks on the paper of Mr. Kiuyver: “Series derived from

H(m),, 66.
me

the series =

— Frep. Scuun: “On an eXpression for the genus of an algebraic plane curve
with higher singularities’. 107.

— Krep. Scuun: “On the curves of a pencil touching an algebraie plane curve
with higher singularities’, 112.

— J. ©. Kuuyver: “Evaluation of two definite integrals”. 201,

CONTENTS: XIT

Mathomatics. Jan pve Vrirs; “Lhe congruence of the conics situated ou the cubic
surfaces of a pencil”, 264,

~— W. A. Verstuys: “The relation between the radius of curvature of a twisted
curve in a point P of the curve and the radius of curvature in P of the section
of its developable with its osculating plane in point P”. 277.

— Jan pe Vries: “A congruence of order two and class two formed by conics”. 311.

— P. H. Scuoure; “On the equation determining the angles of two poydimensional
spaces”. 340.

— J. CarprnaaL: “The locus of the principal axes of a pencil of quadratic
surfaces”, 34].

-- W. Karreyn: “The values of some definite integrals connected with Bessel
functions”. 375.

— D. J. Korrewee and D. pe Lance: “Multiple umbilics as singularities of the
first order of exception on point-general surfaces”. 386.

— P. H. Scuoure: “The formulae of GuLprn in polydimensional space”, 487.

— W. Kapreyn: “On a series of Brssen functions”. 494.

— J. CarpinaaL: “The equations by which the locus of the principal axes of a
pencil of quadratic surfaces is determined”. 532.

— P. H. Scuoure: “On non-linear systems of spherical spaces touching one
another’. 562.

— JAN DE Vrins: “On a special tetraedal complex”. 572.

— JAN DE Vries: “On a group of complexes with rational cones of the com-
plex”. 577.

— KR. Meumxe: “On moments of inertia and moments of an arbitrary order in
spaces of arbitrary high rank’’. 596.

— Jan ve Vrirs: “A group of algebraic complexes of rays’’, 627.

— Jan ve Vries: “On nets of algebraic plane curves”. 63].

— Jan ve Vries: “On linear systems of algebraic plane curves”. 711.

— Jan ve Vries: “Some characteristic numbers of an algebraic surface’. 716,

— K. Bes: “The equation of order nine representing the locus of the principal
axes of a pencil of quadratic surfaces”, 721.

MEASUREMENT (On the) of very low temperatures, V. The expansion coefficient of
Jena- and Thiiringer glass between + 16° and — 182°C. 674. VII. Comparison
of the platinum thermometer with the hydrogen thermometer, 290, VIII, Comparison
of the resistance of gold wire with that of platinum wire. 300.

MECHANISM (The) of the salicylacid synthese. 63.

MEHMKE (k.). On moments of inertia and moments of an arbitrary order in spaces
of arbitrary .high rank. 596.

MEILINK (B.). On the measurement of very low temperatures. VII. Comparison
of the platinum thermometer with the hydrogen thermometer. 290. VIII. Compa-
rison of the resistance of gold wire with that of platinum wire. 300,

MELTING (On the) of floating ice. 459.

METALLIC Bopres (The motion of electrons in). 1, 438, IL, 585. LIL. 684.

XIV CONTENTS

Meteorology. J. P. van per Stok: “On a twenty-six-day period in daily means of

the barometric height”. 18 ‘
— C. Easton: “Oscillations of the solar activity and the climate”. 368.

METHOD (On a new) of damping oscillatory deflections of a galvanometer. 315.

METHYL CHLORIDE and carbon dioxide (The validity of the law of corresponding states
for mixtures of). 285. 377.

METHYL IODIDE (On the system pyridine and). 468. ;

Microbiology. M. W. Bezertnck: “An obligative anaerobic fermentation sar
cina”. 580. ‘

MINERALS in the rocks (On the composition of the silicates in the soil which have

been formed from the disintegration of the). 329.
MiscrBILity (On some phenomena which can occur in the case of partial) of two

liquids, one of them being anomalous, specially water. 517.
— (On the different forms and transformations of the boundary-curves in the case

of partial) of two liquids. 636.
— (On) in the solid aggregate condition and isomorphy with carbon compounds, 658,

mixTuREs (On the function for multiple), 94.

— of normal substances a exact expression for the course of the svmodal curves
and of their plaitpoints for all temperatures, in the case of). 646. =
MOLL (J. w.) presents a paper of Miss Tine Tammes: “On the influence of nutrition
on the fluctuating variability of some plants”. 398. =
— presents the dissertation of B. Sypkens: “On the nuclear division of Fritillaria
imperialis L”. 412.
MOLL VAN CHARANTE (J.). Sulphoisobutyric acid and some of its dagiviiiead 27.
MoNASCUS purpureus Went and Monascus Barkeri Dang. (On the development of the
perithecium .of). 83. --
MONTAGNE (P. J.). On intramolecular atomic rearrangements in benzpinacones, 271.
MoTION of electrons (The) in metallic bodies. I. 438. II. 585. IIL. 684. -
MUSKENS (L. J. J.). Degenerations in the central nervous system after removal-of
the Flocculus cerebelli. 282. =
NEBULAE (On the apparent distribution of the). 117.
— (The) considered in relation to the galactic system. 125.
NERVE-CELLS (On the branchings of the) in repose and. after fatigue. 599.
NERVE-STIMULATION (On artificial and natural) and the quantity of energy involved. 147.
NERVOUS system (Degenerations in the central) after removal of the Floceulus
cerebelli. 282.
NITRATION (The) of disubstituted benzenes. 266.
NOVA PERSEI (Dispersion bands in the spectra of 3 Orionis and). 823.
NUCLEAR DIVISION (Gn the) of Fritillaria imperialis L. 412. cae
NuMBERS (Some characteristic) of an algebraic surface. 716.
NUTRITION (Gn the influence of) on the fluctuating variability of some plants. 398.
OLIE JR. (J.). The transformation of the phenylpotassium sulphate into p. phenol-”
sulphonate of potassium. 328. - - tk

C O.N T) EN 8; XV

ONNES (H. KAMERLINGH.). vV. Kamertincit Onnes (IL).

orper of exception (Multiple umbilics as singularities of the first) on point-general
. surfaces, 386.

ORGAN-FORMATION (An investigation on polarity and) with Caulerpa prolifera, 420.

ORGAN-PIPES (On the relative sensitiveness of the human ear for tones of different.
pitch, measured by means of), 549.

ortonis (Dispersion bands in the spectra of 3) and Nova Persei. 323.
ORTHONITROBENZYLTOLUIDINE (On). 666.
OSCILLATIONS of the solar activity and the climate, 368.

OSCILLATORY DEFLECTIONS (On a new method of damping) of a galyanometer. 315.

OSMOTIC PRESSURE (On the) of the blood and urine of fishes. 537.

— (A formula for the) in concentrated solutions whose vapour follows the gas laws. 723.
— (Kinetic derivation of vaN ’r -Horr’s law for the) in a dilute solution. 729.
— and thermodynamic potential, 741.

OUDEMANS (G. a. J. a.). On Leptostroma austriacum Oud., a hitherto unknown
Leptostromacea living on the needles of Pinus austriaca; and on Hymenopsis
Typhae (Fuck.) Sace., a hitherto insufficiently described Tuberculariacea, occurring
on the withered leafsheaths of Typha latifolia. 206.

- — On Sclerotiopsis pityophila (Corda) Oud., a Sphaeropsidea occurring on the
needles of Pinus silvestris. 211.

OUDEMANS (J. A. C,) presents a paper of S. Box: “The connection between the
primary triangulation of South Sumatra and that of the West Coast of
Sumatra’. 453,

— Determinations of latitude and azimuth, made in 1896—99, 482.
— A short account of the determination of the longitude of St. Denis (Island of
Réunion) executed in 1874, 602.

OxYDATION (On the intramolecular) of a SH-group bound to benzol by an ortho-
standing NO,-group. 63.

PEKELHARING (c. A.) presents a paper of Dr. M. C. Dexuuizen; “On the
osmotie pressure of the blood and urine of fishes”. 537.

— presents a paper of Dr. W. Hurskamp: “On the presence of fibringlobulin in
fibrinogen solutions”. 610.

PENCIL (The locus of the principal axes of a) of quadratic surfaces. 341.

— (The equations -by which the locus of the principal axes of a) of quadratic
surfaces is determined. 532.

— (The equation of order nine representing the locus of the principal axes of a)
of quadratic surfaces, 721.

PERIOD (On a twenty-six-day) in daily means of the barometric height. 18.

PERITHECIUM (On the development of the) of Monascus purpureus Went and Monascus
Barkeri Dang. 83.-

PHALANGISTA VULPINA (On the genital cords of). 87.

puasEs (The derivation of the formula which gives the relation between the concen-
tration of coexisting) for binary mixtures. 156,

XVI CONTENTS.

PHENOMENA (On the) appearing when in a binary system the pluitpointeurve meets the
solubilityeurve. (3rd communication). 177.
— (On some) which can occur in the case of partial miscibility of two liquids,
one of them being anomalous, specially water. 517.
PHENYLPOTASSIUM SULPHATE (The transformation of the) into p. phenolsulphonate
of potassium, 328.

Physics. B. M. van Datrsen: ‘On the function + for multiple mixtures”. 94.

-- W. H. Junius: “Dispersion bands in absorptionspectra”. 134,

— W.H. Junius: “Spectroheliographic results explained by anomalous dispersion”. 140.

— J. D. van per Waats: “The derivation of the formula which gives the relation
between the concentration of coexisting phases for binary mixtures”. 156.

_— G. C. Gerrits: “On Px-curves of mixtures of acetone and ethylether and of
carbontetrachloride and acetone at 0°C”. 162.

— A, Smirs: “On the phenomena appearing when in a binary system the plait-
pointeurve meets the solubilityeurve”. (3rd communication). 177.

— H. Kamertinca Onnes and C. Zakrzewsk1: ‘Contributions to the knowledge
of VAN DER Waats’ p-surface. IX. The conditions of coexistence of binary mixtures
of normal substances according to the law of corresponding states”. 222.

— H. Kamertines Onnes and C. Zaxrzewski: “The determination of the conditions
of coexistence of vapour and liquid phases of mixtures of gases at low tempera-
tures’. 233.

— H. KameriincH Onnes and C. Zaxrzewski: “The validity of the law of cor-
responding states for mixtures of methyl chloride and carbon dioxide”. 285. 377.

— B. Merninx: “On the measurement of very low temperatures. VII. Com-
parison of the platinum thermometer with the hydrogen thermometer’. 290.
VIII. “Comparison of the resistance of gold wire with that of platinum wire.” 300.

— W. H. Juuius: “Dispersion bands in the spectra of 3 Orionis and Nova Persei”. 323.

— A. SommerFELD: “Simplified deduction of the field and the forces of an electron,
moving in any given way”. 346.

— P. Zeeman and J. Guest: “Double refraction near the components of absorption
lines magnetically split into several components.” 435.

—H. A. Lorentz: “The motion of electrons in metallic bodies”. 1. 438. IL.
585. LIT. 684.

— Cu. M. van Devenver: “On the melting of floating ice”. 459.

— J. BE. Verscuarretr: “The influence of admixtures on the critical phenomena
of simple substances and the explanation of TricHNeER’s experiments”. 474.

— J. D. van per Waats: “The transformation of a branch plait into a main plait
and vice versa”. 621.

— H. Kamertrscu Onnes and W. Hevuse: “On the measurement of very low
temperatures. V. The expansion coefficient of Jena- and Thiiringer glass between
+ 16° and- 182° C”. 674.

— Pu. Kounstamm: * A formula for the osmotic pressure in concentrated solutions

whose vapour follows the gas laws”. 723.

CONTENTS. xVII

Physiology. Pu. Kounstamm ; “Kinetic derivation of van ‘r Horr’s law for the osmotic
pressure in a dilute solution”, 729.
— Pu. Kouxstamm: “Osmotic pressure and thermodynamic potential”. 741,
— Artuur W. Gray: “Application of the Baroscope to the determination of the
densities of gases and vapors’. 770.
— H. Zwaarpemaker: “On artificial and natural nerve-stimulation and the
quantity of energy involved”. 147,
— L. J. J. Muskens: ‘“Degenerations in the central nervous system after removal
of the Floeculus cerebelli”. 282.
— W. E:nrnoven: “On a new method of damping oscillatory deflections of a
galvanometer’. 315.
— M. C, Dexuvzen : “On the osmotic pressure of the blood and urine of fishes”. 537.
— H. Zwaarpemaker Cz.: “On the relative sensitiveness of the human ear for
tones of different pitch, measured by means of organpipes”. 549,
— 8. J. pe Lance: “On the branchings of the nerve-cells in repose and after
fatigue”, 599,
— W. Hurskame: “On the presence of fibringlobulin in fibrinogen solutions’. 610,
PLACE (7,) presents a paper of Prof. J. W. Lancrnaan: “On the form of the
Trunk-myotome”. 34,
— presents a paper of E, pg Vrizs: “Note on the Ganglion vomeronasale”. 704,
— presents a paper of J. W. van Bisseick: “Note on the innervation of the
Trunk-myotome”. 708.
PLAITPOINTCURVE (On the phenomena appearing when in a binary system the) meets
the solubilitycurve. 177.
pLaiTpornts (An exact expression for the course of the spinodal curves and of their)
for all temperatures, in the case of mixtures of normal substances. 646.
pLants (On the influence of nutrition on the fluctuating variability of some). 398.
PLATINUM WIRE (Comparison of the resistance of gold wire with that of). 300.
PoLaRIty (An investigation on) and organ-formation with Caulerpa prolifera, 420.
potpers (On the origin of the fresh-water in the subsoil of a few shallow). 53.
POLYDIMENSIONAI SPACE (The formulae of GuLDIN in). 487.
POLYDIMENSIONAL spaces (On the equation determining the angles of two). 340.
porassiuM (The transformation of the phenylpotassiumsulphate into p-phenolsulphonate
of). 328.
POTENTIAL (Osmotic pressure and thermodynamic). 741.
PYRIDINE and methyl iodide (On the system). 468.
quapratic surraces (Ihe locus of the principal axes of a pencil of). 341.
— (The equations by which the locus of the principal axes of a pencil of) is
determined. 532.
— (The equation of order nine representing the locus of the principal axes of a
pencil of). 721.
RAbIUS of curvature (The relation between the) of a twisted curve in a point P o
the curve and the radius of curvature in P of the section of its developable

with its osculating plane in point P. 277.
Proceedings Royal Acad. Amsterdam, Vol. VIL. a4

XVIII CONTENTS,

RATIONAL CONES of the complex (On a group of complexes with), 577.

rays (A group of algebraic complexes of). 627.

REACTION (The) of FRrepEL and Crarts. 470.

REFRACTION (Double) near the components of absorption-lines magnetically split into
several components, 435.

REVERSION (An interesting case of). 90.

ROOZEBOOM (H. W. BAKHUTIS). vy. Bakuuts RoozeBoom (H. W.).

SALICYLACID synthese (The mechanism of the). 63.

SANDE BAKHUYZEN (H. G. VAN DE) presents a paper of C. Easton: “On
the apparent distribution of the nebulae”. 117.

— presents a paper of C, Easton: “The nebulae considered in relation to the
galactic system”. 125.

— presents a paper of J. Wrrper: “A new method of interpolation with com-
pensation applied to the reduction of the corrections and the rates of the
standardolock of the Observatory at Leyden, Hohwii 17, determined by the
observations with the transitcircle in 1903”. 241.

— presents a paper of J. WEEDER: “Approximate formulae of a high degree of
accuracy for the relations of the triangles in the determination of an elliptic
orbit from three observations’. 752.

SARCINA (An obligative anaerobic fermentation). 580.
SCHOUTE (Pp. H.). On the equation determining the angles of two polydimensional
spaces. 340.

— The formulae of GunpIN in polydimensional space. 487.

— On non-linear systems of spherical spaces touching one another. 562.

— presents a paper of Prof. R. MeumKe: “On moments of inertia and moments of
an arbitrary order in spaces of arbitrary high rank”. 596.

SCHUH (FRED.). On an expression for the class of an algebraic plane curve with
higher singularities, 42.

— On an expression for the genus of an algebraic plane curve with higher singu-
larities. 107.

— On the curves of a pencil touching an algebraic plane curve with higher sin-
gularities. 112.

SCLEROTIOPSIS PITYOPHILA (Corda) Oud. (On), a SEES occurring on the
needles of Pinus silvestris. 21].

sERIES derived from the series oi) 66 66

stLIcatEs (On the composition of the) in the soil which have been formed from the
disintegration of the minerals in the rocks. 329.

s1LIcon (The preparation of) and its chloride. 189.

SMITS (A.). On the phenomena appearing when in a binary system the plaitpoint-
curve meets the solubilityeurve. (8rd communication). 177.

soLar activity (Oscillations of the) and the climate, 368.

SOLUBILITY cuRVE (On the phenomena appearing when in a binary system the plait-
pointcurve meets the). 177.

CONTENTS. XIX

SOLVENTS (On the latent heat of mixing for associating). 174.

SOMMERFELD (a.). Simplified deduction of the field and the forces of an electron,
moving in any given way. 346,

seEcTRA (Dispersion bands in the) of 3 Orionis and Nova Persei. 323.

SPECTROHELIOGRAPHIC results explained by anomalous dispersion. 140.

SPHAEROPSIDEA (On Sclerotiopsis pityophila (Corda) Oud., A) occurring on the needles
of Pinus silvestris. 211.

SPHERICAL SPACES (On non-linear systems of) touching one another. 562.

STANDARDCLOcCK (A new method of interpolation with compensation applied to the
reduction of the corrections and the rates of the) of the Observatory at Leyden,
Hohwii 17, determined by the observations with the transitcircle in 1903. 241.

ST. DENIS (Island of Réunion) (A short account of the determination of the longitude
of) executed in 1874. 602.

STEREOISOMERIC g-and @-trithioacet (The intramolecular transformation in the) and
a-and #-trithiobenzaldehydes, 329.

$TOK (J, P. VAN DER). On a twenty-six-day period in daily means of the barometric
height. 18,

SULPHOISOBUTYRIC ACID and some of its derivatives. 275.

suMATRA (The connection between the primary triangulation of South-Sumatra and that
of the West-Coast of). 453.

suRFACE (Contributions of the knowledge of van per Waats’y-). IX. The conditions
of coexistence of binary mixtures-of normal substances according to the law of
corresponding states, 222.

suRFACES (Multiple umbilics as singularities of the first order of exception on point-

general). 386,
suyVER (J. ¥F.). The intramolecular transformation in the stereoisomeric g- and

B-trithioacet and @- und #-trithiobenzaldehydes. 329.
SYPKENS (B.). On the nuclear division of Fritillaria imperialis L. 412.
sysTEMs (On non-linear) of spherical spaces touching one another. 562.
— (On linear) of algebraic plane curves. 711,
TAMMES (TINE). On the influence of nutrition on the fluctuating variability of
some plants. 398.
TARSIUS SPECTRUM (On the development of the brain in). 331.
TEICHNER’s experiments (The influence of admixtures on the critical phenomena
of simple substances and the explanation of), 474.
TEMPERATURES (The determination of the conditions of coexistence of vapour and
liquid phases of mixtures of gases at low). 283.
= (On the measurement of very low). V. The expansioncoefficient of Jena- and
Thiiringer glass between ++ 16° and — 182° ©, 674, VII. Comparison of the
platinum thermometer with the hydrogen thermometer, 290, VILLI. Comparison of
the resistance of gold wire with that of platinum wire. 300.
— (An exact expression for the course of the spinodal curves and of their plait-
points for all), in the case of mixiures of normal substances. 646.
TETRAEDAL COMPLEX (On a special), 572.

xXx Cro N PEaes:

TETRAMETHYLPIPERIDINE (On zz'-). 270. :

THERMODYNAMIC POTENTIAL (Osmotic pressure and). 741.

THERMOMETER (Comparison of the platinum thermometer with the hydrogen). 290.

THREE-PHASE LINes (Critical terminating points in) with solid phases in binary systems
which present two liquid layers. 556.

TOLUIDINE (On the preparation of pure o-) and a method for ascertaining its purity. 395.
vones of different pitch (On the relative sensitiveness of the human ear for), measured
by means of organ pipes. 549.
TRIANGLES (Approximate formulae of a high degree of accuracy for the) in the deter-
mination of an elliptic orbit from three observations. 752.
TRIANGULATION (The connection between the primary) of South-Sumatra and that of
the West-Coast of Sumatra. 453.
TRINITROVERATROL (On). 462.
rritaroaceT (The intramolecular transformation in the stereoisomeric 2-and 8-) and
a-and £-thrithiobenzaldehydes. 329.
TRUNK-MYOTOME (On the form of the). 34.
— (Note on the innervation of the). 708.
TUBERCULARIACEA (On Hymenopsis Typhae (Fuck.) Sace.. a hitherto insufficiently
described), occurring on the withered leafsheaths of Typha latifolia. 206.
TYMSTRA Bz, (s.). On W. Marckwatp’s asymmetric synthesis of optically active
valeric acid. 465.
— and ©, A. Lospry DE Bruyn. The mechanism of the salicylacid synthese. 63.
umBriics (Multiple) as singularities of the first order of exception on point-general
surfaces. 386.
URINE of fishes (On the osmotic pressure of the blood and). 537.
yaLeric actp (On W. Marckwatp’s asymmetric synthesis of optically active). 465,
vapors (Application of the Baroscope to the determination of the densities of gases
and). 770.
vapour (The determination of the conditions of coexistence of) and liquid phases of
mixtures of gases at low temperatures. 233.
— (A formula for the osmotic pressure in concentrated solutions whose) follows
the gas laws. 723.
VARIABILITY of some plants (On the influence of nutrition on the fluctuating). 398.
VERSCHAFFELT (J. ¥.). The influence of admixtures on the critical phenomena
of simple substances and the explanation of Tercuner’s experiments. 474.
VERSLUYS (vw. A.). The relation between the radius of curvature of a twisted curve
in a point P of the curve and the radius of curvature in P of the section of
its developable with its osculating plane in point P. 277.
viscosity (The) of the system hydrazine and water. 329.
visser (a. w.). A few observations on autocatalysis and the transformation of
y-hydroxy-acids, with and without addition of other acids, conceived as an
ion-reaction, 760.
VRIES (£, DE). Note on the Ganglion vomeronasale. 704.

CONTENTS. xXxI

VRIES (HUGO DE) presents a paper of Prof. J. M. Janse: “An investigation on
polarity and organ-formation with Caulerpa prolifera”, 420.

VRIES (JAN DE). The congruence of the conics situated on the cubic surfaces of
a pencil. 264,

— A congruence of order two and class two formed by conics. 311.
— On a special tetraedal complex. 572.

— On a group of complexes with rational cones of the complex. 577.
— A group of algebraic complexes of rays. 627.

— On nets of algebraic plane curves. 631.

— On linear systems of algebraic plane curves. 711.

— Some characteristic numbers of an algebraic surface. 716.

WAALS (VAN DER) g-surface (Contributions to the knowledge of). 1X. The conditions
of coexistence of binary mixtures of normal substances according to the law of
corresponding states. 222.

WAALS (J. D. VAN DER) presents a paper of B. M. van DAtrsen: “On the

function - for multiple mixtures”. 94,
i

— The derivation of the formula which gives the relation between the concentra-
tion of coexisting phases for binary mixtures. 156.

— presents a paper of G. C, Gnrrirs: “On Px-curves of mixtures of acetone and
ethylether and of carbontetrachloride and acetone at O°C’. 162.

— presents a paper of Dr. Cu. M. van Deventer: “On the melting of floating
ice”. 459.

— The transformation of a branch plait into a main plait and vice versa, 621.

— presents a paper of Dr. Pu. Kounstamm: “A formula for the osmotic pressure
in concentrated solutions whose vapour follows the gas laws.” 723.

— presents a paper of Dr, Pa, KounstamM: “Kinetic derivation of van “2 Horr’s
law for the osmotic pressure in a dilute solution”. 729.

— presents a paper of Dr, Pa, Kounsrama: “Osmotic pressure and thermodynamic
potential”. 741.

water (On the origin of the fresh-) in the subsoil of a few shallow polders. 63.

— (The viscosity of the system hydrazine and), 329,

— of our sea-provinces (Some considerations on the conclusions arrived at in the
communication made by Prof. Eve. Dusors, entitled: Some facts leading to trace
out the motion and the origin of the underground). 45.

WEEDER (J.). A new method of interpolation with compensation applied to the
reduction of the corrections and the rates of the standardclock of the Obserya-
tory at Leyden, Hohwii 17, determined by the observations with the transiteircle
in 1908, 241.

— Approximate formulae of a high degree of accuracy for the relations of the
triangles in the determination of an elliptic orbit from three observations. 752.

WENT (fA. F.C.) presents a paper of H. P Kuyeer: “On the development of
the perithecium of Monascus purpureus Went and Monascus Barkeri Dang.” 83,

XXII CONTENTS,

WIND (c. H.) presents a paper of Dr. C. Easton: “Oscillations of the solar activity
and the climate.” 363.
WINKLER (c.) presents a paper of Dr. L. J. J. Muskens: ‘Degenerations in the
central nervous system after removal of the Flocculus cerebelli.” 282.
— presents a paper of Dr. 8S. J. pe Lance: “On the branchings of the nerve-cells
in repose and after fatigue.” 599,
ZAKRZEWSKI (c.) and H. Kameritnen Onnegs. Contributions to the knowledge of
VAN DER Waals’ y surface. JX. The conditions of coexistence of binary mixtures
of normal substances according to the law of corresponding states. 222.
— The determination of the conditions of coexistence of vapour and liquid phases
of mixtures of gases at low temperatures. 233.
— The validity of the law of corresponding states for mixtures of methylchloride
and carbon dioxide. 285. 377.
ZEEMAN (v.) and J. Gerest. Double refraction near the components of absorption
lines magnetically split into several components. 435.
ZIEHEN (TH.). On the development of the brain in Tarsius spectrum. 331.
Zoology. P. P. C. Hogk: “An interesting case of reversion.” 90.
— Tu. Zrenen: “On the development of the brain in Tarsius spectrum.” 331.
ZWAARDEMAKER (H.). On artificial and natural nerve-stimulation and the
quantity of energy involved, 147.
— On the relative sensitiveness of the human ear for tones of different pitch,
measured by means of organ pipes. 549.

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