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KONINKLIJKE AKADEMIE
VAN WETENSCHAPPEN
-- TE AMSTERDAM -:-

Ree CEE DINGS OF THE
eee PION OF SCIENCES

VOLUME Ix

JOHANNES MULLER :—: AMSTERDAM
: ae OLY 1907 >: :

(Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige
Afdeeling van 26 Mei 1906 tot 24 November 1906. Dl. XV.)

eN TEN TS.

<<>>
Page
Proceedings of the Meeting of May 26 1906 : 1
> >» » > » June 30 » 2 RS es eats Pee
> ne > » September 29 » Be ee tied at ae eee
» 1 > » October 27 » pe ed eR ee 249
> >» >» > » November 24 > Lente 6 oe 319

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday May 26, 1906.

DoS

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 26 Mei 1906, Dl. XY).

CG Oke i fr NL: S-

A. Smits: “On the introduction of the conception of the solubility of metal ions with
electromotive equilibrium”. (Communicated by Prof. H. W. Baxuuis Roozesoom), p. 2.

A. Smits: “On the course of the P,7-curves for constant concentration for the equilibrium
solid-fluid”. (Communicated by Prof. J. D. van DER WAatrs), p. 9.

J. Moxy van Cuarante: “The formation of salicylic acid from sodium phenolate’. (Commu-
nicated by Prof. A. P. N. Francuimonr), p. 20.

F. M. Jarcer: “On the crystal-forms of the 2,4 Dinitroaniline-derivatiyes, substituted in the
NH,-group”. (Communicated by Prof. P. van RompBureGn), p. 23.

F. M. Jarcer: “On a new case of form-analogy and miscibility of position-isomeric benzene-
derivatives, and on the crystalforms of the six Nitrodibromobenzenes”. (Communicated by Prof.
A. F. HoLiEeman), p. 26.

H. J. Hampurcer and SvanrE ArrueEntus: “On the nature of precipitin-reaction”, p. 33.

J. Stem: “Observations of the total solar eclipse of August 30, 1905 at Tortosa (Spain)”.
(Communicated by Prof. H. G. van DE SanDE BakHUYZEN), p. 45.

J. J. van Laan: “On the osmotic pressure of solutions of non-electrolytes, in connection with
the deviations from the laws of ideal gases”. (Communicated by Prof. H. W. Bakuuis Roozx-
BOOM), p. 53.

Proceedings Royal Acad. Amsterdam. Vol. IX.

(2)

Chemistry. — “On the introduction of the conception of the solu-
bility of metal ions with electromotive equilibrium.” By Dr.

A. Smits. (Communicated by Prof. H. W. Baxuuis Roozesoom).

(Communicated in the meeting of April 27, 1906.)

If a bar of NaCl is placed in pure water or in a dilute solution,
the NaCl-molecules will pass into the surrounding liquid, till an
equilibrium has been established ; then the molecular thermodynamic
potential of the NaCl in the bar has become equal to that of the
NaCl in the solution.

As known, this equilibrium of saturation, represented by the equation:

UNaCl = H'Nacl
is characterized by the fact that per second an equal number of
molecules pass from the bar into the solution, as from the solution
into the bar.

We shall call this equilibrium a purely chemical equilibrium. It
is true that in solution the NaCl-molecules split up partially into
particles charged either with positive or negative electricity, which
are in equilibrium with the unsplit molecules, but for the hetero-
geneous equilibrium solid-liquid under consideration this is not of
direct importance.

If, however, we immerge a metal e. g. Zw into a solution of a
salt of this metal, e.g. ZuSO,, we observe a phenomenon strongly
deviating from the one just discussed, which according to our present
ideas may be accounted for by the fact that a metal does not send
out into the solution electrically neutral molecules as a salt, but
exclusively zons with a positive charge.

If the particles emitted by the bar of zine were electrically neutral,
then the zinc would continue to be dissolved till the molecular
thermodynamic potential of the zine in the bar of zinc had become
equal to that of the zine in the solution, in which case the equation :

Uen = Ezn
would hold.

This, however, not being the case, and the emitted /n-particles
being electro-positive, an equilibrium is reached /ong before the
thermodynamic potential of the zine-particles with the positive electric
charge in the solution has become equal to that of the zine in
the bar of zine with the negative electric charge. That in spite of
this an equilibrium is possible, is due to the fact that an electrical
phenomenon acts in conjunction with the chemical phenomenon,

(3)

The zine emitting positive Zn-ions, the surrounding solution becomes
electro-positive, and the zinc itself electro-negative. As known, this
gives rise to the formation of a so-called electric double-layer in
the bounding-layer between the metal and the electrolyte, consisting
‘of positive Zn-ions on the side of the electrolyte and an equivalent
amount of negative electricity or electrons in the metal.

By the formation of this electric double-layer an electric potential
difference between metal and electrolyte is brought about, which at
first increases, but very soon becomes constant. This takes place
when the potential difference has become great enough to prevent
the further solution of the Zn-ions.

In order to compute the potential difference between the metal
and the solution, we shall apply the principle of the virtual dis-
placement, as has been done before by Mr. van Laar.')

If we have ‘to do with a purely chemical equilibrium then with
virtual displacement of this equilibrium the sum of the changes ot
molecular potential will be = 0, which is expressed by the equa-
tion of equilibrium :

= (u, dn,) = 0.

If the equilibrium is a purely electrical equilibrium then with
a virtual displacement of this equilibrium the sum of the changes
of electric energy will be = 0.

If however we have an equilibrium that is neither purely. che-
mical, nor purely electrical, but a combination of the two, as is
the case with electromotive equilibrium, then with virtual displace-
ment of this equilibrium, the sum of the changes of the molecular-
potential + the sum of the changes of the electric energy will
have to be = 0.

o
If we represent the mol. potential of the Zn-ions by w., in case

of electromotive equilibrium, we know that a is much smaller
than wz, or the mol. potential of the zinc in the bar of zinc.

If we now suppose that a Zn-ion emitted by the zine virtually
carries a quantity of electricity de from the metal towards the solu-
tion, then this quantity of electricity being carried by a ponderable

de
quantity — when » = valency of the metal and ¢ = the charge
VE

of a univalent ion, the increase of the thermodynamic potential
during this process will be equal to

1) Chem. Weekbl. N°. 41, 1905.
1*

which increase is negative, because fz, > len.

In the virtual displacement of the quantity of electricity de
from the metal towards the solution the change of the thermodyna-
mic potential is not the only one that has taken place during this
process. .

If we call the electric potential of the solution V, and that of the
zine V,, , we know that in the above case V, >V, and Ve -—VnmA
indicates the potential difference of the electrolyte and the metal.
With the virtual displacement of the quantity of electricity de from
the metal to the electrolyte this quantity has undergone an electrical
potential increase A, and so the electric energy has increased
with Ade.

From the principle of virtual displacement follows that with electro-
motive equilibrium

oe
a ep A= 0) oe ee ee
VE
or
a
A eee ee
VDE

Now we know that the mol. thermodyn. potential of a substance

may be split up as follows:
ap |- RP

where in diluted states of matter uw’ may be called a function of
the temperature alone.

In non-diluted states however, uw’ depends also somewhat on the
concentration. ;

If we now apply this splitting up also to equation (2), we get:

“Pp
(Wen =a [ten) +R TinwG

VDE

(3)

where C' represents the concentration of the Zn-ions in the electrolyte.
If we now put:

Pen Ss hen
BT

we may say of this A that for diluted states of matter it will only

=n... «5

(5)

depend on the temperature, and will therefore be a constant at
constant temperature.
From equation (3), (4) follows

ee aw wth ea eB)

Mr. van Laar already pointed out that this equation, already

derived by him in the same way is identical with that derived by

ee pe P

Nernst A = —— Jn —, in which therefore — stands instead of
VE Pp Pp

rie P represents the ‘“elektrolytische Lésungstension” of the metal,

and p the “osmotic pressure” of the metal-ions in the solution.

Rejecting the osmotic phenomenon as basis for the derivation of
the different physico-chemical laws, we must, as an inevitable conse-
quence of this, also abandon the osmotic idea “elektrolytische Losungs-
tension” introduced by Nernst.

The principal purpose of this paper is to prove that there is
not any reason to look upon this as a disadvantage, for, when we
seek the physical meaning of the quantity A’ in equation (5), it can
be so simply and sharply defined, that when we take the theory of
the thermodynamic poiential as foundation, we do not lose anything,
but gain in every respect.

In order to arrive at the physical meaning of the quantity A, we
put for a moment

C= K-
from which follows
== 0,

From this follows that there is a theoretical possibility to give
such a concentration to the metal-ions in a solution that when we
immerge the corresponding metal in it, neither the metalnor the solution
gets electrically charged.

How we must imagine this condition is shown by equation (2).
Let us put there A =O, then follows from this for an arbitrary metal

En = Un
or in words the molecular potential of the metal in the bar is equal
to that of the metal-ions in the solution.
So .it appears that we have here to do with an equilibrium
which is perfectly comparable with that between the NaCl in the
bar NaCl, and the salt in the solution.

(6)
The only difference is this that the molecules of a salt in solution
are neutrally electric, whereas the metal particles in solution are

charged with positive electricity, hence the physical meaning of

—
the equation un = um is simply this that in absence of a potential

difference, per second an equal number of metal particles are dissolved
as there are deposited.

If we express this in the most current terms, we may say, that
when C' = K the metal-ions have reached their concentration of
saturation, and that K therefore represents the solubility of the
metal-ions.

To prevent confusion, it will be necessary to point out that .the
fact that the dissolved metal-particles in equilibrium with the solid
metal have an electric charge, is attended by peculiarities which
are met with in no other department.

Thus it will appear presently that in every solution of copper-
sulphate which is not extremely diluted, the concentration of the
copper-ions is supersaturated with respect to copper. Yet such a
copper-sulphate-solution is in a_ perfectly stable condition, because
the copper-ions constitute a part of the following homogeneous
equilibrium, ;

CuSO, 2 Cu” + 80,"

which is perfectly stable as long as the solution is unsaturate or is
just saturate with CwSO,-molecules.

If we now, however, insert a copper bar into the solution, the
condition changes, because the Cu-ions which were at first only in
equilibrium with the CwSQO,-mols and with the SO,'-ions, must now
also get into equilibrium with the copper bar, and, the concentration
of the Cuw-ions with respect to copper being strongly supersaturate,
the Cu-ions will immediately deposit on the copper, till the further
depositing is prevented in consequence of the appearance of a double
layer.

We shall further see that in the most concentrated solution of a
zine-salt the concentration of the zine-ions always remains below the
concentration of saturation, which appears immediately when we
immerge a zinc-bar into such a solution; the zine emits zine parti-
cles with a positive charge into the solution, till the appearance of
the electric double layer puts a stop to the phenomenon of solution.

In order to find the values of K for different metals we make
use of the observed potential difference with a definite value of C,

Cr)

We know the potential difference at 18° and with normal con-
centration of the ions, i.e. when solutions of 1 gr. aeq. per liter of
water are used. These potential differences are called electrode
potentials, and will be denoted here by Ao.

If we express the concentration in the most rational measure, viz.
in the number of gr. molecules dissolved substance divided by the
total number of gr. molecules, we may write for the concentration
of 1 gr. eq. per liter

1
55,9 yp + 1

in which » represents the valency of the metal. In this it has been
further assumed, that the dissociation is total, and the association of
the water molecules has not been taken into account.

If we now write the equation for the electrode potential of an
arbitrary metal, we get:

RE Te
A,=—ln
VE s |
05,0» + 1

BE
A, = —In K (35,5 » + 1)
ve

If we use ordinary logarithms for the calculation, we get:
A pee K (55,5 v + 1)
SS eS 00,0 VD
iach eee
If we now express # in electrical measure, then

0,000198
= —_____ T log-K (55,5 » + 1)
Y

and for ¢=18 or 7’ — 291°
0,0578

A log K (55,5 » + 1)

If we now calculate the quantity log K by means of this equation
from the observed values of A,, we get the following. (See table p. 8).

In the succession in which the metals are written down here, the
value of A, decreases and with it the value of log KX.

For the metals down to Ye (#¢ included) log K is greater than
zero, so K greater than 1.

Now we know that C for a solution is always smaller than 1;
hence A will always be larger than C for the metals mentioned,
and as K denotes the concentration of saturation of the metal-ions,

(8 )
Values of log A at 18°.

je Ao | log K nae Ao log K
ape (+ 2,92) | (44877 +) Co | — 0,04 panies 1,805 2
Na | (» 254) | ( 4019) Nie >» 0,049; — 4,872 2
Ba (> 254) | ( 42.92 X 2) Sm | <» 008 | <— 2492
Sr (> 2,49) | ( 42,06 > 2) PL » 0,13 — 3972
Ca (> 2.98) | ( 3842 X 2) I » 0,28 = 66
Mg » 2,26 38,07 *K 2 Cus » 0,61 — 41,58 & 2
AL” » 1,00 16,56 X 3 Bie | <» 067 | <— 12333
Mn | » 0,80 12.81 <2 Ly," » 1,03 —- 18.84 <2
Zn" > 049 - 745 XX 2 Ay’ » 1,05 — 19,92
Ca” >» O14 1,39 <2 Pa » 4,07 — 19,03 < 2
Fe » 0,063 0,065>< 2 Pt » 114 — 20,62 4
Th >» 0,01 | — 0245>< 2 Au ae 1,36 — 2697 & 3

the metal-ions will not yet have reached their concentration of
saturation even in the most concentrated solutions of the corresponding
metal-salts. Hence, when the corresponding metal is immerged, metal
ions will be dissolved, in consequence of which the solution will be
charged with positive and the metal with negative electricity.

Theoretically the case, in which A would always be smaller than
C, can of course not occur. If /og K is smaller than zero, so K
smaller than 1, then the theoretical possibility is given to make the
potential difference between the metal and the corresponding salt
solution reverse its sign, which reversal of sign of course takes
place through zero. Whether it will be possible to realize this,
depends on the solubility of the salt.

If we now take the metal copper as an example, we see that for
this metal K has the very small value of 10-23. On account of this
very small value of A, C is greater than A in nearly all copper-
salt-solutions, or in other words the concentration of the Cwu-ions is
greater than the concentration of saturation. Hence copper-ions are
deposited on a copper bar, when it is immerged, in consequence
which the bar gets charged with positive, and the solution with
negative electricity.

But however small A may be, it will nearly always be possible to

1) The values of a. between parentheses have been calculated from the quan-
: Pp q
lity of heat.

(9)

make C smaller than XK. In a copper-salt-solution e.g. this can very
easily be done, as is known, by addition of ACV, which in consequence
of the formation of the complex-ions [Cu,(CV), |", causes copper-ions
to be extracted from the solution. The solution, which at first hada
negative charge compared with the metal copper, loses this charge
completely by the addition of ACN, and receives then a positive charge.

In the above I think I have demonstrated the expediency of
replacing the vague idea “elektrolytische Lisungs-tension’” by the
sharply defined idea solubility of metal ions.

Amsterdam, April 1906. Anorg. Chem. Lab. of the University.

Physics. — “On the course of the P,T-curves for constant concentra-
tion for the equilibrium solid-fluid.” By Dr. A. Sirs. (Commu-
nicated by Prof. J. D. van per Waa1s.)

(Communicated in the meeting of April 27, 1906),

In connection with my recent investigations it seemed desirable
to me to examine the hidden connection between the sublimation
and melting-point curves for constant concentration, more particularly
when the solid substance is a dissociable compound of two com-
ponents. This investigation offered some difficulties, which I,
however, succeeded in solving by means of data furnished by a
recent course of lectures giving by Prof. van per Waats. Though
his results will be published afterwards, Prof. vAN DER Wats allowed
me, with a view to the investigations which are in progress, to use
that part that was required for my purpose.

In his papers published in 1903 in connection with the investi-
gation on the system ether-anthraquinone') VAN DER WaAAtLs also
discussed the P, 7-lines for constant z for the equilibrium between
solid-fluid *), and more particularly those for concentrations in the
immediate neighbourhood of the points p and g, where saturated
solutions reach their critical condition.

Then it appeared that the particularity of the case involved also
particularities for the P, 7-line, so that the course of the P, 7-line
as it would be in the usual case, was not discussed.

1) These Proc. VI p. 171 and p, 484 Zeitschr. f. phys. Chem, 51, 193 and 52,
387 (1905). ;
*) These Proc. VI p. 280 and p. 357,

(19°)

If we start from the differential equation in p,c and T derived
by vAN pER Waats (Cont. II, 112).
075

Vsr dp = (#_ — “f) (; : ) day +

x? ¢) PT

we get from this for constant z that

si ar (1)
ear. =. Ql).

Gp
V pe ee eet eee he
On yg (2)
or
d W,
ey ee eee em ce AB)
aT }ar Vs
If we now multiply numerator and denominator by P as_ will
vy
prove necessary for simplifying the discussion, we get:
0?
d a . Wsf
ap fe ie ee ot Be ee co
dT ) rf Ow Vy
dy

In order to derive the course of the P, 7-lines from this equation,
the loci must be indicated of the points for which the numerator, -
resp. the denominator — zero, and at the same time the sign of
these quantities within and outside these loci must be ascertained.

In the v,.2-fig. 1 the lines ab and cd denote the two connodal
lines at a definite temperature. The line PsQs whose «= 2, the
concentration of the solid compound AZ cuts these connodal lines
and separates the v,x-figure into two parts, which call for a separate
discussion.

If Ps denotes the concentration and the volume of the solid com-
pound at a definite temperature, then the isobar MQRDD RUN
of the pressure of Ps will cut the connodal lines in two points Q
and (', which points indicate the fluid phases coexisting with the
solid substance AS, and therefore will represent a pair of nodes.

The points for which Za 0 oF, - =0 are situated where
the isobar has a vertical tangent, so in the points D and D’ as
vAN DER Waats') showed already before. In JD the isobar passes
through the minimum pressure of the mixture whose 7=2p, and
so it has there an element in common with the isotherm of this
concentration. In D' however, the isobar passes through the maxi-

1) These Proc. 1V p, 455,

e#)

mum pressure of the mixture whose z= zp, and will therefore have
an element in common with the isotherm of the concentration «rp.

d*y
As for the sign of aa we way remark that it is positive outside

U
the points D and D' and negative inside them.

The ordinary case being supposed in the diagram, viz. V; << Vy,
we may draw two tangents to the above mentioned isobar from the
point P, with the points of contact A and f’. These points of con-
tact now, indicate the points where the quantity V,,=0, as vAN
DER Waals‘) showed.

This quantity is represented by the equation :

Vig = (V, — Vp) — («3 — zp) oe ae fae

and denotes the decrease of volume per molecular quantity when
an infinitely small quantity of the solid phase passes into the coex-
isting fluid phase at constant pressure and temperature.

For the case that the coexisting phase is a vapour phase, V’sp is
negative, but this quantity can also be positive, and when the pres-
sure is made to pass through all values, there is certainly once
reversal of sign, for the case V, > V, even twice.

To elucidate this Prof. van per Waaus called attention to the
geometrical meaning of Vs;.

Let us call the coordinates of the fluid phases Q' coexisting with
P,, Vy and Xy and let us draw a tangent to the isobar in
Then P, Pf will be equal to V.,- if P is the point where this
tangent cuts the line drawn parallel to the axis of v through 7.

If the point P’ lies above P,, V.y is negative, and if 1” lies under
P,, then Vy is positive. For the case that the tangent to the isobar
passes through P,, which is the case for the points Rand f', V.-=0-

In this way it is very easy to see that for the points outside those
for which V.r¢=0, the value of Vy is negative, and for the points
within them, Vy is positive, but this latter holds only till the points
D and D' have been reached, where V,r=. Between D and LD,
V.y is again negative. The transition from positive to negative takes
therefore place through o.

As each of the lines of equal pressure furnishes points where

1) These Proc. VI, p. 234.

(12)

07%)
Ov?
we obtain loci of these points, indicated by lines.
As, however, we simplify the discussion, as vAN DER WaAAats has
0?

=0 and V,-=0, when connecting the corresponding points

shown, when we consider the quantity Vr instead of the

quantity V.-, because this product can never become infinitely great
and is yet zero when V.,,-=0, the locus of the points where
07) |
Ovf?

We know then too that this quantity on the left of the line ot

the compound is negative outside this locus, and positive within it.
2

0
Further the locus of — = 0 is indicated, and we see that these
v

V.s=0 is given in fig. 1.

two lines intersect at the point where they pass through the line of

the compound.

In his lectures vAN DER Waats has lately proved in the following
2

way that this must necessarily be so: If we write for _ ra os
of
0°w -
| gee a gs TS er : 6
( f) v7? Si (zs vf) dv dup ( )
we see that when this quantity = 0, and when at the same time
Gis Bf 3
Oy
| ee Ie — 0
("s D Say?
or
Oy
Ove? ee

I, too, had already arrived at the conclusion that in the left half of
our diagram the two loci mentioned had interchanged places, by
assuming that there existed a three-phase equilibrium also on the
right, and by drawing the corresponding isobar M,Q,D,R,R,'D,'Q,'N'.
It appears then that here the points AR, and f,' lie within the points

D, and D,', which points to a reversed situation (compared with
07 07w :

the left half) of the loci ai V.e—=O and Md = 0(. VAN pER WaALs
Ove . Ov/

has also given this graphical proof.
2

0
As for the sign of the quantity —
Vv

on the right of the line

(13 )

of the compound also there it is negative outside, positive inside
the first mentioned locus.

Before proceeding to my real subject, I shall, for the sake of
completeness, first call attention to the fact that the spinodal curve,
for which the equation:

0? 0x0v 07g

2 ——U ee es ee 7

Oz? 0?) = 02?) ? 7
Ov?

: . ; w
holds, lies entirely ontside the locus | O. Van per Waats’?)

z
proved this in the following way :

; Oy
On the spinodel curve and =— must both be positive, and so

Ov?
dw? 0? Op . A
also fa Sot ee 5, 8 positive outside the line for which
0? : ;
5 ?— 0, the spinodal line will always have to lie outside the curve
Y
Oy
dv?
That the spinodal curve which coming from the left, runs between
2 0? 0?
ee 6 ot — 0. cits ‘the Ine for
Ov? Ov? Ove

Vss=O0 on the left of the line of the compound in two points gq,
and q, which will be discussed afterwards, follows from this, that

3 v=0

0° )
on the line of the compound ee V7 = 90 coincides with
uf Ove

0
and that the line = always lies within the spinodal line, whereas
2

0
on the right of the line of the compound ee Vy = 0 lies within
:

: Seller 9

i
the line re

When we start from the maximum temperature of sublimation,
we get now v,z-lines which have been indicated by 7,, 7,, 7, and
T, in fig. 1 for the equilibria between solid-fluid according to the
equation *)

Oy Ow
dey _ [SP yates tO

ee eee ClCstltij$j#SOB
day 07h ay (°)

Ov
1) loc cit.
*) These Proc. VI, p. 489.

( 14 )

The v,z-curve denoted by 7), relating to the maximum temperature
of sublimation, consists of two branches, which pass continuously
into each other. The points of intersection with the connodal line
ab indicate the vapour phases and those with the connodal line cd
the liquid phases. In this way we get two pairs of fluid phases which
can coexist with the solid compound at the same temperature.

At the place where the two branches of the v,z-line cut the locus

07) Ov
du, af o ws =s 0D is
Witb increase of temperature these branches draw nearer to each
other, and when they would touch, intersection takes place; this is
here supposed to take place for the v,a-line denoted by 7’. This

point of intersection is the point q,, it lies therefore both on the
2
ah Vag =U.

If we now proceed to higher temperatures, detachment takes
place, and the v,a-figure consists of two separate branches, one
of which, viz. the vapour branch is closed. This case is represented
by the v,z-line 7;, for which it is also assumed, that this temperature is
the minimum-melting point of the compound, which follows from the
fact that the liquid branch of the v,z-line 7,, simultaneously cuts
the connodal line cd and the line of the compound.

With rise of temperature the closed v,x-line contracts, and the
corresponding liquid branch descends. The points of intersection of
the closed vapour branch and the liquid branch with the connodal
curves draw nearer and nearer to each other, and at a certain
temperature the two branches will show contact. The closed vapour
branch touches the connodal curve-ad and the liquid branch the
connodal curve cd. This is represented by the v,a-figure 7’,, which
represents the condition at the maximum-threephase-temperature, at
which the points of contact on the connodal curves and the point
for the solid substance must lie in one line.

At higher temperature no three phase equilibrium is possible any
longer, and both the closed vapour branch and the liquid branch
have got detached from the connodal curves. The liquid branch
descends lower and lower, and the closed branch contracts more and
more, and vanishes as a point in g,, where the upper branch of the

spinodal curve and on the curve

0? fy

spinodal curve and the curve = Vir = 0 intersect.
fie

If we now also indicate the locus of the points where ~ Wee
f

(15 )

the peculiarities of the course of the P7Z-lines may easily be derived
by means of the foregoing.
For the determination of the last mentioned locus, we start from

the equation:
OE,
Ws =| Pp 4. + Vs/ -+- (Esp)v . P . ~ (9)
Of / Tx

The factor of Vr, being naturally positive and (¢,;), being always
negative, W;, can only be equal to zero in a point w where Vs; is

°
2

positive, so between the loci where V,,=0 and i 0.
v

Further it is now easy to understand that at the same time
with Vy the quantity W,, will become infinitely great, there where

07yp
= 0. In order to avoid this complication vAN Der Waats has
7

0?
multiplied the quantity Ws, by ~ as equation (4) shows; the
i,

obtained product never becomes infinitely great now.

If we multiply equation (9) by - , we get:
;
7p 0&; 07
—.. Wy = Vee feat ag i 0
pe P+ aaa lae tate: 0
Now we know that the locus for - . Wsr = 0 will have to lie
If
0? 0? pies
between that for a Vs¢ = 9 and for eee = 0, as drawn in
dvs? Ove
. a ay 3 ert
fig. 1, which compels us to make dep W.f = 9 and abe V74="

intersect on the line of the compound.

That this must really be so, is easily seen, when we bear in mind,
2

0°yp er
that on the line of the compound the locus where ——| = 0 coincides
if
07)
0v,?
equation (10) it follows immediately that at the same point also

0? . .
z . Wy = 0. In this way we arrive at the conclusion, that the three loci

with that where .V;¢= 9, from which in connection with

Ov;
07 ve O2w

dp
a ee Ve 0
: Ov;? f Ov;?

W.f = 0 will intersect on the line
Ov,

( 16 )

| 92
of the compound, and that therefore the loci — Vey = 0 and
-

Oy
Ove?
the line of the compound.

By means of equation (10) we understand now easily that the
2

. Wey = 0 will interchange places on the left and the right of

sign of the quantity Wsy must be negative outside the locus

Duy?
0°
Io?

As connecting link for the transition to the P,7-lines we might
discuss the |’,7-lines; for this purpose we should then have to make
use of the following equation (Cont. II, 106)

. We = 0, and positive within it.

[ig Hoe lt [Oty

dT
+- (#5 a dxf — (€sf)o TT
By taking 2 constant we derive from this

, dvp\ — (€sf)v
ai bs
aoe: oS it ax dw fdvy

: (+) <= ck
©
2 ee F
Ov? if

I shall, however, not enter into a discussion of the V 7-lines because
it is to be seen even without this connecting link, what the course
of the P,7-lines must be.

0°?y
f- Our

Oy Oy ;
oof Vig = X, and — u/ . Wye

and if we indicate what the signs are of these quantities in the
different regions on the left and the right of the line of the com-
pound, and where these quantities become = 0, we get the following :

If now for simplication we call

left right
A X . ¢ ee xX =s,
astern =f x, x i=
x, + 4,— x.=0 A At Xx 0
Kt+B+ Kit % +
2 aes %—X + 4
X,—xX,— * X,—xXx,— *

of the compound, we obtain a curve as given by GF'FD in fig. 2.

(17 )

dp oy
ar ae xe . . . - . . . : (4a)

If now led by equation

we draw the P, 7-line for a concentration on the /e/# of the curve

As we have assumed in our diagram, that the vapour-tension

of A is the greatest and of B the smallest, whereas that of A B is

intermediate, we cut now that branch of the three phase line of the
compound, which has a maximum.

cy ea
eT
aos
Sh
f= &
| ee
f ae D
he
é
&
a»
e,
‘iP 1B
t
FY;
t
Z
’
/
‘
if ft
,
‘
ae
ne ait
,
pv oy 5
,
’
r)
nite G
‘
of é
PT ’
has? 4 Cf /
- 4
4 eo
4 s
s va
1 a
i
oomn
ig. 2?
Fig. 2.

This intersection takes place in the points F” and F, about which
it may be observed, that F”" lies at a higher temperature than F. This
situation can, however, also be reversed, and as appears from the
diagram, the transition takes place at a concentration somewhat to
the left of that of the compound. We see further, that the inter-
mediate piece, which continuously joins the line of sublimation GF”
to the melting-point curve F D, has a maximum and a minimum
(points where A,—0), about which the isotherm teaches us, that,
when we are not in the immediate neighbourhood of the critical
State, they are very far apart and that the minimum lies at a

negative pressure.
2

_ Proceedings Royal Acad. Amsterdam. Vol. IX.

: ( 18 )

It is also noteworthy about this figure, that when following the
P, T-line, starting with the point G resp. with the point D, we
first meet with a point, where the tangent is vertical, and (place
where X, = 0) after that with a point where it is horizontal.

If we now consider a concentration on the right of the line of
the compound, the P, 7-line corresponding with this will cut the
other continually ascending branch of the three phase line of the com-
pound, and by means of equation (4a) and the scheme for the
reversal of sign of X, and X, preceding it, we obtain a curve as
indicated by G, F,' F, D,. The situation of the loci X,=0O and
X,=0 being different on the right from that on the left, this
P, T-line differs from that just discussed. When now, starting from
the point G, resp. D,, we follow the P,7-line, we meet jirst with
a point, where it is vertical, so we have just the reverse of the
preceding case. About the situation of the points F' and f, we
may point out, that /’,' always lies at lower temperature than F;.

The loci X,—0O and X,=—O intersecting on the line of the
compound, the P,7-line for the concentration of the compound will
have to give to a certain extent the transition-case between the two
lines discussed.

What happens when we approach the curve of the compound, we
see immediately from fig. 1. The distance between the loci A; =
and X, =O becoming smaller and smaller, the points of contact of
the vertical and horizontal tangents will draw nearer and nearer,
which prepares us for what happens when we have arrived at the
line of the compound. We see from the scheme for the signs of
X, and XX, that when the loci X,=0O and X,=0 have coincided,
the signs of Y, and_X, reverse simultaneously, on account of which

dT
with what we know about the course of the P,7-lines somewhat
to the right and the left of the curve of the compound we are led
to the conclusion, that the ,7-line for the concentration of the
compound will have two cusps, each formed by two branches with
a common tangent. I have not been able to decide whether these
points will be cusps of the first or the second kind. The former
has been assumed in the diagram.

It is further noteworthy for this P,7-line that, as Van DER WAALS *)
already demonstrated before, both the line of sublimation and the
melting-point line must touch the three-phase line, so that the P, 7-line

dp - -
#63 retains the same sign, viz. remains positive. Combining this
:
sh

1) Verslag 21 April 1897, 482.

ae _- € 43)

sa
= ad
-
>

; = = concentration of the compound assumes a shape, as given

“It it were cattle to Sa the degree of association of the com-
ee ound smaller and smaller, the points 7,’ and /, would move to
fa “lower pressure and higher temperature. Moreover these two points
and the neighbouring point of intersection of the melting-point and
~ sublimation branches would draw nearer and nearer to each other,
till with perfect absence of dissociation these three pet would
_ have coincided.
_ Another peculiarity will present itself for the case that we have
“Sa a three-phase-line as described by me before, viz. with two maxima

= ‘and one minimum *), for then there is a point where ag= 27) on

| one and then it is Saini. to be seen that in consequence of
at

the coinciding of the points F’ and F, we get for this concentration
a P, T-line, as represented in fig. 3, which curve has the form of
a loop.

Amsterdam, April 1906. Anorg. Chem. Lab. of the University.

- a ‘These Proc. VIII, p. 200.

3 In this point the¥direction of the three:

( 20)

Chemistry. — “The formation of salicylic acid from sodium
phenolate.” By Dr. J. Motu van Cuarants. (Communicated
by Prof. A. P. N. FRancuronr).

(Communicated in the meeting of April 27, 1906).

The communication from Losry bE Bryn and Tr stra read at
the meeting of 28 May 1904 and their subsequent article in the
Recueil 23 385 induced me to make this research. Their theory,
and particularly the proofs given in support do not satisfy me and
as, in consequence of other work, I had formed an idea of the
reaction I made some experiments in that direction.

According to my idea, an additive product of sodium phenolate
with sodium phenylearbonate, or what amounts to the same an
additive product of two mols of sodium phenolate with one mol. of
carbondioxide C,H,OC(ONa),OC,H, might be the substance which
undergoes the intramolecular transformation to the salicylic acid

OH derivative and then forms, dependent on the tem-

A : perature, sodium salicylate and sodium phenolate

CoH, ONa or else phenol and basic sodium salicylate. This

CONa view is supported by previous observations of

Ss various chemists and has been partially accepted
OC,H, also by CtatsEn *).

As Lopry DE Bruyn and Tismstra give no analytical figures in
their paper it did not seem to me impossible that the phenolsodium-
o-carboxylic acid obtained by them might be the substance formed
by intramolecular transformation of my supposed additive product.
I, therefore, took up their method of working, OH
prepared sodium phenylearbonate in the usual
manner, from sodium phenolate and carbon dioxide, C,H, ONa
and heated this to 100° in a sealed tube for 100 Ae
hours. On opening the tube considerable pressure was
observed. This pressure was always fonnd when OO
the experiments were repeated. The gas liberated proved to consist
entirely of carbondioxide and amounted to */,—'/, of that present
in the sodium phenylearbonate. If we argue that the sodium phenyl-
carbonate under these circumstances is partiaily resolved into carbon
dioxide and sodium phenolate the latter compound ought to be present
or else the splitting up might give carbon dioxide and my supposed

1) B. B. (1905) 38 p. 714.

(21)

intermediary product (C,H,O), C(ONa),. In the first case it is strange
that during the cooling of the tube, which often was left for a few
days, the carbon dioxide is not greatly reabsorbed. Those substances
had now to be searched for in the product of the reaction. On
treatment with ether a fair amount of phenol was extracted althoueh
moisture was as much as possible excluded. It was then brought
into contact with cold, dry acetone, by which it was partially dissolved,
but with evolution of gas and elevation of temperature. From the
clear solution, petroleum ether precipitated a substance which, after
having been redissolved and reprecipitated a few times in the same
manner, formed small white needles containing acetone which efflo-
resced on exposure to the air. On analysis, this compound proved
to be sodium salicylate with one mol. of acetone. As an ebullioscopic
determination in acetone, according to Lanpsprrcer, did not give the
expected molecular weight, sodium salicylate was dissolved in acetone
and precipitated with petroleum ether and a quite identical product
was obtained as proved both by analysis and determination of the
molecular weight. Both products, after being dried at 100°, yielded
no appreciable amount of salicylic ester when heated with methyl
iodide.
The amount of sodium salicylate obtained by heating sodium
phenylearbenate in the manner indicated was, however, very trifling.
I suspected that the evolution of gas noticed in the treatment with
acetone, and which was identified as pure carbondioxide without
any admixture, was caused by the presence of unchanged sodium
phenylearbonate, so that, therefore, the reaction was not completed,
and that the tube after being heated must still contain a mixture of
unchanged sodium phenylearbonate, sodium phenolate, sodium sali-
cylate and free phenol, besides the said additive product (C,H,O),

OH
C(ONa), and the salicylic acid derivative possibly C,H, ONa
formed from this. I now thought it of great

importance to first study the behaviour of acetone CONa
with these substances as far as they are known. aoe
: OC, H,

Sodium phenolate dissolves in boiling acetone, from which it
crystallises on cooling in soft, almost white needles, several ¢.m. long,
which contain one mol. of acetone. They lose this acetone, in vacuo,
over sulphuric acid. At the ordinary temperature acetone dissolves
only 0,1 °/,.

Sodium phenylcarbonate placed in carefully dried acetone gives off

(22)

earbon dioxide with a slight elevation of temperature. The quantity
amounts to about '/, of the carbon dioxide actually present, at least
if account is taken of the comparatively large solubility of that gas
in acetone. The acetone, or if the mixture is extracted with ether,
also the ether, contains a quantity of phenol corresponding with the
total amount obtainable from the sodium phenylearbonate. The undis-
solved mass consists of a mixture of neutral and acid sodium
carbonate, nearly, or exactly in equivalent proportions. The decom-
position of 3 C,H,OCOONa to 3 C,H,OH + CO, + NaHCO, + Na,CO,
“requires 2 mols. of water. As the experiments however, have been
made in a specially constructed apparatus into which no moisture
or moist air could enter, with extremely carefully dried acetone, we
are bound to admit that this water has been generated by the acetone,
and we may, therefore, expect a condensation product of the acetone
which, however, could not be isolated, owing to the small quantities
of materials used in the experiments. It seems strange that in
this reaction the evolution of carbon dioxide is so extraordinarily
violent.

Sodium salicylate dissolves in acetone from which it crystallises,
with or without addition of petroleum ether, in small needles, which
may contain one mol. of acetone of crystallisation. In different deter-
minations the acetone content was found to vary from one-half to
a full molecule. At 16° it dissolves in about 21 parts of acetone.

Disodium salicylate was prepared by adding an (95°/,) alcoholic
solution of salicylic acid to a concentrated solution of sodium
ethoxide in alcohol of the saine strength. After a few moments it
crystallises in delicate, white needles. By boiling with acetone in
which it is entirely insoluble it may be freed from admixed mono-
sodium salicylate.

The behaviour of acetone with these substances now being known,
the experiment of heating the sodium phenylearbonate for 100 hours
was once more repeated, without giving, however any further results.
A portion was treated with acetone in the same apparatus which
had been used for the sodium phenylearbonate. A quantity of carbon
dioxide was collected corresponding with an amount of unchanged
sodium phenylearbonate representing 50—60°/, of the reaction-
product. Another portion was extracted with ether and yielded about
20 °/, of phenol whilst, finally, a small amount of sodium salicylate
was also found. The residue which had been extracted with ether
and acetone contained sodium carbonate but no disodium-salieylate.
It, however, contained phenol, probably from sodium phenolate.

It seems strange there is such a large quantity of free phenol

( 23 )

in the heated sodium phenylearbonaie, and as no disodium-salicylate
has been found it cannot have been caused by the formation of
that compound.

I have not been able to find the looked for additive product ;
perhaps it has been decomposed by acetone in the same manner as
sodium phenylcarbonate. The results obtained show in my opinion
that the formation of salicylic acid from sodium phenylearbonate is
not so simple as is generally imagined.

A more detailed account of research will appear in the “Recueil”.

Chemistry. — “On the crystal-forms of the 2,4-Dinitroaniline-deri-
vatives, substituted in the NH,-group”’. By Dr. F. M. Jancer.

(Communicated by Prof. P. van Romeuren).

Communicated in the meeting of April 27, 1906).
to) if b /

More than a year ago I made an investigation as to the form-
relation of a series of position-isomeric Dinitroaniline-derivatives *).
On that oceasion it was shown how these substances exhibit, from
a crystallonomic point of view, a remarkable analogy which reveals
clearly the morphotropous influence of the hydrocarbon -residues,
substituted in the NH,-group.

Among the compounds then investigated, there were already a few
1-2-4- Dinitroaniline-derivatives kindly presented to me by Messrs.
van RompurcH and Francuimont. Through the agency of Prof.
van RompureH and Dr. A. Mutper, | have now received a series of
other derivatives of 2,4-Dinitroaniline which in the happiest manner
complete my former publications. I wish to thank these gentlemen
once more for their kindness. I will describe and illustrate all these
derivatives in a more detailed article in the Zeits. f. Kryst.

For the present I will merely give a survey of the results obtained,
which have been collected in the annexed table.

I have chosen such a form-symbolic, that the morphotropous rela-
tion of the great majority of these substances is clearly shown. They
all possess the same family-character which is shown in the values
of the axial relations and the topic parameters. Only a few of these
substances show no simple relationship with the other ones.

1) Jarcer, Ueber morphotropische Beziehungen bei den in der Amino-Gruppe
substituierten Nitro-Anilinen; Zeits. f. Kryst. (1905). 40. 113—146.

Name of the compound

EE SS

*) On the isomorphism and the complete miscibility of this compound with p-Nitrosodiethylanil

4-2-Nitro-Aniline.
4-4-Nitro-Aniline.
4-9-4-Dinitro—-Aniline.

4-2-4-6-Trinitro-Aniline.

4-4-Nitro-Diethyl-A.

4-2-4-Dinitro-Methyl-A.
4-9-4-Dinitro-Ethyl-A.
{-2-4-Dinitro-Dimethyl-A.
4-2-4-6-Trinitro-Dimethyl-A.
4-2-4-Dinitro-Methyl-Ethyl-A.
4-2-4-Dinitro-Diethyl-A.
4-2-4-, + 1-3-4-Dinitro-Diethyl-A.
(Double compound.)
4-2-4-6-Trinitro-Diethyl-A.
{-2-4-Dinitro-Ethyl-n-Propyl-A.
4-2-4-6-Trinitro-Ethyl-Isopropyl-A.
4-2-4-Dinitro-Isopropyl-A.
4-2-4-Dinitro Dipropyl-A.
4-2-4-6-Trinitro-Dipropyl-A.
{-2-4-Dinitro-Isobutyl-A,.
4-2-4-6-Trinitro-Isobutyl-A
4-2-4-Dinitro-Diisobutyl-A.
4-2-4-Dinitro-Allyl-A.
4-2. 4-Dinitro-Methyl-Phenyl-A.
4-2-4-Dinitro Ethyl-Phenyl-A.
4 -2-4- D.nitro-Benzyl-A.
4-2 4-Dinitro-Methyl-Benzyl-A.
4-2-4-Dinitro-Ethyl- Benzyl-A.
4-2-4-Dinitro-Phenyl-Benzyl-A.

4-2-4 (6-Trivitro-Ethyl-Nitraniline.

1-2-4-6-Trinitr »-Tsopropyl-Nitraniline.

4-2-3-4-6-Tetranitro-Methyl-Nitraniline.

Survey 0

sep ee la ie
ght.) © state.) .
790 | 438 95.70
146° | 138 96.03
1g90 | 183 | 143.30
acoe | 998 | 199.39
78° | 19% | 4162.07
1790 | 197 | 195.94
40 | ot | 145.44
g7o | a1 | 4142 95
o4e | 956 | 4165.05
sgo | 995 | 457.45
go> | 939 | 173.94
59° | 478 | 4(364.02)
164° | 98% | 199.44
55° | 953 | 4189.43
409° | 998 | 211.80
950 | 995 |. 453.79
4g? | 967 | 202.50
igse | 312 | 997.93
go> | 939 | 472.70
gso | 98% | 196.53
119° | 995 | 250.24
76° | 993 | 157.93
1660 | 973 | 104.46
950 | 987 | 240.48
11¢° | 973 | 487.50
1440 | 997 | 204.44
73° | 301 | 219 87
1680 | 349 | 250.00
96° | 301 | 183.09
1ose | 315 | 201.53
14g0 | 332 | 489.74

Axial-Elements:
1: b : © = 1.3667 :1: 1.1585.
:b:¢ = 2.0350 :4: 1.4220; = 88°10!
bse = 1.9896 :4:4 4088; @—85° 1)’ |
b:¢ = 1.6560 :4: 1.5208; ¢—80°47}'
:e = 1.0342 :1: 0.9894; 2 = 80°34’
© = 1.2286 :1: 0.9707; 2 = 83°98
2 = 1 51 21; 0.9745
a = 330954! g — 83099’ y= 75°74.
eo — 17.9154 74: 1.0803.
:¢ = 1.2936 21: 1.3831.
te = 1.4497 21: 4.6639.
a= 75°46! =p =99933' += 68°57’
¢ = 1.204 :1: 1.1513
bse = 1.3435 :1:1.3013; = 86°39!
© = 1.1750 24: 0 9462; 2 = 86°98"
2 ¢ = 1.0535 :4: 0.9297.
:e¢ = 2.0162 24.
2=75°0' g—99°7! 7—141%6Y

“fon
Proce

Base

——
- ¢

OAD =A: A 5790:
418°43' 2 —104°33'
—— 4. 0191 >4-. 0.9246.

Se ooo lL: fu: 9055

419946) p—1411°0! += 102°35'
== 0 7104':4 0.3591; P=—S85°34}!
= 0.7325 -4: 0 3470.

40717 :4; 0.9124; . A= 63°53!
= 1.0251 :1: 0.9632

TA163' B=11190' 7—116°40'
ads 24> A_6968 ~ P= 864!

0.4933 :1: 0.6586; = 78°6}!

7= S12"!

: 1.3087; = 84°!
241.3645; = 64021}!
p= 86°23)
: 0.9368; 7 = 78°33!

1
7 1
— 1.4487 : 4.
| 1
4: 4.4712; == 76°37’

zedings (1905) p. 658,

eee

Topic Parameters:

ao a

-J

.3635 : 3
.5406 : 3
.8206 : 3.
Joi 5S
-6240 : 5.
.8090 : 4.7
B33 ee
.8035.: 4.

.0686 : 6.5

.9480 : 5.802

5382

5890
8650
TAO
.9891
.3129
.0310

6856

( 26 )
Crystallography. — “On a new case of form-analogy and misci-
bility of position-isomeric benzene-derivatives, and on the crystal-

forms of the six Nitrodibromobenzenes.” By Dr. F. M. Jancrr.
(Communicated by Prof. A. F. HoL Eman.)

(Communicated in the meeting of April 27, 1906).

§ 1. The following contains the investigation of the crystal-forms
exhibited by the six position-isomeric Nitrodibromobenzenes, which
may be expected from the usual structure-representations of benzene.
It has been shown that, in this fully investigated series, there
again exists a miscibility and a form-analogy between two of the
six terms.

The above compounds were kindly presented to me by Prof.
Hotieman, to whom I again express my thanks.

This investigation is connected with that on the isomeric Dichloro-
nitrobenzenes, which has also appeared in these proceedings (1905,

p. 668).

A. Nitro-2-3-Dibromobenzene.
Structure: C,H,.(NO,) .Br .Br ; meltingpoint: 53° C.
(1) (2) (3)
The compound, which is very soluble in most organic solvents,

Fig. 1.

erystallises best from ligroin + ether in small, flat, pale sherry-
coloured needles which generally possess very rudimentary terminating
planes.
Triclino-pinacoidal.
a:b26 = GATS. 2 Tote:

A = 90°80: e045"),

B=110°37' = @ = 110°36?/,

C= SDT)? 89°59*/,'

ll |

—— = =_ - s. =)

( 27 )

The crystals, therefore, show a decided approach to the mono-
clinic system; on account, however, of their optical orientation, they
can only, be credited with a triclinic symmetry. .

The forms observed are: a = {100}, strongly predominant and very
lustrous; 6 = {010}, smaller but yielding good reflexes ; ¢ = {O01},
narrower than a, but very lustrous ; 0 = 141!, well developed and

.very lustrous; » = 441}, smaller but very distinct ; s = 111}, very

narrow but readily measurable.
The habit is elongated towards the b-axis with flattening towards {100}.
Measured: Calculated :
a:b = (100) : (010) =* 90°167/,' =ak
Ger (100):: (001) = ==* 69 23 —
a0 = (£00): (411) —* ae pi

c:0 = (001) : 411) =* 75 47'/, =
b:o = (010): (111) =* 36 6 as
a:o— (100):(441) = 5052 © 50°49!
c:@ — (001): (111) = 56 52 56 43
b:w— O10): (411)=— 46 28 46 35
0: = (111):(111) = 4713 47 29/,
a:s = (100):(411)= 4959 50 49°/,
b:s —(010):(411) = 4548 © 45 527/.
e:s =(0/):d1j)D=—= — 56 4
o:s —(111):(111)— 63 39 63 592/,

Readily cleavable, parallel {100}.

The extinction on {100} amounts to about 26'/, in regard to the
b-axis; in convergent light a hyperbole is visible occupying an eccen-
tric position.

The sp. gr. of the crystals is 2,305 at 8°; the equivalent volume 121.47.

B. Nitro-2-5-Dibromo-Benzene.

Structure: C,H, . (NO,)ji . Bri). Bris); m.p.: 84°,5.

This compound has been previously studied crystallographically by
G- Beis, (Zeits. f.. Kryst. 82, 377). This paper, however, contains
several errors, which render a renewed investigation desirable ; more-
over, another choice of axial (coordinate) planes is required, which
makes the crystals show more analogy with the other triclinic terms
of this series.

The crystals deposited from acetone + ligroin have the form of
small plates flattened towards {001} (figs. 2 and 3). They are pale
yellow and very lustrous.

( 28 )

Triclino-pinacoidal.
4:06 = 1,4909 ei 2,0214.

A= OTe o/s a 90°57*/,"
B= A113°2*/2 B= 113°21'/,'
C= 90°27' ¥== SOT 2:

Forms observed: c= {001}, strongly predominant and reflecting
ideally ; a = {100}, and rv = {101}, usually developed equally broad
and also yielding sharp reflexes; 6 = {010}, smaller, readily measur-
able ; m = {110}, large and lustrous ; p= {113}, mostly narrow but
very lustrous; sometimes as broad as m.

Broad flattened towards {001}. The approach to monoclinic sym-
metry is also plain in this case.

Measured : Calculated:
a:b = (100) : (010) —=*89°33' —-
b<c = (010); (001) =*88 ser —
c:a == (001): (100) =*66 2e7- a
6b: m = (010) = (110) sane =
a:r — (100) : 101) =*43 45 a

c:m — (001) : (410) = 75 46 75°387/,'
a:m = (100) : (110) = 53 33 5350",
c:r = (001) : (101) = 69 37 69 36'/,
7 :m= (101) : (110) = 65 20 65 11
p:m= (113) : (110) = 60 59 60 447/,
r:b = (101): (010) = 89 55 89 22

r: p = (101) : 413) = 50 53 at.

( 29 )

Readily cleavable, parallel m.

The optical orientation is that of Frus, in which his forms {010},
{O01} and {117} assume, respectively, in my project the symbols
{001}, {110} and {010}. It may be remarked that Frxs has incor-
rectly stated the structure and also the melting point. Moreover, his
angles (111): (100) and (4111) : (010) appear to be > 90°. Perhaps it
is owing to this, that the agreement between the calculated and found
values is with him so much more unfavourable than with me. I have
never observed forms {552} and {15.15 . 4!

The sp. gr. at 8° is 2,368; the equiv. volume: 118,66.

Topical axes: y%: pw: w = 5,2190 : 3,5005 : 7,0758.

On comparing the said position-isomeric derivatives, one notices at once
not the great similarity between the two compounds, which, although
constituting a case of direct-isomorphism, still very closely resembles it.

Nitro-2-3-Dibromobenzene.
Triclino-pinacoidal.
26 st Aros. bs L.Ghts

Nitro-2-5-Dibromobenzene.
Triclino-pinacoidal.
G2 bs.6— 1,4909 +1 = 2.0814.

A=90°30' B=110°37' C—90°167/,'
a=90°457/,'B=110°36?/,'y=89°59?/,'
XW: w= 55,2565 :3,5571 : 6,9409.

Ate (2 (1S! C962 e
ka—O0°S7*/,’ B=113°21"/' y=90°2'
X:W: @ = 95,2190 :3,5005: 7,0758.

However:
{100}, {010}, {001}, {101},
{110}, {113}.

However:

{100}, {010}, {001}, 1171},

{111} and {117}
Cleavable parallel {100}. Cleavable parallel {110}.

Habit tabular towards {100}. | Habit tabular towards {001}.

We, therefore, still notice such a difference in habit and cleava-
bility that a direct isomorphism, in the ordinary meaning of the
word, cannot be supposed to be present. There occurs here a case
of isomorphotropism bordering on isomorphism.

Notwithstanding that difference, both substances can form an
interrupted series of mixed crystals, as has been proved by the
determination of the binary melting point curve and also crystallo-
graphically *).

The melting point of the 1-2-3-derivative (53°) is depressed by
addition of the 1-2-5-derivative. The melting point line has also

Forms: Forms:
1

1) The binary melting-curve possesses, — as proved by means of more a exact
determination, — a eutectic point of 52° C. at 2°/, of the higher melting com-
ponent; therefore here the already published melting-diagram is eliminated. There
is a hiatus in the series of mixed-crystals, from + 3% to circa 489 of the 1-2-3-deri-
vative. I shall, however point out, that the possibility of such a hiatus thermody-
namically can be proved, — even in the case of directly-isomorphous substances.

(Added in the English translation).

( 30 )

not, as in the previously detected case of the two tribromotoluenes
(Dissertation, Leyden 1903) a continuous form; the difference is
caused by the lesser degree of form-analogy which these substances
possess in proportion to that of the two said tribromotoluenes.

The third example of miscibility, although partially —, and of
form-analogy of position-isomeric benzene-derivatives*) is particularly
interesting.

Mixed crystals were obtained by me from solutions of both com-
ponents in acetone + ether. ?

They possess the form of fig. 1 and often exhibit the structure of
a sand time-glass or they are formed of layers. With a larger quantity
of the lower-melting derivative, long delicate needles were obtained
which are not readily measurable. The melting points lie between
+ 75° and 844°; 1 will determine again more exactly the mixing limits.

C. Nitro-2-4-Dibromobenzene.

Structure: C,H, . (NO,)a). Brey . Broa); m. p. 61°.6.

Reerystallised from alcohol, the compound forms large crystals
flattened towards a and elongated towards the c-axis. They are of
a sulphur colour.

Triclino-pinacoidal.

C2026 = 159807 22s 11698:

BS VIN == 07 3b
B= tissse fe eng WIS Sh
C= oie ye y= 8733"

Forms observed: a = {100} predominant and
very lustrous; 4 = {010} and c = {001}, equally
broad, both strongly lustrous; p= {110}, narrow
but readily measurable; o = {111}, large and
yielding good reflexes.

The compound has been measured previously,
by Grotn and Bopewie (Berl. Berichte, 7, 1563).
My results agree in the main with theirs; in
the symbols adopted here, their a- and 6-axes
have changed places and the agreement with
the other derivatives of the series is more
Fig. 4. conspicuous.

1) The examples now known are 1-2-3-5-, and 1-2-4-6-Tribromobenzene ;
1-2-3-5-Tribromo-4-6-Dinitro- and 1-2-4-6-Tribromo-3-5-Dinitrotoluene; and 1-2-5-,
and 1-2-3-Nitrodibromobenzene, partially miscible.

(31 )

Measured: Calculated:
a: b= (100) : (010) =* 89°21"/,' =
> ¢ = (100) : (001) =* 66 29"/,' en
(010) : (001) =* 82 467/,' de
p:a = (110) : (100) =* 46 36 ee
c: 0 = (001) : (111) —* 48 42 2s
0: ):
C

p = (111): (410) = 51 43 (cirea) 52° 1
ape (O0LY: (110) == Oh 29. (circa) 100.43!

Cleavable towards {010}; Groru and Boprwia did not find a
distinct plane of cleavage.

Spec. Gr. of the crystals = 2,356, at 8° C., the equiv. vol. = 119,27.
Topic Axes: x: W:w = 5,2365 : 46304 : 5,4166.

Although the analogy of this isomer with the two other triclino-
isomers is plainly visible, the value of a: is here quite different.
“In accordance with this, the derivative melting at 84'/,° Jowers the
melting point of this substance. A mixture of 87°/, 1-2-4- and 13°/,
1-2-5-Nitrodibromobenzene melted at 56°. There seems, however, to
be no question of an isomorphotropous mixing.

D. Nitro-2-6-Dibromobenzene.

Structure: C,H, (NO,)ay . Bre) . Bre);
m.p 82°.

Reerystallised from alcohol the compound
generally forms elongated, brittle needles
which are often flattened towards two
parallel planes.

Monochno-prismatic.

6 O5GTo: 1 20,6257
p = 83°24’.

Forms observed: 6 = {010}, strongly pre-
dominant; g = {011} and o = {111} about
equally strongly developed. The crystals
are mostly flattened towards 6 with incli-
Fig. 5. nation towards the a-axis,

( 32 )

Measured: . Calculated :
i (O14) : O11) —* 63°437/,' —
0:0 = (4111): (111) =* 47 52 =

o:g = (111): 014) =* 74 207/, ne

o:g = (411):(011)— 45 427/, 45°42’
q:6 =(011):(010)= 58 84, 58 8'/,
b:o0 = (010):(111)= 66 6 66 4

_ No distinct plane of cleavage is present. An optical investigation
was quite impossible owing to the opaqueness of the crystals.

Sp. Gr. = 2,211 at 8° C.; the equiv. vol.: 127,09.

Topic parameters: y%: W: © = 4,0397 : 7,1147 : 44516.

E. Nitro-3-5-Dibromobenzene.

Structure: C,H, (NO,)q) . Bris) . Brys); m.p.: 104°,5. The compound
has already been measured by Boprwie (Zeitschr. f. Kryst. 1. 590);
my measurements quite agree with his.

Monoclino-prismatic. |

Bopewie finds a:6:¢c=0,5795:1:0,2839, with 8—=56°12'. Forms:
§1103%, {LOO}, {001} and {011}.

I take 8 = 85°26’ and after exchanging the a-, and c-axis

a:6:.c = 05678 :1 : 04531,

with the forms {011}, {001}, {201} and {211}. Completely cleavable
towards {201}. Strong, negative double refraction.

Sp. Gr. = 2,363 at 8° C.; equiv. vol. = 118,91.

Topic axes: 4: HY: o = 43018 : 7,5761 : 3,6601.

The great analogy in the relation a: of this and of the previous
substance is remarkable; also that of the value of angle ?.

F. Nitro-3-4-Dibromo-Benzene.

Structure C,H, (NO,)1 . Brs). Bry; m.p. 58° C. Has been measured
by Grotn and Bopewie (Berl. Ber. 7.1563). Monoclino-prismatic.

a:4 =0,5773:1 with @ = 78°31’. Forms {001}, {110} and {100},
tabular crystals. Completely cleavable towards {100}, distinctly so
towards {010}. The optical axial plane is {010}; on a both optical
axes (80°) are visible. I found the sp. gr. at 8°C. to be 2,354. The
equivalent volume is therefore 119,34.

( 33 )

I have tried to find a meltingpoint-line of the already described type
in the monoclinic derivatives in which some degree of form-analogy
is noticeable. However, in none of the three binary mixtures this
was the case; the lower melting point was /owered on addition of
the component melting at the higher temperature, without formation
of mixed crystals. For instance :

A mixture of 82,3°/, 1-2-3- and 17,7°/, 1-3-5-Nitrodibromo-benzene
melted at 48'/,° C.

A mixture of 76,5°/, 1-2-6- and 23,5°/, 1-3-5-Nitrodibromo-benzene
ak 69'/.> C.

A mixture of 90,5°/, 1-3-4- and 9,5°/, 1-2-6-Nitrodibromo-benzene
at 54° C.

Moreover, no mixed crystals could be obtained from mixed solutions.

The slight form-analogy with the Nitro-dichloro-benzenes *) investi-
gated by me some time ago is rather remarkable.

Nitro-2-3-Dichloro-Benzene (62° C. rhombic) and Nitro-2-6-Dichloro-
Benzene (71° C. monoclinic) exhibit practically no form-analogy with
the two Dibromo-compounds. There is alsv nothing in the Dichloro-
derivatives corresponding with the isomorphotropous mixture of the
2-3- and 2-5-Dibromo-product. The sole derivatives of both series
which might lead to the idea of a direct isomorphous substitution
of two Cl by two Br-atoms are the Nitro-3-5-Dihalogen-Benzenes
(65° C. and 104°,5 C.); the melting point of the Dichloro-derivative
is indeed elevated by an addition of the Dibromo-derivative.

As a rule, the differences in the crystal-forms of the compounds
of the brominated series are much less than those between the forms
of the chlorinated derivatives — a fact closely connected with the
much greater value which the molecular weight possesses in the
Nitro-Dibromo-Benzenes than in the corresponding Ch/oro-derivatives.

Zaandam, April 1906.

Physiology. — “On the nature of precipitin-reaction.” By Prof.
H. J. Hampurcer and Prof. Svante Arruentus (Stockholm).

(Communicated in the meeting of April 27, 1906).

One of the most remarkable facts discovered during the last years
in the biological department, is most certainly the phenomenon that
when alien substance is brought into the bloodvessels the individual
reacts upon it with the forming of an antibody. By injecting a

1) These Proc. VII, p. 668.

Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 34 )-

toxin into the bloodvessels, the result is, that this is bound and free
antitoxin proceeds. Exriicn explains this as follows. When a toxin
is injected, there are most probably cells which contain a group of
atoms able to bind that alien substance. Now Weicert has stated the
biological law, that when anywhere in the body tissue is destroyed,
the gap usually is filled up with overecompensation. So, it may be
assumed, that when the cell looses free groups of atoms, so many
of these new ones are formed, that they can have no more place
on this cell and now come in free state in circulation. This group
of atoms is the antitoxin corresponding to the toxin.

As a special ease of this general pbenomenon the forming of
precipitin is to be considered.

When a calf is repeatedly injected with horseserum, which ean be
regarded as a toxic liquid for the calf, then after some time it
appears that in the bloodserum of that calf an antitoxin is present.
In taking some bloodserum from this calf and. by adding this to the
horseserum a sediment proceeds. This sediment is nothing else than
the compound of the toxin of the horseserum with the anti-toxin that
had its origin in the body of the calf. We are accustomed to call
this antitoxin precipitin, and the toxin here present in the horseserum,
and which gave cause to the proceeding of precipitin, precipitinogen
substance. The compound of both is called precipitum.

It is very remarkable that such a precipitate proceeds only, when
the precipitin is brought in contact with its own precipitinogen sub-
stance. Indeed by adding the designed calfserum containing preci-
pitin, not to the horseserum but to the serum of another animal, no
precipitate proceeds. In this we have also an expedient to state if in
a liquid (e.g. an extract of blood stain) horseserum is present or not
(UnennutH, WaAssERMANN inter alia). Meanwhile such a calfserum
gives notwithstanding also a precipitum with serum of the ass related
to the horse.

To the same phenomenon the fact is to be brought, that when
a rabbit has been injected with oxenserum, the serum taken from
the rabbit does not only give a precipitate with oxenserum but also
with that of the sheep and the goat, which are both related to the ox.

Some time ago an expedient was given to distinguish also *) serum
proteid from related species of animals by a quantitative way, and
in connection with this a method *) was proposed to determine accu-

1) H. J. Hampurcer, Eine Methode zur Differenzirung von Eiweiss biologisch
verwandter Thierspecies. Deutsche Med. Wochenschr. 1905, 5. 212.

2) H. J. Hampurecer, Zur Untersuchung der quantitativen Verhiltnisse bei der
Pricipitinreaction. Folia haematologica. Il Jahrg, N°, 8,

( 35 )

rately the quantity of precipitate which is formed by the precipitin
reaction. This method also permitted to investigate quite generally
the conditions which rule the formation of precipitate from the two
components.

Immediately two facts had pushed themselves forward by a preli-
minary study which were also stated in another way by EISENBERG ')
and AscoLi’).’

1. That when to a fixed quantity of calfserum *) (precipitin =
‘antitoxin) increasing quantities of diluted horseserum (precipitinogen
substance = toxin) were added, the quantity of precipitate increased, in
order to decrease by further admixture of diluted horse serum.

2. that whatever may have been the proportion in which the two
components were added to each other, the clear liquid delivered
from precipitate always give a new precipitate with each of the
components separately. This leads to the conclusion that here is
question of an equilibrium reaction in the sense as it has been
stated and explained for the first time by Arruenius and Mapsen ‘).

This conclusion has become also the starting point of the now
following researches of which the purpose was to investigate by
quantitative way the principal conditions by which precipitin reaction
is ruled.

Methods of investigation.

To a fixed quantity of calfserum °) (precipitin = antitoxin) increas-
ing quantities of diluted horse-serum (precipitinogen substance =

1) Eisenperc. Beitrige zur Kenntniss der specifischen Pracipitationsvorginge
Bulletin de |’Acad. d. Sciences de Cracovie. Class. d. Sciences Mathem. et nat.
p. 289.

2) Ascou. Zur Kenntnis der Pracipitinewirkung. Miinchener Med. Wochenschr.
XLIX Jahrg. 8. 398.

3) They used sera of other animals.

4) ARruentus und Mapsen. Physical chemistry to toxins and antitoxins. Fest-
skrift ved indvielsen of Statens Serum Institut. Kjobenhavn 1902; Zeitschr. f.
physik. Chemie 44, 1903, S. 7.

In many treatises the authors have continued these investigations; compare e.g.
sill :

Arruenws. Die Anwendung der physikalischen Chemie auf die Serumtherapie.
Vortrag gehalten im Kaiser]. Gesundheitsamt zu Berlin am 22 Sept. 1903. Arbeiten
aus dem Kaiserl. Gesundheitsamt 20, 1903.

Arruenius. Die Anwend. der physik. Chemie auf die Serumtherapeutischen Fragen.
Festschrift f. Botrzmann 1904. Leipzig, J. A. Barru.

5) To make it easy for the reader, we speak here only of calfserum and horse-
serum, Compare the third note on this page.

( 36 )

toxin) are added. There upon the mixtures are heated for one
hour at 37° and then centrifugated in funnelshaped tubes of which
the capillary neck was fused at the bottom. The in 100 equal volumes
calibrated capillary portion contains 0.02 or 0.04 ¢.c. The centri-
fugating is continued till the volume of the precipitate has become
constant °)..

Experiment with calf-horse serum.

As it was of importance, at all events for the first series of proofs,
to dispose of a great quantity of serum containing precipitin, a large
animal was taken to be injected. Dr. M. H. J. P. THomassen at Utrecht
was so kind to inject at the Governement Veterinaryschool there,
a large calf several times with fresh horse serum and to prepare
the serum out of the blood drawn under asceptic precautions.

The serum used for the following series of experiments was
collected Nov. 28, 1905, sent to Groningen and there preserved in
ice. On the day of the following experiment January 25, 1906,
the liquid was still completely clear and free from lower organisms ;
there was only on the bottom a thin layer of sediment, which
naturally was carefully left behind at the removing of the liquid. ”)

The horseserum used for the proof in question was fresh and
50 times diluted with a sterile NaCl-solution of 1°/).

Each time two parallel proofs were taken as a control. The
capillary portion of the funnel shaped tubes used for this experiment
had a ealibrated content of 0.04 ec. Each division of the tubes thus
corresponded to 0.0004 ce.

To this series of experiments another was connected in which the
quantity of diluted horseserum was constant, but increasing quan-
tities of calfserum were used.

From the first table it appears, that when to 1 cc. calfserum
increasing quantities of diluted horseserum are added, the quantity
of precipitate rises. When more horseserum is added as is the case
in the second table, the quantity of precipitate descends. This appears
from the following.

1) Compare Folia haematologica |.c, for further particulars of the method.
2) Fuller details of other proofs taken on other days with calf-horseserum, also
of experiments with serum obtained by injecting rabbits with pig-, oxen-, sheep-

and goat-serum will be communicated elsewhere,
‘

(37)

TABLE L
1 ce of the mixture of 1 ce. Volume of the precipitate, after centri-

The quantity

calfserum (precipitin or | fugating for: of precipitate
: found in 1ce.
serum containing anti- of the mixtures
calculated for
toxin) +... ec. horse- the ¢ofal quan-

tity of the
serum 1/,, (precipitino- mixed compo-
‘ ee nents according

gen or toxin containing to the last

| observation.

serum. th.-$h. - 3h. -$h. - $h.- 20m. - 15m.)

cal SS a ae

ow ee. horseserum '/,, | 4 — 1/, — not to be measured accurately
0.04
3 » » | 4—¥— vd Dd » » »
|
=) » >| $— 3— 3— 3— yo 3 O08
— » » See ee 3.08
0.4 :
3? » » Ae) —— a) 10.5
ae » me — 44 40 — 10 — 40° 10: —-10 | 10.5
= » » 26 — 23 — 20 — 18 — 17 — 17 — 17 | 18.4
= > oo Sa ae aia | me by Gee 1 ae C2 MT
0.2
wad » ibe oe ha Ht DAE.
ae A B96 9k = 90 — ot = oy — 9 23.4
0.13 » » » 43 — 43 — 39 — 34 — 32 — 32 — 32 36.2
0.13 » » » | 48 — 43 — 39 — 34 — 32 — 32 — 32 | 36.2
0.15 » » » Be 6 — 4 a a | 39.
0.45 » » » ee A —— | 39.4
0.18 » » » 6d — 1 — 54 —" 48 — 2 — 43 — £8 |! 50.7
0.18 » » » 65 61 — 54 — 48 — 49 — B— 488 50.7
022°» » » 65 6a A Ay Ay 4 54
O22» » » 6506 45 45 — 45 54
O95.» » » hes 5D 33) HS 66.3
0.25 » » » i9— 1a — 69) — 5S Spe 1559 es S- 66.3
|
Ox3'*-y » » Sree 62 See 74.1
G3 2 » » Se 30) —— 10 — 62 ——58 = 57 — 52 74.41

( 38 )

So e. g. the quantity of precipilate when 0.3° cc. horse serum is added to
1 ce. calfserum, is 74.1 (table I). But when, as may be read in the second table
0.5 cc. horse serum is added to 0.9 cc. calfserum the precipitate has a volume

TABLE I.
|
1 cc of the Volume of the precipitate, after centrifu- | The quantity of
mixture of 0.5 cc gating for iy rr fous
h = ‘mixtures calculated
5 ic sane for the total
quantity of the
Yeo --. cc ‘mixed components,
se taerk so according to the
bh. = hoe gh ahs ph. 20m, -15m,| Sete
oe aS
0.4 cc calfserum, | 1.— $ — _ not to be measured accurately
0.1 » » 4—4 — » » » » » |
0.3 » » 2— 2— 2— 2— 2— 2— 2] 1.6
0.3 » » Q— 2— 2— F— 2— 2-- BZ} 1.6
0.5 » » 6G 5 = BS ba oe 5
0.5 » » y aes eee sy ee kad ale
O27 > D 48 — 36 — 32 — 28 — 2% — 2 — 25) 30
0.7 » D | 00 88 ee eee 30
9.9 » D) 84 — 6 — 57 — 50 — 43 — 43 — 48) 51.6
0.9 » » 81 — 63 — 55 — 49 — 43 — 43 — 43 | 51.6
|
1.4 » » | 95 — 81 — 67 — 58 — 52 — 50 — 50) 80
1A > >) | 0k — 8) 68k Bee ee 80
| )
1.3 » D 62 — 79 — 66 — 59 — 59 — 55 — 355) 99
1.3 » » | 97 — 80 — 69 — 60 — 59 — 55 — 55) 99
1.5 » » 96 — 84 — 74 — 6 — 62 — 59 — 59} 118
3 ow » 95 — 84 — 73 — 64 — 62 — 59 — 59 118
1.9» » | 90 — 75 —, 65 — 55 — 53 — 541 — S51] 122.4
|
1.9 » » 89 — 75 — 6 — 55 — 538 — St — S51 | 122.4

( 39 )
of 51.6. If instead of 0.9 calfserum 1 cc. was used the quantity of horseserum

: es | : ;
would necessarily have amounted to 0,5 X 9 9 = 0,59 cc. So it appears that by the

addition of 0.3 cc. horseserum to 1 cc. calfserum the precipitate amounts to 74.1 and
by the addition of 0.55 cc. horseserum but to 37.3 }),

This decrease must be attributed partly to the solubility of the
precipitum in NaCl-solution, a solubility which is felt the more strongly
as a greater quantity of diluted horseserum is added. (Compare also
Fol. Haematol |. c.).

So we see that the clear liquid above the precipitate contains,
besides free precipitin and free precipitinogen substance, as has already
been stated, also dissolved precipitate.

These three substances must form a variable equilibrium, which
according to the rule of GuLDBEere and Waace is to be expressed
by the following relation.

Concentration of the free precipitinogen subst. >< Concentr.
of the precipitin =k, <Concentr. of the dissolved precipitate . . . . (I)

in this &, is the constant of reaction.

Meanwhile it appears from the experiment, that a greater quantity
of precipitate must be dissolved than corresponds with this equation,
or to express it more clearly, than corresponds with the conception
that the solubility of the precipitate in NaCl solution is the only fact
by which the quantity of sediment decreases.

To take away the difficulty, the hypothesis was made that still
another portion of the precipitate forms a dissolvable compound with
free precipitinogen substance (of horseserum) and that we have
here a case analogical to the reaction of CaH,O, with CO,. As is
known CaH,O, is precipitated by CO,, but by addition of more CO,
the sediment of CaCO, decreases again, while CO, with CaCO, forms
a dissolvable substance.

As will soon be seen, a very satisfactory conformity between
calculated and observed quantity of precipitate is obtained through
this hypothesis, which could afterwards be experimentally affirmed.

Let us now try, reckoning both with the solubility of the precipi-
tate in NaCl-solution and with the forming of a dissolvable mixture
of precipitate with precipitinogen substance, to precise more closely
equation I.

1) The hyperbolic form of the precipitate curve with ncreasing quantity of horse-
serum may still appear from the following series of experiments taken on another
day (Table III). This series has not been used for the following calculation.

( 40 )
TABLE IIL.

1ce of the mixture | Volume of the precipitate after centrifu-

|
| ,
The quantity of «
| precipitate found

gating for: in 1 cc. of the

of 1cc calfserum

ich mixtures calculated
+ +++ cc horse- for the {otal
quantity of the

ET mixed components

according to the

Phe ea bh 0m. — iesden| eee

0.4 ce horseser. 1/,,| 38 — 38 — 28 — 244 — 923 — 23 — 23 | 2.3
Oe» » » | 40 — 22 — 29 — % — 23 — 23 — 2] 2.3
0.2 » D » | 66 — 54 — 48 — 44 — 49 — 42 — 42 | 50.4
0.2 » » » | 59 — 50 — 45 — 48 — 4H — HW — 41 50.4
0.3 » » » | 88 — 69 — 65 — 56 — 55 — 55 — 55] 71.5
0.3 » » » | 87 — 68 — 65 — 56 — 55 — 55 — 55] 71.5

0.6 » » ye | kee a7 ee ee 70.4
0.6 » > »| 71 — 57 — 53 — 47 — 4B — 48 — 48 68.8

1 » » 90 25 — 10 — 4 aa eee 26
4.2% » Qe De 4
4.2» » duces Sy eee eee ee, S|
1.4» » » not to be measured

» »

—
>=
u
zy
zy
zy
y

( 41)

Firstly we shall try to find an expression for the three substances
occurring in the clear liquid which stands above the precipitate: for
the free precipitinogen substance, for the free precipitin which it
contains and for the quantity of dissolved precipitate.

Nirstly the quantity of free preipitinogen substance. Let A be the
total quantity of that substance used for an experiment. To determine
how much of this is still present in the liquid in free state, it is to
be determined how much is bound. Bound is:

1. a certain quantum to form the precipitate which is present in
solid condition. If we set down as a rule that 1 mol. precipitum
proceeds from 1 mol. precipitinogen substance and 1 mol. precipitin,
then the wanted precipitinogen substance will be expressed by P, if
the molecular quantity precipitate also amounts to P.

2. a quantity pV when p represents the percentage of the quantity
of dissolved precipitate and V the total volume of the liquid.

3. a quantity necessary to form the compound of precipitate- and
precipitinogen substance. Admitting that 1 mol. of this compound
proceeds from 1 mol. precipitate and 1 mol. precipitinogen substance
and then that y of this compound is present, then together 27 must
be charged, while in each of the two components y mol. precipitinogen
substance is present, so that the quantity of precipitinogen substance,
which is left in free state, amounts to

A—P—pV—2y.
So when the volume of the liquid is V, the concentration of the
free precipitinogen substance =
A—P—pV—2y
ER ee es
It is possible to calculate in the same way the concentration of
the free precipitin.
If B is the total quantity of precipitin, which is used for the
experiment, then there is to be subtracted from this:

1st. a quantity P for the same reason as is given at the calcula-
tion of the free precipitinogen substance (see above).

2d. a quantity pV, likewise as explained there.

3d, a quantity necessary to form the compound precipitate-preci-
pitinogen substance. While in this compound but 1 mol. precipitin
is present, only 1y is to be charged. So that the quantity of pre-
cipitin which remains in free state, amounts to b—P—p )—y.

While the volume of the liquid amounts to V7, the concentration
of the free precipitin is =

( 42 )

B—P—pV—y
- otis. 5 Se

As for the concentration of dissolved precipitate in the third place,
this must be expressed by

pV
F ieee ce ee
So the equation (1) becomes:
A—P—pV—2y ge ee py
V Sar Sey

or
(A—P—pV—2y) (B—P—pV--y) =k, pv’. .. (5)
Now one more equation, expressing the reaction according to
which precipitate combines with precipitinogen substance. This is
to be written down as follows.

Concentration free precipitinogen substance X concentr. dissolved precipitate

= ky concentr. compound precipitinogen subst. — precipitate.
A—P—pV—2y ja y
V SNe
or
(A—P—pV—2y)p=ky. ..... (6)

By putting shortly P-+-pV = P’ and by substituting the value of
y of equation (6) into equation (5) we obtain

A—P' A— Pp.
A—P'—2p B—P'—p a= kp": aca
k,- 2p k,+2p

In this equation are known:
ist. A, the total quantity of precipitinogen substance (diluted
horseserum added) ;

2nd, $B, the total quantity of precipitin (calfserum) used ;

3-4. |, the volume of the liquid resulting from the mixing of
the two components ;

4h P, the quantity of solid precipitate directly observed.

Unknown are:

1st. p, the quantity in percentages of precipitate which is dissolved
(so p represents the solubility of the precipitate) ;

2nd, f,, the constant for reaction of the formation of precipitate;

3°, &,, the constant for reaction of the formation of the com-
pound precipitate-precipitinogen substance ;

4th. Pp’, this is however P+pV and therefore known as soon

as p has become known.

As equation (7) contains 3 unknown quantities three observations
will be necessary to determine them.

When we introduce then the so found values in the other experi-
ments and calculating the quantity of precipitate, it appears that the
calculated quantities correspond in very satisfactory way with those
which are observed.

Let us observe that to avoid superfluous zeros 1 ce calfserum
(5) is taken = 100.

While as appears from the experiments in the case in question
1 ce calfserum is equivalent to nearly */, ce horseserum 1:50, 1 ce
horseserum 1:50, that is A, obtains a value of 300.

0,04
So, where in the first experiment Sa CC: horse serum was used

0,04
A obtains a value of re < 300'= 4.

In the experiment, where on 1 ce. calfserum 0,8 cc. horseserum
was used, with a value B=100, A becomes 0,3 « 300 = 90.

Let us now combine the two tables to one by calculating for the
second table how much */,, horseserum is used on 1 cal//serwm..

We see that the comformity between the determined and caleu-
lated precipitate (col. IIT and 1V) is very satisfactory. The average
of the discrepancy amounts to 1.3.

This result deserves our attention not only in view of the know-
ledge of the precipitin reaction as such, but also from a more
general point of view, this reaction belonging to the great group
of the toxin-antitoxin reactions.

Till now, in studying the last, we were obliged to deduce the
equilibrium conditions from the toxins, that is to say by determining
the toxic action which was left by the gradual saturation of the
toxin by increasing quantities of antitoxin, but with the precipitin-
reaction the equilibrium conditions may be deduced from the quantity
of the formed toxin-antitoxin compound.

And not only that, but owing to the fact that the compound forms
a precipitum, the quantity of this may be fixed in an accurate and
direct way by simple measurement, thus without the aid of red blood
corpuscles or of injecting-experiments in animals.

So there is good reason to expect that a further study of the
precipitin-reaction will facilitate too the insight in other toxin-anti-
toxin reactions.

(32)

TABLE IV.
I Seog I III IV Vv
Acc. calfserum, B = 100.
Used quantity| Used quantity | Determined Calculated Difference
of of volumes of the | volumes of the between
horseserum korseserum precipitate precipitate III and IV.
1/5, (on Acc. expressed in 1 cc. of the | in 1 cc. of the
calf serum). in the just mixtures. mixtures.

accepted units

ese ee —===—=—==

0.013 cc. 4 not to be measu- 0.2

0.027 » 8 eo 3.9 40.9
0.05 » 45 10 he: siatode 408
0.08 » oy 17 17.8 E088
0.4 » 30 21 23.6 + 2.6
0.13» 39 32 29.7 ie ois
0.15 » 45 34 34 0
0.18 » 54 43 40.1 — 2.9
0.2 » 60 45 43.9 a
0.25 » fo tee 52 52.1 S04
0.266 » 79 51 53.6 +. 2.6
0.294 » 88.3 55 57.1 42.4
03." 90 57 57.5 +0.5
0.33 » 100 59 58.9 = 04
0.385 » | 445.4 ae 57.4 49.4
0.457 » 137 50 51.3 Cage
0557 » 167 43 41.3 = AT
0.13 » O14 25 26.8 + 1.8
4 » 300 5 5.5 +. 0.5
4.67 » 500 2 0 =.9

RESUME.

We may resume our results as follows.
By mixing precipitin and precipitinogen substance (to compare
resp. with antitoxin and toxin) an equilibrium reaction proceeds

( 45 )

obeying to the law of GuLDBeRe and Waacee. By this equilibrium
reaction part of the precipitin molecules combines with the corre-
sponding quantity of molecules precipitinogen substance, while by the
side of this compound a certain quantity of each of the two components
remains in free state. The compound is partly precipitated and partly
remains dissolved. How much remains dissolved depends for the
greater part on the salt solution which is present, for the sediment
is soluble in Na Cl-solution.

Besides this equilibrium reaction there is still another which
consists in this, that part of the precipitate combines with free
precipitinogen substance to a soluble compound. This reaction
too obeys the law of GuLpBerGc and Waace. The case is to be
compared with the precipitation of Ca(OH), by CO,. By excess of
CO, a part of the resulting CaCO, is transformed in a soluble
bicarbonate. So CaH,O, takes the function of the precipitin and
CO, that of the precipitinogen substance.

Astronomy. — “Observations of the total solar eclipse of August 30,
1905 at Tortosa (Spain).” By J. Stein S.J. (Communicated
by Prof. H. G. vAN DE SANDE BakHUyzEN.)

At the invitation of Mr. R. Cirera S. J., director of the new
“Observatorio del Ebro” I went to Tertosa towards the end of
June 1905 in order to take part in the observation of the total solar
eclipse. I was charged with making the measurements of the common
chords of the sun and moon at the beginning and at the end of the
eclipse and had also to determine the moments of the four contacts.
The results might perhaps contribute to the correction of the relative
places of the sun and moon.

The determination of the co-ordinates was much facilitated by the
circumstance that the signals of the three ‘points Espina, Gordo and
Montsia of the Spanish triangulation were visible at this place. The
measurements of the angles with a theodolite yielded the following
results :

g = 40°49' 13.43 ; A=1™ 58518 east of Greenwich.

In these results the spheroidal shape of the earth is accurately
taken into account. Later measurements made by Mr. J. Usacu
gave the same results. Electric time-signals, directly telegraphed from
the Madrid observatory, gave for the longitude: 1™ 58*.8 east of
Greenwich. As the most probable value we have adopted 1™ 58*.5,

( 46 )

the mean value of the two determinations. As a test 30 other deter-
minations of latitude have been made with an instrument temporarily
adjusted for Talcott observations, from which I derived as mean
value: g = 40°49' 14".8. The height above the sea-level is 55 meters.

The instrument at my disposal’ for the eclipse observation was a
new equatorial of Maimuart (Paris), 2™.40 focal length and 16 em.
aperture, provided with an eye-piece with a double micrometer.
I have determined the screw value of one of the two screws from
18 transits of cireumpolar stars near the meridian. | found for it:

R, = 60.3534 + 0".0117;

the value of the other screw was determined by measuring the
intervals by means of the first:
A, 2. OO010 2 sta. =

The observatory possesses a good sidereal clock, the rate of which
had been carefully determined during four months by means of star
transits. In the night of 29—80 August, Mr. B. Brrnory, a clever
observer had observed 20 clock-stars, so that the accuracy of the
determination of the clock-error left nothing to be desired.

During the phase observations the object-glass was reduced to
25 mm. by means of a screen of pasteboard. The eye-piece with
a power of 30 was provided with a blue glass. The observations
of the chords were continued as long as was allowed by the field of
view of the eye-piece, which was more than 20' in diameter. At my
signal “top” the moments of the observations were noted by Mr. Bexpa,
who was seated in front of a mean time standard clock, which
before, during. and after, the observations was compared with the
sidereal elock; another assistant recorded the micrometer readings.

During the beginning and the end of the eclipse the sky in the
neighbourhood of the sun was perfectly clear, so that I could per-
form the measurements of the chords undisturbed, although now
and then I met with difficulties owing to irregularities in the rate
of the driving clock. From some minutes before, until after, totality
the sun was covered with light clouds, yet the moments of contact
could be recorded with sufficient accuracy.

In the derivation of the results I have taken the solar parallax
= 8".80; for the rest I have borrowed the constants from the public-
ation “Observatorio Astronomico de Madrid. Memoria sobre el eclipse
total de Sol del dia 30 de Agosto de 1905”. They are:

Mean radius of the sun &, = 15'59'.63 (Auwers)

Fe m » > moon 7, = 15'32".83 (KuEsTNER and BATTERMANN)

Parallax of the moon y= 57’ 2:68

( 47 )
OBSERVATIONS.

he 2m 6
Pirst comtac t: 41 55 39 .4 (mean time of Greenwich.)

Length of the chords (corrected for refraction)

41 56 28 .2 294.93
Sy Be be | 390.24

57 35 .2 437 22

58 20 .0 507.74
59 8 .2 566.98

59 38 .9 608 . 94
1276.9. .2 642.58
4:25 .0 721.69
249 .9 798.82
418 .3 876.43
457 .0 906.12
544 3 935.04

6 15 <9 959.75

6 53 .2 983 .94

7 18-49 1004.93
Sint 2 1030 37

8 43 .3 1052.50
923 .3 1078.17
9.49 .1 1096.89

10 16 .4 1106.16
10 42 .2 4124.37

fi 9273 4138.90

44 20%7:4 1144.49
4456-53 4160.37
12 24 .3 41178 82

hi) m8
Second contact: PAGS 2

Third contact: AAG) 7.2

Length of the chords
he Wise Ss "
215 53 .0 1297.92

( 48 )

Length of the chords
i

bo
ae
~~]
ey]
s1
aU)

4256 94
18 4.5 1232.27
18 25 .3 4219.81
48 42 .5 4209.54
149 13 .3 4193.25
49 38 .2 1181.49
20 45 .0 4157.42
2.5 33 4129.77
2 28 .3 4117.78
22 1 .0 1095.75
22 35 .3 1073.82
23 4 1054.40
23 21 .3 1041 52
23 54 .3 1620.90
24:36 .0 993 .28
ya ae Ae 973.04
25 35 .3 950.47
26 2.3 £20.28
26 29 .3 903.24
26 52 .3 880 81
27 13 .3 863.90
27 36 .2 845.44
8 11:6 819 14
28 43 .3 779.01
29 5 .d 762.98
29 38 .6 726.38
30 2 .3 697.40
30 22 .3 677.17
30 52 .3 637.13
31 414 .8 610.37
31 40 .6 573.84
32 4.5 538 .62
32 42 .6 480 78
33 3.3 437 .21
33 13 .3 406 .92

1 ‘
Fourth contact; 2 34 44 .7.

( 49 )

Right ascension of the sun, Aug. 30,12 M.T. Gr. ag = 158°10'44".94

Declination eRe ty Riess + do= 9° 9'33'19
Right ascension of the moon ,, __,, # a@¢ = 157°42'47".95

(HANsEN-NEWCOMB),.
Declination ee oe oe ae) - do = 9°53’ 3'.48

(HANsEN-Newcoms).

Each observation gives an equation of condition for the determin-

ation of the corrections A of the elements of the sun and moon.
Let these corrections be successively

AR, Ar, Aag, Aa, Ade, Ady Ax,

then we obtain by comparing the observed distances and chords

with those computed the following equations: (the coefficients have
been rounded off to two decimals).

EQUATIONS OF THE CHORDS.
I. Observations after the first contact. Obs.—Comp.
" "

+7.98 AR +7.97 Ar 47.1408 —3.20A3 1.67 a =+50.71 —10.36
4550 ,, 45.49 ,,- 44.88 enact ° 44.49 a. Seles as
4.88 , +4.87 , 44.30 ,, 1.93 , 44.01, +37.59 4+ 0-70
4410 , +4409, $3.56 , 1.60 , 1081, 432.40 414.7
eee aa, tt 10 4.39, 10.69 , -E 99.
ee et ete ee DC EO.GA.
eee tt As oe 1. C4, 0:59 $22:
eee eee oe ee = LOT | +b
19.50 , +248 , 42.06 , 0.92 , 40.43 , +46.66 — 1.12
412.99 , 19.9%, +4. , —0.82 ,, 40.38 ,, +20.03 + 4.06
49.9 , 429.18, +1.76 , —0.79 ,, 40.36 ,, +19.60 + 4.28
42.43 , $2.10, +1.68 4, —0.75 ,,° 40.34 ,, +413.42
49.08 , 42.05, $1.63 , 0.73 , +0.33 , +4+15.60 +1.4
ope 4. ST, «6070 | 40.81 4, © 444.57

+H14.99 , 44.95 ,, 44.53 , —0.69 , $0.30, +17.85 + 4.4
44.94 , 41.90, +148 , —0.66 , 40.299 ,, +45.73 + 2.82
H4.89 , 41.85, 31.43 , 0.64 , 40.27 ,, +441.49 — 0.97
44.85 , -H.8i, 34.38 ,, —0.62 , 40.26, +442.97 + 0.87
44.82, +4.78,, 1.35 ,, —0.60 , +40. , 416.44 + 4.68
44.80 , 4.76 , $1.32 , —0.59 , 40.2 , -+440.08 — 1.56
ewes io. +100 5 —058 , 10.2% , -H3.51 41.19
eee ne te 057. 4 COR 2.97 4 1.77
1 , 11.69, 34.9% , —0.56 , 40.93,, +4 9.37 — 1.70
44.0, 31.67, +4.3 , 0.55 , 0.22, 40.37 — 0.50
44.69 , 34.65, $4.21 ,, —O.54 ,, $0.22, +42.71 + 2.06

Proceedings Royal Acad, Amsterdam, Vol. IX.

‘ t
44.52 AR +4.46 ar—0.99 Aw +0.49 AS 40.16 ar —=— 5.39 4 0.51

44.57

44.61
1.62

Il.

( 50 )

Observations before the last contact. Obs.—Comp.

" U

oO (—9

—1 05 ~.,.,. 0.52 .. OS 5 = 5. Ope cer
S409 %° 24058. Je Peo 419-255 eee ee
LAO fo CG AL > O Gaigigoe te) 7 Sn epee
ae (ares ess ee 3 (Og ee >

24 dao eer, eee, Se 4g ee
B56) 3051.45 1 eee
ss SRE RD OO AIS eae ss bf
oy 5 OR Jalgten + groan ee
1 0482" 5 0 se Sa
4905) SROs re 4 aen) Ae 77s ae eee
Oe ee ei ate eee ese
oo 40's SS 8 ee
4 yy) e5 eae 2 ore =e
Mog 1050. ey. hee ee
1.50" “S, -4b0:98 5 epee, Whe ae
wo 095 -a.90- ga ee
4.60. 4 4058 ,- 40-30 9) -— £88 ee
1.64 ~ , 40.80 4) olga 413.98 2 ee
69 3! 30.829 3) 332 = aa
A .h <= 2055, S084 =. ee ae
41.79. 087 40.33 = a ee
AS ls 0.9 ba 0 eee
i GS = Ses SB a
8.0 3. 40008. S20 abs ee
8.08; 5; A Pe ene 3 4 Glag eee
» 4406-5 0ae 4, | 4708 “Snes
2.095, Sd oo eajae a
9.37 |, habe os, 18 3. ee

—2,.52 ae je Ab ae
9.65 5 4 | 405s) 15.97 sae
—9:81 °°, [ASR S6 o> Poe | Sg 'a6: nea

—3.00 6 ab 0.63 a
—3.39 , 44:64 4° 40.71 ,, —18.h1, ae
eM ole S076. 96 ee
—3.92 ~ 44.85.45 -40.81-.,' —38)e7 =e

(51 )

Equations of the contacts
I AR+ Ar + 0.903 Aa—o — 0.405 Ad&—o = + 3".78
TI AR— Ar — 0.9668 Aac_a — 0.2007 Adhi—o +
4+ 0.0004 A?a_o — 0.0036 AaAd + 0.0091 A*h_o = — 6".52°)
HII AR— Ar + 0.3085 Ago — 0.9489 Adi_o +

— + 0.0104 A?a—o + 0.0068 AaAd + 0.0012 A*h—o = 4+ 4".02
IV AR+ Ar— 0.889 Aq—o + 0.435 Adio = — 11.18.

A mere glance at the equations derived from the distances of the
chords shows the impossibility to derive from them all the unknown
quantities. On account of the proportionality of the coefficients
Wwe may use one single equation instead of the first 25 equations
after the 18* contact; the same for the 35 others. In order to diminish
the weight of the observations immediately after the first and before

the last contact —- when the chord is less sharply defined and varies
rapidly — I have formed the two normal equations not according

to the method of least squares but simply by addition.

We obtain the following equations:

68.1(/AR+ Ar)+56.2Aa—25.2Ad—-+ 489".46 — 0.35(A R-Ar) —- 12.9Az
—81.6(AR+ Ar)+65.1Aa-31.6Ad=+397".87-+0.24(AR-Ar)+12.8Ax
whence :

AR + Ar=-4 1".05 — 0.015 Ad — 0.003 (AR — Ar) — 0.16 Ax.

Aa= + 7.428 + 0.465 Ad — 0.001 (AR — Ar) — 0.02 Az.
Neglecting the last terms, we find for the result from the equations
derived from the length of the chords:
AR+Ar=+4+1".05 — 0.015 Adio
Aaeo = 4+ 7.428 + 0.465 Ad_o.
From the equations of the 2°¢ and 38'¢ contact we derive:
Aa_© = + 7".793 + 0.464 Ado.
Aaeo = + 7.13 + 0.667 (AR — Ar)
Ady_©@ = — 1".43 + 1.437 (AR — Ar).
And lastly the equations of the 1s* and 4 contact yield:
Aa—o = + 8".35 + 0".468 Adi—oO
[AR + Ar = — 38".78]

The latter result for AR -+ Ar, which differs entirely from that
found above is little reliable. We can entirely account for it by
assuming that the first contact has been observed too late and the
last contact too early. It can hardly be doubted that the 1s* contact

1) It is not allowed (as it is generally done) to neglect the quadratic terms
in the equations of the 224 and 3°4 contact, because the corrections Az and jd, as

compared with the distance between the centre of the sun and that of the moon,
(in this case 46") are too large.

( 52)

is recorded too late because the eclipse began earlier than was expected
and in consequence took me by surprise. As an evidence that the
time of last contact was given too late there is an instantaneous
photograph of the sun (diameter = 10cm.) taken at the very moment
when I gave the signal ‘top’. This plate shows a small impression
on the limb of the sun.

To enable me to compare the obtained results, Messrs. Ta. Wor
and J. D. Lucas kindly put at my disposal the results of their highly
interesting observations of the 2°¢ and the 3 contact, made at Tortosa
by means of sensitive selenium elements. (See for this Astron. Nachr.
N°. 4071). They found:

beginning of totality 1" 16™ 15s,6
end me be ses ee es B
which yield the following equations :
AR — Ar— 0.9650 Aa—o — 0.2117 Ado + 0.0004 Ata_o —

— 0.0039 Aa Ad + 0.0092 A?h_¢ = — 5".73
AR — Ar + 0.3063 Aa_o — 0.9493 Adio + 0.0105 Ata_o +

+ 0.0069 Aa Ad + 0.0012 A?*h&_O© = + 4".10
whence

(A)

Aa_o = + 6".42 + 0.653 (AR — Ar)
Adio = — 1".76 + 1.404 (AR — Ar).
When we subtract the two equations A from each other we get :
Aa_O = + 7.238 + 0.465 Ad_o,
which agrees exceedingly well with the result of the chord equations
Aa = + 7'.428 + 0.465 Ad; but it also appears that it is impossible
to determine Aa, Ad and AR—Arv separately from the combination
of the contact and chord equations.

In the derivation of the final result we have accorded the same
weight = 1 to the results of the chord measurements and to those
of the contact determinations made by Wu1ir—Luwucas, and the weight
4 to my observations of the 2.4 and 3"4 contact. Thus we find, leaving ~
out of account the first and the fourth contact :

AR + Ar= + 1".07 — 0.02 (AR—Ar)
Au_o = + 6".66 + 0.66 (AR—Ar)
Ad_O = — 1".65 + 1.42 (AR—Ar).

The last column of the chord equations contains the deviations in
the sense of observation — computation, which remain when we sub-
stitute these numerical values. The mean error of the first 25 obser-
vations (excluding the first) amounts to + 2."53; that of the last 35
(excluding the last) is + 2."21.

(53 )

Chemistry. —. “On the osmotic pressure of solutions of non-electro-
lytes, in connection with the deviations from the laws of ideal
gases.” By J. J. van Laar. (Communicated by Prof. H. W.
Bakuuis RoozeBoom.)

Communicated in the meeting of April 27, 1906).

1. By H. N. Morse and J. C. W. Frazer') very accurate
experiments were recently made on the determination of the osmotic
pressure of dilute sugar solutions in water. The solutions had a
concentration up to 1-normal, and as ¢ is then about */,, [the
association factor of the water is viz. at 18° C. about 1,65, so that
in 1 L. of water about 55,6: 1,65 = 34 Gr.mol. of water (simple
and cemplex molecules) are present], the difference between the
exact expression — /og(1—.) and the approximate value « | formula
(2)| is not yet appreciable. It is however not so with the difference
between the molecular volume of the solution v = (1 — x) v, + 2,
(v supposed to be a linear function of 2, about which more presently)
and the molecular volume of the solvent v,, when v, (the molecular
volume of the dissolved sugar) cannot be put equal to v,. We shall see
that this difference for 1-normal solutions amounts to 19°/,, so that by
means of the experiments we can very well ascertain, if we have to
make use of v or of v,. And these have really taught us, that the osmotic
pressures measured agree (and even with very great accuracy) with
‘the calculated values, on/y when v, is put in the numerator, and
not v. This harmonizes therefore perfectly with what I have repeatedly
asserted since 1894). (What I have called above v, for the sake of
symmetry, was formerly always indicated by v,). Not the molecular
volume therefore of the whole solution, but the molecular volume
of the solvent m the solution. And this deprives those of their last
support, who*in spite of all evidence (for not the dissolved substance,
but the so/vent brings about that pressure) persist in trying to explain
the osmotic pressure by a pressure of the molecules of the dissolved
substance comparable with the gas pressure. If such a thing could
be thought of, v should be taken into consideration and not v,, for
the molecules of the dissolved substance move in the who/e volume v
and not in the volume v,, which is perfectly fictitious with regard
1) Amer. Chem. Journ. 34, 1905, p. 1—99. See also the extensive abstract
N? 274 in the Phys. Chem. Centralblatt IIIf (1906).

2) See inter alia my previous paper on this subject in These Proceedings, May
27, 1905, p. 49. (Some remarks on Dr. Pu. Kounstamm’s last papers).

(54 )

to the solution, which would be equal to v only when v, happened
to be equal to 7.

2. In order to compare the results, found by Morse and Frazer,
more closely with those for the osmotic pressure already given by
me in 1894, we shall return to its derivation for a moment, chiefly
in order to ascertain on what limiting suppositions this formula holds.

With equilibrium between the pure solvent (concentration 0,
pressure p,) and the solvent in the solution (concentration c, pres-
sure p) [ the dissolved substance is nowhere in equilibrium, for it
is supposed that there is a membrane impenetrable to it] the molecular
thermodynamic potentials must have the same value. Hence ’) :

Ut, (2, p) = Hy, (9; Po)
Now in general :

OZ
Bb, = = C,— 96,4 RT loge, ,
On,
ln kT (log PA AY De = Cs == and
1
00
6, = =; © being given by
On,

O = | pdv — pu — RT Zn, . log = n,.

For binary mixtures of normal substances we may now introduce
the variable z and we obtain ({n, is now =—1, so that the term
with /og =n, vanishes), as may be supposed as known:

0 0
po Cf nee + p ape e al + RT log(l1— x), . (1)
; Oa Ox

when @ is written for f pdv by way of abbreviation.

This expression is perfectly accurate for the above mentioned
mixtures. For the further caleulation we now introduce the idea

“ideal” mixtures. They are such as for which the influence of the
2

pas, and

two components inter se may be neglected. Then

0?v
becomes a /inear function of a. But also — 0, so that v becomes
av

1) The following derivation is only different in form from the cited one in these
Proceedings.

( 55 )

also a linear function of x. We shall further demonstrate this in
§ 6, and show that in the case of such mixtures:

/ a. o is a linear function of x
| b. v 3? > »> > > >
Cc.

a
| b 29> 9) >? >> >> 99
d. the heat of mixing is = 0,

so that we may say: ideal mixtures are such for which the heat
of mixing is practically = 0, or with which no appreciable contraction
of volume takes place, when 1—wz Gr.mol. of one component is
mixed with x Gr.mol. of the second.

The conditions a, 6, ¢ and d are simultaneously fulfilled, when
the critical pressures of the two components are by approximation
of the same value.

wo -
3. For w— xz — we may now write w,, as »=(1— z)o, +

0a:
Ou

Ow w dv
—' 5 —? 3 Te é é I— T— —2,,
/ai 3) /,# =) | In the same way v —a v,

0? 0
+ 2w,, when a =U: | Otherwise evidently wo —2« = =o, —
&

and we get:
u, (@,p) = C, — w, + pe, + RT log (1 — 2)
u, (0, Pp.) = C, meet + Pos

always when v, and w, are supposed to be independent of the
pressure. For else w, and v, would have another value at the
pressure p than at the pressure p,. We must therefore also suppose
that our liquids are mmcompressible. But there is not the slightest
objection to this supposition for ordinary liquids far from the critical
temperature (and there is only question of such liquids in discussions
on the osmotic pressure). Only when « draws near to 1, and so the
Osmotic pressure would approach to o, v, (and so also w,) must no
longer be supposed to be independent of p.
By equating these two last equations, we get:

pr, + RT log (1 — z) = p,r,,

hence

( 56 )

ee lg
Re Peni Pol a Aa eae 5 patie adie aD

the expression already derived by me in 1894. *)

1) Cf. Z. f. Ph. Ch. 15, 1894; Arch. Teyler 1898; Lehrbuch der math. Chemie,
1901; Arch. Teyler 1903; Chem. Weekbl. 1905, N°. 9; These Proceedings, June
21, 1905.

In the original Dutch paper another note followed, which Mr. van Laar has
replaced by the following in the English translation.

A conversation with Dr. Kounstamm suggested the following observations to me.

db
Dr. Kounstamm finds (These Proceedings, May 27, 1905) the quantity Uo ae

in the denominator of the expression for x. This is quite correct, and harmonizes
perfectly with the general expression, which according to equation (1) on p. 54,
would also have been found by me for non-linear variability of v.

Then we should viz. have:

Ov Jee Se ae ( — 0
p(e—#5- = Pe dn = — BL log (12) + (o# 57) — On

: (63) :
where, when calculating @ — x — by means of van per Waats’ equation of state,

Oa
0 (v—b :
& =O) appears, in consequence of which p (v —2x

also a term — px ) occurs

in the first member.

; 2 ae Ov db
Now it is of no importance whether v is diminished by ges or by. z as

a da’

0 (v—b)
ae approaches to O both for small and for very large values of p. I therefore
obtained a correction term in the denominator, in connection with the size of the
molecules of exactly the same value as Dr. Konnstamm. That this did not always clearly
appear in my previous papers, is due to the fact that I then always introduced the

approximation v — x <—
Pp ag

=v, which was perfectly justifiable for my purpose.

(0°v

F dv 1/ 9
or as V— © =%,— 1,2?
Ox : a Oe

) — etc., this is sufficiently accurate for prac-
1

tical purposes. (for ideal mixtures, where v is a linear function of <, it is of
course quite accurate).

Yet in a so early paper as the one cited by K. of 1894 (Z. f. Ph. Ch. 15,
p. 464) it is clearly to be seen that the result obtained by me agrees perfectly
with his. For it says (line 4 from the top) that va’ (the index a’ is there always

Ov
used for the liquid phase) =~ _ But this is in the z-notation nothing but
Na!

v—ax-=—, the physical meaning of which is: the molecular volume of the water

Ox
in the solution with the concentration 2.

The phrase occurring on page 466: “und niemals etwa va’ — b im Sinne etc.”
refers there to the well-known attempts of Ewan and others. The same is the
case with the phrase in the paper on non-diluted solutions in the Ch. Weekblad
of June 7th 1905: “Ook heeft men getracht, etc.” (p. 5).

0 v

(57 )

We repeat once more: this expression holds from «—0 to «—
near 1, when the following conditions are satisfied :

a. the solution is an ideal binary mixture of normal components:

6. the solution is practically imcompressible.

Then (2) represents the additional pressure on the solution, in order
to repel the penetrating water (the so-called ‘osmotic’ pressure).

As however in all the experiments made up to now water was
the solvent, hence an anomalous substance, (2) must not be applied to
solutions in water without reservation. It is, however, easy to show
that the influence of the association does not play a part before
the term with 2’ (justas the influence of the two components inter
se), so that in the above experiments, where xz? may undoubtedly
be neglected (cf. §1), formula (2) may certainly be used.

Let us, however, first reduce it to a form more practical for use.

4. Let us write (2) for this purpose :

i i RT
ee kk =) — 2 (1 + 4 2), Fo eehey eee)

1 1

which is more than sufficient for solutions up to 1-normal. Let us further
assume that ¢ Gr. mol. are dissolved in 1000 Gr. H,O (called by
Morsg and Frazer “weight-normal solutions’), then :

nom Mk: c

— 84fe 1+’

when we put */,,c = c (84 = 55,6:1,65 is the number of Gr. mol.
H,O in 1000 Gr. at 18° C; ef. § 1).

We find then:
Reh: ¢. L4y é
O == = ae
vy, lie ae ye ns

or when we restrict ourselves to terms of the second degree with
respect to c’:

RT ed eat
nx =-——e (I — '/,c) = — — (Il — */,, ¢).
v, v, 34

In this A= 82,13 (cc.M., Atm.), and v, —1001,4:34 cM’ at

: RT
18°. For =

we therefore,find at 18° C.:

VY
RT 82,13 « 291,04
ee A 1 on 87:
o4v, 1001,4
hence
m1g° = 23,87 c(1—0,015c) Atm. . . . . (26)

We see from the calculation, as we already observed above, that

( 58 )

the influence of the association of the solvent is only appreciable in
the term with c’. If water were a normal liquid, we should have
had 7/,,, ¢ instead of '/,, c = 0,015 c. (4 c’ would then be = */,X1/,,,c)-

Let us now consider what the last expression would have become
for 2,,0, when not v, had occurred in the denominator, but the
molecular volume of the solution v.

When ¢c Gr. mol. are dissolved in 1000 Gr. H,O, then the
total volume will be (at 18°) 1001,4+190c¢ ccM. [For 1 Gr.
mol. = 342,2 Gr. of melted sugar occupies a volume of 190 ccM.
at 18° (density = 1,8)].

Altogether there are now 34-+c Gr. mol., hence the molecular
volume of the solution will be:

_ 1001,44+190¢ 10014 140,19¢
Pe lestele) oF Be esses
For v7, we found however above:

_ 1001,4
ee
so that the value of 7,,° with v in the denominator instead of »v,
would have become:

1+ 0,03 ¢

W.0 = 93.87 p (luis
Fist et aaa 02 Sn aa
ise;
1+ 0,015
x, ,° = 23,87 ¢ pT ay
14 0,19¢

For (weight)normal solutions (¢ =1) we should therefore have
found instead of 2,,° = 23,87 (1 — 0,015) = 23,51 Atm., 2,,°
1,015
1,19

Now Morse and Frazer found 24,52 Atm., which is considerably
nearer the theoretical value 23,51 Atm. (with v, in the denominator)
than near the inaccurate expression with v in the denominator ’).

So it is out of the question that the molecules of the dissolved
substance should exert a certain pressure comparable with the gas-
pressure, for then the volume of the solution as such, viz. v, would
have to be taken into account, and not the in that solution perfectly
fictitious molecular volume of the solvent 2.

== 23,01 X = 20,36 Atm.

5. But there is more. We shall viz. derive the expression for
the pressure which would be exerted by the dissolved molecules,

1) With 0,5-normal the two values would have been 11,85 and 10,98 Atm.,
whereas 12,08 Atm. has been found experimentally.

(59 )

when they, according to the inaccurate interpretation of the osmotic
pressure, could move free and undisturbed throughout the space of
the solution.
VAN DER WAALS’ equation of state, viz.
RT a

gives for the rarefied gas-state:
Br » aad «i RT b a: kT
Pp —_ — — —— | — —[{ l 4. a= ——s =),
v | v—b v v v v

: 1
when we again content ourselves with terms of the degree —.
z=

Let us now write:
a

ae eee
RT uf
then
RT *)
) i a Seay 2
Vv Vv

where v now represents the volume, in which 1 Gr. mol. of the
dissolved substance moves. This volume is however evidently (ef.
also § 4):

__ 1001,4 + 190

’

or ‘
1001,4

(1 + 0,19 e),

so that we get:
& BL ¢ 1 yc
* = 10081 (1 ..0,19 6) 1001,4 (1 + 0,19 c)
Y

ar
or as ——_— = 23,87 is (c.f. §4), and with y’ = i001:

1001,4

— 23,87 Bee ee
, Coe ne (°)

and this is an altogether different expression from (2°). Not only is
v, replaced by v (which gives rise to the factor 1 + 0,19 c), but-
we also find 1 — y'c instead of 1—0,0J5c. In this y' is different
for every dissolved substance, dependent on the values of a and ,
whereas the coefficient 0,015 has the same value for all substances
dissolved in water, independent of the nature of the dissolved substance
(c.f. § 4). Also the coefficient 0,19 depends on the dissolved substance
on its molecular volume). Moreover y' depends also on 7’on account

( 60 )

of a: RT. Except with H,, where y is negative at the ordinary tempe-
rature, y is everywhere positive. But at higher temperatures its value
is reversed, and becomes negative.

So, when comparing (2°) and (3), we see clearly, that it is out
of the question that the so-called osmotie pressure should follow the
gas Jaws. Only with c =O this would be the case, but for all other

values of ¢ the deviation for the osmotic pressure is altogether different

from that for the gas pressure. This is still more clearly pronounced,
when we compare the original formulae. For the osmotic pressure
viz. the equation

yi fae i
x = — (— log (A—2z)) = — «(1 + 4, a?
Vv, + Y%;

holds; for the gas pressure on the other hand:

RT
(>) vU

so that the deviations from the gas laws (at the ordinary tempe-
ratures) are even in opposite sense from the deviations of the osmotic
pressure for non-diluted soiutions.

In view of these facts it is in my opinion no longer possible to
uphold the old conception of the osmotic pressure as arising in
consequence of a pressure of the molecules of the dissolved substance
comparable with the gas pressure. The molecules of the dissolved
substance have nothing to do with the osmotic pressure except in
so far as they reduce the water in the solutions to another state of
concentration (less concentrated), which causes the pure water (concen-
tration 1) to move towards the water in the solution (concentration
1—a) in consequence of the zmpulse of diffusion. On account of

; : ted
this a current, of which the equivalent of pressure = —(-log (1-2)),
: =

arises in the transition layer near the semi-permeable membrane,
which current can only be checked by a counterpressure on the
solution of equal value: the so-called osmotic pressure.

This is in my opinion the on/y correct interpretation of the osmotic
pressure.

As I already observed on former occasions, we might just as well
speak of an “osmotic” temperature, when the impulse of diffusion
is not checked by pressure on the solution, but by cooling it. For
at different temperatures the temperature functions C, (ef § 2) are
no longer the same in the two members of

(4, (2, T) == i, (0, Ty),

( 61 )

whereas the terms pv are now the same. In this case 7’ would have
to be < 7;,, because the temperature exerts an opposite influence
on the change of uw from the pressure.

In consequence of the term RT’ log (1 — x), , (x) will be < uw, (0). u

Ou
must therefore be increased. Now. =v,, hence positive, whereas
P

Wy) ;
== (=) = — (e, + pr,), so negative. So the value of m, («), which is

too small in consequence of x, can again be made equal to that
of yu, (0), either by mcrease of pressure (“osmotic pressure), or by
lowering of the temperature (‘“osmotic” temperature).

It would, however, be advisable to banish the idea “osmotic
pressure” altogether from theoretical chemistry, and only speak of
it, when such differences of pressure are actually met with in case
of semi-permeable walls (cell-walls, and such like).

6. Appendix. Proof of some properties, mentioned in § 2.
a. In a previous paper in these Proceedings (April 1905) I
Ov

derived for the perfectly accurate expression | equation (4), p. 651):

db 1 (v—b)*da

Oz 1 22/,(v>—b)? —
RT v’

db d
With — =f and — Va, in which 8 = 6, — 6, and a= Va,—ya,,
wv

dx d.
this becomes :
2af/a (v—b)?

dee REP?
ae 1 2%/e @— 5)"
es a

: , db
And now we see at once, that this passes into 2 or —. when
¢ &

BYa=avr,

aya. x
For then in the numerator becomes equal to ¢/, in the de-

2 9

: Ov db '
nominator. But when ———,, then also —- —O, as
Oz dx 0a?

is a linear function of z.

2b
ae = (0, and v
dz?

( 62 )
[We above derived the condition BY/a=eav from the general
Ov ae
’ Ox
would immediately follow from this by differentiation, and then “

0
expression for — If we knew this condition beforehand
Hi

0
would not be necessary to start from the general expression for =
b. On p. 651 [equation (5)] of the paper cited the perfectly
general expression :
7@ 2 (av—BYa)?
da? yp? 1 2th @= 5)"

at
; °O
was derived for eu, which becomes therefore =O, when again
&
0°O 07v
BYa=av. Now O= | pdv — pp=w—pvr. And as oe and 5,7 are
& «

3
F) ~- will be = O, in other words wo
1h;

both —O when av= Bya, also

is a linear function of z.

ce. The heat of dilution. It is given by the formula

L,= Be Sp vale: 3) —* (0)
oT

This is viz. the so-called differential heat of dilution per Gr. mol.
of the solvent when dn Gr. mol. solvent ie a are added to
m+n

a solution consisting of m Gr. mol. dissolved substance and » Gr.
mol. solvent.
This becomes [see equation (1)]:

O-r Ov |
Ip=— 055] 5 —(o- 9) + p(o— #3) +0, —pn| |

0 0
If = 0, then w— 2 a @,; and vy — x ~ will be = v,, when
On? Ow Ow
0?
53 0. But then ZL, = 0. q.e.d.

And hence also the total heat of mixing will be = 0, when z Gr.
mol. of the 2"¢ component are mixed with 1—wz Gr. mol. of the
1st component.

d. The peculiarities mentioned in §2 under a, 6 and d, which

( 63 )

characterize the so-called zdeal mixtures, are therefore all satisfied
when
BYa=eav.
This yields:
B[V a, + xa] = a[b, + 28],

when it is permissible — for liquids far from the critical temperature
— to replace v by 6. Hence we get:
BYa,=ab,,
or
(6, hei b,) Ya, = (Ya, se Va,) b,,
or also
ba, = 6, a.,
hence
Ya, Va, ;
b, ee b, :

from which we see, that the case of ideal mixtures occurs, when
the critical pressures of the components have the same value.
e. Finally
%fa\ 2b, Va, —b, Ya,)?
da? 5) 6°

a . . .
so we see that also z will be a linear function of z, when

b, Va, = b, Ya, or p, = p,. In this way also c of § 2 has been proved.

(June 21, 1906).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday June 30, 1906.

DEG

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 30 Juni 1906, Dl. XV).

Ca, a eA TS.

L. E. J. Brouwer: “Polydimensional Vectordistributions”. (Communicated by Prof. D. J.
KorrtTEewEsG), p. 66.

F. M. Jarcer: “On the fatty esters of Cholesterol-and Phytosterol, and on the anisotropous
liquid phases of the Cholesterol-derivatives” (Communicated by Prof. A. P. N. Francuimonr),
p- 78.

F. M. Jancer: “Researches on the thermic and electric conductivity power of crystallised
conductors” I. (Communicated by Prof. H. A. Lorentz), p. 89.

H. W. Bakuuis Roozesoom: “Three-phaselines in chloralalcoholate and anilinehydrochloride”,
p: 99:

H. Haca: On the polarisation of Roéntgen rays’, p. 104.

P. van Rompurcu: “Triformin (Glyceryl triformate)’, p. 109.

P. van RompBurGH and W. van Dorssen: “On some derivatives of 1-3-5-hexatriene’’, p, 111.

L. E. J. Brouwer: “The force field of the non-Euclidean spaces with negative curvature’.
(Communicated by Prof. D. J. Korrewee), p. 116.

A. Paxnekork: “The luminosity of stars of different types of spectrum”. (Communicated by
Prof. H. G. vAN DE SANDE BAkuvuyZEN), p. 134.

Errata, p. 148.

Cor

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 66 )

Mathematics. — “Polydimensional Vectordistributions’’.*) By L. E. J.
Brouwer. (Communicated by Prof. D. J. Korrrwec.)

Let us call the plane space in which to operate S,; we suppose
in it a rectangular system of coordinates in which a C, represents
a coordinatespace of p dimensions. Let a ?X-distribution be given
in S,; ie. let in each point of S, a p-dimensional system of vectors
be given: By XA, 2°. a, we understand the vector component parallel

to C, indicated by the indices, whilst as positive sense is assumed
the one corresponding to the indicatrix indicated by the sequence
of the indices. By interchanging two of the indices the sense of the
indicatrix changes, hence the sign of the vectorcomponent.

Theorem I. The integral of »X in SS, over an arbitrary curved
bilateral closed S, is equal to the integral of »+! Y over an arbitrary
curved S,41, enclosed by S, as a boundary, in which ?+!Y is
determined by

OX,

Brae
i “Qo “Gs Ip+1
Ya, Pentre Te — a
ae ho er i p=, Ox,
Nn

SS Ea

n p+l
where for each of the terms of the second member the indicatrix
(<4, q+ + @y, Mg) has the same sense as (@, @,...@ ,41). We call

the vector Y the jirst derivative of 1X.

Proof. We suppose the limited space S,41 to be provided with
curvilinear coordinates w,...u,41 determined as intersection of curved
C,’s, i.e. curved coordinatespaces of p-dimensions. We suppose the
system of curvilinear coordinates to be inside the boundary without
singularities and the boundary with respect to those coordinates to
be everywhere convex.

The integral element of »+'Y becomes when expressed in differen-
tial quotients of »X:

Ofc,

nea aa ped ey ae
qa" °g, P
> a —— : : du,. . « dup4i.
— q .
pl %qy 7p Vaz, Otay
Ou es Oup+1

1) The Dutch original contains a few errors (see Erratum at the end of Ver-
slagen 31 Juni 1906), which have been rectified in this translation.

( 67 )

We now unite all terms containing one of the components of
pX, e.g. Xio3...p- We then find:

Opt Ox, 0xp
Ou, ae all Ow,
Oe a oe : ; :
See eE | re dy 41 -|-
Oa p41 . . .
Oa pti Ox, 0a,
Oup+1 ie a Ouy 41
0x, +2 Oz, dz~ |
a: Se |
ox : a =
ie aria du, < .. dtp +
Oxp49 . . .
Otp4o O02, 0p
Ou, +1 Oup+i be ee Oty +1 |

+ ...(n—>p terms).

If we add to these the following terms with the value 0:
02, Ox, 0x,

Ou, Ou, Ou,
ox : ‘ ;
Beene : 3 E a ditp4i +
Ox, ‘ f
On, Oz, dx,
gee Ga ae
Ox, Ox, dz,
Ou, cg Oe Ou,
0X93... ; ‘
eae fa tng +
Oz, Ox, du,
Dee Cues Oup i

+... (p terms),

the n-terms can be summed up as:

0Xj03 Pp Ow 0x,
Pg LE Ree ie enter) —— du
Ou, st Ou, ae
0X93... p Ow Oty
eae d 1 d pa || en Le ale ee Ce wee: lu J
Oup+ ee Ou41 i Ouy +1 |

( 68 )

Let us suppose this determinant to be developed according to the
first column, let us then integrate partially each of the terms of the
development according to the differential quotient of Xj93,..: p> appearing
in it; there will remain under the (p-—+ 1)-fold integration sign
p(p-+1) terms neutralizing each other two by two. Thus for instance:

as they transform themselves into one another by interchangement
of two rows of the matrix-determinant.
So the p-fold integral remains only, giving under the integration sign

ee Ory |
— dl, «1.4... . —— du
Ou, ; Ou,
Xi23...p | ‘
+ ] sy 1 eg
——- du, ; . Ay}
Ou, 41 ert Ott, hi

to be integrated over the boundary, whilst in a definite point of that
boundary the /'® term of the first column gets the sign ++ when
for the coordinate wp the point lies on the positive side of the boundary.

Let us now find the integral of Xj23,..) over the boundary and
let us for the moment suppose ourselves on the part of it lying for all

( 69 )

ws on the positive side. The indicatrix is in the sense uv, wu, ... uv, 4, and
if we integrate Xjo3...p successively over the components of the
elements of boundary according to the curved C)’s we find:

dx, Ox
duty, ee tae duy,
| Ou, Ux,
. Aagsss we ; ;
4 = Z
[* nate Ox
1 J |
du, : Y du,
| Ux P Ux p
P |

where (@,414,--- 4) = (1423...p(p+1)); so that we can write
as well

| | Ox Ox
a See a dis
| Ou, Ou,
as FB Ox
dug—1 diig—1
Ug—l Ug—l
: f X23 ++p 3 3 |
wv & )
q=1,2...(P+1) | Augti - dugti |
Ug+l Ug+l
| |
dx, Ly
| du, 44 dees ||
| Duy 41 : Ou) 41 - +
Ow Ow
| | ee ae = ie |
du. :, Ou, |
orf Xias..y : 3 | .
: .
> Ox 0a |
eee: a oe dupt1 - |
0 UpHl P+ On byl p+

If we now move to other parts of the boundary we shall conti-
nually see, where we pass a limit of projection with réspect to one
of the coordinates w, the projection of the indicatrix on the relative
curved Cy change in sense.

So in an arbitrary point of the boundary the integral is found in the
same way as on the entirely positive side; we shall find only, that
for each coordinate w, for which we are on the negative side, the
corresponding term under the sign = will have to be taken nega-
tively, by which we shall have shown the equality of the p-fold

( 70 ) F

integral of »X over the boundary and the (p-+ 1) fold integral of
Rar over the bounded Sp +1.

We can also imagine the scalar values of »X set off along the
normal-S,,_,,’s. As such the integral over an arbitrary curved bilateral
closed S,—, can be reduced to an (x —-p- 1)-dimensional vector
over a curved S,—,+41, bounded by S,_,. If again we set off the
scalar values of that vector along its normal-S,—1, the veetor P—1Z
appears, which we shall call the second derivative of »X. For the
component vectors of ?—'Z we find:

Se! q x ial
ln. ee, ree

i= 0x
Sq ge pei” <n *q
The particularity may appear that one of the derivatives becomes 0.
> . : aw mM xr
If the first derivative of an "X is zero we shall speak ofan ,,_1X,

‘fe ‘ Mm >
if the second is zero of an 4X.

. . / . +1 yg
Theorem 2. The first derivative of a ?X is a” pA, the second a

—I x, - ; 2 >
‘ »X; in other words the process of the first derivation as well as

that of the second applied twice in succession gives zero.

The demonstration is simple analytically, but also geometrically the
theorem is proved as follows:

Find the integral of the first derivative of ”X over a closed S, 41,
then we can substitute for the addition given by an S,4: element
the integral of »X along the bounding S, of that element. Along
the entire S,:1 each element of those S, boundaries is counted twice
with opposite indicatrix, so that the integral must vanish.

The analogous property for the second derivative is apparent, when
we evaluate the integral of the normalvector over a closed S,—,41.

By total derivative we shall understand the sum of the first and
second derivatives and we shall represent the operation of total
derivation by vy.

h=n )
a
Theorem 3. 7? = — s ;
z 0x)?
h=

Proof. In the first place it is clear from theorem 2 that the
vector vy’ is again a ’X. Let us find its component Xj. ».
The first derivative supplies the following terms

( 71 )

Pe =x oe oP
iocts oh ee ’
q=p +1

where
u=p

0X 12...(u—1)(u +1)...p
—-\ +>
Poin == So

(+ sign for (ug 12... (u—1)(« +1)... pJ= 1 ~))
OXie...p

Oxy
So
u=p q="

0? Xo19..(u—1)lu-+1)...p
i > = ts Oa, Vag oa
u=1 g=p+1
(- sien for (ug 12...(u—1)(u+1)...p) = (71 ~?))
g=n
0? Xj9., P
ae 02,"
q—ptl
The eu derivative supplies the terms

: OZ 10. (u—1)(u+1)...p
= St
’, > 02,

(+ sien for (w12...(w—1)(w+1) .. p) = Giz =. p)

or for (qu1...(u—1) (w+1) ... p) = (g 12 .-P)) ;

aC OXo12.. (u—1)(u+1)...p
where Z12...(u—1)(u+1)...p = == i a ae
ey

g=pt1

= (- sign for (w 12 ...(w—1) (uv+-1)... p) = (12 ~P).
Ly
u—p qn

07 Xo12 ..(u—1)(u-+l)..p
a

02,0ag
u—!- -g—p-1

= sien for (qul ..(w—1)(u+1)..p) = (q12. ”))

up
_
0° X12

Ee ae as

ef |

( 72)
The terms under the sign SZ of 7, are annulled by those of
T.,, so that only

0? Xj. plies ES ’

ar Se Ox ;?

te
is left.

Corollary. If a vectordistribution PV is given, then the vector-

wa Var : ‘ ;
distribution = Dyn” integrated over the entire space, has for
C4 72 — Tr
second derivative V. (if 4,7"*—! expresses the surface of the
x—I_sphere in 5S,).

The theorem also holds for a distribution of sums of vectors of
various: numbers of dimensions, e.g. quaternions.

We shall say that a vectordistribution has the potential property
when its scalar values satisfy the demands of vanishing at infinity,
which must be put to a scalar potential function in S,.1) And in
the following we shall suppose that the vectordistribution from
which we start possesses the potential property. Then holds good:

Theorem 4. A vectordistribution V is determined by its total
derivative of the second order.

For, each of the scalar values of V is uniformly determined by
the scalar values of (7? /’, from which it is derived by the operation

Y dv
[ pee 2)r2—2 ,

Theorem 5. A vectordistribution is determined uniformly by its
total derivative of the first order.

For, from the first total derivative follows the second, from whieh
according to the preceding theorem the vector itself.

We shall say that a vectordistribution has the jield property, if
the scalar values of the total derivative of the first order satisfy the
demands which must be put to an agens distribution of a scalar
potential function in ,. And in the following we shall suppose
that the vectordistribution under consideration possesses the field
property. Then we have:

Theorem 6. Each vectordistribution is to be regarded as a total

') Generally the condition is put: the function must become infinitesimal of
order n—-2 with respect to the reciprocal value of the distance from the origin.
We can, however, prove, that the being infinitesimal only is sufficient.

( 73 )

derivative, in other words each vectordistribution has a potential
and that potential is uniformly determined by it.

Proof. Wet V be the given distribution, then

ar V V.dv
ae hin(n —2)r"—2
miepoeniar ror viv V, or /(VP)—V lV, or VPS.
Farther follows out of the field property of V7, that Pi is uniformly
determined as V—? of VV, so as 7 of V. So P has clearly the
potential property; it need, however, not have the field pruperty.
N.B. A distribution not to be regarded here, because it has not
the field property, though it has the potential property, is e. g. the
fictitious force field of a single agens point in S,. For, here we
have not a potential vanishing at infinity and as such deter-
mined uniformly. The magnetic field in S, has field property and
also all the fields of a single agens point in S, and higher spaces.
Let us call \'/ V the first derivative of PV and \2/ V the second;
we can then break up ?V into

Bd
Vf ms \ v a Ee AP Bay

(n—2)r"—2 \/ 1 pti
and
\7/V.dv ep a Py
M ke (2 —2)r—* ia etal \/ P, Pe r

From the preceding follows immediately :

Theorem 7. Each A has as potential a » V. Hach se V has
as potential ea, ;

We can indicate of the ae V the elementary distribution, i.e. that
particular ee V of which the arbitrary S, integral must be taken to

P
obtain the most general p+. V.
po. 543
For, the general »+1V is \2/ of the general V, so it is the

general |S, integral of the \2/ of an isolated (p + 1)-dimensional
vector, which, as is easily seen geometrically, consists of equal vectors
in the surface of a ?sphere with infinitesimal radius described round
the point of the given isolated vector in the Rp4i of the vector.

P eee Z
In like manner the general,—1V is the \1/ of the general ?~' V,

so it is the general’ S, integral of the \1/ of an isolated ?—' vector,
consisting of equal ?vectors normal to the surface of an "~? sphere with
infinitesimal radius described round the point of the’ given isolated
vector in the #,~p+:, normal to that vector.

From this follows:
Theorem 8. The general ?V is an arbitrary integral of elemen-
tary fields #, and £,, where:
p—!

es Spa Zdv Poot : ; :
| DF , where » Z consists of the »—! vectors in the
ky, (n—2)r"— ken (n—2)9"—2

surface of an infinitesinal P-'sphere Sp., . . . . . . (A)

p+l
p+l

Ydv
ae J f= where , Y consists of the ?+! vectors normal
(n— n—2

to the surface of an infinitesinal "~P—'sphere Spy. . . . (2)

For the rest the fields #, and /, must be of a perfectly identical
structure at finite distance from their origin; for two fields /, and £,
with the same origin must be able to be summed up to an isolated
Pvector in that point.

We can call the spheres Sp, and Sp. with their indicatrices the
elementary vorter systems Vo, and Vo-. A field is then uniformly
determined by its elementary vortex systems and can be regarded
as caused by those vortex systems.

We shall now apply the theory to some examples.

The force field in S,.
The field E,. The elementary sphere Sp, becomes here two points
lying quite close to each other, the vortex system Voz passes into
two equal and opposite scalar values placed in those two points. It

. , OS E é
furnishes a scalar potential ¥ in which y denotes the angle of the

radiusvector with the S, of Vo., i.e. the line connecting the two
points. The elementary field is the (first) derivative of the potential
(the gradient); it is the field of an agens double point in two di-
mensions.

The field E,. The elementary sphere Sp, again consists of two
points lying in close vicinity, the elementary vortex system Vo, has
in those two points two equal and opposite planivectors. The plani-
vector potential (determined by a sealar value) here again becomes

COs fp

~ 3 80 the field itself is obtained by allowing all the vectors of

(75)

a field £, to rotate 90°. As on the other hand it has to be of an
identical structure to /, outside the origin we may call the field
E, resp. H, ‘dual to itself”.

In our space the field £, can be realized as that of a plane,
infinitely long and narrow magnetic band with poles along the edges ;
the field “#, as that of two infinitely long parallel straight electric
currents, close together and directed oppositely.

The planivector (vortex) field in S,.

The field E,. The elementary sphere Sp. isa circlet, the elementary
vortex system Vo, a current along it. It furnishes a linevector

sin
potential == directed along the circles which project themselves

on the plane of Vo. as circles concentric to Vo,, and where @ is
the angle of the radiusvector with the normal plane of Vo-. The
field is the first derivative (rotation) of this potential.

The field E,. Tie elementary sphere Sp, is again a circlet, the
elementary vortex system Vo, assumes in the points of that circlet
equal ‘vectors normal to it. The *V-potential consists of the *V’s
normal to the potential vectors of a field £,; the field ZL, is thus
obtained by taking the normal planes of all planivectors ofa field £,.
As on the other hand /, and £, are of the same identical structure
outside the origin, we can say here again, that the field LZ, resp. EZ,
is dual to itself.

So we can regard the vortex field in S, as caused by elementary
circular currents of two kinds; two equal currents of a different kind
cause vortex fields of equal structure, but one field is perfectly
normal to the other.

So if of a field the two generating systems of currents are
identical, it consists of isosceles double-vortices.

The force field mn S,.
The field E,. Voz gives a double point, causing a scalar
cos ; “ .
potential aes. where g is the angle of the radiusvector with the
-

axis of the double point; the derivative (gradient) gives the wellknown
field of an elementary magnet.

The field E,. Vo, consists of equal planivectors normal to a
small circular current. If we represent the planivector potential by the

. . . sin ‘ .
linevector normal to it, we shall find for that linevector ——~ directed
Tr

( 76 )

along the circles, which project themselves on the plane of Vo-
as circles concentric to Vo-, and where ¢ is the angle of the radius-
vector with the normal on the circular current. The field /, is the
second derivative of the planivector potential, i.e. the rotation of
the normal linevector.

According to what was derived before the field /, of a small
circular current is outside the origin equal to the field #, of an
elementary magnet normal to the current.

In this way we have deduced the principle that an arbitrary
force field can be regarded as generated by elementary magnets and
elementary circuits. A finite continuous agglomeration of elementary
magnets furnishes a system of finite magnets; a finite continuous
agglomeration of elementary circuits furnishes a system of finite
closed currents, i.e. of finite dimensions; the linear length of the
separate currents may be infinite.

Of course according to theorem 6 we can also construct the

1
scalar potential out of that of single agens points ee the second
oy

derivative of the field), and the vector potential out of that of rectilinear
1

elements of current (perpendicular tor x the first derivative of the
x

field), but the fictitious ‘field of a rectilinear element of current” has
everywhere rotation, so it is the real field of a rather complicated
distribution of current. A field having as its only current a rectilinear
element of current, is not only physically but also mathematic-
ally impossible. A field of a single agens point though physically
perhaps equally impossible, is mathematically just possible in the
Euclidean space in consequence of its infinite dimensions, as the
field of a magnet of which one pole is removed at infinite distance.

In hyperbolic space also the field of a single agens point is
possible for the same reason, but in elliptic and in spherical space
being finite it has become as impossible as the field of a rectilinear
element of current. The way in which Scuerine (G6ttinger Nachr.
1870, 1873; compare also Fresporr Diss. Gottingen 1873; Oprrz
Diss. Gottingen 1881) and Kuiniine (Crelle’s Journ. 1885) construet
the potential of elliptic space, starting from the supposition that
as unity of field must be possible the field of a single agens point,
leads to absurd consequences, to which Kiem (Vorlesungen iiber
Nicht-Euklidische Geometrie) has referred, without, however, proposing
an improvement. To construct the potential of the elliptic and
spherical spaces nothing but the field of a double point must be
assumed as unity of field, which would lead us too far in this

paper but will be treated more in details in a following com-
munication.

With the force field in S, the vortex field in S, dual to it has
been treated at the same time. It is an integral of vortex fields as
they run round the force lines of an elementary magnet and as
they run round the induction lines of an elementary circuit.

The force field in S, .

The field E,. Voz again gives a double point, which furnishes a
Ch eee. ; 3
scalar potential =a where @ is the angle between radiusvector and
pr ;

axis of the double point; its gradient gives what we might call the
field of an elementary magnet in 5,.

The field E,. Vo, consists of equal planivectors normal to a
small *—*sphere Sp,. To find the planivector potential in a point
P, we call the perpendicular to the S,—; in which Sp, is lying
OL, and the plane LOP the “meridian plane” of P; we call
gy the angle LOP and OQ the perpendicular to OZ drawn in
the meridian plane. We then see that all planivectors of Vo, have
in common with that meridian plane the direction OZ, so they can be
decomposed each into two components, one lying in the meridian
plane and the other cutting that meridian plane at right angles. The
latter components, when divided by the 2— 2»¢ power of their
distance to P, and placed in P, neutralize each other two by two;
and the former consist of pairs of equal and opposite planivectors
directed parallel to the meridian plane and at infinitely small distance
from each other according to the direction OQ. These cause in P

: : : : bet sin &p

a planivector potential lying in the meridian plane =c——. The
: pn—

field H, is of this potential the VY = \2/, and outside the origin is

identical to the field of an elementary magnet along OL.

The force field in S, can be regarded as if caused 1st. by magnets,
2°d. by vortex systems consisting of the plane vortices erected normal
to a small *—®sphere. We can also take as the cause the spheres
themselves with their indicatrices and say that the field is formed
by magnets and vortex spheres of »—2 dimensions (as in S, the
cause is found in the closed electric current instead of in the vortices
round about: it).

Here also fields of a single plane vortex element are impossible.
Yet we can speak of the fictitious ‘field of a single vortex” although

( 78 )

that really has a vortex i.e. a rotation vector everywhere in space.
We can say namely: ;

If of a force field in each point the divergence (a scalar) and the
rotation (a planivector) are given, then it is the V of a potential :
ik sali eal a2 ate ies ai this formula takes the field as an

k,(n — 2)r7—2 k,(n — 2) r*—*
integral of fictitious fields of agens points and of single vortices.

Crystallography. — “On the fatty esters of Cholesterol and
Phytosterol, and on the anisotropous liquid phases of the
Cholesterol-derwatives.”’ By Dr. F. M. Jagger. (Communicated
by Prof. A. P. N. FRANcHIMONT.)

(Communicated in the meeting of May 26, 1906).

§ 1. Several years ago I observed that phytosterol obtained from
rape-seed-oil suffers an elevation of the melting point by a small
addition of cholesterol. The small quantity of the first named sub-
stance at my disposal and other circumstances prevented me from
going further into the matter.

My attention was again called to this subject by some very
meritorious publications of BomerR*) on the meltingpoint-elevations
of phytoterol by cholesterol and also of cholesterol-acetate by phyto-
sterol-acetate. Apart from the fact that the crystallographic data
from ©. Micce led me to the conclusion, that there existed here an
uninterrupted miscibility between heterosymmetric components, a
further investigation of the binary meltingpoint-line of the two
acetates appeared to me very desirable, as the ideas of BOmER on
this point are not always clear; this is all the more important, as
we know that BoOmer based on these melting point elevations a
method for detecting the adulteration of animal with vegetable fats.
My further object was to ascertain in how far the introduction of
fatty acid-residues into the molecule of cholesterol would modify the
behaviour of the esters in regard to the phenomenon of the optically-
ansotropous liquid phases, first noticed with the acetate, propionate
and benzoate, with an increasing carbon-content of the acids. Finally
1 wished to ascertain whether there was question of a_ similar
meltingpoint-elevation as with the acetates in the other terms of the
series too.

1!) Bomer, Zeit. Nahr. u. Genussm. (1898). 21, 81; (1901). 865, 1070; the last
paper \with Winter) contains a complete literature reference to which | refer.

( ey

§ 2. In the first place the esters of cholesterol and phytosterol
had to be prepared.

The cholesterol used, after being repeatedly recrystallised from
absolute alcohol + ether, melted sharply at 149°.2. The phyto-
sterol was prepared by Merck, by Hessk’s') method from Calabar
fat, and also recrystallised. It melted at 137°. A microscopic test did
not reveal in either specimen any inhomogeneous parts.

First of all, I undertook the crystallographic investigation of the two
substances. The result agrees completely with the data given by Muces,
to which I refer. I have not, up to the present, obtained any measu-
rable crystals; on account of the optical properties, cholesterol can
possess only triclinic, and phytosterol only monoclinic symmetry.

Although an expert crystallographer will have no difficulty in
microscopically distinguishing between the two substances, the crystals
deposited from solvents are, however, so much alike that a less expe-
rienced analyst may easily make a mistake. I, therefore, thought it
of practical importance to find a better way for their identification
with the microscope.

This was found to be a very simple matter, if the crystals are
allowed to form on the object-glass by fusion and solidification,
instead of being deposited from solvents. Figs. 1 and 2 show the
way in which the solidification of the two substances takes place.

S vf)
SS

Fig. 1. Fig. 2.
Cholesterol, Phytosierol,
fused and then solidified. fused and solidified by cooling.

Phytosterol crystallises in conglomerate spherolites. Between crossed
nicols they exhibit a vivid display of colours and each of them is

‘) Hesse, Annal. der Chemie, 192. 175.

( 80 )

traversed by a dark cross, so that the whole conveys the impression
of adjacent interference images of monaxial crystals, viewed perpen-
dicularly to the axis and without circular polarisation. The charac-
ter of the apparently simple crystals is optically negative.
Cholesterol, however, presents a quite different image. When melted
on an object-glass, the substance
contracts and forms small droplets,
which are scattered sporadically and,
on solidification, look like little nug-
gets with scaly edges, which mostly
exhibit the white of the higher order.
That the microscopical distinction
in this manner is much safer than
by Miicer’s method, may be seen from
fig. 3 where phytosterol and choleste-
rol are represented as seen under the

Fig. 3. microscope, after being crystallised
Phytosterol and Cholesterol from from alcohol. A is cholesterol, B phy-

§ 3. Of the fatty esters, I have prepared the acetates, propionates,
butyrates and isobutyrates by heating the two alcohols with the pure
acid-anhydride in a reflux apparatus. A two or three hours heating
with a small flame, and in the case of the cholesterol, preferably in
a dark room, gives a very good yield. When cold, the mass was
freed from excess of acid by means of sodium hydrocarbonate, and
then recrystallised from alcohol + ether, afterwards from ethy]
acetate + ligroin, or a mixture of acetone and ligroin, until the melt-
ingpoint was constant. Generally, I used equal parts by weight of
the alcohol and the acidanhydride.

The jformiates, valerates, isovalerates, capronates, caprylates and
caprinates were prepared by means of the pure anhydrous acids.
These (valeric, caprylic and capric acids) were prepared synthetically
by Kantpaum; the isovaleric acid and also the anhydrous formic
acid were sold commercially as pure acids “KanLBAum”. Generally,
a six hours heating of the aleohol with a little more than its own
weight of the acid sufficed to obtain a fairly good yield. Owing,
however, to the many recrystallisations required the loss in substance
is much greater than with the above described method of preparing.

Both series of esters erystallise well. The phytosterol-esters in soft,
flexible, glittering scales; the formiate and the valerates present some
difficulties in the crystallisation, as they obstinately retain a trace of

( 81 )

an adhesive by-product which it is difficult to remove. The choles-
terol-esters give much nicer crystals; the formiate, acetate and ben-
zoate have been measured macroscopically ; the other derivatives
erystallise in delicate needles or very thin scaly crystals which are
not measurable; I hope yet to be able to obtain the butyrate in-a
measurable form '). In the case of the caprylate, the purification was
much assisted by the great tendency of the product to crystallise.
The purification of the capric ester was, however, much more diffi-
cult; at last, this has also been obtained in a pure state even in
beautiful, colourless, plate-shaped crystals, from boiling ligroin *).

The phytosterol-esters retain their white colour on exposure to the
light; the cholesterol-esters gradually turn yellowish but may be
bleached again by recrystallisation.

The determination of the melting points, and in the case of the
cholesterol-esters, also that of the transition-temperatures: solid —
anisotropous-liqiid, was always executed in such manner, that the
thermometer was placed in the substance, which entirely surrounded
the mercury-reservoir. Not having at my disposal a thermostat, |
have not used the graphic construction of the cooling-curve, in
the determinations, but simply determined the temperature at which
the new phases first occur when the outer bath gets gradually warmer.

As regards the analysis of the esters, nothing or little can be
learned from an elementary analysis in this case, where the formulae
of cholesterol and phytosterol are still doubtful, and where the
molecules contain from 28 to 37 carbon-atoms. I have therefore
rested content with saponifying a small quantity of the esters with
alcoholic potassium hydroxide, which each time liberated the cholesterol
or phytosterol with the known melting points. On acidifying the
alkaline solution with hydrochloric acid, the fatty acids could be
identified by their characteristic odour.

The esters were called pure, when the melting points, and in the
ease of cholesterol-esters, both temperatures, remained constant on
further recrystallisation.

1) I have even succeeded lately in obtaining the formiate in large transparent
crystals from a mixture of ligroin, ethyl acetate and a little aleohol.

2) The crystals of the caprinate are long, flat needles. They form monoclinic
individuals, which are elongated parallel to the b-axis, and flattened towards {O01}
The angle 8 is 88° & 89°; there are also the forms: {100} and 101}; measured:
(100) : (101) =+20.°. The optic axial plane is (010}; inclined dispersion: p> »
round the first bissectria. Negative double refraction. On {004} there is one optical
axis visible about the limits of the field. The crystals are curved-plane.

)

Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 82 )

§ 4. I give in the following tables the temperatures observed
etc.) Next to my data are placed those of Bémer as far as he has
published them. The temperatures in | | will be discussed more in
detail later on.

[.. FATTY ESTERS OF CHOLESTEROL. |

| Le | i, | ts | BomeEr’s data:
|
|

Chol. Formiate | — | [+ 909] | 9695 — | 286°:
sieAveiate © |. =e 290°} 112°.8 — | 419°5 |
Poy" Propionate | 8820.9) aegeo ate 96° | 114° |
| » -sButyrate | 96°4 | 107°.3 — 96° | 408° |
|» Isobutyrate = a 1269.5 ae
| » -n-Valerate | 91°.8 99°.2 _ — —
_ » Iso- valerate — [+ 109°] | 110°.6 _ --
| » Capronate 910.2 100°.1 — _— _
| » Caprylate _ [+ 101°] | 106°.4 = -
| » Caprinte 82°.2 90°.6 — _ —
|» Benzoate | 44595 | 478°.5 5 146° | 1780.5
| » Phtalate *) = — — — | 182°.5
|
» Stearinate *) — ~ — 65°

Benzoates and phthalates although not being fatty esters, have nevertheless
been included.

1) According to Scuénpeck, Diss. Marburg. (1900).

2) According to BémeEr loco cit.

3) According to Bertuetor. It is as yet undecided, whether liquid crystals are
| present here ; ; perhaps this case is analogous with that of the caprylate.

|
The temperatures in [] cannot be determined accurately; see text.

§ 5. Most striking with these remarkable substances are the splen-

1) It should be observed that in these substances three temperatures should be:
considered, namely 1. transition: solid — anisotropous-liquid ; 2. transition: aniso-
tropous-liquid — isotropous-liquid ; 3. transition: solid — isotropous-liquid.

This distinction has been retained, particularly on account of the cases of labile,
liquid crystals discovered here.

( 83 )

did colour-phenomena observed during the cooling of the clear,
isotropous, fused mass to its temperature of solidification, and also
during the heating in the reverse way. These colour phenomena are
caused by interference of the incident light, every time the turbid
anisotropous liquid-phase occurs, or passes into the isotropous liquid.
During this last transition we notice while stirring with the ther-
mometer, the ‘‘oily slides” formerly described by Remnirzmr, until the
temperature ¢, has been exceeded. These colours also occur when
the solid phase deposits from the anisotropous liquid, therefore below
t,. The most brilliant, unrivalled violet and blue colour display is
shown by the butyrate and normal valerate, also very well by the
capronate and caprinate.

The temperatures in { | ¢, answer to anisotropous liquid phases
which are dabile in regard to the isotropous liquid, and which double-
refracting liquids are, therefore, only realisable in undercooled fused
material, Of this case, which is comparable with the monotropism, as
distinguished by Lexmann from the case of enantiotropous transfor-
mations, the acetate is the only known example up to the present. Now
the number of cases is increased by three, namely the formate, the
caprylate and without any doubt also the csovalerate, to which I will
refer presently. Cholesterol-formiate and caprylate melt therefore,
perfectly sharply to a clear liquid at, respectively 967/,° and 106.°2.
If, however, the clear liquid is suddenly cooled in cold water,
one notices the appearance of the turbid, anisotropous, more-labile
phase, accompanied by the said colour phenomena. The acetate in
particular exhibits them with great splendour. It is quite possible
that many organic compounds which are described as ‘melting
sharply”, belong to this category and on being cooled suddenly
possess a double-refracting liquid phase, even although this may last
only a moment. The phenomenon of liquid crystals would then be
more general than is usually believed.

Prof. LeHmann, to whom I have forwarded a little of the cholesterol-
esters, has been able to fully verify my observations. This investigator
has, in addition, also found that cholesterol-caprinate may probably
exhibit two anisotropous hquid phases. Although, personally, | never
noticed more than one single phase, and Prof. LenMann’s determinations
are only given provisionally, this case would certainly have to be
regarded as one of the most remarkable phenomena which may be
expected in a homogeneous body, particularly because the percep-
tibility of those fo phases implies that they would not be miscible
in all proportions with each other.

( 84 )

§ 6. The behaviour of cholesterol-isobutyrate is a very remarkable
one. Microscopic and macroscopic investigation shows absolutely nothing
of an anisotropous liquid phase, not even on sudden cooling and this
in spite of the fact that the normal butyrate gives the phenomenon
with great splendour. This differently-behaving ester has been prepared
from the same bulk of cholesterol as was used for preparing the
other esters. The cause of the difference can, therefore, be found only
in the structure of the fatty acid-residue, which contrary to that of
the other esters, is branched.

All this induced me, to prepare ‘the analogous ester of isovaleric
acid ; perhaps it might be shown also here that the branching of the
carbon-chain of the acid destroys the phenomenon of the anisotro-
pous liquid phase. At first T thought this was indeed the case, but
a more accurate observation showed that in the rapid cooling there
occurs, if only for an indivisible moment, a labile anisotropous
liquid; the duration, however, is so short that, for a long time, I
was in doubt whether this phase ought to be called stable or labile
as in the case of the formiate and caprylate! Even though the carbon-
branching does not cause a total abrogation of the phenomenon of
liquid crystals, the realisable traject appears to become so much smaller
by that branching, that it almost approaches to zero, and the expected
phase is, moreover, even still labile. From all this I think we may
conclude, as has been stated more than once by otbers, that the oceur-
rence of the liquid phases is indeed a inherent property of the
matter, which cannot be explained by the presence of foreign admix-
tures etc. (TAMMANN ¢. S.).

§ 7. We now give the melting points of the analogous phytosterol-
esters which, with one exception, do not exhibit the phenomenon of
the double-refracting liquids. As the phytosterols from different vege-
table fats seem to differ from each other, and as BomeEr does not
mention any phytosterol esters from Calabar-fat in particular, | have
indicated in the second column only the /mzts within which the melting
points of the various esters prepared by him from diverse oils, vary.
(See table following page.)

From a comparison of the two tables it will be seen that the lowering
of the melting point of phytosterol by the introduction of fatty aeid-
residues of increasing carbon-content, takes place much more rapidly
than with cholesterol. On the other hand, the succession of the melting
points of the acetate, propionate, butyrate and n-valerate is more
regular than with the cholesterol-derivates.

All phytosterol-esters share with phytosterol itself the great ten-

( 85 )

I]. FATTY ESTERS OF PHYTOSTEROL.

ae Seana i

BoMER:

—

Phytosterol-Formiate 110° 103°—113° |

Phytosterol-Acetate 129° 1 123°—135° |

Phytosterol-Propionate 405235 | 1049°—116° |

Phytosterol-Butyrate 91°.2 85°— 90°

| Phytosterul-Isobutyrate 5 Whee — |
| Phytosterol-norm.-Valerate Pe Ge eh, i= 30° =
| Phytosterol-Isovalerate 1009.4 =

dency to crystallise from the melted mass in sphaerolites; with an
increasing carbon-content of the fatty acid-residue, these seem gene-
rally to become smaller in circumference.

_ The formiate crystallises particularly beautifully; this substance
possesses, moreover, two solid modifications, as has been also stated
by Prof. Leamany, who is of opinion that these two correspond with
the two solid phases of the cholesterol-derivative. In the phytosterol-
ester the sphaerolite-form is the more-labile one.

On the other hand, when recrystallised from monobromonaphthalene
or almond-oil, they form under the microscope well-formed needle-
shaped crystals which, however, are always minute. Probably, we
are dealing in all these cases with polymorphism. I have also often
observed whimsical groroths and dendritics.

A difficulty occurred in the determination of the melting point of
the normal valerate. It melts, over a range of temperature at about
67°.1, but if the mass is allowed to cool until solidified, the ester
fuses to a clear liquid when heated to 30°. This behaviour is quite
analogous to that observed with a few glycerides of the higher fatty
acids, for instance with Tri/aurin and Trimyristin by ScuHey. *)

After half an hour the melting point had risen again to 53*/,° and
after 24 hours to 67°. After 24 hours, small white sphaerolites had
deposited in the previously coherent, scaly and slightly double-retrac-
ting layer on the object glass, which exhibited the dark cross of the
phytosterol. In order to explain this phenomenon, I think I must
assume a dimorphism of the solid substance. Moreover, liquid crystals
are formed here, as has also been observed by Prof. LEHMANN.

1) Scuey. Dissertatie, Leiden (1899) p. 51, 54.

( 86 )

According to Prof. LeHmany, normal phytosterol-valerate forms
very beautiful liquid erystals, which are analogous to those of chole-
sterol-oleate ; like these they are not formed until the fused mass is
undercooled. Consequently, the anisotropous liquid phase is here also
labile in regard to the isotropous one.

I do not think it at all improbable that the changes in the melting
points observed by Scuey with his higher tryglicerides also owe
their origin to the occurrence of labile, double-refracting liquid phases.
A further investigation is certainly desirable.

§ 8. We now arrive at the discussion of the mutual behaviour
of both series of fatty esters in regard to each other.

It has been sufficiently proved by Bémer that the meltingpoint-
line of cholesterol and of phytosterol is a rising line. In connection
with Miecr’s and my own erystal determinations we should have
here indeed a gradual mixing between heterosymmetric components !
In mixtures which contain about 3 parts of cholesterol to 1 part,
of phytosterol, the microscopical research appears to point to a new
solid phase, which seems to crystallise in trigonal prisms. This com-
pound (?) also occurs with a larger proportion of cholesterol *). Whether
we must conclude that there is a miscibility of this new kind of
crystal with both components, or whether an eventual transformation
in the solid mixing phases proceeds so slowly that a transition
point in the meltingpoint-line escapes observation, cannot be decided
at present.

The matter is of more interest with the esters of both substances.
According to BomeER?) the formiates give a meltingpoint-line with a
eutectic point; the acetates, however, a continuously rising melting
point-line.

The method of experimenting and the theoretical interpretation is,
however, rather ambiguous, as Boer prepares mixed solutions of
the components, allows these to crystallise and determines the melt-
ingpoint of the solid phase first deposited. By his statement of the
proportion of the components in the solution used, he also gives an
incomplete and confusing idea of the connection between the melting-
point and the concentration.

Although a rising of the binary meltingpoint-line may, of course,
be ascertained in this manner quite as well as by other means
— and Bomer’s merit certainly lies in the discovery of the fact

1) Compare Boner, Z. f. Nahr. u. Gen. M. (1901) 546.

*) Boer, Z. f. Nahr. u. Gen. Mitt. (1901) 1070. In connection with the dimor-
phism of the formiates, a mixing series with a blank is however very probable in
this case.

(87 )

itself — the determination of the binary meltingpoint-line must be
reckoned faulty as soon as it is to render quantitative services, which
is of importance for the analysis of butter; for if the meltingpoint-
curve is accurately known, the quantity of phytosterol added may
be calculated from the elevation of the melting point of the cholesterol
acetate. I have, therefore, now determined the binary melting point
line in the proper manner. (Fig. 4).

Fig. 4.
Cholesterol-, and Phytosterol-Acetate.

Although the curve takes an upward course it still deviates con-
siderably from the straight line which connects the two melting-
points. As the course of the curve from 40°/, cholesterol-acetate to
0°/, is nearly horizontal, it follows that the composition of mixtures
can be verified by the melting-point, when the admixture of phytosterol
in the animal fat does not exceed 60°/,. The results are the most
accurate when the quantity of phytosterol-ester*) amounts to 2°/,—40°/,.
In practice, this method is therefore applicable in most cases. The
cholesterol-acetate used in these experiments melted at 112.°8; the
phytosterol-acetate at 129.°2.

A mixture of 90 °, Cnol. Acet. 4-10 °/, Phyt. Acet. melts at 117°

» » Moo. eo DS » +20 » » » es 1805

» » 73.3» » » + 26.7» » » » ‘» 4122.°5
60 » » » +40 » » » ee ae |
» » 44 » » + 576» » » » pvp 128°

» » 20-2 » » +80 » » » oa, 220.
D) » Dy AG sD Px OO. oS » ‘3 42903

2
s
7

1) It should be observed that although Boémer, in several parts of his paper,
recommends the said method for qualitative purposes only, it is plain enough in
other parts that he considers the process suitable for quantitative determinations
in the case of small concentrations. In his interpretation of the melting point line
this is, however. not the case, for his experiments give no explanation as to the
mixing proportion of the components in mixtures of definite observed melting point.
Quantitative determinations are only rendered possible by a complete knowledge of
the binary melting point line. When the concentration of cholesterol-acetate is
0,5 — 19/,, the meltingpoint-is practically not altered; when it is 2°/, however,
the amount is easy to determine.

( 88 )

Probably, a ease of isomorphotropous relation occurs here with
the acetates; both esters are, probably, monoclinic, although this is
not quite certain for the cholesterol-ester. This is pseudotetragonal and

according to Von ZepHarovicH: monoclinic, with 8 = 73°38’;
according to Oprrmayer: triclinic, with ?—106°17', a—90°20', y—90°6',
while the axial relations are 1,85: 1: 1,75.

The phytosterol-ester has been approximately measured microsco-
pically by BrykircH and seems to possess a monoclinic or at least a
triclinic symmetry with monoclinic limit-value. In my opinion both
compounds are certainly nof isomorphous. In any case it might be
possible that even though a direct isomorphism does not exist in
the two ester-series, there are other terms which exhibit isomor-
photropous miscibility in an analogous manner, as found forthe ace-
tates by Boner. I have extended the research so as to include: the
isovalerates ; the result however. is negative and the case of the
acetic esters seems to be the only one in this series.

The following instance may be quoted :

31.8°/, cholesterol-butyrate ++ 68,2°/, phytosterol-butyrate indicate
for ¢, 81° and for ¢, 83° ete. ete.

For the formiates, the lowering had been already observed by BOmer ;
other esters, also those of the iso-acids behave in an analogous
marner: at both sides of the melting-diagram occurs a lowering of
the initial melting points. It is, however, highly probable that in
some, perhaps in all cases, there exists an isod7morphotiropous mixing
with a blank in the series of the mixed crystals.

The anisotropous liquid phase of cholesterol-esters gives rise in this
case to anisotropous liquid mixed erystals. I just wish to observe that
for some of the lower-melting esters, such as the butyrate, capronate, _
caprinate, normal valerate, ete., the temperature ¢, for these mixed
crystals may be brought to about 50° or 60° or lower and this creates
an opportunity for studying liquid mixed crystals at such tempera-
tures, which greatly facilitates microscopical experiments.

In all probability, I shall shortly undertake such a study of these
substances. Of theoretical importance is also the possibility, to which
Prof. Baknurs Roozesoom called my attention, that in those substances
where ¢, answers to the more-labile condition, the at first more labile
liquid mixed erystals, on being mixed with a foreign substance,
become, finally, stable in regard to the isotropous fused mass. Expe-
riments with these preparations, in this sense, will be undertaken
elsewhere. Perhaps, a study of the low-melting derivatives or else a
similar study of the low-melting liquid, mixed crystals by means of
the ultra-microscope might yield something of importance.

Zaindam, May 1906.

( 89 )

Physics. — “Researches on the thermic and electric conductivity
power of crystallised conductors.” 1. By Dr. F. M. Javcer.
(Communicated by Prof. H. A. Lorenrz).

(Communicated in the meeting of May 26, 1906).

1. Of late years, it has been attempted from various sides to
find, by theoretical means, a connection between the phenomena of
the thermie and electric conductivity of metallic conductors, and this
with the aid of the more and more advancing electron theory.

In 1900 papers were published successively by P. Drupg’), J. J.
THomson *) and EK. Rigcke*) and last year by H. A. Lorentz *).

One of the remarkable results of these researches is this, that
the said theory has brought to light that the quotient of the electric
and thermic conductivity poweg of all metals, independent of their
particular chemical nature, is a constant, directly proportional to the
absolute temperature.

When we assume that the electrons in such a metal can move
freely with a velocity depending on the temperature, such as happens
with the molecules in ideal gases and also that these electrons
only strike against the much heavier metallic atoms, so that in other
words, their mutual collision is neglected, whilst both kinds of
particles are considered as perfectly elastic globes, the quotient
of the thermic conductivity power 4 and the electric conductivity
power o may be indeed represented by a constant, proportional to
the absolute temperature 7’.

The theories of Drupe and Lorentz only differ as to the ab-

Fi 2
solute value of the quotient; according to Drups — = a5) fH
G

OF
T have the above cited meaning, whilst @ is a constant and e

represents the electric charge of the electron.
By means of a method originated by Koutravuscu, JAzcer and

; 2 8 fa?
according to Lorentz ——= — e T. In these expressions 4, 6 and
my)

Ae 3s
DiessELHoRST have determined experimentally the values for — with
o

1) P. Drupe, Ann. Phys. (1900). 1. 566; 3. 369.

2, J. J. Tuomson, Rapport du Congrés de physique Paris (1900). 3. 138.

3) E. Rrecke, Ann. Phys. Chem. (1898). 66. 353, 545, 1199; Ann. Pliys. (1900).
2. 835.

*) H. A. Lorentz, Proc. 1905, Vol. VII, p. 438, 585, 684.

(90 )

various metals'). The agreement between theory and observation is
in most cases quite satisfactory, only here and there, as in the case
of bismuth ?), the difference is more considerable. From their meas-
urements for silver at 15°, the value 47 & 10° may be deduced in

. a .
C.G.S. units, for the expression ——-. (Compare Lorentz, loco cit.
é

p. 505); according to Drupr’s formula: 38 X 10°.

§ 2. I hope, shortly, to furnish an experimental contribution
towards these theories by means of a series of determinations of an
analogous character, but more in particular with crystallised con-
ductors, and in the different directions of those crystal-phases.

If we take the most common case in which may be traced three
mutual perpendicular, thermic and electric main directions in such
erystals, the propounded theories render it fairly probable for all
such conducting erystals that:

Az — Ay — Ae and therefore also: Az: Ay : Az == Oy : Gy : Gy.
as ae 3 E

In conducting crystals, the directions of a greater electric con-
ductivity should, therefore, not only be those of a greater thermic
conductivity, but, theoretically, the quotient of the electric main-
conductivities should be numerically equal to that of the thermic
main-conductivities.

Up to the present but little is known of such data. The best
investigated case is that of a slightly titaniferous Haemitate of

—

1) W. Jagger und Dressetuorst, Berl. Sitz. Ber. (1899). 719 etc. Comp. Reincanum,
Ann. Phys. (1900) 2, 398.

A
2) With Al, Gu, Ag, Ni, Zn, the value of ze at 18° varies between 636 x 108
and 699 108; with Cd, Pb, Sn, Pt, Pd between 706 108 and 754 >< 108; with
a
Fe between 802 and 832 > 105, therefore already more. With bismuth = at

18° — 962 & 105. Whilst in the case of the other metals mentioned the values of

A :
— at 100° and at 18° are in the average proportion of 1,3:1, with bismuth the
O

proportion is only 1.12. In their experiments, Jaeger and DirssetHorst employed
little rods, and bearing in mind the great tendency of bismuth to crystallise, their
results with this metal cannot be taken as quite decisive, as the values of the
electric and thermic conductivity power in the chief directions of crystallised bis-
muth differ very considerably.

( 91 )

Swedish origin which has been investigated by H. BAckstrém and
K. AnestréM') as to its thermic and electric conductivity power.
In this ditrigonal mineral, they found for the quotient of the thermic
conductivity power in the direction of the chief axis (c) and in that
perpendicular to it (a) at 50°:

Aa
eh
7)

c
For the quotient of the electric resistances w at the same tempe-
rature they found:

dagen 1.78, and, therefore: pee 1.78:
Wa O-

From this it follows that in the case of the said conductor, the
theory agrees with the observations as to the relation between the
conductivity powers only qualitatively, but not quantitatively, and
— contrary to the usually occurring deviations — the proportion of
the quantities A is smaller than that of the quantities o.

JANNETTAZ’S empirical rule, according to which the conductivity for
heat in crystals is greatest parallel to the directions of the more
complete planes of cleavage, applies here only in so far as haematite
which does not possess a distinct plane of cleavage, may still be
separated best along the base {111} (Mintimer), that is to say parallel
to the plane of the directions indicated above with a.

§ 3. In order to enrich somewhat our knowledge in this respect
the plan was conceived to investigate in a series of determinations
the thermic and electric conductivity-power of some higher and also
of some lower-symmetrical crystalline conductors, and, if possible,
of metals also. For the moment, I intend to determine the quotient
of the conductivities in the different main directions, and afterwards
perhaps to measure those conductivities themselves in an absolute
degree.

I. On the thermic and electric conductivities in crystallised Bismuth
and in Haematite.

Measurements of the thermic and electric conductivity of bismuth
are already known.

Marrrvccr*) determined the thermie conductivity, by the well-

1)-H. Backstrom and K. Anasrrim, Ofvers. K. Vetensk. Akad. Férh. (1888)
No. 8, 533; BAcksrrém, ibid. (1894), No. 10, 545.
2) Marreuccr, Ann. Chim. et Phys. (3). 43. 467. (1855).

( 92)

well-known method of INGENHovsz, by measurement of the length
of the melted off waxy layer which was put on the surface of
cylindrical rods of bismuth, cut // and 1 to the main axis, whilst
the one end was plunged into mercury heated at 150°. For the

average value of the quotient of the main conductivities — perpen-
dicular and normal to the main axis — he found the value 1,08.

JaNNeTTAZ’s rule applies in this case, because the complete cleavability
of ditrigonal bismuth takes place along {111} (Miter), therefore,
perpendicularly to the main axis. JAnneTTaz') has applied the
SéNaRMoNT method to bismuth. He states that in bismuth the ellipses
have a great eccentricity but he did not take, however, exact
measurements.

A short time ago, Lownps?*) has again applied the SfnarmMontT method
to bismuth. He finds for the quotient of the demi-ellipsoidal axes
1.19 and, therefore for the quotient of the conductivities 1.42.

The last research is from Prrrot*). By the S£NaRMoNT method
he finds as the axial quotient of the ellipses about 1.17 and conse-
quently for the quotient of the conductivities J and // axis 1.368,
which agrees fairly well with the figure found by Lownps. Secondly,
Perrot determined the said quotient by a method proposed by
C. Soret, which had been previously recommended by THoULET‘),
namely, by measuring the time which elapses between the moments
when two substances with known melting points #, and &, placed
at a given distance at different sides of a block of the substance
under examination begin to melt. As indices were used; a-Naphtyla-
mine (#=50° C.), o-Nitroaniline (9 = 66° C.), and Naphthalene
ie — io ©.).

As the mean of all the observations, Perrot finds as the quotient
of the main conductivities 1,3683, which agrees perfectly with his
result obtained by SENarMont’s method.

He, however, rightly observes that this concordance between the
two results is quite an accidental one, and that the method of THouLET
and Soret must not be considered to hold in all cases. The proof
thereof has been given by CatL.er in a theoretical paper ;°*) the

agreement is caused here by the accidental sma// value of a quotient

% in which 7 represents the thickness of the little plate of bismuth

1) Jannertaz, Ann. de chim. phys. 29. 39. (1873).

2) L. Lownps, Phil. Magaz. V. 152. (1903).

8) L. Perrot, Archiv. d. Science phys. et nat. Généve (1904. (4). 18, 445.
*) Tourer, Ann. de Chim. Phys. (5). 26. 261. (1882)

6) C. Catter, Archiv. de Scienc. phys. et nat. Genéve (1904). (4). 18. 457.

( 93 )
and h and & the coefficients of external and internal conductivity.

§ 4. 1 have endeavoured to determine the quotient of the chief
conductivities by the method proposed by W. Vorer.

As is wellknown, this method is based on the measurement of
the angle, formed by the two isotherms at the line of demarcation
between two little plates which have been joined to an artificial
twin, when the heat current proceeds along the line of demarcation.
If a, and a, are the two chief conductivities of a plate of bismuth
cut parallel to the crystallographic main axis, and if the angle which
the two main directions form with the line of demarcation equals
45°, then according to a former formula’):

A, on &
—————1 U0) Go SS
"eb 2

§ 5. The bismuth used was kindly furnished to me by Dr. F. L.
Perrot, to whom [| again wish to express my hearty thanks.

The prism investigated by me is the one which Dr. Prrror in
his publications’) indicates with J/, and for which, according to

v
Sénarmont’s method, he found for — the value 1,390. The prism
rE.
given to Dr. vaN Everpincen yielded in the same manner for
2
~ the value 1,408.

Two plates were cut parallel to the crystallographic axis, in two
directions forming an angle of 90° and these were joined to twin
plates with g = 45°.

It soon appeared that in this case the Voie method *) was attended
by special difficulties which, as Prof. Voicr informed me, is generally
the case with metals. First of all, it is difficult to find a coherent
coating of elaidic acid + wax; generally the fused mixture on the
polished surface forms droplets instead of congealing to an even
layer. Secondly, the isotherms are generally curved and their form
presents all kinds of irregularities, which are most likely caused
by the great specific conductivity of the metals, in connection with
the peculiarity just mentioned. On the advice of Prof. Vorier I
first covered the metallic surface with a very thin coating of
varnish; this dissolves in the fused acid, and causes in many Cases
a better cohesion, but even this plan did not yield very good results.

1) These Proceedings. (1906). March p. 797.

2) p. 4, note 10.
3) Voiet, Gdltinger Nachr. (1896). Heft 3, p. 1—16; ibid. (1897). Heft 2. l—o

( 94 )

However, at last, I succeeded in getting a satisfactory coating of
the surface by substituting for white wax the ‘ordinary, yellow
bees-wax. This contains an adhesive substance probably derived
from the honey, and, when mixed in the proper proportion with
elaidie acid it yields the desired surface coating.

I have also coated‘) the bottoms of the plate and the sides, except
those which stand _| on the line of demarcation with a thick layer
of varnish mixed with mercury iodide and copper iodide. During the
operation the heating was continued to incipient darkening (about 70’).

The plates should have a rectangular or square form, as otherwise
the isotherms generally become curved.

It is further essential to heat rapidly and to raise the copper bolt
to a fairly high temperature; the isotherms then possess a more
straight form and give more constant values for e.

I executed the measurements on the double object table of a
LEHMANN’s crystallisation microscope on an object glass wrapped in
thick washleather, to prevent the too rapid cooling and solidification
of the coating.

After numerous failures, I succeeded at last in obtaining a long
series of constant values. As the mean of 30 observations, I found
€ = 22°12’ and therefore:

a
“* — 1,489;
i

“Cc

§ 6. The value now found is somewhat greater than that found
by Perrror. I thought it would be interesting to find out in how
far a similar deviation was present in other cases, and whether when
compared with the results obtained by the methods of S#narmonrt,
JaneTtaz and RogntGEN, it has always the same direction.

In fact, the investigation of many minerals has shown me that
all values obtained previously, are smaller than those obtained by
the process described here.

I was inclined at first to believe that these differences were still
greater than those which are communicated here. Alihough a more
extended research, including some plates kindly lent to me by Prof.
Voict, showed that these differences are not so serious as I suspected,
at first the deviation exists always in the same direction.

For instance, I measured the angle ¢ of a plate of an Apatite-
crystal from Stillup in Tyrol and found this to be 17°. From the

) Ricuarz’s method of experimenting (Naturw. Rundschau, 17, 478 (1902)) did
not give sufliciently sharply defined isotherms and was therefore not applied.

(95 )

position of the isotherms it also follows that 2. > 4 so that — —1.35.
In a quartz-plate obtained from Prof. Voier I found pete 304°,

A e - :
therefore — = 1,75. In a plate of Antimonite from Skikoku in Japan

a
cut parallel to the plane {010}, - was found to be even much larger
“a

than 1,74, which value is deduced from the experiments of SENArMoN?T
and JANNETTAZ as they find for the quotient of the demi ellipsoidal
axes 1.32.

For Apatite they find similarly 1,08, for quartz 1,73, whilst
TucHscuMipt determined the heat-conductivity of the latter mineral
according to Werser’s method in absolute degree. His experiments

he
give the value 1,646 for the quotient a
a

The deviations are always such that if 2, >, the values of the

: A
quotient = turn out to be larger when Voier’s method is employed
2
instead that of DE SENARMONT. The method employed here is, however,

so sound in principle, and is so much less liable to experimental
errors, that it certainly deserves the preference over the other processes.

Finally, a sample of Haematite from Elba was examined as to
its conducting power. A plate cut parallel to the c-axis was found
not to be homogeneous and to contain gas-bubbles. I repeatedly
measured the angles « of a beautifully polished preparation of Prof.
Voict, and found fairly constantly 104°, whilst the position of the
isotherms showed that 4a was again larger than 4,.

: f Ps
For the Haematite we thus obtain the value: = = 1,202. ‘The

Le
value found by BAckstrémM and Anestrém for their mineral with the
aid of CuHRISTIANSEN’s method was 1,12. In this case the deviation
also occurs in the above sense.

From the experiments communicated we find for the quotient

A A
%a:%- in both crystal phases, if by this is meant (<) (5) the
a c
values :

With Bismuth : A 4-438:

f ‘ Ke
With Haematite: — — 1,480.

da ;
In this my measurements of — are combined with the best value

( 96 )

; : 6.
found by van EverDINGEN‘) with Prrrot’s prism, namely — = 1,68,
. Oc

and with the value found by the Swedish investigators for haematite:
1,78 at 50° C.

7. If there were a _ perfect concordance between theory and

observation, we should have in both cases = = 1. The said values
*c

1,128 and 1,480 are, therefore, in a certain sense a measure for

the extent of the divergence between the observation and the con-

clusion which is rendered probable by the electron theory.

In the first place it will be observed that the agreement is much’
better with dismuth than with haematite. However, this may be
expected if we consider that the theory has been proposed, in the
first instance, for metallic conductors. The influence of the peculiar
nature of the ovide when compared with the true metal is shown
very plainly in this case.

The question may be raised whether, perhaps, there may be
shown to exist some connection between the crystal structure and

: . : . 4a
the chemical nature on one side, and the given values of — on the
%e

other side.

Such a connection would have some significance because it may
be, probably, a guide for the detection of special factors situated in
the crystalline structure, which stand in the way of a complete
agrement of electron theory and observations.

§ 8. First of all, it must be observed that we are easily led to
compare the structures of the two phases. Both substances inves-
tigated erystallise ditrigonally and have an analogous axial quotient;
for bismuth: a:c¢ — 1:1,3035 (G. Rosr); for haematite a:e =
1:1,3654 (Mexczrr). In both substances, the habit is that of the
rhomboid, which in each of them approaches very closely to the
regular hexahedron. The characteristic angle @ is 87°34’ for bismuth
for haematite 85°42’. Particularly in bismuth the pseudo-cubic
construction is very distinct; the pianes of complete cleavage which
answer the forms {111} and {144} approach by their combination the
regular octahedron in a high degree. Although haematite does not

1) van Everpincen, Archives Néerland. (1901) 371; Versl. Akad. v. Wet. (1895—
1900); Comm. Phys. Lab. Leiden, 19, 26, 37, 40 and 61. See Archiv. Néerl.

p. 452; rods No. 1 and No. 5.

( 97 )

possess a periect plane of cleavage, it may be cleaved in any case
along {111} with testaceous plane of separation. It admits of no doubt
that the elementary parallelepipeds of the two crystal structures are
in both phases pseudo-cubic rhombohedral configurations and the
question then rises in what proportion are the molecular dimensions
of those cells in both crystals ?

If, in all crystal-phases, we imagine the whole space divided
into volume-units in such a manner that each of those, everywhere
joined, mutually congruent, for instance cubic elements, just contains
a single chemical molecule, it then follows that in different crystals

¢ M
the size of those volume elements is proportionate to Fe in which
€

M represents the molecular weight of the substances and d the
sp.gr. of the crystals. If, now, in each crystal phase the content
of the elementary cells of the structure is supposed to be equal to

M
this equivalent-volume or the dimensions of those cells will be reduced

for all crystals to a same length unit, namely all to the length
of a cubic-side belonging to the volume-element of a crystal phase,
whose density is expressed by the same number as its molecular

if
weight ; then in that particular case = = 1. If we now calculate -
the dimensions of such an elementary parallelopiped of a Bravais
M
structure whose content equals the quotient = and whose sides are

in proportion to the crystal parameters a:4:c, the dimensions
% wand w» thus found will be the so-called topic parameters of the
phase which, after having been introduced by Brckr and MuTHMaNn
independently of each other, have already rendered great services
in the mutual comparison of chemically-different crystal-phases. In
the particular case, that the elementary cells of the crystal-structure
possess a rhombohedral form, as is the case with ditrigonal crystals,
the parameters y, w and w become equal to each other (=e). The
relations applying in this case are

Soult

ab sin —
V 3 ' ee 2

e=—| —. , ] > “with sn Saale eae
sin? a.sin A 2 Sin @

If now these calculations are executed with the values holding
here: Bi= 207,5; Fe,O, = 159,64; dgj = 9,851 (PrerRoT); dre, 0, = 4,98,
then

Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 98 )

Vpi = 21,064 and Va, 0, = 32,06,
and with the aid of the given relations and the values for @ and A
we find for each phase: *)

Qn; 2,7641

Or, 0, 3,1853

If now we just compare these values for the sides of the rhom-
bohedral elementary cells of the crystal structure with those of the

% :
quotients — in the two phases, they curiously enough show the
Xe

following relation :

Ors x
(=) : (=) =. Sea wee.
%c / Fes Os tc) Bi \ Fest, eee

Allowing for experimental errors, the agreement is all that can
be desired: in the first term of the equation the value is exactly:
1.312, in the last term: 1,328.

x
In our case the quotient may therefore be written for both
Xe

phases in the form: C.9’, in which C is a constant independent of
the particular chemical nature of the phase.

Instead of the relation

0,7: 0,", perhaps 0,’ sin a, : 9,” sin a, = 1.305
is still more satisfactory. These expressions, however, represent
nothing else but the surface of the elementary mazes of the three
chief planes of the trigonal molecule structure, for these are in our
case squares whose flat axis =a. The quotient in the two
phases should then be directly proportional to the reticular density
of the main net-planes of Bravais’s structures.

A choice between this and the above conception cannot yet be
made, because «, and a, differ too little from 90°. Moreover, a further
investigation of other crystals will show whether we bave to do
here with something more than a mere accidental agreement. Similar
investigations also with lower-symmetric conductors are at this
moment in process and will, I hope, be shortly the subject of further
communications.

Zaandam, May 1906.

1) For bismuth z= 87°:34' and A=87°40': for haematite 2 = 85°42’ and
A=86°0'. The angle A is the supplement of the right angle on the polar axes
of the rhombohedral cells and z is the flat angle enclosed between the polar axes.

( 99 )

Chemistry. — “Three-phaselines in chloralalcoholate and «aniline-
hydrochloride’. By Prof. H. W. Baxuuis Roozesoom.

It is now 20 years since the study of the dissociation pheno-
mena of various solid compounds of water and gases enabled me
to find experimentally the peculiar form of that three-phaseline which
shows the connection between temperature and pressure for binary
mixtures in which occurs a solid compound in presence of solution
and vapour, The general significance of that line was deduced,
thermodynamically, by van per Waats and the frequency of its
occurrence was proved afterwards by the study of many other
systems.

That this three-phaseline is so frequently noticed in practice in
the study of dissociable compounds is due to the circumstance that,
in the majority of the most commonly occurring cases, the volatility
of the two components or of one of them, is so small, that at the
least dissociation of the compound both liquid and vapour occur in
its presence.

In the later investigations, which have led to a more complete
survey of the many equilibria which are possible between solid
liquid and gaseous phases, pressure measurements have been
somewhat discarded. When, however, the survey as to the connec-
tion of all these equilibria in binary mixtures got more and more
completed and could be shown in a representation in space on
three axes of concentration, temperature and pressure, the want was
felt to determine for some equilibria, theoretically and also experi-
mentally, the connection between temperature and pressure, in order
to fill up the existing voids.

Of late, the course and the connection of several p,f-lines, have
been again studied by vAN DER Waats, Suits and myself either
qualitatively or qualitative-quantitatively.

To the lines, which formerly had hardly been studied, belonged
the equilibria lines which are followed, when, with a constant
volume, the compound is exposed to change of temperature in presence
of vapour only. They can be readily determined experimentally only
when the volatility of the least volatile component is not too small.
STORTENBEKER at one time made an attempt at this in his investigation
of the compounds of iodine with chlorine, but did not succeed in
obtaining satisfactory data.

In the second place it was desirable to find some experimental
confirmation for the peculiar form of the three-phaseline of a
compound, recently deduced by Smits for the case in which-a

Hi pe 8

( 100 )

minimum occurs in the pressure of the liquid mixtures of its
components.

Mr. Lropoip has now succeeded in giving experimental contributions
in regard to both questions, by means of a series of very accurately
conducted researches where chloralalcoholate and anilinehydrochloride
occur as solid compounds.

Solid compounds which yield two perceptibly volatile components
(such as PCl,, NH,.H,S, PH,.HCl, CO,.2 NH, etc.) have been investi-
gated previously, but either merely as to their condition of dissociation
in the gaseous form, or as to the equilibrium of solid in presence
of gaseous mixtures of different concentration at constant temperature;
but liquids occur only at higher pressures, so that the course of
the three-phase lines had never been studied.

These two compounds were selected because in their melting points
neither temperature nor pressure were too high. Moreover, the diffe-
rence in volatility of the two components in the first example (chloral
-+ alconol) was much smaller than in the second (aniline + hydrogen
chloride). It was also safe to conclude from the data of both com-
pounds that the liquid mixtures of their components would show a
minimum pressure.

mee

eas
ie
a

ES
Bee OS Ne a a) of 2

ae hae

pe
ff
mes ©
eee
BEAD
Ee
Se
i

ale aN
ee bse

( 101 )

This last point was ascertained first of all by a determination of
the boiling point lines, in which a maximum must occur. In both
cases this was found to exist and to be situated at the side of the
least volatile component, respectively chloral or aniline.

The investigation of the three-phase lines showed first of all that
these possess the expected form in which two maxima and one
minimum of pressure occur.

In the first system (Fig 1) CFD is the three-phase line, T and T,
are the respective maxima for the vapour pressure of solutions with
excess of either alcohol or chloral and saturated with chloralaleo-
holate; the minimum is situated very close to the melting point F.

In the second system (anilinehydrochloride Fig. 2) the first maxi-
mum, im presence of excess of HCl is situated at such an elevated

Yam

( 102 )

pressure that this has not been determined, the second T, at a
moderate pressure is situated at the side of the aniline. The minimum

T, is situated at the same side and is removed further from the
melting point than in Fig. 1.
7, minimum F melting point
p16 -eM- 22.5 cM.
t 1975 199°2

The determination of these lines and also that of the equilibria-
lines for compound + vapour or liquid -++ vapour which also occur
in both figures can only take place on either side of point F’, for
in measuring the pressures, we can only have in the apparatus
a larger, or smaller, excess of either component. Moreover, it is
possible to fill the apparatus with the compound in a dry and pure
condition. In the case of the compounds employed, this was attained
by preparing very pure crystals by repeated sublimation in vacuo.

In the second example, the sublimation line ZG of aniline hydro-
chloride was thus determined. On this line then follows the piece
GF of the three-phase line, because beyond G, no vapour can exist
which has the same composition as the compound, except in the
presence of some excess of HCl, so that a little liquid is formed
with a slight excess of aniline. If, however, the apparatus is properly
filled with the compound so that there remains but little space for
the vapour then the three-phase line G may be traced to very near
the melting point /, where one passes on to the line /’A, for the
equilibrium of the fused compound with its vapour.

We have here, therefore, the first experimental confirmation of
the normal succession of the p,é-lines when those are determined
with a pure compound which dissociates more or less.

Theoretically, the minimum 7’ in the three-phaseline must be
situated at the left of the terminal point G of the sublimationline.
The difference here, although small, is yet perfectly distinct:

fA G
p. dbteM. 16.5 cM.
pels Wipe 198°

In the case of chloralalcoholate the points 7’, and G both coincide
so nearly with /’ that this point is practically undistinguishable from
the triple point of a non-dissociating compound, both ZL and FA,
or their metastable prolongation /’A' appear to intersect in /’, Moreover,
the investigation of the melting point line proved that chloralalco-
holate in a melted. condition is but little dissociated.

( 103 )

In both compounds the p,é-lines have also been determined with
excess of chloral or aniline. A very small quantity of these suffices
to cause the occurrence of liquid in presence of the compound at
temperatures far below the melting point and we then move on the
lowest branch of the three-phaseline.

In the case of a slight excess of chloral (Fig. 1) this was followed
from D over 7, to /’, just a little below the melting point, and from
there one passed on to the liquid-vapour line F,A,, which was
situated a little above FA.

In the case of a slight excess of aniline the piece D7T,7,GF,
could be similarly followed (Fig. 2). In this occurred the minimum 7\,
whilst the piece G/’, coincided entirely with the corresponding part
of GF, which had already been determined in the experiment with
the pure compound. Just below /’ the compound had disappeared
entirely and one passed on to the liquid-vapour line F,A,, which,
unlike that in Fig. 1, was situated below FA.

If the excess of the component is very trifling, liquid is formed
only at higher temperatures of the three-phaseline, and below this
temperature a sublimationline is determined, with excess of the
component in the vapour, which line must, therefore, be situated
higher than the pure sublimationline.

With chloralaleoholate a similar line LE (Fig 1) was determined,
situated decidedly above LF. At £, liquid occurred and a portion of
the three-phaseline LF’ was followed up to a point situated so
closely to / that the liquid-vapourline, which was then followed, was
situated scarcely above FA.

The excess of chloral was, therefore, exceedingly small, but in
spite of this, B# was situated distinctly above LF. The position of
LE depends, in a large measure, on the gas-volume above the solid
compound, as this determines the extra pressure of the excess of
the component, which is totally contained in the same; so long as
no liquid occurs. It appeared, in fact, to be an extremely difficult
matter to prepare chloralalecoholate in such a state of purity that
it exhibited the lowest imaginable sublimationline LF, which meets
the three-phaseline in F.

Similar sublimation lines may also occur with mixtures containing
excess of alcohol. But also in this case, even with a very small
excess of alcohol we _ shall retain liquid even at low temperatures
and, therefore, obtain branch C7F of the three-phaseline. Such hap-
pens, for instance, always when we use crystals of the compound
_which have been crystallised from excess of alcohol. They then
contain sufficient mother-liquor.

( 104 )

We then notice the peculiar phenomenon that the compound is
apparently quite solid till close to the melting point and we find
for the vapour pressure the curve C7'F, whilst the superfused liquid
gives the vapour pressureline /’A, which is situated much lower.
Ramsay has found this previously without being able to give an
explanation, as the situation of the three-phaseline was unknown
at that period. ;

In the case of anilinehydrochloride, it was not difficult, on
account of the great volatility of HCl, to determine sublimationlines
when an excess of this component was present. In Fig. 2 two such
lines are determined BF and b,L,. From £, the three-phaseline was
followed over the piece /,H, afterwards the liquid-vapourline #7, /,.
From Z£ also successively HH and HJ. With a still smaller excess
of hydrogen chloride we should have stopped even nearer to /' on
the three-phaseline.

In the case of chloralaleoholate we noticed also the phenomenon
that-a solid substance which dissociates after fusion may, when
heated not too slowly, be heated above its meltingpoint, a case lately
observed by Day and ALLEN on melting complex silicates, but which
had also been noticed with the simply constituted chloralhydrate.

An instance of the third type of a three-phaseline where the
maximum and minimum have disappeared in the lower branch of
the three-phase line has not been noticed as yet.

The two types now found will, however, be noticed frequently
with other dissociable compounds such as those mentioned above,
and therefore enable us to better understand the general behaviour
of such substances.

Physics. — “On the polarisation of Rontgen rays.” By Prof. H. Haga.

In vol. 204 of the Phil. Trans. Royal Soc. of London p. 467,
1905 BarkLA communicates experiments which he considers as a
decisive proof that the rays emitted by a Réyrcen bulb are partially
polarised, in agreement with a prediction of BLonpLor founded upon
the way in which these rays are generated.

In these experiments BarkLA examined the secondary rays emitted
by air or by some solids: paper, aluminium, copper, tin, by means
of the rate of discharge of electroscopes. In two directions perpen-
dicular to one another and both of them perpendicular to the direction
of the primary rays, he found a maximum and a minimum for the
action of the secondary rays emitted by air, paper and aluminium,

( 105 )

The difference between the maximum and minimum amounted to
about 20°/,.

I had tried to examine the same question by a somewhat different
method. A pencil of RénTcGEN rays passed through a tube in the
direction ef its axis, without touching the wall of the tube. A photo-
graphic film, bent cylindrically, covered the inner wall of the tube
in order to investigate whether the secondary rays emitied by the
air particles showed a greater action in one direction than in another.
I obtained a negative result and communicated this fact to Barkua,
who advised me to take carbon as a very strong radiator for secon-
dary rays. I then made the following arrangement. .

—

, |

Let S, (fig. 1) be the front side of a thick-walled leaden box,
in which the R6énremn bulb is placed; SS, and S, brass plates
10 x 10¢.m. large and 4 m.m. thick. Their distance is 15 ¢.m. and
they are immovably fastened to the upper side of an iron beam. In
the middle of these plates apertures of 12 m.m. diameter were made.
A metal cylinder A is fastened to the back side of S,; a brass tube
B provided with two rings Rk, and R, slides into it °*).

An ebonite disk # in which a carbon bar is fastened fits in
tube 6. This bar is 6 ¢.m. long and has a diameter of 14 m.m. At
one end it has been turned off conically over a length of 2 c.m.

1) Fig. 1 and 2 are drawn at about half their real size,

( 106 )

The aperture in S, was closed by a disk of black paper; the back
side of A was closed by a metal cover, which might be screwed off.
The dimensions were chosen in such a way, that the boundary
of the beam of ROénTGEN rays, which passed through the apertures
in S,, S,and/S,, lay between the outer side of the carbon bar and
the inner side of the tube B. The photographic film covering the
inside of B was therefore protected against the direct RONTGEN rays.
If we accept Barka’s supposition on the way in which the
secondary beams are generated in bodies of small atomic weight,
and if the axis of the primary beam perfectly coincided with that
of the carbon bar, then a total or partial polarisation of the RONTGEN
rays would give rise to two maxima of photographic action on
diametrically opposite parts of the film and between them two
minima would be found. From the direction of the axis of the cathode
rays the place of these maxima and minima might be deduced.

A very easy method proved to exist for testing whether the primary
beam passed symmetrically through the tube 6 or not. If namely
the inner surface of cover D was coated by a photographie plate or
film, which therefore is perpendicular to the axis of the carbon bar
then we see after developing a sharply defined bright ring between
the dark images of the carbon bar and of the ebonite disk. This ring
could also be observed on the fluorescent screen — but in this case
of course as a dark one, — and the RoéntcEn bulb could easily
be placed in such a way, that this ring was concentric with the
images of the carbon bar and of the ebonite disk.

This ring proved to be due to the rays that diverged from the
anticathode but did not pass through the carbon bar perfectly parallel
to the axis and left it again on the sides; these rays proved to
be incapable of penetrating the ebonite, but were totally absorbed by
this substance; when the ebonite disk was replaced by a carbon one,
then the ring disappeared; it is therefore a very interesting instance
of the selective absorption of ROnTGEN rays’).

When in this way the symmetrical passage of the RONTGEN rays
had been obtained, then the two maxima and minima never appeared,
neither with short nor with long duration of the experiment, though
a strong photographic action was often perceptible on the film. Such
an action could for instance already be observed after one hour’s
exposure, if an induction-coil of 30 cm. striking distance was used
with a turbine interruptor. A storage battery of 65 volts was used;

1) Take for this experiment the: above described arrangement, but a carbon bar
of 1 cm. diameter and 4 cm. long.

( 107 )

the current strength amounted to 7 ampéres; the ROn7rGEN bulb was
soft’.

Sometimes I obtained one maximum only or an irregular action
on the film, but this was only the case with an asymmetric position
of the apparatus.

From these experiments we may deduce: 1* that the primary
RONTGEN rays are polarised at the utmost only to a very slight
amount, and 2™4 that possibly an asymmetry in the arrangement
caused the maxima and minima observed in the experiments of
BarKLA, who did not observe at the same time in two diametrical
opposite directions.

With nearly the same arrangement I repeated Bark1a’s experiments
on the polarisation of secondary rays, which he has shown also by
means of electroscopes and described Proc. Roy. Soc. Series A vol.
77, p. 247, 1906.

/\
P

Lig. 2

Let the arrow (fig. 2) indicate the direction of incidence of the
RONTGEN rays on the carbon plate KX large 88 em. and thick
12 mm. The secondary rays emitted by this plate could pass through
the brass tube G, which was fastened to S,. This tube was 6 cm.
long and on the frontside it was provided with a brass plate with
an aperture of 5 mm. It was placed within the leaden case at 8 cm.
distance from the middle of the carbon plate; leaden screens protected
the tube against the direct action of the primary rays. In these
experiments the above mentioned induction-coil was used with a

( 108 )

Wenunett interruptor; the voltage of the battery amounted to 65 Volts
and the current to 7 Amperes. A very good photo was obtained in
30 hours and it shows very clearly two maxima and two minima,
the distance between the centra of the maxima is exactly half the
inner circumference of the tube, and it may be deduced from their
position that they are caused by the tertiary rays emitted by the
conic surface of the carbon bar.

In this experiment the centre of the anticathode, the axis of the
carbon bar and the centre of the carbon plate lay in one horizontal
plane, and the axis of the cathode rays was in one vertical plane
with the centre of the carbon plate; the axes of the primary and
the secondary beams were perpendicular to one another. According
to BarKLA’s supposition we must expect that with this arrangement
the maximum of the action of the tertiary rays will be found in
the horizontal plane above mentioned. In my experiment this sup-
position really proved to be confirmed. In order to know what part
of the photographic film lay in this plane, a small side-tube F was
adjusted to the outside of cylinder A, and this tube / was placed
in an horizontal position during the experiment. A metal tube with
a narrow axial hole fitted in tube /, so that in the dark room,
after taking away a small caoutchouc stopper which closed £, I
could prick a small hole in the film with a long needle through
this metal tube and through small apertures in the walls of A and
B. This hole was found exactly in the middle of one of the maxima.

So this experiment confirms by a photographic method exactly
what Barkia had found by means of his electroscopes and it proves
that the secondary rays emitted by the carbon are polarised.

In some of his experiments BarkLa pointed out the close agreement
in character of primary and secondary ROnTeEN rays; in my experi-
ments also this agreement was proved by the radiogram obtained on
the film placed in cover D. Not only did the secondary rays act
on the film after having passed through the carbon bar of 6 em.,
but also the bright ring was clearly to be seen, which proves that
ebonite absorbs all secondary rays which have passed through carbon *).
The ring was not so sharply defined as in the experiments with
primary rays; this fact finds a natural explanation in the different
size of the sources of the radiation: in the case of the primary rays
the source is a very small part of the anticathode, in the case of
the secondary rays it is the rather large part of the carbon plate
which emits ‘rays through the apertures in G and S,.

1) The ri * was perfectly concentric: the arrangement proved therefore to be
exactly symmetrical.

( 109 )

This agreement makes it already very probable that the Ronrens
rays also consist in éransversal vibrations; these experiments however
yield a firmer proof for this thesis. If namely we accept the suppo-
sition of Barkia as to the way of generation of secondary rays in
bodies with a small atomic weight, then it may easily be shown,
that the supposition of a longitudinal vibration of the primary Ronraun
rays would, in the experiment discussed here, lead to a maximum
action of the tertiary rays in a vertica/ plane and not in an /ori-
zontal plane, as was the case.

Groningen, Physical Laboratory of the University.

Chemistry. — “Triformin (Glyceryl triformate)’. By Prof. P. van
RoMBURGH.

Many years ago I was engayed in studying the action of oxalic
acid on glycerol ') and then showed that in the preparation of formic
acid by Lorin’s method diformin is produced as an intermediate product.

Even then I made efforts to prepare triformin, which seemed to
me of some importance as it is the most simple representative of the
fats, by heating the diformin with anhydrous oxalic acid, but I was
not successful at the time. Afterwards Lorin *) repeated these last
experiments with very large quantities of anhydrous oxalic acid and
stated that the formic acid content finally rises to 75°/,, but he does
not mention any successful efforts to isolate the triformin.

Since my first investigations, I have not ceased efforts to gain my
object. I confirmed Lort’s statements that on using very large
quantities of anhydrous oxalic acid, the formic acid content of the
residue may be increased and I thought that the desired product
might be obtained after all by a prolonged action.

Repeated efforts have not, however, had the desired result, although
a formin with a high formic acid content was produced from which
could be obtained, by fractional distillation in vacuo, a triformin still
containing a few percent of the di-compound.

I will only mention a few series of experiments which I
made at Buitenzorg, first with Dr. Nanninea and afterwards with
Dr. Lone. In the first, a product was obtained which had a sp.gr. 1.809
at 25°, and gave on titration 76.6°/, of formic acid, whilst pure
triformin requires 78.4°/,. The deficiency points to the presence of
fully 10°/, of diformin in the product obtained.

1) Compt. Rend. 93 (1881) 847.
2) Compt. Rend. 100 (1885) 282.

( 110 )

In the other, the diformin, was treated daily, during a month,
with a large quantity of anhydrous oxalic acid, but even then the
result was not more favourable.

The difficulty in preparing large quantities of perfectly anhydrous
oxalic acid coupled with the fact that carbon monoxide is formed
in the reaction, which necessitates a formation of water from the
formic acid, satisfactorily explains the fact that the reaction does not
proceed in the manner desired. A complete separation of di- and triformin
cannot be effected in vacuo as the boiling points of the two compounds
differ but little. .

I, therefore, had recourse to the action of anhydrous formic acid
on. diformin. I prepared the anhydrous acid by distilling the strong
acid with sulphuric acid in vacuo and subsequent treatment with
anhydrous copper sulphate. Even now I did not succeed in preparing
the triformin in a perfectly pure condition, for on titration it always
gave values indicating the presence of some 10°/, of diformin.

Afterwards, when 100°/, formic acid had become a cheap com-
mercial product, I repeated these experiments on the larger scale,
but, although the percentage of diformin decreased, a pure triformin
was not obtained.

I had also tried repeatedly to obtain a crystallised product by
refrigeration but in vain until at last, by cooling a formin with
high formic acid .content in liquetied ammonia for a long time, I
was fortunate enough to notice a small crystal being formed in
the very viscous mass. By allowing the temperature to rise gradually
and stirring all the while with a glass rod, I succeeded in almost
completely solidifying the contents of the tube. If now the crystals
are drained at O° and pressed at low temperature between filter
paper and if the said process is then repeated a few times, we
obtain, finally, a perfectly colourless product melting at 18°, which on
being titrated gave the amount of formic acid required by triformin.

The sp. gr. of the fused product ‘at 18° is 1.320.

ji tl
MR. 35.22; calculated 35.32.

The pure product when once fused, solidifies on cooling with great
difficulty unless it is inoculated with a trace of the crystallised
substance. On rapid crystallisation needles are obtained, on slow
crystallisation large compact crystals are formed.

In vacuo if may be distilled unaltered; the boiling point is 163° at
38™™, On distillation at the ordinary pressure it is but very slightly
decomposed. The boiling point is then 266°. A product contaminated

(141 )

with diformin, however, cannot be distilled under those circum-
stances, but is decomposed with evolution of carbon monoxide and
dioxide and formation of allyl formate.

If triformin is heated slowly a decided evolution of gas is noticed
at 210° but in order to prolong this, the temperature must rise
gradually. The gas evolved consists of about equal volumes of carbon
monoxide and dioxide. The distillate contains as chief product ally!
formate, some formic acid, and further, small quantities of ally]
alcohol. In the flask a little glycerol is left’).

Triformin is but slowly saponified in the cold by water in which
it is insoluble, but on warming saponification takes place rapidly.

Ammonia acts with formation of glycerol and formamide. With
amines, substituted formamides are formed, which fact I communi-
cated previously ’*).

The properties described show that triformin, the simplest fat,
differs considerably in its properties from the triglycerol esters of
the higher fatty acids.

Chemistry. — “On some derivatives of 1-3-5-hexatriene’. By
Prof. P. van Romspuren and Mr. W. van Dorssen.

In the meeting of Dec. 30 1905 it was communicated that, by
heating the diformate of s-divinylglycol we had succeeded, in pre-
paring a hydrocarbon of the composition C,H, to which we gave
the formula:

CH, = CH — CH = CH — CH = CH,.

Since then, this hydrocarbon has been prepared in a somewhat
larger quantity, and after repeated distillation over metallic sodium,
50 grams could be fractionated in a Laprensvre flask in an atmosphere
of carbon dioxide.

The main portion now boiled between 77°—78°.5 (corr.; pressure
764.4 mm.).

Sp. GY.13.5 0.749
ND1i3.5 1.4884

Again, a small quantity of a product with a higher sp. gr. anda
larger index of refraction could be isolated.

_ 1) This decomposition of triformin “has induced me to study the behaviour Of
the formates of different glycols and polyhydric alcohols on heating. Investigations
have been in progress for some time in my laboratory.

2) Meeting 30 Sept. 1905.

( 1439

In the first place the action of bromine on the hydrocarbon was
studied.

If to the hydrocarbon previously diluted with chloroform we add
drop by drop, while agitating vigorously with a Wirt stirrer, a
solution of bromine in the same solvent, the temperature being — 10°,
the bromine is absorbed instantly and as soon as one molecule has
been taken up the liquid turns yellow when more is added. If at
that point the addition of bromine is stopped and the chloroform
distilled off in vacuo, a crystalline product is left saturated with an
oily substance. By subjecting it to pressure and by recrystallisation
from petroleum ether of low boiling point, fine colourless crystals
are obtained which melt sharply at 85°.5—86° ’).

A bromine determination according to Limpic gave 66.84°/,, C,H,Br,
requiring 66.65°/,.

A second bromine additive product, namely, a tetrabromide was
obtained by the action of bromine in chloroform solution at O° in -
sunlight; towards the end, the bromine is but slowly absorbed. The
chloroform is removed by distillation in vacuo and the product
formed is recrystallised from methyl alcohol. The melting point les
at 114°—115° and does not alter by recrystallisation. Analysis showed
that four atoms of bromine had been absorbed.

Found: Br: 80.20. Calculated for C,H,Br, 79.99.

A third bromine additive product was found for the first time in
the bromine which had been used in the preparation of the hydro-
carbon to retain any hexatriene carried over by the escaping gases.
Afterwards it was prepared by adding 3 mols of bromine to the
hydrocarbon diluted with 4 vol. of chloroform at 0° and then heating
the mixture at 60° for 8 hours. The reaction is then not quite com-
pleted and a mixture is obtained of tetra- and hexabromide from
which the latter can be obtained, by means of ethyl acetate, as a
substance melting at 163°.5—164°.

Found: Br. 85.76. Calculated for C,H,Br, 85.71.

On closer investigation, the dibromide appeared to be identical
with a bromide obtained by Griner') from s. divinyl glycol with
phosphorus tribromide; of which he gives the melting point as
84°.5—85°. A product prepared according to GrRineR melted at
85°.5—86° and caused no lowering of the meltingpoint when added
to the dibromide of the hydrocarbon.

Griner obtained, by addition of bromine to the dibromide prepared
from his glycol, a tetrabromide melting at 112° together with a

1) Not at 89° as stated erroneously in the previous communication,

(143)

product melting at 108°—109°, which he considers to be a geome-
trical isomer. On preparing *) the tetrabromide according to Griner the
sole product obtained was that melting at 112°, which proved identical
with the tetrabromine additive product prepared from the hydro-
carbon, as described above. For a mixture of these two bromides
exhibited the same meltingpoint as the two substances separately.
Prolonged action of bromine on the tetrabromide according to
Griner, yielded, finally, the hexabromide melting at 163°—164°, which
is identical with the one prepared from the hydrocarbon.
The bromine derivatives described coupled with the results. of
GRINER prove that our hydrocarbon has indeed the formula given above.
According to THIELE’s views on conjugated double bonds we might
have expected from the addition of two atoms of bromine to our
hexatriene the formation of a substance with the formula
CH,Br — CH = CH —CH =>CH—CH,Br. . . (1)
or
CH,Br — CH = CH — CHBr —CH=CH,. . . (2)
from the first of which, on subsequent addition of two bromine
atoms the following tetrabromide would be formed.

CH,Br — CHBr — HC = CH — CHBr—CH,Br. . . (3)
As, however, the dibromide obtained is identical with that prepared

from s. divinyl glycol, to which, on account of its mode of formation,
we must attribute the formula

CH, = CH —CHBr— CHBr—CH=CH,. . . (4)
(unless, what seems not improbable considering certain facts observed, a
bromide of the formula (1) or (2) should have really formed by
an intramolecular displacement of atoms) the rule of TureELE would
not apply in this case of two conjugated systems.

Experiments to regenerate the glycol from the dibromide have
not as yet led to satisfactory results, so that the last word in this
matter has not yet been said. The investigation, however, is being
continued.

Meanwhile, it seems remarkable that only the first molecule of
bromine is readily absorbed by a substance like this hexatriene, which
contains the double bond three times.

By means of the method of Sapatier and SENDERENS, hexatriene
may be readily made to combine with 6 atoms of hydrogen. If its

1) Ann. chim. phys. [6] 26. (1892) 381.

2) Investigations on a larger scale will have to decide whether an isomer, melt-
ing at 108°, really occurs there as a byproduct which then exerts but a very
slight influence on the melting point of the other product.

8
Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 114 )

vapour mixed with hydrogen is passed at 125°—130° over nickel
reduced to a low temperature, the hydrogen is eagerly absorbed and
a product with a lower boiling point is obtained, which, however,
contains small quantities of unsaturated compounds (perhaps also
cyclic ones). In order to remove these, the product was treated
with bromine and after removal of the excess and further purification
it was fractionated. As a main fraction, there was obtained a liquid
boiling at 68°.5—69°.5 at 759.7 mm.
Sp. gr-,,0 = 0,6907 np == 1.3919.

Although the boiling point agrees with that of the expected hexane
the sp. gr. and the refraction differ still too much from the values
found for hexane by Britunt and by Eykman’).

Therefore, the product obtained from hexatriene was shaken for
some time with fresh portions of fuming sulphuric acid until this
was no longer coloured. After this treatment were obtained
one fraction of

B. p. 69°—70°, Sp. gr.,, 0.6718 np,, 1.88250.
- and another of
B. p. 69°.7—70°5, Sp. gr.,, 0.6720, np,, 1.38239.
An n-hexane prepared in the laboratory, according to Briiuu *) by
Mr. ScHERINGA gave the following values
B. p. 69°, Sp. gr.,, 0.664 np,, 1.3792
whilst an n-hexane prepared, from diallyl according to SABATIER and
SENDERENS, by Mr. SINNIGE gave
B. p. 68.5°—70,° Sp. gr.,, 0.6716, np,, 1.38211.

It is, therefore, evident that the hexane obtained by SaBatimr’s, and
SENDERENS process still contains very small traces of impurities.

There cannot, however, exist any doubt that 1-3-5-hexatriene
absorbs 6 atoms of hydrogen with formation of normal hexane.

Of greater importance, however, for the knowledge of the new
hydrocarbon is the reduction by means of sodium and absolute
alcohol.

If, as a rule, unsaturated hydrocarbons are not likely to take up
hydrogen under these circumstances, it becomes a different matter
when a conjugated system is present. Now, in 1-3-5-hexatriene, two
conjugated systems are found and we might therefore expect the
occurrence of a 2-4-hexadiene :

CH,—CH—CH—CH=CH—CH,

1) Briint (B.B. 27, (1894) 1066) finds Sp. gr.o9 = 0.6603, nDoyp = 1.38734,
Eyxman (R. 14, (1881) 187) Sp.gr.44 = 0.6652 npd\, = 1.87725.
2) Ann, 200. 183.

(115 )

or, of a 2-5-hexadiene :
CH,—CH=CH—-CH,—CH=CH
The first, still having a conjugated system can again absorb two
atoms of hydrogen and then yield hexene 3.
CH,—CH,—CH=CH—CH,—CH

ist the other one cannot be hydrogenated any farther a

The results obtained seem to point out that both reactions have
indeed taken place simultaneously, and that the final product of the
hydrogenation is a mixture of hexadiene with hexene.

10 grams of 1-3-5-hexatriene were treated with 100 grams of
boiling absolute alcohol and 15 grams of metallic sodium. After
the sodium had dissolved, a current of steam was passed, which
caused the ready separation of the hydrocarbon formed, which,
however, still contained some alcohol. After redistillation, the
hydrocarbon was washed with water, dried over calcium chloride
and distilled over metallic sodium.

At 75°.5 it commenced to boil and the temperature then slowly
rose to 81°. The liquid was collected in two fractions.

fraction I. B.p. 75°.5—78°, Sp.gr.,, 0.73826 ny,, = 1.4532

Ra | Weis és Riernny 6 Beri a — , = 1.4665

These fractions were again united and once more treated with
sodium and alcohol. But after purification and drying no liquid of
constant boiling point was obtained, for it now commenced to boil
at 72°.5, the temperature rising to 80°. The main fraction now
possessed the following constants:

B.p. 72°.5—74°, Sp.gr.,, 0.7146 np,, 1.4205

The fraction 75°—80 gave np,, 1.4351.

An elementary analysis of the fraction boiling at 72°.5—74° gave
the following result:

Found Calculated for C,H,, ~° Calculated for C,H,,
C 87.06 or.7 85.6
BH 13.32 12:3 14.4

The fraction investigated consists, therefore, probably of a mixture
of C,H,, and C,H,,. The quantity collected was insufficient to effect
another separation. We hope to be able to repeat these experiments
on a larger scale as soon as we shall have again at our disposal a
liberal supply of the very costly primary material. —

Utrecht, Org. Chem. Lab. University.
1) If CH,=CH—CH,—CH,—CH=CH, should be formed, this will not readily

absorb more hydrogen either.

: ( 116 )

Mathematics. — “The force field of the non-Huchdean spaces
with negative curvature’. By Mr. L. E. J. Brouwer. (Commu-

nicated by Prof. D. J. KorTEwEe).

A. The hyperbolic Spy.

I. Let us suppose a rectangular system of coordinates to be placed
thus that ds = VA? du? + B? dv? + C? dw?, and let us assume a line-
vector distribution X with components X,, X,, X», then the integral
of X along a closed curve is equal to that of the planivector Y over
an arbitrary surface bounded by it; here the components of Y are
determined by:

1 (d(X%B) a 28 ©) Cy
BC | Ww

For, if we assume on the bounded surface curvilinear coordinates
€ and 4, with respect to which the boundary is convex, the surface
integral is

ae dv 0 (X, B) _ 9 (Xw C)
SG: x ae) Ow Ov ) a5 a

Joining in this ae? = terms containing X, C and adding and
0(X,C) dw dw
eee 3 : ay we obtain:

0 (X» C w Ow
fosan| (Xv C) dw O(Xw C) du

Y,==

subtracting

Oy UE ae aye
Integrating this partially, the first term with respect to 7, the second

to §, we shall get HE X, Cdw along the boundary, giving with the

integrals be X, Bdv and he X, Adu analogous to them the line integral

of X along the boundary.

In accordance with the terminology given before (see Procee-
dings of this Meeting p. 66—78)') we call the planivector Y the
first derivative of _X.

1) The method given there derived from the indicatrix of a convex boundary
that for the bounded space by frent-position of a point of the interior ; and the method
understood by the vector Xpgr... a vector with indicatrix opqr.... We can however
determine the indicatrix of the bounded space also by post-position of a point of the
interior with respect to the indicatrix of the boundary; and moreover assign to
the vector Xpgr... the indicatrix pgr...o. We then find:

( 117 )

Analogously we find quite simply as second derivative the scalar:

a a
2= 35

According to the usual way of .. the first derivative is
the rotation vector and the second the divergency.

. 1 Yr . e . .
II. If X is to be a 2X, i.e. a second derivative of a planivector
=, we must have:

1 (0% B) 3 wee 2 ony

a
a Be te dn
and it is easy to see that for this is necessary and sufficient
A= 0.

Ill. If X is to be a oX, i.e. a first derivative (gradient) of a
scalar distribution g, we must have:
dg mh OD aoe
poe CCB SC
and it is easy to see, that to this end it will be necessary and
sufficient that
¥ = 6:

IV. It is easy to indicate (comp. Scurrinc, Géttinger Nachrichten,

1870) the 0X, of which the divergency is an isolated scalar value in
the origin.
It is directed according to the radius vector and is equal to:
1

. 9
sinh?r

when we put the space constant = 1 *).

OX, ae
n Ip
Veycty...0t x a ey

p p+l me 02x
poe q.

rr ae pHi

Ty+1 p+
1b: Se 17
oS Buta,
“gh ars

These last definitions include the well known divergency of a vector, and the
gradient of a potential also as regards the sign; hence in the following we shall
start from it and we have taken from this the extension to non-Euclidean spaces.

2) For another space constant we have but to substitute in the following formulae
r
— for r.

( 118 )

It is the first derivative of a scalar distribution :
ST Seats
and has in the origin an isolated divergency of 4.

V. In future we shall suppose that X has the field property and
shall understand by it, that it vanishes at infinity in such a manner
that in the direction of the radius vector it becomes of lower order

1 : ; : : :

than — and in the direction perpendicular to the radius vector of
i

lower order than e~”.

For a 0X this means that it is derived from a scalar distribution,
having the potential property, i.e. the property of vanishing at infinity.

Now the theorem of Gruen holds for two scalar distributions (comp.
Fresporr, diss. Gottingen, 1873):

dw 2 = op Nae
fox w-fov y.dr= fpf ao— fy g.dtr
(=f {grad. ~, grad. w} i.) :

If now g and w» both vanish at infinity whilst at the same time
lim. p we? = 0, then the surface integrals disappear, when we apply
the theorem of Green to a sphere with infinite radius and

fo Viy.de= fw. 7? ody,

integrated over the whole space, is left. |

Let us now take an arbitrary potential function for g and
—1l1-+4cothr for yw, where r represents the distance to a point P
taken arbitrarily, then these functions will satisfy the conditions of
vanishing at infinity and lim. gy we? = 0, so that we find:

An pp = [ (— 1+ coh) yg de

So, if we put —1- cothr=F(r), we have:

oe
Be. 27 oX
xey{ Ye Fad... Sa

VI. We now see that there is no vector distribution with the field
property, which has in finite nowhere rotation and nowhere diver-
eency. For, such a vector distribution would have to have a potential,
having nowhere rotation, but that potential would have to be every-
where 0 according to the formula, so also its derived vector,

( 119 )

From this ensues: a vector field is determined uniformly by its
rotation and its divergency.

VII. So, if we can indicate elementary distributions of divergency
and of rotation, the corresponding vector fields are elementary fields,
i. e. the arbitrary vector field is an arbitrary space-integral of such
fields.

_ For such elementary fields we find thus analogously as in a Euclidean
space (l.c. p. 74 seq.):

1. a field £,, of which the second derivative consists of two
equal and opposite scalar values, close to each other.

2. a field #,, of which the first derivative consists of equal
planivectors in the points of a small circular current and perpendicular
to that same current.

At finite distance from their origin the fields H, and EH, are here
again of the same identical structure.

VIII. To indicate the field 4, we take a system of spherical
coordinates and the double point in the origin along the axis of the
system. Then the field #, is the derivative of a potential:

cos —~

It can be regarded as the sum of two fictitious “fields of a
single agenspoint’, formed as a derivative of a potential — 1 + cothr,
which have however in reality still complementary agens at infinity.

IX. The field H, of a small circular current lying in the equator
plane in the origin is outside the origin identical to the above
field #,. Every line of force however, is now a closed vector
circuit with a line integral of 4 along itself. We shall find of this
field #, a planivector potential, lying in the meridian plane and
independent of the azimuth.

In order to find this in a point P with a radius vector 7 and
spherical polar distance g we have but to divide the total current
between the meridian plane of P and a following meridian plane
with difference of azimuth d%, passing between P and the positive
axis of revolution, by the element of the parallel circle through P
over dd. For, if ds is an arbitrary line element through /P in the
meridian plane making with the direction of force an angle fF, if dh
is the element of the parallel circle, = the above mentioned current
and H the vector potential under consideration, we find:

d= — dh. Xds sin F,

( 120-)

whilst the condition for A is:
d(Hdh) = dh ds X sin Fr.

ey
So we have but to take = for H.
(2

To find = we integrate the current of force within the meridian zone
through the spherical surface through P. The force component perpen-

shr

Z 2 P co
dicular to that spherical surface is 2 st oe aan therefore
sinh 7

sinh®r

* coshr
==: ak decor . sinhr dp . sinhr sin p dd = dd coth r . sin? gp.

So:
a a ars Bee ee Q:
dh — sinhrsin gd} — sinh?r

X. From this ensues, that if two arbitrary vectors of strength unity are

given in different points along whose connecting line we apply a third

coshr :
vector = — 5» the volume product of these three vectors, i.e. the
volume of the parallelepipedon having these vectors as edges taken
with proper sign, represents the linevector potential according to the
first (second) vector, caused by an elementary magnet with moment
unity according to the second (first) vector.

To find that volume product, we have first to transfer the two
given vectors to a selfsame point of their connecting line, each
one parallel to itself, i.e. in the plane which it determines with that
connecting line, along which the transference is done, and maintaining
the same angle with that connecting line.

The volume product w(S,,5,) is a symmetric function of the two
vectors unity of which we know that with integration of S, along
a closed curve s, it represents the current of force of a magnet unity
according to S, through s,, in other words the negative reciprocal energy
of a magnet unity in the direction of S, and a magnetic scale with
intensity unity within s,, in other words the force in the direction
of S, by a magnetic scale with intensity unity within s,, in other
words the force in the direction of S, by a current with intensity
unity along s,. So we can regard yw (S,, S,) as a force in the direction
of S, by an element of current unity in the direction of §,.

With this we have found for the foree of an element of current
with intensity unity in the origin in the direction of the axis of the

system of coordinates ;

( 121 )

coshr_
——— sing,
sinh?r

directed perpendicular to the meridian plane.

XI. For the fictitious field of an element of current (having mean-
while everywhere current, i. e. rotation) introduced in this way we
shall find a linevector potential V, everywhere “parallel” (see above
under § X) to the element of current and the scalar value of which
is a function of 7 only.

Let us call that scalar value U, and let us regard a small elemen-
tary rectangle in the meridian plane bounded by radii vectores from
the origin and by circles round the origin, then the line integral of
V round that rectangle is:

0 0
— a tu ain gy sinh r dg} dr — ap {U cos p dr} dy.

This must be equal to the current of force through the small
rectangle:
coshr . }
——— snp. sinh r dg. dr,
sinh?r
from which we derive the following differential equation of U
with respect to 7:
)
U —— {U sinh r} = coth r,

Or
the solution of which is:
U = cosech r — 4 rsech? kr +c. sech? 4 r.

Let us take c=0, we shall then find as vector potential V of
an element of current unity /:

cosech r — 4 rsech* 4 r= F, (r),
directed parallel to #.

Let us now apply in an arbitrary point of space a vector G, then
the vector V has the property that, when integrated in G along an
elementary circuit whose plane is perpendicular to G, it indicates
the force in the direction of G, caused by the element of current
H, or likewise the vector potential in the direction of / caused by
an elementary magnet with intensity unity in the direction of G.

So, if we call of two vectors unity Hand F’ the potential x (Z, F),
the symmetric function F,(r, cos ¢, where 7 represents the distance
of the points of application of the two vectors and g their angle
after parallel transference to a selfsame point of their connecting line,
we know that this function x gives, by integration of e.g. # over

( 122 )

a closed curve e not only the negative energy of a magnetic scale
with intensity unity bounded by e in the field of an element of
current unity F, but also the component along F of the vector
potential caused by a current unity along e.

From this ensues for the vector V of an element of current,
that when the element of current is integrated to a closed current
it becomes the vector potential of that current determined uniformly
on account of its flux property.

So really the vector potential of a cae i.e. of a field of currents
is obtained as an integral of the vectors V of the elements of current.

XII. We can now write that in an arbitrary point:

Sa!
ae FG) ie, 2 2 ee
Tt
where we first transfer in a parallel manner the vector elements
of the integral to the point under consideration and then sum up.
Let us now consider an arbitrary force field as if caused by its two
derivatives (the magnets and currents), we can then represent to our-
selves, that both derivatives, propagating themselves according to a
function of the distance vanishing at infinity, generate the potential
of the field.
The field X is namely the total derivative of the potential :

i Fart f A Wey cee
4x An

The extinguishment of the scalar potential is greater than that of the
vector potential ; for, the former becomes at great distances of order e—?",
the latter of order re—”. Farther the latter proves not to decrease
continuously from o to O, but at the outset it passes quickly
through O to negative, it then reaches a negative maximum and
then according to an extinguishment re—” it tends as a negative (i.e.
directed oppositely to the generating element of current) vector to zero.

XI]. The particularity found in Euclidean _ spaces, _ that
1

F, (vr) = F, (r) = —, is founded upon this, that in Euclidean spaces
pa

the operation of twice total derivation is found to be alike for scalar
distributions and vector distributions of any dimensions (l.c. p. 70).

Not so in non-Euclidean spaces; e.g. in the hyperbolic Sp, we
find for the Y’* of a scalar distribution w in an arbitrary point

( 123 )

when choosing that point as centre of a system of RieMANN normal
coordinates

(i e. a system such that ds =

Vda? + dy? + dz?
2? + y? a 2?
+

07% O7u O7u
Ne ide ~(S+5et5a)

but as ‘7? of a vector distribution with components X, Y and Z,
we find for the z-component X,2:

xX.— ox aN, «OFX
ee | =. Ox? cg Oy? =)

The hyperbolic Sp,.

I. As first derivative Y of a vector distribution NX we find a
_ planivector Sata by a scalar value:
0(X, A) 0 (X B)
AB ale |
As second derivative Z we Piha the scalar :
1 (0(X%,B) , 0(X, A) |
AB Ou F Ov ;

Il. If X is to be a 2X, i.e. a second derivative ofa planivector
with scalar value w we must have:
Ow Ow
x, = —- = ; xy = —,;
r Bov 7, Ae
to which end is necessary and sufficient: Z= 0.

a ws 1 = - 2 .
If X is to be a oX, i.e. a first derivative of a scalar g we must
have:

Og Og
Ds = — ; Xy — — ’
Adu Bov

to which end is necessary and sufficient: Y = 0.

Ill. The }N, of which the divergency is an isolated scalar value
in the origin, becomes a vector distribution in the direction of the
radius vector :

1
sinh r

It is the first derivative of a scalar distribution :

Leoth 3 r.

( 124 )

The divergency in the origin of this field is 2z.

The scalar distribution /coth4r has thus the potential property.
(This was not the case for the field of a single agens point in the
Euclidean Sp,).

IV. In the following we presuppose again for the given vector
distribution the field property (which remains equally defined for 2
and for » dimensions as for 3 dimensions); no vector field is possible
that has nowhere rotation and nowhere divergency; so each vector
field is determined by its rotation and its divergency and we have
first of all for a gradient distribution:

yX = = l coth 4r dt,

0X = + ee

V. For the field #, of an agens double point we find the gradient
of the potential :
cos —
‘sinh
It can be broken up into “fields of a single agens point” formed
as a derivative of a potential /coth }r.

VI. Identical outside the origin to the above field £, is the field
E, of a double point of rotation, whose axis is perpendicular to the
axis of the agens double point of the field /,. For that field #, we
find as scalar value of the planivector potential in a point P the total
current of foree between P and the axis of the agens double point,
that is:

sin & coth r.

So if are given a vector unity V and a scalar unity S and if
we apply along their connecting line a vector cothr, the volume
product w of V, S and the vector along the connecting line is the
scalar value of the planivector potential in S by a magnet unity
in the direction of JV.

Of w we know that when summing up S out of a positive scalar
unity S, and a negative S, it represents the current of force of a
magnet unity in the direction of V passing between S, and S,, in
other words the negative reciprocal energy of a magnet unity in the
direction of V and a magnetie strip S,.S, with intensity unity, in

( 125 )

other words the force in the direction of V by a couple of rotation
S,—>8S,. So we can regard w as the force in the direction of V by
an isolated rotation in S. So that we must take as fictitious “force
field of an element of rotation unity”

coth r,
directed perpendicularly to the radius vector. In reality, however, this
force field has rotation everywhere in Sp,.

VII. Let us now find the sealar value U, function of 7, which we
must assign to a planivector potential, that the ‘field of an element
of rotation unity” be its second derivative. We must have:

— = — conf:
dr

U = lcosech r.

And we find for an arbitrary oe :

5 ay. ot
5, \a/ ef ~ l cosech r dz,

1 WW aX
Kay {LE a, ae Rae cl Ge” “Cerny

And an arbitrary vector field X is the total derivative of the potential

Wx Wx
J Groatf Sao

VIII. We may now wonder that here in Sp, we do not find
Ff, and F, to be identical, as the two derivatives and the two
potentials of a vectordistribution are perfectly dually related to each
other in the hyperbolic Sp, as well as in the Euclidean Sp,. The
difference, however, is in the principle of the field property, which
postulates a vanishing at infinity for the scalar potential, not for the
planivector potential; and from the preceding the latter appears
not to vanish, so with the postulation of the field property the duality
is broken.

But on the other hand that postulation in Sp, lacks the reasonable
basis which it possesses in spaces of more dimensions. For, when
putting it we remember the condition that the total energy of a
field may not become infinite. As soon as we have in the infinity
of Spx forces of order e—, this furnishes in a spherical layer with
thickness dr and infinite radius described round the origin as centre an
energy of order e—2" & é&"—Dr dr = e—3y dr; which for n = 3 would

( 126 )

give when integrated with respect to 7 an infinite energy at infinity of
Spr. So for n> 3 are excluded hy the field property only vector distri-

butions which cannot have physical meaning.

For »=2 however the postulation lacks its right of existence ;
more sense has the condition (equivalent for 2 > 2 to the field pro-
perty) that for given rotation and divergency the vector distribution
must have a minimum energy. Under these conditions we shall once
more consider the field and we shall find back there too the duality
with regard to both derivatives and both potentials.

IX. Let us consider first of all distributions with divergency only
and let us find the potential function giving a minimum energy for
given \7’.

We consider the hyperbolical Sp, as a conform representation of
a part of a Euclidean Sp, bounded by a circle; if we then apply
in corresponding points of the representation the same potential, we
retain equal energies and equal divergencies in corresponding plane
elements. So the problem runs:

Which potential gives within a given curve (in this case a circle)
in the Euclidean Sp, under given divergency distribution a minimum
energy ?

According to the theorem of Green we have for this:

Ou Ou Odu 0.du 4
Lf oz (5 )ar= fx. eet t= [us — = .d0— [ug du. dt,

so that, as \(/?du is 0 everywhere within the boundary curve, the
necessary and sufficient condition for the vanishing of the variation
of the energy is:

u =O, along the boundary curve.

For the general vector distribution with divergency only in the
hyperbolical Sp, we thus find under the condition of minimum
energy also, that the potential at infinity must be 0. So we find it, just
as under the postulation of the field property, composed of fields £,,
cos p

inh » ~
The lines of force of this field #, have the equation,

derived from a potential - :

sin ~ coth r = ¢.

Only a part of the lines of force (in the Euclidean plane all of
them) form a loop; the other pass into infinity. None of the equi-
potential lines, however, pass into infinity; they are closed and are
all enclosed by the circle at infinity as the line of O-potential.

(127 )

The same holds for the arbitrary oX; of the lines of force one
part goes to infinity; the potential lines however are closed.

X. If we now have to find the field with rotation only, giving
for given rotation distribution a minimum energy, it follows from a
consideration of the rotation as divergency of the normal vector, that the
scalar value of the planivector potential at infinity must be 0, and the

u : : .
general 2X is composed of fields £,, derived from a planivector
sin ~

potential

(whilst we found under the postulation of the field
r

property sin ¢ cothr).

In contrast to higher hyperbolical spaces and to any Euclidean
and elliptic spaces the fields #, and #, cannot be summed up here
to a single isolated vector.

For this field Z, and likewise for the arbitrary 3X the lines of
force (at the same time planivector potential lines) are clused curves.

XL We have now found

Wx
1 —_ 2/ 0
a 7 f SE booth 3 ve

ipo
x= Wf Yi lL coth 4 r dr.
: 27

And from this ensues that also the general vector distribution X
having under given rotation and divergency a minimum energy is
equal to:

\a7 x ==
Xdiv. -- Oe wf aa l coth 37 dt + vf

For, if V is an arbitrary distribution without divergency and without
rotation in finite, it is derived from a scalar potential function, so it
has (according to § VIII) no reciprocal energy with Xqiv.; neither
(as according to §IX all lines of force of X;.. are closed curves
and a flux of exclusively closed vector tubes has no reciprocal
energy with a gradient distribution) with X,o:.; so that the energy
of Xaiv. + Xro.-+ V is larger than that of Xaiv. 4+ Xrot.-

So finally we have for the general vector distribution of minimum

energy XX:
Xx
Ka Vf SS looth br de

Lcoth 4 r dt.

( 128 )

C. The hyperbolic Spy.

I. Let us suppose a system of rectangular coordinates, so that
ds. = V A,u,? Se = Agila

and let us suppose a linevector distribution XY with components
X,...X,, then the integral of X along a closed curve is equal to
that of a planivector Y over an arbitrary surface bounded by it,
in which the components of Y are determined by:

a 1 [Oe 2)

= ae ————}.

Y is the first derivative or rotation of X.

Further the starting vector current of X over a closed curved
Spr is equal to the integral of the scalar Z over the bounded
volume of that Spn—1; here

1 0X, Agee

n
setebayeeey Ot,

Z is the second derivative or dwergency of X.

Oatz, Viz,

ae ee tg
Ul. If X is to be a 2X, i.e. a second derivative of a planivector
=, we must have:

I (isa Sp ergs
: 1 5: (eat a Jae
a sb a So, a a a en eel
te Aa ena 0.
n 1
The necessary and sufficient condition for this is:

Pi:

. a . it Y . . .
If YX is to be a 9X,:i.e. a first derivative of a scalar g, we must
have:

0g
p ener a
A, Oz
The necessary and sufficient condition for this is:
i sy UY

Hi, The ae which has as divergency an isolated scalar value in
the origin (comp. Opitz., Diss. Géttingen, 1881), is directed along
the radius vector, and if we put the space constant equal to 1 is
equal to

1
sink?

It is the first derivative of a sealar distribution

( 129 )

an
ar: =
sino —

~ i
and it has in the origin an isolated divergency of kh, (if &,7"—! ex-
presses the spherical surface of the Euclidean space Sp,).

IV. For two scalar distributions g and w the theorem of Grenn
holds (comp. Opitz., l.c.):

Fy 0
fox ~ dO,z—1 — gV'* wb . dt, ={vF - dO, —fvvrg - Up

(= S(V g, Vv). i)

If at infinity g and w both become O whilst at the same time
lim pw e"—lir — 0,
then for an ”—'sphere with infinite radius the surface integrals dis-
appear and we have left

fv 2 UNF apt. Inn = [wp s Wg. dtp,

integrated over the whole space.

If here we take an arbitrary potential function for g and w, (7)
for y, where r represents the distance to an arbitrarily chosen
point P — these functions satisfying together the conditions of the
formula — we have:

ks Pp = Jun (RiseNA @ 5 deat

If thus we postulate for the vector distributions under consideration
the field property (which remains defined just as for Sp,) we have,

if we put w, (r)—F, (r), for an arbitrary me

1
oX = w [wer Sie a eT

from which we deduce (compare A § VI) that there is no vector
field which has in finite nowhere rotation nor divergency; so that
a vector field is uniformly determined by its rotation and its divergency.

V. So a vectorfield is an arbitrary integral of :

1. Fields £,, of which the second derivative consists of two
equal and opposite scalar values close to each other.

2. Fields £,, of which the first derivative consists of planivectors
distributed regularly in the points of a small "—*sphere and perpen-
dicular to that “—*sphere.

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 130 )

At finite distance from their origin the fields #, and £, are of
identical structure.

VI. In order to indicate the field #, we assume a spherical
system of coordinates') and the double point in the origin along
the first axis of the system. Then the field , is the derivative of
a potential :

cos &~

sinhn—)

The lines of force of this field run in the meridian plane. It can
be regarded as the sum of two fictitious “fields of a single agens-
point” constructed as derivative ofa potential w,(r) to which, however,
must be assigned still complementary agens at infinity.

Vil. The field /£, of a small vortex--—sphere according to the
space perpendicular to the axis of the double point just considered is
identical outside the origin to the field /#,. Each line of force is now
however a closed vector tube with a line integral £, along itself.
We shall find for this field #, a planivector potential H, lying in
the meridian plane and dependent only on rand g. It appears then

simply that this H is a LX.

Let ¢ be an (m—2)-dimensional element in the ~—2 coordinates
existing besides r and g, then it defines for each r and g an element
on the surface of an "—?-sphere of a size dh = ce sinh”"—2r sin"—g,
and for the entire Sp, what may be called a “meridian zone”.

We then obtain for the current of force =, passing inside a
meridian zone between the axis of the system and a point P with
coordinates 7 and g, if ds represents an arbitrary line element
through P in the meridian plane under an angle fF with the direction
of force :

d= = dh. X ds sin F,
whilst we can easily find as necessary and sufficient condition for H:
d (Hdh) = dh. ds. X sin F ;

j s
so we have but to take = for H.
L

1) By this we understand in Spx a system which with the aid of a rectangular
system of numbered axes determines a point by 1. 7, its distance to the origin,
2. g, the angle of the radius vector with 2X), 3. the angle of the projection of
the radius vector on the coordinate space Xq...Xn with X5, 4. the angle of
the projection of the last projection on the coordinate space X3...Xn with X3;
etc. The plane through the X-direction and the radius vector we call the meri-
dian plane.

(1383
To find = we integrate the current of force inside the meridian
zone passing through the "—'spherical surface through P between the
axis of the system and P. As we have (n—1) cos ced for the
Sule Tr
force component perpendicular to that spherical surface we find:

: b
= f(—tjoory
0

cosh ' ; : :
ar sinh r dg .cé sinh "—*r sin"—2g = ce sin ™—y cothr.
sinh ™r
>>) coshr
H=— = —— sing.

h sinh"—|p

VIII. If thus are given in different points a line vector L
unity and an "—2vector W unity and if we put along their con-

necting line a line vector , then the volume product w of LZ, WV

sinh™—|p
and the vector along the connecting line is the *—*vector potential
in the direction of W caused by an elementary magnet with moment
unity in the direction of Z.

We know of w(Z,W) that with integration of W along a closed
curved Sp,—2 Q it represents the current of force of a magnet unity
in the direction of Z through Q, in other words the negative reci-
procal energy of a magnet unity in the direction of Z and a
magnetic *—'scale with intensity unity, bounded by Q, in other words
the force in the direction of Z by a magnetic *—'scale bounded
by Q, in other words the force in the direction of Z by a vortex
system, regularly distributed over Q and perpendicular to Q. So we
can regard w(Z,W) as the force in the direction of L by a vortex
unity, perpendicular to JV. With this we have found for the force
of a plane vortex with intensity unity in the origin:

coshr

sinh "—'r pie
directed parallel to the operating vortex element and perpendicular
to the ‘meridian plane’, if now we understand by that plane the
projecting plane on the vortex element; whilst @ is here the angle of
the radiusvector with the Sp,—s perpendicular to the vortex element.

IX. For the fictitious field of a vortex element in the origin intro-
duced in this way (which meanwhile has vorticity every where in space)
we shall find a planivector potential, directed everywhere “parallel”
to the vortex element and of which the scalar value U is a function
of r only.

Let us suppose a point to be determined by its azimuth parallel

( 182 )

to the vortex element and then farther in the Sp*—! of constant azi-
muth by a system of spherical coordinates, of which we take the
first axis in the “meridian plane” (see above under § VIII), and in
the plane of the vortex element, the second in the meridian plane
perpendicular to the first, and the rest arbitrarily; let us understand
meanwhile by @ here the angle of the radius vector with the Spa
perpendicular to the vortex element; let further ¢ be an (x—3)-dimen-
sional element in the n—3 last coordinates, then this defines for
each r and ~ an element on the surface of an "—%sphere, of a size

dk = ce sinh "—%r cos "—8@.

We then consider a small elementary rectangle in the meridian
plane bounded by radii vectores out of the origin and circles about
the origin and a Sp,—1 element consisting of the elements dk erected
in each point of this small elementary rectangle. Applying to this
Spn—i-element the reduction of an (—2)-fold integral along the boundary
to a (n—1)-fold integral over the volume according to the definition
of second derivative, we find:

0
a ip {U cos p . dr. cé sinh *—%r cos "—8g} dp —

0
ria {U sing . sinh r dg . ce sinh "—8r cos "—3g} dr =
r -

, coshr
= cé sinh ®—3r cos"—3p . sinh r dg . dr. ———— sin g.
sinh ®—|p
dU cosh r
(a—2) U — sini 20 cosh ee
dr sinh %—2p
dU ? ek cosh r
Bp td ee ee
The solution of this equation is:
1 1
LB oo As Yeas —2(n—2)1 oth P31 ee NR SRO SAS
i— ae cosh —2(n—2) kp foot n—3lp dtr + (nD sink a

So we find as planivector potential V of a-plane vortex:

1 1 2
— —_______—._ | coth"—}r .dir=F, (r),

(n—2) sinh"—27 2"—8 cosh %n—2)hp

directed parallel to that plane vortex.

Let us now call / the *—2vector, perpendicular to the plane vortex,
the field of which we have examined, and let us also set off the
vector potential V as an"~*vector; let us then bring in an arbitrary
point of space a line vector G; then the vector V has the property

( 133 )

that when integrated in G along a small curved closed Spn—2 in a
Spn—1 perpendicular to G, it indicates the force in the direction of
G caused by the current element JZ, or also the vector potential
in the direction of , caused by an elementary magnet with
intensity unity in the direction of G.

Let us now call the potential y(Z,/) of two "—*vectors unity
HE, F the symmetric function F,(r)cosy, where r represents the
distance of the points of application of both vectors and g their angle
after parallel transference to one and the same point of their con-
necting line, then we know that this function x gives, when e.g. Lis
integrated over a closed curved Sp,—2 which we shall cal] e, not
only the negative energy of a magnetic "—'scale with intensity unity
bounded by e in the field of a vortex unity perpendicular to F' but
also the component along / of the vector potential caused by a
system of vortices about e with intensity unity.

From this ensues again for the vector potential V of a vortex
element, that when the vortex element is integrated to a system of
vortices about a closed curved Sp,—2 it becomes the vector potential
determined according to § VI of that vortex Sp,—9; so that the

° A 1 ° E ;
vector potential of an arbitrary »X is obtained as integral of the
vectors V of its vortex elements, in other words:

7 2X
=o i eda tcn. ib: is rae EE}

where for each point the vector elements of the integral are first
brought over to that point parallel to themselves and there are
summed up.

X. So let us consider an arbitrary force field as if caused by its
two derivatives (the magnets and the vortex systems), we can then
imagine that both derivatives are propagated through the space
according to a function of the distance vanishing at infinity, causing
thereby the potential of the field.

For, the field X is the total derivative of the potential:

ul sf FP. (r) de 4 + — F, (v) de.

The extinguishment of the scalar potential is the stronger, as it is
at great distances of order e—®—)", the vector potential only of
ureer fe 0-2) |

( 134 )

Astronomy. — “The luminosity of stars of different types of
spectrum.” By Dr. A. PanneKkorx. (Communicated by Prof.
H. G. vAN DE SANDE BAKHUYZEN).

The investigation of the spectra of stars which showed that, with
a few exceptions, they can be arranged in a regular series, has led
to the general opinion that they represent different stages of develop-
ment gone through by each star successively. VogeE.’s classification
in three types is considered as a natural system because these types
represent the hottest and earliest, the further advanced, and the
coolest stage. This, however, does not hold for the subdivisions :
the difference in aspect of the lines, the standard in this case, does
not correspond to the different stages of development mentioned above.
Much more artificial is the classification with letters, which Pickrrine
has adopted in his Draper Catalogue; it arose from the practical
want to classify the thousands of stellar spectra photographed with
the objective prism. After we have allowed for the indistinctness
of the spectra which, arising from insufficient dispersion and brightness,
influenced this classification, the natural affinity between the spectra
will appear and then this classification has the advantage over that of
Vocer that the 2"¢ type is subdivided. The natural groups that can
be distinguished are: class A: the great majority of the white stars
(Sirius type), Vocr1’s Ia; class B: the smaller number of those stars
distinguished by the lines of helium, called Orion stars, VocrL’s Id.
In the continuous series the latter ought to go before the first type
and therefore they are sometimes called type 0. Class F forms the
transition to the second type (Procyon); class G is the type of the
sun and Capella (the E stars are the indistinct representatives of this
class); class K contains the redder stars of the 2¢ type, which ap-
proach to the 3¢ type, such as Arcturus (PickeriInG reckons among
them the H and I as indistinct representatives). The 34 type is
called in the Draper Catalogue class M.

The continuity of the stellar spectra is still more evident in the
classification given by Miss A. Maury. (Annals Harv. Coll. Obs. Bd. 28).
Miss Maury arranges the larger number of the stellar spectra in 20
consecutive classes, and accepts groups intermediate to these. The
classes I—IV are the Orion stars, VI—VIII constitute the first type,
IX—XI the transition to the 24 type, XIJI—XIV the 24 type
itself such as the sun, XV corresponds to the redder Arcturus stars,
XVII—XX constitute the third type. If we consider that from class
I to Il a group of lines is gradually falling out, namely the hydrogen
lines of the other series, which are characteristic of the Wolf-Rayet

(135 )

stars or the so-called fifth type stars (Voern IId), it is obvious that
we must place these stars at the head of the series, as it has also
been done by Miss Cannon in her investigation of the southern
spectra (H. C.O. Ann. Bd. 28) ').

Some of these stars show a relative intensity of the metallic
lines different from that of the ordinary stellar spectra; Voarr and
ScHEINER have found this before in a Cygni and a Persei (Public.
Potsdam Bd. 7, part 2). Maury found representatives of this group
in almost all the classes from III to XIII, and classed them in a
parallel series designated by Ilc—xXIlIc, in contradistinction to
which the great majority are called a stars.

According to the most widely spread opinion a star goes succes-
sively through all these progressive stages of development. It com-
mences as an extremely tenuous mass of gas which grows hotter by
contraction, and after having reached a maximum temperature de-
creases in temperature while the contraction goes on. Before the
maximum temperature is reached, there is a maximum emission of
light; past the maximum temperature the brightness rapidly decreases
owing to the joint causes: fall of temperature and decrease in volume.
That the first type stars are hotter than the stars of the second type
may be taken for certain on the strength of their white colour ;
whether the maximum temperature occurs here or in the Orion
stars is however uncertain.

This development of a tenuous mass of gas into a dense and cold
body, of which the temperature first increases and then decreases is
in harmony with the laws of physics. In how far, however, the
different spectral types correspond to the phases of this evolution is
a mere hypothesis, a more or less probable conjecture; for an actual
transition of a star from one type into the other has not yet been

1) According to Camppetu’s results (Astronomy and Astrophysics XIII, p. 443),
the characteristic lines of the Wolf-Rayet stars must be distinguished in two groups
and according to the relative intensity of the two groups these stars must
be arranged in a progressive series. One group consists of the first secondary
series and the first line of the principal series of hydrogen: H@' 5414, Hy’ 4542,
H5' 4261, principal line 4686); it is that group which in Maury’s classes I—III
occurs as dark lines and vanishes and which in the classes towards the other
side (class Oe—Ob Cannon) is together with the ordinary H lines more and more
reversed into emission lines. The other group, which as compared with the
hydrogen lines becomes gradually stronger from this point, consists of broad
bands of unknown origin of which the middle portions according to Cannon’s
measurements of yVelorum have the wavelengths 5807, 5692, 5594, 5470, 4654,
4443. The brightest band is 4654; its relative intensity as compared with the
Hline 4689 gradually increases in the series: 4, 47, 5, 48, 42 (Camppett’s
star numbers).

( 136 )

observed. The hypothesis may be indirectly tested by investigating
the brightness of the stars. To answer to a development as sketched
here the brightness of a star must first increase then decrease; the
mean apparent brightness of stars, reduced to the same distances
from our solar system must vary with the spectral class in such a
way that a maximum is reached where the greatest brightness is
found while the apparent brightness decreases in the following stages
of development.

§ 2. For these investigations we cannot make use of directly mea-
sured parallaxes as a general measure for the distance because of the
small number that have been determined. Another measure will
be found in the proper motions of the stars when we assume that
the real linear velocity is the same for different spectral classes. In
1892 W. H. S. Monck applied this method to the Bradley-stars in
the Draper Catalogue’). He found that the proper motions of the
B stars were the smallest, then followed those of the A stars; much
larger are the mean proper motions of the F stars?) which also con-
siderably surpasses that of the G, H and K stars and that of the
M stars. He thence concluded that these F stars (the 2¢ type stars
which approach to the 1s* type) are nearest to us and therefore have
a smaller radiating power than the more yellow and redder stars
of the 2¢ type. ‘Researches on binary stars seem to establish that
this is not due to smaller average mass and it would therefore appear,
that these stars are of the dullest or least light-giving class — more
so not only than the Arcturian stars but than those of the type of
Antares or Betelgeux” (p. 878). This result does not agree with the
current opinion that the G, K and M stars have successively developed
from the F stars by contraction and cooling. .

It is, however, confirmed by a newly appeared investigation of
Kynar Hrrtzsprunc: Zur Strahlung der Sterne*), where Maury’s
classification of the spectra has been followed. He finds for the
mean magnitude, reduced to the proper motion 0",01, the values
given in the following table where I have added the corresponding
proper motions belonging to the magnitude 4.0.

Here also appears that for the magnitude 4,0 the proper motion
is largest and hence the brightness smallest for the classes XII and

1) Astronomy and Astrophysics XI p. 874.

2) He constantly calls them incorrectly “Capellan stars’? because in the Dr. Cat.
Capella is called F, though this star properly belongs to the sun and the G stars.

5) Zeitschrift fiir wissenschaftliche Photographie Bd. III. S. 429.

( 137 )

eee

i
Spectrum Magn. for | P. M. for

Maury ioc c.| P. M. 0” 01 | Magn. 4.0
22 a ees tg Oe

| | i

ad P| ea 0.012
V—VI BoA phe a 0.045
VII—VII A 8.05 0.065
= x1 F 9.06 0.103
XI—XIII Fc | 4.93 | 0.979
XMI—XIr) | G 7 93 0.064
XV K 9.38 0.119
XV—XVI ae Ae Ti 0.057
XVIT—X VIII Near.” 18:98 0.072

XIII that form the transition from F to G; for the later stages of
development the brightness again increases.

§ 3. A better measure than the proper motion for the mean
distance of a group of stars is the parallactic motion. This investiga-
tion was rendered easy by means of N° 9 of the “Publications of
the astronomical Laboratory at Groningen”, where the components
t and v of the proper motion are computed with the further anxiliary
quantities for all the Bradley-stars. Let t and v be the components of
the proper motion at right angles with and in the direction of the
antapex, 2 the spherical distance of the star-apex, then

. __ fvsind
1 SS sind
is the parallactic motion for a group of stars, i.e. the velocity of the
solar system divided by the mean distance of the group. The mean
of the other component Sr is, at a random distribution of the
directions, equal to half the mean linear velocity divided by the
distance.

The mean magnitudes of the different groups are also different.
Because we here especially wish to derive conclusions about the
brightness, and as both the magnitude and the proper motion depend
on the distance the computation was made after the reduction to

1) The Roman figures in italics in Maury’s classification designate the transition
to one class higher.

( 138 )

magnitude 4.0; that is to say, we have imagined that every star
was replaced by one which in velocity and in brightness was perfectly
identical with the real one, but placed at such a distance that its
apparent magnitude was 4.0. If the ratio in which we then increase
the proper motion is

os 100.2 (m—4)
we have
= pv sina a pr
a A | = ,
at = sin? 2 = n

In this computation we have used Mauvry’s classes as a basis. We
have excluded 61 Cygni on account of its extraordinary great parallax,
while instead of the whole group of Ursa Major (8 y d¢ $) we have taken
only one star (¢). In the following table are combined the results
of the two computations.

Spectrum | Typical al | mean} mean | ee
MAURY | Dr. Cat. star | m a Ee: oe 4 740
or "
I—IlI B ¢ Orionis 33° | 3.57 | 0.007 0. ‘018 0. 007 0. O13"
IV—V B—A 7 Orionis 48 | 4.31 | 0.011} 0.035 | 0.014 | 0.036
VI—VIII A Sirius 93 | 3.92] 0.040] 0.054] 0.038 | 0.064
IX—XII F Procyon 94 | 4.44; 0.089 , 0.153; 0.095 | 0.136
XIII—XIV G Capella 69 | 4.08 | 0.444 | 0.157] 0.460} 0.199
XV K Arcturus |} 101 | 3.90] 0.423 0.419] 0 1290}. 0.096
XVI—XX M Betelgeuze | 61 | 3.85 | 0.049 | 0.068 | 0.050] 0.061

In both the series of results the phenomenon found by Monck and
HrrtzsPpruUNG manifests itself clearly. I have not, however, used
these numbers t49 and g4o, but have modified them first, because it
was not until the computation was completed that I became ac-
quainted with Hrrtzsprune’s remark that the above mentioned ¢ stars
show a very special behaviour; their proper motions and parallaxes
are so much smaller than those of the a stars of the same classes
that they must be considered as quite a separate group of much
greater brilliancy and lying at a much larger distance’). We have

‘; In his list of parallaxes Herrzsprune puts the question whether perhaps the
bright southern star « Carinae (Canopus) belongs to the ¢ stars; but he finds no
indication for this except in its immeasurably small parallax and small proper
motion. In her study of the southern spectra Miss Cannon has paid no regard

Class | * | 74.0 | 14.0 a/4
I | 5 | 0.009 | 0.022 0.8
II | 13 | 005 009 | 4.4
ll | 14 | 006 | 0.8
IV | 18 | O14 8 | 1-2
IV | 16 | O16 oa 0.7
V | 41 | 009 a 0.4
VI | 16 030 ocs | 0.9
VII | 30 040 ~ 09
Vill | 4A | al 55 | 1.6
i | 25 | 050 | 0 | 1.6
xX | 16 | 070 474 | Oc
XI | 92 | 103 a aoa
XII | 23 | 170 ea a}
XIII | 18 | 997 0 | 1.7
XIV | 21 | 192 a des
XIV 20 | m7 | aA 6.2
XVA | 26 | oh | us 3.2
XV B | 35 | oa 070 | 3.0
XVC | 40 | 9 | 087; 14
XVI 19 049 071 | 1.4
XVII 19 049 032. |, 2.4
XVIII 16 a 075 | 1.3
Sie eee 7 | ea 078 | 1.5

to the difference between the @ and the c stars. Yet all the same this question
may be answered in the affirmative; on both spectrograms of this star occur-
ring in her work, we see very distinctly the line 4053.8, which in Capella and
Sirius is absent and which is a typical line for the ¢ stars. Hence follows that
a Carinae is indeed a ¢ star,

( 140 )

therefore repeated the computation after exclusion of the c and the
ac stars.

The table (see p. 139) contains the results for all the classes of Matry
separately ; class XV_ is divided into three subdivisions: XV A are
those whose spectra agree with that of a Boétis, XV C are those which
agree with the redder a Cassiopeiae, while XV 6 embraces all those
that cannot with certainty be classed among one of the other two
groups.

The values for ty9 and go differ very little from those of the
preceding table. If we take the value of the velocity of the solar
system = 4.2 earth’s distances from the sun, the g’s divided by 4.2
yield the mean parallax of stars of different spectral classes for the
magnitude 4.0 (20,4). Reversely, we derive from the q’s the relative
brightness of these stellar types, for which we have here taken the
number which expresses how many times the brightness exceeds
that of magnitude 4.0 when placed ata distance for which g = 0".10,
hence with the parallax 0".024. Finally the last column 2t/q contains
the relation between the mean linear velocities of the group of stars
and our solar system.

In the following table we have combined these values in the same
way as before.

Spectrum | Typical | | | L for

Maury | Dr. eat star . | — of ot 040 Gx
oe a, -

I—IlI B = Orionis 32 | 0.005°; 0 014 | 0.0033 | 51 08
IV—V B—A 7 Orionis 45 0.013 | 0.036 | 0.0086 yw 0.7
VI—VIII A Sirius 87 0.040 | 0.063 | 0.015 2.5 1.3
IX—xXII | F Procyon 86 0.101 | 0.144 | 0.034 | 0.50 | 44
XIN—XIV G Capella 59 0 182 | 0.294 | 0.0353 0.20::) hae
XV K Arcturus | 101 0.120 | 0 096 | 0.023 UB 2.5
X VI—XX M Betelgeuze | 61 0.050 | 0.061 | 0.015 2.7 1.6

§ 4. Conclusions from this table. The numbers of the last column
are not constant but show a systematic variation. Hence the mean
linear velocity is not constant for all kinds of stars but increases
as further stages of development in the spectral series are reached.
(Whether the decrease for the 3'¢ type, class M, is real must for
the present be left out of consideration). That the linear speed of the
Orion stars is small is known and appears moreover from the

( 141 )

radial velocities. While Campsern found 19.9 kilometres for the
velocity of the solar motion, and 34 kilometres for the mean velocity
of all the stars, Frost and Apams derived from the radial velocities
of 20 Orion stars measured by them, after having applied the correction
for the solar motion: 7.0 kilometres as mean value’), hence for the
actual mean speed in space 14 kilometres, whence follows the ratio
0.7 for 2t/q. Hence the Orion stars are the particularly slow ones and
the Arcturian stars (class XV) are those which move with the greatest
speed.

§ 5. When we look at the values of gio or those of 249 or
Lo.1o, derived from them, we find, as we proceed in the series of
development from the earliest Orion stars to the Capella or solar
type G, that the brightness constantly decreases. That g was larger for
the 24 type as a whole than for the first (the Orion stars included)
has long been known; some time ago Kaprnyn derived from
the entire Bradley-Draper material that on an average the 24 type
stars (F GK) are 2,7 times as near and hence 7 times as faint as
the 1st type stars (A and B). This result perfectly agrees with the
ordinary theory of evolution according to which the 2¢ type arises
from the 1st type through contraction and cooling.

A look at the subdivisions shows us first of all that the Orion
stars greatly surpass the A stars in brightness, and also that among
the Orion stars those which represent the earliest stage greatly
surpass again in brightness those of the later stages. As compared
with the solar type G the Sirius stars are 12 times, the stars which
form the transition to the Orion stars 38 times and lastly the ¢ Orionis
type 250 times as bright. This result is in good harmony with the
hypothesis that one star goes successively through the different con-
ditions from class I to class XIV; we then must accept that the
density becomes less as we come to the lower classes. Whether the
temperature of the Orion stars is higher than that of the Sirius stars
or lower cannot be derived from this result; even in the latter case
it may be that the larger surface more than counterbalances the
effect of smaller radiation. This must be decided by photometric
measurements of the spectra. As the Wolf-Rayet stars follow next
to class I, an investigation of their proper motion, promised by
Kaprryn, will be of special interest.

Past the G stars, the solar type of the series, the brightness again
increases. The values obtained here for g confirm in this respect the
results of Monck and Herrzsprunc.

1) Publications Yerkes Observatory. Vol. II. p. 105.

( 142 )

Against the evidence of the g’s only one objection can be made,
namely that these classes K and M might have a proper motion
in common with the sun, so that gq would not be a good measure
for the distance. A priori this objection is improbable but it may be
tested by material, which, though otherwise of small value, may for
this kind of investigations yield very valuable conclusions on this
point, namely the directly measured parallaxes. HERTzsPRUNG gives
mean values of the measured parallaxes reduced to magnitude 0,0;
by the side of these we have given the values for somewhat different
groups derived from our 24.9:

Observed 2.0 Derived from g 20.0

ny 0".0255 (6) SEE 0".021

iy Vi 0.106; =45) ie 0 .054

Vi— yall 0 .153 (40) VI—Vill 70094
PEI 0 .226") (6) LX Xe ad

FON x 0 .442 (2)

x1Y 0 DoTiatG) XHIl—X1V-0 2335

XV 0.514, 48) XV 0 14

XVI Ox, 43) XVI—XX .0 .096

XVII—XVIII 0 .115 (3)

In general Hertzsprune’s numbers are somewhat larger, this can
be easily explained by the circumstance that many parallaxes measured
in consequence of their large proper motions will probably be above
the mean. It appears sufficiently clear from this, at any rate, that
alsu the directly measured parallaxes markedly point at an increase
of brightness past class XIV, and that there is not the least ground
to assume for the other groups a motion in common with the sun.

It is therefore beyond doubt that the K and M stars have a
greater intrinsic brilliancy than the F and G stars. Moncxk derives
from this fact that they have a greater radiating power, because
about the same value for the masses is derived from the double stars.

That the latter cannot be derived from the double stars will
appear hereafter. Moreover Monck’s conclusion of the greater radiating
power of the K and M stars is unacceptable. In incandescent bodies
this radiating power depends on the temperature of the radiating
layers and of the atmospheric absorptions. With unimpaired radiance
a greater amount of radiation is accompanied with bluer light (because
the maximum of radiation is displaced towards the smaller wave-
lengths) as both are caused by the higher temperature. The general
absorption by an atmosphere is also largest for the smaller wave-
lengths, so that when after absorption the percentage of the remain-

( 143 )

ing light is less, the colour of the radiated light will be redder.

Therefore it is beyond doubt that a redder colour corresponds at
any rate with a less degree of radiance per unit of surface.

Then only one explanation remains: the K and M stars (the redder
2nd type stars like Arcturus and the 34 type) possess on an average
a much larger surface and volume than the other 2»¢ type stars of
the classes F and G. This result is at variance with the usual
representation of stellar evolution according to which the redder K
and later the M stars are developed from the yellow-white F and G
stars by further contraction and cooling.

§ 6. A further examination of the constitution of these stars shows
us that it is improbable that they should possess a very small
density; the low temperature, the strongly absorbing vapours point
to a stage of high condensation. These circumstances lead to expect
greater (with regard to the F and G stars) rather than less density.
From the larger volumes it then follows that the K and WM. stars
have much larger masses than the F’s and G's. This result is the
more remarkable in connection with the conclusion derived above
about their greater mean velocity. If the stars of our stellar system
form a group in the sense that their velocities within the group
depend on their mutual attraction, we may expect that on an
average the velocities will be the greater as the masses are smaller.
No difficulty from this arises for the Orion stars with small speed,
because the same circumstances which allow us to ascribe to them
a mass equal to that of the A, F and G stars, enable us likewise
to ascribe to them a larger mass. The K stars which have both
a greater mass and a greater velocity are characterized by this
thesis as belonging to a separate group, which through whatever
reason must originally have been endowed with a greater velocity.
Arcturus with its immeasurably small parallax and large proper
motion is therefore through its enormously great linear velocity and
extraordinary luminosity an exaggerated type of this entire class, of
which it is the brightest representative. Therefore it would be worth
while to investigate separately the systematic motions of the K stars
which hitherto have been classed without distinction with the F and
G stars as 2>4 type.

If this result with regard to the greater masses of the K and M
stars should not be confirmed, the only remaining possibility is the
supposition that the density of these star ts extremely small. In this
case their masses might be equal to that of other stars and they
may represent stages of evolution of the same bodies. Where

( 144 )

they ought to be placed in the series of evolution remains a riddle.
There is a regular continuity in the series F—G—K—M;; and accord-
ing as we suppose the development to take place in one direction
or in the other we find in the transition G—K either cooling accom-
panied with expansion, or heating accompanied with contraction. The
puzzling side of this hypothesis can also be expressed in the follow-
ing way: while in the natural development of the celestial bodies,
as we conceive it, the temperature has a maximum but the density
continuously increases, the values obtained here would according
to this interpretation point at a maximum density in the spectral
classes F and G.

In Vol. XI of Astronomy and Astrophysics Maunprer has drawn
attention to several circumstances, which indicate that the spectral
type rather marks a difference in constitution than difference in the
staze of development. ‘‘There seems to me but one way of recon-
ciling all these different circumstances, viz.: to suppose that spectrum
type does not primarily or usually denote epoch of stellar life, but
rather a fundamental difference of chemical constitution” *). One of
the most important of these facts is that the various stars of the
Pleiades, which widely differ in brightness and, as they are lying at
the same distance from the sun, also in actual volume show yet
the same spectrum. The result found here confirms his supposition.

One might feel inclined to look for a certain relation between
these K and M stars and the ¢ stars, which, according to Hertzsprune,
have also a much greater luminosity, hence either less density or
greater mass than the similar a stars; and the more so as these ¢ stars
reach no further than class XIII. Yet to us this seems improbable;
the K stars are numerous, they constitute 20°/, of all the stars,
while the cstars are rare. Moreover the spectra of all the K stars
are with regard to the relative intensity of the metallic lines perfectly
identical with the astars of preceding classes such as the sun and
Capella. Therefore it as yet remains undecided to which other
spectra we have to look for other phases in the K star lives and
to which spectra for those in the c star lives. The c stars, except a
few, are all situated in or near the Milky Way : this characteristic feature
they have in common with the Wolf-Rayet stars and also with the
4th type of Succut (Vogel’s IJId), although these spectra have no lines
in common which would suggest any relation between them.

§ 7. The constitution found here for the Arcturian stars among
the third type stars may perhaps be tested by means of other

1) Stars of the first and second types of spectrum. p. 150.

( 145 )

data, namely by those derived from the double stars. The optically
double stars cannot however teach us anything about the masses of
the stars themselves as will appear from the following consideration
{also occurring in “The Stars’ by Nrwcoms). Let us suppose that a
binary system is nm times as near to us, while all its dimensions
become n times as small, but that the density and the radiation
remain the same. Then the mass will diminish in the propor-
tion of n* to 1, the major axis of the orbit @ in the proportion
of n to 1 and hence the time of revolution remains the same;
the luminosity becomes m’* times as small, therefore the apparent
brightness remains the same as well as the apparent dimensions of
the orbit, in other words: it will appear to us exactly as it was
before. Hence the mass cannot be found independently of the
distance. Let @ be the angular semi-major axis, J/ the mass, P
the time of revolution, d the density, 2 the radiating power, 2 the

parallax and @ the radius of the spherical volume of the star, then
3

a
we shall have: 2°M = =" the mass J/ is a constant value x 0°,

the apparent brightness H/ is a constant x 2°04. Eliminating from
this the parallax and the radius, we find
| pe 38
i — € ra
a J

Thus from the known quantities: elements of orbit and brightness,
we derive a relation between the physical quantities: density
and radiating power, independently of the mathematical dimen-
sions. This relation has been derived repeatedly. In the paper

3/ 3
cited before Maunprer gives values for the density 6= (7) °F,
in the supposition of equal values of 2; he found for the Sirius stars
(ist type) 0,0211, for the solar stars (all of the 2°4 type) 0,3026,
hence 14 times as large on an average; we can also say that
when we assume the same density the radiating power of the
Sirius stars would be 6 times as large; the exact expression would
be that the quotient 4°/d? is 200 times as Jarge for the Sirius stars
as for the solar stars.

In a different form the same calculation has been made by
Hertzsprune by means of ArrKeEn’s list of binary system elements °).
By means of — 2,5 log H=m he introduces into his formula the
stellar magnitudes ; if we put in the logarithmical form

1) Lick Observatory Bulletin Nr. 84.
10

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 146 )

3 log H + 4 log P — 6 log a = const. +- 3 log 4 — 2 logd
m—?°/, log P+ 5loga= My
then we have m, = const. — 2,5 log 4 + °/, log 4.

If we arrange the values of mm, after the spectra according to the
Draper Catalogue (for the Southern stars taking CANNon; according
to the brightest component @ Centauri was reckoned to belong to
class G), we find as mean values:

Class A —2.92 (9 stars — 4.60 to — 1.09)
F —1.32(19 ,, —3.61 ,, +014)
, GandE—049(/1 ,, —1.60 ,, + 1.28)

The 3 stars of the type K (with H) give — 4.88 (y-Leonis),
—1.05 and + 0,87, hence. differ so widely that no valuable result is
to be derived from them. To the extraordinarily high value for
z°/d? given by y Leonis attention has repeatedly been drawn.
While for a great number of stars of the other classes the extreme.
values of m, differ by 3.5 magnitudes we find that y Leonis differs
by 5. magnitudes from the mean of the two other values, that is to
say: its radiating power is a hundred times as large, or its density
is a thousand times as small as for these other stars. For the classes
A and F we find that 2*/d? is 640 and 8 times respectively as large
as for class G; conclusions about class K as a whole, such as are especially
wanted here, cannot be derived from it. It may be that an investi-
gation of binary systems with partially known orbit motion (for which
we should require auxiliary hypotheses) would yield more results.

About the mass itself, however, something may be derived from
the spectroscopic binary systems. The elements derived from obser-
vation asini and P directly yield M sin *z; as it is improbable that
there should be any relation between the type of spectrum and the
angle between the orbit and the line of sight we may accept the
mean of sin*i to be equal for all groups. For systems of which only
one component is visible, the element derived from observation
contains another unknown quantity, viz. the relation 8 of the mass
of the invisible to. that of the visible star. If a is the semi major
axis of the orbit of the visible star round the common centre of
gravity, we have

a® sin *i : Ree
= ni

‘A p

Se ht
(1+)?
It is not perfectly certain, of course, that on an average p is the

same for all classes of spectrum; if this is not the case the M’s.
> sin *4
may behave somewhat different from the values of — computed

here.

( 147 )

Unfortunately, of the great number of spectroscopic double stars
discovered as yet (in Lick Observatory Bulletin N°. 79 a number
of 147 is given) the orbit elements of only very few are known.
They give, arranged according to their spectra:

Group II—IV (Bb) Group VI—VIII (A)
Orion type Sirius type
o Persei 0.61. 6B Aurigae 0.56
y Orionis 2.01 o Ursae (3.41) *)
d Orionis 0.60 Algol 0.72
6 Lyrae 7.85 @ Androm. 0.36 *)
a Virginis 0.33 a, Gemin. 0.002
V Puppis 34.2
Group XII—XIV a (F—G) Group XII—XIV ac
Solar type e@ Ursae min. 0.00001
a Aurigae 0.185 $Geminorum 0.0023
xy Draconis 0.120 7 Aquilae 0.0029
(W Sagittarii- 0.005) J Cephei 0.0031
(X Sagittarii 0.001)
e Pegasi 0.117 Group XV (K)
4 Pegasi 0.254 6 Herculis 0.061

Of the K stars only one representative occurs here, so neither
this material offers anything that could help us to test the results
obtained about this stellar type. But all the same, some remarkable
conclusions may be derived from this table. It appears here that
notwithstanding their small number the Orion stars evidently surpass
the others in mass, while the Sirius stars seem also to have a some-
what greater mass than the solar stars. Very striking, however,
is the small mass of the ¢ stars approaching towards a. Hence the
c stars combine a very great luminosity with a very small mass, and
consequently their density must be excessively small. If it should be
not merely accidental that the three regularly variable stars of short
period, occurring in Mavry, all happen to show c characteristics
and a real connection should exist between this particularity of
spectrum and the variability, we may reasonably include into the

1) In the case of ¢ Ursae a has been taken equal to the semi major axis of the
relative orbit; hence this number is proportionally too large by an unknown

number of times.
2) Assumed period 100 days, veloeity in orbit 32.5 kilometres.

( 148 )

group W and X Sagittarii which also yield small values; as has
been remarked, for the southern stars no distinction is made between
the a and the ¢ stars’).

We may expect that within a few years our knowledge of the
orbits of the spectroscopic double stars will have augmented consi-
derably. Then it will be possible to derive conclusions like those
found here from much more abundant material, and also to arrive
at some certainty about the mean mass of the K stars. With regard
to the latter our results show at any rate that in investigations on
grouping of stars and stellar motions it will be necessary not to
consider the 2¢ type as one whole, but always to consider the
F and G stars apart from the redder K stars.

1) In this connection may be mentioned that in 1891 the author thought he
detected a variability of 2 Ursae minoris with a period of a little less than 4 days.
The small amplitude and the great influence of biased opinions on estimations of
brightness after ARGELANDER’s method in cases of short periods of almost a full
number of days, made it impossible to obtain certainty in either a positive or a
negative sense. CAmpBELL’s discovery that it is a spectroscopic binary system with
a period of 34235 14m makes me think that it has not been wholly an illusion.

Buk OR AT wk:

In the Proceedings of the Meeting of June, 1905, p. 81:
line 7 from top, read: “cooled by conduction of heat’,
3 JO 4,8 o4 . dors? “iin 2) TV sread <P ae
In Plate V belonging to Communication N°. 83 from the physical
laboratory at Leiden, Proceedings of the Meeting of February 1905,
p- 502, the vacuum glass B’, has been drawn 18 em. too long.

(August 21, 1906).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday September 29, 1906.

DOG

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 29 September 1906, Dl. XV).

eee Daw Ls.

M. NieuwEenuvis-von UEXKULL-GULDENBAND: “On the harmful consequences of the secretion
of sugar with some myrmecophilous plants”. (Communicated by Prof. J. W. Mort), p. 150.

H. Kameruiwen Onnes: “Methods and apparatus used in the cryogenic Laboratory at Leiden.
X. How to obtain baths of constant and uniform temperature by means of liquid hydrogen”,
p. 156. (With 3 plates); XI. “The purification of hydrogen for the cycle”, p.171.(With 1 plate);
XII. “Cryostat especially for temperatures from — 252° to —259°”, p. 173. (With 1 plate); XIII.
“The preparation of liquid air by means of the cascade process”, p. 177. (With 1 plate); XIV.
“Preparation of pure hydrogen through distillation of less pure hydrogen”, p. 179.

H. Kameriixcu Onnes and C. A. Cromme in. “On the measurement of very low temperatures
1X. Comparison of a thermo element constantin-steel with the hydrogen thermometer”. p. 180.

H. Kameriiwcu Onnes and J. Cray: “On the measurement of very low temperatures X.
Coefficient of expansion of Jena glass and of platinum between + 16° and —182°, p. 199. XI.
A comparison of the platinum resistance thermometer with the hydrogen thermometer, p. 207.
XII. Comparison of the platinum resistance thermometer with the gold resistance thermometer”,
p. 213. :

Jan DE Vriks: “Quadratic complexes of revolution”, p. 217.

J. K. A. Wertueim Satomonson: “A few remarks concerning the method of the true and
false cases”. (Communicated by Prof. C. W1NKLER), p. 222.

J. J. van Laar: “The shape of the spinodal and plaitpoint curves for binary mixtures of
normal substances. 4th Communication: The longitudinal plait’. (Communicated by Prof.
H. A. Lorentz), p. 226. (With 1 plate).

di
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 150 )

Botany. — “On the harmful consequences of the secretion of sugar
with some myrmecophilous plants.” By Mrs. M. Nrevwenuuis-
VON UEXKULL-GULDENBAND. Ph. D. (Communicated by Prof.
J. W. Mott).

(Communicated in the meeting of June 30, 1906).

During my residence of about eight months at Buitenzorg in 1901
I occupied myself chiefly with an investigation of the structure and
peculiarities of the sugar-secreting myrmecophilous plants. The results
of these observations, extending over some 70 plants, are inconsistent
with the opinion expressed by DeE.pino, Kerner, TRELEASE, Burck
and many others, that the extrafloral secretion of sugar by plants
would serve to attract ants which in return would protect the plants
against various harmful animals.

For I was unable to observe in a single instance that the secretion
of sugar is useful to the plant; on the other hand it appeared to
me that the ants feed on the sugar, but that, instead of being useful
at the same time, they injure the plant indirectly by introducing
and rearing lice; moreover the extrafloral nectaries attract not only
ants but also numbers of beetles, bugs, larvae, etc. and these are
not content with the sugar alone, but at the same time eat the
nectaries themselves and often consume the leaves and flowers to no
small extent.

In about one third of the plants, investigated with this purpose,
the secretion of sugar in this way certainly does much harm; with
another third the plants experience only little harm by attracting
the undesirable visitors, while with the last third no indication at
all could be found that by secreting sugar they were worse off than
other plants.

Of those that were indirectly injured by secreting sugar I here
only mention a few examples out of the many which I shall consider
more extensively elsewhere.

Spathoglottis plicata Bl. is a common orchid in the Indian archi-
pelago. In the environs of Buitenzorg it is e.g. found on the Salak,
and it is used in the Botanical Garden to set off the beds in the
orchid quarter. Its leaves (all basal leaves) have a length of as much
as 1.20 M., according to SmitH, they are narrow, have a long point
and are folded lengthwise; their inflorescence is erect, reaches a
height of about 2 metres and bears at its extremity, in the axils
of coloured bracts, a number of flowers, the colour of which varies
from red violet to white. The bracts and perianth leaves have blunt,

(151 )

thick and darker coloured points. On the inflorescences two kinds
of ants always abound, one large and one small species. Even when
the flower-buds are- still closed the ants are already found on the
bracts and no sooner are the flowers open than the ants also attack
the perianth leaves. It appeared that sugar was secreted as a bait here.

In order to prove this the flowers were placed for some time
under a damp glass bell-jar; after a few hours by means of FEauine’s
reagent sugar could be proved to be present in the liquid secreted
by the leaves at the exterior side. I could find no special organs
for this secretion, however; probably the secretion is an internal one
the product being brought out by the epiderm or the stomata.

It was already known to Dertrino that some orchids secrete sugar
on the perianth; the remarkable point with the just mentioned
Spathoglottis is that the ants have such an injurious influence on it.
Whereas namely the small species remains on the flowers and is
content with the sugar there secreted, the big species also descends
to the basal-leaves and attacks these also, often to such an extent
that only a skeleton of them remains. These harmful big ants are
not expelled at all by the much more numerous small ones. It
further appeared most clearly that the secretion of sugar was the
reason indeed why such important organs as the leaves were eaten
by the big species. The proof was namely afforded by those plants
that had finished flowering and bore fruit; with these secretion of
sugar took place no longer and the leaves, which were produced in
this period, remained consequently uninjured. So it was the secretion
of sugar during the flowering period which attracted the ants, while
the leaves as such were no sufficient bait.

A second instance of the great harm that may be caused to the
plants themselves by the secretion of sugar, is seen with various
tree- and shrublike Malvaceae. In the Botanical Garden stands an
unnamed tree, a Malvacea from Indo-China. This not only has
nectaries on the leaves and calyx, but also offers the ants a very
suitable dwelling-place in the stipules, which occur in pairs and are
bent towards each other. The spaces formed in this way are indeed
inhabited by ants, but not by so many as might be expected. The
reason is that in spite of the abundance of nectaries they find no
sufficient food, since on these trees a species of bugs occurs which
not only consume the secreted sugar but also eat the nectaries them-
selves. These bugs moreover injure the leaves to such an extent
that the tree suffers from it, as may be seen by a cursory examina-
tion. The same may be stated of a tree named “Malvacea Karato”
and of some other species of this family.

411*

( 152 )

In order to prove that the secretion of sugar by attracting harmful
insects is indeed injurious to these trees it would still be necessary
to show that they remain uninjured when the secretion of sugar
does not take place. This proof is readily afforded by some other
Malvaceae.

Two shrublike Malvaceae of common occurrence in India, namely
Hibiscus rosa sinensis L. and Hibiscus tiliaceus [. have nectaries
on their leaves. They are not frequented by ants or other harmful
insects, however, because in the nectaries, as far as my observations
go, a fungus always occurs, which may be recognised already from
the outside by its black colour. This fungus prevents the secretion
of sugar, and the nectaries cease to have an attraction for insects’
which otherwise would be harmful to the plant. These shrubs by
their healthy appearance contrast strongly with the above mentioned
plants in the Malvaceae quarter, which are frequented by ants and
other insects.

On account of the circumstance that the extrafloral nectaries are
found chiefly on and near the inflorescences, Burck proposed the
hypothesis, that in some cases they would serve to attract ants into
the neighbourhood of the flowers in order to protect these against
bees and wasps, which would bore them and rob honey. But even
with the plants investigated by him I could find no confirmation of
his hypothesis. First the nectaries only rarely occur on the inflores-
cences exclusively; also the plants mentioned by him as proof as:
Thunbergia grandiflora Roxb., Gmelina asiatica L., and Gmelina
bracteata, Nycticalos macrosyphon and Nycticalos Thomsonii cannot
serve as examples, since these plants also on their vegetative parts
such as leaves and stems possess nectaries, which according to him
are not present there or are not mentioned. In regard to the so-called
“food-bodies’” (Burck’sche K®6rperchen) on the calyx of Thunbergia
grandiflora, it appeared to me that these are no “‘food-bodies” at
all, but ordinary sugar-secreting deformed hairs which I also found
on the bracts, leaves and leaf-stalks of this plant.

-Furtber it appeared to me that the number of bored flowers stands
in no relation to the number of nectaries occurring on the calyx,
as should be the case according to Burcx. It is much more dependent
on external factors, as e.g. the more or less free situation of the
plants, the weather ete.

As an example the creeper Bignonia Chamberlaynii may be men-
tioned. Of this plant on many days only 1,6 °/, of the fallen flowers
appeared not to have been bored by Xylocopa coerulea, although
numerous ants always occur on the nectaries of the calyx.

( 153 )

An example of the fact that the more or less free situation in
fluences the number of perforations of the flowers is found in two
species of the genus Faradaya, both having nectaries on the calyx
and the leaves. With Paradaya papuana Scuerr., which stands in
the Botanical Garden at Buitenzorg surrounded by many other rich|;
flowering plants, the flowers are often perforated by a boring wasp;
of the fallen flowers only 1 °/, was undamaged. This was different
with another still unnamed species of the same genus which, as fai
as the nectaries were concerned, showed no difference with the
former and grew at some distance from it in a less open site. Its
branches hung partly to the ground and bore far fewer flowers
than Faradaya papuana. Now of this three 19,3 °/, of the flowers
remained unperforated.

And in regard to the weather it appeared that the number of
bored flowers closely depends on it. After a sunny day a much
larger number of flowers had been bored the next morning than
when rain had prevented the insects from flying out. This was
e.g. very conspicuous with [pomoea carnea Jacg., a shrub having
nectaries as well on the leaves as on the calyx, the latter being
bored by Vespa analis and two Xylocopas. Collected in the morning
without regard to the weather of the preceding day 90°/, of the
fallen flowers were bored; after rainy days 57 °/, of the flowers
were damaged and after sunny days even 99,1 °/, were bored.

From this appears most clearly how little value must be assigned
to statistical data about the perforation of flowers and about their
being eventually protected by ants if not at the same time all other
circumstances which may influence the results have been taken into
account.

When trying to fix the part, either favourable or otherwise, played
by insects with regard to a plant, one meets with greater difficulties
in the tropics than e.g. in Middle Europe, because the vegetative period
lasts so much longer. So one may meet an abundance of definite
insects during one part of that period which are not found during
another part. This special difficulty of the question whether special
arrangements in a plant form an adaptation to a definite animal
species is still enhanced in a botanical garden by the circumstance
that there nearly all the plants are in a more or less uncommon
site or surroundings. Yet here also the matual behaviour of the animals
frequenting the plants may be investigated as well as their behaviour
towards the plants themselves, while the results enable us to draw
some justified conclusion as to the mutual relations in the natural
sites of these plants. I took this point of view when I began my

( 154 )

investigation and among others put myself the following questions
to which the here briefly mentioned answers were obtained:

1. On what parts of plants is extrafloral secretion of sugar found ?

In the cases examined by me | found secretion of sugar on the
branches, leaves, stipules, bracts of different kind, peduncles and
pedicels, ovaries and the inner and outer side of calyx and corolla,
in each of these organs separately or in a great number of different
combinations. The most commonly occurring of these combinations
were: a. on leaf-sheaths and calyx together, 5. on the leaf-blade only
ce. on the leaf-stalks, peduncle and calyx. Of other combinations I
only found from one “to three examples each.

2. Does the structure or place of the nectaries clearly indicate
that they are made for receiving ants?

Except in a few cases (as the nectaries occurring in the closely
assembled flowers of Gmelina asiatica Scheff. on that side of the
calyx, that is turned away from the axis of the inflorescence) this
question must be decidedly answered negatively. Although it seems
as if the very common cup shape of the nectaries were eminently
suitable for storing the secreted honey, yet on the lower side of the
leaves these nectaries are for the greater part found with their
opening turned downward. I remind the reader of the two large,
also downwardly directed cup-shaped nectaries at the base of the
side leaves of some species of Erythrina.

The frequent occurrence of nectaries on the calyces, which only
in the budding period secrete honey, seems to indicate that these
buds require special protection. But inconsistent with this view is
the fact that sometimes, according to my observations, only half of
the flowers has nectaries in the calyces (e.g. Spathodea campanulata
BEAUV.).

With many species of Smz/ax only part of the branches attracts
ants and these are branches that carry no flowers and so, according
to the prevailing conception, would least require protection. It is
difficult to make the idea of the protection of the flowers agree with
the fact that nectaries occur on the inner and outer side of the upper
edge of the tube of the corolla of Nycticalos macrosyphon, Spathodea
serrulata and others. Attracting ants to the entrance of the corolla,
which is the very place where the animals causing cross-fertilisation
have to enter, has certainly to be called unpractical from the biolo-
gist’s point of view.

Against the conception that these plants should require protection,
also the fact pleads that exactly with young plants, where protection
would be most necessary, these baits for protective ants are absent.

(155 )

A short time ago Utz’) has drawn attention to this as a result of
his investigation of American plants.

3. Is sugar secreted in al/ nectaries?

This is not the case; in some nectaries I could detect no secretion
even after they had stayed for a long time under a bell-jar; this
was the case e.g. with the leaves of Gmelina asiatica. Consequently
they are not frequented by ants, although these insects always occur
on the similarly shaped but strongly secreting nectaries of the calyx.

The quantity of the secreted substances moreover fluctuates with
the same nectaries of the same plant and depends on many external
and internal influences.

4. Are all the products secreted by the nectaries always and
eagerly consumed by the ants?

Evidently this also is not always the case, for whereas the necta-
ries of some plants are constantly frequented by ants, with others
the nectaries so to say overflow, witbout a single animal visiting
them. (So with some species of Passi/lora).

5. At what age of the organs do the nectaries secrete sugar?

As a rule the nectaries of the inflorescences cease to secrete as
soon as the flowers are opened; those of the leaves even only
functionate in the youngest stages of development.

6. Are the ants that frequent the plants with nectaries hostile
towards other visitors?

Although I daily watched the behaviour of the ants with the
extrafloral nectaries for hours, I have never observed that they
hindered other animals in any way. On the Luffa species one may
see the ants at the nectaries peacefully busy by the side ofa species
of beetles which does great damage to the plant by eating leaves
and buds.

The results of my investigations of some wild plants in Java in
their natural sites agreed entirely with those obtained in the Buiten-
zorg Botanical Garden.

Exactly those species of ants that occur on the so-called ‘ant-
plants” of the Indian archipelago, seem to belong to the harmless
ones; the dangerous species with powerful mouth-apparatus, e.g.
those which are called semut ranggrang in West Java and according
to Dr. VorprrMAN are used by the Malay for defending Mango trees
against beetles, are carnivorous. So these ants have to be specially
allured by hanging animal food (dead leguans) in the trees to be
protected.

1) Eneter’s Bot. Jahrbiicher. Heft II], Bd. 37, 1906.

( 156 )

What the real meaning is of the often so highly differentiated
organs as many extrafloral nectaries are and of the secretion of sugar
which they present in most cases, can only be settled by new
investigations which however will have to bear not only on the
biology but also on the physiology of the plant. |

Physics. — “Methods and apparatus used in the cryogenic labora-
tory at Leiden. X. How to obtain baths of constant and
uniform temperature by means of liquid hydrogen.” By Prof.
H. Kameruinen Onngs. Communication N°. 94/ from the Physical
Laboratory at Leiden.

(Communicated in the meeting of 28 May, 1906).

§ 1. Introduction. Communication N°. 14 of Dec. ’94 treated of

the results I had obtained after I had employed regenerators for
the cascade method, and especially discussed the way how to obtain
a permanent bath of liquid oxygen to be used in measurements at
the then observed lowest temperatures. At the end of that paper I
expressed the hope to be able to construct a cycle of hydrogen
similar to that of oxygen. A mere continuation of the cascade method
would not do. By means of Jiquid oxygen or nitrogen, even when
they evaporate in vacuo, we practically cannot reach the critical
temperature of hydrogen; for the liquefaction of this gas we had
therefore to avail ourselves of cooling by adiabatic expansion.
In Comm. N°. 23 of Jan.’96 1 made some remarks on what could
be derived from van DER Waats’ law of corresponding states for the
liquefaction of hydrogen following this method. I had found that an
apparatus to liquefy hydrogen beginning with — 210°C. might be
constructed almost after the same model as an apparatus that had
proved suitable for the liquefaction of oxygen beginning with ordinary
temperatures and without any further frigorific agents. My efforts,
however, to obtain an apparatus for isentropic cooling by combining
to a regenerator the outlet- and inflow-tubes of a smali expansion
motor, fed with compressed gas, had failed. Therefore I directed
my attention towards the then newly published (1896) application
of the Joune-KeLvin process (LinpE’s apparatus for liquefying air
and Drwar’s jet of hydrogen to solidify oxygen).

Though the process of LinpE was the most promising, because he
had succeeded with his apparatus to obtain liquid air statically, yet it
was evident that only the principle of this method could be followed.

( 157 )

The cooling of an apparatus of dimensions like the first of Linpr
(weight 1300 kilogrammes) by means of liquid air (oxygen) evapo-
rating in vacuo could not be thought of. And yet, according to
what has been said above, this had to be our starting point.

It rather lay to hand to magnify the spiral (enclosed in a vacuum
glass) such as Dewar had used for his jet of hydrogen to solidify
oxygen, and so to get an apparatus with which air could be liquefied,
and which could then serve as a pattern for an apparatus to liquefy
hydrogen. It was indeed a similar construction with which in 1898
Dewar had statically liquefied hydrogen for the first time. About the
installation which apparently afterwards enabled Dewar to collect
large quantities of liquid hydrogen nothing further has come to my
knowledge.

The arrangement of the Leiden hydrogen circulation is based on
Dewar’s principle to place the regenerator spiral into a vacuum
glass (1896). As to the regenerator spiral itself Hampson’s apparatus
for liquefying air (1896) has been followed because it appeared that
the proportions of this spiral have been chosen very favourably, and
with its small dimensions and small weight it is exceedingly fit,
according to the thesis menticned above, to serve as a model for a
regenerator spiral to liquefy hydrogen of about — 205° at expansion
from a higher to the ordinary pressure. The other physicists, who
after Dewar have occupied themselves with liquid hydrogen, —
Travers 1900 and 1904, Ouszewsk1 1902, 1904 and 1905 (the latter
rather with a view to obtain small quantities in a short time with
simple accessories) — have also built their apparatus after this model. -

The Leiden hydrogen liquefactor for constant use has enough
peculiar features to occupy a position of its own as an independent
construction by the side of the apparatus of TRAvErs and OLszEwskI,
which do not satisfy the requirements for the Leiden measurements.
Moreover I was the first to pronounce the principle according to which
this apparatus is built and from which follows that the regenerator
spiral fed with hydrogen that has been cooled by liquid oxygen (air)
evaporating at a given low pressure, must lead to the goal.

The problem of making a circulation in order to maintain a bath
of liquid hydrogen —- and of this problem the arrangement of the
liquefactor for constant use (which, tested with nitrogen, has really
proved efficient) is only a part — has not yet been treated by others.

That also at Leiden we had to wait a long time for its solution
cannot be wondered at when we consider the high demands which,
I held, had to be satisfied by this cycle. For with a view to the
intended measurements I thought it necessary to pour a bath of

( 158 )

1.5 liter into the cryostat (described in VIII of the series “Methods
and apparatus used in the Cryogenic Laboratory” of these commu-
nications) and to keep it to within 0°.01 at a uniform and constant
temperature. The requirements were therefore very much higher than
they had formerly been for the bath of liquid oxygen. These require-
ments could by no means be fulfilled before 1 had the disposal of
a vacuum pump (mentioned as early as Jan. ’96 in Comm. N°. 23),
(comp. Comm. N°. 83, March ’03), suitable to evaporate in a short
time large quantities of liquid air at a pressure of a few centimeters,
and before I possessed compressors for constant working with ex-
tremely pure hydrogen. With the former instrument and the com-
pressors, described in § 3, the liquefactor, described in § 2, delivers
3 a 4 liters of liquid hydrogen per hour. Thus I was able to bring
to this assembly (28 May ’06) 4 liters of liquid hydrogen prepared
at Leiden the day before and to use it in several experiments.

Our installation proved quite satisfactory for operations with the
afore mentioned cryostat. After we had succeeded in making with
it some measurements in liquid hydrogen boiling under ordinary
and under reduced pressure the vacuum glass of the cryostat cracked
and only by mere accidence the measuring apparatus were spared.
Therefore we have constructed another modified cryostat, to be
described in XII, which besides insuring the safety of the measuring
apparatus has the advantage of using less liquid hydrogen than the
cryostat, described in VIII (Comm. N°. 94¢, June ’05). This new
cryostat entirely satisfies the requirements; the temperature is kept
constant to within 0°,01. It is noteworthy that while the measure-
ments are being made the cryostat shows in no way that we are
working with a bath of no less than 1.5 liter of liquid hydrogen.

I wish to express thanks to Mr. G. J. Fim, mechanist at the
cryogenic laboratory, for his intelligent assistance. Under his super-
vision the liquefactor and cryostat, to be described in the following
sections, and also other accessories have been built upon my direc-
tion in the workshop of the laboratory.

§ 2. The hydrogen liquefactor for constant use.
a. The apparatus does not yet entirely realize the original design‘).

1) It might be improved by dividing the regenerator spiral in several successive
coils, each opening into the next with its own expansion-cock, where the pressures
are regulated according to the temperatures. Compare the theory of cooling with
the Joute-Ketvin process and the liquefying by means of the Linpe process given
by vAN per Waats in the meeting of Jan. 1900.

( 159 )

The latter is represented schematically by fig. 1 on Pl. I and
hardly requires further explanation. The compressed pyerogen Z0es
successively through the regenerator coils D,, D,, D,, D,, C, B, A.
B is immersed partially in a bath of liquid air fliiehi being admitted
through P, evaporates at a very low pressure; D,, D,, C and A
are surrounded by hydrogen expanding at the cock M. and D, and D,
by the vapours from the airbath in /. As, however, we can dispose
of more liquid air than we want for a sufficient cooling of the admitted
hydrogen, and the vacuum pump (comp. Comm. N°. 83, March ’03)
has a greater capacity than is required to draw off the evaporating
air’) at reduced pressure, even when we sacrifice the regenerator
working of the spirals D,, D,, D, and D,, we have for simplicity
not yet added the double forecooling regenerator D, by means
of which a large quantity of liquid air will be economized, and hence
the apparatus consists only of one forecooling regenerator C, the
refrigerator / with cooling spiral 6 and the principal regenerator
A in the vacuum glass # with a collecting vessel L, placed in
the case V, which forms one complete whole with the case U.

b. The principal regenerator, Pl. I fig. 2, consists of 4 windings
of copper tubing, 2.4 m.m. in internal diameter and 3.8 m.m. in external
diameter, wound close to each other and then pushed together, indicated
by A,, A,, A, and A,, (number of layers 81; length of each tube
20 M.). As in the ethylene regenerator (Comm. N°. 14, Dec. ’94, and
description of Marutas °*), fig. 1 /’) and in the methy] chloride regenerator
(Comm. N°. 87, March ’04, Pl. I) the windings are wound from the
centre of the cylinder to the circumference and again from the circum-
ference to the centre round the cock-carrying tube J/,, and are enve-
loped together in flannel and fit the vacuum glass Z, (the inner
and outer walls are marked with f,, and £,,). Thence the liquid
hydrogen flows at #, into the collecting vessel Z,. At M,, the
four coils are united to one channel which (comp. cock 7’ in fig. 3
of Marutas’ description l.c.) is shut by the pivot point J/,, moved
by the handle M,,. The packing | M, hermetically closes the tube
M, at the top, where it is not exposed to cooling (comp. Maruias’
description l.c.). The hydrogen escapes at the side exactly as at
the ethylene cock JZ, fig. 2 in Maruias’ description l.c., through 6
openings J/,, and is prevented | from rising or circulating by the
Screens. Le | ae

The new-silver refrigerator case F, is suspended in the new-

1) When using oxygen we might avail ourselves of cooling down to a lower
temperature, which then must be carried out in two steps (comp. § 40).
2) Le laboratoire cryogene de Leyde, Rev. Gen. d. Sc. Avril 1896.

( 160 )

silver case U,, from which it is insulated by flannel U,,. A float F,,
indicates the level of the liquid air, of which the inflow is regulated
through the cock P,, with pivot P,, and packing P, identical with
the cock mentioned above, except that the glass tube with cock is
replaced by a new-silver one /,.

The evaporated air is drawn off through a stout copper tube F,
(comp. § 44). The 2 outlet tubes 5,, and B,, of the spiral B,, and B,,
(each 23 windings, internal diameter of tube 3.6 m.m., external diameter
5,8 m.m., length of each 6 M.) are soldered in the bottom. The
two inflow tubes 5,, and 4,, are soldered in the new-silver cover,
on which the glass tube /, covering the index F,, of the cork
float /',, are fastened with sealing wax (comp. for nitrogen Comm,
N°, 83 IV, March ’03, Pl. VID.

d. The forecooling regenerator spiral C,, C,, C,, and C, is
wound in + windings like A, wrapped in flannel and enclosed in the
cylinder of the new-silver case U,. The four windings (internal diam.
of the tubing 2.4m.m., external diam. 3.8 m.m., number of layers 81,
length of each tube 20M.) branch off at the soldered piece C,, from
the tube C,,, soldered in the cover of U,. They unite to the two
tubes C,a and C,b through which the hydrogen is led to the refri-
gerator. The axis of this spiral is a thin-walled new-silver tube C,
shut at the top.

The hydrogen blown off is expelled through the tube U,.

e. The liquid hydrogen is collected in a new-silver reservoir Z,,
fitting the vacuum glass Z,, which by means ofa little wooden block
V, rests on the wood-covered bottom of the insulated case V,, which
is coated internally with paper V,, and capoc V,,. Thanks to LZ,
the danger of bursting for the vacuum glass is less than when the
hydrogen should flow directly from #, into the glass L,,. This beaker
moreover prevents rapid evaporation in case the glass should burst
(comp. § 1).

The level of the liquid hydrogen is indicated by a float L,,,,
which by means of a silk cord £Z,,, slung over the pulleys Z,,
and £,, is balanced by an iron weight Z,,, moving in a glass
tube V,,, which can also be pulled up and down with a magnet
from outside. The float is a box Z,, of very thin new-silver, the
hook JZ,,, is a bent capillary tube open at both ends and soldered
in the cover. The glass V,, fits by means of india rubber on the
cylinder V,,, which is connected with the case by means of a thin-
walled new-silver tube V,,.

The hydrogen is drawn off through the new-silver siphon tube J,,,

which is continued as the double-walled tube V,, ,,,, leading

( 161 )

towards the delivery cock ,,. Here, as at the ethylene cock
(description of Marnias l.c. fig. 2), the packing N, and the screw-
thread are in the portion that is not cooled. The pin V,, made of a new-
silver tube, passes through the cock-carrying tube N,. Both the outlet
tube V, and the delivery cock N, are surrounded by a portion of
the cold hydrogen vapours, which to this end are forced to escape
between the double wall of the tube through J,,, and along Kha
(Kd on PI. II). The outer wall J,,,, V;,, of the double-walled tube is
insulated from the side tube V,, at the case V,, by means of wool.

The glass Z is covered with a felt cover Z,, fitted at the bottom
with a sheet of nickel-paper to prevent radiation towards the liquid
hydrogen. This cover fits tightly on the lower end #, of EH and
rests on the tube JV,,, and the pulley-case Z,,.

jf. We still have to describe the various safety arrangements to
prevent the apparatus from bursting when the cock J/ should sud-
denly admit too much gas, as might occur when the opening has
been blocked by frozen impurities in the gas, which suddenly let
loose or when one of the tubes breaks down owing to the same
blocking or an other cause.

For this purpose serves in the first place the wide glass tube
W,, which ends below mercury. The quantity of gas which of a sudden
escapes, and the great force with which the mercury is sometimes
flung away rendered it necessary to make a case W,, with several
screens W,, all of varnished card-board to collect the mercury and
to reconduct it into the glass W, (where a sufficient quantity of it must
be present for filling the tube during the exhaustion).

If the pressure in the reservoir rises higher than that for which the
safety tube is designed, the thin-walled india rubber tube V,,, which is
drawn over the perforated brass cylinder wall V,, (separated from
it by a thin sheet of tissue-paper), breaks. The safety apparatus is
connected with the case V, by a wide new-silver tube V,,.

In order to avoid impurities in the hydrogen in the liquefactor
through diffusion of air the india rubber cylinder V,,, that is drawn
over the rings V,,, and V,,, after being exhausted is filled through
the cock V,, with hydrogen under excess of pressure; during the
exhaust the india rubber cylinder V,,, is pressed against the india
rubber wall V,,.

An arrangement of an entirely identical construction protects the
ease U,, which encloses the principal regenerator, and the case U,
which encloses the forecooling regenerator C.

As to the protection against pressure which may occur in conse-
quence of evaporation of air, it was sufficient to protect the refri-

( 162 )

gerator space & by means of the tube Y opening below mercury.

g. In protecting the different parts against heat from the sur-
rounding atmosphere, care has been taken that those surfaces of
which the temperature might fall below the boiling point of air and
which are not sufficiently protected by the conduction from less
cooled parts, should not come into contact with air but only with
hydrogen. The refrigerator vessel ', for instance, is surrounded
with the hydrogen which fills the cases U and V; hydrogen is also
to be found in the space between the vacuum glass Z and the wall
of the case V; and lastly a side tube V,, and V,, branches off
from the case V in order to surround with hydrogen the double-walled
siphon tube V,,, V,,, and the double walled cock N,, .V,,,.

The new-silver case V, from which the vacuum glass Z is insulated
by layers of paper V,, and the refrigerator vessel # by a layer of
flannel, and in the same way the new-silver case U, are further pro-
tected from conduction of heat from outside by separate wrappings
of capoe V,,, packed within a card-board cover V’,, pasted together.
To prevent condensation of water vapour, the air in this enclosed space
communicates with the atmosphere by means of a drying tube t.dr
filled with pieces of sodium hydroxide, as in the ethylene- and
methyl chloride regenerators (comp. above sub 6).

The air-tight connection between the case U and the case V is
effected by the india rubber ring Ua, which fits on the glass and
on the strengthened rims U,, and V,, of the new-silver cases. India
rubber of somewhat larger dimensions can only be used for tightening
purposes when it is not cooled. In this case the conduetion along the
new-silver wall, which is insulated from the vacuum glass by layers
of paper, is so slight that the ring-shaped strengthened rims remain
at the ordinary temperature and the closure can be effected by a
stout stretched india rubber ring. When the india rubber is only
pressed on the glass this closure is not perfectly tight; therefore the
whole connection is surrounded with an atmosphere of almost pure
hydrogen, which is obtained and maintained by the india rubber ring
Ue, which fits tightly on U, and V, and which is filled with hydrogen
under excess of pressure through the cock Ud. Thanks to the small
conduction of heat of new-silver no cooling is to be feared for the
connections of V,, and U,, no more than for the packings of the
cocks M, and N,.

h. The cases V and U are joined and form one firm whole by
the three rods Ub with the screw-fastenings U,, and V,,. The vacuum
glass £,, held by the india rubber ring Ua, rests with a wooden

ring /, and a new-silver cylinder U,, against the refrigerator vessel /’.

( 163 )

The whole construction can stand exhaustion, which is necessary
to fill the apparatus with pure hydrogen. After the case U, of which the
parts U, and U, are connected together by beams, and the case V
are mounted separately, the vacuum glass / is placed in position
and the case V is connected with the case U. The entire lique-
factor is suspended from the ceiling by means of some rods and is
particularly supported by the stout outlet tube J’, for air and the
outlet tube U, for hydrogen. ,

Plate II represents the circulation schematically: the pieces of appa-
ratus in their true proportions, the connections only schematically.
The liquefactor is designated by the letters iq. The compressed hydro-
gen is admitted through ‘Ac, the hydrogen blown off is let out
through Khd or Khe.

7. Before the apparatus is set working it is filled with pure
hydrogen (the cock J being open) by means of exhaustion and
admission of pure hydrogen along Ac. In the drying tubes Da and
®b the pure hydrogen is freed from any traces of moisture which
it might have absorbed.

§ 3. The compressors and the gasometers.

a. The hydrogen is put under high pressure by means of two
compressors in each of which the compression is brought about in
two steps.

While other physicists use compressors with water injection running
at great speed of the same kind as I have formerly arranged for
operations with pure gas (comp. Comm. N°. 14 of Dec. ’94, § 10,
and N°. 51, Sept. ’99, § 3), I have used for the hydrogen circulation
slowly running compressors (see Pl. Il © at 110 and 9 at 80 revo-
lutions per minute) which are lubricated with oil. To enable
constant working with hydrogen the highest degree of purity of
the gas is required. For if air is mixed with the gas it is deposited in
the regenerator spiral and when some quantity of it is collected there
it will freeze and melt alternately through the unavoidable variations
of temperature in different parts of the spiral, so that even small quanti-
ties, taking into consideration that the melted air flows downward,
necessarily must cause blocking. And such small quantities of air may
easily come in through the large quantity of injection water which is
necessary for the above mentioned compressors with water injection
or may penetrate into the pieces of apparatus ~lich are required
when the same injection water is repeatedly used. Lastly the chance
of losing gas is much smaller with the last mentioned compressors

( 164 )

and the manipulation much easier. These compressors are made very
carefully by the BurckHARDT company at Basel.

In the first compressor (© Pl. II, displacing 20 M’* per hour) the
gas is raised in the first cylinder (double-acting with slide) from
1 to 5 and in the second cylinder (plunger and valves) from 5 to 25
atmospheres; in the second compressor ) (plunger and valves) in
the first cylinder from 25 to 50 and in the second from 50 to 250
atmospheres. After each compression the gas is led through a cooling
spiral. With the two first cooling spirals (those of © Pl. II) an oil-
separator is connected.

Safety-valves lead from each reservoir back to the delivery;
moreover the packings are shut off with oil-holders (Comm. N°. 14
794 and N°. 83, Pl. VIII). The hydrogen that might escape from
the packing at $ is collected.

b. The high pressure compressor forces the hydrogen through two
steel drying tubes Da and Db filled with pieces of sodium hydroxide
(comp. § 2, 7, and Pl. II), of which the first also acts like an air-
chamber for the regenerator spiral. As in all the operations the gas
(comp. ¢) originally is almost dry and comes only into contact with
oil, we need only now and then run off a small quantity of
concentrated sodium hydroxide solution.

c. For the usual working the compressors suck the gas from
gasometers. If these should float on water the separation of the water
vapour, which is inevitably taken along by the large quantities of gas
displaced, which constantly come into contact with water, would give
rise to great difficulties in the compression. Therefore we have used
for this purpose two zinced gasometers, Gaz a and Gaz 6, Pl. U, with
tinned welds (holding each 1 M.*) floating upon oi *), which formerly
(comp. Comm. N°. 14, Dec. 94) have been arranged for collecting
ethylene ’).

The cock Kpa (Kpb) is immersed in oil; likewise the connection
of the glass tube, through which the oil of the gasholder can be
visibly sucked up till it is above the cock, with the cover are immersed
in oil. The india rubber outlet tube and the connection with the

1) The drawing sufficiently represents the construction which has been followed
for economizing oil. The gasometers can be placed outside the laboratory and
therefore they are protected by a cover of galvanized iron and curtains of tarred
canvas, which can be drawn round them.

2) Formerly it was of the utmost importance that ethylene could be kept pure
and dry in the gasometers. But now the purifying of ethylene through freezing in
liquid air (comp. Comm. N°. 94e IX § 1) has become a very simple operation and
weldless reservoirs for the storage of the compressed gas are obtainable in all
dimensions.

(165 )

copper exhaust tube are surrounded by a second india rubber tube
filled with glycerine. From the cock onward the conduction can be
exhausted; to prevent the tube from collapsing during the exhaust
a steel spiral has been placed in it. A float with valve Aph (Kp)
prevents the oil from being drawn over into the apparatus.

- Besides these gasometers we dispose of two other gasometers holding
5 M* each to collect hydrogen of a less degree of purity. They
are built following the same system as the zinced gasometers for the
economizing of liquid, carefully riveted and caulked and float on
a solution of calcium chloride. The oil-gasholders serve only for the
storage of very pure hydrogen and this only while the apparatus is
working.

During the rest of the time the pure hydrogen is kept in the
known steel bottles shown on PI. Il at Xtha. When we wish to
liquefy hydrogen, this is blown off into the gasometer through Kg (Khe,
Kpe and Kpb for instance to (raz 5), after this gasometer, which has
been left standing filled with hydrogen, is washed out on purpose with
pure hydrogen. When we stop working the hydrogen by means of
© and is repumped along Apf and Kpe through Ka and Kf
into the reservoirs Rha.

The gasometers may be connected with the pumps or the liquefactor
either separately or together. The former is especially required when
the cryostat is worked (comp. XII) and for the purification of
hydrogen (comp. XIV).

§ 4. The cooling by means of liqud air.

a. The liquid air is sucked into the refrigerator vessel / (PI. I),
which by As (Pl. II) is coupled to the vacuumpump %, along the
tube Pb connected with the siphon of a vacuum bottle %a con-
taining liquid air.

This has been filled by catching the jet of liquid air from the
apparatus (PI.IV, fig. 2) in which it is prepared (comp. XIII), into
the open glass (see the annexed fig. 1) and is kept, covered with
a loose felt stopper m (fig. 1). To siphon the liquid air into the
apparatus, where it is to be used, the stopper is replaced by a cap
A (fig. 1) with 3 tubes; one of these d is designed to raise the
pressure uf the bottle with a small handpump, the other c is connected
to a small mercury manometer, and the third 6 reaches down
to the bottom, so that the liquid gas can be let out. (When
the bottle is used for other liquid gases, d is used for the outlet
of the vapours and ¢ for the admission of the liquid gas). One of
the first two tubes reaches as far as the neck. It may also be used

12
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 166 )

to conduct liquid air from a larger stock into the bottle. With the
cap a closed glass tube & is connected, in which an index of a
cork float dr indicates the height of the liquid.

The caps, as shown in fig. 1, were formerly blown of glass and
the three tubes were fastened into it by means of india rubber. After-
wards the cap /,, as shown in fig. 2, with the three tubes and with
a double wall h, of very thin new-silver have been soldered to form
one whole, which is fastened on the bottle with an india rubber
ring &. The space between the walls is filled with capoc /, and the
whole piece rests on the neck of the bottle by means of a wooden
block 7. After it is placed on the bottle the cap is wrapped round
with wool.

With a view to the transport the vacuum glass is placed in a
eard-board box with fibre packing.

When the siphon is not used it is closed with a piece of india
rubber tubing, fitted with a small stopper. When we wish to

( 167 )

siphon over, this stopper is removed and the inflow tube P% (PI. I)
is connected with the siphon-tube } (fig. 2) with a piece of india
rubber tubing. To prevent breaking of the india rubber, which through
the cold has become brittle, the new-silver tubes are arranged so
that they fit into each other, hence the india rubber is not strained
so much.

The admission of liquid air into the refrigerator vessel is further
regulated with the cock P, Pl. I. When the float indicates that the
reservoir is almost empty, another reservoir is put in its place.

The cock Ks is regulated according to the readings on the
mercury manometer tube Y.

5. The air is caused to evaporate at a pressure of 15 mm., which
is possible because a BurckHarpt-Welss-pump % PI. II is used as
vacuumpump.

The vacuumpump is the same as that used in measurements with
the cryostat containing a bath at — 217° (comp. Comm. No. 94? June ’05)
and has been arranged to this end as described in Comm. No. 83
V. March ’03. The letters at § on Pl. II have the same meaning as
on Pl. VIII of Comm. N°. 83. As has been described in Comm.
No. 94¢ VIII, June ’05, this vacuumpump %, displacing 360 M* per
hour, is exhausted by a small vacuumpump, displacing 20 M* per hour’)
(indicated by # on PI. II).

§5. How the liquefactor is set working.

a. When the apparatus is filled with pure hydrogen, as described
in § 2, and when air evaporating under low pressure is let into the
refrigerator, for convenience the hydrogen, admitted through © and 4
Pl. II along Ke, is caused to stream through during some time
with wide open cock ©, Pl. I, for the forecooling of the whole
apparatus. Then the cock J is regulated so that the pressure in
the regenerator spiral rises slowly. It is quite possible for the appa-
ratus to deliver liquid hydrogen at 100 atm., it has done so at 70 atm.
As a rule, however, the pressure is kept between 180 and 200 atm.
because then the efficiency is some times larger *). The liquefactor
then delivers about 4 liters liquid hydrogen per hour. Part of the
hydrogen is allowed to escape along Aha Pl. I fig. 2 (Kd PI. Il)
for the forecooling of the siphon N,, Pl. I and the cock JN.

As soon as liquid hydrogen begins to separate we perceive that the

1) When we use oxygen (comp. § 2 note 2), and a pressure as low as a
few mm should be required, forecooling is required in the second refrigerator
like F, where oxygen evaporate$ under low pressure, for instance towards §.

*) v. p. Waats has shown the way how to compute this (comp. note 1 § 2).

12%

( 168 )

cock M must be tightened a little more in order to keep the pressure
within the same limits. |

When liquid hydrogen collects in Z rime is seen on the tube
No: Pl. I, fig. 2 near the cock JN.

b. The gaseous hydrogen escapes along Khd (PI. Il) to © and
to one or to both gasholders. When liquid hydrogen separates, the
compressor © receives, besides the hydrogen escaping from the
liquefactor, a quantity of hydrogen from the gasholders along Apa
and Kpb. New pure hydrogen is then admitted from Jtha, Pl. Il,
along Kg.

c. The float (Z,,, Pl. I) does not begin to indicate until a fairly
large quantity of liquid hydrogen is collected.

§ 6. The siphoning of liquid hydrogen and the demonstration of
liquid and solid hydrogen.

a. When the float Z,,,, Pl. 1, shows that the glass is filled to the
top (this usually happens an hour after the liquefactor is set working)
the hydrogen is siphoned into the vacuum glasses Hydr a, Hydr 6
ete., Pl. II, which are connected behind each other so that the cold
hydrogen vapour, which is led through them, cools them successively
before they are filled. When one is full the next is moved 6ne
place further.

They are fitted with caps of the same description as the bottles
for siphoning liquid air, figs. 1 and 2 in the text of §4. Pl. Ill
represents on a larger scale 2 bottles coupled behind each other and
a third which has been filled, all as on PI. II, in side- and top-elevation.
The evaporated hydrogen escapes along d', and d", and further along
K, (see Pl. II) to the gasholder. The letters of the figures have the
same meaning as in fig. 2; for the explamation I refer to the de-
scription of that figure in § 4.

The conduction of heat in the thin new-silver is so little that
the new-silver tubes can be soldered in the caps h, and that they
are sufficiently protected by a double wall h,, of new-silver with
a layer of capoe between, which is again thickly enveloped in
wool.

It has occurred that the india rubber ring 4’ has burst through
the great fall of temperature, but in general the use of india rubber
has afforded no difficulties, and hence the somewhat less simple
construction, which would lie to hand, and through which we avoid
cooling of the india rubber at the place where it must fit, has not
yet been made.

b. If we desire to see the jet of liquid hydrogen flowing from

( 169 )

the cock NW, Pl. I, we connect with the tube NV, and the india
rubber tube d,; instead of the
silvered flasks of Pl. II and
Pl. Ill, a transparent vacuum

Ww. b :
: cylinder fig. 3a, closed by an
india rubber ring with a new-
ee silver cap with inlet tube. After
$ the cock is opened the india
rpbber outflow tube d, covers

with rime and becomes as hard
as glass; soon the first drops in
spheroidal state are seen splash-
ing on the bottom of the glass and
the lively liquid fills the glass. If,
as shown by fig. 3d, a glass cover
is placed on the top, the glass
may be left standing in the

Fig. 3 open air without the air con-
densing into it, which would hasten the evaporation. In the same
manner I have sometimes filled non-silvered vacuum flasks holding
1 diter, where the liquid hydrogen boils vividly just as in the glass
mentioned before. The evaporation is of course much less and the
vising of the bubbles stops when the vacuum glass or the vacuum
flask is placed in liquid air.

ee en)

i a 16: Ie
is eee as Were es J

To demonstrate the pouring of hydrogen
from one open vessel into the other, I use
a glass, cap round which a collar of thin
india rubber sheet is bound (comp. the
accompanying fig. +). The flask from which
and the glass into which we want to pour,
the latter after being filled with liquid air
and quickly turned down and up again
(if this is not done quickly a blue deposit
of H,O from the air will come in), are placed

Fig. 4 under the cap, which fills with hydrogen and
hence remains transparent, then with the india rubber round the neck
of the bottle and round the glass we take hold of the two, each in
one hand. Through the cap we can observe the pouring. The escaping
hydrogen rises in the air as clouds.

In order to keep the half filled glass clear it is covered, under
the pouring off cap, with a glass cap, and so it can be takeu
away from the pouring off cap.

( 170 )

c. It is very instructive to see what happens when we proceed
' to remove this cap and the glass is tilted over a little. Above the
level of the liquid hydrogen thick snowy clouds of solid air are
_ formed, the minute solid particles drop on the bottom through the
extremely light hydrogen (specific weight ‘/,,), there they collect toa
_ white pulver which, when the hydrogen is shaken, behaves as heavy
sand would behave in water. When tbe hydrogen is evaporated that
sand soon melts down to liquid air’).

d. Solid hydrogen may be easily demonstrated when we place
the glass, fig. 3a, under a bell as fig. 3c in which a wire can
be moved up and down (for instance by fastening it into an india
rubber tube) and connect the bell with the airpump. A starch-like
white cake is soon formed, which can be moved up and down
with the wire.

e. To fill a vacuum flask as shown on PI. III we first cool it
by washing it out with liquid air. The connection at J,, Pl. I fig. 2
and Pl. Ill, is brought about simply by drawing a piece of india
rubber tubing V,, over the new-silver tubes WV, and C; fitting into
each other, round which flannel is swaddled. This again is enveloped
in loose wool. When some bottles are connected they are filled with
pure hydrogen through the tube 6, of Hydr. a after repeated
exhaustion and care is also taken that each newly connected bottle
is filled with pure hydrogen and that no air can enter the apparatus
while the connections are being made.

When from the indications of the float Z,,, (Pl. I, fig. 2) we
conclude that a bottle is full, it is disconnected, but as long as the
liquid hydrogen is kept in this glass the evaporating hydrogen is
allowed to escape into the gasholder, as is represented by PI. III for
Hydr. c. The disconnection at NV, is simply effected by taking off
the flannel band C,, heating the piece of india rubber tubing N,,
(unvoleanized) with one’s fingers (or with a pair of pinchers arranged
to this end) till it becomes soft again and can be shoved from. the
tube J,.

§ 7. Transport to the cryostat, closure of the cycle.
a. The vacuum glasses filled with liquid hydrogen (see Hydr. d
on Pl. I) are transported to the room where the cryostat &7 is mounted

1) All this has been demonstrated by me at the meeting of 28 May. To show
the small specific weight of hydrogen I held a very thin-walled glass bulb, which
sinks only a little in ether (as a massive glass ball in mercury), suspended by a
thin thread in the glass with liquid hydrogen, where it fell like a massive glass
ball in water and tapped on the bottom.

(4745

into which the hydrogen is siphoned. To this end the tube 4", of Pl. II]
is connected (again by a piece of india rubber tubing, enveloped in
flannel and wool) to the inflow tube a, of the cryostat and the
tube d, to an inflow tube of pure hydrogen under pressure, which is
admitted from Xhc, Pl. U, along Awa. With all these connections and
disconnections care must be taken that there should always be an
excess of pressure in the tubes that are to be connected, that the
disconnected tubes should be immediately closed with stoppers
but that first the apparatus after having been exhausted should prelimi-
narily be filled with pure hydrogen. The liquid hydrogen is not
admitted into the cryostat €7 until the latter has been cooled —
coupled in another way (see the dotted line on Pl. Il) — by means
vf pure hydrogen which has been led from the through a cooling
tube immersed in liquid air. This refrigerator is of a similar construe-
“tion as the nitrogen condenser Pl. VII of Comm. N°. 83 (March ’08).
Instead of Nliqg should be read H, and instead of Oz lig, Aér liq,
which is siphoned from the vacuum flask 2c. (comp. § 6).

During the siphoning of the liquid hydrogen into Gr the rapidity
of the influx is regulated after a mercury manometer, which is con-
nected with the tube c on the cap h, Pl. III (comp. fig. 2 of § 4).

b. From the cryostat the evaporated hydrogen escapes along Y,,
into the compressor &, Pl. H, which can also serve as vacuumpump
and which precautiously through 9 and Af at the dotted connection Kf
stores the gas, which might contain minute impurities, in the separate
reservoir Xhd; or it escapes along Y,, and Kpe or Kpd into the gas-
holders Gaz a or Gaz b.

XI. The purification of hydrogen for the cycle.

a. This subject has been treated in Comm. N°. 94d IX. To be
able always to obtain pure hydrogen, to make up for inevitable
losses, and lastly to be freed from the fear of losing pure hydrogen,
which perhaps might deter us from undertaking some experiments, a
permanent arrangement for the purification has been made after the
principle laid down in IX. The apparatus for the purification is
represented on Pl. IV and is also to be found on PI. II at 3.

The impure hydrogen from thd is admitted through An and along
a drying tube into a regenerator tube (see Pl. IV) consisting of two
tubes enclosing each other concentrically, of which the outer a serves
for the inflow, the inner 6 for the outlet. Outside the apparatus
a and 6 are separated as a, and 0,, within the apparatus from the
point c downwards a is continued as a, and subsequently as the spiral

( #2)

a, to terminate at the top of the separating cylinder d, from which:
the gas escapes through 4,, and the impurities separated from the
hydrogen as liquid escape along eand Km (comp. Pl. I). The liquid -
air, with which the cooling tube and the separating cylinder are cooled,
is admitted along / and the cock m (and drawn from the vacuum glass

Wb, Pl. Il); a float dr indicates the level of the liquid air. The eva- - cc

porating air is drawn off by the vacuumpump § (PI. I) along At.
The refrigerator vessel p is protected against heat from outside by
a double wall qg of new-silver with eapoe v packed between, of
which the lower end is immersed in a vacuum glass 7, while the
whole is surrounded with a layer of capoe enclosed in a varnished
cover of card-board pasted together in the same way as for the
hydrogen liquefactor. The glass tube Y, opening below mercury,
serves among others to read the pressure under which the i cc
takes place. mele’

The cock Am is turned so that some more bottles of known’
capacity are collected of the blown- off gas than, according to the
analysis, would be formed by the impurities present in the gas. In
this way the purity of the hydrogen is brought to */,,°/,. It is led
along A7/ to the gasholders, and compressed by © and § in :thd.

6. A second purification is effected in the following manner. When
we have operated with the liquefactor with pure hydrogen we
always, after the experiments are finished, admit a portion of this
not yet quite pure gas into the apparatus. After some time, usually
after 4 liters of liquid hydrogen are formed, the cock is blocked. As
soon as it becomes necessary to move this repeatedly to and fro
— Travers and OLszewski say that this is constantly necessary but
I consider it as a sign that the apparatus is about to get more and
more disordered — the work is suspended and the cock M (PI. I)
closed, after which 2D, and ®, (Pl. I) are blown off to the gas-
holders along K, and K,, and A, is shut. The liquid hydrogen,
after being siphoned, is allowed to evaporate and to pass over into
the gasholder for pure hydrogen. The impurities are found when,
with J/ and X, closed, we return to the ordinary temperature and
analyze the gas, which in D has come to high pressure. ;

If necessary, the purified hydrogen is once more subjected to this
process.

When, after the liquefactor with pure hydrogen has been wieleaa .
we go on admitting a quantity of preliminarily purified hydrogen of
1/30 °/, and take care that the impurities are removed, we gradually
obtain and maintain without trouble a sufficient quantity of pure
hydrogen.

(173 )

XII. Cryostat especially for temperatures from
— 252° to — 259°.

§ 1. The principle. In X § 1 I have said that we succeeded in
pouring into the cryostat of Comm. N°. 94¢ VIII a bath of liquid
hydrogen, maintaining it there and making measurements in it, but
then the vacuum glass cracked. By mere chance it happened
that the measuring apparatus which contained the work of several
series of measurements came forth uninjured after removal of the
sherds and fragments of the vacuum glass. With the arrangement
which I am going to describe now we need not be afraid of an adversity
as was imminent then. Now the bath of liquid hydrogen is protected
against heat from outside by its own vapour. The new apparatus
reminds us in many respects of that which I used to obtain a bath
of liquid oxygen when the vacuum glasses were not yet known;
_the case of the cryostat then used has even been sacrificed in
order to construct the apparatus described now.

The principal cause of the cracking of vacuum glasses, which I
have pointed out in several communications as a danger for placing
precious pieces of apparatus into them are the great stresses
caused by the great differences in temperature between the inner
and the outer wall and which are added to the stresses which
exist already in consequence of the vacuum. To the influence of
those stresses it was to be ascribed, for instance, that only through
the insertion of a metal spring the vacuum tubes (described in Comm.
N°. 85, April °05) could resist the cooling with liquid air. It some-
times happens that a vacuum flask used for liquid air cracks without
apparent cause and with the same cooling the wide vacuum cylinders
are still less trustworthy than the flasks. At the much stronger cooling
with liquid hydrogen the danger of cracking increases still. Habit
makes us inclined to forget dangers, yet we should rather wonder
that a glass as used for the cryostat of Comm. N°. 94? VIII filled
with liquid hydrogen does not crack than that it does.

In the new cryostat of Pl. V the cause of the cracking of the vacuum
glass has been removed as much as possible and ‘in case it should
break in spite of this we have prevented that the measuring apparatus
in the bath should be injured. The hydrogen is not poured directly
into the vacuum glass B’,, but into a glass beaker Ba, placed in the
vacuum glass (comp. Comm. N°. 23, Jan.’96 at the end of § 4) but
separated from it by a new-silver case, which forms, as it were,
a lining (see X, LZ Pl. I). Further the evaporated hydrogen is led
along the outer wall of the vacuum glass B’,,. To be able to work

(174)

also at reduced pressure and to prevent any admixtures of air from
entering into the pure hydrogen used, the whole bath has been placed
in a stout cylindrical copper case Ub, which can be exhausted.

This cryostat is especially fit for hydrogen, yet may profitably
replace those described till now, at least when it is not necessary
that we should see what takes place inside the bath. A modified pattern,
where this has become possible, in the same way as in the cryostat
with liquid oxygen of Comm. N°. 14, Dec. ’94, I hope to describe
erelong.

In the eryostat now to be described, as in the former, the meas-
uring apparatus, without our changing anything in the mounting
of them, will go through the whole range of temperatures from
— 23° to — 90° with methyl chloride, from — 103° to — 160° with
ethylene, from — 183° to — 217° with oxygen and from — 252°
to — 259° with hydrogen (only for the temperatures between — 160°
and — 180° we still require methane).

§ 2. Description.

a. The new cryostat is represented on Pl. V. The letters, in so
far as the parts have the same signification, are the same as for the
descriptions of the other cryostats; modified parts are designated by
new accents and new parts by analogous letters, so that the expla-
nations of Comms. N°. 83, N°. 94¢ and N°. 94¢ on the attainment
of uniform and constant temperatures, to which I shall refer for
the rest, can serve also bere. Pl. Il shows how the cryostat is
inserted into the hydrogen cycle. In chapter X § 7 is described how
the liquid hydrogen is led into the cryostat. Especially for the regu-
lation of the temperature this plate should be compared with Pl. VI
of Comm. N°. 83, March ’03. Instead of Bu Vac on the latter plate,
the compressor © serves as vacuumpump here (see Pl. II of the
present paper).

6. The measuring apparatus (as on the plate of Comm. N*. 94¢
VII I have represented here the comparison of a’ thermoelement
with a resistance thermometer) are placed within the protecting
cylinder §, of the stirring apparatus. This is held in its place by 4+
glass tubes &,, fitted with caps of copper tubing §,, and &,, at the
ends of the rods.

The beaker La, containing the bath of liquid hydrogen, is supported
by a new-silver cylinder £a,, in the cylindrical rim Ba, of which
the glass fits exactly; the beaker is held in its place by 4 flat, thin,
new-silver suspension bands running downwards from Ba, and
uniting below the bottom of La. The ring Ba, is the cylinder Ba,

(175 )

continued, with which it is connected by six strengthened supporting
ribs Ba,. At the top it is strengthened by a brass rim Ba, with a
protruding part, against which presses the upper rim Ua of the
ease U. On Ba, rests the cover .V',, in which a stopper is placed
carrying the measuring apparatus. The india rubber band effects
the closure (comp. also Comm. Nos. 83, 94° and 944).

c. In the case U the vacuumglass 4',, of which the inner wall
B',, is protected by the thin new-silver cup 5d, is suspended by
bands LZ’, and supported by the wooden block Z’',. The card-board
cover 5’, forces the evaporated hydrogen, which escapes between
the interstices of the supporting ridges, over the paste-board screen
B',,, with notches 5',,, along the way indicated by arrows, to escape
at 7’,. The case is lined with felt, covered with nickel paper (comp.
Comm. N°. 14, Dec. ’94, and Comm. N°. 51, Sept. 799).

d. The keeping of liquid hydrogen within an enclosed space, o1
which the walls have for a great part a much higher temperature
than the critical temperature of hydrogen, involves special safety
arrangements. That this was no needless precaution appeared when
the vacuum glass cracked unexpectedly (comp. X § 1) and of a quantity
of more than 1,5 liter of liquid hydrogen nothing was to be seen
after a few seconds. Now this disappearance is equivalent with the
sudden formation of some hundreds of liters of gas, which would explode
the case if no ample opportunity of escape were offered to the gas
as soon as the pressure rises a little above the atmospheric.

In the new cryostat I have avoided this danger in the same way
as at the time when I first poured off a bath of liquid oxygen within
a closed apparatus (comp. Comm. N°. 14, Dec. 794).

The bottom of the case U is made a safety valve of very
large dimensions; as cover W, of perforated copper with strengthened
ridges it fits into the cylindrical case Ub, which is strengthened
with the rim W. Over the external side of this cover (as in the
safety tubes for the hydrogen Jiquefactor) a thin india rubber sheet
W, — separated from the copper by a sheet of paper — is stretched,
which at the least excess of pressure swells and bursts, while moreover
the entire vacuum glass or pieces of it, if they should be forced out
of the case, push the cover JV, in front of them without resistance.
As the airtight fit of the sheet of india rubber W, on the ring W
is not trustworthy and diffusion through contact of the india-rubber
with the air must be prevented, it is surrounded with hydrogen ;
this is done by filling the india rubber cylinder Wa, drawn over
the supporting ring Ud, and the auxiliary cover Wb, with hydrogen
along We.

(176 )

The cords Wd serve to press the auxiliary cover WO with a certain
force against the safety sheet, namely by so much as the excess
of pressure amounts to, which for one reason or other we want
to admit into the case. To prevent the india rubber from cooling
down, for then the arrangement would no longer satisfy the requi-
rements, the lower end of the case is lengthened by the cylindrical
piece Ub, which between the rim U6, and the principal body of
the case is made of new-silver to prevent the cooling of the lower
rim. The entire: lower part is stuffed with layers of felt and wool
while also a copper flange Ub, by conduction of heat from outside
protects the lower wall from cooling.

e. The hydrogen is admitted through the new-silver tube a, on
which the siphon tube of a vacuumglass (X § 7) is connected with
a piece of india rubber tubing a, (which otherwise is closed with
a stopper a,, comp. X § 4a). The new-silver tube is put into the
new-silver side piece Ud, which is soldered on the case and, being
stuffed with capoe held back by a paper tube Ue, carries at the
end a piece of cork Uf for support. When the vacuum glass B,
with the case U are placed round the beaker La, the tube a, is
pulled back a little. When subsequently the case is fastened in its
position the tube is pushed forward until a ridge on a, is checked
by a notch in Ud, so that its end projects into the beaker Ba
and the hydrogen can flow into it. The india rubber tube a, forms
the closure on a, and Ud.

§ 3. Remarks on the measurements with the cryostat.

In chapter X §7 I have communicated how the preliminary cooling
is obtained. In one of the experiments, for instance, 3 liters of liquid
air were used for it and the temperature was diminished to —110°.
Then hydrogen was very carefully siphoned into the cryostat under
constant stirring; a quantity of 5 liters was sufficient to obtain a bath
of 1.5 liter. About 0.2 liter per hour evaporated after this. During the
reduction of the pressure to about 60 m.m. + 0.2 liter evaporated,
and then the evaporation remained about the same. The temperature
could be kept constant to within 0.01° in the way described in the
former papers. The temperature curves obtained were no less regular
than those of Pl. HI in Comm. N°. 83 (Febr. and March ’08).

If the pressure is reduced down to 54 m.m. the tapping noise of
the valves of the stirring apparatus becomes duller. This isa warning _
that solid hydrogen begins to deposit.

( 177 )
XII. The preparation of liquid air by means of the cascade process.

§ 1. ficiency of the regenerative cascade method. In none of
the communications there was as yet occasion to treat more in
detail of the preparation of Jiquid air by the Leiden cascade pro-
cess. In the description of the preparation of liquid oxygen (in Comm.
N°, 24, Dec. ’94) I have said that especially the ethylene refri-
gerator had been constructed very carefully, and that the principle
after which various cycles operating in the regenerative cascade can
be made was embodied there.

When the new methyl chloride circulation (comp. Comm. N°, 87,
March ’04) was ready and the inadequate methyl chloride refrigerator
was replaced by one constructed after the model of the ethylene
boiling vessel with application of the experience gained, it was possible
to prepare a much larger quantity of liquid oxygen (10 liters per
hour easily) with the same ethylene boiling vessel. This quantity
will still increase when the regenerator in the ethylene boiling vessel
will be enlarged so much as our experience with the new methyl
chloride regenerator has again taught to be desirable and when the
exhaust tube of the ethylene boiling vessel will have been replaced
by one of greater width than could be used originally. The intro-
duction of a nitrous oxide and of a methane cycle, which in ’94
stood foremost on our programme, has dropped into the background
especially when, also for other reasons (in order to obtain the tem-
peratures mentioned at the end of XII § 1), it appeared desirable to
procure vacuumpumps of greater displacing capacity (’96) and these,
being arranged for operations with pure gases (described in Comm.
N®. 83, March ’03) had become fit to be introduced into the ethylene
and the methyl chloride cycles (while in general for the cryostats
these two cycles were sufficient, ef. the end of XII § 1). Larger
quantities of oxygen could be used in consequence, for which (as
mentioned in °94) a BroruErHoop compressor was employed (comp.
the deseription of the installation for operations with pure gas in
Comm. N°. 51 § 3, Sept. ’99). A picture of the cascade method in
this stage of development accompanies a description of the cryogenic
laboratory by H.H. Francis Hynpman in “Engineering” 4 Mrech ’04.

This picture represents how the oxygen cycle is used to maintain
the circulation in the nitrogen cycle, described in Comm. N°. 83,
March 1903. In the same way as nitrogen we also liquefy air with
the oxygen cycle. When it is drawn off the liquid air streams from
the tube in a considerable jet; about 9 liters of liquid air are collected
per hour, so that in one day we can easily prepare half a hectoliter.

(178 )

Liquid air has striking advantages above liquid oxygen when
we have to store large quantities or when with the gas liquefied in
the cryogenic laboratory we must cool instruments in other rooms.
Only where constant temperatures are aimed at pure oxygen or
nitrogen will be preferred for refrigerating purposes, and even then
the liquid air can be the intermediate agent, for we need only lead
the gases mentioned throngh a cooling tube immersed in liquid air in
order to liquefy nearly as much of it as the quantity of air evaporated
amounts to. And so the permanent stock of liquid air maintained
in the Physical Laboratory has gradually increased, so that for several
years liquid air has been immediately sent off on application both
at home and abroad.

§ 2. The airliquefactor. The apparatus for the preparation of liquid
air by means of liquid oxygen is in principle identical with that
serving for nitrogen, but of larger dimensions (see Pl. VI).

Identical letters designate corresponding parts of the apparatus
represented (Comm. N°. 83, Pl. VII) for the liquefaction of nitrogen.
To liquefy air the ordinary atmospheric air, after being freed by a
solution of sodium hydroxide from carbon dioxide, is compressed to 10
atmospheres in the spiral Rgf/, Pl. VI fig. 1. This spiral branches
off from the tube Rg,, in the soldered piece ftg,, and carries four
branches Rg,, Rg,, Rg, and Rq,. Each of these tubes has an internal
diameter of 3.5 mm., an external diameter of 5.8 mm., and is 22 M.
long. The spiral is wound in 63 layers in the same way as the regenerator
spiral of the hydrogen liquefactor (comp. X )and, lined with flannel, it fits
the new-silver tube p,, round which it is drawn in the new silver ease p.
The four windings are united below to one soldered piece to the
spiral Rf, 8M. long, which is immersed in a bath of liquid oxygen
and whence the liquid air flows through /, into the collecting
apparatus (see fig. 2). This is placed by the side of the principal
apparatus (see fig. 2) and contains the collecting vessel 7,, where
the liquid air is separated and whence it is drawn through the
siphon. The collecting glass is fitted with a float dr. During work
we can see it rising regularly at a fairly rapid rate.

§ 3. Further improvements. The regenerative cascade might still
be modified in many points before the principle is fully realized and
before one improvement or other, made for one of the cycles, has been
introduced also in the others and the efficiency is grown to a maximum ;
but this problem is rather of a technical nature. We prefer to spend
the time at our disposal on other problems, as enough liquid air is

KAMERLINGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden.
XIII. The preparation of liquid air by means of the cascade process.
Pl. VI.

WA) IROL

(22
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( 179 )

produced by the regenerative cascade. Enough but not too much,
because for operations with liquid hydrogen (comp. X) and also for
other experimentations in the realm of cryogenic work it is very
important that we should dispose of such a relatively abundant
stock of liquid air as is produced by the Leiden cascade.

XIV. Preparation of pure hydrogen through distillation
of less pure hydrogen.

It was obvious that we could obtain pure hydrogen for the
replenishment of the thermometers and piezometers*) when we distil
liquid hydrogen at reduced pressure *), and then evaporate the very
pure liquid thus obtained. Therefore the following apparatus has been
constructed (fig. 5).

A vacuum glass A is connected with the liquefactor (see Pl. 1
and III at \,) or with a storage bottle, exhausted and filled with

liquid hydrogen as indicated in X §7. Then C'(exhausted beforehand)
in the vacuum glass £6 is filled several times out of A, and the

vacuum glass 6 is connected with B, to the liquefactor and exhausted
like A and also filled with liquid hydrogen and connected with the
ordinary airpump at £6, so that the hydrogen boils in B at 60 m.m.
Then hydrogen is distilled over along c, into the reservoir C, we

1) In Comm. N°. 94e (June ’05) I have mentioned that a purification through
compression combined with cooling might be useful in the case of hydrogen even
after the latter in the generating apparatus (Comm. N°. 27, May 96 and N°. 60,
Sept. 1900) had been led over phosphorous pentoxide. I said so especially with a
view to the absorption of water vapour as, with due working, the gas — at least
to an appreciable vapour tension — cannot contain anything but H,O and SO,H;:
How completely the water vapour can be freed in this manner appears from a
calculation of Dr. W. H. Keesom, for which he made use of the formula of ScHEEL
(Verh. D. phys. Ges. 7, p. 391, 1905) and from which follows for the pressure of
water vapour (above ice) at —180°C. 10-15 mm., so that water is entirely held back
if the gas remains long enough in the apparatus. This holds for all substances of which
the boiling point is higher than that of water (SO; vapours, grease-vapours etc.).
The operation is therefore also desirable to keep back these substances. As to a gas
which is mixed only with water there will remain, wher it is led in a stream of 3 liters
per hour through a tube of 2 cm. in diameter and 8 cm. in length over phosphorous
pentoxide, no more than 1 m.gr. impurity per 40000 liters (Mortey, Amer. Journ.
of Sc. (3) 34 p. 149, 1887). This quantity of 1 m.gr. is probably only for a
small part water (Morey, Journ. de chim. phys. 3, p. 241, 1905). Therefore the
operation mentioned would not be absolutely necessary at least with regard to
water vapour when a sufficient contact with the phosphorous pentoxide were
ensured. But in this way the uncertainty, which remains on this point, is removed.

*) This application follows obviously from what has been suggested .-by Dewar,
Proc. Chem, Soc. 15, p. 71, 1899.

( 180 )

shut c, and disconnect the india rubber tube at a@ and remove the
whole apparatus to the measuring apparatus which is to be filled
with pure hydrogen; to this end the apparatus is connected with the
mercury pump, intended for this purpose, at c,. To take care that
the hydrogen in # should evaporate but slowly and the quantity
in C should not be lost before we begin to fill the pieces of appa-
ratus, B is placed in a vacuum glass with liquid air.

Physics. — “On the measurement of very low temperatures. IX.
Comparison of a thermo-element constantin-steel with the hydrogen
thermometer.” By Prof. H. KameruincoH Onnes and C. A.
CroMMELIN. Communication N° 957 from the Physical Labora-
tory at Leiden.

(Communicated in the meeting of June 30, 1906).

§ 1. Introduction. The measurements communicated in this paper
form part of a series, which was undertaken long ago with a view
to obtain data about the trustworthiness of the determination of
low temperatures which are as far as possible independent and
intercomparable. Therefore the plan had been made to compare
a thermo-element'), a gold- and a platinum-resistance thermometer?)

—

1) Comp. comms. N°. 27 and 89. (Proc. Roy. Ac. May 1896, June 1896, and

Feb. 1904).
2) Comp. comms. N°. 77 and 93. (Idem Febr. 1902 and Oct. 1904).

( 181 )

each individually with two gas thermometers and also with each
other, while the deviation of the gas thermometer would be determined
by means of a differential thermometer’). Nitrogen had originally
been chosen by the side of hydrogen, afterwards nitrogen has been
replaced by helium. Because all these measurements have often
been repeated on account of constant improvements, only those figures
have been given which refer to the gold- and the platinum-resistance
thermometer *), and these, for which others will be substituted in
Comm. N°. 95°, are only of interest in so far as they show that the
method followed can lead to the desired accuracy. The results obtained
with regard to the above-mentioned thermo-element do not yet satisfy
our requirements in all respects; yet all the same it appeared desirable
to publish them even if it was only because the temperature deter-
minations for some measurements, which will erelong be discussed,
have been made with this thermo-element.

§ 2. Comparisons made by other observers.

a. Constantin-iron elements have been compared with a hydrogen
thermometer only by Hotsorn and Wren’) and Lapensure and
Kricen *). The calibration of the two former investigators is based
on a comparison at two points viz. in solid carbon dioxide and
alcohol (for which — 78°.3 is given) and in liquid air (for which
they found —189°1). They hold that the temperature can be
represented by the formula

t=—aE+ bE’

and record that at an observation for testing purpose in boiling
oxygen (— 183°.2 at 760 m.m. mercury pressure) a good harmony
was obtained.

LaprNBurG and Kriicen deem Horzorn and WIen’s formula unsatis-
factory and propose

t=ab.-| be +. cE’.

They compare the thermo-element with the hydrogen thermometer
_at 3 points, viz. solid carbon dioxide with alcohol, boiling ethylene
and liquid air. As a control they have determined the melting point
of ether (— 112°) and have found a deviation of 1 deg. With this
they rest satisfied.

1) Comp. comm. N°. 94c. (Idem June 1905).
*) Comp. comm. N°. 93. (Idem Oct. 1904).
*) Silz.ber. Ac. Berlin. Bd. 30, p. 673, 1896, and Wied. Ann. Bd. 59, p. 213. 1896.
4) Chem. Ber. Bd. 32, p. 1818. 1899.

13
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 182 )

Rotue’) could only arrive at an indirect comparison with the hydrogen
thermometer. He compared his thermo-elements constantin-iron at
— 79° with the alcohol thermometer which Wiese and BOorTcHer ?)
had connected with the gas thermometer and at — 191° with a
platinum-resistance thermometer which at about the same tempera-
ture had been compared with the hydrogen thermometer in the
Phys. Techn. Reichsanstalt by Ho.porn and DirTENnBerGEr °*).

The thermostat left much to be desired; temperature deviations
from 0°.4 to 0°.7 occurred within ten minutes (comp. for this § 7).
As Rorue confined himself to two points, he had to rest content with
a quadratic formula and he computed the same formula as HOoLBorN
and WIEN.

From the values communicated for other temperatures we can
only derive that the mutual differences between the deviations of the
different thermo-elements constantin-iron and constantin-copper from
their quadratic formulae could amount to some tenths of a degree.
Nothing is revealed with regard to the agreement with the hydrogen
thermometer. This investigation has no further relation to the problem
considered here.

5. Among the thermo-elements of other composition we mention
that of WrosBLewski *), who compared his new-silver-copper element
at + 100° (water), —103° (ethylene boiling under atmospheric pressure)
and —131° (ethylene boiling under reduced pressure) with a
hydrogen thermometer and derived thence a cubic formula for ¢.
He tested it by means of a determination of the boiling points of
oxygen and nitrogen and found an agreement with the hydrogen
thermometer to within O°.4. As, however, Wrosirewski found for
the boiling point of pure oxygen at a pressure of 750 m.m. —181°.5,
no value can be attached to the agreement given by him.

Drwar’s °) investigation of the element platinum-silver was for the
time being only intended to find out whether this element was suited
for measurements of temperatures at — 250° and lower (where the
sensitiveness of the resistance thermometer greatly diminishes), and
has been confined to the proof that this really was the case.

c. To our knowledge no investigation has therefore been made as
yet, which like that considered in our paper, allows us to judge in

’) Ztschr. fiir Instrumentenk. Bd. 22 p. 14 and 33. 1902.
ete 3 ; Bd. 10 p. 16. 1890.

8) Drude’s Ann. Bd. 6 p. 242. 1901.

4) Sitzungsber. Ac. Wien Vol. 91. p. 667. 1885.

5) Proc. R. §. Vol. 76, p. 317. 1905,

( 183 )

how far thermoelements are suitable for the accurate determination
of low temperatures (for instance to within /,, precise), and also
by what formula and with how many points of calibration any
temperature in a given range can be determined to within this
amount.

§ 3. Modifications in the thermo-elements and auxiliary apparatus.

We shall consider some modifications and improvements which
have not been described in § 1 of Comm. N°. 89. The first two
(a and 6) have not yet been applied to the element with which
the following measurements are made, but they have afterwards been
applied to other elements and so they are mentioned for the sake
of completeness.

a. If we consider that the thermo-element in different measurements
is not always used under the same circumstances, e.g. is not immersed
in the bath to the same depth etc., and that even if this is the case, the
time during which this is done at a constant temperature will not always
be so long that in either case the same distribution of the temperature
will be brought about in the metallic parts of the element, it will prove
of the greatest importance that care should be taken, that the tem-
perature of the juncture, given by the electromotive force, differs
as little as possible and at any rate very little from that of the
surface of the copper protecting block, that is to say that of the
bath.

The construction of the place of contact shown by fig. 1
is a better warrant for this than that on Pl. lof Comm.
N°. 89. The wires a and & are soldered on the bottoms
of small holes c, bored in the protecting block and are
insulated each by a thin-walled glass tube. If the con-
struction of Pl. I Comm. N°. 89 is not carried out as it
should be (whether this has succeeded will appear when we
saw through trial pieces) and consequently the juncture is
a little removed from the upper surface of the block, it may
be easily calculated that, owing to conduction of heat along
the wires while the thermo-element is immersed in
liquid oxygen a difference in temperature of as much
as one degree may exist between the place of contact
and the block. When the elements are used under

Fig. 1. other circumstances, this difference in temperature will
have another value and hence an uncertainty will come into the
determination of the temperature of the block. Perhaps that also a
_ retardation in the indications of the element will be observed.

13*

( 184 )

Although this construction (fig. 1) (for which a block of greater
thickness is required than for that of Comm. N°. 89, Pl. I) has not
been applied to the element used, we need not fear uncertainties on
this point thanks to the very careful construction of the latter.

b. When temperatures below —253° have to be determined we
might fill the apparatus with helium instead of hydrogen as men-
tioned in § 1 of Comm. N°. 89.

c. The glass tubes of the mercury commutators, described in’
Comm. N°. 27, are not fixed in corks (see Pl. IV, fig. 4, £) but in
paraffin, so as to obtain perfect insulation, which, as experience has
taught, is not guaranteed by the glass wall. The tubes are continued
beyond the sealing places of the platinum wires ¢, c, c,and ¢,, (as shown

Fig. 2 and 3.

by figs. 2 and 3) to avoid breaking of the platinum wires as
formerly frequently happened.

d. The platinum wires of the Weston-elements have been amal-
gamized by boiling with mercury (which method has since that time
been replaced by the method with the electric current *)). The elements
themselves have kept good through all these years.

e. In spite of all the precautions which have been described in
Comm. N°. 89, thermo-electromotive forces still remain in the wires,
which with the great differences of temperature between various
points of one wire must doubtlessly amount to a measurable quantity.
When, however, care is taken that the circumstances under which
the element is used with respect to the temperature along the wires
are about the same as: for the calibration, a definite value of the
electromotive forces will answer to a definite temperature of the
copper block. We do not aim at an accurate determination of the
electromotive force of the combination of the metals which at the

1) Comp. Jazcer, Die Normalelemente, p. 57.

(185 )

juncture are in contact with each other, but we only require that
a definite electromotive force for a definite temperature of the bath
in which the element is immersed should be accurately indicated.
(for the rest comp. § 9).

In order to lessen the influence of the conduction of heat along
the wire at the juncture we shall for the new elements destined for
taking the temperature of a liquid bath make a trial with the insertion
into the glass tube at 2 c.m. above the copper rim of the copper
block of a copper tube, 5 ¢.m. long, which is soldered on either
side of the glass tube and remains over its whole length immersed
in the liquid.

§ 4. Precautions at the measurements of the electromotive forces.

a. The apparatus and connections which have been described in
§3 of Comm. N°. 89 have been mounted entirely on paraffin, with
which also the enveloping portions of the apparatus are insulated.
Only the wires running between the different rooms stretched on
porcelain insulators, of which the high insulation-resistance has
repeatedly been tested, have no paraffin-insulation. The ice-pots are
hanging on porcelain insulators. As a matter of course, all parts of
the installation have been carefully examined as to their insulation
before they are used.

6. The necessity of continually packing together the ice in the
ice-pots has been argued before in Comm. N°. 89.

c. The plug-commutators are of copper. All contacts between different
metals in the connection have been carefully protected from variations
of temperature by packing of wool or cotton-wool, from which they
are insulated by paraffin in card-board boxes. This was only omitted
at the contact places of the copper leads with the brass clips of the
resistance boxes. To secure to the Weston-elements an invariable
temperature, the latter have also been carefully packed. The accu-
mulator is placed in a wooden box.

d. With regard to the testing elements, care has been taken that
the steam left the boiling apparatus (comp. Comm. N°. 27, § 8) at
a given constant rate.

e. Before a measurement is started we investigate by short-cir-
cuiting in the copper commutators in the conductions, leading from
the thermo-elements and the Weston-battery to the connections, whether
all electromotive forces in the connections are so small and constant
(not more, than some microvolts), that elimination through the reversal

( 186 )

of the several commutators may be considered as perfectly certain.

§ 5. The control of the thermo-elements.

It appeared :

a. that when the four places of contact were packed in ice, the electro-
motive force of the element amounted to less than one microvolt;

b. that the changing of the two places of contact constantin-steel,
so that they were alternately placed in the cryostat, indicated only a
very small difference in electromotive force. Care is taken, however,
that always the same limb is placed into the cryostat ;

c. that while the place of contact was moved up and down in
the bath no difference could be perceived in the reading (hence
the difference of temperature certainly < 0°.02).

All this proves that the electromotive forces which are raised
in the element outside the places of contact, are exceedingly small.

§ 6. Corrections and calculations of the determinations of the
electromotwe forces.

a. In the following sections Rk, AR, A’ have the meaning which has
been explained in Comm. N°. 89 § 3. #,, #, and H’ signify the
electromotive forces of the observation-element, the comparison-element
and the Weston-battery respectively. If we have obtained #, FR, and
R’ it follows that:

Rw Ry _,
era ome ee a
As a test we use:
z E'
io R! ‘4

5. In order to find &, we read on the stops of the resistance box
R', (in the branch of small resistance), and A» (in the branch of
great resistance) which are switched in parallel to form &,.

a. To none of the resistance boxes temperature corrections had
to be applied (nor to those given by A, and R’ either).

8. To R', we sometimes had to add the connecting resistance of
the stops.

y. To R', is added the correction to international ohms according
to the calibration table of the Phys. Techn. Reichsanstalt.

Jd. To k",, is added the amount required to render the compen-
sation complete, which amount is derived from the deflections on

( 187 )

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(188 )

‘the scale of the galvanometer at two values of R", (see tables I
and IV). .

c. In order to find R,, R'. and R"., which with regard to R, have
‘a similar meaning as FR’, and R",, with regard to Ry, are treated
like R', and R&R", concerning the corrections a, B, ¥ and Jd. The
thence derived result A. holds for the temperature at which the
water boils in the boiling apparatus at the barometric height B
existing there during the observation.

«. R"’, is corrected to the value which it would have at a pres-
sure of 760 m.m. mercury at the sealevel in a northern latitude of 45°.

d. To find R' the corrections mentioned sub y and d are applied
to the invariable resistance L’.

e. E', referring to the temperature ¢ of the Weston-battery, is
derived from Jancer’s table *).

§ 7. Survey of a measurement. Table I contains all the readings
which serve for a measurement of the electromotive force namely
for that at — 217° (comp. § 8). We suppose that during the short
time required for the different readings (comp. § 3 of comm. N°. 89)
the electromotive force of the accumulator (comp. § 4, c) remains
constant. We further convince ourselves that the temperature in the
boiling apparatus of the comparison-element has remained sufficiently
constant and that we have succeeded *) in keeping the temperature
of the bath in the cryostat constant to within 0°.01 *) (see table I).

In exactly the same way we have obtained on the same day of
observation the values for the electromotive forces which are combined
in table III.

From the preceding survey it appears that the measurements can
be made with the desired precision even at — 217°. At — 258° the
sensitiveness of the element constantin-steel is considerably less
thah at — 217°. It seems to us of interest to give also for this very
low temperature a complete survey of the readings and adjustments
so that the reader may judge of -what has been attained there
(see Table IV).

1) Jarcer, Die Normalelemente 1902. p. 118.

2) Comp. Comm. N®. 83, § 5 and Pl. III.

3) Together with the readings we have also recorded the temperature of the
room (¢k) and of the galvanometer (¢;); these are of interest in case one should
later, in connection with the sensitiveness, desire to know the resistance of the
galvanometer and the conducting wires during the observation. lor the notation of
the combination P;-+ Q of the comparison-elements we refer to Comm. N’. 89 § 2.

( 189 )

From table I directly follows
TABLE II.
Corrections and results.

Observation-element. Comparison-element. Weston-elements.

corr. 6 #',,=+ 0.001 2 | corr. § R', =+ 0.001 n
corr. ¥ R’,,= + 0.0080 Q | corr. y KR’, =—0.00015 0 | corr. y R’;=—2.4 9
corr.3 R”,,=+179 0 corr. 5 R", =4-1490 corr. $ R';=+0.64.9

Re, =50.3163 0

barom.hght.45°N.B.=76.21cM.
corr. € R'”, =— 0.0373 9

Final results.

R= 53.6404 2 Re =50.2787 0 R'=7998.3 0,

[== 4.808
E'=1.0187 volt.

— |

P.y= 6.8312 milliv. E, = 6.4037 milliv. |

4u 3! | |
EES — OOOO
TABLE III.
| Ey | E,
| |
6.8312 6.4037
6.8308 6.4039
6.8310 6.4038
Aas ee
Mean | 6.8310 | 6.4038

§ 8. The temperatures.

a. The thermo-element is placed in a cryostat, as represented on
the plate of Comm. N°. 94¢ but there a piezometer takes the place
which in our measurements was occupied by a hydrogen thermo-
meter. To promote a uniform distribution of the temperature in the

( 190 )

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( 191 )
iA BL EV.

Corrections and results.

Observation-element. Comparison-element, Weston-elements.

corr. £. R',, = +0.001 o | corr.s. Rk’, =+0.001 0

corr. 7. R',, =-+ 0.00537 | corr. 7. R’, = + 0.0084 9 corr. ». Rh’; =—2.49
corr. ¢. RK", =-++ 200. corr. 6. R= — 209 n corr. ¢. R'; =+0.89

RM”, = 50.4133 9

Bar.hght. 45° N.B, =76.82 cM.
corr. ¢ Rl”, =— 0.14592

Final results.

R,, = 55.9981 o | R, = 50.2644 9 | R' = 7998.4 o

> t?’—=18°.5
' =1.0187 volt.

Ey = 7.1321 milliv. E,= 6.4075 milliv.
Qn24’

bath a tube is mounted symmetrically with the thermo-element, and
has the same shape and dimensions as the latter. Comp. also Comm.
N°’. 94° § 1.. For the attainment of a constant and uniform tem-
perature with this cryostat we refer to Comm. N°. 94¢ and the
Comms. quoted there. The temperature was regulated by means of
a resistance thermometer. For the two measurements in liquid hydrogen
we have made use of the cryostat described in Comm. N°. 94/,

6. With a bath of liquid methyl chloride we have obtained the
temperatures — 30°, —59° and — 88°; with ethylene — 103°,
— 140° and —159°; with oxygen — 183°, —195°, —205° —213°
and — 217°; with hydrogen — 253° and — 259°.

c. The temperatures are read on the scale of the hydrogen thermo-
meter described in Comms. N°. 27 and N°. 60. On the measurements
with this apparatus at low temperatures another communication will
erelong be published.

( 192 )

§ 9. Results.

Column I of the following table VI contains the numbers of the
measurements, column II the dates, column III the temperatures
measured directly with the hydrogen thermometer, column IV the
electromotive forces — /#,, in millivolts, column V the number of
observations, column VI the greatest deviations in the different deter-
minations of #,, of which the appertaining /,, is the mean, column VII
the same reduced to degrees.

TABLE VI.
CALIBRATION OF THE THERMO-ELEMENT
CONSTANTIN-STEEL.

I | II | Il i at Vv | VI ee VII
20 | 97 Oct. 05 — 58.753 2.3995 | 3 | 0.0006 0.016
24 30 Oct. 05 — 88.140 3.4895 | 3 29 81
47 | 8 July 05 — 103.833 4.0229 | 3 56 168
16 | 7 July 05 — 139.854 5.1469 | 8 6 oA
48 | 6 Oct. 05 — 139.873 5.1469 | 4 12 MA
49 | 6 Oct. 05 — 158.834 5.6645 | 3 15 59
41 27 June 05 — [182.692] | 6.9997 | 3 40 MG
98 | 2 Mrch, 06 —'195.178 6.4717 | 4° 28 450
42 | 99 June 05 — [204.535] | 6.6382 | 3 3! 186
27 | 2 Mrch. 06 — 204.694 6.6361 4 26 156
44 | 30 June 05 | —[212.832] | 6.7683 | 3 8 56
43 | 6 July 05 — 242.868 6.7668 | 3 45 106
299 | 3 Mrch. 06 — 7.44 6.8291 3 AA 412
45 | 6 July 05 — 27.416 6.8310 | 3 4 32
30 | 5 May 06 — 252.93 7.4315 | 4 47 39
34 5 May 06 — 259.24 7.1585 | 4 = =

The observations

11, 12 and 14 are uncertain because in those

cases the hydrogen thermometer had a very narrow capillary tube
so that the equilibrium was not sufficiently secured. According to
other simultaneous observations (Comm. N°. 95° at this meeting),
which have later been repeated, the correction for N°. 11 is probably
— 0°.058. The two other ones have been used unaltered,

( 193 )

' The mean deviation of /#, for the different days from the mean
value, and also the mean largest deviation of the values of 2 found
on one day amounts to 3 microvolts, which amount shows that in
the observation of the comparison-element the necessary care has
not been bestowed on one or other detail, which has not been
explained as yet. We must come to this conclusion because the
observation-element yields for this mean only 1,8 microvolt.

§ 10. Indirect determinations.

In order to arrive at the most suitable representation of LZ, as a
function of ¢, it was desirable not only to make use of the obser-
vations communicated in § 9 but also to avail ourselves of a large
number of indirect measurements, obtained through simultaneous
observations of the thermo-element and a platinum-resistance thermo-
meter, the latter having been directly compared with the hydrogen
thermometer (comp. Comm. N°. 95%, this meeting).

These numbers have been combined in table VII where the columns
contain the same items as in the preceding table, except that here
the temperatures are derived from resistance measurements.

TABLE VII.
INDIRECT CALIBRATION OF THE THERMO-ELEMENT
CONSTANTIN-STEEL.

I | I | IL IV V | VI | VII
22 | 43 Dec. 05 — ofses | 1.9503 | 3 | 0.005 0.012
% | 44 Dec. 05 — 58.748 2.3980 | 4 6 16
23 | 13 Dec. 05 — 88.161 3.4802 | 3 6 17
4 | 23 Jan. 05 — 103.576 4.0100 | 5 9 oy
3 | 30Jan.05 | [— 192.604) | 6.9970 | 4 32 147
5 | 46 Mrch. 05 | (— 182.828] | 6.2340 | 3 13 60
4 | 2 Febr. 05 — 195.435 6.4730 | 3 20 407
6 | 417 Mich.05 | — 195.964 6.4814 | 5 10 53
7 | 30 Mrch. 05 | — 204.895 6.6397 | 3 55 330
26 | 26 Jan. 06 — 212.765 6.7637 | 4 33 933
8 | 3 April 05 — 212.940 6.7686 | 4 15 106
25 | 95 Jan. 06 — 217.832 6.83176 | 4 29 932

tS ee a | ee

( 194 )

§ 11. Representation of the observations by a formula.
a. It was obvious that the formula of AVENARIUs:

t a
E=f 00 --b (<3)
can give a sufficient agreement for a very limited range only. If,
for instance, the parabola is drawn through 0°, —140° and —258°,
we find:
a= + 4.7448
b= -+ 0.76117. |
In this case the deviation at —204° amounts to no less than 7°.
If we confine ourselves to a smaller range and draw the parabola
through 0°, —88° and —183°, we find:
a= 4.4501
b= + 0.57008,
while at —140° the deviation still amounts to 1°.3.
Such a representation is therefore entirely unsatisfactory.
6. With a cubic formula of the form

=e t A ae :

=a ag + (si5 +e (=)

we can naturally attain a better agreement. If, for instance, we
draw this cubic parabola through 0°, —88°, —159° and —253°,
we find:

a= + 4.2069
b= +. 0.158
c= — 0.1544

and the deviation at —204° is 0°.94. A cubic formula confined to
the range from 0° to —183°, gave at —148° a deviation of 0°.34. *)
A cubic formula for 7, expressed in # (comp. § 2), gives much larger
deviations. *)

e. A formula, proposed by SransrieLD*) for temperatures above
0°, of the form

1) As we are going to press we become acquainted with the observations of
Honter (Journ. of phys. chem. Vol. 10, p. 319, 1906) who supposes that, by
means of a quadratic formula determined by the points —79° and —183°, he can
determine temperatures at —122° to within 0°.1. How this result can be made
to agree with ours remains as yet unexplained.

2) After the publication of the original Dutch paper we have taken to hand
the calculation after the method exposed in § 12 of a formula of the following form:

he t h ee t : t :
at Tap (sa) + °(sen) + * (arn

We hope to give the results at the next meeting.
3) Phil. Mag. Ser. 5, Vol. 46, p. 73, 1898,

( 195 )

H=aT+blogT+ ¢,

where TZ represents the absolute temperature, proved absolutely
useless.

d. We have tried to obtain a better agreement with the observa-
tions by means of a formula of five terms with respect to powers
of ¢t. To this end we have tried two forms:

t nN mea 7
naa, +0(z) +e(z15) a a cnet

and

t t 2 t 3 5

First the constants of the two equations are determined so that
the equations satisfy the temperatures —59°, —140°, —159°, —183°
and — 213°. (A) indicated at — 253° a deviation of 113.1 micro-
volts, (B) a deviation of 91.8 microvolts. We have preferred the
equation (4) and then have sought an equation (LIV) which would
represent as well as possible the temperature range from 0° to — 217°,
two equations (61 and AIll) which would moreover show a not too
large deviation at — 253°, for one of which (ASIII) a large deviation
was allowed at — 217°, while for the other (AI) the deviations are
distributed more equally over all temperatures, and lastly an equation
(BID which, besides —253°, would also include —259°.

§ 12. Calculation of the coefficients in the formula of five
ivurms. The coefficients have first been derived from 5 temperatures
distributed as equally as possible over the range of temperatures,
and then corrected with respect to all the others without a rigorous
application, however, of the method of least squares.

In order to facilitate this adjustment we have made use of a
method indicated by Dr. E. F. van pp Sanpr BakuuyzEN in which
instead of the 5 unknown coefticients 5 other unknown values are
introduced which depend linearly on the former’). For these are
chosen the exact values of / for the five observations used originally,
or rather the differences between these values and their values found
to the first approximation.

Five auxiliary calculations reveal to us the influence of small
variations of the new unknown value on the representation of the
other observations and by means of these an approximate adjustment

1) Also when we rigorously apply the method of least squares this substitution
will probably facilitate the calculation.

( 196 )

may be much more easily brought about than by operating directly
with the variations of the original coefficients *).

After the first preliminary formula was calculated all the 28
observations have subsequently been represented. The values thus
found are designated by f,. The deviations of the observed values
from those derived from this first formula are given in column Il
of table VIII under the heading W—R#,. The deviations from the
temperatures in the immediate neighbourhood of each other have
been averaged to normal differences and are combined in column
IV under the heading (W—-R,). .

These deviations have served as a basis for an adjustment under-
taken according to the principles discussed above.

It yielded the following results:
leaving — 253° and — 259° out of consideration we find as co-
efficients of the equation (4) (comp. § 11):

a,= + 4.32044 e,=-+ 0,011197

6, = + 0,388466 f, = — 0,0044688). . . . (BID)

c, = — 0.024019
If we only leave out of consideration — 259° we find for the
coefficients of equation (B) the two following sets (comp. § 11):
a, — + 4.33049 ¢,=- 0,053261
b, = + 0.436676 f,= +0,003898) . . . (BIID

c, = + 0,048091
and
a, = + 4.35603 e, = + 0,108459
b, = + 0,581588 7, = + 0,01186382) . . . . . (BD)

c, = + 0,157678
If we include in the equation all the temperatures, also that of
the liquid hydrogen boiling under reduced pressure, we find for the
coefficients of the equation (5)
a, = +4.35905 e,—=-+0,111619 |
b- = + 0,542848 f/f, = + 0,01821380, . . - (BrP)
c, = + 0,172014 |
The deviations from the observations shown by these different equa-

tions are found under (W—R,) (W—R,) (W—R,) and (W—R,) in
columns V, VI, VII and VIII of table VIII.

1) When the polynomial used contains successive powers of the variable beginning
with the first power, that influence is determined by the interpolation-coefficients
of LAGRANGE.

THERMO-ELEMENT CONSTANTIN-STEEL.

—103.576
—103.833
—139.851
—139.873
—158.831
[—182. 604]
[—182.750|
[—182. 898)
—195 .135
195.178
—195.261
[—204 535]
—204. 694
—204 895
2.765
[212.839]
—12..868
—212.940
—217.414
—17.416
—217.832
—252.93
259.24

( 197 )

TABLE VIII.
DEVIATIONS OF THE CALIBRATION-FORMULAE FOR THE

VIII

VI VII
W-R, | (W—R,) | (w-R,)| (W—R, | (W—R, | (W—R)
—0.0080 | —0.0080 | —o 0930 | —o 0032 | —0.0013 | —0.0011
elt Pie 26) oe) 4 46h
see a
| + Seeeerag) ee 299: | 0.T Sg | By
oe
0
| + ih Seg ay Ai RS 9 SERS at MR
bp 145
Meee | l- l+
ae
0 eresimay i a9 | 4o|—
a
eo a 3) 93/4. 33/4 ~-3s
ae
is
= OS ee ee ba ee
i 076
+ (34 |
ce Se (ee ee
a is |
Poe
i
emer on] 99/4 6B lt le
4+ 38
rl
— 3
tf a|-s|- w/o].
ee
0 ee oF) a 80 | O90
meee sy | 68} 37} 90/-+ 490

Proceedings Royal Acad. Amsterdam, Vol. IX.

( 198 )

To observation 11 of this table we have applied the correction
mentioned at table VI. To the observations 17 and 7 we have
accorded half the weight on account of the large deviation from the
single determinations mutually (comp. tables VI and VII) ’).

§ 18. Conclusion. For the mean error of the final result for one
temperature (when this is taken equal for all temperatures) we find
by comparison with the formula found:

microv.
R, + 2.8
R, + 38.2
R, + 2.6 (2.1 when leaving also out of account — 217°)
R, £138

The mean error of the result of one day, according to the mutual
agreement of the partial results, is:

+ 2.9 microvolts,

whence we derive for the mean error of one temperature, supposing
that on an average two daily results are averaged to one final result :
+ 2.0 microvolts.

(2 microvolts agree at — 29° with 0°.05, at -- 217° with 0°.16).

Hence it seems that we may represent the electromotive force of
the thermo-element constantin-steel between O° and — 217° by the
five-terms formula to within 2 microvolts. For the calibration to — 217°
we therefore require measurements at at least 5 temperatures ’).

The representation including the temperatures of liquid hydrogen
is much less satisfactory ; for the mean error would be found according
to this representation + 3.2 microvolts, agreeing with 0°.075 at
— 29° and 0°.74 at — 252° and — 259°.

In order to include the hydrogen temperatures into the formula a
6 term will therefore probably be required.

But for measurements at the very lowest temperatures the element
constantin-steel is hardly suitable (comp. § 7).

In conclusion we wish to express hearty thanks to Miss T. C.
Joutus and Messrs. C. Braak and J. Cray for their assistance in
this investigation.

1) In the calculations for observations 3, 11 and 5 are used temperatures 0°,081
lower than the observed ones. A repetition of the calculation with the true values
has not been undertaken, as it would affect only slightly the results, the more
because the observations are uncertain.

2) If the four term formula (comp. footnote 2 § 11) should prove for this inter-
val as sufficient as the five term formula, this number would be reduced to four.

( 199.)

Physics. — On the measurement of very low temperatures. X. |
Coefficient of expansion of Jena glass and of platinum between
+ 16? and —182°.” By Prof. H. KameriincH Onnes and
J. Cuay. Communication N°. 95° from the Physical Laboratory
at Leiden.
(Communicated in the meeting of June 30, 1906).

§ 1. Zitroduction.
The difference between the coefficients @ and 4 in the expansion

4
Yee BB vs
Sitar a |

and k, and k, in the formula for the cubic expansion

oh eed
=+,[1+| haa th (5) a |

‘between O° and —182° found by Kameriincn Onnes and Hevse
(comp. Comm. N°. 85, June ’03, see Proceedings of April 05) and
those found by Wisse and Borrcuer and Tuirsen and Scueen for
temperatures above 0° made it desirable that the strong increase of
6 at low temperatures should be rendered indubitabie by more
accurate measurements *).

In the first place we have made use of more accurate determi-
nations of the variation of the resistance of platinum wires with the
temperature (comp. Comm. N°. 95°, this meeting) in order to substitute
more accurate temperatures for those given in Comm. N°. 85, which
served only for the calculation of a preliminary formula, and then
to calculate by means of them new values for a and 6 which
better represent the results of the measurements than those given in
Comm. N°. 85.

By means of the formula

W,—= W, (1 + 0,00390972 t — 0,0,9861 2°),
which holds for the kind of platinum wire used in Comm. N°. 85,
we have arrived at the following corrections:

in table IV read — 87°14 instead of — 87°,87

and —181°,42 __,, »» —182°,99
in table V read — 86°,98 _,, » — 87°,71
and —181°,22 _,, »» —182°,79

formula for the linear expansion / = /, E +

1) That the coefficient of expansion becomes smaller at lower temperatures
is shown by J. Zaxrzewskt by measuremenis down to — 103°. This agrees with the
fact that the expansion of most substances above 0° is represented by a quadratic
formula with a positive value of 6! Our investigation refers to the question whether
b itself will increase with lower temperatures.

14°

( 200 )

Thence follows

Jena giass 16"! Tee bo 907 |
k, = 23848 k,= 272. | one
Thiringer glass (n°. 50) a= 920 b=120 \ :
k, = 2761 k, — 362.

Secondly it remained uncertain whether the mean temperatures
of the ends were exactly identical with those found after the method
laid down in § 4 of that Comm. The execution of the control-
determination as described in Comm. N°. 85 § 4 (comp. § 4 of this paper)
proved that in this respect the method left nothing to be desired.

Moreover, availing ourselves of the experience acquired at former
determinations, we have once more measured the expansion of the
same rod of Jena glass and have reached about the same results
which, owing to the greater care bestowed on them, are even more
reliable. |

Lastly it was of importance to decide whether the great increase
of 4 at low temperatures also occurred with other solid substances
and might therefore be considered as a property of the solid state
of several amorphous substances. Therefore and because it was
desirable also for other reasons to know the expansion of platinum
we have measured the expansion of a platinum rod in the same
way as that of the glass rod. Also with platinum we have found
the same strong increase of 6, when this is calculated for the same
interval at lower temperatures, so that cubic equations for the lengths
of both substances must be used when we want to represent the
expansion as far as — 182°.

After these measurements were finished Scuegrt, (Zeitschr. f. Instr.
April 1906 p. 119) published his result that the expansion of pla-
tinum from —190° to O° is smaller than follows from the quadratic
formula for the expansion above 100°. For the expansion from + 16°
to —190° Scunen finds — 1641 per meter, while — 1687 w would
follow from our measurements. But he thinks that with a small
modification in the coefficients of the quadratic formula his observa-
tions can be made to harmonize with those above 100°. Our result,
however, points evidently at a larger value of 6 below O°. |

The necessity of adopting a cubic formula with a negative coeffi-
cient of ¢ may be considered as being in harmony with the
negative expansion of amorphous quartz found by Scneer (I. c.)
between —190° and 16° when we consider the values of a and 6
in a quadratic formula for the expansion of this substance between
0° and + 250°.

( 201 )

A more detailed investigation of these questions ought to be made
of course with more accurate means. It lies at hand to use the
method of Fizeau. Many years ago one of us (K.O.), during a
visit at Jena, discussed with Prof. Punrricn the possibility of placing
a dilatometer of Asse into the Leiden cryostat, but the means
of procuring the apparatus are lacking as yet. Meanwhile the
investigation following this method has been taken in hand at the
Reichsanstalt*). A cryostat like the Leiden one, which allows of
keeping a temperature constant to 0,01° for a considerable time,
would probably prove a very suitable apparatus for this investigation.

Travers, SeNTER and Jaquerop’) give for the coefficient of expan-
sion of a not further determined kind of glass between 0° and — 190°
the value 0,0000218. From the mean coefficient of expansion from
0° to 100° we conclude that this glass probably is identical with
our Thiiringer glass.

The mean coefficient of expansion between 0° and — 190° for
Thiiringer glass found at Leiden in 1903 is 0,00002074.

§ 2. Measurement of the coefficient of expansion of Jena glass and
of platinum between 0° and — 182°.

The rod of Jena glass used was the same as that of Comm. N°. 85,

At the extremities of the platinum tube of 85 c.m. length glass ends
were soldered of the same kind as the Jena rod. For the determina-
tion of the mean temperature of the ends thin platinum wire was
wound round these extremities which wire at either end passed over
into two platinum conducting wires and was enveloped in layers
of paper in order to diminish as much as possible the exterior
conduction of heat.

The temperature of the middle portion of the Jena rod was also
determined by means of a platinum wire wound round it as in
Comm. N°. 85. The rod was further enveloped in thin paper
pasted together with fishglue, and to test the insulation the resistance
was measured on purpose before and after the pasting. The tempera-
ture of the bath was determined halfway the height of the bath
by means of the thermo-element constantin-steel (comp. Comm. N°. 95a,
this meeting).

This temperature was adopted as the mean temperature of
the platinum tube, which was entirely surrounded with the liquid
gas and was only at its extremities in contact with the much less

1) Henning, afterwards Scuect, Zeitschr. f. Instrk. April 1905, p. 104 and April 1906,
p- 118. Ranpaut, Phys. Review 20, p.10, 1905 has consiructed a similar apparatus,
*) Travers, SENTER and JAgueroD, Phil. Trans. A 200.

(.202 )

conducting pieces of glass, which partly projected out of the bath.
~The scale (comp. Comm. N°. 85) was wrapped round with a
thick layer of wool enclosed in card-board of which the seams had
been pasted together as much as possible. The temperature of the
room was kept as constant as possible by artificial heating and cooling
with melting ice,:so that the temperatures of the scale vary only
slightly.

They were read on three thermometers at the bottom, in the
middle and at the top.

- The seale and the points of the glass rods were illuminated by
mirrors reflecting daylight or are-light, which had been reflected by
paper and thus rendered diffuse.

The vacuum tube (comp. Comm. N°. 85) has been replaced by a
new one during the measurements. The evacuation with the latter
had succeeded better. So much liquid gas was economized. For the
measurement with liquid oxygen we required with the

)B ‘first tube 1'/, liter per hour and */, liter with the
— second. Of N,O we used with the first only */, liter
| ) per 1*/, hour. }
| In order to prevent as much as possible irregularities
in the mean temperature the bath has been filled as
high as possible, while dry air was continually blown
against the projecting points. They were just kept free”
from ice. In two extreme cases which had been chosen
on purpose — the bath replenished with oxygen as high
as possible and the points covered with ice, and the
bath with the float at its lowest point and the point
| entirely free from ice — the difference of the mean
temperature of the ends was 10 degrees, corresponding
to a difference in length of 4 microns. The greatest
difference which has occurred in the observations has
certainly been smaller and hence the entire uncertainty ~
of the length cannot have surpassed 2 microns.

At the lower extremities the difference is still smaller.
‘| All this holds with regard to oxygen, in nitrous
ie oxide such variations in the distribution of the tem-
eee?| perature can be entirely neglected.

With some measurements we have observed that the

he length of the rods, when they had regained their

q ordinary temperature after cooling, first exceeded the
original length, but after two days it decreased again —
Fig. 1. to that value.

( 203 )

The cause of those deviations has not been explained. In a ease
where a particularly large deviation had been stated which did not
altogether return to zero, it appeared, when the points were un-
wrapped, that a rift had come into the glass.

To see whether a thermical hysteresis had come into play a
thermometerbulb (see fig. 1) with a fine capillary tube was filled
with mercury. First the level of the mercury was compared with
an accurate thermometer at the temperature of the room ina water-
bath in a vacuum glass. Then the apparatus was turned upside down
so that the mercury passed into the reservoir 4, which is a little
greater than A. Subsequently A and also a part of the stem was cooled
down during 3 hours in liquid air in a sloping position so that thanks
to the capillary being bent near 6 no mercury could flow back

TABLE I. — JENA GLASS 16™.
Date | Time sige ql; | ie | WV, K. | = | d
16 Dec. | 24.35 | 15.7 | 1026.285 | 1026.280 | 40.620 15.9
1904 | 3450 | 160 286 .279 | 40.786 17.0
4h.22| 16.3 292 .290 | 40.845 17.4
=
20 Dec | 14.50 | 15.3 | 1025.574 | 1025.559 | s. 3.503 5.021 40.6
. 24410} 15.4 .560 .950 |m.25.029 38.28 | — 86.78
24.30 | 15.4 O71 .561 | 7. 6.300 1348 1;=22A
21 Dec. | 34.15 | 14.6 | 1026.308 | 1026.291 |m.40.523 15.4
3h.45 | 14.7 .299 . 284 15.4
4h. AS} 414.7 . 308 -289 |m 40.583 15.6
22 Dec. |104 50 | 15.0 | 1025.408 | 1025.091 | s. 2.405 5.021 1,=30.8
4124.45} 45.0 442 .095 |m. 9.880 38.28 |—181.48
124.50 | 15.0 115 .098 |i 5.005 7.191 4;=18.0
23 Dec. |124.30 | 15.8 | 1026.344 | 1026.341 |m.40.606 15.6
oh. 15.6 339 309 15.2
34.30} 15.6 330 .300 |m.40.537 15.2
41 Jan. | 34.40 | 15.4 1026.288 | 1026.278 | 40.634 15.9

1905 | 44.30] 415.5 291 280 | 40.703 16.4

( 204 )

to A. When A had regained the temperature of the room the
mercury was passed again trom £ into A and the apparatus
replaced into the same waterbath as before. The deviation of the
level of the mercury was of the same order as the reading error of
the thermometer, about 0.003°. A perceptible thermical hysteresis
therefore we do not find.

me | eae eee ik = PLATINUM. _
D ii icokok W. W
ate Time sEate 146° t 4 s 2
46 Dee. 5h 59 16.5 1027.460 | 1027.461 A720
4904 16.4 41027.461 | 1027.459 47-0
17 Dec. ') 1h 4D 16 6 4026 .620 | 4026.630
9h 45 16.3 1026 .618 622
1015 4620 613 617
19 Dec. Sh 14.8 1027 .459 | 1027.442 4525
81 30 14.8 457 | 1027.440 $55
——
20 ‘ec. 3h Ayes 4026 .627 | 1626.630 | s3.475| 4.993 °40.2
3h 30 Aer | 630 633 | m — 86.32
L=
3h 55 15.4 631 635 |77.575| 8.653 ‘31.5
21 Dee. 4h 40 14.7 1027.460 | 1027.444 1525
5410 14.9 459 444 Ae S
6h 14.8 459 449, sey
=
22 Dec. 10% 40 4523 1025.963 | 1025.951 | 52.440} 4.993 “28.9
11410 A523 1025 .973 961 | m —182.6
o\, ——
1h 45 14.9 1025.964 947 115.649] 8.653 "48.5
23 Dec, 11h 25 Aer 1027.434 | 1027.436 15.0
15.6 440 44A 15.0
45 a7 440 449 afar.
3 Febr. Qh 15.4 1027.463 | 1027.459 55D
15.4 459 455 45 2

1) Journ. Chem. Soc. 63. p. 135, 1893,

( 205 )

In table II (p. 204) the temperatures are used which are found
with the thermo-element. A control-measurement with the thermo-
element placed in the same vacuum tube without rod gave for the
temperature in nitrous oxide — 87°,3 instead of — 86°,32.

The mean value of the two determinations is used for the calculation.

Another reason for the measurement of the temperature of the
bath with a thermo-element as a control was the large difference
between the mean temperature found by us and the boiling point of
nitrous oxide — 89° given by Ramsay and SHIELDs').

As we are going to press we find that Hunrpr’) has given
— 86°.2 for that temperature.

§ 3. Results.

Jena glass 161II a 835 b 117
k, 2505 k, 358.
Platinum a 9053 b 49,4 | bee
} k, 2716 k, 148,4.
As regards platinum:
Benoit finds irom, .-0° tos. S0- @ 890.1 6. 12,1
SCHEEL from 20° to 100° a 880,6 6 19,5

Honporn and Day from 0° to 1000° a 886,8 6 13,24

As to the differences between the values obtained now and those
of Comm. N°. 85 (comp. § 1), we must remark that these are almost
entirely due to the differences in the determinations of temperature.
The uncertainties of the latter, however, do not influence in the
least the conclusion about 6 and the necessity of a cubic formula.

There is every reason to try to combine our determinations on
Jena glass above and below O° in such a cubic formula. Taking into
account also the previous determination 242.10—° as the mean cubic
coefficient from 0° to 100° (Comm. N°. 60, Sept. 1900, § 20) we
find in the formula for the linear expansion below 0° and in the
corresponding one for the cubic expansion

t= H[1+ |e 300 +" (roo) + (yoo) |]

Jena glass 16I1II a' 789,4 k', 2368,1
b' 39,5 k', 120,2
c’ — 28,8 as 86,2

4) With this measurement in N,O we have not obtained a temperature deter-
mination with the thermo-element. This determination is not included in the
calculation. It is mentioned here on account of the agreement with the determi-
nation of 20 Dec., which for the rest has been made under the same circum-
stances.

*) Journ. Phys, Chem. May 1906, p. 356.

( 206 )

§ 4. Control-experiment.

The ends of the Jena glass rod were . subsequently cut off and
sealed together with a short intermediate rod. This short stick was
placed in a glass of the same width as the vacuum tube with the
same stopper and so short that the points projected in the same

TABLE Ill. — JENA GLASS ENDS.
Date | Temp. | oF | L | W, | W | |
scale t 716° t 7 Ss a
=}
42 April 1905
404415 45.4 | 297.684 | 297 683 15.4
Ah 686 685 15.4
444 43 15.4 | 927.684 | 997.682 45.5
45.4 681 679 45.5
N,0 :
=
3h 50 45.4 | 297.533 | 297 536 | s 3.473 5.024 42.3
4h 4 15.4 543 5A :
—
Ah 52 45.4 550 548 | i 5.490 7.491 32.3
43 April 47.4 | 927.677 | 297.681 47.4
14 April 16.2 | 297.675 | 227.676 15.9
40h 10
O
; =
2150 48.4 | 997.474 | 997.482 | s 1.94 5.021 35.5
=
4hQQ 18.9 482 494 | i 4.683 7.491 "8.9
45 April 46.6 | 927.795 | 297°727 45.7
Mn 16.6 724 726 16.0
4h 20 16.4 | 227.706 708 45.8
4h 46 16.4 mM 3 16.0
“46 April 44.4 | 227.706 | 227.702 43.6
“47 April 44.2 | 927.682} 227.678 14.0

(207)

manner as those of the rods in the vacuum glass. Now we have
taken only a double glass filled with wool, enveloped in a card-board
funnel and tube for letting out the cold vapours.

The measurements are given in table IIT.

The 4’s found in the experiment are of the same order of magnitude
as those found with the long rods. The calculation with the coefficients
a and 6 found in $2 yields:

Ly,o0 = 227,547 while we have found Ly,o9 = 227,544
Lo, = 227.487 a ee Lo, = 227,488.

In conclusion we wish to express hearty thanks to Miss T. C.

Jouies and Miss A. SiLievis for their assistance in this investigation.

Physics. — “On the measurement of very low temperatures. XI. A
comparison of the platinum resistance thermometer with the
hydrogen thermometer.” By Prof. H. Kameruincu ONNes and
J. Cray. Communication N°. 95° from the Physical Laboratory
at Leiden.

(Communicated in the meeting of June 30, 1906).

§ 1. Introduction. The following investigation has been started
in Comms. N°. 77 and N°. 93 VII of B. Mumiyk as a part of the
more extensive investigation on the thermometry at low temperatures
spoken of in Comm. N°. 95%. In those communications the part of
the investigation bearing on the electrical measurements was chiefly
considered.

The hydrogen thermometer was then (comp. Comm. N°. 93 § 10)
and has also this time been arranged in the same way as in Comm.
N°. 60. Afterwards it appeared, however, that at the time the thermo-
meter did not contain pure hydrogen, but that it was contaminated by
air. The modifications which are consequently required in tables
V and VI of Comm. N°’. 93 and which particularly relate to the very
lowest temperatures, will be deait with in a separate communication.

Here we shall discuss a new comparison for which also the filling
with hydrogen has been performed with better observance of all the
precautions mentioned in Comm. N°. 60.

We have particularly tried to prove the existence of the point of
inflection which may be expected in the curve (comp. § 6) represent-
ing the resistance as a function of the temperature, especially with
regard to the supposition that the resistance reaches a minimum at
very low temperatures, increases again at still lower temperatures
and even. becomes infinite at the absolute temperature O (comp.

( 208 )

Suppl. N°. 9, Febr. ’04). And this has been done especially because
temperature measurements W ith the resistance thermometer are so
accurate and so simple.

From the point of view of thermometry it is important to know
what formula represents with a given accuracy the resistance of a
platinum wire for a certain range, and how many points must be
chosen for the calibration in this range.

In Comm. N°. 93 §10 the conclusion has been drawn that between
0? and —4180° a quadratic formula cannot represent the observa-
tions more accurately than to 0°.15, and that if for that range a
higher degree of accuracy is required, we want a comparison with
the hydrogen thermometer at more than two points, and that for
temperatures below —-197° a separate investigation is required. In
the investigation considered here the temperatures below — 180° are
particularly studied ; the investigation also embraces the temperatures
which can be reached with liquid hydrogen.

It is of great importance to know whether the thermometer when
it has been used during a longer time at low temperatures would
retain the same resistance. We hope to be able later to return to
this question. Here we may remark that with a view to this question
the wire was annealed before the calibration. Also the differences
between the platinum wires, which were furnished at different times
by Herarvs, will be considered in a following paper.

§ 2. Investigations by others. Since the appearance of Comm.
N°. 93 there has still been published on this subject the investigation
of Travers and Gwyer’). They have determined two points. They
had. not at their disposal sufficient cryostats such as we had for
keeping the temperatures constant. About the question just mentioned :
how to obtain a resistance thermometer which to a certain degree
of accuracy indicates all temperatures in a given range, their paper
contains no data.

‘§ 3. Modification in the arrangement of the resistances. The
variation of the zero of the gold wire, mentioned in Comm. N°. 93
VIII, made us doubt whether the plates of mica between the metallic
parts secured a complete insulation, and also the movability of one of the
glass cylinders made us decide upon a modification in the construction
of the resistances, which proved highly satisfactory and of which we

1) Travers and Gwyer. Z. f. Phys. Chem. LIl, 4, 1905. The wire of which
the calibration is given by Otszewsx1, 1905, Drude’s Ann. Bd. 17, p. 990, is appa.
rently according to himself no platinum wire. (Comp. also § 6, note 1).

( 209 )

have availed ourselves already in the regulation of the temperatures
in the investigation mentioned in Comm. N°. 942.

- A difficulty adheres to this arrangement which we cannot pass by
unnoticed. Owing to the manner in which this thermometer has been
mounted it cannot be immersed in acid. Therefore an apparatus
consisting entirely of platinum and glass remains desirable. A similar
installation has indeed been realized. A description of it will later
be given. The figures given here exclusively refer to the thermometer
described in Comm. N°. 94¢ (p. 210).

Care has been taken that the two pairs of conducting wires were
identical. Thus the measurement of the resistance is performed in
a much shorter time so that both for the regulation of the tem-
perature in the cryostat and, under favourable circumstances, for the
measurement the very same resistance thermometer can be used.

§ 4. The temperatures.

The temperatures were obtained in the cryostat, described in Comm.
N°. 94¢, by means of liquid methyl chloride —39°, —59°, —88°, of
liquid ethylene — 103°, — 140°, — 159°, of liquid oxygen — 182°,
— 195°, — 205°, — 212°, — 217°, by means of liquid hydrogen
— 252° and — 259°. The measurements were made with the hydrogen
thermometer as mentioned in § 1.

§ 5. Results for the platinum wire. These results are laid down
in table I (p. 210).

The observations marked with [ | are uncertain on account of the
cause mentioned in Comm. N°. 952 § 10 and are not used in the
derivation and the adjustment of the formulae. For the meaning of
W—R4z, in the column “remarks” I refer to § 6.

§ 6. Representation by a formula.

a. We have said in § 1 that the quadratic formula‘) was insuffi-
cient even for the range from 0° to —180°.

If a quadratic formula is laid through — 103° and — 182°, we
find :

) The correction of Cattenpar, used at low temperatures by Travers and
Gwver, Z. f. Phys. Chem. LII, 4, 1905 comes also to a quadratic formula.
Dickson’s quadratic formula, Phil. Mag. June 1898, is of a different nature but
did not prove satisfactory either; comp. Dewar Proc. R. Soc. 64, p. 227, 1898.

The calibration of a platinum thermometer through two fixed points is still

often applied when no hydrogen thermometer is available (for instance BesTeLMEYER
Drude’s Ann. 13, p. 968, ’04).

COMPARISON BETWEEN THE PLATINUM RESISTANCE
THERMOMETER AND THE HYDROGEN THERMOMETER.

(2105

DA Boe oe

a hydrogentherm.| measured | Remarks
0° 0° | 137 884 i mean cf 5 measurements,
27 Oct bh. — 29.80 421 .587
705
2h. 50 — 58.75 105.640
30 Oct. 3 h. 50 — 88.14 89.277
05
8 July 10h. 12 — 103.83 80.448
05
26 Oct. Hieh: 20 — 139.87 59.914
05
7 July 4h. 25 — 139.85 59.920
05
26 Oct. 3 h. 16 — 158.83 48 .929
0d
27 June 1 b. 40 [— 182.69] 34.861 W—R4,—— 0.061
05
30 June 11h. 0 — 182.75 34 858
06
27 June 3h. 50 [-- 195.30] 27.598 W—R 4, =-+-0.082
0d
2 March 3h. 35 — 195.18 27 .595
706
29 June 11h. 6 [— 204.53) 22.016 W—Ryy —— 0,110
05
2 March 1h. 30 — 204.69 22.018
06
39 June 3h. O [— 212.83] 17.255 W—R 4, =— 0.082
O05
5 July 5h. 53 — 212.87 17 290
05
5 July 3 h. 20 — 217.4 14.763
05
3 March 10h. O — 27.41 14.770
05 |
5 May AD |e | — 252.93 | 1.963
06 '
5 May 5h. 7; —- 259.24 1 444

Nee
| Temperature |Resistance

( 211 )

t 3
oe 9097 — 0,009862
os }1 +08 0 (a :) (35) f

For instance at — 139° it gives IV—R: + 0,084. A straight line
may be drawn through — 182°, — 195°, — 204° and — 212° and
then — 217° deviates from it by 0°,25 towards the side opposite to
— 158°. Hence the existence of a point of imflection is certain
(comp. sub d). Therefore it is evident that a quadratic formula will
not be sufficient for lower temperatures.

6. But also a cubic formula, even when we leave out of account
the hydrogen temperatures, appears to be of no use.

For the cubic formula through the points —88°,14, —158°,83,
—204°,69, we obtain:

t 3
—— —0,0,7367 0,0 ‘
1. = W, }1+ 0,393008 — (CZ =) a 58880( 55 7 |

It gives for instance at —182° a deviation of -—0,110, at — 217°
a deviation of + 0,322 °*)

c. In consequence of difficulties experienced with formulae in
ascending powers of ¢, we have used formulae with reciprocal powers
of the absolute temperatures (comp. the supposition mentioned in § 1
- that the resistance becomes infinite at the absolute zero).

Three of these have been investigated :

eg 2S +5 (sia) + a( sca) +4( a — ara) Rearing 7s
W, 100 100 100 273,09
watt ati) (is) CF)
W, 100 100 100 273,09
=e 102 104 -
fA] grana| ©
Ws —lta oo +? (soo) + (55 +4(=—sea0) +
W, 100 100 100 YER
10° 108
ae ‘(Ge + er, ara.) a)

We shall also try a formula with a term = instead of 7

For the first we have sought a preliminary set of constants which
was subsequently corrected after the approximate method indicated
by Dr. E. F. van pe SanpE BakuuyzEN (comp. Comm. N°. 95a) in
two different ways. First we have obtained a set of constants A,
with which a satisfactory accurate agreement was reached down to
— 217°, a rather large deviation at — 252° and a moderate deviation
at — 259°. Column JW—R4, of table II contains the deviations.
Secondly we have obtained a set of constants which yielded a fairly

’) These values deviate slighily from those communicated in the original.

( 212 )

accurate agreement including — 252°, but a large deviation at — 259°,
These are given in table II under the heading W— Ryzrz.

Lastly we have obtained a preliminary solution B which fairly
represents all temperatures including — 252° and — 259° and from
which the deviations are given in table II under W—Afg, and a
solution of the form C which agrees only to — 252° and to which
W—Reg relates.

The constants of the formulae under consideration are :

mre 2

a i 0.399625 |- 0.400966 + 0.412793 | --0. 40082
i |— 0.0002575|+- 0.001159 |4+ 0.013812 | +0.001557
c |-+ 0.0049442+ 0.0062417|+ 0.012683 | 0.00557
d | 0.019380 FE: 0.026458 |4+- 0.056221 | +0.01975
| a Boas —0.16504

EAD ode

COMPARISON BETWEEN THE PLATINUM RESISTANCE
THERMOMETER AND THE HYDROGEN THERMOMETER.

| I

‘Number | Resistance | |

Temperature | of obser-
the hydrogen| with tho| PS! | Mar | Ran | Pe |
thermometer.) hydrogen | jin o
therm. |
l
0° | 137.884 | 0 0 | 0 0
— 29.80 3 124 .587 + 0.025 | -+ 0.066 | + 0.210 | + 0.063
— 58.75 3 105.640 + 0.0114 | — 0.014 | + 0.153 | + 0.048
— 88.14 4 89.277 — 0012} — 0.050 | — 0.001 | + 0.008
— 103.83 3 80.448 — 0.023 | — 0.061 | — 0.075 | -— 0.015
— 139.87 3* 59.914 + 0.004 | — 0.005 | — 0.082 | — 0.005
— 158.83 3 48 929 + 0.023 | + 0.044 0 + 0.008
— 182.75 2 34.858 — 0.029 | + 0.027; + 0.083 | — 0.035
— 195.18 2 27 .595 + 0.009 | + 0.061 | -+ 0.148 | + 0 007
— 204.69 1 22.018 — 0.04) + 0.012} + 0.100 | — 0.014
— 212.87 3 17.290 — 0.024 | — 0.065; — 0.001 | — 0.031
— 217.41 Ae 14.766 + 0.028} — 0.048 | +- 0.270; + 0.007
— 252.93 2 1.963 + 2.422 + 0.057 | — 0.001 0
— 259.% 4 4.444 | 4+0.199}] — 4.201 0

( 213 )

In those cases where the W—R have been derived f.om two deter-
minations the values in the 2.4 column are marked with an *?).

If we derive from the differences between the observed and
the computed values as far as —217° the mean error of an obser-
vation by means of Ay;, this mean error is expressed in resistance
+ 0,025 2, in temperature + 0?,044.

The mean error of an observation of the hydrogen thermometer,
as to the accidental errors, amounts to 0°,02 corresponding in resist-
ance to + 0,010 &, while that of the determination of the resistance
may be left out of consideration. We cannot decide as yet in how
far the greater value of the differences between the observations and
the formula is due to half systematic errors or to the formula.

For the point of inflection in the curve representing the resistance
as a function of the temperature we find according to B — 180°’).

In conclusion we wish to express hearty thanks to Miss T. C.
Jottes and Mr. C. Braak for their assistance in this investigation.

Physics. — “On the measurement of very low temperatures. XII.
Comparison of the platinum resistance thermometer with the
gold resistance thermometer. By Prof. H. KamertincH OnNes
and J. Cray. Communication N°. 95¢ from the Physical labora-
tory at Leiden.

(Communicated in the meeting of June 30, 1906).

§ 1. Jntroduction. From the investigation of Comm. N°. 93, Oct.
04, VIL it was derived that as a metal for resistance thermometers
at low temperatures gold would be preferable to platinum on
account of the shape of the curve which indicates the relation
between the resistance and the temperature.

Pure gold seems also better suited because, owing to the signifi-
cation of this metal as a minting material, the utmost care has been
bestowed on it for reaching the highest degree of purity and the
quantity of admixtures in not perfectly pure gold can be exactly
determined. The continuation to low temperatures of the measurements
described in Comm. N°. 93 VIII — which had to be repeated
because, although MEwinx’s investigation just mentioned had proved
the usefulness of the method, a different value for the resistance .

1) The deviations of the last two lines differ a little from the original Dutch
paper.
®) Owing to e being negative (B) gives no minimum; a term like that with ¢
does not contradict, however, the supposition w=o at 7T’'=O (§ 1) as the formula
holds only as far as —259°, :
15
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 214 )

had been found before and after the exposure of the wire to low
acquired a special value through this peculiarity

temperatures
of gold.

As will appear from what follows, the pomt of inflection of the
resistance as a function of the temperature must lie much lower for
gold than for platinum.’ Our favourable opinion about gold as a
thermometric substance was confirmed with regard to temperatures
to a little below —217°. With respect to the lower temperatures
our opinion is still uncertain. A minimum of resistance seems not
to be far off at —259°.

§ 2. The apparatus and the measurements. About the measure-
ments we can only remark that they are performed entirely according
to the methods discussed in Comm. N°. 98.

The pure gold was furnished through the friendly care of Dr.
C. Hortsema. It has been drawn to a wire of 0,1 mm. in diameter
by HERagvs.

The gold wire was wound upon 2 cylinders, it was about 18 m.
in length and its resistance at 0° was 51,915 Ohms. The tempera-
tures were reached in the cryostat of Comm. N°. 94¢ as in the
investigation in Comm. N°. 95°.

The determinations of temperature were made by means of the
resistance of the platinum wire of Comm. N°. 95°. The zero
determinations before and after the measurements at low fempera-
tures agreed to perfection (this agreement had left something to be
desired in the measurements dealt with in Comm. N°. 93).

The measurements were made partly directly by means of the
differential galvanometer, partly indirectly by comparing the gold resis-
tance with a platinum resistance, which itself had been compared
with the originally calibrated platinum resistance (comp. Comm.
N°. 95°).

§ 3. The Results, obtained after-the direct and the indirect method
are given in column 3 of table II and indicated by d and 7
respectively.

For the observations the cryostat was brought to the desired
temperature by regulating it so that the resistance of the platinum
wire had a value corresponding to this temperature, and by keeping
this temperature of the bath constant during the measurements of
the resistance of the gold wire. The temperatures given in table III
are the temperatures on the hydrogen thermometer according to the
observations of Comm. N°. 95¢ belonging to the resistance of the
platinum thermometer.

( 215 )

TABLE Ill.
CALIBRATION OF THE GOLD RESISTANCE THERMOMETER.

Temperature Observed :

Date. W—R W—R W—R
resistance. | gold resistance. d BI Bil

1906 | 0 51.915 d 0 0 | 0
4 Febr. 5 b.57| — 28.96 AG 137 — 0.002 | —0.018 | + 0.029
» 3h. 4 — 58.58 40 326: + 92 + 442 |) + 46
» 12h, 25| — 87.43 34. 640 i Pe Fe WDB hae 2
42 June 2h. 20] — 103.82 31.432 d eet Oh ES A ee oe
>» Wh, — 139.86 24.984 d So: ie Oe ras (0 ee ee
17 Jan. 3h. 20) — 459.44 90.394 i a OR a | a ae
4 June 41 hb. 50) — 182.75 15.559 d oo 6b AR ae ae a8
>» 5h, 8| — 195.48 12.980 d Bee Sey ett toe Se oe an
aw a — 204.69 10.966 d eo Ohh iae P18 op
o° tEh. — 29.87 9.203 d ae iit a egle
42 Jan. 41 hb. — 216.25 8.460 i eT a, ae
48 May 4h. 10| —- 252.88 2.364 d aS BN gd ge ee a peel Wah ee
i 6K — %9.18 2.047 d eer = OTT Te Me Ob

In order to agree with Dewar, we ought to have found for the
resistance of the gold wire at the boiling point of hydrogen 1.7082
instead of 2.364 2. Also the further decrease of the resistance found
by Dewar’) in hydrogen evaporating under a pressure of 30 mM. is
greater than that was found by us. We may remark that this latter
decrease of the resistance according to bim would belong to a decrease of
4 degrees on the gas thermometer, and that we in accordance with
Travers, SENTER and Jaquerop?) found a difference in temperature
of 6,°3 between the boiling point of hydrogen at a pressure of
760 m.m. and of 60 m.m. (preliminary measurements).

§ 4. Representation of the variation of the gold resistance by a
formula. As to this we refer to what has been said in Comm.

1) Dewar, Proc. Roy. Soc. Vol. 68 p. 360. 1901.
®) Travers, SenteR and Jaguerop, Phil. Transact. A. 200.
Proc. Roy. Soc. Vol. 68, p. 361, 1901.

( 216 )

N°. 95°, XII. § 6. The resistance of the gold wire can be represented
fairly well as far as — 217° as a function of the temperature by a
formula of the form A.

W, t
Gr = 1+ 0,89070 + 0.017936 Ee

0,0085684 0.0080999 70 (A)
= 100 + 100.~»=— 273.09 J ©

This formula A is not oa to include the hydrogen temperatures.
For the deviations W—R, comp. table III.
We have therefore made use of a formula £, and

W, t ae LoS
po ee —— } + 0,0102335 | —_} +

qe 09

00052211 }( 12° pa
ie 273,09

is in good harmony down to — 258°, a

We _ 1 + 0,894548 (sa) + 0,0200118 ry, |
W, 100

100 ~—-100 |

T 278 “=

100 100

—(sai

se 100
+0 wich —0 268011 (> zs) | (B I)

(BIL)

'
gives a fair harmony also at —- 259°’).

The deviations are given under the headings W—Rg,and W—Rp yy
in columns 5 and 6 of table III. The mean error of an observation
with respect to the comparison with formula BJ is + 0,017 2 in
resistance and + 0°,09 in temperature. Formula BJ gives for the
point of inflection of the gold resistance — 220°.

+ 0,0102889 € 3) + 0,0229106 (>

— 0,00094614 (=)

Mathematics. — “Quadratic complexes of revolution.” By Prof.
JAN DE VRIES.

§ 14. When the rays of a complex can be arranged in reguli of
hyperboloids of revolution with the same axis, then the complex can
bear revolving about that axis. If such a complex of revolution 2
contains also the second regulus of each of the indicated hyperboloids,
then it is symmetric with respect to each plane through its axis

1) The coefficients of the formulae and the values of the deviations, found at
a renewed calculation, differ slightly from those given in the original Dutch paper.

( 217 )

and it can be distinguished as a symmetric complex of revolution.
This is the ease with the complexes of tangents of surfaces of
revolution.

We determine the general equation of the quadratic complexes of
revolution with axis OZ in the coordinates of rays

—— . ! —— — f —— ee »!
Pi = & — .& , Ps —_ y 7] ‘ P: —_—_ «~ ~y
' ' !

p,m yz —zy So 28 22, Peary — ye.
By substitution of
i=, Pr, > P,P, top, = Py
P= p,— BPE > Ps PPT OPS Pe = Pes
(where’a?’+ 8?=1) in the general quadratic equation we easily
find that the equation of an 2 can contain terms only with
(P.* + Ps')> (Pa +Ps’)s Por Por (Pi Ps—Ps Ps) and (Pp, Ps + Ps Ps)-

As the latter combination can be replaced by — p, p, in consequence
of a wellknown identity we tind for @ the equation
A(p,?+-p.")+ Bp,?+2Cp,p.+ Dp. + Ep. +P, )+2F(Pips—PsPs)—9- (1)
If C=O, equation (1) does not change when wz is replaced by
— «x; so it represents a symmetrical complex.
The coordinates of rays
gq, Suu : gq, =v—v : g,=w—w ,
gq, vu —w' , gq; = wu —w' , J, = uv’ — w',
where uw, v and w represent the coordinates of planes are connected
with the coordinates p by the wellknown relations

Pits =P 2? Vs = Ps * Vs = Pa? Qi — Ps? Ws = Po? Qs"
So 2 can also be represented by
Eq? +927) +-Dqy? +20 G96 +B? + A947 +957) +24 (92% —19s)=9- - (2)

This equation is found out of (1) by exchanging p,; and qi, and
ona, ©, DEF and £, D; C, B, A, — F.

§ 2. The cone of the complex of the point (v’,7/, 2’) has as
equation :

A(e—#) + AQy—y'P + Be—2/)* +2€ (y'x—e'y)(z—2') + Diy'e—a'y)? +
+E (e'y-y'2zl +E (2'a-2'2z)? + 2F (a-2')(a'2z—-z'x) + 2F (y—y')(y'z-z'y)=0. (8)

In order to find the equation of the singular surface we regard
the cones of the complex whose vertices Jie in XOZ and note the
condition expressing that the section of such a cone and XOY
breaks up into two right lines. After suppression of the factor 2’

which is to be rejected and substitution of z* + y?= 7’ for 2’, we
find the equation

( 28)

D(AE — F%)rt + (AE + BD — C? — PF?) (Ez? — ae ag
4+ B(Eet—2Fe+ AP =0... . (4)

As this can be decomposed into two factors oe se form
Ir + M (Ez? —2Fz-+ A), the singular surface = consists of two
quadratic surfaces of revolution.

These touch each other in the cyclic points /, and J, of the plane
XOY and in the points B, and B, on OZ determined by

J pss ihe |

The two surfaces cut each other according to the four isotropic

right lines indicated by the equations
a7? = 0 and M2" —- 22 | A= 0... 22 ee

If 2 is symmetric (C= 0) the two parts of the singular surface

have as equations
(AE — F*) (w? + 4?) + B(Et —2Fz2+ A4)=0, . . (6)
D(a? yt) elie! OBA = 0 oa

If we find B=O and D=—O, then = breaks up into the four
planes (5) and @ is a particular tetraedal complez.

Out of (8) it is easy to find that the cones of the complex of the
points B,, B,, 7, and J, break up into pencils of rays to be counted
double.

These points shall be called bisingular.

§ 3. The rays of the complex resting on a straight line 7 touch
a surface which is the locus of the vertices of the cones of the
complex touched by 7. This axial surface is in general of order four
and of class four and possesses eight nodes. *)

We shall determine the axial surface of OZ. The points of inter-
section (0, 0, 2’) of an arbitrary cone of the complex with OZ are
indicated by the equation

[E(#? + y*) + Blz? — 2[F (a? + y’) + Bel2' + [A (2? + y’) + Be*] = 0.
This has two equal roots if

(AE — F*) (a+ y*) + B(Ex —2 Fe + Aa? +y)=0 . (8)

So the axial surface of OZ consists of the two isotropic planes

through the axis and a quadratic surface of revolution which might

be called the meridian surface. If 2 is symmetrical, it forms part
of the singular surface as is proved out of (6).

Also the axial surface of the right line 7. lying at infinity in

XOY breaks up into two planes, and a quadratic surface. Its

1) Sturm, Liniengeometrie Ill, p. 3 and 6.

(219 )

equation is found most easily by regarding the rays of the complex
normal to XOZ. From «=2’, z=2’ ensues p,=0, p,=0,
Pp, =D» Ps =—9, Pp, —— zp,. By substitution in (1) we find

(A + De? + E2* — 2 Fz)p,?=0,
and from this for the indicated surface

Diet? +y?)+ E?—2Fe+A=0 .... (9)

For the symmetrical complex this parallel surface is according to
(7) the second sheet of the singular surface.

The planes of the pencils of rays of the bisingular points B,, B,
form the lacking part of the axial surface of /.. We can show this
by determining the equation of the axial surface of the right line
z’=0, y’ =6, and by putting in it =o. We then find

(Ez? — 2 Fz + A){D (e? + y*) + E2?—2 Fz + A}=0 . (10)

The meridian surface, the parallel surface, and the two parts
of the singular surface belong to a.selfsame pencil, having the skew
quadilateral B,J, B,J, as basis.

If in the equation of the cone of the complex the sum of the
coefficients of 2’, y? and 2° is equal to zero, then the edges form 2
triplets of mutually perpendicular rays. The vertices of the ¢triortho-
gonal (equilateral) cones of the complex belonging to 2 form the
surface of revolution

(D+ EB) («? + y?) + 2Fe? —4Fz24 (24+ B)=0. . (11)

Jt has two circles in common with each of the parts of >. These
contain the vertices of the cones of the complex which break up
into two perpendicular planes.

§ 4. The distance /, from a right line to OZ is determined by

Pa”
ee ee Ge ag ss ea)
Seer.
the angle 4 between a ray and XOY by
Ps
ea (13)
seat 2.
So the condition /, tang 4—= a furnishes the complex
PsP = a(p,* + Pp,*) - Pama siges 1)

Here we have a simple example of a symmetrical complex of
revolution.

The equation
eo lg 1-7) at a vee 5 > €15)

( 220)

determines a complex 2 whose rays form with the axis a constant
angle, so they cut a circle lying at infinity.
The equation
pF pF 4 pli. oa ee
furnishes a complex 2, whose rays cut the circle 2*-+ y? =a’.

For XOY euts each cone of the complex according to this circle.
If 7 represents the distance from a ray O then
2 Pa eee
Pi + PF Ps
If NOY is displaced along a distance c in its normal direction,
p, and p, pass into (p,— cp,) and (p, + cp,). So for the distance
/, from a ray to the point (0,0, c) we have
1? — (py Pe eBid ote SOUP Sea) ie ete ag (18)
’ Pi + Pa’ + Ps”
If in this equation we substitute —c for c we shall find a relation
for the distance 7, from the ray to point (0, 0, —c).
The equation

a,l,* + 4,1’ =8
furnishes a complex 2 with the equation
(a, + @,) c? — B}(p,? + p.”) — Bp,” + (a, + 4) (Pa? 1 Ps 1 Po’) +
+ 2 (a, — @,)¢(p, Pp; — PsP.) = 9- : caterer oT (19)
This symmetrical complex is very extensively and elementarily
treated by J. Nevsere (Wiskundige Opgaven, IX, p. 334—341, and
Annaes da Academia Polytechnica do Porto, 1, p. 187—150). The
special case a,/, + a, 1, = 0 was treated by F. Corry (Mathesis, IV,
p. 177—179, 241—243).
For /, =J/, we find simply .
ee | ee re
This complex contains the rays at equal distances from two fixed
points. As c does not occur in the equation the fixed points may
be replaced by any couple of points on the axis having O as centre’).

§ 5. When there is a displacement in the direction of OZ the
coordinates of rays p,, P,, Pp, and p,, do not change whilst we obtain
Pr=Pithp, and pp=p, — hp

so
P: Ps + Pa Ps = Pi Pa 1 Pa Po
The forms (p,2-+p,”) and (p,p, — Pp, P,) are now not invariant.

1) This complex is tetraedral. See Sturm, Liniengeometrie, I, p. 364.

( 221 )

When in equation (1) of the complex 2 the coefficients H and F
are zero, the complex 2 is displaced in itself by each helicoidal
movement with axis OZ..This complex can be called helicoidal.

The singular surface has as equation

(BD — C) (2? + y) + AB=0;... . . (21)
so it consists of a cylinder of revolution and the donble laid plane
at infinity.

§ 6. By homographic transformation the complex 2 can be changed
into a quadratic complex with four real bisingular points.

If we take these as vertices of a tetrahedron of coordinates
O,0,0,0,, it is not difficult to show that the equation of such a complex
has the form

Ap’, + B ps, + 2 CPis Pas + 2 DPis Pas + 2 EPs Pas = 9. (22)
If we again introduce the condition that the section of the
cone of the complex with one of the coordinate planes consists of
two right lines we find after some reduction for the singular surface
A(D-E)y,*y,’+2{ AB-(C-D) (C-E)}4,9,9sy,t+B(D-E)y,*¥,2=0 « (23)
So this consists of two quadratic surfaces, which have the four
right lines O,O0,, O,0,, O,O0,; and O,O, in common.
_ For A=0, 4=0 the complex proves to be tetraedral.
For D= FE the equation is reducible to
eae Pa 2 (C — Dy p,,p,, = 9,
and indicates two linear complexes.
For the axial surfaces of the edges O,O, and O,0, we find
a, 0,{2A0,2,+(D—E)a,27,{}=0.. . . (24)
and
eee eee, () — BE) ee} 0. 5. . (25)
For a point (0,y,,0,y,) of the edge O,O, the cone of the complex
is represented by |
Ay, # + 2(C — B)y,y,2,2, + By,22,7=0;. ~ (26)
so it consists of two planes through O,0,.

This proves that the edges O,0,, O,0,, O0,0;, 0,0, are double
rays of the complex °*).

1) See Sturm, Liniengeometrie Mi, pp. 416 and 417.

( 222)

Physiology. — “A few remarks concerning the method of the true
and false cases.” By Prof. J. K. A. WertHem SALoMONsoN.
(Communicated by Prof. C. WINKLER.)

The method of the true and false cases was indicated by FECHNER
and used in his psychophysical investigations. He applied this method
in different ways: first to determine the measure of precision
(Pricisionsmasz) when observing difference-thresholds, afterwards to
determine these difference-thresholds.

Already in the course of his first experiences arose the difficulty
that not only correct and incorrect answers were obtained, corre-
sponding with the “true” and “false” cases, but that also dubious
eases occurred, in which the observer could not make sure as to
the kind of difference existing between two stimuli, or whether there
did exist any difference at all. Frcuner himself, and many other
investigators after him, have tried in different ways to find a solution
to this difficulty. What ought to be done with these dubious cases?

Frcuner has indicated several methods, which he subjected to an
elaborate criticism. Finally he concluded that the method to be
preferred to all others was that one, in which the dubious cases
were distributed equally amongst the false and the true cases. If
e.g. he found w true cases, v false cases and ¢ dubious cases, he
calculated his measure of precision as if there had been w+ 3¢
true cases and 4¢-+ v false cases.

Furthermore he showed that a method, employed especially by
American experimental physiologists, in which the reagent is urged
always to state a result, even if he remains in doubt, practically
means the same thing as an equal distribution of the ¢ cases amongst
the true and the false cases.

FecuNer still worked out another method, by means of which
the threshold value was first calculated from the true cases, then
from both the true and dubious cases, whilst the final result was
obtained with the aid of both threshold values.

A most elegant method to calculate the results of the method of
the false and true cases has been pointed out by G. E. Méuter,
starting from this view, that as a matter of necessity the three groups
of cases must be present, and that they have equal claims to exist;
that the number of cases belonging to each of these groups in any
case, are equally governed by the well-known law of errors. From
the figures for the true false and dubious cases the thresh Ctes

may afterwards be calculated.

I need not mention some other methods, e.g. that of Foucav.t,

( 223 )

that of Jastrow, because the method of Foucavtr is certainly in-
correct (as has been demonstrated among others by G. E. Mixer),
whilst that of Jastrow is not quite free of arbitrariness.

Against all these different ways of using the method of the false
and true cases, I must raise a fundamental objection, which I will
try to elucidate here.

Whenever two stimuli of different physical intensity are brought
to act on one of the organs of the senses, either the reagent will
be able to give some information as to the difference between these
stimuli, or he will not be able to do so. If he cannot give any
information, then we have before us a dubious case, if on the con-
trary he is able to give some information, this information may
either be correct, — this constituting a true case — or it may be
incorrect, when we shall have a false case.

If the experiment is repeated a sufficient number of times, we
shall have obtained at last a certain number of true cases w, of
false cases v and of dubious cases ¢.

Generally it is admitted that the reagent has indeed perceived
correctly w times, that he has been mistaken v times, that he
was in doubt ¢ times. If this premiss were correct, FECHNER’s or
G. E. Miiurr’s views might be correct too. This however is not the
case. An error has already slipped into the premiss, as will become
evident furtheron.

No difference of opinion exists as to the dubious cases. To
this category belong first those cases, where the reagent got the
impression of positive equality, and next those cases, where he
did not perceive any difference, and consequently was in doubt.
Together they embrace such cases only, in which a greater or lesser
or even infinitesimal physical difference was not perceived.

Neither need any difference of opinion exist as regards the false
cases. In these cases a stimulus has been acting on the organs of
the senses, and information was given about the effect, but on account
of a series of circumstances, independent of the will of the reagent,
his judgment was not in accordance with the physical cause. The
. physical cause therefore has not been perceived, but accidental cir-
cumstances led the reagent to believe that he was able to emit a
judgment, though this judgment, accidentally, was an incorrect one.

And now we are approaching the gist of the argument. If it be
possible, that amongst a series of experiments a certain number
occur, in which the reagent really does not perceive the physical
cause, but is yet induced by chance to emit a judgment which proves
to be an incorrect one, then there ought to be also a number of

(224 )

cases, in which likewise the physical cause is not perceived, in
which however by chance a judgment is emitted, though this
time a correct one. These facts being dependent on circumstances
beyond our will, the chances are equal that either a wrong or a
right judgment may be given. If therefore we had v false cases, we
may reasonably admit the existence of v cases, in which practically
the physical cause has not been perceived, and where yet a judgment,
this time a correct one, has been given. These v cases however have
been recorded amongst the true cases, though they cannot be
admitted as cases of correct perception: it is only in 2w—v cases that
we may suppose the physical cause to have been really and correctly
perceived; in all other cases, in 2v +7 cases therefore, there has
been no perception of the real difference of the stimuli.

In this way we have only to consider two possibilities, constitu-
ting the perceiwed and non-perceived cases, the number of which
I will indicate by § and y. The supposition that we may apply
the principles of the calculus of probability to them, is justified a
priori.

This supposition is changed into a certainty, if we apply the
mathematical relations, stated by FeEcHNerR to exist between the
numbers of true and false cases.

As is well known, Frcunrer added to the number of true eases,
obtained by the experiment, one half of the dubious cases: he
used therefore in his calculation a rectified number of true cases
w =w-+kt. In the same manner he corrected the number of false
cases by adding to them likewise one half of the dubious cases :
v=v — 4 f

In calculating the number of my perceived cases, I get § = w—,
whilst the number of non-perceived cases is represented by x= ¢-+ 2v.
Evidently I may also express the number of perceived cases by

§ = w'—v'.
As Fercuner has given for the relative value of the corrected
number of true cases the expression :
; Dh
sat = Eb oe feve
wtttv n eae hg

0
and for the corrected relative number of false cases the expression:
Dh
!
x v+tht 1
a ae a =} ——= Je"d
w+t+tv n Va

( 225 )
we obtain from these immediately for § and x the two relations :

and

We find therefore that the way of dealing with the true, dubious
-and false cases as proposed by me, allows us to use FEcHNER’s well-
known tables.

I wish to lay some stress here on the fact, that G. E. Méuzr’s
formulae give the same result, saving only the well-known dif-
ference in the integral-limits: these latter being 0 and (S,+D) hy.
I need scarcely add that my remarks do not touch in the least
the question about “thresholdvalue” between Frcuner and G. E.
Mérier. |

It is. evident, that the result of the calculation of a sufficiently
extensive series of experiments according to the principles, given in
my remarks should give numbers, closely related to those either of
Frecuner or of G. E. Méiier — depending on the limits of inte-
gration. Stilt I wish to draw special attention to the fact that the
formulae of G. E. Mitier about the true, false and dubious cases
are rather the statistical representation of a series of nearly identical
psychological processes, whilst the opinion professed by me on the
method of the false and true cases, represents a pure physiological
view.

Finally my remarks show, that Carret, and FULLERTON’s way of
applying the method of the true and false cases is less arbitrary
than it seems to be at first sight. They take for the thresholdvalue the
difference of stimuli with which the corrected number of true cases
attains 75°/,. Such being the case, § and x are both = 50°/,. They
consider therefore the thresholdvalue to be a difference between two
stimuli such, that there is an equal chance of this difference being
perceived or not.

( 226 )

Chemistry. — “The shape of the spinodal and plaitpoint curves
for binary mixtures of normal substances.” (Fourth communi-
cation: The longitudinal plait.) By J. J. van Laar. (Com-
municated by Prof. H. A. Lorentz.)

1. In order to facilitate the survey of what has been discussed
by me up to now, I shall shortly resume what has been communi-
cated on this subject in four papers in These Proceedings and in
two papers in the Arch. Teyler.

a. In the first paper in These Proceedings (22 April 1905) the
equation:

| po

RT = — [« (1—2) (ev—BY a)? + afv—b)*] «we CO)
was derived for the spinodal lines for mixtures of normal substances,
on the supposition that @ and 6 are independent of v and 7’, and
that a,, =Va,a,, while

(av—By/a)* [(1—2) »—8a (1—#)8] +

1 Ya (o—0| 8(ee AV) (av—28)/a) + oe | —0 (2)
was found for the v,a-projection of the plaitpoint line, when
a Va,—VYa, and B=6b,—4,.

b. In the second paper in These Proceedings (27 May 1905) the
shape of these lines for different cases was subjected to a closer examina-
tion. For the simplification of the calculations B=0, i.e. b,=6,, was
assumed, so that then the proportion 6 of the critical temperatures of
the two components is equal to the proportion 2 of the two critical

a b Yt

*=g, —=w,— =t (where 7, is the
v diy

“third” critical temperature, i. e. the plaitpoint temperature for

v=), the two preceding equations become:
t= 4w [#(1—2) + (y+ 2)? (l—w)*] . - - . (la)

p +2) (1—o) (1—3
(p +2) (oF Ce) ory (2a)

It now appeared that the plaitpoint curve has a double point,
when gy = 1,43, ie. d= a = 2,89. If 6 > 2,89, the (abnormal) case
of fig. 1 (loc. cit.) presents itself (construed for ¢= 1,0=(1-+ 4
if on the other hand 6 < 2,89, we find the (normal) case of fig. 2
(loc. cit.) (construed for g = 2, 6 = 2*/,).

At the same time the possibility was pointed out of the appearance
of a third case (tig. 3, loc. cit.), in which the branch of the plaitpoint

pressures. If we then put
a

(1—22) + 3 + #) (1—o)* +

( 227)

line running from C, to C, was twice touched by a spinodal line.
Here also the branch C,A is touched by a spinodal line fin the first
two cases this took place only once, either (in fig. 1, loc. cit.) on the
branch C,A (A is the point z=0, v4), or (in fig. 2 loe. cit.) on
the branch C,A (C, is the before-mentioned third critical point].

So it appeared that a// the abnormal cases found by KuxENen may
already appear for mixtures of perfectly normal substances.

It is certainly of importance for the theory of the critical phenomena
that the existence of two different branches of the plaitpoint curve
has been ascertained, because now numerous phenomena, also in
connection _with different “critical mixing points’ may be easily
explained.

c. In the third paper in These Proceedings (June 24, 1905)') the
equation :

1 (d 1 Life eee
b=7(Z)= OY a [OY sh Avz)-1)

was derived for the molecular increase of the lower critical temperature

for the quite general case a, = ai, 6, So. whieh equation is reduced

to the very simple expression
(ESS st | eae 0 ei en 2k eee (3°)

for the case x = 1(p,=>p,).

This formula was confirmed by some observations of CeENTNERSZWER
and BicHNER.

d. The fourth paper appeared in the Archives Teyler of Nov. 1905.
Now the restricting supposition §—0 (see 6) was relinquished for the
determination of the double point of the plaitpoint line, and the quite

general case a, a,, +, = 6, was considered. This gave rise to very

=
> Ps
intricate calculations, but finally expressions were derived from which

oi
for every value of d= aa the corresponding value of Beds, and
1 Py
also the values of z and v in the double point can be calculated.

Besides the special case 6 = 2 (see J) also the case 2 =1 was
examined, and it was found that then the double point exists for
6 = 9,90. This point lies then on the line v= 0.

') The three papers mentioned have together been published in the Arch. Néerl, of
Nov. 1908.

( 228 )

e. The fifth paper (These Proceedings, Dec. 30, 1905) *) contained
the condition for a minimum critical (plaitpoint) temperature, and
that for a maximum vapour pressure at higher temperatures (i. e.
when at lower temperatures the three-phase-pressure is greater than
the vapour pressures of the components). For the first condition
was found:

4nxnYx
cS (3V a 1)? : (4)
for the second:
Fy 4
ae (5)

which conditions, therefore, do not always include each other *).

After this the connodal relations for the three principal types were
discussed. in connection with what had already been written before
by Korrewse (Arch. Néerl. 1891) and later by van per Waats (These
Proceedings, March 25, 1905). The successive transformations of main
and branch plait were now thrown into relief in connection with the
shape of the plaitpoint line, and its splitting wp into two branches as
examined by me.

J. In the szvth paper (Arch. Teyler of May 1906) the connodal
relations mentioned were first treated somewhat more fully, in which
also the p,a-diagrams were given. There it was proved, that the
points #,, A, and R',, where the spinodal lines touch the plaitpoint
line, are cusps in the p,7-diagram.

Then a graphical representation was plotted of the corresponding
values of @ and za for the double point in the plaitpoint line, in
connection with the calculations mentioned under d.

Both the graphical representation and the corresponding table are
here reproduced. The results are of sufficient importance to justify
a short discussion.

We can, namely, characterize all possible pairs of substances by
the values of @ and 2, and finally it will only depend on these
values, which of the three main types will appear. To understand
this better, it is of importance to examine for what combination
(7,6) one type passes into another. As to the transition of type I
to If (Ill), it is exactly those combinations for which the plaitpoint
line has a double point. In fig. 1 (see the plate) every point of the

') Inserted in the Arch. Néerl. of May 1906.

2) These results were afterwards confirmed by Verscuarrett (These Proceedings
March 31, 1906; cf. also the footnote on p. 749 of the English translation).

(229 )

plane denotes a combination (4,2), to which every time a certain
pair of substances will answer.

= = aa? | a | */o
ite ae
4,00 | 7,00 en O13 0,96 en 0,040 . 2,57 en 2,57
1,19 7,21 » 0,13 0,94 » 0,036 2,49 » 2,60
4,74 6,26 » 013 0,84 » 0,025 | 226 » 268
1,88 5,76 » 0,13 0,78 » 0,024 218 » 2,74
2,04 5,42 » 0,12 0,72 » 0,018 244 » 274
2,22 4,94 » 0,12 0,63 » 0,014 2,02 » 2,79
2,89 2,89 » 0,12 0,24 » 0,003 1,73 » 2,87
9,90 1,00 » 0,14 0,01 » 0,001 1,00 » 2,95
00 — » Of1 — » 0,000 — » 3,00

In the said figure the line C”’APB denotes the corresponding
values of 0 and za from 6=0 to 0=9,9. For C’ 0=0, x=9,
for A 6=1, x=7,5; with 6 = 2,22 corresponds a = 4,94. (Case
& = G@ or a,=<a,); for P r=6=2,89 (Case x =O orb, = 5,);
for 5b 6=9,9, = 1. For values of 6 > 9,9 the double point would
lie on the side of the line v= 6, where v <b. It appears from the
figs. 23, 24 and 25 of the said paper, that then the line BD (a = 1)
forms the line of demarcation between type I and II (III). For
starting from a point, where 7< 1 (however little) and @ is com-
paratively low, where therefore we are undoubtedly in region II (IID,
we see clearly that we cannot leave this region, when with this
value of a that of 6 is made to increase. For we can never pass
to type I, when not for realizable values of v (so < 4) a double
point is reached, and now a simple consideration (see the paper
cited) teaches, that for <1 a double point would always answer
to a value of v< b.

Now it is clear that 0 = 0,7 =9 is the same aO=a0,x2='/);
that 6 = a = 2,89 is identical with 0 = x = 1/29 = 0,35; ete., ete.
(the two components have simply been interchanged), so that the
line CA’ will correspond with the line C’A, while A’ B’ corresponds
with AB. If we now consider only values of 6 which are i Bie
in other words we always assume 7, > 7,, we may say that the

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 230 )

region of the normal type II (III) is practically bounded by the
lines ABD, AA’ and A’C. On the right of ABD we have the
abnormal type I (C,H, + CH,OH, ether + H,O); on the left of A’C
we have also the type I. But whereas in the first region of I the
branches of the plaitpoint line are C,C, and C,A, they are CC,
and C,B (see figs. 23—25 loc. cit.) in the second region. It is namely
easy to show, (loc. cit.), that for 7 >1 the branches of the plait-
point line are either C,C, and C,A (type Il and II), or C,A and
C,C, (type I, while for «<1 these branches are C,C, and C,B
(type II and III) or C,B and C,C, (type fl). The line w= 1 divides
therefore the region of type II (III) into two portions, where we
shall resp. find the shape of the plaitpoint line branches mentioned
(viz. for 6>>1). But in practice it will most likely never happen,
that with @>>1 a value of 2 corresponds which is much smaller
than 1, for a higher critical pressure goes generally together witha
higher critical temperature. We may therefore say that with a given
value of a the abnormal type I is found when 6 is comparatively
large {larger than the double point (of the plaitpoint line) value of
6|, whereas the normal type I (or Ill) appears when 6 is compara-
tively small (smaller than the said double point value).

It is now of the greatest importance to examine, when type II
passes into III, where the plaitpoint line C,C, is twice touched by
a spinodal line (in A, and &,’). This investigauon forms the con-
clusion of the last paper in the Arch. Teyler.

The calculations get, however, so exceedingly intricate, that they

>
proved practically unfeasible for the general case a, a4, Oe Z by.

Only the special cases 8 = 0 (6, = 6, or T= 6) and a — 1 admitted
of calculation, though even then the latter was still pretty complicated.

Then it appeared, that for 80 the region of type III is exactly
—(, that it simultaneously appears and disappears in the double
point P, where 7=6=>= 2,89. But in the case 71 the region
lies between 6—4,44 and 6=9,9 (the double point). This is
therefore QB in fig. 1; i.e. for values of 6>>1 and < 4,44 we
find type II (see fig. 2°); for 6=4,44 (in Q) the plaitpoint line
gets a point of inflection (see fig. 2°), whereas from 6 = 4,44 to
6—9,9 we meet with type ILI (fig. 2) with two points R, and R,’,
where the spinodal lines touches the plaitpoint line. This type
disappears in the double point ?, where 6=9,9 and A, and &,’
coincide in P (fig. 2¢), and passes for values of 6 >9,9 into type I
(fig. 22). We point out, that the figs. 2*—2¢ represent an intermediate
case (i.e. between 7 =6 and a =1, see fig. 1), for in the case of

( 231 )

ma—1 the branch AR,C, would coincide with AB (v= d). There-
fore the special value 4,44 has been replaced by 6, (the value of
6 in Q) and the value 9,9 by 46, (the value of @ in P).

Of the curve which separates type Il from type III we know as
yet only the points P and Q (see fig. 1) and the further course of
this line is still quite unknown, for which reason we have denoted
it by a dotted line.

In any case the investigations, described in the Arch. Teyler have
proved, that this very abnormal type III zs possible for mixtures of
normal substances. If the critical pressures of the two components
are the same (7=—1), then we meet with this type when @ lies
between 4,4 and 9,9. The critical temperatures must therefore lie
far apart, but not so far (see fig. 1) as would be necessary for the
appearance of type I.

We shall once more emphatically point out that the nwmeric results
of our investigation will naturally be modified, when @ is not assumed
to be independent of v and 7’, or when one of the two components
should be associating substances. This will cause the types II and I
to make their appearance earlier than has been derived above (i. e.
with lower values of 6 with for the rest equal values of 2), but
that qualitatively everything will remain unchanged. This appears
already from the fact that the substitution of the quite general assumption

b, - 6, for the simplified assumption 6, = 4, (in the first paper in

the Arch. Teyler) has made no change is the existence of a double
point in the plaitpoint line with certain corresponding values of @
and a, and that also the calculations for the limits of type III
(in the second paper in the Arch. Teyler) may be carried out tor

the quite general case 6, = 6,. So the phenomena remain qualitatively

the same for very different pairs of values of 6, and /,, and will
therefore not change essentially either, when one definite pair of
values, holding e.g. for the critical circumstances of one of the com-
ponents, is subjected to changes, whether by association, or by other
causes, when v or 7’ change — no more as e.g. the critical pheno-
mena for a simple substance will essentially change when b is no
longer constant, but is supposed to be dependent of v and 7) or
when that substance forms complex molecules.

The longitudinal Plait.

2. In former papers it has been demonstrated that in the neigh-
bourhood of C, a minimum plaitpoint temperature makes its appearance

( 232 )

both with type I in the line C,C, and with type II in the line C,A,
and that therefore with decrease of temperature a separate plait
begins to detach itself starting from C;, at a definite temperature
T, (the plaitpoint temperature in C,), which plait will merge into
the main plait (or its branch plait) later on in an homogeneous
double point. The consequence of this is, that with type I e.g. at
lower temperatures the main plait will always be open towards
the side of the small volumes, so that increase of pressure will never
‘cause the two split phases to coincide.

Let us however specially consider the case of type II. Here the
usual course, inter alia described in the last cited paper in the
Proceedings of Dec. 30, 1905, is this. At a certain temperature,
passing from higher to lower temperatures, a spinodal curve touches
the branch of the plaitpoint line AC, in R,. In the well-known way
a closed connodal curve begins to form within the connodal line
proper, which closed curve gets outside the original connodal curve
at lower temperatures, and gives rise to a new (branch) plait, and at
the same time to a three phase equilibrium (figs. 8¢ and 3°). In many
eases this branch plait has already appeared before the plait which
starts from C,, begins to develop at somewhat lower temperature.
Later on the two branches coincide (at the minimum temperature
in J), and then form again a continued branch plait (fig. 3°). *).

Now for the special case 6,0, the point D lies always very
near C, (see the paper in these Proceedings referred to under 0.
in § 1). If then e.g. 7./7,= 2), then 7,,/7;= 096; whee
represents the temperature in the minimum at D. The real longi-°
tudinal plait round C, exists then only at very high pressures
(fig. 34), while the open plait of fig. 3° can hardly be called a
longitudinal plait, but is much sooner to be considered as the
branch plait of the transverse plait which has joined the original
longitudinal plait. Increase of pressure makes here always the two
coexisting liquid phases approach each other, unless with very high
pressures, when these phases diverge again.

the

The calculation proves that in the quite general case 6, = b,

point D may get much nearer in the neighbourhood of £&,, and also
that the temperature in the plaitpoint C, may be comparatively bigh,
so that in opposition to what has been represented in fig. 37 the
longitudinal plait has already long existed round C, before a three
phase equilibrium has formed at M (fig.4* and 4°). The meeting

') In this and some other figures the spinodal curves seem to touch in the
homogeneous double point D, instead of to intersect, as they should.

( 233 )

of this longitudinal plait, which has then already greatly extended,
with the branch plait takes place much more in the neighbourhood
of the line 1,2 of the three phase triangle, so that after the meeting
the plait assumes the shape drawn in fig. 4°, which makes it for
the greater part retain its proper character of longitudinal plait. So
at first increase of pressure makes the phases approach each other
(this portion may be exceedingly small, but as a rule it will exist);
then further increase of pressure makes the phases 1 and 2 again
diverge, till z, and x2, approach to limiting values at p = o, without
the longitudinal plait ever closing again — as was formerly considered
possible [ef. inter alia van per Waats, Cont. II, p. 190 (1900)].
For in consequence of the minimum at VD the longitudinal plait
always encloses the point C,. Only at temperatures higher than 77,
at which the longitudinal plait does not yet exist, there can be
question of homogeneity till the highest pressures. But then the
plaitpoint P belongs to the branch plait of the transverse plait, and
not to the longitudinal plait. This is indicated among others by
fig. 3", after the closed connodal curve in MW has broken through
the connodal curve proper of the transverse plait; or by fig. 3°,
before a longitudinal plait has developed round C,,.

Of course we may also meet with the case, that the plait round
C, coincides with the branch plait at the moment that the latter
with its plaitpoint just leaves the transverse plait, as shown in fig. 5%,
but this involves necessarily a relation between @ and 2, and is
therefore always a very special case. Then the branch plait happens
to leave the transverse plait exactly in the minimum at D. After
the meeting the plait shows the shape as traced in fig. 5°. Now
increase of pressure causes the two phases 1 and 2 to diverge from
the beginning.

But the longitudinal plait round C, may also meet the connodal
line of the transverse plait, before the closed connodal line has got
outside the transverse plait (fig. 6). Then the three phase equilibrium
does not develop, as in fig. 4", at the transverse plait (from which
a branch plait issues), but at the /ongitudinal plait round C,. The
latter penetrates then further into the transverse plait, till its meets
the isolated closed connodal curve in DP (fig. 6°), after which the
confluence with it takes place in the unrealizable region (fig. 6°).

This plait is then the longitudinal plait proper, of which there is
generally question with mixtures of substances which are not miscible
in all proportions. But we should bear in mind that just as well
the above treated case of fig. 4 may present itself, with that of fig. 5
as transition case.

( 234 )

The calculation teaches that the transition case presents itself when
the proportion 4 of the critical temperatures of the two components
is in the neighbourhood of 1, and the proportion a of the critical
pressures is at the same time pretty large.

A clear representation of these different relations is also given by
the two p, 7-diagrams of fig. 7 and fig. 77. (The temperature of C,
is there assumed to be lower than that of #,, but it may just as
well be higher). The plaitpoints p’ on the part #,A below the cusp
are the unrealizable plaitpoints (see also figs. 3—6); the plaitpoints
p on the part 2, before J also (then the isolated closed connodal
curve has not yet got outside the main plait); the plaitpoints P
beyond JW are all realizable.

So after the above we arrive at the conclusion that in all cases
in which a distinct longitudinal plait appears of the shape as in
figs. 4° or 6¢ (so when the minimum J lies near &,), the critical
mixing point JM of the three phases need not always lie on the
longitudinal plait (see fig.4%), and also that the longitudinal plait
with its plaitpoint P will not always coincide with the transverse
plait itself, but it ean also coincide with the branch plait of the
transverse plait, so that at that moment no three phase equilibrium,
i.e. no vapour phase is found (see fig. 4°). The two liquid phases
1 and 2, however, coincide in this case.

The case drawn in figs. 5¢ and 5° remains of course an exception,
and the conditions for its occurrence may be calculated (see above).
But this calculation, as well as that which in general indicates the
situation of the points R,, D and M, will be published elsewhere
(in the Arch. Teyler). It is, however, self-evident that the above
general considerations are by no means dependent on these special
calculations.

It is perhaps not superfluous to call attention to the fact that the
concentration a, of the vapour phase is neither in fig. 4%, nor in
fig. 5 or 67, the same as the concentration of the two coinciding
liquid phases a9, as VAN DER Lee wrongly believes to have shown
in his Thesis for the doctorate (1898), [see p. 66—69, 73—74 and
Thesis III; also vAN per Waats, Cont. II, p. 181 (1900)]. Now we
know namely, that when «, lies between v, and wv, at lower tem-
peratures, this need not continue to be so till v, and 2, have coincided.
The latter would be quite accidental; in general one of the maxima,
e.g. in the p,a-line, which lie in the unstable region between 2, and
z,, will get outside the plait before 7, and «x, have coincided.
Cf. the figs. 12« to 12/ in my Paper in These Proceedings of March
25 1905 and §8 p. 669—670, and also the footuote on p. 665.

( 235 )

Already in a previous paper (These Proceedings June 27 1903) 1
had elaborately demonstrated this, and somewhat later (These Proceed-
ings 31 Oct. 1903) Kuenen arrived at the same opinion independently
of me.') And in 1900 Scureinemakers (Z. f. Ph. Ch. 85,p. 462—470)
had experimentally demonstrated that one maximum leaves the
longitudinal plait for exactly the same mixture (phenol and water),
for which van per Lex thought he could theoretically prove, that
a Die.)

Finally I shall just point out that in the peculiar shape of the
p,T-diagram of the plaitpoint line (fig. 7) in the neighbourhood of
the point D, and in the fact that the two critical moments represented
by figs.4* and 4° (as D and MM in general do not coincide) do not
coincide, the clue may be found for the explanation of a highly
puzzling and as yet unexplained phenomenon, which has been observed
as well by Gururin as by Rorumunp [Z. f. Ph. Ch. 26, p. 446
(1898)|*) in their experiments, viz. the appearance and disappearance
of a distinct cloudiness when the mixture is heated above the
“critical temperature of mixing’, which cloudiness often continued
to exist up to 10° above this temperature.

1) C.f. also KueNEN: Theorie der Verdampfung und Verfliissigung von Gemischen.
Leipzig 1906, p. 170, note.

*) For the rest the assumption x3 = 2,2 at the point M leads, as the calcula-
tions teach, not only to strange, but to highly absurd conclusions.

8) C.f. also FRieDLANDER, Ueber merkwiirdige Erscheinungen in der Umgebung
des kritischen Punktes. Z. f. Ph, Ch. 38, p. 385 (1901).

(October 25, 1906).

Fo Swi: nih rd FE i sae Te
et See Bee ht Oks ics ee as set Pee
‘ oe

bal LAA.

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

. PROCEEDINGS OF THE MEETING
of Saturday October 27, 1906.

DOS

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 27 October 1906, Dl. XV).

CO PE SN Ts.

L. E. J. Brouwer: “The force-field of the non-Euclidean spaces with positive curvature’.
(Communicated by Prof. D. J. Korrewse), p. 250.

W. van Bemmeten: “On magnetic disturbances as recorded at Batavia”, p. 266.

J. J. Buankxsma: “Nitration of meta-substituted phenols”. (Communicated by Prof. A. F.
HOLLEMAN), p. 278.

A. F. Hotiteman and H. A. Sirks: “The six isomeric dinitrobenzoic acids”, p. 280.

A. F. Hotieman and J. Huismyca: “On the nitration of phthalic acid and isophthalic acid”,
p. 286.

A. PayyEKOEK: “The relation between the spectra and the colours of the stars”. (Commu-
nicated by Prof. H. G. van pk SanpE BakKHUuYZEN), p. 292.

R. A. Weerman: “Action of potassium hypochlorite on cinnamide”. (Communicated by Prof.
S. A. HooGEwERrFF), p. 303.

J. A. C. Oupemans: “Mutual occultations and eclipses of the satellites of Jupiter in 1908”,
p. 304, (With one plate).

H. Eyssrorx: “On the Amboceptors of an anti-streptococcus serum”. (Communicated by
Prof C. H. H. Sproncx), p. 336.

W. HL Junius: “Arbitrary distribution of light in dispersion bands, and its bearing on spec-
troscopy and astrophysics”, p. 343. (With 2 plates).

F. M. Jazcer: “On a substance which possesses numerous different liquid phases of which three
at least are stable in regard to the isotropous liquid”. (Communicated by Prof H. W. Baxuuis
RoozEsoom), p. 359.

H. W. Baxuuis Roozesoom: “The behaviour of the halogens towards each other”, p. 363.

W. A. = “Second communication on the Pliicker equivalents of a cyclic point of a
twisted curve”. (Communicated by Prof. P. H. Scnoure), p. 364.

H. Kameruincu Onnes and C. Braak: “On the measurement of very low temperatures.
XIII. Determinations with the hydrogen thermometer”, p. 367. (With one plate).

Errata, p. 378.

a
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 250 )

Mathematics. — “The force field of the non-Euchdean spaces
with positive curvature’ by Mr. L. E. J. Brouwer. (Commu-
nicated by Prof. D. J. KorTewse).

(Communicated in the meeting of September 29, 1906).

D'). The spherical Sp,.

I. The theorems under C $ I and II hold invariably for the sphe-
rical and elliptical Sp,’s. But on account of the finiteness of these
spaces we need not postulate a limiting field property for the
following developments. We shall first consider the spherical spaces.

Firstly we remark for the general linevector distribution of the
spherical Spa that the total sum of the divergency is 0; for the
outgoing vectorcurrents out of the different space-elements destroy

each other. This proves already that as elementary oX we can but
take the field of a double point.

Scuerinc (Gottinger Nachrichten 1873), and Kinune (Crelle’s Journal,
1885) give as elementary gradient field the derivative of the potential

fon

f ae ;
function |——— = ? (r).”)
sint—|

re
But the derivative of this field consists of two equal and opposite
divergencies in two opposite points; and it is clear that an arbitrary
integral of such fields always keeps equal and opposite divergencies
in the opposite points, so it cannot furnish the general divergency-
distribution limited only to a total divergency sum = 0.

Il. If we apply for a spherical Sp, the theorem of Green to the
whole space (i. e. to the two halves, in which it is divided by an
arbitrary closed Sp,—1, together), doing this particularly for a scalar
function g which we presuppose to have nowhere divergency and
a scalar function having only in two arbitrary pots P, and P,
equal and opposite divergencies and nowhere else (such functions
we shall deduce in the following), we then find

Pp, ae P p, == 0,
i.o. w. g is a constant, the points P, and /P, being taken arbitrarily.

1) A,B and C refer to: “The force field of the non-Euclidean spaces with nega-
tive curvature”. (See these Proceedings, June 30, 1906).
2) We pul the space constant =1, just as we did in the hyperbolic spaces.

( 254 )

So there is no =x possible with nowhere divergency, thus no x
having nowhere rotation and nowhere divergency, and from this
ensues :

A linevector distribution in a spherical Sp, is determined uniformly
by its rotation and its divergency.

Ill. The general vector distribution in a spherical Sp, must thus
be obtainable again as an arbitrary integral of:

1. fields 2#,, whose second derivative consists of two equal and
opposite scalar values close to each other.

2. fields H,, whose first derivative consists of planivectors distri-
buted regularly in the points of a small "~*sphere and perpendicular
to that *—*sphere.

At finite distance from their origin the fields #, and EF, have
an identical structure.

IV. For the spherical Sp, there exists a simple way to find
the field #, namely conform representation by stereographic pro-
jection of a Euclidean plane with a doublepoint potential, which
double point is situated in the tangential point of the sphere and the
plane. If we introduce on both surfaces as coordinates the distance
to the double point and the angle of the radiusvector with the
doublepoint-axis — in the plane @ and g, on the sphere rand gy —
we have:

4 o0=—tan}r.

The potential in the plane: 7 becomes on the sphere:

4 cos p cot 3 7r.

This potential shows nothing particular in the centre of projection
on the sphere, so it is really the potential to be found of a single
double point, the field #,. (If we place in the opposite point of
the double point an other double point in such a way that the

unequal poles correspond as opposite points, we find as potential

cos
4 cos p (cot 4 r+tan}r) = — , Which is the Schering potential of a

double point).

V. Here too we can meanwhile break up the field of a double
point into two fictitious “fields of a single agens point”; for this

we have but to take {cot hrdr—=— loin =F (9) so that for an

Li?

( 252 )
arbitrary gradient distribution holds

fe W 0X FO as 2 a eee

The ‘‘field of a single agens eae has however divergencies every-
where on the sphere.

1
0 aX: — Aw

VI. Out of the field H, we deduce in an analogous way as under
B § VI the field #, of a rotation double point normal to the agens-
doublepoint of the field £,. As scalar value of the planivector potential
we find there:

4 sin g cot $7,
as we had io expect, col atete dual to the scalar potential of the

field £,.
As fictitious force field of a unity-rotationelement we deduce out

of this (in the manner of B § VI):

4 cott7,
directed normally to the radiusvector. For the rest this force field has
rotation everywhere in Sp,. :

VII. Out of this we find (comp. under B § VII) for the scalar
value of the planivector potential of a rotation-element:

fi cot 4 rdr— F(r),

so that for an arbitrary =e

x= Wf WE ya a_i

And an arbitrary vectorfield is the 7 of a potential:

f * Fr () dr.

E. The spherical Sp,.

I. The purpose is in the first place to find #,; we shall compose
it of some singular potential functions with simple divergency distri-
butions, and which are easy to construct.

Let us suppose a principal *sphere 6B with poles P, and P,, and
on B a principal circle C with poles Q, and Q, debormnniag on B
meridian circles M cutting C in points #.

( 253 )

We can construct in the first place out of the ScueriNe potential
the potential of two double points, in P, and P,, the positive
poles of which are both directed towards Q, (so that in opposite
points unequal poles correspond). Let us determine a point S of the
hypersphere by the distance PS=rand “ QPS=g (where for P
and Q the index 1 or 2 must be taken according to S lying with
P, or with P, on the same side of 5), then this potential (a) becomes

cos ~

sin? r
where the sign + (—) must be taken for the half hyperspheres
between P, (P,) and BS.

This field has no other divergency but that of the double points
F, and F..

If we now reverse the sign of the potential in the half hyper-
sphere on the side of P,, we obtain the potential (@):

COs @

sin? r
The divergency of this consists in the first place of two double
points, one directed in P, towards Q, and one directed in P, towards
Q, (so that now in two opposite points equal poles correspond) ;
and then of a magnetic scale (indeed a potential discontinuity) in
sphere 6 varying in intensity according to cos gy.

Il. By the side of this we wish to find a potential, the divergency
of which consists of only such a magnetic scale in sphere B with
an intensity proportional to cosg. Now a field of a magnetic scale
in £ with an intensity varying according to an other zonal sphe-
rical harmonic, is easy to find. Let us namely take in each “meridian
sphere” PQH# as potential of a point S the angle PHS=4a— / QHS
(P and Q to be provided with indices in the way indicated above
according to the place of S) = tan— {cos g tan r}, then we have such
a potential: in the hypersphere it is a zonal spherical harmonic about
PQ as axis; on the sphere B it has its only divergency in the
shape of a magnetic scale, the intensity of which varies according to
a zonal spherical harmonic with pole Q.

Let us now take in turns all the points of the sphere B as pole
Q' of such a potential function, and let us integrate all those poten-
tials over the solid angle about P each potential being multiplied by
cos Q'Q, then according to a wellknown theorem on spherical har-
monics the integral is a zonal harmonic of form cos gf (r), where

ety — cf cos p . tan {cos gy tanr}dm, (dw representing the element

( 254 )

of the solid angle about P), whilst this integral field has as only
divergency a magnetic scale in 6 with intensity proportional to
COS @.

Effecting the integration we obtain :

T(r) = 22 a sin ~ cos ~ tan—! {cos g tan r} dg.
0

’

F(t) = 2% | — cotr + E

sin? r
and for the corresponding potential function (y) we find:

aE A |.
sin? r

III. If we take the difference of the field (8) multiplied by 4 and

2m cos ~

the field (y) multiplied by Te the magnetic scale in B disappears
7

and we have left the field (d):

Fs 4

—_—r
cos p )2

: + cot r| ,
mz | sin?r

which field has as only divergeney two double points in P, and P,
of which in the opposite points equal poles correspond.

The sum of this field (d) and the field (a) multiplied by 4 must
now give a field having as divergency a single double point with
unity-moment in P,, i. o. w. the field £,.

We therefore find on the half hypersphere between P, and B:

Tr
+ cotr

sin ?

1
— cos ~
5 4

and on the half hypersphere between P, and B:
—?r
= + cot r
is

or if we define on both halves the coordinates 7 and g¢ according to
P, and P, Q, we obtain the following expression holding for both halves:

1
ae cos 9
a Pp

sin

Sin

1 xn—r vette
= | = + cot r| = wy cov.
x r

IV. To break up this field into two fictitious “fields of a single
agens point” (having however divergency along the whole hypersphere)

we take for the latter fv (r) dr = F, (r).

( 255 )

Then for an arbitrary gradient distribution holds:

a
x= YW [LH oa, meen soot oe OY ER)

V. The field £, of a circular current according to the equator
plane in the origin, is identical outside the origin to the above’ field
E,; but now each force line is closed, and has a line integral of
4m along itself.

According to the method of A §IX we find of this field 4, the
planivector potential H in the meridian plane and independent of the
azimuth.

We find when writing 7—r=8:

1
= = — sin*g (1 + Boot r) dd.
x

vanishing along all principal circles in the opposite point.

From which we deduce for the force of an element of current
with unity-intensity in the origin directed according to the axis of
the spherical system of coordinates :

1. 1+ Beotr
— sin —Y ——_——_
‘4 sin rT

directed normally to the meridianplane.

VI. From this we deduce as in A § XI a vector potential V of
an element of current parallel to that element of current and a
function of r only. For the scalar value U of that vector potential
we have the differential equation :

0 0
—~ {Using sinrdg) dr — — \Ucosgdr}) dg=
Or 0g
1 1 t
x sin P
Or:
0 ® 1
U— 5, |Usins =— (1+ Bootr),
or x

of which the solution is

a + — pen ae

cos* ir zg l{cos*tr smr

c 1 +B =

( 256 )

We choose c= 0, and we find as vector potential V of a unity-
element of current:
il

tftp

—

= #,'@).

cos? kr sinr
directed parallel to the element of current. The function £’, (r) vanishes
in the opposite point.

For an arbitrary flux now holds:

|
x
1x a wy fa rap)ide 2 ee
An
And finally the arbitrary vector field X is the V of the potential:

Wiese +f yaaa yu yi ie
An An

F. The spherical Spn.

I. To find the field #, we set to work in an analogous way as
for the spherical Sp,. The principal sphere L becomes here a
n—Isphere B; the principal circle C’ of the points H a principal
sphere C of the points H.

For the potential (a) is found:

cos ~

sint—1 p?
for the potential (@):
COOP
sint—l p’
this field (3) has in the sphere 6 a magnetic *—'scale. _

The potential (y) is integrated out of fields tan—'{ cos g tanr}
according to cos g, the first zonal *—!spherical harmonic on B. This
integration furnishes when dw represents the element of the n-dimen-
sional solid angle about P:

cos p f (r),
where :
Tv
K(r) = | cosytan—| | cos gtanr | dw = baa rns cos ptan—} cogean ig=
kn—1 : tan 7 dp
= —— | sin"g ——___—__
n—1 1+ tan *r cos *
0

(k, defined as under C § III). °

( 257 )
Putting under the sign of the integral a factor sin *y tan *r outside
1
the brackets and, by regarding that factor as 7 (1-++'cos *y tan *r),
Tr

writing the integral as sum of two integrals to the former of
which the same division in two is applied, ete., we find, if we write

fons r dr = Sh:

0
n—l1 r : ;
a Pe) sin "7 = — sin "—2r cos r Sp—9 — sin "—4r cos r Sy_4...-
kn—1
... — sin *r cos r S, 4- 2 (1 — cos 7)
(for 2 even)
= — sin"—°r cos r Sp_9 — sin "—4r cos 7 Sn—4.0es

..e-— sinrcosrS, +2r

(for 2 odd)
er ue ea A
— a fri n—|y dr = (n—1) S, 2 rin n—Ip.dr,
. . 0 0
; (for 2 even)
—1)(n—8).... : ; /
— oa rin n—|p dr = (n—1) Sof sin n—|p dr,
Ge 2e— tyr
0 0
(for n odd)
If we write &, for 2.”7.2.2.2...., to nm factors, we have
n kn
Siem ie ak Eee and el == Sa
(n—2) (n—4).... kn

Therefore :

Ff (r) sin 817 = by fin n—ly dr,

0
and the potential (y) becomes :

cos ~ ’
kn = sin "—|r dr.
sin "—|p

Il. We find the field (#) by taking difference of field (8) multi-

1 1
plied by 4 and field (y) by Bee.” i. e.
: n On—1 1

( 258 )

; =
S,—1 — | sin ®—r dr al, sin "—|p dr
cos @ cos ~ Z
Sp—1

sin "—lp ~ sin 2—Ip ~ Seen

This field has as only divergency two double points, in P, and
P,, of which equal poles correspond in the opposite points. The field
E, is then obtained by adding to it the field (a) multiplied by 3
We find on the half sphere between P, and B:

i cos @
— .— sin "—|r dr.
Sr—1 sin ®—|y

On the half "sphere between P, and B:

1 cos
— — # sin "—|y dr.
Sr—-1 sin™—|p

Or, if we define on both halves the coordinates r and ¢ according to
P, and P,Q,, we arrive at the expression holding for both halves:

: fo n—I|y dr — Wn (r .) cos @.

1 cos ~

S,—1 sin ®—|p

III. For the potential of the fictitious “field of a single agens
point” we find:

furs, o.
And for the arbitrary gradient distribution holds :

1 me W oX
Kay f So node... 1. ©

Of the divergency distribution of F, (r) in points ofa general posi-
tion we know that, taken for two completely arbitrary centra
(fictitious agens points) with opposite sign and then summed up,
it furnishes 0: so on one side that distribution is independent of
the position of the centre and on the other side it lies geome-
trically equivalent with respect to all points; so it is a constant.
But if the function /, (r) has constant divergency in points of general
position it satisfies a differential equation putting the divergency
constant. In this is therefore a second means to determine the func-
tion /, and out of this the field £,.

The differential equation becomes :

( 259 )

E sin t— lp ys =csin"—|p (77
om oe ee a, )

' dF, :
sin "Ip , oy =o f sin n—Ip dr,

aF, if sin "—|y dr
— c . — +...

dr sin "—\p
If the field #, is to be composed out of the function /, (r) then
the opposite point of the centre may not have a finite outgoing

vector current; we therefore put na sin"—|r dr = 0, 80 that we get

dF, c ;
= — —— | sin"—!r dr,

dr sin "|p
S

which corresponds to the above result.

IV. The field H, of a smal] vortex "—*sphere according to Sp,—1,
perpendicular to the axis of the just considered double point, is iden-
tical to that field #, outside the origin; but now each force line is
closed and has a line integral £, along itself,

According to the method of C § VII we shall find of this field
£, the planivector potential H, lying in the meridian plane and depen-

dent only on r and @;so that it isa x We find:
dh = ce sin "—°p sin "2g,

Force in 7-direction :

Tw
fen r—lp dp
1 coir ;

es | ae a — (n—1) cos @ . w, (r.)

(n—1) cos @

?
= =f @-1 COS P Wy (7) . c& sin "—2r sin "2 . sin r dy =
0
= WnP . cé sin ™—|¢ sin "—lg,

~ a
H = — = @n (1) sin r sin y = yn (7) sin Q.
From this ensues for the force of a plane vortex element with
unity-intensity in the origin :

Xn (7) sin gp,

( 260 )

directed parallel to the acting vortex element and projecting itself on
that plane according to the tangent to a concentric circle; whilst »
is the angle of the radiusvector with the Sp,—2 perpendicular to
the vortex element.

V. In the same way as in C § IX we deduce from this the
planivector potential V of a vortex element directed everywhere
parallel to the vortex element and of which the scalar value is a
function of 7 only. That scalar value U of that vector potential is here
determined by the differential equation :

0
aa = | U cos p . dr. cé sin "—3r cos 19) dg —
Op

)
eos | sin sin» dip sin 2 co Mg | ar =
7
= Yn (7) sin p . sin r dp . dr . cé sin "—3r cos "3g.
dU
(n—2) U — re 5 (n—2) U cos r = Y (r) sin r.
,

dU
oa — (n—2) Utgk r= — yn (r).-

il
U = eanaae forrear * In (r) dr,
2

a function vanishing in the opposite point, which we put = Ff, (7).
We then find for an arbitrary flux :

ae
1 se \l/ oX
<< wf — F(t) dt 333 =e ee

And taking an arbitrary vector field to be caused by its two deri-
vatives (the magnets and the vortex systems) propagating themselves
through space as a potential according to a function of the distance
vanishing in the opposite point, we find :

aA | [Arnot [YA roar :

G. The Elliptic Spp.

Also for the elliptic Sp, the derivative of an arbitrary linevector
distribution is an integral of elementary vortex systems Vo, and
Vo.,, which are respectively the first and the second derivative of

( 261 )

an isolated line vector. For elementary 5X we shall thus have to put the
field of a divergency double point.

d
The Schering elementary potential f. z

stn "—Ip

=p (r) is here a plu-

rivalent function (comp. Kier, Vorlesungen iiber Nicht-Euklidische
Geometrie II, p. 208, 209); it must thus be regarded as senseless.

II. The unilateral elliptic Sp, is enclosed by a plane Sp,—1,
regarded [twice with opposite normal direction, as a bilateral singly
connected Sp,-segment by a bilateral closed Sp,1. If we apply to
the Sp, enclosed in this way the theorem of Green for a scalar
function g having nowhere divergency, and for one having in two
arbitrary points P, and P, equal and opposite divergencies and
fartheron nowhere (such a function will prove to exist in the follo-
wing), we shall find :

Pp, ea Pp, == 0,
i. 0. Ww. g is a constant, the points P, and P, being arbitrarily chosen.

So no )X is possible having nowhere divergency, so no ee having
nowhere rotation and nowhere divergency; and from this ensues:

A linevector distribution in an elliptical Spa is uniformly deter-
mined by its rotation and its divergency.

Ill. So we consider :

1. the field £,, with as second derivative two equal and opposite
scalar values quite close together.

2. the field #, with as first derivative planivectors regularly distri-
buted in the points of a small *—*sphere and perpendicular to that
small "—*sphere.

At finite distance from their origin the fields H, and £, are of
identical structure.

IV. To find the potential of the field #, we shall represent it
uni-bivalently “on the spherical Sp,; the representation will have as
divergency two doublepoints in opposite points, where equal poles
correspond as opposite points; it will thus be the field (d), deduced
under # $II, multiplied by 2:

( 262 )

Yam
sin *—|r dr
LL i r =
ee = dn (r) cos ~
san *—'r $ Sn—1

In the field corresponding to this in the elliptic space, all force lines
move from the positive to the negative pole of the double point; a
part cuts the pole Sp,—; of the origin: these force lines are unilateral
in the meridian plane; the remaining do: not cut it; these are bilateral
in the meridian plane.

The two boundary force lines forming together a double point in
the pole Sp,—i, have the equation :

oT
sin ™—l@ {sin "—!r + (n—1) cot rf sin n—ly an} =a FE

The Sp,-1 of zero potential consists of the pole Sp,—: and the
equator Sp,—: of the double point; its line of intersection with the
meridian plane has a double point in the force lines doublepoint. All
potential curves in the meridian plane are bilateral.

V. For the fictitious “field of a single agens point’ the potential is

are A, (r)dr. It is rational to let it become O in the pole Sp,-1; so

we, find:
am
fr (7) dr= F,(),

and for the arbitrary gradient distribution holds:

oX = 1 es Fi\deoo 2: ooh eee

We could also have found F, . out of the differential equation
(7) of # § III, which it must satisfy on the same grounds as have
been asserted there. For the elliptic Sp; also we find:

aF, if sin "—|p dr
— c

dr sin "—|p

But here in the pole Sp,-1, lying symmetrically with respect to
the centre of the field, the force, thus fo ™—Irdr must be 0; so

that we find:

( 263 )

Yom

dF c :
= i sin n—]p dr.
dr sin %—Ip

VI. In the usual way we deduce the 1X, which is planivector
potential of the field 4,.

dh = cé sin "—*r sin "—? gp.
Force in r-direction:

Wat
sin "—|p dr
2 2Zcotr -r
on ee ee es | = (n—] ’ A
(n—1) cos @ eat coy a ei | (n—1) cos @ . Un (r)

?
= =fo-» COS P. Un (7). c& sin "—°r sin "—@ . sin rdgy =
0
= Ut, (7). ce sin *—!¢ sin "—¢.

AS temas (7) sin 7 sin | = Xp (7) sin @.

From which ensues for the force of a plane vortex element with
unity-intensity in the origin:

Xn (7) sin GY,

directed parallel to the acting vortex element and projecting itself
on its plane according to the tangent to a concentric circle; @ is
here the angle of the radiusvector with the Sp,—s perpendicular to
the vortex element.

VII. Here too a planivector potential of a vortex element can be
deduced, but we cannot speak of a direction propagated parallel
to itself, that direction not being uniformly determined in elliptic
space; after a circuit along a straight line it is transferred into
the symmetrical position with respect to the normal plane on the
straight line.

But we can obtain a vector potential determined uniformly, by
taking that of two antipodic vortex elements in the spherical Sp, (in
their *sphere the two indicatrices are then oppositely directed).

The vector potential in a point of the elliptic Sp, then lies in the
space through that point and the vortex element; if we regard the
plane of the element as equator plane in that space then the plani-
vector potential V is normal to the meridian plane: it consists of:

( 264 )

1. a component U, normal to the radiusvector, according to the
formula:

U. 1
eet cos ease fee 2(n--2) $9» Yn (7) dr +
=

:
= feos 2) ry, (7) ar
sin 2(n—2) — : 27+ kn (”)
™—r
2. a component U, through the radiusvector, according to the
formula :

U. i
zs = ee fos 2in—2) Lr xn (r) dr —
27

sing cos %An—2)

1
SS 2(n—2) 1
LP
If we regard this planivector potential as function of the vortex
element and the coordinates with respect to the vortex element and
represent that function by G,, then

x=wf 2 CARVE Ak . 4. ae

holds for an arbitrary flux in the aa Spn-

And regarding an arbitrary vector field as caused by the two
derivatives (the magnets and the vortex systems) propagating them-
selves through the space to a potential, we write:

oh (ese (7%)
ky

Vill. In particular for the elliptic Sp, the results are:
Potential of an agens double point:

paag
sin *r dr
cone 209 (baa) |
sin? r 4 S, aX sin ?7
or if. we put }7—r=y:
2 cos ~ y )
rae eh carmel ce det tL
. mu sin “Pr

Equation of the boundary lines of force:

( 265 )

sin*p (1 + ycotr) = +1.
Potential of a single agens point:

— .y- cor.

Vector potential of an elementary circular current:

ae 1+ ycotr
— sin gp. - ;
x sin rT

So also force of an element of current:
2 1+ y cotr

— snp.
x

sin rT

Linevector potential of an element of current:

cosp| 48° x +7
cos* tr smr_ sinrir

4 ie 27r—zx 4 7?

according to the radiusvector :

normal to the radiusvector:

xz (cos? ir sin 7 sin? rl
IX. For the elliptic plane we find:
Potential of an agens double point:

cos ~p cot r.

Equation of the boundary lines of force:
sng—=+tsinr, or o=| °

Potential of a single agens point:
— lsinr.
Scalar value of the planivector potential of a double point of rotation:
sin —p

sin rf
Thus also force of a rotation element:
sin —
Planivector potential of a rotation element :
leot 4 r.

We notice that the duality of both potentials and both derivatives
existing for the spherical Sp,, has disappeared again in these results.
The reason of this is that for the representation on the sphere a
divergency in the elliptic plane becomes two equal divergencies in
opposite points with equal signs; a rotation two equal rotations in
opposite points with different signs; for the latter we do not find
the analogous potential as for the former; the latter can be found
here according to the Schering potential formula.

With this is connected immediately that in the elliptic plane the
field of a single rotation (in contrast to that of a single divergency)
has as such possibility of existence, so it can be regarded as unity

18
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 266 )

of field. That field consists of forces touching concentric circles and

creat ;
= 8in ©

Postscript. In the formula for vector fields in hyperbolic spaces:
eyes Nive
\ P(yar + (2 F,(r) dt

nothing for the moment results from the deduction but that to \27_X
and \1/ X also must be counted the contributions furnished by infi-
nity. From the field property ensues, however, immediately that the
effect of these contributions disappears in finite, so that under the
integral sign we have but to read \2/_X and \1/ X in finite.

For the \1/ at infinity pro surface-unity of the infinitely great
sphere is < order e-”; the potential-effect of this in finite becomes
<order -re— "—*)" %— e-" = re—™—r; so the force-effect < order
e—™—lr; so the force-effect: of the entire infinitely great spherical
surface is infinitesimal.

Pot. xX =

if
And the \2/ at infinity pro surface-unity is << order — ; it fur-
% :
; nt yk 1
nishes a potential-effect in finite < order e—"—l)r, —, thus a force-
7

1 ;
effect << order e~—"—")". — ; so the force-effect, caused by the infi-
Tr .

nite, remains < order —.
dh :

The reasoning does not hold for the force field of the hyperbolical
Sp, in the second interpretation (see under B § VIID), but it is in
the nature of that interpretation itself that the derivatives at infinity

are indicated as such, therefore also counted.

Meteorology. — “On Magnetic Disturbances as recorded at Batavia.”
By Dr. W. van BruMMELEN.

(Communicated in the meeiing of September 29, 1906).

Some months ago Mr. Maunpur of the Greenwich-Observatory
addressed a request to the Batavia Observatory to provide him with
a list of magnetic storms recorded at Batavia with a view of testing
his results as to the influence of the synodie rotation of the sun to
the occurrence of disturbances.

Mr. Mavnper concludes from an inspection of the disturbances

( 267 )

recorded at Greenwich (and also at Toronto) that they show a
tendency to recur after a synodic rotation of tle sun and that some-
times even two and more returns occur. His conclusion is:

“Our magnetic disturbances have their origin in the sun. The solar
“action which gives rise to them does not act equally in all directions,
“but along narrow, well defined streams, not necessarily truly radial.
“These streams arise from active areas of limited extent. These active
“areas are not only the source of our magnetic disturbances, but
“are also the seats of the formation of sun-spots.”

As soon as I could find the necessary leisure I prepared a list for
the period 1880—1899, containing 1149 disturbances and immediately
after made some statistical calculations based on them.

A discussion of such statistical results is always better made by
the author of the list, than by another person for whom it is impos-
sible to consult the original sheets.

Though intending to publish the list, statistics and some repro-
ductions in full, I wish to give a preliminary account of my results,
because these questions are now of actual importance.

Rules followed in preparing the list.

An exact definition of what is understood by the~ expression
“magnetic storm” has never been given; certain features however
are characteristic to it, viz:

1. The sudden commencement.

2. The postturbation.

3. The increased agitation.

Concerning the second, which I ealled the postturbation *), the
well known fact may be remarked, that during’a storm the mean
level of the components of the force changes, till a maximum digres-
sion is reached, and afterwards returns slowly to its old value.

In 1895 I called attention to this phenomenon and investigated
its distribution over the earth.

This research enabled me to give the following description of the
postturbation.

During a magnetic storm a force appears contrary to the earth’s
ordinary magnetic force, with this difference, that its horizontal com-
ponent is directed along the meridians of the regular part of the
earth's magnetism, consequently not pointing to the magnetic pole, but
to the mean magnetic axis of the earth.

') Cf. Meteorologische Zeitschrift 1895, p. 321. Terrestrisch MagnetismeI p. 95,
IE 115, V 1238, VIN 153.

18*

( 268 )

In accordance with this description, during the earlier part of a
storm the horizontal force diminishes, the vertical force increases,
and during the latter part these forces resume slowly their original
values. The characteristic features sub 1 and 2 either do not neces-
sarily attend every storm, or if so, they do not show themselves
clearly enough to enable us to decide definitely whether a succession
of waves in a curve must be considered as a storm or not.

On the contrary the increased agitation is an essential feature
and has therefore been adopted by me as a criterion.

Unfortunately it is impossible to establish the lowest level aoe
which the never absent agitation may be called a storm, because
the agitation is not only determined by the amplitude of the waves,
but also by their steepness and frequency.

To eliminate as much as possible the bad consequences which
necessarily attend a personal judgment, the list has been prepared:

1. by one person;

2. in as short a time as possible;

3. from the aspect of the curves for one component only (in
easu the horizontal intensity, which in Batavia is most lable
to disturbance);

4. for a period with nearly constant scale-value of the curves
(1 mm. = + 0,00005 C.G:S.);

For each storm has been noted:

1. the hour of commencement;
ae eS ex piranon:

3. i >» >», Maximum;

4. the intensity.

Mr. Mavunper calls a storm with a sudden start an S-storm
analogously I will call one with a gradual beginning a G-storm.
In the case of a sudden impulse the time of beginning is given to
the tenth of an hour; in that of gradual increase of agitation only
by entire hours.

The hour of beginning of a G-storm is not easy to fix. I have
chosen for it the time of the very beginning of the increased agita-
tion, and not the moment in which the agitation begins to show an
unmistakable disturbance character.

Afterwards it became clear I had shown a decided preference for
the even hours, which may be accounted for by the fact that only
the even hours are marked on the diagrams.

To eliminate this discordance I have added the numbers of G-storms

(269 )
commencing at the odd hours for one half to the preceding and for
the other half to the following hour.

Becanse a storm as a rule expires gradually, it is often impossible
to give the exact moment it is past. If doubtful I have always taken
the longest time for its duration; hence many days following a great
storm are reckoned as being disturbed, which otherwise would have
passed as undisturbed.

For the time of the maximum I have taken the moment of maxi-
mum agitation, which does not always correspond with the hour
of maximum postturbation.

I believe the hour at which the mean H-force reaches its lowest
level is a better time-measure for the storm-maximum, but to determine
it a large amount of measuring and calculating is required, the
change in level being often entirely hidden by the ordinary solar-
diurnal variation.

The intensity of the storm has been given after a scale of four
degrees: 1 small; 2 = moderate; 3 = active; 4— very active.

It is not possible to give a definition of this scale of intensity in
words, the reproduction of typical cases would be required for this.

Hourly distribution of the beginning of’ storms.

It is a known fact, that the starting impulse is felt simultaneonsly
all over the earth. The Greenwich and Batavia lists furnished me
with 53 cases of corresponding impulses, which, if the simultaneity
is perfect, must enable us to derive the difference in longitude of
the two observatories.

I find in 6 cases 7212™
eee eet eo
eens eG

Mean 7» 7™15s

True difference 757™19s,

It it very remarkable indeed to derive so large a difference of
longitude with an error of 4 seconds only, from 53 cases measured
roughly to 0.1 hour.

The simultaneity should involve an equal hourly distribution if
every S-impulse were felt over the whole earth. As this is not the
case, which is proved by the lists of Greenwich and Batavia, it is
easy to understand that the Batavia-impulses show indeed an unequal
hourly distribution. We find them more frequent at 6"'and 10" a. m,
and 7" p.m.

( 270 )

Hourly distribution of S-impulses.

Number Number
Hour | Hour

in %/ | in °%/g

Sa a eg
0 a.m 4A 12 4,7
4 2.5 13 5.0
2 3.0 14 3.3
3 2.2 15 3.9
4 4.4 16 4,4
5 | 3.9 17 3.6
6 6.3 18 4A
7 ie ae
8 Sep 20 3.6

9 5.8 24 | ee ee
10 6.1 22... | "3.6
dA | 5.0 | 23 | 3.3

This same distribution we find again in the case of the G-storms,
but much more pronounced; a principal maximum at about 8" a. m.,
and a secondary one at 6" p.m.

Accordingly the hour of commencement of the G-disturbances is
dependent upon the position of the station with respect to the sun,
and it seems, that the hours most appropriate for the development
of a G-disturbance also favour the development of an S-impulse.

Hourly distribution of the commencement of G-storms (in °/,).

noon :
0 2 4 6 8 10 -42: 44- 16. 18 20. 22

Hour

17:9 -T.4 4.7 6108.0 6b Fed

Intensity: 4 6:0 6.05.1 ‘Gute
» 2 4.5 4.9 4.2 5.7 20.8 164. 7.3 5.6 5.4 9.2 84 7.6
» 3 and 4 7.4 3.5 43-75 185 43,4 3:9 55 5.1.9.1) 4175

All

on
—S
or
—
—
or)
for)
w
_
oO
~J

16.5 6.8 5.2;5.6 8.7 .8.3 | 7.&

( 274.)

Hourly distribrution of the maximum (in °/,).

| é noon
Hour | 0 2 4 G SOROS A ie . AG 1S. 20-1. 22
Intensity 4 | Men ont 5... 1-8 4.7.12.8 12.6 5.3 6-4 1641-46.9.16.2
2 » 2 | aoe yO Aa eS Oar 2.0710.8 13.5 45.2
s » 3and4 | fie ee) 244-6 0-8: 4.0: -7-2-6.8 12.0 13.6 17.2 49.2
n i ia =
| All 44.4.7.4 4.91.594 8.6 9.2 4. 71.6: 29.7 13.3: 16.3

Intensily 1 | 12.3 16.7 10.9 5.8 4.3 13.8 5 8 2.9 2.9 5.1 10.1 9.4
ui » 2| 11.3 T1 3.65.63.3 8.5 11.38.5 6.1 5.2 14.1 14.9
: » Zand 4) 12.2 9.3 5.83.23.5 7.7 9.0 6.4 8.0 7.7 10 317.0
: : — =
All | 114.910.2 6.0463.6 9.2 9.26.5 6 6.3 11.6 14.8

These hourly numbers show for each intensity, and for both kinds
of storms the same, strongly marked distribution over the hours of
the day.

Thus the development of agitation during a storm is dependent
on the position of the sun relatively to the station in a manner
which is the same for S- and G-storms.

The period has a principal maximum at 10% p. m. and a secondary
one at noon; and being compared to the diurnal periodicity of the
commencement of G-storms, it is evident, that: On the hours when
the chance for a maximum-agitation begins to increase, we may expect
most storms to take a start.

_ Hence we may come to the following supposition.

The susceptibihty of the earth’s magnetic field to magnetic agitation
is lable to a diurnal and semidiurnal periodicity. Whatever may be
the origin of the merease of agitation, sudden or gradual, this period-
icity remains the same.

This was the same thing, that was revealed to me by the inspec-
tion of the hundreds of curves in preparing the list.

The agitation rises at about 85 a. m. after some hours of great
calm and reaches a maximum at about noon. A second period of
calm, less quiet however than in the early morning, is reached in

the afternoon, and a second rise follows till a maximum is attained
shortly before midnight.

(2735

The day-waves however are smaller and shorter, the night-waves
larger and longer and also more regular in shape. These regular
night-waves are often restricted to one large wave, very suitable
for the study of these waves.

Hourly distribution of the end of the storm.

Number | Number

Hour | Hour
of cases of cases
Oa.m.| 163 | 12 66
2 172 14 60
4 204 16 43
6 140 18 50
8 60 20 46
10 36 22 4Y

Quite in agreement with the above mentioned conclusions, the
curve representing the diurnal periodicity of the final-hour is nearly
the reverse of that for the maximum.

Evidently the hour 0 (the end of the day) has been strongly
favoured.

Resuming we may according to the Batavia disturbance-record
draw the following conclusions :

1st. the origin of S-storms is cosmical ;

24, the origin of G-storms may be also cosmical, but the com-
mencement 1s dependent on the local hour;

3°, the development of all storms, concerning the agitation, is in
the same way dependent on the local hour.

Storms and sunspots.

Iu the following table the year has been reckoned from April 15
till April 1st of the following year, with the exception of 1882, the
diagrams for the months Dec. ’82, Jan.—March ’83 missing. For
1880—’83 the yearly numbers have been increased in proportion
to the number of missing record days.

(273 )

Numbers in °/,.

Intensity

Sunspot -

Year 4 2 | 3 and 4 | All

number
s | G Ss | G | s | G Ss | G

1880/81 37.5 eae | Aco 6.6 7°42] 2.38) 4.6) 5.1
81/82 5 a) fee oe? |} 8.0) oo) 41:2) 9.3) 0.8] 526
82 70.8 foe) ace | goed ie i.6. 143-0) 6.1) 7.9] 5.2
83/84 68.8 Peor icabedo oe | 6.9 | 5.3) 7.4) 6.2
84/85 59.5 fe eee OLS Gla | 8.302641 8.21 5.7
85/86 45.7 Pay eise oes | S01 40.6 | 6.91 9.0} 3.6
86/87 19.6 ee tol eee Oe 1 429 | 7.6] 41.6) 4.5
87/88 11.6 Beige tous. A oe! 4.6: 3.3) 5.6
88/89 6.4 aed ALG | ACT | - 255 | 3.4 | 3.8) 4.5
89/90 9.9 1S fF. 49.) 5.3 | -3.3 1] 1.2 | 4.6] 4:6] 4.4
90/91 13.0 face) a0) 223°) 93.8 4. O36.) 3A | 3.8] 5.2
91/92 | 41.4 B-8 les-6 | 5a 5.4) 6:9 |) 8.41 4:71 5.5
92/93 74.5 9.5 | 6.4] 8.3] 3.3|42.4| 4.6)10.3|] 4.6
93/94 85.2 Bee Oe M20 | 3.81 44.8 |. 5.3 | 10.3 | 4.8
94/95 74.2 eee ore t| 2.1 heed | 6:9.) 5:7) Gib
95/96 57.4 6.8] 5.3] 5.34 6.9] 5.0] 8.4] 54] 6.6
96/97 38.7 Pees oad | OF | 2.5 )-3.4 1) 3.8 | 4.6
97/98 26.5 Poets ot Oto 5.4), 05.1 |- 7.6 | 3.5 | 5.7
98/99 22.9 aio owe 4.5 ) 5.9 | 41.9 | a8) |. (2.0) 6:5

From these numbers it appears that those for the G-storms show
no correspondence with the sunspot-numbers, also that those for the
S-storms show a correspondence which is emphasised according as
the intensity increases, and finally that the S-storms show a maximum
when the G-storms have a minimum and the reverse.

This latter fact is apparently caused by the circumstance of the
storms hiding each other, the G-storms being eclipsed by the S-storms
in a higher degree during greater activity of the sun, than the S-
storms by the G-storms. Indeed a simple inspection of the diagrams

( 274 )

shows that the agitation of G-storms is greater during a sunspot
maximum, than in minimum-years. Also in maximum-years the S-
storms of intensity 1, are hidden by their stronger brothers to such
an extent, that the eleven-yearly periodicity is nearly the reverse
for them.

Annual distribution of S- and G-storms.
(Only the uninterrupted period April 1, 1883—April1, 1899
has been considered).

| Numbers
Month | ear

ee
January 31 | 54
February 31 53
March 29 60
April 24 57
May 24 61
June 27 51
July 31 | 61
August 29 47
September 32 55
October 31 64
November 22 58
December 18 58

A strong difference in behaviour between G- and S-storms can be
noticed. The G-storms have no annual periodicity as to their frequency,
* whereas the S-storms show a strong one.

This points, just like the daily periodicity of commencement, either
to a different origin, or to a changing tendency of the development
of the S-impulse during the day and year.

Comparison with Greenwich-storms.

_ Maunprr derives from the reproduction of storms published in the.
volumes of the Greenwich Observations a maximum at. 6" p. m. and

( 275 )

from the original recording-sheets on the contrary at 1" p.m. The
cause of this discrepancy he finds in the manner he looked for the
commencement. He writes: ‘the times when the phases of diurnal
disturbance are most strongly marked are naturally most often taken
as the times of commencement.”

At Greenwich these phases of agitation are most prominent at
41" p.m. and 6° p.m.

As I assumed for the hour of beginning the first increase of
agitation it is clear my times of beginning are on an average much
earlier.

Thus the difference shown by the hourly distribution of commen-
cement between the Greenwich- and the Batavia-list, may be ascribed
chiefly to difference of interpretation.

As appears from the figures given above, compared with those for
Greenwich the annual periodicity is quite the same for both the
northerly and the equatorial stations, which differ no less than 60
degrees in latitude. But the Greenwich dates, quoted from a complete
magnetic calendar, prepared by Mr. Enis and extending from 1848
to 1902 give no separation of G- and S-storms. Thus it is not
possible to decide whether at Greenwich the G-storms lack an annual
periodicity in their frequency.

The impulse at the start.

The material at present at my disposal for investigating the features
of this phenomenon in other places on earth, is very small.

Notwithstanding this I may conclude: that this phenomenon is of
great constancy in features all over the earth, and consequently a
phenomenon of great interest, which might teach us much about the
manner the S-storms reach the earth.

Description of its features for some places.

Greenwich. According to the reduced reproductions of disturbance-
curves published in the volumes of the Greenwich-Observations, -
the impulse consists of a sudden movement in H, D and Z, instantly
followed by the reversed movement, the latter being considerably
greater. The direction of the movement is always the same.

I have measured 34 cases and have found on an average:

HAD AH AZ
25yW - + 77y + 39y (1 y= 0.00001 C.G.S.).

Batavia. The preceding impulse is missing for H and Z, only for

D it is often present.

( 276 )

Here also the direction of the movement is constant.

35 cases for the years 1891 and ’92 gave on an average for the

magnitude of digression HAD=9yW; AH=—-+45y; AZ——16y.

duration __,, Ws 3.0 re: 5 min.; 12 min.

Though the movement of H and D are not sudden in absolute
sense, that for Z is too gradual to justify the application of the word
sudden to it.

The reproductions of disturbance-curves for Potsdam and Zi Ka Wei
also show some cases of the preceding impulse. At both stations the
direction of the movement is remarkably constant. In the publications
of the Cape Hoorn observations (1882/83) I found three cases exposing
also a constant direction.

Summarizing them, we have:

Station
Potsdam
Greenwich
Zi Ka Wei
Batavia
Cape Hoorn

a odie ia a a=
44un440
eral

Consequently with one exception for D and one for Z we find that:
the commencing impulse of the S-storms is the reverse of the vector
of postturbation, being deflected however to the West of it.

Suppositions concerning the origin of disturbance.

The hypothesis on the existence of defined conical streams of
electric energy, which strike the earth, though not quite new, has
obtained increased plausibility by MaunpEr’s results. From the statistics
based on the record of disturbances at Batavia it might be concluded
that it is chiefly the S-storms that find their origin in the sudden
encounter of the earth with such a stream.

And as the earth is first struck at its sunset-arc, it is not impos-
sible that the G-storms, which begin by preference shortly after sun-
rise and have no annual periodicity in their frequency as the S-
storms have, are only partly caused by these encounters.

When in the case of the streams we admit that energy progresses
from the sun in the form of negative electrons, we might think the
G-storms find their origin by electrified particles being propagated
by the light-pressure according to the theory set forth by Sv. ARRHENIUS.

Further we may suppose, that when the earth has received a
charge the following development of the storm is the same as it is

(277)

dependent on the local hour only. Arrnenius has already given an
explanation of the nocturnal maximum.

In recent times it has often been attempted to explain magnetic
fluctuations by the movement of electric charge through the higher
layers of the atmosphere. (ScHUSTER, VAN BrzoLp, ScHMIDT, BIGELOW).

The remarkable analogies which are met with in many cases
between the streamfield of atmospherical circulations and the fields
of magnetic fluctuations, lead to such speculations.

I believe it is allowed to hazard analogous speculations concerning
the cause of the beginning of impulse and postturbation.

We may suppose the streams to contain negative electrons. When
they strike the earth the outer layers will be charged with negative
electricity. These outer layers do not rotate in 24 hours, but in a
longer time increasing with their height.

So a countercurrent of E—W direction charged with negative
electrons will originate, causing an increase of H and a decrease
of Z. The electrons, however, on entering the magnetic field of the
earth, will follow the lines of force towards the magnetic south pole
(the positive pole). The movement of negative electrons along the
lines of force has been fairly well proved, as is well known, by the
aurorarays.

By this movement, the current of electricity will become NE—SW
and a westerly deflecting S-impulse will be the consequence.

The sudden charge of the extreme layers of the atmosphere with
negative electricity, will attract the positive ions, with which the
high layers may be supposed to be charged, to still higher layers.

These positive ions will thus enter into a faster moving counter
current, and a positive charged counter current will be the conse-
quence.

These ions will move along the lines of force towards the north,
but much slower than the negative electrons, and therefore the
resulting deflection of the magnetic force caused by such a + current
viz. a force contrary to the ordinary one, will be of no appreciable
magnitude. It is conceivable that the effect, which accordingly is in
the same sense as the postturbation, will develop in a more gradual
manner than the commencing impulse of the S-storms; moreover
we may understand that it disappears still more gradually in propor-
tion as the negative electrons again leave the earth or are neutralised
by positive ions.

Only we should expect the current to follow the latitude-parallels
and accordingly the vectors of postturbation to point to the true
south and not to the southerly end of the earth’s mean magnetic axis.

Perhaps we may find an explanation for this fact in the influence
no doubt exerted by the earth’s mean magnetic field and the
distribution of positive ions in the atmosphere.

These speculations are indeed very rough, but they have one great
advantage, viz. to avoid the difficulty, raised by Lord Ketvin, of
allowing an expenditure of the sun’s energy causing magnetic disturb-
ances, much too great to be admitted.

Curer (Terr. Magnet. X, p. 9) points to the fact, that also MAUNDER’s
defined streams require far too great an expenditure of energy.

According to my opinion we have only to deal with the charge
received at the moment of the impulse, and by accepting an inter-
mittent emission of the sun’s energy, it is not necessary to integrate
it over the entire time between one or more returns of the stream.

Part of the energy is also supplied by the rotation-energy of the
earth; and it is curious to remark, that by such an influence the
rotation of the earth would be lengthened for a minute fraction
during a magnetic storm.

Chemistry. — “Nitration of meta-substituted phenols”. By Dr. J. J.
BianksMA. (Communicated by Prof. HonLEman).

(Communicated in the meeting of September 30, 1906).

Some years ago’) I pointed out that by nitration of meta-nitro-
phenol and of 3-5-dinitrophenol tetra- and pentanitrophenol are formed.
This showed that the NO,-groups in the m-position do not prevent
the further substitution of the H-atoms in the o- and p-position by
other groups. I have now endeavoured to increase these two cases
by a few more and have therefore examined the behaviour of some
m-substituted phenols which contain, besides a NO,-group in the
m-position, a second group in the m-position, namely of

C,H, . OH . NO, . (CH,,OH,OCH,,OC,H,,Cl,Br) 1.3.5.

Of these phenols the 5-nitro-m-cresol?) and the monomethylether
of 5-nitroresorcinol *) were known. The still unknown phenols were
made as follows:

The 5-nitroresorcinol (m.p. 158°) from its above cited monomethy1-
ether by heating for five hours at 160° with (30°/,) HCl, or by
reduction of 3-5-dinitrophenol with ammonium sulphide to 5-nitro-

1) These Proce. Febr. 22, 1902. Rec. 21. 241.

2) Nevite en WINTHER Ber. 15. 2986.
’ 8) H. VERMEULEN Rec. 25. 26,

( 279 )

3-aminophenol (m.p. 165°) and substitution of the NH,-group in this
substance by OH.

The monoethylether of 5-nitroresorcinol (m.p. 80°) was prepared
(quite analogous to the methylether) from 5-nitro-3-aminophenetol ;
the 3-Cl (Br) 5-nitrophenol was obtained by substituting the NH,-group
in the 5-nitro-3-aminoanisol by Cl(Br) according to SanpMrEYER and
then heating the 3-Cl (Br)-5-nitroanisol so obtained m.p. (101°°) and
88°); with HCl as directed. We then obtain, in addition to CH,Cl,
the desired product 3-Cl (Br) 5-nitrophenol (m.p. 147° and 145°).

The 3-5-substituted phenols so obtained readily assimilate three
atoms of bromine on treatment with bromine water and three nitro-
groups are introduced on nitration with H NO, (Sp. gr. 1.52) and
H,S0O,. These last compounds, which all contain four NO,-groups
resemble picric acid, tetra- and pentanitrophenol. From a mixture of
nitric and sulphuric acids they crystallise as colourless crystals which
are turned yellow by a small quantity of water; the compounds
have a bitter taste, an acid reaction and communicate a strong yellow
colour to organic tissues (the skin), others strongly attach the skin and
all are possessed of explosive properties owing to the presence of
four NO,-groups ’*).

OH OH OH OH

a Pa as

| [1580 | 147°
so f/f is Be ee od. Aoctiocs NO, oe
| Ya Ya a

OH
NO. \NO, NOW \NO, NO,” \NO, NO, Sea
175° 152° fadise 447°
NO /CHs NO3\ ae NO\ Coe NO, Cl(Br)
No, noe.
with ye vio with 3 with |NH.C.H;

OH
x \w iG \vo Lo en ae \vo;
HO CH, HO OH NH.\ //NH, C,H;HN NHC,H;
we Xe ee Se

NO;

In this scheme are given only the melting points of the as yet
unknown compounds.
Tetranitro-m. cresol yields on boiling with water trinitroorcinol;

1) 91° according to pe Kock Rec. 20, 113.
*) A comparative research as to these properties in the different compounds
has not yet been instituted.

( 280 )

in the same manner, tetranitroresorcinol *) yields trinitrophloroglucinol ;
tetranitrochloro- and bromophenol also yield trinitrophloroglucinol on
boiling with water or, more readily, with Na,CO, solution. By the
action of NH, or NH, C, H, ete. in alcoholic solution various other
products are obtained, such as those substances included in the scheme
which have been obtained previously from pentanitrophenol’). We
also see that water or alcohol cannot serve as a solvent for the
purpose of recrystallising these compounds but that chloroform or
carbon tetrachioride may be used.

If, in the above cited 3-5-substituted phenols the OH-group is sub-
stituted by OCH, it is not possible to introduce three nitro-groups.
For instance the dimethylether of 5-nitroresorcinol yields two iso-
meric trinitroresorcinoldimethylethers (principally those with the
melting point 195°, just as in the nitration of 5-nitro-m-xylene)*);
similarly, the methylether of 5-nitro-m-cresol (m.p. 70°) yields the
methylethers of three isomeric trinitro-m-cresols, principally the
compound with m.p. 139°. The constitution of these substances is
not yet determined.

Amsterdam, September 1906.

Chemistry. — Prof. HoLieman presents a communication from him-
self and Dr. H. A. Sirks: “ The siz isomeric dinitrobenzoic acids.”

(Communicated in the meeting of September 29, 1906).

Complete sets of isomeric benzene derivatives C,H, A,B have been
studied but little up to the present; yet, for a closer understanding
of those derivatives, it must be deemed of great importance to subject
the six possible isomers of which such sets consist, to a comparative
investigation. A contribution hereto is the investigation of the six
isomeric dinitrobenzoic acids which Dr. Srxs has executed under
my directions.

The considerations which guided me in the choice of this series

') According to Henriques (Ann. Chem. 215, 335), tetranitroresorcinol (m.p. 166°)
is formed by the nitration of 2-5-dinitrophenol. In Bemstein’s manual (vol. II, 926)
a reasonable doubt is thrown on the correctness of this observation. The sub-
stance obtained has probably been an impure trinitroresorcinol formed by the
action of water on the primary formed tetranitrophenol. (Rec. 21, 258).

2) Rec. 21, 264.

5) Rec. 25, 165.

( 281 )

of isomers were the following. Firstly, all six isomers were known,
although the mode of preparation of some of them left much to be
desired. Secondly, this series gave an opportunity to test V. Mryer’s
“ester rule” with a much more extensive material than hitherto and
to study what influence is exercised by the presence of two groups
present in the different positions in the core, on the esterification
velocity, and to compare this with that velocity in the monosub-
stituted benzoic acids. Thirdly, the dissociation constants of these acids
could be subjected to a comparative research and their values con-
nected with those of the esterification constants. Finally, the melting
points and sp. gr. of the acids and their esters could be investigated
in their relation to these same constants in other such series.

The six dinitrobenzoic acids were prepared as follows. The sym-
metric acid 1,3,5, (1 always indicates the position of the carboxyl
group) was obtained by nitration of benzoic acid or of m-nitroben-
zoic acid. All the others were prepared by oxidation of the correspon-
ding dinitrotoluenes. This oxidation was carried out partly by per-
manganate in sulphuric acid solution, partly by prolonged boiling
with nitric acid (sp. gr. 1.4) in a reflux apparatus.

We had to prepare ourselves three of the dinitrotoluenes, namely,
(1,3,4), (1,3,6) and (1,3,2), (CH, on 1); (4.2,4) and (1,2,6) are com-
mercial articles whilst (1,3,5) was not wanted because the orre-
sponding acid, as already stated, was readily accessible by direct
nitration of benzoic acid. As will be seen the three dinitrotoluenes
which had to be prepared are all derivatives of m-nitrotoluene and
it was, therefore, tried which of those might be obtained by a further
nitration of the same.

m-Nitrotoluene, which may now be obtained from pg HAgn in a
pure condition and at a reasonable price was, therefore, treated with
a mixture of nitric and sulphuric acids at 50°. On cooling the
nitration-product a considerable amount of 1,3,4 dinitrotoluene crys-
tallised out, which could be stil! further increased by fractionated
distillation in vacuo of the liquid portion ; the highest fractions always
became solid and again yielded this dinitrotoluene, so that finally
about 65 grms. of dinitrotoluene (1,3,4) were obtained from 100 grms.
of m-nitrotoluene,

As the fractions with a lower: boiling point, although almost free
from dinitrotoluene (1,3,4), did not solidify on cooling, it was thought
probable that they might contain, besides a little of the above dinitro-
toluene, more than one of the other isomers, whose formation in the
nitration of m-nitrotoluene is theoretically possible. If we consider

19
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 282 )

that, in the many cases which I have investigated, the presence of
14°/, of an isomer causes about 0.5° depression in the melting point,
the fact that the oil did not solidify till considerably below 0° and
again melted at a slight elevation of temperature whilst the pure
isomers did not liquefy till 60° or above, cannot be explained by
the presence of relatively small quantities of 1,3,4-dinitrotoluene in
presence of one other constituent, but it must be supposed to consist
of a ternary system. This was verified when the fractionation was
continued still further; soon, the fractions with the lowest boiling
points began to solidify on cooling, or slowly even at the ordinary
temperature, and the solidified substance proved to be 1,2,3 — dini-
trotoluene. The fractionation combined with the freezing of the
different fractions then caused the isolation of a third isomer namely
1,3,6 — dinitrotoluene, so that the three isomeric dinitrotoluenes
desired had thus all been obtained by the nitration of m-nitroto-
luene. The fourth possible isomer (1,3,5) could not be observed even
after continued fractionation and freezing.

As regards the relative quantities in which the three isomers,
detected in the nitration product, are formed, it may be mentioned
that this product consists of more than one half of 1, 3, 4-dinitro-
toluene, whilst (1,2,3) seems to occur in larger quantities than
(1, 3,6), as the isolation of the latter in sufficient quantity gave the
most trouble.

The corrected solidifying points of the dinitrotoluenes (the sixth,
symmetric One was prepared by Brmstxm’s method A. 158, 341 in
order to complete the series) were determined as follows. Those of
the dinitrobenzoic acids and of their ethyl esters are also included
in the subjoined table.

Dinitrvtoluenes | 58.3 92.6 59.3 50.2 65.2 70.4

Dinitrobenzoic acids | 163.3 206.8 204.4 479.0 | 206.4 480.9

Ethyl esters 71.0 92.9 | 88.4 68.8 74.7 40.2

The specific gravities of the dinitrotoluenes and the ethyl esters
were determined by means of Eykman’s picnometer at 111°.0 with
the following result :

( 283 )

= oe on 4 | toluenes
3

esters

Jel Dig Oe
3.4 ! 4.9594 | 4.9791
3.5 | 1.9772 | 1.2935
2.3 | 1.2625 | 1.2895
2.5 | 4.2890 | 1.2859
2.4 | 4.2860 | 4.2858
2.6 1.9923 | 429993

Water at 4° as unity. Corrected for upward atmospheric pressure
and for expansion of glass.

Conductivity power. This was determined in the usual manner
with a Wheatstone-bridge and telephone at 25° and at 40°. As the
acids are soluble in water with difficulty v= 100 or 200 was taken
as initial concentration; the end concentration was v = 800 or 1600.
In the subjoined table the dissociation constants are shown.

3.4 | 3.5 2s [20 2.4 2.0

Dinitrobenzoic acids

at 40°

l
at 25° 0.103 0.163 | 1.44 2.64 |
0.171 | 0.177 | 1.38 a

On comparing these figures it is at once evident that the acids
with ortho-placed nitro-group possess a much greater dissociation
constant than the other two, so that in this respect, they may be
divided into two groups. In the acids without an ortho-placed nitro-
group, the value of the dissociation constant is fairly well the same.
In the other four, the position of the second group seems to cause
fairly large differences. That second group increases the said con-
stant most when it is also placed ortho: in N/,,, solution 2-6-dinitro-
benzoic acid is ionised already to the extent of 90 °/,. Again, a NO,-group
in the para-position increases the dissociation constant more than one
in the meta-position; and for the two acids 2,3 and 2,5 which both
-have the second group in the meta-position, K is considerably larger
for 2,5, therefore for the non-vicinal acid than for the vicinal one, so
that here an influence is exercised, not only by the position of the

19*

( 284 )

eroups in itself, but also by their position in regard to each other.
It also follows that OstwaLp’s method for the calculation of the
dissociation constants of disubstituted acids from those of the mono-
substituted acids cannot be correct as is apparent from the sub-
joined table:

Dinitrobenzoic acid
K calculated | K found
CO,H on 4
oe
3.4 0.23 0.16
3.5 0.20 0.16
2.4 4A 3.8
2.5 3.6 2.6
2.3 3.6 1.4
2.6 64 8.1

In the two vicinal acids 2,3 and 2,6 the deviations from the cal-
culated value are particularly large, as I have previously shown for
other vicinal substituted acids (Rec. 20, 363).

In view of the comparison of the figures for the dissociation
constants of these acids and for their esterification constants, it seemed
desirable to have also an opinion as to the molecular conductivity
of these acids in alcoholic solution. They were, therefore dissolved in
95 vol. °/, alcohol to a N./,,, solution and the conductivity power
of those liquids was determined at 25°. The subjoined table shows
the values found and also those of the aqueous solutions of the
same concentration and temperature :

dinitrobenzoic sid 3.4 | aS | 2.3 | 2.5 | 2.6 | 2.4
SS LL.
9 in alcoh. sol. | A | | 1.45 1.75 | 2.25 2.7 | 2.9

go) in aqueous sol. | 161.5 |

162.5 be pe — | 335.5

from which it appears that also in alcoholic solution the acids with
an ortho-placed nitro-group are more ionised than the others.
Esterification velocity. The method followed was that of GoLDscHMIDT,
who dissolved the acid in a large excess of alcohol and used hydro-
chloric acid as catalyzer. The alcoholic hydrochloric acid used here

( 285 )

was 0,455 normal. Kept at the ordinary temperature it did not
change its titre perceptibly for many months. As Gotpscumipt showed
that the constants are proportionate to the concentration of the
catalyzer, they were all recalculated to a concentration of normal
hydrochloric acid. Owing to the large excess of alcohol the equation
for unimolecular reaction could be applied. The velocity measure-
ments were executed at 25°, 40° and 50°. At these last two tempera-
tures, the titre of the alcoholic acid very slowly receded (formation
of ethylchloride) and a correction had, therefore, to be applied. The
strength of the alcohol used was 98.2 °/, by volume.

In order to be able to compare not only the esterification-constants
E of the dinitrobenzoic acids with each other but also with those of
benzoic acid and its mononitroderivatives, the constants for those acids
were determined at 25° under exactly the same circumstances as
in the case of the dinitroacids. The results obtained are shown
in the subjoined table:

Acids | E at 25°} E at 40°; E at 50°

benzoic acid 0.0132 | _ _
m. NO, » 0.0074 | -— =
OF io2 » 0.0010 oo —
3.4 dinitro » 0.0086 0.033 0.077
3.5 » » 0.0053 0.028 0.060
Tes DS » 0.0005 0.0025 0.0071
2D » 0.0003 0.0027 0.0076
2.4 » » 0.0002 0.0017 0.0056
2.6 » ») unmeasurably small

As will be seen, E is by far the largest for benzoic acid and each
subsequent substitution decreases its value.

On perusing this table it is at once evident that in the dinitroben-
zoic acids two groups can be distinguished: Those with an ortho-
placed nitro-group have a much sma//er constant than the other two.
Whilst therefore the dissociation constant for acids with an ortho-
placed nitro-group is the largest their esterification constant is the
smallest. As shown from the subjoined table, this phenomenon
proceeds quite parallel; the acids whose dissociation constant is
greatest have the smallest esterification constant and vice versa.

( 286 )

Dinitrobenzoic acids | diss. const. at 40° | esterif. const at 40°

a EE a ES EE

3.4 O-171 0.033
Do 0.4177 0.028
2.3 1.38 0.0025
2.5 2.16 0.0027
2.4 3.20 0.0017
2.6 7.6 < 0.0001

On perusing. the literature we have found that this regularity
does not exist in this series of dinitrobenzoic acids only, but is observed
in a comparatively large number of cases. The strongest acids
are the most slowly esterified. This might lead us to the conclusion
that in the esterification by alcoholic hydrochloric acid it is not the
ionised but the unsplit molecules of the acids which take part in
the reaction.

A more detailed account of this investigation will appear in the
Recueil.

Amsterdam ees:
, Sept. 1906, Laboratory of the University.
Groningen
Chemistry. — Prof. Ho1ieman presents a communication from

himself and Dr. J. Huisinca. “On the nitration of phthahe
acid and isophthatc acid”.

(Communicated in the meeting of September 29, 1906).

Of phthalic acid, two isomeric monoderivatives are possible, both
of which are known particularly by a research of Minuer (A. 208,
233). Isophthalic acid can yield three isomeric mononitro-acids. Of
these, the symmetric acid, which is yielded in the largest quantity
during the nitration, is well known. As to the mononitrated by-
products formed, the literature contains a difference of opinion ; in
any case, there is only made mention of one second mononitro-acid
whose structure has remained doubtful.

The investigation of the nitration of phthalic and isophthalie acid
was taken up by us in order to determine the relative amount of
the isomers simultaneously formed, as in the case of the mononitro-

( 287 )

phthalic acids only a rough approximation (by Mrmer) was known,
whilst in the case of the mononitroisophthalic acids it had yet to
be ascertained which isomers are formed there from.

We commenced by preparing the five mononitro-acids derived
from phtalic acid and isophthalic acid in a perfectly pure condition.
In the case of the e- and £-nitrophthalic acids no difficulties were
encountered, as the directions of Misr, save a few unimportant
modifications, could be entirely followed. The acids were therefore
obtained by nitration of phthalic acid and separation of the isomers.

The symmetric nitro-isophthalic acid was prepared by nitration
of isophthalic acid. It crystallises with 1 mol. of H,O and melts at
255—256° whilst it is stated in the literature that it crystallises
with 1'/, mol. of H,O and melts at 248°. At first we hoped that
the other two nitroisophthalic acids might be obtained from the
motherliquors of this acid. It was, therefore, necessary to obtain the
isophthalic acid in a perfectly pure condition, as otherwise it would
be doubtful whether the byproducts formed were really derived from
isophthalic acid. By oxidation of pure m-xylene (from KAHLBaAUM) an
isophthalic acid was obtained which still contained terephthalic acid
which could be removed by preparing the barium salts.

The motherliquors of the symmetric nitro-isophthalic acid appeared,
however, to contain such a small quantity of the byproducts that
the preparation of the nitro-acids (1, 3,2) and (1, 3,4) was out of
the question. These were therefore, prepared as follows:

Preparation of asymmetric nitro-isophthalic acid (A, 3, 4). On
cautious nitration of m-xylene at 0° with nitric acid of sp. gr. 1.48
a mixture is formed of mono- and dinitroxylene which still contains
unchanged m-xylene. This, on distillation with water vapour, passes
over first and when drops of the distillate begin to sink to the
bottom of the receiver the latter is changed and the distillation is
continued until crystals of dinitroxylene become visible in the con-
denser. 100 gr. of xylene gave about 85 gr. of mononitroxylol (4, 3, 4).

After rectification of this mononitroxylene (b. p. 238°) it was
oxidised in alkaline solution with a slight excess of permanganate ;
20 gr. yielded 12 a 13 gr. of acid which, however, consisted of a
mixture of nitrotoluylic acid and nitro-isophthalic which could be
separated by crystallisation from water. In this way, the as. nitro-
isophthalic acid was obtained with a melting point of 245°. In water
it is much more soluble than the symmetric acid, namely to the
extent of about 1°/,. at 25°. Unlike the symmetric acid, it crystal-
lises without water of crystallisation in small, fairly thick, plate-like
erystals. It is very readily soluble in hot water, alcohol and ether,

( 988 )

Preparation of the vicinal nitro-isophthalic acid (1, 3, 2). GrevInex has
observed that in the nitration of m-xylene with nitric and sulphuric acid

NO, \,

there is formed, besides the symmetric dinitro-m-xylene |

Vion

NO,
CH;
| abe: /\xO: 3
as main product, also the vicinal isomer | | - . On reduction
CH;
NO,
with hydrogen sulphide both dinitroxylenes pass into nitro-xylidenes
CH;

which are comparatively easy to separate. The nitro-xylidene | Ne

J Us

NoH
yields by elimination of the NH,-group vicinal nitro-m-xylene. Whilst
however, GREVINGK states that he obtained a yield of 25°/, of vicinal
nitroxylidene we have never obtained more than a few per cent of
the same so that the preparation of vicinal nitro-m-xylene in this
manner is a very tedious one, at least when large quantities. are
required. When it appeared that the “fabrique de produits chimiques
de Thann et Mulhouse” exported this nitroxylene, the oxidation,
although to some extent with material of our own manufacture, has
been mainly carried out with the commercial product. This oxidation
was also done with permanganate in alkaline solution. The vicinal
nitro-isophthalic acid is a compound soluble with great difficulty in
cold, but fairly soluble in hot water, crystallising in small beautiful,
shining needles, which melt at 300°. It crystallises without any water
of crystallisation and is readily soluble in alcohol and ether, from
which it is again deposited in small needles.

The three possible mononitroisophthalic acids having now been
obtained, we could take in hand the problem to ascertain the nature
of the byproduct formed in the nitration of isopthalic acid. After the
bulk of the nitroisophtalic acid formed had been removed by crystal-
lisation, a residue was left which was far more soluble in. water
than this acid, which pointed to the presence of the asymmetric
nitro-acid and which, indeed, could be separated by fractional crystal-
lisation. We will see presently how it was ascertained that the
nitration product was really only a mixture of the symmetric and
the asymmetric acid.

As in the determination of the relative quantities in which the
nitration products are formed, use was made of solubility determina

( 289 )

tions, we first give the solubilities in water at 25° of the five
nitrophthalic acids, in parts per 100.
a-nitrophtalic acid 8-nitrophthalie acid
2.048 very soluble
symmetric nitroisophthalic acid
with water of crystallisation. Asymmetric nitroisophthalic acid
0.157 0.967

Vicinal nitroisophthalic acid
0.216

Quantitative nitration of phthalic acid. This was done with abso-
lute nitric acid. It appeared that it proceeded very slowly even at
30°, and therefore the phthalic acid was left in contact with six
times the quantity of nitric acid. for three weeks. After dilution
with water the acid was expelled by heating on a waterbath or else
evaporated over burnt lime. The solid residue was then reduced to
a fine powder and freed from the last traces of nitric acid by pro-
longed heating at 110°. As under the said circumstances the mono-
nitrophthalic acids are not nitrated any further, it could be ascer-
tained by titration whether all the phthalic acid had been converted
into the mononitro-acid; the product had but a very slight yellow
colour so that a contamination could be quite neglected. Of the pro-
duct, now ready for analysis, different quantities were weighed and
each time introduced into 100 c.c. of water, and after adding an
excess of a-nitrophthalic acid they were placed in the shaking appa-
ratus. The amount of acid dissolved was determined by titration and
from these figures the content in S-acid was calculated by making
use of a table which had been constructed previously and in which
was indicated which #-nitroacid contents correspond with a definite
titre of a solution so obtained. As the mean of four very concor-
dant observations it was found that in the nitration of phthalic acid
with absolute nitric acid at 30° is formed:

49.5 °/, a- and 50.5 °/, @-nitrophthalic acid.
The quantitatwe nitration of isophthalhe -acid was done in the
same manner as that of phthalic acid; here also, a few weeks were
required for the complete nitration at 30°. The contamination with
yellow impurities could again be quite neglected as a but very faintly
coloured nitration product was obtained. This nitration product so
obtained contains the anhydrous symmetrical nitroisophthalic acid, so
that in the solubility determinations by which its composition was deter-
mined the hydrated acid had to be employed as the anhydrous acid
takes up water but very slowly and has a greater solubility. The

( 290 )

determination of the total amount of byproduct showed that this
had formed to the extent of 3.1 °/, only. The qualitative investigation
had shown already that this contains the asymmetric acid, and that
it consists of this solely was proved in the following manner. If the
3.1 °/, found were indeed simply asymmetric acid, a solution,
obtained by shaking 100 grams of water with excess of symmetric
and vicinal acid -+ 1 gram of nitration product (containing 0.031
gram of asymmetric acid), ought to have the same titre as a solution
obtained by shaking 100 grams of water with excess of both acids
+ 0.031 gram of asymmetric acid. If on the other hand the nitration
product also contained vicinal acid, therefore less than 0.031 gram
of asymmetric acid, the titre ought to have been found less. This
however, was not the case, which shows that the asymmetric acid
is the sole byproduct. The result, therefore, is that in the nitration
of isophthalic acid with absolute nitric acid at 30° there is formed :

96.9°/, of symmetric and 3.1°/, of asymmetric nitroisophtalic acid.

If we compare the above results with that of the nitration of
benzoic acid where (at 30°) is formed 22.3°/, ortho-, 76,5 °/, meta-

and 1.2°/, paranitrobenzoic acid the following is noticed.
COgII
/1\ COH
As in phthalic acid 5 al the positions 3 and 6 are meta in

il
regard to the one carboxyl and ortho in regard to the other and
the positions 4 and 5 are also meta in regard to the one carboxyl
but para in regard to the other it might be expected from my
theories that the a-acid (the vicinal) is the main product and
the f-acid the byproduct, because in the latter the nitro-group
must be directed by one of the carboxyles towards para and
because p-nitrobenzoic acid is formed only in very small quantity
in the nitration of benzoic acid. As regards the isophthalie acid

COsH
URS it might be expected that the chief product will be sym-
; seo. Metric acid but that there will also be byproducts (4, 3, 2)
Noy and (1, 3,4) the first in the largest quantity, although it
should be remembered that a nitro-group seems to meet with great
resistance if it must take a position between two other groups.

As regards the nitration of isophthalic acid the result of the above
investigation is fairly satisfactory, although the total absence of the
vicinai nitroisophthalie acid is somewhat remarkable. In the case of
phthalic acid this is true in a less degree as about equal quantities
are formed of the two possible isomers.

In his dissertation, Dr. Huistnca has now endeavoured to calculate,

( 291 )

more accurately than before, from the relative proportion in which
the isomers C,H,AC and C,H,BC are formed by the introduction of
C in C,H,A or C,H,B, in what proportion the isomers C,H,ABC
are formed by the introduction of C in C,H,AB. He observes first
of all that in a substance C,H,A there are two ortho and two meta
positions against one para position so that if the relation of the
isomers is as C,H,AC p:q:r (ortho, meta, para) this relation for
each of the ortho and meta positions and for the para position will
be */,p:"/,qir

He further gives the preference to an addition of these figures of
proportion instead of a multiplication, which had been used by me up
to the present in the prediction of these isomers. He prefers the
addition because he considers the figures of proportion to be proportional
to the directing forces which are exercised by the groups A and B
on the other positions of the core and that the cooperation of such
directing forces on one H-atom should be represented by a sum.
But only the proportion of those directing forces are known and
not their absolute value; the foree which, in the nitration of
nitrobenzene, pushes the NO,-group towards the m-position may be
of quite a different order than the force which in the {nitration
of benzoic acid directs the same group towards the m-position.
Therefore the figures. which represent the directing forces (or are
proportionate to the same) of two different groups cannot always
be simply added together; this then will be permissible only when
the two substituents present are equal.

As an example of his method of calculation the following may be
mentioned. As in the nitration of bromobenzene 37.6°/, ortho, 62.1°/,
para and 0.3°/, of meta nitrobromobenzene is formed, the substitution
in the different positions of the benzene core takes place in the

Br
18.8 J \188
proportion | |; for the proportion in which the isomers are
O15 0.15

62.1
formed in the nitration of o-dibromobenzene the calculation gives

Br
18.8 + oe
|

621 O.15\ 7 18.8 + 0.15
62.1 +.0.15

0-dibromobenzene and 23.3°/, vicinal whilst the experiment gave
81.3 °/, asymmetric and 18.3° , vicinal.

It cannot be denied that in a number of cases this method of
calculation gives figures which approach to the experimental ones a

or 62.25: 18.95; or 76.7 °/, asymmetric nitro-

( 292 )

good deal more than those obtained formerly when the undivided
figures were simply multiplied. But on the other hand there are
other cases, particularly those in which a metasubstituted substance
is nitrated, where this calculation does not agree with the experiment
by a long way. If we take into account the figures of proportion for the
single positions we obtain as a rule a much better approach to
the figures observed by means of the products than with the sums,
even in the case where the two substituents present are unequal, when
Huisinea’s method of calculation cannot be applied. The proof there-
of is laid down in the subjoined table which gives the figures of
proportion in which the isomeric nitroderiva tives are formed from
the substances at the top of the columns, with the figures obtained
from both the sums and the products.

sum, 18: 82 15:85 |93.3:76.7| 19:81 | 55.6:44.4| *88 :62
*totalquantity byproduct.

CO, H: Br ortho

o, H: Cl ortho|\CO, H: Cl meta CO, H: Br meta

16.0:84.0 19.7:80.3
17.7:82.3 23.3 :76.7

A fuller account of this investigation will appear in the Recueil.
Amsterdam, org. lab. Univ. 1906.

11.4:88.6
23.3: 76.7

8.7: 91.3
17.7:82.3

found

product

Astronomy. — “The relation between the spectra and the colours
of the stars.’ By Dr. A. Pannexorx. (Communicated by
Prof. H. G. vAN pg SANDE BAKHUYZEN).

(Communicated in the meeting of September 29, 1906).

The close relation between a star’s colour and its spectrum has
long been known. The stars of the 1st, 24 and 34 types are usually
called the white, the yellow and the red stars, although accurately
spoken the colour of the so-called yellow stars is a very whitish
unsaturated yellow colour and that of the so-called red stars is deep
yellow mixed with very little red. In a paper read at Dusseldorf ?) in

*) Die ‘Farben der Gestirne. Mittheilungen der V. A. P. Jahrg. 10. S. 117.

( 293 )

1900 we showed that in the different glowing conditions the colours
must succeed each other in this order. If for a given high tempera-
ture we accept the colour to be white, we find that with decrease
of temperature the colours in the triangular diagram of colours make
a curve which from white first goes directly to yellow of 2587 but
which, as the colour becomes deeper, bends towards the red and
‘corresponds to light of greater wavelength. With increase of tem-
perature, on the contrary, the line of colour runs from white to
the opposite side, to the blue of 2 466.

Because the colours which are produced by white light after having
been subjected to different degrees of atmospheric absorption, also
follow about this same line, we may expect that the colours of
the self-luminous celestial bodies will in general lie on this line
or near it; they are determined on this line by one coordinate, one
number. This renders it comprehensible why on the one hand the
designation by means of letters and words, or the measurement with
ZOLLNER’S colorimeter, which produces quite different colours, has
given so few satisfactory results, and on the other hand why the scale
of Scummpt, who designates the colours by one series of figures,
where 0 is white, 4 yellow, and 10 red has proved to be the best to
work with. After this method has been drawn up the best and most
complete list of stellar colours, published in 1900 by H. Ostuorr at
Cologne, in the A. N. Bd. 153 (Nr. 8657—58). This list in which
the colours of all stars to the 5 magnitude are given, down to a
tenth class of colour, and which was the fruit of systematic estimates
during 14 years, enables us to accurately determine the relation
between spectrum and colour.

In a former paper’) we remarked that we did not know where
in the continuous series of spectra of the Oriontype and the first
type we have to look for the highest temperature or at any rate the
greatest luminosity. We may assume that it will be there where the
colour is whitest; the spectral-photometric measurements, to which
we have alluded in that paper, are still wanting, but for this purpose
we can also avail ourselves advantageously of estimates of colour ;
this has been the reason for the investigation of which the results
follow here.

In this case where we required a specification of the spectra, as
detailed as possible, to serve as an argument for the colour, we
have naturally used again Mavry’s classes. In order, however, to
determine a mean colour for each class we must correct the colours

1) The luminosity of stars of different types of spectrum. Proceedings of June
30 1906 p. 134.

(294 )

observed for two modifying influences, viz. the influence of the
brightness and that of the altitude above the horizon. Quantitatively
nothing is known about the values of these influences; experiments
of OstHorr himself to determine the influence of the brightness
have as yet yielded few results. Therefore we must derive them
here from the material of stellar colours themselves, which serve
for our investigation; this may be done in the very probable assump-
tion that the real colour within each spectral class is an almost
constant value and is independent of brightness.

§ 2. The stars of Osrnorr’s list which occur in the spectral cata-
logue of Maury, were arranged according to their classes and then
(excluding those which are marked c¢, ac, C, P or L, as was always
done in this investigation) always taking together some classes, we
classified them according to their brightness and combined their
magnitudes and classes to mean values. These mean values must
show the influence of the brightness on the colour; they are given
in the following tables:

Classe IJI— VI Classe VII—VIII Classe [IX—XII
Mg. Col. Mg. Col. Mg. Col.
1.78 1.46 (5) Olay Hee (8) 10; 39 3
2.80 2.27 (6) 2.4 1.83 (6) 2.69 2.97 (9)
3.35 1.96 (5) 3.17 2.59 (7) 3.18 38.06 (8)
3.70 2.86 (7) 3.95 2.57 (6) 3.65 3.73 (10)
4.00 9.47 (8) 8.82 2.95 (6) 3.85 3.40 (8)
4.15 2.91 (7) 4.00 2.86 (5) 4.10 8.69 (9)
4.50 2.60 (9) 4.10 2.60 (7) 4.29 4.17 (7)
4.95 2.42. (11) 4.20 2.50 (5) 4.65 3.79 (8)
4.36 2.96 (5) 5.10 3.34 (9)
4.62 2.72 (A) ?
4.96 2.66 (5)
Classe XIJI—XIV Classe XV Classe XVI—XVIII
Mg. Col. Mg. Col. Mo” «Col:
0.2 3.4 (1) 0.7, 4.0 (2) 0.95 6.45 (2)
3.07 4.71 (7) 2.12 5.50 (6) 250 6.40 (6)
3.54 4.61 (7) 2.92 5.66 (9) $.22 6.65 (6)
3.98 4.72 (9) 3.37 5.74 (9) 3.72 6.65 (4)
424 4.88 (8) 3.55 5.46 (9) 4.15 6.75 (6)
4.84 4.88 (8) 3.75 5.71 (8) 4.63 7.07 (7)
3.90 5.55 (10) 4.88 7.22 (9)
4.00 5.70 (7) 5.28 7.228)
4.14 5.85 (11)
4.45 6.08 (6)

4.87 6.48 (7)

( 295 )

In all these series we clearly see an increasing deepening of colour
with decreasing brightness. We have tried to represent the colour as
a linear function of the magnitude; and by a graphical method
we found:

Cl. WI—VI c= 2.15 + 0.35 (m — 3)
VII—VIII 2.27 + 0.36 e
IX—XIl 3.17 + 0.39 2
,, XITI—XIV 4.45 + 0.42 3
SA's 5.47 + 0.39 -
XVI—XVIII_ 6.60 + 0.20 S

Thus we find about the same coefficient in all groups except in
the last. The value of the coefficients is chiefly determined by the
difference between the observed colours of the very bright stars of
the 1st magnitude and of the greater number of those of the 34 and
4th magnitudes. In order to make the coefficient of the last group
agree with the others, it is necessary to assume for the apparent
colour of @ Tauri and a Orionis 5.6 instead of the real estimates
6,4 and 6,5. It does not do, however, to assume such a large error
for these bright and often observed stars; therefore we must for
the present accept the discordant coefficient of the red stars as real,
although it is difficult at the present to account for it.

If now we combine the results of the five first groups by arranging
the deviation of each observed value of c from the constant for the
group (the value of c form = 38), according to brightness and deriving
thence mean values we find:

m c—c, C, C, O—C, O—C,
a 1.03 116 OUt +-.07 Ba,
£6 - 0.63 —0.54 —0.47 —09 ae |
2.91 -+0.02 +0.04 0.02 =o 4.04
3.73 +£0.32 =O 3f 20.27 + 01 + 05
412 +0.48 +040 10.39 + 08 + 09
4.73 +£0.50 +0.52 +0.60 — 02 50

A linear relation c = c, + 0,34 (m — 3) yields the computed values
given under C, and the differences obs.-comp. 0 — C,. These are
distributed systematically and show the existence of a non-linear
relation. A curve, which represents as well as possible the mean
values, gives the computed values C, and the differences, obs.-comp.
O— C,. For a greater brightness the curve gives a greater variation of
the colour with the luminosity and for fainter stars a smaller one. In
all the six groups, except the fifth and the sixth, we remark that

( 296 )

the last values, which hold for the faintest magnitudes, show a
decrease in the colour figures with regard to the preceding ones.

This phenomenon may be accounted for by the existence of the
colourless perception of faint sources of light. In faint stars we do
not see any colour at all; there the perception of colour disappears
almost .entirely and there remains only a colourless (i. e. whitish)
impression of light. With stars which approach this limit, the
impression of colour will be mixed up to a high degree with the
colourless impression, and therefore they appear paler and will be
indicated by a lower figure. As for the redder stars this colourless
impression is relatively much weaker, the paleness of colour for
these stars occurs only with a much less degree of brightness; in
this manner we explain why the 5 and 6 groups do not show
this decrease. Whether in these cases the phenomenon occurs with
fainter stars cannot be decided because Maury’s spectral catalogue
does not contain fainter stars.

For the practical purpose of reducing the observed colours to one
brightness it is about the same which of the two relations is
adopted, as long as we keep within certain limits of brightness, for
instance between the magnitudes 1 and 5. To facilitate the reduction
we have made use of the linear formula given above for the 5 first
groups (down to class XV included) while for the redder classes
0,20 has been adopted as the coefficient of brightness.

To explain the long known phenomenon that the colour deepens
with decreasing brightness as is shown in the tables on p.
Hetmnottz in his Physiologische Optik has given a theory called
“Theorie der kiirzesten Linien im Farbensystem”. In the diagram
of colours in space, where each impression of light is represented
by a point of which the 3 coordinates represent the quantities of the
elemental colours, red, green, blue, the lines of equal colours are
not straight radii through the origin, but curved lines which with
increasing distance from the origin bend more and more towards
the axes and so diverge more and more from one radius which is
straight and represents the ‘“Principalfarbe”’. Hence in the triangle
of colours the points of equal colour diverge the more from the
principal colour and run in curved lines towards the sides and the
vertices as the triangle of colours is removed farther from the origin,
and thus represents a greater brightness. HELMHOLTZ gives as principal
colour a certain “yellow-white’ to which with extremely great
intensity all colours seem to approach. Therefore colours which
lie on the blue side of this principal colour must become bluer
by fading. |

( 297 )

This does notagree with what we have found here, in the supposition
that Hetmnortz’s “yellow-white” is also yellow-white in our scale, i. e.
is also represented by a positive number in Scumipt’s scale. We
also find here with the whitest stars that when they become fainter
the colour becomes more yellow to just the same degree as with
the yellower stars. Now the expression “yellow-white” is vague, but
if we consider that what is called white in the scale of Scumipr is
whiter, that is to say bluer than the light of Sirius, and that the
solar light, the standard for white for ordinary optical considerations,
if weakened to the brightness of a star, in the scale of Scumipr
would be called 3 a 4 (Capella 3, 4), then the principal colour,
if Hetmno.tz’s theory is true, instead of being yellow-white would
still lie on the blue side of the Sirius light.

§ 3. After the colours had thus been reduced to the brightness
3,0, they had still to be freed from the influence of the atmosphere,
which makes them redder. This cannot be done with the desired
accuracy, because neither time nor altitude are given along with the
observations. The influence at high and mean altitudes is probably
very small, and the observer is sure to have taken care that most
of the stars were observed at a proper altitude (for instance between
30° and 60°). Therefore this correction is only practically important
for the few southern stars which always remain near the horizon ;
in these cases it will be possible to represent the variation of colour
by a correction depending on the declination. Instead of the declina-
tion of the star we have taken the declination of the B. D.-zone
which Ostuorr has added to his catalogue.

For each spectral class we have determined mean colour-values
for all stars north of the equator, and for the stars south of the
equator we have formed the deviations from these class-means which
then were arranged according to their declination and combined to
mean values for groups of stars. We have excluded, however, those
classes in which too few northern stars occurred, namely I, II and III.

The means found are:

Zone Deviation n. Curve Zone Deviation n. Curve
Oe O56.) oh 20h. | 090. 0.14 5 ++ 0.26
ee ee) ee 0G 10.2" , 410.35. 4. -- 32
eee Gee 09-159. 1038 6 +. 57
=50 4050 5 + 12 |—150 4117 6 + 79
eGo eae ty) 189 = 6|=6 9.98. 6, 1.32
Seo =0.05 5 4-22 |
20

Proceedings Royal Acad. Amsterdam, Vol. IX.

( 298 )

Through these values we have drawn a curve which from the
equator towards the southern declinations ascends steeper and steeper
and which gives the values of the last column. According to this
curve we have applied the following corrections, for

zone 1° 2°-5° 6°-8° 9°-10° 11° 12° 13° 14° 15° 16° 17° 18° South
neg. corr.0 0,1 0,2° 0,3 0O,f 0,5 0,6 0,7 0,8 0,9 1,1 1,3

We may assume that by these corrections the variation of colour
due to atmospheric absorption has at least for the greater part been
eliminated.

§ 4. After the two corrections (§ 2 and § 3) had been applied we
could determine for all spectral classes the mean values of the colour;
they are given in the following table. Class XV was again subdivided
into 3 classes according as the spectrum agreed with « Bootis (A)
or with @ Cassiopeiae (C’) or was not accompanied by any such
remark; the result shows indeed that here class C’ is considerably
redder than class A while the 4b’s lie between the two.

Class _—- Colour Number Class Colour Number

I 2.47 6 XII 3.68 17
Il 2.36 10 XU 4.12 13
IIL 2.30 9 XIV 4.45 12
IV 1.94 14 XIV 5.09 9
LYE 1.62 10 XVA 5.18 18
a’ 7a Mi 9 XVB 5.35 26
Vi 2.16 10 XVC 5 55 31
Vil 2.27 23 XV 6.34 5
Vit 2.37 o4 XVI 6.47 ia
IX 2.64 20 AINE 6.80 15
xX 3.11 14 XVUI 6.74 15
XI 3.40 {) XIX 6.67 6
XI 3.41 4

The deviations of the separate values from these mean values
give, as a measure for the accuracy of the results, for the mean
error of a colour-number, / 0,20 = 0,45; the real accuracy will be
ereater, however, and the mean error smaller because in these values
are also included the errors of the adopted corrections for brightness
and declination, the errors which may have been made by Maury while
classifying each star in a definite class, and also the real deviations
of the single stars from their class-means.

( 299 )

‘With 9 stars (out of 355) the deviation exceeds a unit of colour:
the reduced colours are here:

B Can. maj. I11 1,2 « Hydrae XIII 5,2 y Persei XVB 6,8
o, Cygni IX 1,4 w Persei pA ares ws: 11 Urs.min. XVB 6,6
JS Delphini IX 3,8 0, Cygni XIV 6,5 5 Orionis XVII 7,9

In this investigation we have, as it was said before, excluded the
c- and ac-stars, the LZ (bright lines), the P (peculiar spectra) and
the C' (composed spectra). It is important to examine the ¢ and the
ac-stars among them more closely in order to see whether they show
a distinct difference in colour from the a-stars of the same class-
number. In the mean 11 ac-stars give a deviation of + 0,1 (from
+ 0,5 to —0,3), and 12 c-stars + 0,7; so these last ones are a little
redder than the a-stars. Here, however, the great individual deviations
are very striking; the extreme values are:

o Cassiop XIII + 2,5; x, Orionis HI + 1,8 ;4H Camelop VJ + 2,0;
3H Camelop V/ + 1,5; 4 Leonis VIT — 0,3 ; 8 Orionis Via.

The differences are very great, but no regularity can be detected.

§ 5. The results found solve a problem which in my former paper
remained unsolved, namely where in the continuous series of spectral
classes shall we have to look for the maximum of radiating power.
The colour-numbers show very distinctly a fall in the first classes, a
minimum between the 4 and the 5 class and then a continual
rise. The stars which in order of evolution directly follow on
y Orionis (u Aurigae, uw Hydrae, uw Herculis) have the whitest colours ;
both the earlier and the later stages of evolution are yellower ;
classes I and II agree in colour best with class VIII. Therefore, in
so far as we are entitled to derive the entire radiation from the
colour, the maximum of radiating power lies between the 4" and
the 5* class.

The mean colour-numbers for each of the groups formed before are:

Gl TSW }%2:35
IV—V_ 1.87
VI—VIIE_ 2.30
IX—XII 3.20
XHI—XIV 4.58
eV, = 5:43
XVI—XIX 6.66

20*

( 300 )

Is it possible to derive from these numbers, even though only
approximately, values for the radiating power per unit of surface?
The two influences mentioned above which determine the colours of
the self-luminous celestial bodies may be subjected to a calculation,
if we disregard the specific properties of the composing substances
and treat them as abstract theoretical cases. In other words we can
investigate the radiation of a perfectly black body and in the absorp-
tion neglect the selective absorption in lines and bands in order only
to examine the general absorption. As a first approximation this may
be deemed sufficient.

In this calculation we have made use of the measurements of
A. Konic on the relative quantities of the elemental colours red,
ereen and blue as functions of the wavelength in white sunlight. Ir
for an other source of light we know the relation of the brightness
with regard to the former source as a function of the wavelength, we
ean calculate the quantities of the red, green and blue in this second
source of light. If we call the numbers of Konic & (4), G (4), B (a),
which are chosen so that

f®e da = 1000 uk G (2) da = 1000 Ap B (a) da = 1000

and if #(4) represents the brightness of another source of light, then
ib f(a) R(a) da [ro G(a)da and [7@ B(a) da

represent the quantities of R, G, and B occurring in this light. As
the impression of brightness of a source of light is almost proportional
to the quantity of red, this calculation gives at the same time a
measure for the optical brightness.

The radiation of a black body may be represented by :

—

5 ies ada
where 7’ is the absolute temperature and @ and c constants. For two
sources of light of different temperatures the relation of the imten-
sities is:

1 1

if b= e(F. -z) and 5'= 0.43 6. As unit for 4 we adopt 0,001 mm;
0

T, is supposed to be given, then 6’ is a function of the variable

temperature 7’ only and may be called the degree of glowing with

regard to the glowing of a body at a temperature 7. If we adopt

( 301 )
for 6' different values (¢ = 15000 about)'), we can calculate for
each of them the brightness and colour of the light, as well as the
temperature 7. We then find for the degrees of glowing +1, 0
and —1

’=+1 69200 R + 68100 G + 175800 B

0 1000 R + 1000 G;]+ £41000 B

— 1 17,7 kR-+ 15,7 G + 6,3 B

If we represent the colour contained in a total quantity of light
of 1000 by the quantities R, G, B and the brightness by magnitudes,
we shall find for

Bot L Col = 221K 218G4562R Br=+44,6 My.

s=—1 Col. = 445 R + 396G + 160 B Br. = — 4,4 Mg.

Thus the first colour, may be described as a mixture of 654 white
and 347 of a blue consisting of 3 Land 344 4, hence corresponding
in tint to 4 466; the second colour is a mixture of 480 white and
©21 of a yellow consisting of 285 & and 236 G, hence corresponding
to the wavelength 4 587. A degree of glowing 6’ = — 2, corre-
sponding in colour almost with the light of petroleum, involves a
decrease in brightness of 8,6 magnitudes.

For the calculation of the atmospheric absorption we have assumed
that the general absorption in a gas is inversely proportional to
the fourth power of the wavelength. For a layer of gas adopted
arbitrarily, which after a comparison with MUiLuur’s spectral-photo-
metric measurements appeared to correspond to 1,05 atmosphere,
we have calculated /(4) and thence found for the remaining quantity
of light, the initial quantity being 1000 & + 1000 G+ 1000 B:

783 R+ 771 G4 571 B,
or reduced to 1000 as the sum,

368 R + 363 G + 269 B;
the brightness is then 0,783 of the original brightness or is diminished
by 0,27 magn.

The colouring due to the absorption by 1.05 atmospheres is almost
equal to that brought about by a diminishing of the degree of glowing
of '/,. For the latter yields

257 R+ 248G +1184 B
henee when reduced to a sum of 1000
372 R+ 361G+ 267 B

1) In the paper read at Dusseldorf (see note p. 292) wrong temperatures are
given because the difference between b and b' was overlooked. The temperatures
16000°, 7500°, 5000’, 3750°, 3000°C do not differ inter se 1, but only 0.43
in degree of glowing.

( 302 )

which is nearly identical with the value above. Here, however, the
brightness is diminished to 0.257 of the original, hence by 1.48
magnitude.

Therefore it appears here that these two different causes produce
similar colours, but that they correspond to an entirely different decrease
of brightness. When comparing the two we may say that atmospheric
absorption is more apt to redden, a decrease of temperature more
apt to fade the light. Therefore it is impossible to derive the
radiating power from the colour only, as we do not know to what
degree each of the two influences, temperature and absorption, is at
work in the different spectral classes. Perhaps that one day accurate
speciral-photometric measurements will enable us to separate the
two influences, for they give a different distribution of intensity over
the spectrum. For the log. of the brightness of different 2 with regard
to 2500 we find

A= 650 = 600550500 © 450 - 400
with abs. 1.05 atm. + 0.114 + 0.083 + 0.051 0.000 —0 084 —0.231
with glowing — */, + 0.154 + 0.111 + 0.061 0.000 —0.074 —.0166

For the latter the decrease in intensity from the red to the violet
is more regular, for the former the decrease is slower for the greater
and more rapid for the smaller wave-lengths.

These calculations show that it is not strictly true that, as has
been said in the preceding paper, a redder colour must necessarily
involve a smaller radiating power. Where we have two influences
which in different ways bear on the colour and the brightness, the
possibility exists that-a redder colour may be accompanied by a
greater radiating power, namely when one source of light has a
much higher temperature and at the same time a greater atmospheric
absorption than the other. An increase of the degree of glowing of
+ */, combined with an absorption of 2 atmospheres gives such a
case according to the figures given above.

Herein we have therefore a new possibility to account for the
peculiarities found in the X stars, namely by assuming that, as
compared with the G stars, they have a much higher temperature,
which causes a stronger radiation, and which by very strong atmos-
pheric absorption, is only little faded but greatly reddened. We
must add, however, that this explanation seems little probable to us
as the band-absorption, which begins at the A-stars and which is
characteristic for the J stars (the 34 type) indicates a lower tem-
perature,

( 303 )

Chemistry. — “Action of Potassium hypochlorite on Cinnamide’’.
By Dr. R. A. Weerman (Communicated by Prof. Hoocrwerrr).

(Communicated in the meeting of September 29, 1906).

From the experiments of Bavcke') on propiolamide and those of
FREUNDIER *), VAN Lince*) and Jrrrreys ‘*) on cinnamide it appears
that in the case of these unsaturated acids, the HormMann reaction
to prepare an amine from an amide by means of a halogen and an
alkali does not succeed.

As to the non success we may form two hypotheses: first of all
that the double bond *) prevents the intramolecular rearrangement of
atoms which must be assumed in the HOFMANN reaction, and secondly
that the amine supposed to be formed, in this case C,H,CH = CH NH,,
suffers decomposition under the said circumstances. °).

The first, however, is not the case as from cinnamide may be
prepared the urea derivative:

C,H,C# = CH _NH

C,H,Cu = Ca —Co—Nu
where consequently one-half of the amide has undergone the trans-
formation.

This being a case of an unsaturated amide, it is necessary to make
use of the modification proposed by Hoocewrrrr and van Dorp and
not to work with free halogen. Further the hypochlorite solution
must not contain any free alkali; on account of the insolubility of
cinnamide and the consequent inertness, an alcoholic solution is
employed.

Although at first sight it appears strange that in alcoholic solution
the urea derivative is formed and not the urethane, this may be
explained by the experiments of Srimexirz and Earte‘), which show
that isocyanates react very readily with halogen-amides °*).

1) Rec. 15, 123.

2) Butt [3] 17, 420.

8) Dissertation vAN Linge, Bazel 1896.

4) Am. Chem. Journ. 22, 43.

5) On account of the great analogy existing between the Lossen transformation
of hydroxamic acids and the Horsann reaction, this first supposition was not very
probable, as Tuiere had prepared from the acylated cinnamohydroxamic acid the
urethane C,H;CH =CH—N4¥—CO,C,H;. A second indication, though less conclusive,
in the more distant analogy between the Beckmann rearrangement and the
Hormayn reaction was the formation of isochinolin from the oxime of cinnamaldehyde.
(Ber. 27, 1954).

6) Tutete, Ann. 309. 197.

7) Am. Chem. Journ. 30, 412. C 1904, I, 239.

8) This is the reason why, in the preparation of urethanes according to JerFReys,
the sodium ethoxide should be added all at once.

( 304 )

In order to prepare the urea derivative, the cinnamide is dissolved
in eight times its weight of 96 pCt. alcohol, and when cooled to the
temperature of the room the hypochlorite solution, prepared according
to GRAEBE*), is slowly dropped in, the free alkali being neutralised
with 2N hydrochloric acid immediately. before .use. For every 2
mols. of amide, 1 mol. of potassium hypochlorite should be added.
The liquid gets warm, and very soon a crystalline mass composed
of very slender needles is deposited. After a few hours the mass is
collected at the pump; this does not go very readily on account of
the fine state of division. The yellowish mass is treated with hot
alcohol and then washed with water. A fairly pure urea derivative
is thus obtained (m. p. about 218). By recrystallisation once or twice
from glacial acetic acid it is obtained pure in needles (m. p. 225—226).

0,1733 grm. yielded 0,0894 grm. H,O and 0,4682 grm. CO,
0.1654 _,, 53 0.1863 4, 2g) Ae Se
0,1654 _,, » 13,9 CC.N at 194° and 765 m.M.
Found 73,68 5,78
pct. C pCt. H 9,70 pCt. N
73,66 5,85 |
Theory C,,H,,N,O,: 73,95 pCt. C 5,51 pCt. H 9,59 pCt. N
The compound is insoluble at a low temperature in water, ligroin,
alcohol, methyl alcohol, ether, carbon disulphide and benzene; at the
boiling temperature slightly soluble in alcohol and benzene and freely
so in glacial acetic acid, chloroform and acetone. It is insoluble in
alkalis or acids.
Chemical Laboratory, Technical High School, Delft.

Astronomy. “Mutual occultations and eclipses of the satellites of
Jupiter in 1908. By Prof. J. A. C. OvupEmans.

(Communicated in the meeting of September 29, 1906).

N.B. In the present communication the four satellites of Jupiter, known
since 1608, have been denoted by I, II, HI and IV in accordance
with their mean distances from the planet. The further letters and
f indicate whether the satellite is near or far, i.e. whether it is in
that half of the orbit which is nearest to or furthest from the Earth.
The jovicentric longitudes as well as the geocentric amplitudes are
counted in “signs” and “degrees”’, the latter beginning from the superior

1) Ber. 35, 2753.

( 305 )

geocentric conjunction. Eastern elongation, denoted by ¢.e, has an
amplitude of 3s, western elongation, w.e, one of 9s .

Not to interrupt the text unnecessarily, all particulars have found a
place at the end of the paper.

FIRST PART. OCCULTATIONS.

In the numbers 3846 and 3857 of the Astronomische Nachrichten
we find two communications relative to observations of the occul-
tation of one satellite of Jupiter by another. The first (1) is by
Mr. Pu. Favura at Landstuhl, dated 8 December 1902, with post-
scripts of 29 December 1902 and 14 January 1903. The other (2)
by Mr. A. A. Nuzanp at Utrecht, dated 27 Februar y 1908.

Fautu notes in addition that Houzrav, in his Vademecum, p. 666
mentions a couple of similar observations (3), and further that STANLEY
WILLIAMS, on the 27%" March 1885 at 125 20™, saw the third satellite
pass the first in such a way that the two satellites combined had a
pear-shaped appearance. (4)

The -satellites of Jupiter move in orbits but little inclined to the
plane of Jupiter’s equator. Laptace assumed a fixed plane for each
satellite; the plane of the satellite’s orbit bas a constant inclination
on this fixed plane, whereas the line of intersection, the line of
the nodes, has a slow retrograde motion. The inclinations of the fixed
planes on the plane of Jupiter’s equator amount only to a few
minutes; their intersection with the plane of Jupiter’s orbit is identical
with the line of the nodes of the equator. The value generally
adopted for the inclination of the latter plane on the orbit of Jupiter
is 3°4’, whereas the longitude of the ascending node, which therefore
is also that of the fixed planes, is at present about 315}°.

In order to be able to assign the time at which, as seen from the
Karth, an occultation of one satellite by another is possible, it is necessary
to know the longitude of the ascending node and the inclination of
the mean fixed plane on the orbit of the Earth. At the time that
the mean fixed plane, prolonged, passes through the Earth, occultations
of one satellite by another may be observed. As Jupiter completes
a revolution around the sun in nearly 12 years, these times will
succeed each other after periods of six years. Jupiter will pass
alternately through the ascending and the descending node of the plane
which passes through the centre of the sun parallel to the mean
fixed plane.

It follows that, as occultations of one satellite by another have
been observed in 1902, we must expect that these phenomena will
be again visible in 1908 (5).

( 306 )

To facilitate these observations I thought it desirable to calculate
in advance the conjunctions of any two satellites for the most favourable
part of 1908.

We have to consider that while formerly the orbits of the
satellites were determined by repeatedly measuring the distances
and their angles of position relatively to the planet, this method is
now replaced by the measurement of the distances and the angles
of position of the satellites relative to each other (especially with
the heliometer) (6). For observations during a moderate interval the
periodic times of the satellites may be assumed to be accurately
known. Admitting this, if, leaving out of consideration KEpPLErs
third law, we introduce the major axis of each satellite as an unknown
quantity, the total number of such unknowns will be six for each
orbit at a determined time. If, as was done by BgssEL at Kéningsberg
in 1834—39, and by Scuur at Gottingen in 1874—1880, the distance
and the angle of position between the planet and the satellite are
measured, we get two equations with six unknown quantities. If
however we measure the distance and the angle of position of
two satellites relative to each other, the number of unknown
quantities in these equations is doubled and thus becomes 12. If
finally all the combinations two by two, are observed, as was done
by Git and Finnay at the Observatory of the Cape, we get a great
number of equations with a total of 24 unknown quantities. These
equations must then be solved by the method of least squares.
This number becomes 29 if we add the masses of the satellites, (only
to be found by the perturbations caused by one satellite in the
motion of the others,) and the compression of Jupiter (7), given by
the retrogradation of the lines of the Nodes on the fixed planes.

Now the observation of an occultation, even of a conjunction with-
out an occultation, can be made by everybody possessing a telescope
of sufficient power. Such an observation also furnishes two equations
between the unknown quantities, at least if, for a non central occul-
tation or a simple conjunction, the difference in latitude is measured
at the filar micrometer. This consideration engaged me to compute
in advance the time of these conjunctions for the most favourable
part of 1908. If by experience we find that this preliminary work
leads to valuable results, it might be worth while to continue it for
some future period, for instance for 1914.

For the moment at which the mean fixed plane passes through
the centre of the Earth, I find, 1908 July 8, 195,6 Mean Time at
Greenwich, (5).

This date, it is to be regretted, is very unfavourable. For on that

( 307 )

day Jupiter culminates at Greenwich at 210" M. T., its declination
being 16°48’°5 North, whereas the Sun’s declination is 22°30’ North.
From these data I find for the 8% of July, for Utrecht, duly making
allowance for refraction : |
Setting of the upper limb of the sun at 8°20" mean time,
hs ,, Jupiter aoe O°

So there is but a poor chance for an observation of the computed
occultation at Utrecht. For southern observatories it is somewhat
better. At the Cape for instance, we have;

Sunset at 55 5™ mean time,
Setting of Jupiter ,, 7 25 ce ee

' We thus find that on July 8, 1908, at Utrecht, the setting of the
sun precedes that of Jupiter by 1°24™5; at the Cape by 220”.

We have computed all the conjunctions of the satellites of Jupiter
which will occur between 31 May and 20 July 1908. In what follows
a short account is given of the way which led to our results.

In the Nautical Almanac are given the Geocentric Superior Conjunc-
tions; in the Almanac of 1908 they will be found on pp. 504, 505.

To begin with, a separate drawing was made of the four orbits,
which were supposed to be circular, for each interval of two periods
of I (about 85"). On these orbits we plotted the positions of the
satellites for each second hour, making use of divided pasteboard arcs.
The number of hours elapsed since the moment chosen as a starting-
point were noted for each position. The equation of the centre ete.
was neglected.

The scale of this drawing gave 4” to 1mm. The radii, of the
orbits therefore were: for I 27°9 mm.; for Il 4445 mm.; for III
70°9 mm. and for IV 124:7 mm.

The direction from the Zero of I to the common centre of all
the circles showed the direction towards the Earth. Knowing this, we
could easily find for each of the six possible combinations of two of
the satellites, those equal hour numbers, the connecting line of which
is parallel to this direction.

These connecting lines show the approximate times at which, as seen
from the Earth, one of the satellites is in conjunction with another.
The want of parallelism of the real lines joining the Earth with the
satellites, in different parts of their orbits, may safely be disregarded.
The plate annexed to this paper represents, reduced to half the scale,
the drawing for the period of 85 hours, following 12 July 1908,
1152™-3 M. T. Greenwich.

The dotted lines indicate the lines connecting the equal numbers,

( 308 )

Each of them represents a conjunction of two satellites. The corre-
sponding hours read off from the figure are :

6"2 : IVy occulted by III,
21°32 BY; - tees
BoC IV; os ” IL,
ao OU): | ae Pee ©
e725. Tiel. es
TO 2c Tile ees Oe t=

They were added to the instant which must be regarded as the
startingpoint for this figure. The instants of the conjunctions were next
converted into civil time of Paris by the addition of 12°9™21s.
The elongation and the latitude of both the satellites, expressed in
radii of Jupiter, were then computed by the aid of the Tables écliptiques
of Damoisrau, 2nd part. (8). In the case that the elongations did
not perfectly agree, a slight computation led to a more accurate
result for the time of conjunction (9).

In the case that the two satellites moved apparently in opposite
directions, (wbich happens if the one is in the further part of ils
orbit, the other in the nearer part), the correction to the adopted
time was mostly insignificant.

If, on the contrary, they moved the same way (which happens if
both are “far” or if both are “‘near’’, so that the one has to overtake
the other) the correction amounted sometimes to an hour or more.
In every case, in which the correction exceeded 20 minutes, the
computation was repeated with the corrected time. Further below
will be found the list of the results. From May 31 to July 19, ie.
during a period of fifty days, there occur 72 conjunctions. It is to
be regretted that at a determined place of observation but very few
of them will be visible. For only those conjunctions are visible which
occur between sunset and the setting of Jupiter. For Utrecht we have,
in mean time:

Setting of the Setting

upper limb of

of the Sun Jupiter Difference

1908 June 1 810m 11554m 3h44m
ge 8 20 °5 1119 258 °5
eee 8 24 10 44 2 20
July 1 8 24 10 9 1 45

wea | 818 9 34 116

a ae ee 8 59 051 ‘5

( 309 )

For the Cape of Good Hope:

1908 June 14 4059" gh4gm uj gm
5 odd 457-5 846-5 349
Pee 458. 8 16 3 18
July 1 5 2 7 46 2 44
Be 5 | 5 6°5 7 16 2 95
aria 5 5 18 6 47 1 34

The circumstances are thus seen to be considerably more favourable
for a southern than for a northern observatory.

Several of the occultations will not be visible because the common
elongation falls short of unity i.e. of the radius of Jupiter. This is
the case of Nos. 8, 9, 12, 18, 15, 16, 20, 23, 39 and 64. In the first
eight of these cases and in the last one the planet stands between
the two. satellites. In case No. 39 both the satellites I and IV are
covered by the planet ’).

For other conjunctions it may happen that one of the satellites is
invisible because of its being in the shadow of the planet. Such cases are :

(N°. 21), June13 9828" M. T. Grw., II eclipsed,
(N°. 31), ,, 20 1251 rata . Pala
er oletaly: 4.245 15-27 os, ays) Wy ann a
NY AGS on 5 12,19, 3 ee ee 5 ee

If the satellite which at the conjunction is nearest to the Earth
is eclipsed by the planet’s shadow, it might, as seen from our stand-
point, project itself wholly or partially as a black spot on the other
satellite. The case however has not presented itself in our computations.

Possibly the last of the conjunctions just mentioned may really
be visible; for according to the WN. Almanac, the reappearance of
IV from the shadow of the planet takes place at 1257™15s M.
T. Greenwich and the predicted eclipses of this satellite are occasionally
a few minutes in error. A few minutes later, according to the J.
Almanac at 1216™, II enters the dise of Jupiter.

1) According to the Nautical Almanac we have for this night (M. T. of Greenwich):
IV. Occultation Disappearance 10°19",
I. Occultation Disappearance 11 20,
I. Eclipse Reappearance 14 26 273,
IV. Occultation Reappearance 15 13,
IV. Eclipse Disappearance 18 5 6,
IV. Eclipse Reappearance 2252 2.

( 310 )
NO. Taps.

(1) The article of Faurs, abridged, runs thus:

— — — Ausser den in Hovzeav, Vademecum p. 666 aufgefiihrten
Beobachtungen, (vid. below Note 3), kenne ich aus neuerer Zeit nur
einen Fall: Stantey Winiiams sah am 27 Marz 1885 an einem 7 cm.
Rohre mit 102-facher Vergrésserung um 12 20™ den III Trabanten
vor dem I, wobei beide ein birnformiges Objekt bildeten.

— — — In fiinf Wochen konnte ich drei Bedeckungen verfolgen,
wobei auzunehmen ist, dass mir durch schlechte Witterung etwa 10
andere Gelegenheiten entgangen sein mégen, unter denen sicher
einige Bedeckungen vorkommen. Nach meiner Erfahrung kénnen Kon-
junctionen der Jupitermonde unter sich weit genauer beobachtet
werden als Bedeckungen durch Jupiter oder Voriibergange vor ihm.
Somit méchten die hier angegebenen Beispiele Anlass bieten, in den
spiteren Oppositionen Jupiters den durchaus nicht seltenen Bedeck-
ungen oder wenigstens Beriihrungen und sehr nahen Konjunktionen
der Trabanten unter sich mehr Aufmerksamkeit zu schenken, zumal
schon kleine Instrumente zur Wahrnehmung der Phasen einer event.
Bedeckung geniigen. Die Beobachtungen der letzten Zeit sind:

1. Oct. 7; II bedeckt I; die S. Rander beriihren sich und I ragt
im N. etwas hervor. Konj. um 9° 16™ M. E. Z. ’)

2. Oct. 23; II bedeckt III so, dass die Mitte von II nérdlich am
N. Rand von III vorbeigeht; Konjunktion um 8' 7™ 38,5,

3. Nov. 10; III bedeckt I so, dass der S. Rand von III die Mitte
von I streift (gute Luft); Konjunktion um 7’ 33™ 205.

Instrument: 178 mm., Vergrésserung 178 fach.

Landstuhl, 1902 Dez. 8.

P.S. vom 29 Dezember. Am Abend des 24 Dezember gelang
nochmals die Beobachtung einer Bedeckung, bei welcher I tiber IV
hinwegzog. Aus je fiinf vor- und nachher notierten Zeitmomenten
folgen als Mittelwerte 65 24™,25, 24,625, 24™50, 24™625 und
24™,50. Die Konjunktion fand also statt 6 24™ 305.

Der Uhrstand war um 3" mit dem Zeitsignal verglichen worden.
IV Stand ein wenig siidlicher als I, vielleicht um ein Viertel seines
Durchmessers. Die weitaus interessantere Konjunction zwischen II
und IV am 25 Dezember blieb gegenstandslos, weil IV um etwa
zwei Durchmesser voriiberging,

P.S. vom 14 Januar (1903). Heute Abend, am 14 Januar, bewegte

1) i. e. Mittlere Europiische Zeit, 1 later than Greenwich-time.

( 344.)

sich der Trabant III tiber II hinweg. Die sehr schlechte Luft liess
nur den ersten Kontakt auf etwa 6" 2™ feststellen. Um 6" 18™ mochten
sich beide Komponenten so weit getrennt haben, dass dies in einem
weniger schlechten Augenblick bemerkt wurde; um 65 32™, dem
nachsten blickweisen Auftauchen der beiden Lichtpunkte, waren diese
um etwa einen Durchmesser von einander entfernt. Die Bedeckung
war fast genau central. Pu. BP,

(2) Mr. Nistanp writes in N°. 3857 of the Astronomische Nachrichten:

— — — Am 15 Juli 1902 fand eine Konjunktion der Trabanten
II und III statt, welche ich bei guter Luft am Refraktor (Brennweite
319 cm., Oeffnung 26 cm.) mit Vergr. 248 beobachten konnte. Es
wurde III nahezu central von Il bedeckt. Einige Minuten lang blieb
eine feine schwarze Linie zwischen den beiden Scheibchen sichtbar,
welche um 14'40™11s M.Z. Utrecht verschwand and um 14520™31,
wieder erschien; die Konjunktion musz also um 14)15™21s statt-
gefunden haben. Dass diese Trennungslinie vor und nach der Kon-
junktion immer dieselbe Richtung hatte, und zwar scheinbar senk-
recht auf der Bahnebene der Trabanten stand, mag als Beweis dafiir
gelten, dass der Voriibergang wirklich nahezu central gewesen ist.
Dann lasst sich aber aus dieser centralen Passage die Summe der
Durchmesser der Monde II und Ll mit erheblicher Genauigkeit
bestimmen.

Nehme ich fir die mittlere Entfernung %—© die Halbmesser
der Bahnen gleich 177’’,8 und 283’’,6, so finde ich fiir die relative
Bewegung von IT und III zur Beobachtungszeit 13’’,86 pro Stunde.
' Aus der beobachteten Zeitdauer von 10™208 = 05172 folgt dann fiir
die Summe der beiden Durchmesser, 2’’,38. Wird (siehe die Angaben
von Dovue.ass, Astr. Nachr. 3500) fiir das Verhaltniss der Durch-
messer von II und III */,, angenommen, so finde ich, in vorziiglicher
Uebereinstimmung mit den a.a. O. genannten Werten, fiir den Durch-
messer von Il 0’’,87 und von III 1’’,54 (in mittl. Entf.).

Utrecht, 1903 Febr., 27. A. A. NIJLAND.

Remark. As from the observed instants I derived a result slightly
different from that of Mr. Niszanp, this gentleman allowed me to consult
his reduction of the observation. It appeared that, in order to find the
amplitudes, he had combined the preceding geocentric superior conjunc-

- tion with the following transit, from the ingress and egress of which the
inferior conjunction could be derived. A slight error had however been
committed in the computation. After correction the relative motion of
the two satellites was found to be 13'°786 and the sum of the dia-
meters 2'-374, Moreover their proportion was, evidently erroneously and

( 312 )

against the real intention, put at 4 to 11 instead of at 4 to 7. We
thus get for the diameters 0-863 and 1511, which is still in good
agreement with the result of Mr. Nranp. As values have been assumed
for the radii of the orbits which hold for the mean distance of Jupiter
from the sun, these values need no further reduction.

(3) We find in Hovuzxrav, Vademecum (Bruxelles, 1882), p. 666:

On rapporte une occultation du satellite II par le satellite III,
observée a Sommerfeld, prés de Leipzig, par C. Arnoupt, le 1%
novembre 1693, (Wuiston, The longitude discovered by the eclipses,
8°, London, 1738), et une auire du satellite IV, également par le
IlI™e, vue par Lutumer a Hanovre, le 30 octobre 1822 (Nature, 4°,
London; vol. XVII, 1877, p. 148).

1st Remark. The little book of Wuiston here quoted is in the
library of the University at Utrecht, Division P, 8¥°, number 602. We
have turned over the leaves several times, but have not found any
mention of the observation of C. Arnott. It is true that the author,
in § XVIII, recommends the observation of the mutual occultations of
the satellites. He remarks that, if at such an occultation they have
opposite motions, the relative velocity is “doubled”. He mentions the
complaint of Derxam}), that the strong light of Jupiter renders the
observation of these occultations rather difficult. He remarks that, the
interval being equal, their number must be one and a half time as
large as that of the eclipses. Again he mentions that Lynn is the
first who, in the Philosophical Transactions N°. 393, has proposed to
apply these conjunctions to the determination of the longitude, seeing that
they can often be observed with an accuracy of less then half a minute *).
But I do not find the observation of a single occultation nor its prediction.

It needs hardly be said that the conjunctions, visible from places,
the difference in longitude of which is to be determined, are too rare
to be of much importance for the purpose. In accuracy of opservation
they are at all events surpassed by occultations of stars. But they
may well be compared with the eclipses of the satellites of Jupiter and
are indeed superior to them in this respect that they yield a result in
a few minutes which is independent of the optical power of the telescope.
For the eclipses this is only true in the case of the combination of a
disappearance with a reappearance.

2nd Remark. The original account of the observation of LurHmer
was communicated by him to Bope who inserted it in the (Berliner)
Astronomisches Jahrbuch fir 1826, p. 224:

“Am 30 Oct. Ab. 6" 55’ Bedeckung des vierten % Trabanfen vom
dritten.”’

1) Poacenporrr’s Biographisches Worterbuch, (article W. Dernam) gives no
reference to the passage where this complaint is to be found, nor even to any
paper on the observation of the satellites of Jupiter.

2) At least if there were no undulation of the images. See at the end of note 4

( 313 )

If we assume 9°42'= 38m48s East of Greenwich for the longitude
of Hannover, this is = 6h 16m 12s M. T. of Greenwich, at least sup-
posing that at that time it was already usual to give the observations
expressed in mean time.

In Nature, XVIl (Nov. 1877—April 1878) p. 149 (not 148) we find
in “Our Astronomical Column” :

“Jupiter’s SATELLITES. — Amongst the recorded phenomena connected
with the motions of the satellites of Jupiter are several notices of
observed occultations of one satellite by another, and of small stars
by one or other of the satellites.1) The following cases may be men-
tioned: — On the night of November 1, 1693, CurisropH Arnotpr, of
Sommerfeld, near Leipzig, observed an occultation of the second satellite
by the third at 10h 47m apparent time. On October 30, 1822, Luruer,
of Hannover, witnessed an occultation of the fourth satellite by the
third at 6h 55m mean time.

It thus appears that the editor of Natwre also took it for granted
that the statement must be understood to have been made in mean time.

(4) I did not succeed in finding the account of this observation
of Sranney WiniiaAms in any of the journals accessible to me, and
therefore applied to the author, who lives at Hove near Brighton,
for particulars about the place of its publication.

He kindly replied on the 7‘ instant, that the details of his obser-
vation of 27 March 1885 were published both in the 41'* volume
of the “English Mechanic” and in the volume for 1885 of the German
Journal “Svrius’’.

He had moreover the courtesy of communicating to me the original
account of the observation in question. From this account the
following passages may be quoted:

Occultation of satellite I by satellite I11.

1885 March 27, ... . 2’/, inch refractor. Power 102.

11°55" (Greenwich mean time). They are now only just free from
contact. mot like an elongated star with little more than a

black line between the components.

12°00™ to 12504m. After steady gazing I cannot see any certain
separation between the satellites, and therefore with this instrument
and power first contact must have occurred about 12"02™. Definition
is very bad, however, and in a larger telescope there probably might
still be a small separation between the limbs.

. ») It is to be regretted that these “several notices of observed occultations of
one satellite by another” are not more fully quoted.
21
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 314 )

12510™. They now appear as one elongated satellite. At times a
trace of the notches is apparent. :

1220. The elongation is now very nearly at right angles to the
direction of the motion of the satellites, and is so slight as to be
scarcely noticeable in this bad and unsteady definition. I think from
the smallness of the elongation that nearly half satellite I must be
concealed behind III. In this bad definition it is not possible to say
which satellite is in front of the other from the appearance alone.

In his letter Mr. Stani.zy WuniiamMs mentions the remarkable fact
that he too observed on 15 July 1902 the same conjunction which
has been described by Nuanp. His instrument was a reflector of
6'‘/, inch, with a power of 225. The following are the particulars
as communicated :

1902 July 15, 13545™-2. Satellites II and III are in contact. The
one will occult the other. See diagram MC 7°

13°52™. The satellites form one disc, which has the slightest
possible elongation in a north and south direction. Owing to con-
fused seeing this disc always appeared more or less fuzzy, and it is
impossible from the appearance alone to say, which satellite is occult-
ing the other.

13°56™. The combined dise is considerably elongated now.

14°02™-2. Satellites II and If] in contact as in diagram adjoining

noo Wl .

14°04". Satellites clearly separated. The occultation must have
been nearly central. II is a little more south now relative to III,
than it was before occultation. Possibly the slight elongation noted
at 1352™ was not real.

The above times are Greenwich mean times. Satellite III was on
the farther side of its orbit moving east, IJ on the near side moving
west. As the dise of II is larger than that of II, the phenomenon
should be described as a transit of II over or across III, rather than
an occultation of one satellite by the other.

The arithmetical mean of 13545™-2 and 1452™-2 is 13553™-7, which
is 1™:1 earlier than Ni LANp’s observation.

(5) For the numbers which follow we refer to Katsmr’s “Sterren-
hemel’, 4th Edition, p. 707 and following.

In the 4» Vol. of his Mécanique Celeste, p62, Tisseranp, following
SOUILLART, adopts inclinations for the orbits of II and IV, which

( 315 )

respectively exceed those given in the “Sterrenhemel”’ by + 4" and — 8",

According to LEverRIER we have, for the orbit of Jupiter in 1908,0:

Ascending Node = 99°31'56",
Inclination = 11829.

The fixed plane of the first satellite coincides with the plane of
Jupiter’s equator: the longitude of the ascending node on the plane
of Jupiter’s orbit, for the beginning of 1908 is therefore 315°33'35",
the inclination 3° 4' 9".

Furthermore we have for the four fixed planes relative to the
plane of Jupiter’s orbit :

Long. asc. node Inclination
I 315°33' 35" 3° 4' 3”
II 315 33 35 3.34
Epoch 1908.0.
IIl 315 33 35 2 5911
ry 315 33 35 2 3957

For the mean fixed plane of the three first satellites we thus
find: longitude of ascending node on the plane of Jupiter’s orbit
at the beginning of 1908: 315°33' 35", inclination 3° 2' 6".

Moreover we have for the respective fixed planes in 1908, according
to TIssERAND:

Change in
long. ase, node 1000 days Inclination
II 122°-293 — 33°:031 0°28' 9"
Ill 26 “403 — 6 -955 010 44
IV 238 -982 — 1-856 13.51

The effect of these inclinations, however, is but trifling. At the
distance of 90° from the node they produce only deviations

for II of 1°46,
eee) PT | dare!
aes ee os ae CEE

The determination of the position of the fixed planes, as also
that of the planes of the orbits of the satellites relative to these,
will be much improved by the measurements which Ds Sitter at
Groningen is making on photographic plates. Eventual observations
of conjunctions of the satellites, rather even of occultations, will
contribute their part in this determination and will furnish a test
for the adopted values.

» 21*

( 316 )

In the meeting of our section of last March a provisional account
of these measures by DE SITTER was communicated by Messrs J. C.
Kapreyn and E. F. van DE SANDE BAKHUYZEN ').

Our computations were then already too far advanced to keep
them back altogether; but we hope that by the side of these mea-
sures they still may have their use, for this reason that conjunctions
and mutual oceultations of the satellites may well be observed at
several observatories which are not equipped for taking photographs.

From the preceding numbers we find for the position of the fixed
plane relative to the ecliptic (for 1908,0).

Ascending Node 336°48'23" = Q,
Inclination — 2 “4a ==

Now, if Ry, Ly and 8 represent the radius vector, the longitude
and the latitude of Jupiter; A *? L * the radius vector and the longi-

tude of the Earth, (those given in the N. Almanac after correction
for aberration), the condition that the fixed plane must pass through
the Earth is expressed by:

ed =e : rots aes
R,, cos B sin (Loy Q) Roy sin B cot [= ii. sin fe ie

Yf
which is satisfied July 8, 1905 at 19°38™-3. For at that moment
log Ry = 0°728527 log h. = 0007179
Ly, =. 14122390 L. = 286°40' 3'°5

B =+ 05226°:73 2 = 336 4852 -0
so that our equation becomes
1:423706 — 2:204190 = — 0:780484

Similarly we find for the instant at which the same plane passes
through the centre of the sun:

25 April 1908 at 18":5 M. T. Grw.

On both sides of this latter epoch there exists the possibility of
an eclipse of one satellite by another, at the time of the heliocentric
conjunctions. We hope to treat this subject in the second part of this
communication.

1) This provisional account may be considered as a sequel to the thesis of
Mr. be Sirrer. This thesis, maintained by him at Groningen on 17 May 1901, bears the
title: Discuesion of Heliometer-observations of Jupiter's satellites made bysir Davin
Gur K.C.B, and W, H. Fixtay M.A. Further particulars will be given in the
Annals of the Royal Observatury at the Cape of Good Hope.

( 317 )

(6) In 1833—39 Besse, at the Heliometer, measured not only
distances of all the satellites from both limbs of the planet, but also
angles of position of the centre of the planet to III and IV.

His heliometer was the first big instrument of the sort made in
the establishment of FraunHorErR; the objective had an aperture of
70°2 Par. lines and a focal distance of 1131-4 Par. lines = 7 feet
10 inches 3-4 lines, Paris measure, (15°84 and 255-22 ¢.M.). The
mean error of a single observation of distance (which properly was
the mean of eight pointings) appeared to be

for | +0"26, for the mean distance resulting from all the measures, + 0"055

2 VERS), ee re ne . p > 9 . + 0:067
qu) WSS | ae : 2 oe os - + 0:042
aE OAS km » ” Bicabige! 278 , + 0:045
Mean: Orsi, cg . - oe ee ’ + 0°052

Scaur, at Gédttingen, used the heliometers which were made by
Merz at Miinchen for the observation of the transits of Venus in
1874 and 1882. The aperture of the objectives of these instruments
was 34 Par. lines, something less than half that of the heliometer
of Koéningsberg; the focal distance was 33 feet (113-7 cM.).

At these heliometers the reading, instead of being made on the
drums of two micrometers, was made by a microscope at right
angles to two scales fitted to the two halves of the objective. As
however in this way more time was required than for reading the druns
of a micrometer of BessxL’s instrument, Scuur, instead of taking the
mean of eight pointings, was content with the mean of four pointings,
which also make a complete measurement.

The mean errors of each observation obtained by Scuur for a
complete set of four measures was:

far. sek. a=: O34,

“a te O44:
jie 3, OF,
» 1V + 0-42,

Mean: = 39,

a result, which, taking into account the shorter focal distance, may
be considered fairly good. BrsseL as well as Scuur aimed not so
much at the determination of the position of the orbits of the
satellites as at that of the mass of Jupiter.

( 318)

Scuur improved in different respects the reduction of the observations
of the measures made by BrsseL. In consequence, the mean errors
of the single determinations of Brssgl. were considerably lessened.
The numbers quoted just now, became:

for * ta. Ft.
PS a),
lil. 2") 36.
ee fase 1S: be

Mean: + O24.

bP]

As has been mentioned already, Gitt and Finnay, acting on a
suggestion formerly made by Orro Srrvve'), did not measure the
distances and the angles of position of the satellites relative to the
centre of the planet, but relative to each other. (The instrument
at their disposal, a heliometer of Repsotp, aperture 7} inch = 19-05
em., focal distance somewhat over 2 Meter, far surpassed in
perfection all the instruments used up to that time). These observations
can be made with much more precision. The drawback is that the
formation of the equations of condition and their solution become
more complex and absorb much more time. Both the gentlemen
named and Mr. de Sirrer have not been deterred by this conside-
ration. They found + 0"-087, a number considerably less than that
of BrsseL, for the probable error of the measurement of a single
distance. Mr. de Sirrer even finds that the probable error of the
mean distances (the real unknown quantities) does not exceed
+ 0"-020 or + 0"-021.

(7) It may be remarked that Mr. de Sirrer found it expedient
to alter the choice of the unknown quantities. He retained for
each satellite: the longitude in the orbit, the inclination and the
ascending node relative to an adopted position of the fixed plane,
but not the eccentricity nor the position of the perijovium and
the mass. There thus remained as unknown quantities only three
elements of each satellite. On the other hand he introduced corrections
of the coefficients of the perturbations or rather of the periodic
terms, which afterwards must lead to the knowledge of the mass
of the satellites, to that of the eccentricities and of the position of the

1) Vide the first report of Hermann Srruve, in the first supplementary vol. of
the Pulkowa observations, lst page at the bottom.

( 319 )

apsides. He further introduced two unknown quantities, vz. the
constant errors which might vitiate the observations of the two
observers Ginn and Finuay. He thus also obtained a total of 29
unknown quantities. It need not be said that the solution of about
400 equations with so many unknown quantities, is an enormous
labour. Still, owing to the help of some other computers, this labour
has been brought to a happy issue.

We must not enter here into further particulars about this impor-
tant work, though we did not feel justified in omitting to mention it
altogether. I will only ‘remark that it is not sufficient to determine
the position of the planes of the orbits of the satellites for one
epoch; for as was already remarked the position of these planes
changes continually. It seems that these changes may be sufficiently
represented by assuming a regular retrogradation of the line of
intersection with a fixed plane, the inclination remaining the same.

The main cause of this retrogradation is the polar compression of
Jupiter. It is desirable however to establish the amount of this
retrogradation by the observations, and to derive afterwards the
compression by means of this amount. Consequently the position of
the planes of the orbits has to be determined for different epochs.
In this respect too Mr. pr Sirter has done good work, vide the
communication already mentioned, presented in the meeting of last
March by Messrs Kapreyn and EK. F. van DE SANDE BAKHUYZEN.

(8) The same volume, which contains the ecliptic tables of DamotsEau,
contains also in a second part (not mentioned on the title) tables
“nour trouver les configurations des satellites de Jupiter.”

We have contemplated whether it would not be desirable not to
use these tables, unmodified, for our computations. We have therefore
taken note of the investigations of SouiLLarT, ADAMS, Marry, GILL,
Finuay, and de Sitter, but it appeared that such a course would
aggravate our labour very considerably. We would have had to
determine new elements for all the satellites and to compute new
tables. This would have caused considerable retardation, unnecessary
for our purpose, which was no other than to prepare astronomers
for the observation of the conjunctions visible in 1908.

We therefore have based our computations on the tables of DAMoIsEAu,
but we have first examined in how far they represent the observed
conjunctions. The following summary shows not only the difference
between the observation and the tables in the elongations z and a’,
of the two satellites, expressed in radii of Jupiter, but also their
difference in time.

( 320 )

Relative

| Oceul- | | ey
Observer Date ; tation) by |Error) hourly | o¢ able | YY
| of | | motion |
————— aT Le ee ee r
Fauth 1902 Oct 7 | Uy | 1, | 0,025) 1,278 | + 4,2 |+ 0,04
» | » » 93 | m, | om, |oc2 | 14390 | +14 |4 008
> >» Nov.10 | Ir | III, | 0,00 | 0,883 0,0 |+ 0,3
. > Dec. 24 | IVe | 1, | 0,10 | 4,089 | —55 |-+ 0,005
p 1903 Jan. 14 | IL, | U1, | 0,41 | 03'4 | —192 |— 0,05
Nijland | 1902 July 15 | I, | UL, | 008 | 0751 | +64 |— 001
7
Stanley Williams, » » » | » » 0,07 0,751 + 5,3 is 0,01
» » | 188 Mareh o7|-1, | Ill, | 0,00 | 0,992 | 0,0 + 0,01

The observation of Luramer in Hannover, of Octob. 30, 1822 is
not contained in this table. Its calculation yields the result:

Difference cae On81 -+ 0°92

So there is a difference in the amplitudes, of 0°81, = 081
18"37 = 14"9, in the latitudes, of 0°92 —=16"9. Probably the
observation has been made with an unsatisfactory instrument, for it
is impossible to suppose an error of this amount in the tables of
DamoisEav for 1822. The difference in sign of the latitudes y and g/
is explained by the fact that the longitude of the ascending node of
the fixed plane was 10%14°°37, which is intermediate between the
two jovicentric longitudes.

As the two satellites moved in the same direction, the hourly
change of distance was small, viz. O'280. It would thus require
nearly three hours to annul the difference of O'-81.

The remaining conjunctions, however, show a satisfactory accuracy
and we may thus expect that the table, as given below, will serve
its purpose.

As a second test I have computed, by the aid of the second part
of Damoiseau, the two superior conjunctions and the intermediate
inferior conjunctions of II, and I have compared these to those given

( 324 )

in the Nautical Almanac of 1902. The epochs were found a little
earlier, to wit:

superior conjunction of 10 July, 10°46™-9 M.T. Grw. 0™-7 earlier
inferior conjunction (mean

of ingress and egress) 16July 5°4070 ,, ,, O™3 ,,
superior conjunction Settee ot en OMT ,,
all three less than a minute.

Now, as the conjunctions in the Nawtical Almanac have been
calculated by the aid of Damorszav’s tables écliptiques (making allow-
ance for some slight corrections indicated by Apbams) the differences
must be solely due to the fact that in DamoisEav’s second part the
mam terms only of the equations and perturbations have been taken
into account.

The same tables represent as accurately the superior conjunction
of I on January 1, 1908, 14°4™-2 M.T. Grw. = January 2, 2°13™-55
civil time of Paris; the error amounts to 0°07 or O01 linear
measure only, an are traversed by the satellite in O™5.

(On the terms taken into account in the second part of the tables
of Damoisgau vide 3'¢ appendix below).

In his letter Mr. Stantey Witiiams mentions another rare obser-
vation, made as well by himself as by the Spanish observer J. Comas
of Valls, (near Taragona), on 14 August 1891, to wit of the coin-
cidence and of the subsequent separation of the shadows of two
satellites on the planet. He concludes that an eclipse must have
taken place. These phenomena will be treated in the second part of
this communication.

(9) Below follows the table which has served for this computation.
The unit, the radius of Jupiter, is 18°37. SourtLart states that he
found mentioned in the papers of Damoisgau that this number was
borrowed from AraGco. According to Hovuzzav, Araco must have
made the determination by means of the double image micrometer
(an invention made nearly simultaneously by himself and Parson ;
of the latter the observatory at Utrecht possesses a specimen).
Particulars about these measures are not known. The number is
smaller than that found by other astronomers, vide for instance
Hovuzeau, p. 647—650; Sxxn, Astron. Nachr. N°. 3670 (15 Aug. 1900).

Hourly change of the elongation « as a function of the amplitude.

| ae I Il Ill IV ae
r r r r
Os; O°] 05(12), 0°70 895 0,708 | 0,560 | 0,420 | 6s 0° 6s} Om!
| 4 fs ne [- 2
5] 41 | 25 70,891 | 0,705 | 0,558 0,448 | 515 | 25
10 8 7 | 4
| | 40} 41 | 20 90,881 0,697 | 0,554 0,444 | 1015 | 20
47 | 44 10 8
45 | 414 | 15 [0,864 0,683 0.544 0,406 1515 | 45
| 93 18 | 45 4
90} 411 | 10 [0,841 0,665 | 0,526 0,3°5 20015 | 10
| 30 | 23 18 14
|} |95] 44 | 590,814 0,642 | 0,508 0,381 315] 5
| | | 36 29 23 47
1/ o} nt | 0 40,775 | 0,613 | 0,485 0,364 an ee)
| | 42 33 26 20
5 | 410 | 25 [0,733 | 0,580 0,459 0,344 5141 9
48 37 30 92
40] 40 | 20 [0,685 0,543 0,429 0,322 401 4! 20
53 42 33 25
15] 10 | 15 [0,632 0,501 0,396 0,297 15 | 4 | 45
57 46 36 97
90} 40 | 40 0,575 0,455 0,360 0,270 20 | 4 | 40
| 62 49 39 29
21 10 | 540513] | 0,406 0,324 0,244 5414/5
| 66 52 4A 31
2/ 0} 10 | 040,447 0,354 0,280 0,210 8| 0} 4] 0
69 5D 43 32
5] 9 | 25 $0,378 | 0,299 0,237 0,178 | 513]
| 72 57 45 34
140} 9 | 20 [0,306 0,242 0,192 0,144 4013 | 20
| 75 59 47 35
45} 9 | 45 0,234 0,183 0,145 0,109 4513 | 45
| 76 60 48 36
| 901 9 | 10 $0,155 0,123 0,097 0,073 20 | 3 | 40
77 61 48 36
| |9 71 9 | 5 J0078 0,062 0,049 0,037 913] 5
| 78 62 49 37
L 0} 9 | 0 40,000 0,000 0,000 0,000 9| OfSio8

Finally we will give below, vide pp. 334 and 335, two instances
of computation ; one of a case in which the apparent motion of the
two satellites was opposed, the other in which it was in the same
direction.

1st Appendix. What is the maximum duration of the several occul-
tations of one satellite by another ?

We ‘have seen above that it took 19™2 to annul the small
difference of the elongations of O11 (2"°0). This was caused by
the minuteness of the relative motion of the satellites. But in the
case that the hourly motions, which we will denote by wu and w’,

v'—-x

are absolutely equal, the denominator of the fraction ——— is zero.
Ci —=— 5

(t323°3

The case then corresponds to that of the “Station of Venus” and
it is a very ancient problem to compute its epochs.
Let be r and r’ the radii vectores of two satellites; 6 and 6’ the
corresponding amplitudes, then for the occultation :
r sin 0 = 1' sin 6'.
The condition of an equal change of longitude leads to:
ok Mager ao
dt dt
Now, if 7’ and 7” represent the sidereal periods, we have, neglecting
the apparent movement of Jupiter:
deems te HEY
ae ae et Pw wee Wek
consequently :
r—'lz cos @ = r'—"l cos 6’,

from which:

cos? @ — — cos? 6 = — —— sin’ Gh
r is ?
Adding
y?
sin? 9 = — sin? 6’,
Ue
we get
1\ 2
i=— (=)-+ sin? 6
Tr 7 Tr
Therefore, putting =
1
eee ee
a E Suse : :
airs? eae tee
re
and
2
sin? 6 = eat Ss:
w+ue+l

The equality of the hourly changes of the two elongations of
course only lasts for an instant; very soon inequality sets in and
the two satellites begin to separate. Meanwhile it may be long ere
such becomes perceptible at the telescope, only, in a case like the
present, the satellites do not pass each other, but after the conjunction
they have the same position the one to the other as before.

As an example take a conjunction of I and II under the cireum-

( 324 )

stances in question. Let the amplitudes be between 0 and 3 signs,
so that both the satellites, as seen from the Earth, (the head being
turned to the North Pole), are to the left of and both receding from
the planet. Before the conjunction I is to the right of II, but the
motion of I is quicker than that of IL. I will overtake II as soon
as its amplitude is 44°39’, that of II being then 26°14". At the
same time, however, the apparent velocities are equal. Now as I
approaches its greatest elongation it retards its motion much more
considerably than II, the amplitude of which is so much smaller.
The consequence is that, after the conjunction, I is left behind, and
gets again to the right of II as before conjunction.

This case represents a transition between two other cases. 1. If,
under the same circumstances I is somewhat more in advance (has
a greater amplitude), it will pass II, but after a while will be over-
taken by I, which then, as seen from the Earth, passes behind it.
2. If, however, I is somewhat less ahead, it will continue to be
seen to the right of IH, the distance [ — II going through a minimum
but not reaching zero.

Now, in order to answer the question, how long will be the
duration of the occultation counted from the first external contact,
the apparent radii of the satellites must be known. Owing to the
irradiation they are greater at night than in daytime’) as several
observers have actually found. The observations of the satellites of
Jupiter being made nearly exclusively at night time, we will adopt
the apparent radii holding for the night. I took the mean of the
values found by Sere at the giant telescope at Washington on the
one hand and that found by several observers on the other. (I have
taken the values as summarised by Sere himself). For the reduction
to the unit used throughout for these computations, viz the radius
of the equator of Jupiter, this radius is taken = 18":37 in accordance
with DAMoIsEav.

Diameter Radius
I 1"-07 = 0'-058 O'-029
II 0:95 0:052 0 -026
Ill 1'-56° 0-085" 0 0425
IV 1 41 20/076 0-038

1) Vide e.g. T. J. J. See, Observations of the Diameters of the Satellites of
Jupiter, and of Titan, the principal Satellite of Saturn, made with the 26 inch
Refractor of the U. S. Naval Observatory, Washington; 19 Oct. 1901. Astr. Nach-
richten N°. 3764, (21 Jan, 1902).

( 325 )

Therefore ;
Sum of the diameters Sum of the radii
Pied sO 120 O'-055
Issa) none 20-148 0 -0715
I +I1V .. 0:134 0 067
BPEL 22 t= 0-137 0 0685
Ei tok vis 2, 0 128 0 064
Ti+ 1V 0 161 0 -0805

For the mean radii recites we will take two figures more than
did Damorsgav in his tables, and we will adopt for the purpose the values
found by Sovurttart in DamoisEav’s papers, (SOUILLART, second paper,
Mémoires présentés par divers savants a l’Académie des Sciences,
Tome XXX, 2™e Série, 1889; p. 10) ’).

I 60491,
II 96245,
Ili 15°3524,
IV 27-0027.

The result of our computation is, that the time between the first
contact and the central occultation is: —
for I and II 1 and III I and IV II and III II and IV III and IV

45-324, 15-245, 15403, 25263, 15-774, 35-725 ;

between the central occultation and the second contact:

15-204, 4%:161, 15-059, 2190, 42-767, 35-725,
therefore in all

2h-528, 25-406, 25-162, 4h-453, 3-541, 75-450,
or

2532™ 7) 2h24™, 2510, 4h27™, 3532™, (hie

Still even these numbers do not represent the maximum of the
time during which the two satellites may be seen as a single body.
For we can imagine the case that the shortest distance becomes
equal to —(r-+7’), i.e. that between two central conjunctions there

1) According to SOUILLART, DAMOISEAU derived these numbers in the following
way: He adopted the mean distance of IV, in accordance with Pounn’s determi-
nation = 496"0, and took 18'-37 for Jupiter’s semidiameter, so that, by division
Try = 27:00102834. The mean distances of the other satellites were then derived
from the sidereal periods by the application of KeppLer’s third law. But to these
mean distances he added the constant terms preduded in the radu vectores by
the perturbing force.

1 beg leave to remark that 496”"0: 18°37 is not 27:00102834 but 27:000544366.
Happily the 4th, 5th, 6th, 7th and 8 figure have no appreciable influence on our
computations, nor probably on those of SourLuart. For the rest the 2.4 appendix,
further below, may be consulted on such numbers of many decimals.

*) On June 4, 1908, such a conjunction must take place according to our com-
putation. Vide the table further below.

( 326 )

occurs a contact on the other side. In this case the duration will,
very nearly indeed, have to be multiplied by V2. It thus becomes —
for Iand II, IandIII, IandIV, IlandIII, IlandIV, II] andIV

30-574, 35402, 35-057, 6":296, 5-006, 10"-43,
or:

334m, 3b24m, gh 37, 618, 5h On, 10°26".

These numbers hold only for those very rare occasions in which 1°.

the occultation is central and 2"4. the rate of change of the elongation
is equal or nearly so for the two satellites. As soon as there is
some difference of latitude the time during which the two satellites
are seen as a single body is of course smaller.

2-4. Appendix. Investigation of the uncertainty, existing in the

determination of the synodic periods of the satellites.

In his introduction to the Tables Ecliptiques, p. XIX, DeLampre says:
‘Nous n’avons aucune observation d’éclipse antérieure 4 1660”. Now let
us assume that the difference in time between the first eclipse observed
in 1660 and the last observed in 1816, two years before the publica-
tion of these tables, (taking into account also the next ones in 1660
and the preceding ones in 1816) leaves an uncertainty, in the case
of the four satellites, of 20, 30, 40 and 60 seconds, which will be
too favourable rather than too unfavourable. If we divide this un-
certainty by the number of synodic periods in 156 years, to wit
32193, 16032, 7951 and 3401, we get for the uncertainty of a
single period

for I for II for II for IV
Os:00062, Os00188, 080050, 0s:0176.

Therefore, if we find that DeLamBrE gives these periods to 9 places
of decimals of the second, we cannot attach much importance to
the fact.

When Damoisravu, 20 years after DeLAMBRE, published new eclipse-
tables") for the satellites of Jupiter, he adopted the period of I un-

1) The tables of Detampre and DamorseAu were destined mainly to serve for the
prediction, in the astronomical ephemerides, of the eclipses of the satellites caused
by the shadow of Jupiter. It is for this reason that both he and Detamrre, united
all those terms of the perturbations in longitude which have the same argument
at the time of the opposition of the satellites, even though these arguments might
be different for all other points in the orbit. Therefore it becomes necessary once
more to separate these terms as soon as tables have to be computed from which
may be derived the longitude and the radii vectores of the four satellites for any
point of their orbits, tables such as have been given by Besse in his Astrono-
mische Untersuchungen and by Marrn in the Monthly Notices of the Royal
Astronomical Society, Vol. LI, (1891).

( 327 )

changed, but applied the following corrections to the remaining ones:
Il + 0s:005 127 374,
III + 0-029 084 25,
IV — 0-092 654 834,

the amount of which is even respectively nearly 3, nearly 6 and
somewhat over 5 times that of the uncertainties derived just now.
But even if we increase the number of intervening years from 156 to

1
176, our estimated uncertainties are only diminished by about a of

their amount. We thus conclude that these periods can only be con-
sidered to be determined with certainty :

that of I to 3 decimals of the second
>» os Ll, Til and IV to 2 decimals ,, _,, sgh

The Nautical Almanac, which, where it gives the superior conjunc-
tions of the satellites, gives also the synodic periods, wisely confines
itself to three decimals. The use of 9 decimals may therefore provi-
sionally be taken for astronomical humbug. Some other instances of
the same kind might be quoted e.g. the formerly well known con-
stants, 20"-4451 for the aberration and 8"57116 for the parallax of
the sun!

3d Appendix. Meaning of the equations taken into account in
the 24 part of the tables of DamotsEav.

On p. 321 we have referred to the 3'4 appendix for information
as to the equations which have been taken into account for each
satellite in the second part of the tables of Damorszav. We will now
supply this information; we will denote by U, uw, w, wi, ui and
ury the mean longitudes of the sun, of Jupiter and of the four
satellites; by 2, the longitude of the perihelium of Jupiter, by 2' that
of the Earth, by ayy and ayy the perijovia of III and IV; by the
longitude of the ascending node of Jupiter’s equator on its orbit ;
finally by At, Atm and 4;y the longitudes of the ascending nodes
of II, III and IV each on its own fixed plane.

In order to be able to supply the data following below we have
taken the daily motion of the argument of each equation from the
tables in the second part of Damoisrav. This amount was then mul-
tiplied by the synodic period expressed in days; the product thus
obtained was then compared with the factor by which, in the first

( 328 )

part, p.p. (III), (V), (VI) and (VIII) the letter z (the number of
synodie periods) is multiplied.

These daily motions are so nearly equal for several of the equations
of IJ, Il and IV that, in order to make them out, we must take
from the tables the motions for a long interval, e.g. for 10 years,
(duly taking into account the number of periods). These must then
be divided by the number of days (10 years = 3652 or 3653
days). Multiplying this quotient by the synodie period in days, we
get 360° + a fraction. The 360° are of no account ; the fraction
is the factor of 7; we thus recognise which is the equation we have
to deal with. In the preface of the second part of Damoisrau we
look in vain for any information on the subject.

I. For this satellite five terms have. been taken into account.

N°. 1 with an amplitude of 1°16, is the equation of the ee
of a its argument is U—uw,.

. 2, (amplitude 0°29), is the equation caused by the ellipticity

of ee s orbit ; the argument is the mean anomaly of Jupiter u,— 2,

N°. 3 is 180° -++ the mean anomaly of the Earth, U—a'; i
its aid and that of N°. 1 ze. the difference in longitude between
the Sun and Jupiter, we find, in the table of double entry IX, one
term of the geocentric latitude of the satellite.

N’. 4 with an amplitude of 0°45, shows the perturbation caused
by II in the motion of I. The argument iS wj—2wUI.

N°. 5, (amplitude 3°-07) gives the jovicentric latitude of I, neces-
sary to find the second term of the geocentric latitude. The argument
is wj—Aj.

II. Seven terms. N°. 1, 2 and 38 have the same arguments as the
analogous terms for I; the amplitudes of N°. 1 and 2 are halfthose
of I. The term of the latitude to be taken from IX, by the aid of
1 and 3, is of course the same for all the satellites.

N°. 4, (amplitude 1°-06), shows the perturbation caused by III in
the motion of II. The argument is wyy—wnmrt.

N°. 5, 6 and 7 serve for the latitude.
N° 5, (amplitude 3°:05), has the argument wy—aqyv;
: ba Re 0°27), 5) ae ee uyi—Aq ;

|e Ts ( ” 0 ‘03), » » ” uyy —Ajrt.

Ill, Nine terms. Nos. 1, 2 and 3 are the same as for I and II;
the amplitudes of N°. 1 and N°. 2 are 0°29 and 0°:07,

( 329 )
N°. 4, (amplitude 0°07), has the same argument as N°. 4 for I,
but it now shows the perturbation caused by II.

N°. 5, (amplitude 0°15), is the equation of the centre ; argument
UTIT

XIII.

N°. 6, (amplitude 0°-04), has the argument wr—-ry, it thus must
account for a perturbation in III depending on the longitude of the
perijovium of IV.

Nos. 7, 8 and 9, with the amplitudes 2°98, 0°18 and 0°-03,
serve for the latitude. The arguments are respectively wir
and wyyy—Ary.

X, UI— At

IV. Seven terms.

Nos 1, 2 and 8 are similar to those of the preceding satellites.

N° 4, (amplitude 0°83), is the equation of the centre, argument
Uuyy — NIV.

Nos 5, 6 and 7 serve for the latitude. N° 5, (amplitude 2°64)
depends on the mean anomaly of Jupiter; its argument therefore
is u, — %,.

N° 6, (amplitude 0°:24), depends on the argument of the latitude
of the satellite itself; argument wry— Ary.

N° 7, (amplitude 0°04), is a minute perturbation, caused by III;
its argument is wyy—AyIr .

Now in regard to the following table of the computed conjunctions.

The first column contains the ordinal numbers.

The second shows the epoch of the conjunction, accurate to the
nearest minute, expressed in civil time of Paris. This time is reck-
oned from midnight and has been used by Dawmoisgzav in his
tables; it thus represents the direct result of our computations. In
the cases that the computed time was just a certain number of
minutes and a half, the half minute has been set down. By sub-
tracting 12° 9™ or, where necessary, 12" 9™-35, the mean time
of Greenwich was found, which is contained in the third columm.

The 4% and the 5 columm contain the numbers of the occulted
and the occulting satellite. The appended letters /# and n show
whether the satellite is far or near (vide supra p. 304). The satellite
is far if its amplitude is between 9° and 38, near if it is between
3° and 98. Furthermore ee denotes an eastern elongation, for which
the amplitude is about 3° and we a western elongation, for which
the amplitude differs little from 9>.

. 22
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 330 )

At the conjunction the elongations, counted along the orbit of
Jupiter, are equal; they are to be found in the next column. If the
elongation is ++, the satellite, as seen by a northern observer, using
a terrestrial telescope, will be to the left of the planet. Therefore
if he uses an inverting telescope, as is the rule for the observation
of the heavenly bodies, he will see it to the right.

The three following columns contain the ordinates of the two
satellites and their difference; northerly latitudes are positive. The
tenth column shows the duration, which the eclipse would have, if
the conjunctiun were central. In a few cases (Nos. 20, 23, 30, 48,
53 and 64), we find y'=y, consequently y’—y=O. If the tables
were correct these conjunctions would be central. But in testing the
tables by the conjunctions observed by Messrs Fauru, NiLanp and
Srantey Witu1ams the difference of the y’s did not completely agree
with the observations and even a small difference may considerably
change the duration of any eventual occultation. Therefore, not to
fill a column with figures, which, likely enough, may be contra-
dicted by the observations, I omitted the value found by calculation
for the true duration.

We remarked before (p. 308) that, if at all, any conjunction will
be visible at a determined place of observation only for a short
time, viz. between sunset and the setting of Jupiter. As a conse-
quence the list will be of little use, unless observatories distri-
buted over the whole of the earth cooperate in the work. The last
column was added as a help to such cooperation. It contains on
every line an observatory, at which the conjunction of that line
will be visible. It is certainly desirable that other astronomers also, at
observatories in the vicinity, examine whether the phenomenon will
be visible, and, if so, prepare for its observation.

A. C. OUDEMANS. Mutual Occultations and Eclipses of the Satellites of
Jupiter in 1908.”

Starting point: the geocentric superior conjunction of I on July 12, 1908,

at 11h 2m.3 M.T. Greenwich = 115 11.65 M.T. Paris = 234 11™.65 civil time Paris.

308 © 4 2) 20

1
30 168 000 000 ’

Scale Imm, — 8” heliocentric.

0 100 200 300 4.00 500 600 700

Unit: the radius of the earth’s equator’

Proceedings Royal Acad. Amsterdam. Vol. IX.

‘a=

( 331 )

foo U LTS.

Geocentric conjunctions of two satellites in June and July 1908.

22*

g |
. 2
oe oes Mo > S 3
Ci il ° ° o @® & (5) < 3
_ Civil time Mean time = a Ss aS me
BS i seSl| r= | 35 = a a ee 8 visible at
at Paris at Greenwich = Se ae Sus i=
Onul6oa | 3

4 June 5h55m)| 31 May 17646 | | WT | +2r48 | —0r16 | —0r12*|} +0, 03°| 45m | Mt. Hamilton

4p . 18-25 1 June 616 {I | HW, | —6,03 | 40,34 | +0,29 | —0, 02 | 14 Cape

Se.” .4. 55 1 » 134 |I ie IV, | —3,15 | +0,148 | —0,12 | 0,30} 7 Washington

a», 43°16 a Sats Sey lll, II. —8,51 | +0,51 | +0,38 | —0,13 | 60 Madras, H. Kong

2» 14 0 aes? (°F 51 II , IV, | —8,16 | +-0,54 | 40,13 | —0,38 | 41 Madras, H. Kong

meee os AS) 2p 155 | O ie IV, | —8,20 | +-0, 37 | +0,13 | —0, 24 | 10 Madras, H. Kong

as 21 10 ee ee Wa? ee ae eg | if II Lp —4,54 | +0,31 | +0,20 | —0,11 | 41 Utrecht

oe — 3 15 Zee 45 6 Cw Le . —0,335| 0,00 | —0,03 | —0,03 | 4 | At the

SZ > 350 | 2> 4541 |, |, | 0,87] 40,45] 0,00| 0,45] 6 |Jsametime dF

A>» 1444 | 4» 237 |1 | [Et 0,99 | 0,958] 40,035) Two contacts at

ve a eer the same side. II just)
4» 16 0 4 » sy | » » 4, 34 reaches central con-)5
ane junction but then =
ay - 1715 oS oes Be BF » pa sgul 0 23 | —0,19*| +0, 03°|/retraces its steps.
The satellites will be visible as one body during nearly 2} hours.

aoe) 7 6A 4 » 19 42 ee peel Le —6,05 ; +0,33 | -+-0,28 | —0,05 | 12 Sydney

6 » 16 24 6 » 4415 Il, —0, 48 0,00 | —0,01 | —0,01 | 4 At the

ow 4184145) 6 » i6 2 III Ii | 10,82 | —0,06 | —0,17 | —0,41 | 6 sametime o ¥

f and II eclipsed?

9 » 5 6 Ss 34657 IV I, —9,19 | +0,59 | +0,43 | —0,16 | 13 Mt. Hamilton
10 » 42% aes 44G AT TY ft I. +0, 29 | +0,14 | —0,05*} —0,19°| 6 | At the
Memes ss | 9 >» 1744 1 Lp |Z, | —0,58 | 4-0,02 | —0,01 | —0,03 | 5 same time of %
10 » 628 2 »: 18 18,5) If y Y —1,54 | 10,15 ; +0,05 | —0,10| 6 Mt. Hamilton
og TS ST (ee Os ft Ill if +9,01 | —0,32 | —0,52 | —0,20 183 Washington
12 » 1034 144 » 2995 | ee —6,03 | +0,32 | +0, 28 | —0,04 | 11 Wellington
13 > 18 465/13 » 637 |, | 1, | —0,82| +0,01%| 40,015) 0,00) 4 | o %

3 2 II eclipsed
soe 24-37 || 13° > 9 98 | if Ill | +4,34 | —0,07 | —0,16 | —0,09 | 6,5 (Utrecht)
fale 25°66 4D bP 4G Te | H, | —5,99 | 40,32 | +0,29 | —0,03 | 10 (Atl. Ocean)
Meee. 7 5S -| 46a 49044 Il. | ue —0,92 | +0,02 | +0, 92 0,00 | 4 J Y

Civil time

-at Paris Fe

47 June 9b39m

47 » 20 47
18 » 0 33
18 » 7 30
18 » 8 28
49 » 1313
20» A 3
21 » 4 3
A» 41 5
23 » 2 32
2% » 1013
QW » 1237
ee. 4447
26 » 13 43.5
26 » 15 49
27 » 0 27
27 » 13 44
27 » 18 56
27 » 23 24
28 » 1457
30 » 5 5.5
1 July 12 34
4 » 15 40
2» 16 21.5
3» 18 21
wm 6 20'.8; |
a » 4 44 |
5 2 6 25 |

Mean time |

at Greenwich

16 June 21230m
47» 8 38
47 », 12 24
44, me 19-94
AF» 2048
19 » 4 4
20 » 8 54
20 » 12 54
20 » 23 45
22 » 1423
23 » 22 4
24 » 0 27
25 3. “25 |
26 » 1 34
26 » 3 40
26 a 42468
27 435
27 » 6 47
7 6» 64144
28 » 2 48
29 » 16 56
4 July 0 25
Se 3 31
2» 4412
3 6 12
3» 1 .BS.D
4 » 13 35.0)
4 » 18 16 |

n = near
f=tfar

Mo te’ 8)
Dien 5
eS) aes
= flo ~_- —
oe eee
On \é8

Occulted
satellite

40r18
=0,37
—0, 28
—0,16
—0,51
0, 32
+0, 02
—0,11
+0, 24
+0, 32
-L0, 03
40, 20
+40, 35
40, 31
+0, 32
—0, 01
—0,17
—0, 20
+40, 03
+0, 20
-L0, 31
-40, 09
+0, 23
—0, 48
-+0, 31
—0, 85
+40, 03
—0, 12

Occulting
satellite y'

-40r10
—0, 52
—0, 4
—0, 30
—0, 45
+0, 29
+0, 04
—0, 15
40,18
40, 29
-L0, 01
+0, 16
-L0, 30°
+40, 25
+0, 30
+0,
OS
—0, 5
10, 07
+0, 17
+0, 31
+0, 04
+0, 24
—0, 42
+0, 34
—0,79
+0, 10
—0,45

—0r08
—0,15
—0, 16
—0, 14
40, 06
—0, 03
40, 02
—0, 04
— 0,06
—0, 03
—0, 02
—0, 04
—0, 045
—0, 06
—0, 02
40, 04
—0, 14
—0, 25
10, 04
= 6.03
0, 00
—0, 05
—0, 02
40, 06
0, 00
-40, 06
+0, 07
—0, 03

|
ce

Duration of
central occultation

6m
9
10

6,5

Visible at

Wellington
Utrecht
(Atl. Ocean)
Sydney
Sydney 2
Hong Kong
Utrecht
II eclipsed
Hong Kong
Washington
Sydney

Hong Kong
Hong Kong
Madras, H. Kong
Madras

cy fm
Madras, H. Kong )
Cape
(Atl. Ocean)
Madras

Mt. Hamilton
Hong Kong

Central Asia

Cape

Berlin etc.

Washington
IL eclipsed

( 336 )

Pathology. — “On the Amboceptors of an Anti-streptococcus serum.”
By H. Eysprorkx. (From the Pathological Institute of Utrecht).
(Communicated by Prof. C. H. H. SpRonck.)

(Communicated in the meeting of September 29, 1906).

As is known, there exists in the serum of an animal which is
treated with the bloodcorpuscles of an animal of another species, a
substance, which is capable of bringing the bloodcorpuscles of the
second animal to solution with the aid of another substance, which
is already present in normal serum. The first substance, which only
appears in immune-sera, is thermostatic and is named differently by
different investigators, according to the idea which they make of its
influence (Amboceptor of Exruicn, Substance sensibilisatrice of BorDET,
Fixateur of Mercunikorr). The other substance, which normally is
present in all sorts of sera in greater or smaller quantities, is easily
made inactive by heating to 55—56° C. or by being exposed to light.
It has been proved, that the last mentioned substance is identical with
a bactericidal substance, demonstrated by Fopor*) and FLteer ’) in
normal blood-serum, to which is given the name of alexin by BucHnrr.
Next to this name at present the denominations complement (EHRLICH)
and cytase (METCHNIKOFF) are used.

Had Mercunikorr in 1889 already pointed out the analogy between
hemolytic and bacteriolytic processes, later investigations have com-
pletely comfirmed this supposition.

In 1901 Borper and Geneou *) published a method to demonstrate
the presence of a ‘substance sensibilisatrice’ in the serum of an
animal, which was immunized against a certain micro-organism, by
means of a combination with the complement. At the same time
they found, that this amboceptor is specific; for instance, the ambo-
ceptor, present in the bloodserum of animals which were immunized
against cholera spirilla, is indeed active against the cholera spirilla,
but not against other bacteria, such as the typhoid bacilli.

On the other hand one is capable of distinguishing with the aid
of an amboceptor at hand, the micro-organism belonging to it from
others, by means of a combination with the complement.

Using the above mentioned method of Borper—Grneou, BEsREDKA *)
succeeded in pointing out an amboceptor also in an anti-streptococcus

') Deutsche Med. Wochenschrift, 1887, N°. 34, S. 745.
*) Zeitschrift fiir Hygiene, Bd. IV, S. 208.

5) Annales de I'Inst. Pasteur, T. 15, 1901, p. 289.

4) Annales de |’Inst. Pasteur, T. 18, 1904, p. 363.

( 337 )

serum prepared by himself. This serum was obtained from a horse,
which for some time was injected intravenously with a mixture of
6—8 different streptococci, which but for one exception originated
‘immediately, so, without passage through animals, from pathological
processes of man. Besides, he has made use of the presence of an
amboceptor in his serum to investigate, whether it might be possible
to separate different races of streptococci from each other with the
aid of this substance.

Among the principal difficulties, which are still experienced in the
preparation of an anti-streptococcus serum, must be mentioned in the
first place, that the streptococci proceeding directly from patholo-
gical processes of man and being very virulent for him (scarlatina,
erysipelas, septicemia etc.) possess in general for our common test-
animals a comparatively small degree of virulency. By this the pre-
paration of a very powerful serum is somewhat impeded and on the
other hand it is almost impossible to controll the obtained serum.
In the second place the question prevails, whether all streptococci,
cultivated from different processes of disease, must be regarded as
representatives of one aud the same species, and to be taken as
varieties, or that the mutual affinity is much smaller. A solution of
this question in such a sense, that it might be possible to come to
a rational sub-division in the large group of the pathogenic strepto-
cocci, would be of great importance for the bloodserum-therapy.

Some years ago ScHoTTMiLuer *) tried to give a new division, based
on biological grounds instead of the older morphological division in
streptococcus longus and streptococcus brevis (voN LINGELSHEIM ’),
Benrine*). By cultivating different races of streptococci on blood-
agar, he was enabled to discern two types: firstly dark grey
colonies with lucid area, secondly greenish ones without area. The
streptococci, belonging to the first group, are very virulent for man and
are found in erysipelas, septicemia, scarlatina, phlegmon etc., while
those, belonging to the second group, are generally less pathogenic
for man and animals. Therefore ScHoTrmtLierR divides the pathogenic
streptococci as follows: |

1. Streptococcus pyogenes s. erysipelatos.

2. Streptococcus mitior s. viridans.

3. Streptococcus mucosus.

Several other investigators (Eve. FRaEnKeL ‘), SILBERSTROM °),

1) Miinch. Med. Wochenschrift, 1903, N°. 20, S. 849; N°. 21, S. 909.

*) Zeitschrift fiir Hygiene, Bd. X, S. 331.

8) Centralblatt. fiir Bakteriologie, Bd. 12, S. 192.

4) Miinch. Med. Wochenschrift, 1905, N°. 12, S. 548; N°. 39, S. 1869.

5) Centralblatt fiir Bakt., le Abth., Orig., Bd. 41, S. 409.

( 338 )

Baumann?) have latterly come to the same result in an almost
similar way.

BrsREDKA *) on the contrary tried to separate the different strepto-
cocci from each other with the aid of the method of the combination
with the complement. The conclusion to which he comes, is, that
the “substances sensibilisatrices’ present in his serum, are “rigoureu-
sement” specific; that the serum of a horse, immunized with the
streptococcus A, only contains the amboceptor A", which corresponds
with that special streptococcus. Thus he found this amboceptor A"
not only active against the streptococcus A, but also against other
races (B,C), from which Brsrepka decides on the identity or at
least on the near relationship of the above mentioned streptococci
A, B and C.

According to these results, some experiments have been taken by
me, to trace, in how far a separation of the different pathogenic
streptococci is really possible by means of the specific action of the
amboceptors.

The anti-streptococcus serum, which I usec, Prof. Spronck willingly
provided me with, for which I offer him my best thanks as well
as for his further assistance in my work.

The above mentioned serum originated from a horse, which was
injected for a great length of time viz. from Jan. 1905 till July 1906,
with a number of specimens of streptococci and staphylococci of
different origin. These injections, which were subcutaneous, took place
weekly. The quantities used were gradually increased during the
first months; whilst after that on an average 40—60 c¢.C. of a mix-
ture, composed of even parts of a culture in ascitic-bouillon of the
different streptococci and of a bouillon-culture of the staphylococci,
were administered. The mixture was twice heated for half an hour to
55° C. Strepto- as well as staphylococci originated directly from man,
without passage through animals.

That the serum really possesses curative qualities is evident, not
only from observations in the clinical surgery, but also from experi-
ments upon animals. Rabbits, which were injected with a mixture
of strepto- and staphylococci, could be kept alive by administering
comparatively small quantities of the anti-streptococcus serum, whilst
animals used for controll died shortly after.

The method, followed by me, is that of Borper—Gerneou'). For

1) Miinch. Med. Wochenschrift, 1906, N*. 24, S. 1193.
2) |. ¢.
s) lic.

( 339 )

each experiment six tubes were used, which contained consecutively *) :

1. a eed eee Oe complement, °*/, e.C. emulsion of streptococci,

/10 2
‘/, ¢.C. anti-streptococcus serum.
N°. 2: ?/,, ¢.C. compl., '/, c.C. emulsion of str., */, c.C. normal
horse-serum.
N°. 3: 7/,, ¢.C. compl., */, e.C. physiological NaCl, */, c¢.C.
anti-streptococcus serum.

Nees eA compl. “/, ¢C. physiol. NaCl, '/, c.C. normal
horse-serum.

N°. 5: ?/,, c.C. physiol. NaCl, */, c.C. emulsion of sir., '/, c.C.
anti-streptococcus serum.

N°. 6: ?/,, eC. physiol. NaCl, */, e:C. emulsion of str., '/, ¢.C.
normal horse-serum.

The tubes are stirred and then remain at the same temperature as
the room. Afler 3—5 hours to each of the tubes is added */,, ¢.C.
of a mixture, composed of 2 ¢.C. of hemolytic serum and 1 c.C.
corpuscles of a rabbit, which were suspended in physiol. NaCl to
remove the adherent serum. Very soon, mostly within ten minutes
the tubes 2, 3 and 4 distinctly show the phenomenon of hemolyse;
which is naturally not brought about in tubes 5 and 6, the com-
plement being absent. The absence or presence of an amboceptor
in the examined serum is proved by the existence or non-existence
of the hemolyse in the first tube.

It is necessary to repeat all these controll-experiments each time;
firstly, because some streptococci produce a hemolysin at their growth ;
secondly, because bacteria are able to combine the complement with-
out the aid of an amboceptor, although in a much smaller degree.
This may be observed very distinctly in vitro; for instance: in six
tubes successive dilutions of a culture of diphtheria bacilli were made;
to each tube 7/,, ¢c.C. of the complement was added. After three
hours ?/,, ¢.C. of a mixture, composed of 2 ¢.C. of hemolytic serum
(heated to 56° C.) and 1 c.C. corpuscles of a rabbit, suspended in
physiol. NaCl, was added. The result after half an hour was as

1) As complement, the fresh blood-serum of a guinea-pig was used. The strep-
tococci, which were to be examined, were cultivated on LoEFFLER’s coagulated
blood-serum and after 24 hours suspended in physiological NaCl to a homogeneous
emulsion. The antistreptococcus serum was heated in advance for one hour to
56° C., as well as the fresh normal horse-serum, used for controll, and the hemolytic
serum originating from guinea-pigs, which were treated 3 or 4 times with 5 c.G.
of defibrinated blood of rabbits. The physiological NaCl used, was always a solution

f 0,9/).

( 340 )

follows: no hemolyse in the first (least diluted) tube, a little hemo-
lyse in the 2.4 tube, more and more hemolyse in tubes 3, 4 and 5
whilst in the sixth (most diluted) tube it was perfect.

The same experiment was made with different other bacteria with
a similar result.

It may be easily understood, that it sometimes occurs, that no
hemolyse is formed in the first of the tubes, used in the method of
BorpDET—GENGOU, in consequence of a surplus of bacteria, as is seen
by the absence of the hemolyse in the second tube at the same time.
Without the controll-tubes, one might wrongly decide on the presence
of an amboceptor in the examined serum.

In the first place an investigation was made, whether in the anti-
streptococcus-serum, used by me, an amboceptor was present against
some five streptococci used at the immunization. The result was
positive. After this, different other streptococci were investigated.
These streptococci originated directly from different diseases of man,
such as: scarlatine, cholecystitis, septicemia, febris puerperalis, angina,
and had not served at the immunization. Among these streptococci
there were some of patients who during their lifetime had been
injected with the same anti-streptococcus serum, but without success.

The latter streptococci were cultivated from the blood or from the
spleen post mortem. Others were cultivated from patients with whom
the injections of the serum had had a very distinct curative effect.
It was therefore supposed that against the first streptococci no
amboceptor would be found in the anti-streptococcus serum.

The investigation however did not confirm this supposition. Add
streptococci, no matter what their origin, showed a strong combination
with the complement under the influence of the anti-streptococcus serum.

Keeping to the specific of the amboceptors, the conclusion of
BrsreDKA ') might be accepted, regarding all the latter streptococci as
identic or at least closely related to those used at the immunization.
Continued experiments with some pathogenic streptococci originating
from animals, have led to a different interpretation. A streptococcus
was used, which was cultivated from the lungs of a guinea-pig,
which had died spontaneously from pneumonia; further the well-
known streptococcus equi and a couple of other streptococci, which
were cultivated with the Str. equi from pus, originating from horses
suffering from strangles. Also against the latter streptococci, the
presence of an amboceptor in the used serum was an undoubted fact.

Considering that the str. equi by its qualities, apparently from its

Hl.

( 341 )

deviating growth on the usual culture-media, shows very distinct
differences from the other pathogenic streptococci, whether from man
or from animals, the conclusion is at hand, that at least in the anti-
streptococcus serum, used by me, very little of the specific working
of the amboceptor is left. It is however quite possible that all patho-
genic streptococci, originating from man as well as from animals,
are very closely related, by which supposition one might keep at
least to the specific of the amboceptors.

However later experiments have shown that the anti-streptococcus
serum is also active against micro-organisms, which do not belong to
the streptococci viz, pneumococci and meningococci.

By the above is fully shown, tbat the specific action of the ambo-
ceptors in the serum of a horse to which large quantities of strepto-
cocci have been administered for a very long time, has strongly
decreased and made room for a more general action. Probably this
general working might be put to the account of one and the same
amboceptor, although the presence of more amboceptors in the same
cannot be denied.

The above mentioned serum exercises, though in a small degree,
also a distinctly sensitive action on anthrax, typhoid and tubercle bacilli.

From the above it appears, that the method of the combination
with the complement of BorpeEt—GeENcou, is not to be used, if it is
necessary to distinguish nearly related bacteria from each other, which
in other ways are also difficult to separate.

Granted that it must be accepted, that such a diminution of the
specific activity only takes place with sera of animals which have
been treated for a great length of time, so that the specific activity
of the amboceptor is more asserted in proportion to the shorter time
in which the animals are immunized, it is evident here, that there
is no question about a certain method being used, because one never
knows, — and this is also the case with sera of animals which have
only shortly been immunized .— how far the specific action extends.
Even if it may be accepted that the horse, from whom the anti-strepto-
coccus serum originates, is a most favourable test-animal as regards
the forming of anti-bodies, then the above mentioned facts would
remain the same.

Dorper’) has recently found, that the amboceptor, present in the
serum of a horse which has been treated with dysenteria bacilli
(type Suga) during 18 months, next to the action on these bacilli,

1) Annales de I'Inst. Pasteur, T. 19. 1905, p. 753:

( 342 )

also presented the self-same effect against the so-called pseudo- or
para-dysenteria bacilli (type Frexner, Kruse). Asserting the specific
activity of the amboceptor, he decides on “‘l’unité specifique”’ of
the dysenteria bacilli. This conclusion appears to me, looking at the
above, very venturesome.

At the same time it is evident, that we must not attach too much
importance to the presence of an amboceptor in a serum for the
effect of that serum. It is not to be accepted, that the anti-streptococ-
eus serum will have a favourable effect on patients suffering from
pneumonia, typhus, anthrax ete. although a certain effect is to be
observed in vitro against the respective causes of these diseases. I
purposely treated this for anthrax bacilli. Different guinea-pigs of
nearly the same weight received partly a small quantity of anti-
streptococcus serum (2—3c.C.), which contained some anthrax bacilli
(one eye of a deluted twelve hours, old culture on bouillon-agar),
partly normal horse-serum (2—3c.C.) with an equal dose of anthrax
bacilli. A favourable effect of the anti-streptococcus serum compared
to normal serum was never perceptible. The animals died generally
about the same time, within 48 hours.

Yet PrepTETscHENSKY'), who has made such investigations with
rabbits, is of opinion that a favourable effect can be perceived from
anti-diphtheria as well as from anti-streptococcus serum, but the colossal
quantities of serum, which he used, justify the supposition, that here
is Only question of the favourable effect, which, as is known, is already
produced in several cases by the injection of normal horse-serum.

It is therefore not permissible, to ascribe a favourable effect to a
serum by force of the presence of an amboceptor, still less, to base
on this a quantitative method for the determination of the force of
such a serum, such as Konig and WassrrMann’) do with regard toa
meningococcus serum prepared by them. In the meningococcus serum
of Jocumann (E. Merck) the presence of an amboceptor could not
only be clearly discerned against meningococci, but also, naturally
in a smaller degree, against some streptococci.

The question, if such a diminishing of the specifie activity in
relation to a prolonged administering of antigens is known for other
substances in immune-sera too, must be answered in the affirmative.
This is especially the case with regard to the precipitins. It is well known
that it is not possible to obtain them absolutely specific. Thus NurraL *
was able to get a precipitation with the blood-serum of all kinds
1) Centralblatt fiir Bakt., le Abth., Ref., Bd. 38, $, 395.

*) Deutsche Med. Wochenschrift, 1906, n° 16, S. 609.

*) Blood immunity and blood relationship, Cambridge, 1904, p. 74, 135, 409.

C3359)

of mammals even with a very strong precipitin-serum, which was
obtained with and against an arbitrary mammifer-albumen (‘“‘“mamma-
lian reaction’). Hauser’) comes to a similar result; only quantitative
differences remain. 3

Also with relation to the amboceptor such a diminution of the
specific action seems to me sufficiently well pointed out.

Physics. — “Arbitrary distribution of light im dispersion bands, and
its bearing on spectroscopy and astrophysics.” By Prof. W. H.
JULIUS.

In experimental spectroscopy as well as in the application of its
results to astrophysical problems, it is customary to draw conclu-
sions from the appearance and behaviour of spectral lines, as to the
temperature, density and motion cf gases in or near the source of
light.

These conclusions must in many cases be entirely wrong, if the
origin of the dark lines is exclusively sought in absorption and that
of the bright ones exclusively in selective emission, without taking
into account the fact that the distribution of light in the spectrum
is also dependent on the anomalous dispersion of the rays in the
absorbing medium.

It is not in exceptional cases only that this influence makes itself
felt. Of the vapours of many metals it is already known that they
bring about anomalous dispersion with those kinds of light that
belong to the neighbourhood of several of their absorption lines’). In
all these cases the appearance of the absorption lines must to a greater
or less extent be modified by the above mentioned influence, since the
mass of vapour, traversed by the light, is never quite homogeneous.

Hence it is necessary, separately to investigate the effect of dis-
persion on spectral lines; we must try to separate it entirely from
the phenomena of pure emission and absorption.

A first attempt in this direction were the formerly described
experiments with a long sodium flame’), in which a beam of white

1) Miinch. Med. Wochenschrift, 1904, n° 7, S. 289.

*) After Woop, Lummer and Prinesuem, Expert, especially Pucctanti has inves-
tigated the anomalous dispersion of various metallic vapours. In Nuovo Cimento.
Serie V, Vol. IX, p. 303 (1905) Pucctanti describes over a hundred lines, showing
the phenomenon.

5) W. H. Junus, “Dispersion bands in absorption spectra.” Proc. Roy. Acad.
Amst. VII, p. 184—140 (1904).

( 344 )

light alternately travelled along different paths through that flame.
With these relative displacements of beam and flame the rays of the
anomalously dispersed light were much more bent, on account of the
uneven distribution of the sodium vapour, than the other rays of the
spectrum; absorption and emission changed relatively little. The
result was, that the distribution of the light in the neighbourhood of
D, and D, could be made very strongly asymmetrical, which could
easily be explained in all details as the result of curvature of the
rays. The existence of “dispersion bands” was thus proved beyond
doubt.

But the pure effect of emission and absorption was not absolutely
constant in these experiments and concerning the density of the sodium
vapour in the different parts of the flame only conjectures could be
made. Moreover, the whirling ascent of the hot gases caused all rays,
also those which suffered no anomalous dispersion, sensibly to deviate
from the straight line, so that the phenomena were too complicate
and variable to show the effect of dispersion strictly separated from
that of emission and absorption.

So our object was to obtain a mass of vapour as homogeneous as
possible and, besides, an arrangement that would allow us to bring
about arbitrarily, in this vapour, local differences of density in such
a manner, that the average density was not materially altered. The
absorbing power might then be regarded as constant. At the same
time it would be desirable to investigate the vapour at a relatively
low temperature, so that its emission spectrum had not to be
reckoned with.

In a series of fine investigations on the refractive power and the
fluorescence of sodium vapour R. W. Woop’) caused the vapour to
be developed in an electrically heated vacuum tube. It appeared
possible, by adjusting the current, to keep the density of the vapour
very constant. Availing myself of this experience I made the following
arrangement for the investigation of dispersion bands.

Apparatus.

NWN’ (see fig. 1) is a nickel tube of 60 centimetres length, 5.5 ems.
diameter and 0,07 cm. thickness. Its middle part, having a length
of 30 cms., is placed inside an electrical furnace of Hrraxus (pattern
E 3). Over its extremities covers are placed, the edges of which fit
into circular rims, soldered to the tube, and which consequently

1) R. W. Woop, Phil. Mag. [6], 3, p. 128; 6, p, 362,

( 345 )

shut air-tight when the rims are filled with cement. When the
furnace is in action a steady current of water, passing through the
two mantles M and JM’, keeps the ends of the tube cool. Each of
the two caps has a rectangular plate glass window and also, on both
sides of this, openings a and 6 (6' and a’), placed diametrically

Fig. 1.

opposite to each other and provided with short brass tubes, the
‘purpose of which will appear presently. Moreover in one of the two
caps (see also fig, 2) two other short
tubes c_and d are fastened in openings:
through c¢ the porcelain tube of a Lr
CHATELIER pyrometer is fitted air-tight,
while on d a glass cock with mercury
lock is cemented, leading to a mano-
meter and a Geryk air-pump. As soon
as the sodium (a carefully cleaned
piece of about 7 grammes) had been
pushed to the middle of the tube in
a small nickel dish provided with elas-

Fig. 2. tic rings, the tube had been immedi-

ately closed and exhausted.

We shall now describe the arrangement by which inside the mass
of vapour arbitrary inequalities in the density distribution were pro-
duced. It consists of two nickel tubes A and B of 0,5 cm. diameter,
leading from a to a’ and from & to 0’ and so bent that in the heated
middle part of the wide tube they run parallel over a length of 30
centimetres at a distance of only 0.8 ems. In the four openings of
the caps, A and B are fastened air-tight by means of rubber packing,
This kind of connection leaves some play so that by temperature
differences between the wide and the narrow tubes these latter need

( 346 )

not alter their shape through tension. At the same time the rubber
insulates A and B electrically from NN’. The four ends of the
narrow tubes which stick out are kept cool by mantles with streaming
water (these are not represented in the figure).

If now an electric current is passed through A or B, the tempera-
ture of this tube rises a little above that of its surroundings; if an
air-current is passed through it, the temperature falls a little below
that of its surroundings. The intensities of the currents and, conse-
quently, the differences of temperature can in either case be easily
regulated and kept constant for a long time.

I, Fig 3 gives a sketch of the whole arrangement.

The light of the positive carbon Z is concentrated by
K the lens / on a screen Q, having a slit-shaped aper-

ture of adjustable breadth. The lens / forms in the

plane of the slit S of the spectrograph a sharp image
of the diaphragm P. The optical axis of the two lenses
passes through the middle of the tube containing the
sodium vapour, exactly between the two small tubes

| A and B.
Nn’ If now the opening in the diaphragm P has the
it shape of a vertical narrow slit and if its image falls
, al exactly on the slit of the spectograph, then in this latter
i the continuous spectrum of the arec-light appears with
great brightness. If the tube NN’ is not heated, D,
and J, are seen as extremely fine dark lines, attri-
| buted to absorption by the sodium, which is always
| present in the neighbourhood of the carbons. In order
that this phenomenon might always be present in the
| field of view of the spectograph as a comparison

----ObF--- witenédncacut

---OL5

spectrum, also when the tube is heated, a small totally
reflecting prism was placed before part of the slit S,

| to which part of the principal beam of light was led

« q by a simple combination of lenses and mirrors without
passing the electric furnace. So on each photograph that

i =F
was taken the unmodified spectrum of the source is
: also seen.

The spectral arrangement used consists of a plane
diffraction grating 10 cms. diameter (ruled surface 8
by 5 ems.) with 14436 lines to the inch, and two sil-

“a vered mirrors of Zrtss; the collimator mirror has a
Fig. 3. focal distance of 150 ems., the other of 250 cms. Most
of the work was done in the second spectrum.

“06>

( 347 )

When heating the sodium for the first time a pretty large quan-
tity of gas escaped from it (according to Woop hydrogen), which of
course was pumped off. After the apparatus had functionated a
couple of times, the tension within the tube remained for weeks
less than 1 mm. of mercury, also during the heating, which, in the
experiments described in this paper, never went beyond 450°. The
inner wall of NWN’ and also the small tubes A and B are after a
short time covered with a layer of condensed sodium, which favours
the homogeneous development of the vapour in subsequent heatings.
It is remarkable that scarcely any sodium condenses on the parts of
the tube that stick out of the furnace, so that also the windows
remain perfectly clear. The density of saturated sodium vapour at
temperatures between 368° and 420° has been experimentally deter-
mined by Jewrtt'). He gives the following table.

temperature | density

368° | 0 00000009
373 / 0.00000020
376 | 000000035
380 | 000000043

: 385 | 000000103
387 / 000000135
390 ~ 0.00000160
295 - 0.00000270
400 | 0 00000350
406 | 0.00000480
408 | 0.00000543
M2 | 0.00000590
418 | 0.00000714
490 | 0.00000750

These densities are of the same order of magnitude as those of
mercury vapour between 70° and 120°. At 387° the density of

1) F. B. Jewerr, A new Method of determining the Vapour-Density of Metallic
Vapours, and an Experimental Application to the Cases of Sodium and Mercury.
Phil. Mag. [6], 4, p. 546. (1902).

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 348 )

1 :
saturated sodium vapour is about T000 of that of the atmospheric air

at O° and 76 ems.

Observations.

If we now regulate the intensity of the current in the furnace in
such a manner that the thermo-couple indicates a steady temperature
(in many of our experiments 390°), then within the tube the density
of the vapour is not everywhere the same, to be sure, for the
temperature falis from the middle towards the ends, but since the
surfaces of equal temperature are practically perpendicular to the
beam of light, ail rays pass nearly rectilimearly through the vapour.
Accordingly the spectrum is only little changed; the two D-lines
have become somewhat stronger, which we shall, for the present,
ascribe to absorption by the sodium vapour in the tube.

We now blow a feeble current of air through the tube A which
thus is slightly cooled, so that sodium condenses on it, the vapour
density in its neighbourhood diminishing. We soon see the sodium
lines broaden considerably. This cannot be the consequence of in-
creased absorption, since the average vapour density has decreased
a little. The reason is that rays of light with very great refractive
indices are now bent towards q’ (fig. 3), rays with very small indices
towards g; hence in the image of the slit ? which is formed on,
rays belonging to regions on both sides of the D-lines no longer
occur, while yet this image remains perfectly sharp since the course
of all other rays of the spectrum has not been perceptibly altered.
If now at the same time the tube 6 is heated by a current of e.g.
20 Amperes, by which the density gradient in the space between
the tubes is inereased, the breadth of the lines becomes distinctly
greater still. The heat generated in the tube by the current is about
1 calory per second; it is, however, for the greater part conducted
away to the cooled ends of the tube, so that the rise of temperature
can only be small.

By switching a current key and a cock, A and 6 can be made
to suddenly exchange parts, so that Ais heated, 6 cooled. The dark
bands then shrink, pass into sharp D-lines and then expand again,
until, after a few minutes, they have recovered their original breadth.

Fine and sharp, however, the lines in the transition stage are
only if the temperature of the furnace is very constant. If it rises
or sinks the minimal breadth appears to be not so small. In this

( 349 )

case, however, there certainly exist currents in the mass of vapour
which cause the distribution of density to be less regular. Also when
A and #& are at equal temperatures, we sometimes see the sodium
lines slightly broadened; it stands to reason to attribute this also to
refraction in such accidental irregularities.

That spectral lines possess some breadth is commonly aseribed
either to motion of the light-emitting molecules in the line of sight
or to changes in the vibrational period of the electrons by the col-
lisions of the molecules. We now have a third cause: anomalous
dispersion in the absorbing medium. The whole series of phenomena,
observed in our sodium tube, corroborates the opinion that this latter
cause must in general be regarded as by far the most important. It
will appear that this conclusion holds not only for dark but also for
bright spectral lines.

If the slit in the diaphragm P is made much broader towards p’,
this has no influence on the spectrum as long as A and B are at
the surrounding temperature. The D-lines appear as in a, Pl. I. If
now A is cooled below this temperature, 6 raised above it, the
dark D-lines only broaden in the direction of the shorter wave-
lengths, while at the side of the longer wave-lengths the intensity of
the light is even increased, since now also anomalously bent rays
from the radiation field p’ can reach the point S through the slit Q.
(see 8, Pl. I). The spectrum @ passes into y when the temperature
difference between A and JB is made to change its sign or also when the
original temperature difference is maintained and the slitin P is made
much broader towards p instead of towards p’. A small shifting of
the whole diaphragm / (starting from the condition in which it was
when taking 8) so that S falls exactly in the shadow, causes the
spectrum d to appear, which makes the impression of an emission
spectrum of sodium with slightly shifted lines, although it is evidently
only due to rays from the field p’ which have undergone anomalous
dispersion in the vapour.

Let us now return to the diaphragm P with a narrow slit placed
on the optical axis. (A piece of glass coated with tiffoil in which
a slit was cut out, was generally used). The spectrum then shows
broad bands when there is a density gradient between A and B.
If beside the slit an opening is cut in the tinfoil, a group of rays
of definite refractivity (and consequently also of definite wave-lengthis)
is given an opportunity to reach S through Q, and a bright spot is
formed in the dark band, the shape of which depends on the shape
of the opening in the tinfoil, but is by no means identical with it.

23*

( 350 )

If e.g. the opening in the diaphragm has the shape

N of fig. 4, then the spectrum ¢ is obtained. When the
YY density gradient is diminished the figure shrinks, §; if
IN now the density gradient is made to change its sign
| A and to increase, the spectrum proceeds through the

J stages a (gradient exactly zero) and 4 to @.

Fig. 4. The relation between the shape of the opening
in the diaphragm and that of the bright spots in the spectrum might
easily have been foretold from the shape of the

Nie dispersion curve. Having, however, experimentally
ia found the relation between the two figures for a
sid /| \ simple case as the one above, it is not difficult to
a | \\\\. design for any desired distribution of light the shape
| of the required opening in the diaphragm. The
Fig. 5. flower c and its inversion x required the diaphragm,
represented in fig. 5. By reversing the gradient the image ¢ passes into z.
So in this way one may also arbitrarily produce duplications,
reversals, bright or dark ramifications of spectral lines and it would
e.g. be possible faithfully to reproduce all phenomena observed in
this respect in the spectra of sun-spots, faculae or prominences. On
Plate II a number of arbitrary distributions of light have been
collected. They were all produced in sodium vapour of 390° on the
average; a is again the spectrum with equal temperatures of the
tubes A and B. In vr on the dark dispersion band D, a bright
double line is seen, reminding us of the spectrum of the caleium
flocculi of Hair. In the same negative D, also shows a fine double
line, which however is no longer visible in the reproduction. The
spectra g, xy, w imitate the origin of a sun spot and prominence
spectrum; ~ namely represents the spectrum of the quiet solar limb
with radially placed slit; in % a prominence appears and a spot with
phenomena of reversal; w shows all this in a stronger degree. If
now the density. gradient is made to change sign, the image first
shrinks again to @ after which it expands to , in a certain sense
the inversion of y. The remarkable aspect of these gradual changes,
admitting of perfect regulation, is only imperfectly rendered by the
photographs.

The relation between the curvature of the rays and
the density gradient.

The question arises whether it is- probable that circumstances as
were realised in our experiments are also met with in nature, or in

ee Wet bl hee ce =~

wey

( 354 )

common spectroscopical investigations undertaken with entirely diffe-
rent purposes.
We remark in the first place that curiously shaped diaphragm

openings are not absolutely essential for the production of phenomena

as those described above. If e.g. our source of light had a constant,
say circular shape; if on the other hand the direction and magnitude
of the density gradient in our tube had not been so regular, but
very different in various places of the field reproduced by the lens F,
then the D-lines would also have shown all sorts of excrescences,
now determined by the configuration of the density distribution.

In the second place we will try to form some idea of the quan-
titative relations.

The radius of curvature @ of the path of the most deviated rays,
occurring in our photographs, may be easily estimated from the
distance d of the diaphragm to the middle of the furnace, the
distance J of the most distant diaphragm openings to the optical
axis, and the length / of the space in which the incurvation of the
rays is brought about. For:

rd Mar" a

Putting d=1 cm., d=110 ems., /=27 ems. this gives: e=3000 ems.
The average density 4 of the sodium vapour was in this case about
pee of that of the atmospheric air.

1000
Let us see how @ changes with the density gradient.
We always have:

v = rp ° ° ° . ° . ° 2 (1)

if n represents the local index of refraction of the medium for the
; : meee 1)

ray under consideration and x ae the change of this index per cm.
s

in the direction of the centre of curvature. Approximately we have,
for a given kind of light:
n—l1

= consiant = #

n—=RA+1
dn dL
——R

From this follows:

( 352 )

but since for rarefied gases n differs little from unity, even for the
anomalously dispersed rays which we consider, RA may be neglected
with regard to 1 and we may write

For every kind of light @ is consequently inversely proportional
to the density gradient of the vapour in the direction perpendicular
to that of propagation.

An estimate of the magnitude of the density gradient existing, in
our experiments, between A and £, may be obtained in two ways.
It may namely be inferred from the produced difference of tempe-
rature, or from formula (2).

The temperature difference between A and & would have been
pretty easy to determine thermo-electrically ; up to the present,
however, I had no opportunity to make the necessary arrangement.
Besides, the relation between the density distribution in the space,
passed by the rays, and the temperatures of A and £& cannot be so
very simple, since we have to deal not with two parallel planes but
with tubes, from which moreover hang many drops of liquid sodium.

: dh
The second method at once gives an average value of a for
s

n—I
the space passed by the rays. It requires a knowledge of R =

for a kind of ray for which in our experiments also @ has been
determined.

Now Woop (Phil. Mag. [6], 8, p. 319) gives a table for the values
of n for rays from the immediate vicinity of the D-lines. These data,
however, refer to saturated sodium vapour of 644°; but we may
deduce from them the values of x for vapour of 390° by means of
the table which he gives in his paper on page 317.

For, when we heat from 389° to 508’, the refractive power of the
vapour (measured by the number of passing interference fringes of

98
helium light 2 = 5875) becomes moe 11 times greater, and at fur-

0
ther heating from 508° to 644° again othe 12,5 times greater (now

found by interference measurement with light from the mercury line
4= 5461); hence from 390° to 644° the refractive power increases
in ratio of 1 to 11 « 12,5 = 137.

Since now for rays, situated at 0,4 AnGstrOm-unit from the D-

( 353 )

lines!) we have n—1=—= + 0.36, (as the average of three values
taken from Woop’s table on page 319), we ought to have with
sodium vapour at 390° for the same kind of rays

0.
Regent ==) NOR6.
137

The density A at 390° is, according Jewett, 0.0000016, hence
n—1 0.0026
A 0.0000016
Then from formula (2) follows
d& if L

ds Ro 1600 x 3000

ho == L600:

= 0,0000002.

Dispersion bands im the spectra of terrestrial sources.

It is very probable that, when metals evaporate in the electric
arc, values of the density gradient are found in the neighbourhood
of the carbons that are more than a thousand times greater than the
feeble density gradient in our tube with rarefied sodium vapour ’).

The radius of curvature will, therefore, in these cases be over a

thousand times smaller than 30 meters and so may be no more
than a few centimetres or even less. A short path through the vapour
mass is then already sufficient to alter the direction of certain rays
very perceptibly.
_ If now an image of the carbon points is produced on the slit of
a spectroscope, then this is a pure image only as far as it is formed
by rays that have been little refracted in the arc, but the rays which
undergo anomalous dispersion do not contribute to it. Light of this
latter kind, coming from the crater, may be lacking in the image
of the crater and on the other hand penetrate the slit between the
images of the carbon points. Thus in ordinary spectroscopic obser-
vations, not only broadening of absorption lines, but also of emission
lines, must often to a considerable extent be attributed to anomalous
dispersion.

1) The spectrum < in our plate shows that the extremities of the peaks corre-
spond pretty well to light of this wave-length; for they approach the D-lines to
a distance which certainly is no more than 1/;; of the distance of the D-lines
which amounts to 6 Anesrr.-units. For these rays the opening of the diaphragm
was 1 cm. distant from the optical axis.

*) If we e.g. put the vapour density of the metal in the crater, where it boils,
at 0.001, the density of the vapour outside the are at a distance of 1 cm. from
the crater, at 0.0001, then we have already an average gradient 5000 times as
large as that used in our experiments.

When we bear this in mind, many until now mysterious phenomena
will find a ready explanation. So e.g. the fact that Liverye and Dewar’)
saw the sodium lines strongly broadened each time when vapour
was vividly developed after bringing in fresh material, but saw them
become narrower again when the mass came to rest, although the
density of the vapour did not diminish. If by pumping nitrogen
into the evaporated space the pressure was gradually increased, the
lines remained sharp; but if the pressure was suddenly released, they
were broadened. All this becomes clear as soon as one has recog-
nised in the lines dispersion bands, which must be broad when the
density of the absorbing vapour is irregular, but narrow, even with
dense vapour, if only the vapour is evenly spread through the space.

Another instance. According to the investigations of Kayser and
Runce the lines, belonging to the second secondary series in the
spectra of magnesium, calcium, cadmium, zinc, mercury, are always
hazy towards the red and are sharply bordered towards the violet,
whereas lines, belonging to the first secondary series or to other series
are often distinctly more widened towards the violet. With regard
to the spectrum of magnesium they say:*) “Auffallend ist bei mehre-
ren Linien, die wir nach Roth verbreitert gefunden haben, dass sie
im Rowu1anp’schen Atlas ganz scharf sind, und dann stets etwas
kleinere Wellenlainge haben. So haben wir 4703,33, RowLanp 4703,17 ;
wir 5528,75, Rowianp 5528,62. Unscharfe nach Roth verleitet ja
leicht der Linie gréssere Wellenlainge zuzuschreiben; so gross kann
aber der Fehler nicht sein, denn die Row.anp’sche Ablesung liegt
ganz ausserhalb des Randes unserer Linie. Wir wissen daher nicht,
woher diese Differenz rithrt.’”’” Kayser has later*) given an explana-
tion of this fact, based on a combination of reversal with asymme-
trical widening; but a more probable solution is, in my opinion,
to regard the .widened serial lines as dispersion bands.

If we namely assume that, when we proceed from the positive
carbon point, which emits the brightest light, to the middle of the
are, the number of the particles associated with the second secondary
series decreases, then rays coming from the crater and whose wave-
length is slightly greater than that of the said serial lines will be
curved so as to turn their concave side to the carbon point. Their
origin is erroneously supposed to be in the prolongation of their
final direction, so they seem to come from the arc, and one believes

1) Livernc and Dewar, On the reversal of the lines of metallic vapours, Proc.
Roy. Soc. 27, p. 132—136: 28, p. 367—372 (1878—1879).

2) Kayser und Runee, Uber die Spektren der Elemente, IV, S. 13.

1) Kayser. Handbuch der Spektroskopie II, 5. 366.

( 355 )

to see light emitted by the vapour, in which light different wave-
lengths occur, all greater than the exact wave-length of the serial
lines. The observed displaced lines of the second secondary series
are consequently comparableto apparent emission lines of the spectrum
J of our piate I. 3

In this explanation things have been represented as if the light of
these serial lines had to be exclusively attributed to anomalous dis-
persion. Probably however in the majority of cases emission proper
will indeed perceptibly contribute to the formation of the line; the
sharp edge must then appear in the exact place belonging to the
particular wave-length.

How can we now explain that lines of other series are diffuse at
the opposite side? Also this may be explained as the result of ano-
malous dispersion if we assume that of the emission centres of these
other series the density icreases when we move away from the
positive carbon point. In this case namely the rays originating in
the crater, which are concave towards the carbon point and conse-
quently seem to come from the arc, possess shorter wave-lengths
than the serial lines, i.e. the serial lines appear widened towards
the violet. This supposition is not unlikely. For the positive and
negative atomic ions which according to Srark’s theory are formed
in the are by the collision of negative electronic ions, move in opposite
directions under the influence of the electric field; hence the density
gradients will have opposite signs for the two kinds. Series whose
lines are diffuse towards the red and series whose lines flow out
towards the violet would, according to this conception, belong to
atomic ions of opposite signs — a conclusion which at all events
deserves nearer investigation.

The examples given may suffice to show that it is necessary syste-
matically to investigate to what extent the already known spectral
phenomena may be the result of anomalous dispersion. A number
of cases in which the until now neglected principle of ray-curving
has undoubtedly been at the root of the matter are found in Kayszr’s
handbook II, p. 292—298, 304, 306, 348—351, 359—361, 366.

Dispersion bands in the spectra of celestial bodies.

Since almost any peculiarity in the appearance of spectral lines
may be explained by anomalous dispersion if only we are at liberty
to assume the required density distributions, we must ask when
applying this principle to astrophysical phenomena: can the values
of the density gradient for the different absorbing gases in celestial

( 356 )

bodies really be such, that the rays are sufficiently curved to exert
such a distinct influence on the distribution of light in the spectrum?

In former communications’) I showed that the sun e.g. may be
conceived as a gaseous body, the constituents of which are intima-
tely mixed, since all luminous phenomena giving the impression as
if the substances occurring in the sun were separated, may be
brought about in such a gaseous mixture by anomalous dispersion.
We will now try to prove that not only this may be the case, but
that it must be so on account of the most likely distribution of
density.

Let us put the density of our atmosphere at the surface of the

1
earth at 0.001293. At a height of 1050 cms. it is smaller by 760 of

this amount, so that the vertical density evadient is
0.001293
1050 s< 760

The horizontal gradients occurring in the vicinity of depressions

— 16 X 10-10,

i 1
are much smaller; even during storms they are only about oo

of the said value’). Over small distances the density gradient in the
atmosphere may of course occasionally be larger, through local heating
or other causes.

Similar considerations applied to the sun, mutatis mutandis, cannot
lead however to a reliable estimate of the density gradients there
occurring. A principal reason why this is for the present impossible
is found in our inadequate knowledge of the magnitude of the
influence, exerted by radiation pressure on the distribution of matter
in the sun. If there were no radiation pressure, we might presuppose,
as is always done, that at the level of the photosphere gravitation is
28 times as great as on the earth; but it is counteracted by radiation
pressure to a degree, dependent on the size of the particles ; for some
particles it may even be entirely abolished. The radial density gra-
dient must, therefore, in any case be much smaller than one might
be inclined to calculate on the basis of gravitational action only.

Fortunately we possess another means for determining the radial
density gradient in the photoshere, at any rate as far as the order
of magnitude is concerned. According to Scumipt’s theory the photo-
sphere is nothing but a critical sphere the radius of which is equal

') Proc. Roy. Academy Amsterdam, II, p. 575; IV, p. 195; V, p. 162, 589 and
662; VI, p. 270; VIII, p. 184, 140 and 323.

*) Arruenius. Lehrbuch der kosmischen Physik, S. 676.

( 357 )

to the radius of curvature of luminous rays whose path is horizontal
at a point of its surface. This radius of curvature is consequently
@ =7 X 10'° cms., a value which we may introduce into the expres-
sion for the density gradient:

dQ __ 1

ds ~~ Re

The refractive equivalent # for rays that undergo no anomalous
dispersion varies with different substances, to be sure; but in an
approximate calculation we may put A=0,5. Then at the height of
the critical sphere we shall have :

dh = at

ds (05X 7X 100
(this is 50 times less than the density gradient in our atmosphere).
All arguments supporting Scumipt’s explanation of the sun’s limb,
are at the same time in favour of this estimate of the radial density
gradient in the gaseous mixture.

Let us now consider rays that do undergo anomalous dispersion.
In order that e.g. light, the wave-length of which differs but very
little from that of one of the sodium lines, may seem to come from
points situated some are seconds outside the sun’s limb, the radius
of curvature of such anomalously bent rays need only be slightly
smaller than 7 x 10'° cms. Let us e.g. put

ob, -10" ¢:m-.

If we further assume that of the kind of light under consideration
the wave-length is 0.4 Angstrom-units greater than that of D,, then
for this kind of light A’ = 1600, as may be derived from the obser-
vations of Woop and of Jrewert'); we thus find for the density
gradient of the sodium vapour

dL a aoe 1

ds Ro 1600 X 6 & 1010
a quantity, 2900 times smaller than the density gradient of the
gaseous mixture.

— 0.29 < 10-10,

== 0.0001 < 10-10,

1
Hence if only 3000 part of the gaseous mixture consists of sodium

vapour, then, on account of the radial density gradient, the critical
sphere will already seem to be surrounded by a “chromosphere” of
light, this light having a striking resemblance with sodium light. This
kind of light has, so to say, its own critical sphere which is larger
than the critical sphere of the not anomalously refracted light. If the

1) See page 352.

( 358 )

percentage of sodium were larger, the “sodium chromosphere’ would
appear higher.

It is customary to draw conclusions from the size of the chromo-
spheric and flash crescents, observed during a total eclipse with the
prismatic camera, as to the height to which various vapours occur
in the solar atmosphere. According to us this is an unjustified con-
clusion. On the other hand it will be possible to derive from these
observations data concerning the ratio in which these substances are
present in the gaseous mixture, provided that the dispersion curves
of the metallic vapours, at known densities, will first have been
investigated in the laboratory.

Until now we only dealt with the normal radial density gradient.
By convection and vortex motion however irregularities in the density
distribution arise, with gradients of various direction and magnitude.
And since on the sun the resultant of gravitation and radiation
pressure is relatively small, there the irregular density gradients may
sooner than on the earth reach values that approach the radial
gradient or are occasionally larger.

The incurvation of the rays in these irregularities must produce
capriciously shaped sodium prominences, the size of which depends,
among other causes, on the percentage of sodium vapour in the
gaseous mixture.

So the large hydrogen and calcium prominences prove that rela-
tively much hydrogen and calcium vapour is present in the outer
parts of the sun; but perhaps even an amount of a few percents
would already suffice to account for the phenomena °*).

If we justly supposed that non-radially directed density gradients
are of frequent occurrence in the sun, and there disturb the general
radial gradient much more than on the earth, then not only rays
from the marginal region but also rays from the other parts of the
solar disc must sensibly deviate from the straight line. Chiefly con-
cerned are of course the rays that undergo anomalous dispersion.
Every absorption line of the solar spectrum must consequently be
enveloped in a dispersion band.

To be sure, absorption lines of elements which in the gaseous
mixture only occur in a highly rarefied condition, present themselves
as almost sharp lines, since for these substances all density gradients
are much smaller than for the chief constituents, and so the curvature
of the rays from the vicinity of these lines becomes imperceptible.

1) This result would be in accordance with a hypothesis of Scumipr (Phys.
Zeitschr. 4, S. 232 and 341) according to which the chief constituent of the solar
atmosphere would be a very light, until now unknown gas.

Proceedings Royal Acad. Amsterdam. Vo

( 359 )

Also of strongly represented elements some lines may appear sharp,
since not all lines of the same element, with given density, cause
anomalous dispersion in the same degree. Perhaps even there are
absorption lines which under no condition give rise to this pheno-
menon; though this were rather improbable from the point of view
of the theory of light.

Be this as it may, the mentioned limitations do not invalidate our
principal conclusion: that the general interpretation of the solar spectrum
has to be modified. We are obliged to see in FRAvUNHOFER’s lines not
only absorption lines, as Kircnyor does, but chiefly dispersion bands
(or dispersion lines). And that also on the distribution of light in
the stellar spectra refraction has a preponderant influence, cannot be
doubted either.

We must become familiar with the idea that in the neighbourhood
of the celestial bodies the rays of light are in general curved, and
that consequently the whole interstellar space is filled with non-
homogeneous radiation fields*) of different structure for the various
kinds of light.

Chemistry. — “On a substance which possesses numerous *) different
liquid phases of which three at least are stable in regard to
the isotropous liquid.” By Dr. F. M. Jazcrr. (Communicated
by Prof. H. W. Baxxuis Roozesoom).

§ 1. The compound which exhibits the highly remarkable phenomena
to be described, is cholesteryl-cinnamylate: C,,H,,0,C.CH:CHC,H,.

I have prepared this substance by melting together equal quantities
of pure cholesterol and cinnamyl-chloride in a small flask, which
was heated for about two hours in an oilbath at 190°. It is of the
greatest importance, not to exceed this temperature and the time of
heating, as otherwise the liquid mass, which commences to darken,
even under these conditions, yields instead of the desired derivative
a brown resin which in solution exhibits a green fluorescence.

1) Das ungleichmassige Strahlungsfeld und die Dispersionsbanden. Physik. Zeitschr.
6, S. 239—248, 1905.

2) In the Dutch publication, I have said: five. Since that time however, more
extended microscopical observation has taught me, that probably there are an
infinite number of anisotropous liquid phases, no sharply fixed transition being
observed in this manner. The hypothesis, that the transition of the first anisotropous
liquid phase into the isotropous should be continuous, would therefore be made
more probable in this way. However there are observed some irreversibilities by
passing from solid to liquid state and vice-versa, which yet I cannot explain at this
moment.

( 360 )

The solidified mass is dissolved in boiling ether, and the brown
liquid is boiled with animal charcoal for an hour in a reflux-appa-
ratus. To the filtrate is then added absolute alcohol, heated to 40°,
until the liquid gets turbid. On being set aside for a few hours the
ester deposits in small, almost white glittering leaflets. These are
collected at the pump, washed with a little ether-alcohol, and then
recrystallised several times from boiling ethyl acetate, to which each
time some alcohol may be added to precipitate the bulk of the ester.
The pure, quite colourless, beautifully crystallised compound shows
no heterogenous components under the microscope.

§ 2. The following experiments were carried out in the usual manner;
the substance was contained in small, thin-walled testtubes, whilst
surrounded of a cylindrical air-bath, and whilst the thermometer
was placed in the liquid completely which covered the mercury reser-
voir. The temperature of the oilbath was gradually raised with constant
stirring and now the following facts were noticed.

At about 151° the solid mass begins to soften‘) while brilliant
colours appear here and there at the sides, principally green and
violet, with transmitted light the complimentary colours red and
yellow. At about 157° the mass is a thick fluid and strongly doubly
refracting; the ground tone of the phase is orange-red, whilst, on
stirring with the thermometer, the liquid crystals everywhere form
links of lustrous bright green and violet slides. Afterwards, by the
construction of the cooling curve, I determined sharply the tempe-
rature ¢{155°.8 C.] at which the substance solidifies; the break in the
curve is distinct as the heat effect is relatively large and the under-
cooling was prevented by inoculation with a solid particle of the
ester.

The colour of the liquid phase is now but little changed on further
heating; on the other hand its consistency becomes gradually more
and more that of a thin liquid. At 199°.5 it is nearly colourless and
one would expect it to become presently quite clear.

But at that temperature the mass becomes all of a sudden enamel-
white, and rapidly thickens, while still remaining doubly-refracting.
We now observe plainly a separation into two liquid layers which
are here both anisotropous. The interference colours have now
totally disappeared. Then, on heating slowly, the liquid phase becomes
isotropous at 201.3° and quite clear. The isotropous liquid is colourless.

1) Bonpzynskt and Humnicki (Zeitsehr. f. physiol. Chem. 22, 396, (1896), describe
a cinnamylate which as regards solubility etc. agrees with mine, but which melts
at 149°. This is evidently identical with my first temperature of transition,

( 361 )

On cooling, the following phenomena occur: At about 200° the
isotropous liquid becomes turbid, at 198° the doubly-refracting mass
attains its greatest viscosity; at 196° it has already become thinner,
but now at about 198° it again becomes thicker and the whole
appearance of the phase is strikingly altered, although still remaining
doubly refracting. It then seems to pass gradually into the green
and red coloured, doubly refracting liquid phase, which, if we prevent
the undercooliug by inoculation, solidifies at 155°.8.

If the solid substance is melted under Leamann’s crystallisation-
microscope, — where the conditions. of experimenting are naturally
quite others than before, — it seems, that but one liquid phase, the
green and red coloured, is continually changed into the isotropous
one: no sudden changing is observed. On cooling, the aspect of the
anisotropous phase now obtained, is quite different from the first
mentioned.

I also think I must come to the conclusion that the liquid
phase (¢= about 190°) occurring on cooling is perhaps only the
passage to the other three, so that here, three stable liquid phases
might occur. It is very remarkable that the transitions of the
two stable anisotropous phases into the intermediate one appear,
when we work carefully, quite continuous; the viscosity appears
to pass gradually into that of the more stable phases. Remarkable
also is the impossibility to find the temperatures of transition
exactly the same on the rising, or falling, temperature of the
external bath. The values obtained for the initial and final tempe-
rature of each phase-traject vary within narrow limits. The same
is the case when, on melting the solid substance, one wishes
to determine the point where the first softening of the mass takes
place; in the determination of the temperature, intervals such as from
147° to 156° are noticed. The progressive change of the cooling of
isotropous-liquid to solid resembles here in a high degree a process
where a continuous transition exists between the different stadia. It
is as if the labile phase is composed of an entire series of condi-
tions which occur successively to form the connection on one side
between anisotropous and isotropous-liquid. The whole shows much
resemblance to a gradual dissociation and association between more
or less complicated molecule-complexes. It is quite possible that
the transitions solid-liquid occur really continuously instead of
suddenly, in which case an uninterrupted series of labile inter-

mediate conditions — which cannot be realised in most substan-
ces — are passed, some of which intermediate conditions might

be occasionally fixed in those substances which like these chole-

( 362 )

steryl-esters usually display the phenomena of the doubly-refracting
liquid-conditions. All this seems probable to the investigator, the more
so as it has been proved by Lrnmany, that in my other cholesteryl-
esters, even in the case of the caprinate, both or one of the two
anisotropous liquid-phases were always labile and only realizable on
undercooling; some of them, such as the zsobutyrate, only exhibited
their labile anisotropous liquid-phases, when containing some impurities
and not when in a pure condition. With the idea of a gradual
dissociation of compound molecule-complexes into more simple ones,
agrees the fact that the anisotropous liquid phases have never been
known yet to occur after the isotropous ones; this is always the
end-phenomenon, which is accounted for by the fact that a dissocia-
tion of this kind is always found to increase with a rise of temperature.

That the cooling between solid and anisotropous-liquid does not
proceed so suddenly as may be predicted from the great calorific
effect is shown in the case of the cinnamylate from the fact that,
after the solidification, particularly at the side of the test tubes, the
interference-colours, which are characteristic before the transition of
the phases into each other, remain visible for a very long time,
often many hours, then slowly disappear. Even with great enlarge-
ment, no well defined crystals can be discovered in those coloured
parts; the whole conveys the impression of a doubly-refracting, irre-
gular network of solidified liquid droplets, just like the liquid crystals
which present themselves to the eye with the aid af a powerful
enlargement *).

In these obscure phenomena we are bound to notice the more or
less labile and partially realized intermediate stadia in a continuous
transition liquid S$ solid. The view expressed by Leumany, that there
should be present a difference between the kinds of molecules in
the different aggregate conditions, is adopted here with this difference,
that such a difference of association of the molecules is thought quite
compatible with the phenomenon of the continwity of the aggregate
conditions, treated of here.

§ 3. I wish to observe, finally, that cholesteryl-cinnamylate when
subjected frequently to these melting experiments, soon undergoes
a small but gradually increasing decomposition, which shows itself
in the yellow colour of the mass and the fall of the characteristic
temperature-limits.

Zaandam, 26 Oct. ’06

1) A still more dislinct case of this phenomenon has now been found by me
in .-phytosteryl-propionate, which | hope soon to discuss in another communication.

( 363 )

Chemistry. “The behaviour of the halogens towards each other’.
By Prof. H. W. Bakuuis RoozEsoom.

_If the phase-doctrine in its first period was concerned mainly with

By an omission the pagination of the
Proceedings of the Meeting of Saturday
October 27, 1906 begins with page 249
instead of page 237, so pages 238-248
do not exist.

Finally it now appears from an investigation Dy MISS KARSTEN
that Chlorine and Bromine only give mixed crystals on cooling and
that in a connected series, whilst, in agreement with this no indication
for the existence of the compound in the liquid or vapour could be
deduced from the form of the boiling point line.

We, therefore come to the conclusion that IC], is a feeble and
ICl a strong compound. IBr is also a feeble compound and no com-
pound exists between Cl and Br. The combining power is, therefore,

1) Still closer than represented in Fig. 7, p. 540. These proceedings [VIII] 1904.
2) These proceedings VI, p. 331.

24
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 362 )

steryl-esters usually display the phenomena of the doubly-refracting
liquid-conditions. All this seems probable to the investigator, the more
so as it has been proved by Leamany, that in my other cholesteryl-
esters, even in the case of the caprinate, both or one of the two

conaitions, treated of here.

Zaandam, 26 Oct. ’06.

1) A still more distinct case of this phenomenon has now been found by me
in .-phytosteryl-propionate, which | hope soon to diseuss in another communication.

( 363 )

Chemistry. “The behaviour of the halogens towards each other’.
By Prof. H. W. Bakuuis RoozEBoom.

_If the phase-doctrine in its first period was concerned mainly with
the question whether two or more substances in the solid condition
give rise to chemical compounds, or mixed crystals, or remain un-
changed in the presence of each other, lately it has commenced to
draw conclusions from the form of the melting point lines of the
solid mixtures, both for the nature of those solid mixtures and of
the liquid mixtures into which they pass, namely whether, and to
what extent, compounds occur therein:

Likewise, the same questions may be answered in regard to liquid
and vapour from the equilibrium lines for those two phases, namely
boiling point lines or vapour pressure lines.

The three systems of the best known halogens having now been
investigated their mutual behaviour may be surveyed.

As regards chlorine and iodine, SToRTENBEKER had proved in 1888
that no other compounds occur in the solid condition but ICI, and
IC]. He also showed that it is probable that ICI, on melting, liquefies
to a very large extent without dissociation, whilst on the other
hand ICI, is almost entirely dissociated into IC] + Cl,.

Miss Karsten has now added to this research by the determination
of the boiling point lines. This showed that the liquid and the vapour
line approach each other so closely in the vicinity of the composition
ICl*), that the conclusion must be drawn that the dissociation of
IC] is also exceedingly small in the vapour, it being already known
that it is very large in the case of IC\I,.

From the investigation of Merrum TerwocrT?) it has been shown
that Br and I form only one compound BrI which in the solid state
forms mixed crystals both witb Br and I and which on account of
the form of the vapour pressure and boiling point lines is largely
dissociated in the liquid and gaseous states.

Finally it now appears from an investigation by Miss KarsTEn
that Chlorine and Bromine only give mixed crystals on cooling and
that in a connected series, whilst, in agreement with this no indication
for the existence of the compound in the liquid or vapour could be
deduced from the form of the boiling point line.

We, therefore come to the conclusion that ICI, is a feeble and
{Cl a strong compound. IBr is also a feeble compound and no com-
pound exists between Cl and Br. The combining power is, therefore,

1) Still closer than represented in Fig. 7, p. 540. These proceedings [VIII] 1904.

#) These proceedings VI, p. 331.

24
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 364 )

greatest in the most distant elements and greater in Br- I than in
Br + Cl.

From the researches of Moissan and others it follows that Fluorine
yields the compound IF, which is stable even in the vapour-condition.
With Bromine, the compound BrF, is formed but no compound is
formed with Chlorine. This, also, is in harmony with the above result.

As, however, the compounds with Fluorine have not been studied
from the standpoint of the phase-doctrine, there does not exist as yeta
reasonable certainty as to their number or their stability.

Mathematics. — “Second communication on the PuicKer equivalents
of a cyclic point of a twisted curve.” By Dr. W. A. VERsivys.
(Communicated by Prof. P. H. Scuours).

§ 4. If the origin of coordinates is a cyclic point (n,7,m) of a
twisted curve C the coordinates of a point of C lying in the vicinity
of the origin on a branch passing through the origin can be repre-
sented as follows:

eal’,
y = bt + Ob, i tr+l 4 by im trt+2 + ete.,
zo, Mtrt™ 4 ¢, ttt! + ¢, ttt? + ete.

Let g, be the greatest common divisor of m and r, let q, be that
of r and m, qg, that of m and n-+-r and finally g, that of m and
r+ m.

If @. =.= 4%: =% =1 the Puicker equivalents depend only
on n, r and m. In a preceding communication’) I gave the PLUcKER
equivalents for this special case ’).

§ 2. If the 4+ G. C. Divisors g are not all unity, the PLUickER
equivalents of the cyclic point (n,7,m) depend on the values of the
coefficients 4 and c, just as in general for a cyclic point of a plane
curve given by the developments:

2 ve

y = irtm td, te tmtl 4+ d, mtmt? + ete.,
the vanishing of coefficients d influences the number of nodal points
and double tangents equivalent to the cyclic point (n, m) *).

1) Proceedings Royal Acad. Amsterdam, Nov. 1905.

2) The deduction of these equivalents is to be found among others in my treatise :
“Points sing. des courbes gauches données par les equations: x=t, y=irb,
z= itnt+r+m,” inserted in “Archives du Musée Teyler”, série Il, t. X, 1906.

8) A. Brit and M. Noerner. Die Entwicklung der Theorie der algebraischen
Functionen, p. 400. Jahresbericht der Deutschen Mathematiker-Vereinigung, Ill,
1892—93.

( 365 )

If the coefficients c and 6 are not zero, if no special relations
exist between these coefficients and if besides n, 7 and m are greater
than one, the cyclic point (”,7, m) is equivalent to
n —1 stationary points 8 and to

fn —1)(n + 7— 3)+ 9, —1}:2 nodes H.

The osculating plane of the curve C in the cyclic point (n, 7, m)

is equivalent to
m —1 stationary planes @ and to
{(m — 1) (r + m— 3) + gq, —1}:2 double planes G.
The tangent of the curve C in the cyclic point (n,7,m) is equi-
valent to
7 —1 stationary tangents 4, to
fn — 1) (n +r — 3) + gq, —1}:2 double tangents » and to
{7 — 1) (r + m— 3) + q, —1}:2 double generatrices w’ of the
developable O formed by the tangents of the curve C.

§ 3. The cyclic point (n,7,m) of the curve C’ is an n+ r-fold
point of the developable O of which C is the cuspidal curve.

The cyclic point (‘n,7,m) counts for

(n+ r— 2)(n-+7r +m)
points of intersection of the cuspidal curve C with the second polar
surface of O for an arbitrary point.

Through the cyclic point (”,7,m) of the cuspidal curve C’ pass

{n(n + 2r + m— 4) + 9,—q,}: 2
branches of the nodal curve of the developable O.

All these nodal branches touch in the cyclic point (n,7,m) the
tangent of the cuspidal curve C (the «-axis).

They have with this common tangent in the point of contact

(a + 7) (n + Ar + m— 4) + g, — q,}: 2
points in common.

The nodal branches passing through the cyclic point (n, 7, m) all
have in this point as osculating plane the osculating plane z= 0 of
the cuspidal curve C.

These nodal branches have with their osculating plane z= 0 in
the cyclic point (n, 7, m)

(na + r+ m) (n+ 2r+m— 4)+ 9, —49,}:2

points in common.

§ 4. The case of an ordinary stationary plane «, the point of
contact of which is a eyclic point (1,1, 2), shows that through a
24*

( 366 )

cyclic point branches of the nodal curve can pass not touching: in
this point the cuspidal curve.

These intersecting nodal branches exist only when g, >1. If
r->1 the coefficients 5 and c must satisfy special conditions.
If r=1 then through the cyclic point (n,7r,m) of the cuspidal
curve pass either qg,:2, or (g,—1):2 of these nodal intersecting
branches. All intersecting nodal branches have a common tangent
in the plane z=0O if r=1.

§ 5. The case of an ordinary stationary point 8 (2,1,1) shows
that through a cyclic point of the cuspidal curve nodal branches
can pass which have the same tangent, but not the same osculating
plane as the cuspidal curve. These particular nodal branches exist
only when g,>1. If q,>>1 and m=1 these particular nodal
branches are always present. If g, > 1 and also m >1 the coefficients
6 and c must satisfy special conditions. These particular nodal
branches have in the cyclic point (n, 7, m) a common osculating
plane (differing from the plane z= 0) if m= 1.

§ 6. The tangent to C in the cyclic point (n,7,m) is an r-fold
generatrix g on the developable O. The r sheets of the surface O
passing through the generatrix g all touch the osculating plane z = 0
of C in the point (n, r, m).

The generatrix g moreover meets in g—(n-+2r-+™m) points &
a sheet of the surface O, when QO is of order og.

In every point R the generatrix g meets 7 branches of the nodal
curve. These 7 branches form, when m >~, a singularity (7, r, m—r)
and the osculating plane of these nodal branches is the tangent
plane of O along 4.

If m <r these r nodal branches form a singularity (7, m,r— m)
and the osculating plane .of these 7 nodal branches is the tangent
plane of O along the generatrix intersecting g in AR.

If » =m these r nodal branches form a singularity (7, 7, 1).

§ 7. In general the singular generatrix g will meet only nodal
branches in the cyclic point (n,7,m) and in the points A. Ifg, >1
the generatrix g may meet moreover nodal branches arising from
the fact that some of the 7 sheets, which touch each other along g pene-
trate each other. These nodal branches meet g in the same point Q.
If g, >1 and n=1 there is always such a point of intersection Q.
If g, >1 and n >1 the coefficients 6 and c must satisfy some special
conditions if the sheets passing through g are to penetrate each other.

( 367 )

Physics. — “On the measurement of very low temperatures. XII.
Determinations with the hydrogen thermometer.” By Prof. H.
KaMERLINGH OnneS and C. Braak. (Communication N° 95¢

from the Physical Laboratory of Leiden).

§ 1. Introduction.

The results of determinations of low temperatures made with the
bydrogen thermometer, which was described in Comm. N°. 27 (June
1896) and more fully discussed in Comm. N°. 60 (September 1900),
have already frequently been used, but no further particulars
have as yet been given about these determinations themselves. We
give them now in connection with a series of observations made in
1905 and 1906. They have served for the investigations described in
Comm. N° 952 and 95° (June 1906) and further for determinations
of isotherms of hydrogen at low temperatures, which will be discussed
in a following communication. Comprising also measurements on liquid
hydrogen, they extend over the whole of the accessible area of the
lower temperatures. All the precautions which proved necessary in
former years, have been taken. The temperature of the bath, in
which the thermometer was immerged, could be kept constant
to 0°.01 at all temperatures. It was therefore to be expected, that
the accuracy and reliability aimed at in the arrangement of the
thermometers, might to a great extent be reached.') In how far this
is really the case, the following data may show.

§ 2. Arrangement of the thermometer.

There is little to add to Comm. N°.60. The steel capillary con-
necting the thermometer bulb and the manometer, was protected from
breaking by passing a steel wire along it, the ends of which are
soldered to copper hoods, which may be slidden on the steel pieces
ce and ¢ (Plate Il, Comm. N°. 27) at the end of the capillary. The
dimensions of the thermometer reservoir of Comm. N°. 60 (80 ¢.M°.)
did not present any difficulty in our measurements, the bath in the
cryostats (see Comm. N°. 83, 94°, 94¢ and 94/ (May and June
1905 and June 1906)) offering sufficient room besides for the
thermometer and other measuring apparatus, for the stirring appa-
ratus, which works so thoroughly, that no variation of tempe-

1) A complete example of the determination of very low temperatures with the
hydrogen thermometer was as yet not found in the literature. Such an example
follows here. .

( 368 )

rature could be found') at least with the thermo-element’). The
section of the glass capillary forming the stem of the thermometer
was 0,0779 mM?. With regard to the temperature correction (see
§ 4, conclusion), it is desirable that this section is small. It appears
both from calculation and from observation’), that the equilibrium of
pressure between the space near the steelpoint and reservoir is still very
quickly reached with these dimensions of the capillary *), much more
quickly than the equilibrium of the mercury in the two legs of the mano-
meter, which is inter alia also confirmed by the rapidity with which
the thermometer follows fluctuations in the temperature of the bath’).
The determination of the pressure which is exerted on the gas,
may, when the determination applies to very low temperatures, be
simplified and facilitated by following the example of CHappuis‘)
and making the manometer tube serve at the same time as baro-
meter tube. The modification applied for this purpose to the arran-
gement according to Comm. N°. 60 Pl. VI, is represented on PI. I,
which must be substituted for part of Pl. VI belonging to Comm. N°. 60.
By means of an india-rubber tube and a T-piece /, the thermometer
(a, b, c, d, e, h, k) is connected on one side with the manometer
1, to which (see Pl. VI Comm. N°. 60) at m, the reservoir at
constant temperature is attached and at m, the barometer, on the
other side with the barometer tube (v,, 7, (airtrap) 7,). Besides from
the manometer and the barometer joined at m,, the pressure can
now also immediately be read from the difference in level of the
mercury in nm, and in g. We have not availed ourselves of this
means for the determinations discussed in this Communication.

§ 3. The hydrogen.
The filling took place in two different ways:

1) Travers, SenTeR and Jaguerop, (Phil. Trans. Series A, Vol. 200, Part. II, § 6)
who met with greater difficulties when trying to keep the temperature constant in
their measurements, had to prefer a smaller reservoir.

2) A resistance thermometer is more sensible (Cf. Comm. Nos. 95% and 95%),
As soon as one of suitable dimensions will be ready, the experiment will be repeated.

5) Calculation teaches that for reducing a pressure difference of 1 c.M. to one
of 0.01 m.M., the gas flowing through the capillary requires 0.1 sec., the mercury
in the manometer 4 sec. Experiment gives for this time 25 sec. This higher
amount must be due to the influence of the narrowing at the glass cock k.

4) We must be very careful that no narrowings occur.

5) A great deal of time must be given to exhausting the reservoir with the
mercury airpump when filling it, as the equilibrium of reservoir and pump is
established much more slowly than that between dead space and reservoir.

6) Travaux et Mémoires du Bureau International, Tome VI.

( 369 )

a. By means of hydrogen prepared in the apparatus of Comm.
N°. 27 with the improvements described in Comm. N°. 94¢ (June 1905)
§ 2. After having beforehand ascertained whether all junctures of
the apparatus closed perfectly, we maintained moreover all the time
an excess of pressure in the generator, in order to exclude any
impurity from the gas. The expulsion of the air originally present
in the apparatus was continued till it could be present in the gas at
the utmost to an amount of 0.000001.

b. By means of hydrogen prepared as described in Comm.
N°. 947 XIV. In order to apply this more effective mode of
preparation, we must have liquid hydrogen at our disposal. ') In § 7
the equivalence of the first method with the last is demonstrated
for measurements down to — 217°. It is still to be examined whether
systematic errors may result from the application of the first method
of filling, in measurements on liquid hydrogen by the deposition
of impurities, less volatile than hydrogen. ’)

§ 4. The measurements.

The zero point of the thermometer is determined before and after
every set of observations. Both for the zero point and for every
determination of temperature, an average value is derived from
three or four observations. Each of these observations consists of a
reading of the barometer, preceded and followed by a reading of the
manometer. The thermometers, indicating the temperature of the
mercury, Of the scale and of the gas in the manometer spaces are
read at the beginning and at the end of every observation. The tem-
perature of the room is kept as constant and uniform as possible.

The temperature of the thermometer reservoir is taken equal to
that of the bath. This is permissible for the cryostats described in
Comm. N°. 94¢ and Comm. N°. 94/ and the treatment given there.

The temperature of the bath is kept constant by means of the
resistance thermometer, described in Comm. N°. 95¢. In order to
facilitate the survey of the observations, the resistance was adjusted as
accurately to the same value as possible, and by means of signals the
pressure in the cryostat was regulated in such a way, that in the very
sensible galvanometer the mirror made only slight oscillations about

1) The hydrogen in the vacuum glass B (see Comm. N°. 94,7 XIV, fig. 4) proved
to evaporate so slowly, that a period of two hours was left for filling and
exhausting the thermometer again, which previously had been kept exhausted for
a long time, being heated during part of the time (cf. footnote 5 § 2).

*) In a former set of observations deviations were found, which in conjunction
with each other prove that the hydrogen must have been mixed with air.

( 370 )

its position of equilibrium. So far as it proved necessary, a correc-
tion curve was plotted of these oscillations (see Plate II Comm. N°. 83,
December 1902). As a rule, however, these deviations were so slight,
that they could be neglected.

Part of the capillary glass stem of the thermometer has the
same temperature as the liquid bath. The length of this part is
derived from the indications of a float!) in the cryostat, which is
omitted in the drawings, not to render them indistinct. In order
to find the distribution of temperature in the other parts of the
capillary within the cryostat, special determinations are made,
viz. a. by means of a resistance thermometer placed by the side of
the capillary, (see Comm. N°. 83 Plate II*)), 6. with the aid of a
thermoelement, whose place of contact was put at different heights
in the cryostat, the distribution of temperature in the cryostat was
examined for the case that liquid air, liquid ethylene or liquid
hydrogen was used as bath, and finally c. the distribution at other
temperatures of the bath was derived from this distribution. This
may be deemed sufficient, as the volume, the temperature of which

is determined, amounts only to of the reservoir, and as an error

|
3000
of 50° in the mean temperature of the capillary corresponds to
only 0°.01 in the temperature of the bath, while the agreement of
the observations sub a and 6 show that an error of more than 20°

is excluded.

§ 5. Calculation of the temperatures.

The calculation of the zero point is made by reducing the observed
pressure of the gas to that under fixed circumstances, the same as
taken in Comm. N°. 60. Put:

V, the volume of the reservoir at 0°.

u, the volume of that part of the glass capillary that has the same
temperature ¢ as the reservoir. As such is considered the part
immerged in the liquid bath, to which is added 2 cm. of the
part immediately above it.

u,' and w," the volumes of the parts of the glass capillary in the
cryostat outside the bath at temperatures ¢,' and ¢,".

u, the volume of the part of the glass capillary outside the cryostat
(u,') and of the steel capillary at the temperature ¢,.

1) For determinatiouis on liquid hydrogen no float was used. The level of the
liquid in the bath was derived from the volume of the evaporated gas.

2) The lowest part from %, to & 9 with close windings is 9 cm.., the part where
the windings are farther apart (about 20 cm.) reaches up to in the top of the
cryostat.

( 374 )

u, the volume at the steel point of the volumenometer.

8, and 8, the variation of the volume V, caused by the pressure
of the gas.

If Hy is the observed pressure, and H, and uw have the same
meaning as in Comm. N°’. 60, the temperature is found from the
formula:

Vil htthe tBu Ms) se
x ie 1+at, iT 1+ at," ED 1+ at, as aa a

=F) Y, +8, +4 +4, ee ae back)

The change of volume of the glass stem caused by the change of
temperature need not be taken into account, as little as that of w.
That of the thermometer reservoir has been calculated by means of a
quadratic formula, of which the coefficients 4, and /, have the fol-
lowing values: 4, — 23.43 X 10-*, £, = 0.0272 x 10%’).

Put

' "

Us Us u, u, > u }
= pe are 1+ at | 2)
! "” m . —
Mle tatu tus tute ee |—* |
then follows from the above for = ead
tatu —p + +%%,¢
t= ike 3
——— - + (3)
eee =) V, k,
Ary l+at

If the term with ¢ is omitted, we find an approximate value for
the temperature. Now ¢ may be calculated again, while in the term
with ¢ this value is substituted. This approximate calculation is quite
sufficient.

§ 6. Survey of a measurement.

The observations communicated in this §, yielded the temperature
corresponding to the electromotive force of the thermoelement deter-
mined in Table IV and V of Comm. 952 and corresponding to the
resistance measured in the observation given in Table I of Comm.
N°. 95¢ (in the last case even almost simultaneous).

i) These values have been derived from Comm. N°. 95%. They refer to the
determinations made in 1903 on the expansion of glass. If we calculate the tem-
peratures by means of the quadratic and cubic formula derived in the same Comm.
from the observations of 1905, we find but slight differences, which amount
respectively to —0°.014 and — 0°.016 at — 100°, and remain always below
0°.01 at — 200° and lower.

( 372 )

The tables I and II are analogous to those of Comm. N°. 60,
only column X has been added to the former, in which the readings
from the kathetometer scale are noted down. In every measurement
they are always determined, in order to be used, if necessary, as a
control for the readings by means of the standard scale, in connection

with the collimation differences of the telescopes.

TA Boa

DETERMINATION IN A BATH OF LIQUID HYDROGEN.

(ABOUT — 253°). READINGS.

May 5,’06,3.10-3.30) 4 | B

Point y RY Esa a (eS)

lower top | 23.00 | 9.0

meniscus rim | 26.01 | 9.0

Manometer

Barometer
EE EE Sr _

higher top | 15.10 | 7.9

meniscus rim | 18.43 | 7.9

lower top | 21.03 | 8.7

meniscus rim | 22.70 | 8.6
higher top | 25.82 | 9.6
meniscus rim | 27.98 |40.0

lower top 22.98 | 9.3

meniscusrim | 25.99 | 9.3

Manometer

higher top | 15.06 | 7.9

Point 14.76 | 8.0

meniscus rim | 18.42 | 8.0

C | D | BE | .F | as eogey | K
974 | 20.17 | 9.4
975 | 17.86 | 8.4
997 | 22.02 | 9.3 | 15.5 | 45.5 | 15.4
998 | 19.90 | 10.3 15.6 | 15.4 | 24.996
15.5 | 14.3
974 | 20.17 | 9.4
975 | 17.86 | 8.4 | 15.5 89.294
997 | 22.02 | 9.3 | 45.5
998 | 19.90 | 10.3 | 15.5
1058 | 28.07 | 41.3 | 45.7 103.279
4059 | 25.43 | 44.0 | 45.7
997 | 22.02'| 9.3 | 15.4 | 15.5 | 15.4
998 | 19.90 | 10.3 15.4 | 15.2
15.3 | 14.3
974 | 20.17 | 9.4
975 | 17.86 | 8.4 | 15.5
974 | 20.17 | 9.4
975 | 17.86 | 8.4

( 373 )

PA Bi ‘TT.
DETERMINATION IN A BATH OF LIQUID HYDROGEN
(ABOUT — 253°). CORRECTED AND CALCULATED DATA OF
THE OBSERVATION.

| A B' | C' | Dp! | F | F'

., | lower meniscus 296.55 296 .70 14.8 14.9 |
o
= height 1.39 44.9 | 14.8
= | higher meniscus 976.4 | 976.37 | 14.9 | 13.8 | 81.53 | 0.14
=

height 41.46

lower meniscus 297.46 297 .48 14.8
ol
= | height 0.77
=
= | higher meniscus | 1058.87 | 1058.90 | 45.0
=

height 0.83

The correction was applied for the difference in level of barometer
and manometer (cf. also Comm. N°. 60). In this way we find H7,
the pressure of the gas in the thermometer.

a Ay BEE LH,

DETERMINATION IN A BATH OF LIQUID HYDROGEN.
(ABOUT —253°). DATA FOR THE CALCULATION,

u, =0.0105 cm3
#,' =0.0126 » ' #, = — 162°

uq''=0.0140 » gO 0°
u, =0.6990 » a 149.5
ug =0.2320 » | a 14° .9

as —0. 1144 | > |

A p= 81.53 m.m,

V¥,= 82.265 cm’ |
2, = —0.004 » ,2,—-+ 0.0021 cm3
H,=1091.88 mm.

“*= 0.799 cm® |

( 374 )

From the indication of the float the value of u, is found. u,' and
u,' are chosen such that the circumstances are as closely as possible
equal to those for which the distribution of temperature in the
cryostat is determined. We get now the table III, in which A, is
the zero point pressure.

From these data with formula (3), where the value 0,0036627
of Comm. N°. 60*) was assumed for a, follows for the approximate

value of the temperature:

t = — 252°.964 |
and after application of the correction for the quadratic term :
t= — 252°.964 + 0°.035 = — 252°.93.

§ 7. Accuracy of the determinations of the temperature.

In order to arrive at an opinion about the error of the observations
with the hydrogen thermometer, we determine the differences of the
hydrogen temperatures found in different observations in which the
resistance was adjusted to the same value, reduction having been
applied for small differences left.

The mean error of a single determination derived from the diffe-
rences of the readings of the thermometer, which succeed each other
immediately, is on an average + 0°.0074, from which we derive
for the mean error of a temperature + 0°.0043, assuming that on an
average 3 observations have served to determine a temperature. As
a rule no greater deviations than 0°.02 were found between the
separate readings of one determination. Only once, on Oct. 27th ’05
(cf. Comm. N°. 95¢ Tab. I) a difference of 0°.04 occurred. Even at
the lowest temperatures only slight deviations occur. Thus on May
5th ’06 two of the observations in the neighbourhood of the boiling
point of hydrogen (cf. Comm. 957 Tab. VI,, observation N°. 30, and
Comm. N°. 95¢ Tab. I) yielded :

3°20’ — 252°.926

3458’ — 252°.929
the two others with another resistance :

2135/ —= 28 ole

3u 7’ — 252°.866 ”)

Determinations of one and the same temperature on different days

1) From the values of «# found by Cuappuis at different pressures and from
BertHetot’s calculations follows by extrapolation from Cunappuis’ value for
p= 1000 mM. 2 =0.00366262 for »=1090 m.M., from Travers’ value of a for
700 m.M. with the same data z = 0.00366288 for p=1090 m.M.

*) At both these temperatures the indications of the resistance thermometer were
not made use of, but only the pressure in the cryostat was kept constant. That in
spite of this the readings of the thermometer differ so little is owing to the great
purity of the liquid hydrogen in the bath.

“Qn the measurement of very

H. KAMERLINGH ONNES and C. BRAAK.
he hydrogen ther-

low temperatures. XIII. Determinations with t

mometer.”
Plate I.

Proceedings Royal Acad. Amsterdam. Vol. IX.

4
a7
e

is eae
i raat hel
te peel oy

( 375 )

with the same filling of the thermometer yielded the following results:
(cf. Comm, N°. 95¢ Tab. VI and N°. 95° Tab. I) ')

July 7, ’05 — 139°.867
Oct. 26%, ’05 — 139°.873
July 6%, ’05 — 217°.416
March 3'¢, ’06 — 217°.424
June 30%, *06 — 182°.730
July 6t, ’06 — 182°.728

For the deviation of tle determinations on one day from the mean

of the determinations on the two days follows resp.:

0°.003, 0°.004 and 0°.001 so mean 0°.0027,
which harmonizes very well with the mean error derived above for
a single observation*), from which appears at the same time that
different systematic errors are excluded. This justifies at the same
time the supposition from which we started, that the error in the
resistance thermometer may be neglected.

Determinations with different fillings agree very well.

The determinations made on July 6%, ’05 and March 3°4, ’06
with the thermometer filled with electrolytic hydrogen (see § 3) and
those made on June 30%, ’06 with the thermometer filled with
distilled hydrogen, give:

mean of July 6, 05 and March 3'¢, ’06 — 217°.420
June 30%, 06 5%50' — 217°.327
6" 5' — 217°.362 > mean — 217°.345.
6"25' — 217°.347

If the last temperature is reduced to the same resistance as the
first, we find — 217°.400, hence the difference of these values is
0°.020, from which, only one determination being made, we must
conclude, that also with regard to the filling systematic errors are
pretty well excluded down to — 217°.

§ 8. Results.

It appears from the foregoing that with our hydrogen thermometer
determinations of temperature, even at the lowest temperatures,

1) The temperature for June 30‘ ’06 given here differs slightly from that given
in Table I of Comm. N® 95¢, though both refer to the same resistance. This diffe-
rence is due to the fact that in Comm. N°. 95¢ the result of one reading has
been used, and here the mean has been given of more readings.

*) It gives namely for the probable error 0°.0029, so only a trifling difference
with the above.

( 376 )

may without difficulty be effected accurate to 0 if the requisite

precautions are taken. Though it is not certain that the determina-
tions in liquid hydrogen of the last series come up to this accuracy,
as there a systematic error caused by the filling may show its influ-
ence, which does not yet make its appearance at —217°, yet it
lies to hand to suppose, that, at least with the thermometer filled
with distilled hydrogen, also these temperatures may be determined
with the same degree of accuracy.

§ 9. Vapour tension of liquid hydrogen at the melting point.

By sufficiently lowering the pressure over the bath of liquid
hydrogen the temperature was reached at which the hydrogen in the
bath becomes solid. This temperature indicates the limit below which
accurate determinations are no longer possible by the method discus-
sed in this Communication.

It could be accurately determined by a sudden change in the
sound which the valves of the stirrer in the bath bring about. (See
Comm. N°. 947, XII § 3).

It appeared from the indication of the resistance thermometer that
the gas in the hydrogen thermometer had partly deposited. Hence
the pressure in the hydrogen thermometer gives the vapour tension
of liquid hydrogen near the melting point. For this we found:

Hae Doraa ‘i-m.").

§ 10. Reduction on the absolute scale.

The reduction of the readings of the hydrogen thermometer on
the absolute scale by means of the results of determinations of the
isotherms will be discussed in a following Communication.

§ 11. Variations of the zero point pressure of the thermometer.

It is noteworthy that the pressure in the thermometer in determi-
nations of the zero point slowly decreases. This change is strongest
when the thermometer has just been put together and becomes Jess
in course of time. This is very evident when the results of the
determinations made at the beginning of every new period of obser-
vation are compared, so after the thermometer has been left unused
for some time under excess of pressure.

Thus on the fifth of July ’05 shortly after the thermometer had

') For this Travers, Senter and Jaguerop (loc. cit., p. 170) find a value lying
between 49 and 50 m.m. The great difference is probably owing to the inferior
accuracy of these last determinations.

( 377)

been put together we found:
H, = 1093.10 mm.
whereas at the beginning of the two following periods of observation
was found:
on Oct. 13% ’05 Ff —— 1092.11 mm.
on Febr. 26% ’06 fee 1091.93. mm.

The determinations before and after every period of observation
give but slight differences when compared. As a rule the pressure
decreases slightly as in the second of the above-mentioned periods of
observation (March 7% ’06, H, = 1091.83 mm.), sometimes there is
a slight increase, as in the first period of observation (Nov. 2.4 ’05,
AH, = 1092.23 mm.) after observations under low pressure. Before
and after the last series of observations, when shortly after the
thermometer had been filled with distilled hydrogen, determinations
were made at — 183° and — 217°, this difference was particularly
large. The zero point pressure after the measurements was then
0.33 mm. larger than before them.

From earlier observations made with another thermometer the
“same thing appeared.

Thus on Nov. 19% ’02

H, = 1056.04 mm.
was found, and the pressure on June 8 ’04 was

AH, = 1055.48 mm.
while during further measurements up to July 7 ’O4 the pressure
retained a value which within the limits of the errors of observation
remained equal to this.

Cuappuis') found a similar decrease viz. 0.1 mm. in three months
with a zero point pressure of 1 M. of mercury.

Finally a decrease of the normal volume was observed by KuRNEN
and Rosson and by Kzxsom also with the air manometer (see
Comm. N°*. 88 (Oct. 1903) III § 3). The same phenomenon was
recently observed with the auxiliary manometer filled with hydrogen
mentioned in Comm. N°. 78 (March 1902), when it was again
compared with the open standard manometer. This comparison will
be discussed in a following Communication.

The possibility of there being a leak is excluded by the fact that
a final condition is reached with the thermometer.

It lies to hand to attribute the variations of the zero point to an

1) Nouvelles études sur les thermométres 4 gaz, Travaux et Mémoires du Bureau
International. T. XIII p. 32.

( 378 )

absorption which comforms slowly to the pressure. As to the
absorption of the gas in the mercury, its adsorption to the wall
and the interchange of gas with a thin layer between the wall and
the mercury they (and especially the last) may be left out of account,
though they are not rigorously zero. For with manometers, where
no influences but these can exert themselves, the pressure of the
gas is sometimes considerably raised during a long time, and not-
withstanding the variations of the normal volume are much slighter
than with the thermometers.

Consequently we shall rather have to think of a slow dissolving
in and evaporating from the layer of glue, which is applied between
the steel caps and the glass.

EAE AT A.
p. 193 1. 1 from top for : deviation, read : value of the deviations.
1. 2 from top for: largest deviation, read: of the largest
deviations. |
p. 195 1. 8 from bottom and |. 2 from bottom for: values read:
. quantities.
p. 196 1. 7 from top for: from, read : for.
1. 9 from top for: and are combined, read: and these
are given.

p. 198 1. 19 from top for: agree, read: correspond.
1. 6 from bottom in note, for: calculations, read: calculation
of the formulae we used.
l. 6 from top must be omitted: “are used”
p. 211 1. 16 from top for: with, read: containing also.

(November 22, 1906).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING
of Saturday November 24, 1906.

DOG

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 24 November 1906, DI. XV).

SO Bee me WL Ss.

F. Murrer: “On the placentation of Sciurus vulgaris”. (Communicated by Prof. A. A. W.
Hvsrecut), p. 380.

W. Burcx: “On the influence of the nectaries and other sugar-containing tissues in the
flower on the opening of the anthers”. (Communicated by Prof. F. A. F. C. Wenz), p. 390.

A. J. P. vAN DEN Brork: “On the relation of the genital ducts to the genital gland in
marsupials”. (Communicated by Prof. L. Boxk), p. 396.

H. Kameriincu Onnes and C. A. Crommenin: “On the comparison of the thermo-element
constantin-steel with the hydrogen thermometer”, p. 403.

W. Kaprreyy: “On a special class of homogeneous linear differential equations of the second
order’, p. 406.

J. C. Kiurver: “Some formulae concerning the integers less than n and prime to n”, p. 408.

H. J. Zwiers: “Researches on the orbit of the periodic comet Holmes and on the pertur-
bations of its elliptic motion”, IV. (Communicated by Prof. H. G. van DE SanDE BAKHUYZEN),
p- 414.

Frep. Scuvu: “On the locus of the pairs of common points and the envelope of the common
chords of the curves of three pencils” (1st part). (Communicated by Prof. P. H. Scnoure), p. 424.

E. E. Mocenporrr: “On a new empiric spectral formula”. (Communicated by Prof. P. Zeeman),
p. 434.

J. A. C. Oupemans: “Mutual occultations and eclipses of the satellites of Jupiter in 1908”,
2nd part. Eclipses. p. 444. (With two plates).

H. Kameruincu Onnes: “Contributions to the knowledge of the ¢-surface of vaN DER WAALS.
XI. A gas that sinks in a liquid”, p. 459.

25
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 380 )

Zoology. — “On the placentation of Sciurus vulgaris.” By Dr.
F. Mcrier. (Communicated by Prof. A. A. W. Huprecut).

(Communicated in the meeting of September 29, 1906).

I. The very eariiest stages. The ovule of Sciurus under-
goes its first developmental stages in the oviduct. Meanwhile the
bicornuate uterus has prepared itself for the reception of the ovule:
underneath the single layer of epithelium the mucosa, which meso-
metrially remains very thin, has become very strongly thickened,
so that an excentrical T-shaped slit is left open, the transverse part
of which lies closest to the mesometrium. A special arrangement
for the attachment of the ovules can nowhere be detected ; a sub-
epithelial zone is found to be richer in nuclei, however, than the
loose connective tissue, separating this layer from the muscularis.

Il. Preplacentary stages (From the arrival of the ovule
in the uterus until the first formation of the allantoid placenta).

The ovules fix themselves in varying numbers, to the right generally
more than to the left, at about equal distances on the anti-mesometral
(i.e. anti-placentary) uterine wall; they are fixed with their vegetative
poles: A pellucid zone is absent, on the other hand the ovule becomes
surrounded by a mass, formed from glandular secretions of cellular
origin from the uterine wall.

The ovules grow pretty quickly, for the greater part by dilatation
of the umbilical vesicle, which in these stages still forms the principal
part of the ovule. It is remarkable that the area vasculosa remains
so small, so that only entoderm and trophoblast form the wall of the
germinal vesicle over the greater part of the umbilical vesicle.

The uterine wall shows intense activity during this stage. Many
processes take place here in rapid succession and simultaneously.
They all start from the spot where the ovule has settled, and from
this point extend in all directions, successively reaching the spaces
of the uterine horn, left open between the fixations of the ovules,
as also the mesometrally situated parts; all these processes begin
sub-epithelially, gradually penetrating deeper and deeper. These
successive processes thus gradually give rise to dish-shaped layers
of varying structure, surrounding the ovule at the anti-mesometral
side and the character of which is most sharply pronounced in the
points that are at the greatest distance from the mesometrium. By
the extension of the anti-mesometral part of the long end of the

( 381 )

T-shaped slit, a broadening is brought about here, which, progressing
more and more in the mesometral direction, finally produces a space,
the cross-section of which presents a shape like that of a cone,
truneated mesometrally by the old transverse part of the T, and
bordered anti-mesometrally by a circular segment corresponding to
the umbilical vesicle. The ovular chambers, formed in this way,
have originated as the result of growth and extension of the anti-
mesometral uterine wall, as a consequence of which the parts of
the horn that connect them, are implanted at the mesometral side
of the foetal chambers and at the same time are bent in this direction.
The proliferation in the stroma tissue, beginning in the sub-epithelial
layer, squeezes the mouths of the glands asunder. Later the epithelium
in these latter degenerates, the walls agglutinate, finally only
remnants of glands are found in the more normal stroma under the
muscularis.

The processes by which the first formation of dish-shaped layers
takes place (the existence of which is always of a relatively short
duration, however) are the oedematous imbibition of the tissue and
disintegration of cell-elements, accompanying the proliferation of the
sub-epithelial multinuclear zone, the final result being a system of
cavities, separated by thin cell-partitions and filled with the products
of oedema and disintegration of tissue. This layer is externally
surrounded by layers which form the transition to the still normal,
although proliferating tissue, while at the opening they are more
and more separated by products of a later process.

Very remarkable is the appearance at this time of “giant cells”,
plasma-lumps of different size, which assume a dark colour and
contain many giant nuclei with a large nucleolus. Continuous layers
or more isolated groups are found as individual differences, apparently.
They lie mostly superficially, often connected with the trophoblast,
not with the vessels. At first sight one would feel inclined to regard
them as the foetal “suction roots” of the trophoblast, described for
Spermophilus by Ressex. Since all transitions are found between the
mucosa tissue and these elements (in some cases even the transition
having been followed up); since, moreover, they are found on the
ovule, in course of being dissolved in the surrounding mass; since,
on the other hand, in no case an origin from the trophoblast could
be made probable, the giant cells are for this and other reasons in
my opinion to be considered as a degenerative maternal formation,
as a symplasm. They cannot be identified with the “monster cells”
of Minor, ScHOENFELD and others, described for the rabbit, since these
elements are also found in Sciurus, only much later. Finally com-

25*

( 382 )

parative anatomical considerations plead for my opinion (see below).
The uterine epithelium gradually disappears in all places where
the germinal vesicle is in contact with it. There is never question of
proliferation now.

Relatively soon “already (even with a very extensive material the
transitions are difficult to follow) a second stage sets in, in which
by proliferation of the stroma cells, beginning from the surface, a
dish is formed of cubical cells with granulated plasm (decidua),
which extends more and more, while the above described cavities
disappear, probably by resorption under influence of the pressure.
In the mean time the decidua cells at the surface undergo further
alterations and are resorbed (very likely by the umbilical vesicle,
since in this and in the cells of the wall a similar substance can
be found), so that a fine meshy texture is formed by the peripheral
part of the cells remaining; by the pressure this meshy texture is
compressed to a thin layer of lamellar structure, which in its youngest
parts still shows the meshes. Vessels are not or scarcely found in
the decidua. The separation between the decidua and the little or
not changed subdecidual tissue outside it, is the limit to which the
differentiating processes in the wall have progressed, at the same
time approximately the limit, marking how far the agglutination
of the germinal vesicle with the wall has advanced ; it may therefore
be called “differentiation limit’.

At this time the above described giant cells become fewer and
fewer in number, have an increasingly degenerate appearance and
soon disappear altogether. At the mesometral side especially by
proliferation of the epithelium an increase in number and size of
crypts takes place (not of glands).

A second period in these pre-placentary stages is characterised for
the ovule by the origin of the amnion ete. The growing embryo
seeks place in an impression of the upper part of the umbilical
vesicle, which becomes more and more accentuated. At the umbilical
vesicle begins, starting again at the part that is most distant from
the embryo, an outgrowth of the trophoblast cells with their nuclei,
which process also continually advances towards the embryonic pole.

Characteristic for this stage are different processes beginning on
the ovule as well as on the uterus in equatorial bands above the
differentiation limit, and from here also proceeding mesomefrally on
all sides. For the ovule these processes consist in a proliferation .of
the trophoblast which will later cover the outer layer of the amnion
fold; irregularly placed, dark, polygonal cells with distinct borders
appear; on the surface of the trophoblast small cell-heaps rise every

( 382}

where. On the corresponding spot of the uterine wall a progressive
process sets in; first: formation of crypts by proliferation of epithelium,
at the same time thickening of the intermediate stroma; later by
this process a ring has been formed, which everywhere projects a
little into the lumen above the differentiation limit, dividing the cavity
of the foetal chamber into a mesometrally situated placentary part,
and an omphaloid part situated below it, while by this arrangement
the meanwhile completed diplotrophoblast (chorion) with its very
strongly thickened layer of trophoblast bulges out. The hyperplastic
stroma of the projecting ring is everywhere well provided with
crypts.

In the meantime this proliferation process has been closely followed
by a regressive one; the epithelium begins to degenerate, first at
the surface, later deeper and deeper in the formed crypts; plasm
and nuclei become darker, more homogeneous, smaller; later the
pyenotic nuclei dissolve in the plasm and a mass is formed, epithelial
symplasm, in which finally greater and smaller vacuoles are evenly
distributed. Everywhere short ramifications of epithelium have pene-
trated into the stroma, which soon degenerate. Also the stroma
itself undergoes similar alterations later.

Wherever this degeneration has somewhat advanced, a third process
sets in, likewise extending as a band: the thickened trophoblast
penetrates with its ramifications some distance into the crypts, later
also between these into the degenerated mass. Here and there the foetal
mass thereby changes, after its elements have become enlarged and
paler, into a syncytium, the nuclei of which contrast well with those
of the maternal symplasm. In this connecting ring the syncytium
soon disappears again; extension causes the trophoblast with its
hollow ramifications, penetrating into the crypts, to become a single
layer once more; exactly here the area vasculosa still remains for a
time in connection with the trophoblast: everything pleads, in my
opinion, for the hypothesis that this has to be considered as a rudi-
ment, namely of an omphaloid placentation (Sorex, which among the
Insectivora stands nearest the Rodents, shows a distinct omphaloid
placentation).

The products of the crypts and glands, transsudates and symplasm
masses, are shed over the cupola of the diplotrophoblast and probably
are resorbed by this latter.

The vessels in the meantime proliferate strongly in the wall of
the mesometral part of the foetal chamber between the crypts, which
has remained unchanged yet; their wall consists as before of simple
endothelium, without a perivascular sheath.

( 384 )

Ill. Placentary stages (After the beginning of the for-
mation of the allantoid placenta). In the omphaloid part of the foetal
cavity the wall is more and more attenuated by extension and re-
sorption of tissue, although the layers may be recognised as before.
The increase in size of the trophoblast cells of the umbilical vesicle,
which had set in formerly, now leads to the formation of true
“monster cells”, the cellular body of which often shows concentric
rings and other peculiarities, while the big nucleus often lies like a
crescent round a vacuole. This process comes nearer and nearer the
mesometrally situated formations. The entoderm, covering these monster
cells, is very narrow and _ small-celled; where it covers the area
vasculosa, it consists on the other hand of cubical, strong cells. In
the umbilical vesicle a coagulated mass is always present. The large
embryo more and more invaginates the upper part of the umbilical
vesicle. Between the monstercells and the entoderm a sort of cuticle
develops.

The processes, extending in equatorial bands, continally advance
towards the mesometral pole of the foetal chamber, also in the
partitions of the foetal chambers, so that they are more and more
incorporated by these latter. In this manner extremely complicated
pictures are formed, especially in cross-sections.

The dilatation now affects very strongly as well the placentary
part of the foetal chambers as their mutual connecting pieces, so that
the omphaioid part becomes smaller and smaller, while the formerly
existing comb-shaped division between them disappears.

The progressive process finally reaches the mesometral pole of the
placentary space and continually advances further into the connecting
pieces of the foetal chambers: the still intact part of the wall, which
at first had the shape of a cupola, later assumes the form of an 8,
finally reduced to two round planes, which by the proliferation are
more and more limited to the connecting pieces. The progressive
process now forms crypts, which in other places are narrow and
deep, but in the place of the placenta are broad and wide by
dilatation and excessive proliferation of the stroma. The epithelium
has many layers, its surface still rises everywhere in papillae. In
the stroma not all the cells reach their full development as decidua
cells simultaneously, so that a peculiar reticulated aspect is produced.
Also the vessels increase.

In this soil now the degenerative process occurs, again advancing
centripetally towards the mesometral pole. The epithelium becomes
a symplasm, exactly like that described above, but this time more
abundant and, everywhere covering the trophoblast. In the stroma

( 385 )

a conjunctival symplasm is formed, and, as was the case in the
hyperplastic process, not everywhere simultaneously, so that small
partitions of symplasm still surround more healthy groups. Outside
the placentary trophoblast all this goes on until a single mass
of epithelial and conjunctival symplasm is formed; within reach
of the placentary trophoblast, however, the trophoblast has already
penetrated before that time. During this degeneration also vessels
are opened, so that extravasates are not rare now and altered blood
is found against the trophoblast.

The third process by which the trophoblast is connected with
the uterine wall, consists in the formation of hollow, one-layered
invaginations in the erypts, which trophablast papillae are covered
by caps of symplasm; between the crypts the trophoblast is extremely
thin by extension, often irrecognisable, also when later the forma-
tion of giant cells had advanced thus far: these latter are then very
long and narrow.

Finally the placentary trophoblast (which now forms if it were
the keystone of the cupola of the diplotrophoblast and consists of a
distinct basal layer of cells of one cell thickness and an often 20
cells thick layer above it) now lays itself everywhere against the
papillae of the mesometral cupola of the placentary foetal chamber
cavity, which papillae are in progress of being degenerated ; the
trophoblast papillae are likewise still covered by the symplasm,
when between them this has already been resorbed.

Next comes the formation of a foetal syncytium from the super-
ficial layers of the thickened placentary trophoblast, the process
beginning above and centrally and proceeding centrifugally downwards ;
the nuclei which at first were dark and small, become larger and
clearer, contain one big nucleolus and are clearly distinct from all
maternal elements. This syncytium everywhere penetrates into the
maternal tissue in strands, so that an intimate interweaving of
maternal and foetal tissue results, proceeding centripetally into the
papilla. Then everywhere ‘vacuoles’ are formed in this mass
(probably now for the first time at the expense of maternal tissue),
which, when they become larger, bend the basal trophoblast layer
(cytotrophoblast) inwards and finally fill with maternal blood. The
allantois has meanwhile penetrated into the trophoblast papillae and
is divided into small lobes by the growth of pairs of bulges of
the cytotrophoblast. Foetal vessels soon penetrate freely into each
lobule.

The primordium of the placenta as a whole has no round shape,
the edges facing the connecting pieces of the foetal chambers are

( 386 )

concave to these sides, corresponding to the shape of the surface,
here occupied by the progressive and regressive processes.

Gradually all the maternal tissue is replaced by foetal, so that
finally the papillae which at first were entirely maternal, have become
entirely foetal. Now the “vacuoles”, surrounded by foetal syncytium
dilate further (also grow at the expense of a foetal symplasm which
now forms everywhere) and subdivide, a process, accompanied by
constantly increasing separation of the allantoic villi by these
cavities, containing maternal blood; the final result is that papilla-
shaped lobes with secondary lateral lobules are formed, all separated
by allantois-strands with foetal vessels; these are surrounded by the
eytotrophoblast, which in its turn surrounds the “vacuoles” (now
identical with cavities, containing maternal blood), enclosed by a
layer of syncytium which at first is broad, later becomes gradually
narrower. The placenta, originated in this way rests on a substructure
of maternal tissue, composed of the same elements as formerly
(decidua, ete.); the decidua-cells often grow out strongly, while the
border between foetal and maternal tissue is in many places marked
by a narrow streak of symplasm. The formation of trophoblastic
giant cells gradually reaches also the supra-placentary parts, so that
here also the enormous cells (later often free) lie in the maternal
tissue.

Outside the placenta a stage soon is reached in which the progressive
and regressive processes, described above, have attained their extreme
limit. Superficially all has been changed into symplasm, only in the
depth decidua-cells still exist, which meanwhile, since the degene-
ration does not reach to this depth, have become pretty large. The
parts of the mucosa spared by these processes, are only the mucous
membrane of the dilated connecting pieces, now entirely incorporated
in the foetal chambers and whose mucosa, attenuated by extension,
only possesses crypts still, that are squeezed flat, and a rather thick
epithelium which for a part turns into symplasm. Against all these
extra-placentary parts lies the extra-placentary trophoblast, now con-
sisting entirely of giant cells which at present often get loose and
then lie freely amid the decidua.

The embryo has, during its further growth, found place in the
umbilical vesicle which gradually has become entirely invaginated
and whose walls almost touch each other. The edges of the bowl,
thus originated and containing the embryo, are not formed by the
sinus terminalis; this latter lies further down in the inner wall. The
small space in the umbilical vesicle is still filled with coagulating
masses, while the entoderm, covering the area vasculosa, which now

( 387 )

often forms papillae, has still a very healthy appearance. The outer
wall of this bowl never disappears. In its further growth the placenta
reaches the edges of the bowl of the umbilical vesicle, later still it
grows into it and coalesces with the inner wall: the sinus terminalis
then lies halfway the thickness of the placenta, while a fold of the
endoderm seems to have been incorporated into it.

In the last period of pregnancy, from the above described parts,
left free by the progressive and regressive processes, epithelium grows
between the degenerated and the normal part of the mucosa, perhaps
joins with the meanwhile proliferating glandular remains in the
depth: the umbilical vesicle is lifted off from the mucosa. Somewhat
later this begins also all round the placenta, so that at the end of
pregnancy this organ is more or less stalked and after parturition
the greater part of the uterine wall is already provided with anew
epithelium.

Comparative considerations. Among Rodents the in-
vestigation of the times at which various processes and organs of
the ovule (not of the foetus) are found, leads to the following series :
Seiurus — Lepus — Arvicola — Meriones — Mus — Cavia, in which
the first has retained the most primitive’ forms, Lepus in many
respects forms a transition to the last, in which more and more by
new processes coming to the fore, the old, primitive ones are sup-
planted, mixed up and altered, in a word become nearly irrecognisable.
Of this latter fact the study of the literature on the relation of ovule
and uterus in Rodents, gives sufficient evidence ; it also appears here
how great a support is afforded by a comparative anatomical in-
vestigation; even, that various problems cannot be solved without
its assistance.

The progression appears clearly in the pecularities of the umbilical
vesicle in the various animals: in all the upper part is invaginated
into the lower, with Sciurus not until late, with Cavia the process
is among the first; the distal wall always remains with Sciurus,
with Lepus it disappears late, with Cavia already quite at the be-
ginning; the endoderm covers the inner wall already very early in
Seiurus, very late in Mus, never entirely in Cavia.

In the same order the antimesometral fixation and the allantoid
placenta occur earlier and together with these the trophoblast thickening,
which causes them. It is exactly the remarkable pre-placentary
processes which have been so carefully studied with Mus and Cavia,
which by this replacing present the greatest difficulties.

With all Rodents the vegetative ovular pole becomes connected

( 388 )

with the anti-mesometral wall of the uterus. This connection only
ends in Sciurus towards parturition, in Mus and Cavia already very
early, in Lepus at an intermediate stage, by epithelium being pushed
underneath from the connecting pieces of the foetal chambers.

In this fixation the umbilical vesicle is surrounded by proliferating
mucosa tissue which later degenerates and is dissolved and resorbed
by the ovule. The epithelium soon disappears after slight progressive
changes, the stroma changes into decidua by very strong proliferation
which in Mus, Cavia, ete. rises as reflexa round the ovule, corre-
sponding with the smallness of the umbilical vesicle and consequently
of the ovule. In accordance with an existing inclination, in the
order of the above mentioned series, to replace nutrition by stroma
products by maternal blood, the vascularisation of the decidua is
very small in the squirrel, very strong in Cavia and correspondingly
the extravasates, surrounding the ovule are very rare in Sciurus,
common and abundant in Mus and Cavia.

In these processes in Sciurus maternal giant cells appear (sym-
plasm) and later foetal ones, when the former have disappeared.
In Lepus ScHorENFeELD and others found the foetal giant cells
(monster cells) already in earlier stages, in accordance with our
series; all the cells then occurring are by him considered as foetal;
probably, however, the maternal cells occur at the same stage and
part of the described cells are of maternal, symplasmatic origin. With
Mus both were found and distinguished by JrnkiNson at much earlier
stages, Konsrer did not see the foetal ones, DuvaL not the maternal
ones. So they must occur still earlier in Cavia; the foetal ones
are then probably the proliferating ‘‘Gegenpolcellen” of v. SpEx,
which perforate the zone at the vegetative pole; the maternal ones
correspond to the products of the processes in the “Implantationshof”’
of v. Spee. Also the disappearance of these formations takes place
at an increasing rate (By all this it becomes clearer still that the
comparison of Cavia and man by v. Spes, which already from a
phylogenetic point of view is hazardous, must be received with caution).

In the light of the comparative investigation these foetal ‘monster
cells’ may be considered as rudiments of an organ which was
strongly developed in the ancestors of the Rodents.

In Sciurus the mass surrounding the ovule (‘“coagulum’’) consists
especially of tissue products; these become less prominent in the
order of the series and are replaced by blood.

Of the omphaloid placentation, which in Sciurus is already rudi-
mentary, not much can be expected in the other members, although
the study (until now neglected) of the morphology of the extra-

( 389 )

placentary parts of the foetal chamber might perhaps shed light on
this subject.

The now following appearance of the allantoid placenta is found
latest in Seiurus, earliest in Cavia. The tendency, increasing in the
well-known order, to bring about as much as possible a nutrition
without tissue products of the mucosa of the uterus and an allantoidean-
placentary exchange between foetal and maternal blood, causes the
processes, playing a part in placentation, to change: in Sciurus we
still have a very strong hyperplasia of stroma-epithelium, later de-
generation, disintegration and resorption with penetration of the
trophoblast into this mass, all temporarily clearly distinct and rela-
tively slow, in Cavia we find almost exclusively vascular proliferation,
while proliferation and degeneration go hand in hand and the invasion
of the trophoblast follows closely on these, this latter process not
proceeding far and being soon finished (since the object: amener une
hémorrhagie maternelle a étre circonscrite par des tissus foetaux
(DuvaL), is sooner reached). In the other animals all intermediate
stages are found.

The later processes in the development of the placenta are in all
different, although they are alike in principle: subdivision of cavities
respectively vessels, containing maternal and foetal blood. The allantois
remains passive, the foetal mass grows further and further round
the allantois-ramifications, as it penetrates further into the cavity of
the foetal chamber.

The formation of foetal giant cells proceeds with all Rodents over
the whole trophoblast from the vegetative to the placentary pole;
also the decidual cells become larger, so that also the giant cells,
which in all have been found supra-placentary (as JENKINSON already
stated for the Mouse), are partly of maternal, partly of foetal origin ;
with Sciurus the two always remain easy to distinguish.

The more or less isolated place, which according to the statements
of authors, Lepus would in some respects occupy, will perhaps
disappear, when the until now somewhat neglected study of the
pre-placentary period will have been more extensively carried out
(also in regard to the morphology of the foetal chamber).

Finally I have not become convinced that also for the morphology
of the foetal chamber cavities the unity in the structural plan goes
for all Rodents as far as is claimed by Freiscumann; the difference
in the statements I met with, will however perhaps disappear when
all this has been studied with the aid of a more extensive material,
although FLEISCHMANN’s conceptions, for similar reasons, are certainly
incorrect in their present shape.

( 390 )

Botany. — “On the influence of the nectaries and other sugar-
containing tissues in the flower on the opening of the anthers.”
By Dr. W. Burcx. (Communicated by Prof. F. A. F.C. WeEnr.)

(Communicated in the meeting of September 29, 1906).

The consideration that the opening of the anthers is preceded by
a very considerable loss of water") and that with very many plants,
e.g. Compositae, Papilionaceae, Lobeliaceae, Antirrhineae, Rhinantha-
ceae, Fumariaceae and further with all plants, chasmogamous as
well as cleistogamous, which fertilise in the bud, this opening takes
place within a closed flower and consequently cannot be caused by
transpiration to the air, gave rise to the question whether perhaps
the nectaries or other sugar-containing tissues in the flower, which
do not secrete nectar outwardly, have influence on the withdrawal
of water from the anthers.

My surmise that also among the plants whose anthers only burst
after the opening of the flower, some would be found in which this
process is independent of the hygroscopic condition of the air, was
found to be correct. If the flowers are placed under a glass bell-jar,
the air in which is saturated with water-vapour, the anthers of many
plants burst at about the same time as those of flowers which are
put outside the moist space in the open air.

This led me to arranging some experiments, yielding the following
results :

1. If in a flower of Déiervilla (Weigelia) rosea or floribunda,
which is in progress of unfolding itself, one of the stamens is squeezed
by means of a pair of pincers, so that the drainage of water from
the stamen downwards is disturbed, the four anthers whose stamens
have remained intact, spring open, but the fifth remains closed.
With this plant it is not necessary to place the flower in a moist
space; the same result is generally obtained if the flower remains
attached to the plant.

If a flower is placed in the moist space together with the loose

1) This loss of water amounts e.g. with Fritillaria imperialis to 90°/, of the
weight of the anthers, with Ornithogalum wmbellatum to 86 °/o, with Diervilla
floribunda to 87/5, with Aesculus Hippocastanum to 88 °/), with Pyrus japo-
nica to 80"/,, with different cultivated tulips 59—68 °/), etc. With plants whose
anthers barst in the flower, the loss is smaller; the anthers and the pollen remain
moist then. With Oenothera Lamarckiana the loss amounts to 41%, with
Canna hybrida grandiflora to 56%, with Lathyrus latifolius to 24 °/.

( 391 )

anthers of another flower, those which are attached to the flower
spring open; the loose ones don’t. If only the corolla with the
stamens attached to it is placed in the moist space, the anthers open
as well as those of the complete flower. Consequently the nectary
which is found in the middle of the flower at the side of the ovary,
exerts no direct influence on the bursting of the anthers. If further
a stamen is prepared in its full length and placed in the moist space
together with some loose anthers, the anthers of the stamen burst,
whereas the loose anthers remain closed.

From these experiments we infer that the anthers open under the
influence of the stamen whether or not connected with the corolla,
Now an investigation with Ferxuine’s solution shows that as well
the stamen as the whole corolla and even the corollar slips, show
the well-known reaction, indicating glucose.

Of Digitalis purpurea two of the anthers of a flower in the moist
chamber, were separated from the corolla by an incision. The uncut
anthers burst open, but the other two remained closed. A stamen
prepared free over its full length causes the anther to burst in
the moist chamber; loose anthers, on the other hand, remain closed.

An investigation with Frauine’s solution showed that here also the
corolla contains glucose everywhere, but in especially large quan-
tities where the stamens have coalesced with the corolla. Also the
stamens are particularly rich in sugar over their entire length.

Of Oenothera Lamarckiana, the anthers of which burst already in
the bud, a flower-bud was deprived of sepals and petals. One of the
stamens was taken away from the flower in full length; of another
stamen only the anther was removed. These three objects were placed
together in the moist chamber. The anthers of the stamens which
had remained connected with the tube of the calyx and those of the
loose stamen sprang open; the loose anther, however, remained
closed. An examination with FrHLING’s solution gave the same result
as was found above with Digitalis.

Similar experiments were made with the flowers of Antirrhinum
majus L., Lamium album L., Glechoma hederacea L., Salvia argentea
L., Nicotiana afjinis Hort. and sylvestris Comes., and Symphytum
officinale L., which all gave the same results, while with the flowers
of Ajuga reptans L., Stachys sylvatica L., Scrophularia nodosa L.,
Cynoglossum officinale L., Anchusa officinalis L., Echium vulgare L.,
Calceolaria pinnata, Hibiscus esculentus, Anoda lavateroides, Malva
vulgaris Tr., Torenia asiatica, Corydalis lutea De., Colchicum autum-
nale L., Lysimachia vulgaris L., Atropa Belladona \.. and Rhinanthus
major Ehrh. the experiments were restricted to showing that with

( 392 )

all of them the anthers spring open in a space, saturated with water-
vapour. With all these plants the corolla and stamens react very
strongly with FEHLINe’s solution.

These experiments indicate that the water is withdrawn from the anthers
by an osmotic action, having its origin in the glucose-containing tissue.

I remark here that the presence of glucose — in so far as we
may infer it from the precipitate of cuprous oxide after treatment
with Frxunac’s solution — in other parts of the flower than the

nectaries proper and especially in the corolla, is a very common
phenomenon (to which I hope to return later) and that it is not
restricted to those flowers in which stamens and corolla have coalesced.
There is rather question here of a quantitative difference than of a
special property, peculiar to these flowers.

2. With Stellaria media the epipetalous stamens are mostly abor-
tive, while of the episepalous ones only three have remained, as a
rule. These three stamens bear at the base on the outside, a gland,
secreting nectar.

If a flower is placed in the moist chamber and one of the stamens
is injured with the pincers, the anthers of the uninjured stamens
will afterwards burst, but the other remains closed. And when loose
anthers from the flower are placed in the moist chamber, together
with an intact flower, the loose anthers remain closed, while the
anthers of the-flower open. As well the petals as the stamens preci-
pitate cuprous oxide from Frxuine’s solution; also the tissue at the
base of the sepals reacts with it. But the bursting of the anthers
stands in no relation to this; if the petals are removed, this has no
influence on the result of the just mentioned experiment.

The experiment indicates that the water is withdrawn from the
anthers by the osmotic action, proceeding from the nectary.

In this connection it deserves notice that the nectaries of the
epipetalous whorl and also those of the missing stamens of the epise-
palous whorl are abortive together with the stamens. The same is
observed with Cerastium semidecandrum l., C. erectum L. and Holo-
steum umbellatum L.; here also the nectaries of the missing stamens
have disappeared as a rule.

With the Papilionaceae, of which I investigated Lupinus luteus L.,
Lupinus grandifolius L., Lathyrus odoratus L., Lathyrus latifolus L.
and Vicia Faba L., the anthers are known to open already in the
closed flower. The petals precipitate cuprous oxide from FEHLING’s
solution, but exert no influence on the opening of the anthers. Flower
buds of Lathyrus latifolius and Lathyrus odoratus were deprived of

( 393 )

their petals and placed in the moist chamber together with loose
anthers. The loose anthers remained closed, but the others burst open.

In the same way as the flowers of Stedlaria media and the men-
tioned Papilionaceae, behave with respect to the opening of the
anthers in a space, saturated with water-vapour :

Stellaria Holostea L., St. graminea L., Cerastium Biebersteinii C.
arvense L., Cochlearia danica L., Sisymbrium Alliaria Scop., Crambe
hispanica L., Bunias orientalis L., Capsella Bursa pastoris Mnch.,
Hesperis violacea L., H. matronahs L., Thlaspi arvense L., Alyssum
maritimum Lam., and further Lychnis diurna Sibth., Silene inflata
Sm. Galium Mollugo L., Asperula ciliata Rochl., Campanula media
L., C. latifolia L.

With all these plants the bursting of the anthers must, in my
opinion, be ascribed to the influence of the nectaries.

With Hesperis two large nectaries are found at the inner side of
the base of the two short stamens and between these and the four
long stamens. If a flower of Hesperis violacea or H. matronalis L.,
after being deprived of its petals and sepals, is placed in the moist
chamber, nearly always the four long stamens only burst; the other
two remain closed.

It has been repeatedly observed that the secretion of nectar begins
as soon as the stamens open.

In connection with what was stated above, one would be inclined
to infer from this that flow of water from the anther causes the
secretion of nectar. If, however, with Stellaria media, the anthers
are removed before they have discharged water to the nectaries, one
finds all the same the nectaries amply provided with honey, when
the flower opens. The same may be observed in the male flowers
of Aesculus Hippocastanum. In the still nearly closed flowerbud
the nectary is dry yet. When the flower continues to open small
drops of liquid are seen to appear on the surface of the nectary,
still before the anthers extend halfway from the bud. These droplets
increase in size as the anthers approach the moment in which they
open. By weighing it may be proved that the anthers have already
lost part of their original weight when the first droplets of nectar
appear on the surface of the nectary. From this circumstance also
one would be inclined 1o infer that the water of the anthers comes
out again as nectar. When, however, from very young buds, whose
nectary is not moist yet, the anthers are removed, yet at a later stage
of development of the bud, secretion of nectar is found in them as
in buds that have kept their anthers.

( 394 )

With Fritillaria imperialis 1 found the same; but here the secretion
of nectar was not so abundant as in buds, the anthers of which had
not been removed. In my opinion these observations indicate that
the sugar, stored up in the nectaries or other sugar-containing tissues
of the flower, at the moment when it begins to exert its osmotic
action, attracts water not only from the anthers but also from other
parts of its surroundings.

3. With the following plants the anthers remain closed in a space,
saturated with water-vapour. In so far as they possess nectaries,
these latter appeared to exert no influence on the bursting of the
anthers.

ranunculus acris L., R. bulbosus L., Aquilegia vulgaris L.,
Clematis Vitalba L., Chelidonium majus L., Brassica oleracea L..,
Geranium molle L., G. Robertianum L., G. macrorhizum L., Geum
urbanum L., Rubus caesius L., Philadelphus coronarius L., Heracleum
Sphondylium. L., H. ianatum Miche, Aegopodium Podagraria Spr.,
Carum Carvi L., Pimpinella magna L., Valeriana officinalis L.,
Ligustrum vulgare L. Majanthemum bifolium De., and Iris Pseuda-
corus L.

It is remarkable that Brassica oleracea L. forms an exception to
what is otherwise generally observed with the Cruciferae; the position
of the stamens with respect to the nectaries which secrete honey
abundantly, would make us expect that in a moist chamber they
would behave like the others. The same remark holds for the species
of Geranium.

The secretion of nectar in the flower attracted the attention of
various investigators many years before SprenceL published his view
of the matter. Also after SpreneEL, in the first half of the preceding
century, it has many times been the object of investigation. All these
investigators agreed in being convinced that, apart from the signi-
ficance of the honey-seeretion for the fertilisation of the flowers by
the intervention of insects, to which SprenceL had drawn attention,
the sugar-containing tissues and the secreted liquid were still in
another respect useful to the plant.

After Darwin had in 1859 brought to the front again SprRENGEL’s
observations on the biological significance of the various properties
of the flower — which observations were falling more and more
into oblivion — and had accepted their consequences by bringing
them into relation on one hand with his conceptions about the
necessity of cross-fertilisation for the maintenance of the vital energy

( 395 ) *

of the species, on the other hand with the theory of natural selection,
the investigation of still another significance of the nectaries for the
plant was for a long period entirely abandoned.

Not until 1878 this subject was again broached by Bonnigr *) who,
in his extensive paper on the nectaries, in which as well the ana-
tomical as the physiological side of the problem were submitted to
a very extensive investigation, proved that sugar-containing tissues
in the flower and especially in the immediate vicinity of the ovary
are not only found with plants which regularly secrete nectar during
the flowering, but also with such plants as under normal conditions
never secrete:such a liquid. With these plants, which in the literature
on flower biology are called “pollen flowers’, since the insects find
no nectar in them, he found as well sugar-containing tissues as in
the so-called ‘insect flowers”. Even with anemophilous plants he
found “nectaires sans nectar’, e. g. with Avena sativa, Triticum
sativum and Hordeum murinum. A number of plants which under
ordinary conditions of life contain no nectar, he could induce to
nectar-secretion by placing them under conditions, favourable for
this purpose.

At the end of his paper he reminds us that an accumnlation of
reserve materials, wherever a temporary stagnation in the develop-
ment exists, may be considered a very general and well characterised
phenomenon. When a plant stops its further development at the end
of its growing period, it has stored up reserve material in its sub-
terranean, parts and when the seed has finished its development, it
has aceumulated nourishing substances in the endosperm or in the
cotyledons of the embryo. These reserve materials, turned into assi-
‘milable compounds, then serve for the first nutrition of the newly
formed parts.

He then arrives at the conclusion that in the vicinity of the ovary
saccharose is stored up, and that this reserve substance after fertili-
sation and in the same proportion as the fruit develops, passes partly
or entirely into the tissue of the fruit and into the seed, after having
first been changed, under the influence of a soluble ferment, into
assimilable compounds.

Investigation showed me also that the accumulation of saccharose
as a reserve substance in the flower is a very common phenomenon *).

1) Gaston Bonnier. Les nectaires. Etude critique, anatomique et physiologique.
Annales des sciences naturelles. Tome VIII. 1878.

2)On this point see also: Paut Knuru, Uber den Nachweis von Nektarien auf
chemischem Wege. Bot. Centralbl. LXXVI. Band, 1898, p. 76 and Ros. SrAceEr,
chemischer Nachweis von Nektarien bei Pollenblumen und Anemophilen. Beihefte
zum Bot. Centralbl. Band XII. 1901, p. 34.

26
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 396 )

But besides the function, discovered by Bonnier and the signi-
ficance of the secreted nectar for the fertilisation, it has become
clear to me that as well the glucose, formed from saccharose, as
the outwardly secreted nectar, are also in other respects of great
importance to the plant. The observations, here communicated, point
already to one very important function, i. e. to enable the stamens
to bring their pollen to the surface at the right time, dependent of
the hygroscopic condition of the arr.

I hope before long to be able to point out still another function.

The secretion of nectar now appears in another light. The view
that it must be considered as an excretion of ‘‘a waste product of
chemical changes in the sap’ ’), which in the course of time has become
more marked through natural selection, as a useful adaptation for
promoting cross-fertilisation, since this liquid was eagerly taken away
by insects, has to give way to the conception that, preceding any
adaptation, it has in its further development kept pace with the
sexual organs. ;

Anatomy. — “On the relation of the genital ducts to the genitat
gland in marsupials.” By A.J. P.v.D. Bronk. (Communicated
by Prof. L. Bok).

(Communicated in the meeiing of October 27, 1906).

in the following communication the changes will be shortly described
which the cranial extremities of the genital ducts in marsupials
undergo during the development and their relations in regard to the
genital gland. In more than one respect the ontogenetic develop-
ment differs in these animals from what can be observed in other
mammals.

It is especially a series of young marsupials of Dasyurus viverrinus
in successive stadia of development from which the observations are
derived. The preparations of other investigated forms (Didelphys,
Sminthopsis crassicaudata, Phascologale pincillata, Trichosurus vulpe-
cula, Macropus ruficollis) correspond however completely with the
conditions we meet in Dasyurus.

In our description we start from a stadium schematically represented
in figure 1 that still prevails for both sexes, (Dasyurus, Didelphys,
Macropus). The genital gland (Figure 1 #) is situated at the medial

1) Cx. Darwin. Origin of species. Sixth Edition. 1872. Chap. IV, p. 73 and
The effects of Cross and Selffertilisation. Edition 1876. Chap. X, p. 402.

( 397 )

side of the mesonephros and is attached to it by a narrow band
(afterwards the mesorchium or mesovarium) (Fig. 1m). The genital
ducts are developed on their whole length. The Wolffian duct
(w.g.) joins transversal mesonephridial tubules in the mesonephros
but has no connection whatever as yet with the genital gland. The
Miillerian duct (Figure 1 m.g.) commences with an ostium abdomi-
nale (0.a.) and runs as far as the region of the mesonephros is
concerned at the lateral side of the Wolffian duct.

relation of the genital gland and genital ducts
in an indifferent stadium.
k. genital gland.
0.a. Ostium abdominale tubae.
g.s. genital cord.
w.g. Wolffian duct.
m.g. Miilerian duct.
$.u.g. Sinus uro-genitalis.

We firstly will follow the transformations, which appear in the
female sex. The first change is a reduction in the cranial part of
the mesonephros. Here nothing is to be observed that points to
a transformation of the mesonephridial tubules by renovation of the
epithelium. The Wolffian duct meanwhile grows cranially, remains
situated near the Miillerian duct, and moves then, passing archwise
through the mesovarium, to the ovarium, penetrates in it and there
ends blind (Figure 2 w.g.). The condition which issues from this
I have demonstrated in Figure 2 (Dasyurus 40 m.m.).

Only now the reduction of the Wolffian duct begins. This occurs
in such a way, that the medial part disappears; both at the cranial
and at the caudal extremity, a remnant of the duct remains.

The cranial rudiment of the Wolffian duct is then found as a little
tubule blind at both ends, which commences in the ovarium and
can be traced till in the mesovarium. Figure 3 points out this little
tubule as I have found it in several animals (Dasyurus, Smin-
thopsis) (Fig. 3 w.@.).

In how far the remnant of the Wolffian duct has relation to the

26*

( 398 )

little tubules which I described and represented in the mesovarium
of a fullgrown Petrogale penicillata, remains out of discussion here *).

Fig. 2.

Relation of the genital ducts

Ov.

0. d.
mM. g.
W. ¢.

Ts Cs

3.U. U:

to the ovarium. Ov.
Ovarium. 0. d.
Mesovarium. t.
Ostium abdominale tubae. a
Miillerian duct. oa
Wolffian duct. w'.g'.
Transversal combination of both * i

the genital cords.
Sinus uro-genitalis. §.U. J.

IE-k ing
Fig. 3.
Relation of the genital ducts
to the ovarium.

Ovarium.

Mesovarium.

Ostium abdominale tubae.

Tuba Falloppie.

Uterus.

Vagina.

Remnant of the Wolffian ducts.

be] n ” 9 bs]

genital cord.

Transversal combination of both
the genital cords.

Sinus uro-genitalis.

In the male sex the Wolffian duct shows in the development of
its cranial extremity, very much resemblance to that of the female
sex. (Fig. 4 and 5).

During the reduction of the mesonephros the cranial extremity of

1) y. p. Broex, Untersuchungen iiber die weiblichen Geschlechtsorgane der
Beuteltiere. Petrus Camper IIL.

—_

( 399 )

the Wolffian duct grows forth and takes its course archwise through
the mesorchium in the testicle. (Fig. 4 w.¥4.). Here is brought about
in one place (Dasyurus) a connection with the future spermatic tubes,
which are still present in the stadium of solid cords of cells.

The mesonephridial tubules disappear almost quite, so that at a certain
stadium (Dasyurus viverrinus 538 m.m.) the Woffian duct, strongly
grown forth in length, runs twisting through the mass of tissue, which
must be considered as the epididymis, without any appearance of
tubules in the form of the coni vasculosi.

Relation of the genita! ducts Relation of the genital duct
to the testicle. to the testicle.
t. Testicle. t. Testicle.
m. Mesorchium. m. Mesorchium.
m.g. Remnants of the Miillerianduct. m.g. Remnants of the Miillerian duct.
w.g. Wolffian duct (vas deferens). d.a. Glandule part in the epididymis.
g.s. Genital cord. w.g. Wolffian duct (vas deferens).
s.u.g. Sinus uro-genitalis. . g-s. Genital cord.
v.a. Vas aberrans. S.u.g. Sinus uro-genitalis.

Meanwhile the Miillerian duct is for the greater part reduced. The
cranial extremity remains as a remnant of the duct either beginning
with an ostinm abdominale or not, and ending caudally blind in the
epididymis tissue.

( 400 )

The change following on this consists therein that the spermatic
tubes obtain a lumen and combine in one or two places (Didelphys)
with the Wolffian duct grown into the testicle. In the epididymis
a great many cell cords have meanwhile appeared in the course of
the Wolffian duct (Vas epididymidis), out of which cell cords the
little tubules of the epididymis will develop.

Of the Miillerian duct a rest has remained in the tissue of the
epididymis, I have not observed rests of this duct in the form of
hydatids. Neither did 1 find them mentioned in literature.

In the genital gland of the full grown animal I found that the
connection of the testicle and epididymis is formed by a mesorchium,
in which “evidently a single tube forms the communication between
the two parts (Didelphys, Halmaturus). Probably the same holds true
for Hypsiprymnus, where, according to DisseLHorst *), the epididymis
is a spindle shaped swelling in the course of the vas deferens.

About the microscopic structure of the testicle and epididymis I found
in DisskLHorsT the communication that it agrees with that of other
animals, As on this immediately follows: “die Spermatogenese war
in vollem Gange’, it seems to me that this communication relates
more to the structure of epithels of the tubules than to the nature
of the connection of testicle and epididymis.

A comparison with what we find in other mammals shows us the
following.

There now and then is to be observed in the female sex (at least
in man) an excrescence of the cranial extremity of the Wolffian
duet, which then becomes the tubo-parovarian tube, which was first
described by Rorn’) and recognised by Minatkovics*) as a part of
the Wolffian duct. Where however in Marsupials the Wolffian duet
penetrates into the genital gland, the tubo-parovarian tube of man
remains in the Ligamentum latum.

For the male sex the following holds true.

A rete testis, whether it has to be considered as tubules, which
have appeared afterwards, and must be considered as a second
generation of tubuli seminiferi (Cogrt)*‘) or as homologa of the

') KR. Dissetuorsr. Die miinnlichen Geschlechtsorgane der Monotremen und
einiger Marsupialen.

Semon’s Zodlogische Forschungsreisen in Australién und den Malayischen Archipel.
1904. p. 121.

*) Quoted by Mrnarxkovics.

°) Mrmarkovics, Untersuchungen tiber die Entwickelungsgeschichte der Uro-genital-
organe der Cranioten.

Internat. Zeitschrift fiir Anatomie und Histologie. Bd. 2.

4) Corrt, Over de ontwikkeling der geslachtsklier bij de z ogdieren. Diss. Leiden
1898.

( 401 )

“Markstrange”’ of the ovarium (Mimmatkovics), or as tubules of the
mesonephros grown into the tissue of the testicle (KoLLMANy) *) is
not found in marsupials. If, during further development a network
resembling the rete testis, arises in the marsupial testicle, it must
be considered as a part which appears quite secondary.

The connection of the testicle and epididymis is not -caused by
a number of tubules of the mesonephros, transformed to vasa effe-
rentia, but by a single tube which must be considered as a part of the
Wolffian duct. For the conception that the connecting tube really is
the Wolffian duct, the phenomena of development in the female sex
can be cited together with those in the male animals. In the marsu-
pials all the tubules of the mesonephros are reduced to minimal
rests (vasa aberrantia). In the mass of tissue, which represents the
so-called epididymis of these animals, a great number of tubes arise
secondary, which afterwards probably possess as epididymis tubules
the same function as the coni vasculosi in the epididymis of other
mammals.

To explain the differences in the connection of the testicle and
epididymis in marsupials and in other mammals, the following con-
siderations seem to me to be of importance.

About the changes, which the mesonephros undergoes, by its con-
nection with the testis, which connection furnishes the later vasa
efferentia testis, we read the following in the extensive investigations
of Coxrt’): In the proximal part of the Wolffian body where the
Malpighian bodies are connected with the blastem of the rete
testis, we see the glomeruli and the inner epithelium of the capsules
disappearing gradually ; after which the outer walls of these capsules
form the blind extremities the mesonephridial tubules. The epithelium
of the mesonephridial tubules also begins to have another aspect.
Two kinds of processes occur here together: a number of epithelium
cells are pushed out into the lumen and are destroyed, while on the
other hand many new cells are formed (mitosis). With this the cells
get another appearance both as concerns the nucleus and the proto-
plasm. The result is that at last the tubules of the menonephros are
surrounded all over their extent, which formerly was not the case, by
an uniform epithelium, formed by cylindrical cells, the nuclei ranged
regularly at the basis of the cells. Whether the connection of these
tubes with the Wolffian duct always remains unchanged during those
transformations or is perhaps broken off and afterwards re-established
in another place I have not been able to investigate.

1) Kottmann, Lehrbuch der Entwickelungsgeschichte des Menschen.
#)-'Le. qe: 96.

( 402 )

My opinion is that these investigations show that the vasa efferentia
testis must not be considered as simple tubules of the mesonephros,
but newly formed tubules, which use quite or for the greater part
the way given to them by the tubules of the mesonephros. And that
they are able to use this way finds its cause in this, that, according
to Fruix and Biuimr') there is most probably no idea of a functioning
of the mesonephros in monodelphic mammals, even not in the pig,
where if is so strongly developed.

Not so in the didelphic mammals. Here the mesonephros does
not only function embryonally, as is known, but still during the
first period of the individual life. A separation of the mesonephros
in two parts as is found in reptilia does not come about here.

The connection of the genital gland, especially of the testicle and
its duct, the Wolffiian duct, could not, it may be supposed, in the
stadium in which this connection will come about in other animals,
be established in marsupials with the help of tubules of the mesone-
phros, because these had still to fulfill their excretory function.

Instead of this the connection could be established in such a way
that the Wolffian duct grews out cranially and brings about itself
the connection between the gland and its excretory duct.

At last the tubes, which occur secondary and independently ot
the tubules of the mesonephros in the tissue of the epididymis, might
be explained in the same way, i.e. as tubules which have the same
signification as the coni vasculosi, but for the same reason do not
originate on the bottom of tubules of the mesonephros but are
separated from them both locally and temporarily.

Another view may be, that the tube which encroached in the
genital gland, might not be the Wolffian duct but the most cranial
tubule of the mesonephros so that in other words the so-called sexual
part of the mesonephros in marsupials should be reduced. I do not
believe that this conception is true, firstly because no separation
between the tubules can be observed, and secondly because at the
reduction of the mesonephros, as is mentioned above, in marsu-
pials, nothing can be observed, as far as my preparations are con-
cerned, of differences between the tubules of the mesonephros, what
must surely be the case at a transformation of a tubule of the
mesonephros to a connecting duct.

1) Feux und Binter, Die Entwickelung der Harn und Geschlechtsorgane in
Hertwie's Handbuch der vergleichenden und experimentellen Entwickelungsgeschichte
der Wirbeltiere.

( 403 )

Physics. — “Supplement to Communication N°. 954 from the
Physical Laboratory of Leiden, on the comparison of the
thermo-element constantin-steel with the hydrogen thermometer’.
By Prof. H. KamernincH Onnes and C. A. Crommenin.

§ 14. Corrected representation of the observations by a five term
Formula.

As appears from note 1 the calculations in § 12 were made with not
perfectly accurate values of the temperature at — 182° and in the
same way the mean errors were derived from the assumption of
those less accurate values. ’)

If the correct values of those temperatures for the calculations of
the deviations W—R,, W—R,, W—R,, W—R, in Table VIII, are
used, the mean errors in microvolts become :

for formula (BI) + 3.0
(BIB + 3.4
(BIT) + 2.8 (2.5 without — 217°)
CEI) 2.4

instead of
(ily. 25
(BI). + 3.2
(AIT) = 2.6 (2.1 without —- 217°)
(BIV) + 1.8

which would also have been obtained if the observations at — 182°
were. excluded.

Now it was necessary to examine whether a repetition of the
adjustment would diminish these mean errors. It appeared convincingly
that this was not possible to an appreciable degree for (BI), (BID),
(SU). It appeared possible for (LIV) to distribute the errors more
equally. However, this only reduced the sum of squares from 26,57
to 26,14.

Instead of the coefficients a,,b,,¢,,¢,and f, (see § 12) we get then

a’, = + 4.32513 é', = + 0.023276
b', = + 0.409153 t',== — 0.0025269
c', = + 0.0015563

The deviations are given in Table IX under W—R’,.

1) The correction amounted to 0°,081 in temperature or to 1.7 microvolt. in
electromotive force.

( 404 )

§ 15. Representation of the observations by means of a four
term formula.

We have now quite carried out the calculation of a formula of

the form
oe t : t
=*(i) + foe (a5) « -

announced in note 2 of §11, by the method of E. F. v. D. SANDE
Baknvyzen, which proved to facilitate matters greatly again.

Four solutions (C) were found, viz. (CTI), (CII), (CIID) representing
the observations down to — 253°, whereas in (CIV) only agreement
down to — 217° has been sought for.

The coefficients in millivolts are the following :

1 2 3 4
| |
a + 4.30192 | + 4.30571 | + 4.30398 + 4.33031
6) + 0.357902 | + 0.366351 + 0.363681 + 0.421274
Bi — 0.0250934 — 0.0192565 | — 0.020071 | + 0.018683
|

+ 0.0270158 | + 0.0270044 | + 0.035268

+. 0.0257462

The residuals have been given in tenth parts of microvolts in
Table IX under W—Re, W—Reun, W—Rcew, W—Reiv:

Just as with the five term formula, the residual at — 182°
appeared also now greater than the others.

In calculation 3 it was tried to distribute the errors more equally,
but the sum of squares appeared now to have increased.

The mean errors are if we include the observations down to
— 253° for (Cl), (CID, (CH, and only those down to — 217° for
(CIV), for

(Cl) 0

(CID) + 2.9

(CIII) + 3.0

(CIV) + 2.3

If — 182° is excluded, they become :
(Cl) + 2.7

(CII) + 2.6

(CIV) a mes LE

The mean errors of (CTI), (CID, (CIT) must be compared with
those of (BI) and (BIL), those of (CIV) with those of (BIV).

( 405 )

This comparison teaches that the four term formula for the represen-
tation of the observations may be considered to be almest equivalent
to the five term formula, and that therefore (this remark is in har-
mony with note 2 of §13) for the calibration to — 217° the lowest
number of temperatures for which observations are required, amounts
to jour. That three are not sufficient was already proved in § 11.
This appears also clearly, when the mean error is determined, which
rises to + 7.6 microvolts for the three term formula.

ands ty Ei (EX,

DEVIATIONS OF THE CALIBRATION-FORMULAE FOR THE
THERMO-ELEMENT CONSTANTIN-STEEL.

rv. =| Vv | - VE..|-. VII

I | II I | |
No, | f | W—R; | Wey | W Boy aa
| — oof — 12 | + 20 Ea Pe Se oa eee
24 and 90|'— 58.75 | + 46) + 30) + % | 4:99) 4+ 4
Mand 93) — 88.45 uw) + ttt al 4+ 4 ae
1 and 47) — 103.70 | — 6 | — 299 | — 98 | — 30 | — 2
16 and 18, eager ee cog) | — 31 | — 47
19) — 458.83 | — 40 | — 10 | — 10 | — 48 | — 40
3, 11 and 5 | [~ 182.73] 4+ 96 ee ee eerie el eee
4,8 ad6| 15.10 | + 2 | + 03 | 4 mm | 4 19-4} 4 41
12, 27 ‘a — 7 | — 0} — 9 | — 41 | — 19 | — 48
14 18and =898 | - a) pom | ow | as | 4
99, 15 and 5 | — 917.55 | — 15 | — 30 | — 299 | — 37 | — 93
0 | — sae | + 9m oo) 0.) 4-90. | 4450

sg — 9.9% | +485 | +4415 | +14 | 4443 | + 313

( 406 )

Mathematics. — “On a special class of homogeneous linear dif-
ferential equations of the second order”. By Prof. W. Kapreyn.

The differential equation of LEGENDRE

Py dy
(l—#?) — — 2a —~ +ur(n+]lhy=0
z dz
is satisfied by a polynomium P, (2) of the nth degree and by a
function Q,(@) which may be reduced to the form

1
FP, dz
—l

This function however is not determined for real values of the
variable in the interval —1 to +1, the difference on both sides
of this line being 227 P, (q).

In analogy to this we have examined the question: to determine
all homogeneous linear differential equations of the second order of

the form
d*y dy
x dz

where the coefficients are polynomia in 2, which possess the property
that y,(z) being a first particular integral, the second integral may

be written
ne is
Ys = 2

where a and ~# represent two Seal values, supposing moreover that
this integral has a meaning everywhere except on the line of dis-

continuity.
Let
} y i
R(@)= 2 ir, 4? (A) = a By gs (a) = = ty uP
UV 0 0
then we obtain firstly the conditions

1—2

R(x) = («a —a) (e— 8) r (x) = (w7—a) (x — 8) = Op xP

S (a) = R' (x) + («—a) (vB) = hy uP,

If now we put

: ig B
peep? ys" Ce) de® .° 1G; = fe y, (2) dé- 5. G, ={# y, (2) de
a «

( 407 )

and
M = — G,'
N = (e+ 8) G,"—G,"—2G,'
m= — G,!

vt (a+-8) Gij-=G—G;,
the further necessary conditions may be deduced from the equation
ft J = 0

where J and / represent the following polynomia of degree 2—1
es

I= Z(@N+ opi M) a
p=0
d—1
+E [hy + (P+Y epi n + Hypa + pep} mi] ax
p=—9

J—1
= Cr Gp + spi Gp' + tpi Gp)
p=

d—2
+4 SA ae Gp" == Sp+2 G,! sin ty+2 Gp)
p=

ae
1
a 2 > (Tp+i—1 Gr, Spi—l Gy + tyti—1 Gp)
p=0
5 ae ane 2 Crh Gp" + spi Gp + tpi Gp).
p>

From this we may easily deduce that if 42, the most general
differential equation of the second order possessing the property in
question is

d*y dy
el ee ge ee) aeet - e — BW + Ce e)y=0

where a, 8, ¢, and ¢, are arbitrary constants.
When 2= 8 the most general equation may be written

d?y dy
(ea) (@—B) (Q,0-+0,) 5 + (te? +507 +0-+5,) 2 +
+ (t0*-fie-tt,) y = 0

Here however the ten constants must satisfy the following three
conditions

8, + (4+8) 8, + (@?+48+8") t, = 20, + (2+8) 0,
8) — aps, — aB (a+B) t, = — (2+8) e, — 2aBo,
(¢,—s,+29,) G, —t, G, = 0.

( 408 )

Mathematics. — “Some formulae concerning the integers less than
n and prime to n.” By Prof. J. C. KLuyver.

The number g(n) of the integers » less than » and prime to n
can be expressed by means of the divisors d.
We have
gy (rn) = = u(d)d, (dd' = n)

if we denote by u(q) the arithmetical function, which equals 0 if ¢
be divisible by a square, and otherwise equals + 1 or —1, according
to g being a product of an even or of an odd number of prime
numbers.

This equation is a particular case of a more general one, by means
of which certain symmetrical functions of the integers » are expres-
sible as a function of the divisors d.

This general relation may be written as follows *)

k=d'
S/O) = THO,

For the proof we have to observe that, supposing (m, 7”) ~ D, the
term f(m) occurs at the righthand side as often as d in a divisor
of D. Hence the total coefficient of the term 7 (m) becomes

= u(d),
d/D

that is zero if D be greater than unity, and 1 when m is equal to
one of the integers ».
We will consider some simple cases of KRONECKER’s equation.
First, let
f(y) =e.
The equation becomes

: —a. pun 9
Ser J py (d) > etkd = J yu (d) 4 3 3
y d/n oa djn
or because of
= p(d) = 0,
djn
is ern — |
= et = = p(d) ——.
vy d/n erd —s d
If we write
wer?

a | am u erd __ i;

1) Kronecker, Vorlesungen iiber Zablentheorie. I, p. 251.

( 409 )

we may introduce the BrrnovuiiiANn functions Jif), defined by the
equation

a k—

gad ooh , > whfi(9),

e —1 —1

and hence show that
1 [2 lf tr
si— + > aknk-lf | —
» (7 ei nr
By equating the corresponding terms on the two sides we get

B,
= Paes (=) —- (— 1 aoe Ss u(d)d'—2m + l

m! ain

1 B B
— ad) l—Lad + — 27d?— — x‘ +. | ;
ee 2/ 4!

as a first generalisation of the relation
= ¥— —> id)d-.
y djn

Observing that we have

1
> p(dd'—2m $1 = x u(d)a2n—1 ,
djn nim la
there. follows for two integers n and 7’, pe having nos same set
of prime factors,

AG) Peay:
= /'m(2) aC

In the same way an expression for the sum of the /*® powers
of the integers » may be obtained. Expanding both sides of the
equation

Fer=s
e S Wd) i

we find
1
— Svk = D u(d)d*f;,(d’).
| ae djn

Other relations of the same kind, containing trigonometricad functions
are deduced by changing « into 2z7z.
From

e2n IN 1

eee OO) ey

c e2niad ___ |
we find by separating the real ad imaginary parts

= cos 2rav = } sin 2aen T w (d) cot red,
y d/n
= sin 2xaev = sin? xan J ww (d) cot xed.

: |
y [n

( 410 )
In particular the first of these equations gives a simple result if
1 a : E
we put «=— +e, where « is a vanishing quantity. As _the factor
nr

sin 22xn tends to zero with « the whole right-hand side is annulled
but for the term in which d= 7.
So it follows that
2y
= Cos : == (x),

and we have u(v), originally depending upon the prime factors of
n, expressed as a function of the integers prime to 2.
1
Similarly we may put in the second equation «= 5 and write
n
sin HY ad
=> — = = pu (d) cot —

y n d/n 2n

Still another trigonometrical formula may be obtained by the sub-
stitution eae +e. Let D be the greatest common divisor of the
a

integers and gq, so that
n=n,D . Q= O03
then as ¢€ vanishes, we have to retain at the right-hand side only
those terms in which gd is divisible by », or what is the same the
terms for which the complementary divisor d’ divides D.
Hence, we find

= 2arqv t ' ys 1 t
= cos = Sa|— )|/d=—=DZul(a,d)—. (dd' = D)
d'/D d/D d

y 7

Instead of extending the summation over all divisors d of D, it
suffices to take into account only those divisors ¢ of n, that are
prime to ”,. In this way we find

i 1
DZ u(n,d) — = p(n,) D = u(d) —,
d/D d 3 Jd
and as the second side is readily reduced to

gin) fu) £)

we obtain for any integer g, for which we have (n,q) ~ D,

$ ij Paes * p(n)

( 441 )

Concerning the result
~- 20
=' cos —— = pt (n)

a slight remark may be made. To each integer vy a second »v’ = n — v
is conjugated; hence denoting by @, an irreducible fraction << 4 with
the denominator n, we may write
2= cos 270, = u(n),
and also
2 cos 270n = = u(r).
nSg ng
Now for large values of y the fractions 9, will spread themselves
not homogeneously, but still with some regularity more or less all
over the interval 0 — } and there is some reason to expect, that in

the main the positive and the negative terms of the sum + cos 2z@,,
ng
will annul each other, hence the equation
2 cos 2x0, = & p(n)
nSq nSg
is quite consistent with the supposition of voN STERNECK, that as g takes
larger and larger values the absolute value of 2u(n) does not
ah

exceed Vy.

Another set of formulae will be obtained by substituting in
KRONECKER’S equation

T(y) =log\e" —e -).

Thus we get

2riz Qnty fed’ 2nir Qnikd
ren caere = (6 panes
y djn —

2rix 2xiy Qxizd'
¥ tog n—en =e n a8
y d/n

and after some reductions

It - &
= log 2 sin — (v—a) = DJ p(d) log 2 sin = .
y nr ] a

d/n
By repeated differentiations with respect to « we may derive from
this equation further analogies to the formula
y(n) = Suda.
d/n

So for instance we obtain bij differentiating two times

27
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 412 )

1
> —— = — = u(d) d”
y TY 3 djn

and by repeating the process

an By 2m
ee log sin y at = u(d) a"
< | Ss 9 u y Pe = Im din’ ( ) b]

a result included in the still somewhat more general relation

k=o if
nt > > SS] SS Gaye.
* ee oe (s) Fn ue (d)

which is self evident from.
Returning to the equation

Tt v
= log 2 sin y — (v—2x) = J pe (A) log 2 sin a j
) 7. djn d
we obtain as x tends to zero
5
= log 2 sin — = — J pw (A) log d.
y n djn

In order to evaluate the right-hand side, we observe that for
np, p43... we have

d
— {u(d)lgd= — E (1 — eylogp) (1 — evan). | .
d/n dy y=0

So it is seen that, putting
— Su(d) log d = y (n),
djn

the function y(n) is equal to zero for all integers 7 having distinct
prime factors, and that it takes the value /og p, when n is any powe1
of the prime number p.
Hence we may write
_ IY Po
12 sn — = 7 ™),
> n
a result in a different way deduced by KRONECKER ’).
Again in the equation
_ ot — ma\e(d)
IT 2 sin — (v — #) = | 2 sin —
y n d

d/n

n
we will make w tend to ee

If x be odd, all divisors d and d’ are odd also and we have
at once

1) Kronecker, Vorlesungen tiber Zahlentheorie. I, p. 296,

( 413 )

d’—1
i 4) ——-_ & (d)
2 cos — = IT (— 1) 2 = (— 1)39),
y n din
If n= 2m and m be odd, we shall have g (m) = ¢ (%). Half the
numbers x prime to m and less than m will be equal to some
integer », the other half will be of the form »—-m.
Hence we have
22 20% a
112 sin = (— 1) IF2 vin —— = (— 1)8#) 172 sin — ,
y n x n z

and therefore
ITH
IT2 sin —— _
x m ° / (axe
12 Palate (— 1)37(%) —— = (— ie 2G
y n SEL
IT 2 sin —
> n

—n)

Lastly, if m= 2m, and m be even, we shall have (m) = 3 p(n).
Now each of the numbers x prime to m and less than m at the
same time will be equal to some integer y and to one of the dif
ferences vy — m. Reasoning as before we have in this case

n ™m

_ wH ;
n(2 sin =) a (G)-™

nV ‘ _ 2y
IT 2 cos — = (— 1)3?™) = (— 1)ire
y n mee 2)
IT 2 sin —

y nr

27 v 27 2 3 2
112 sin —— = (— 1) n(2 sin =) — (— 1) n(2 sin =)
y z n x

and therefore

From the foregoing we may conclude as follows. If we put
IT 2 cos ates (— 1)? 2),
y n

the arithmetical function (nm) is different from zero only when
n is double the power of any prime number p, in which case we
have 4(n)=/og p.

Again we introduce here the irreducible fractions @, less than 3
with the denominator n; then denoting by J/(q) the least common
multiple of all the integers not surpassing gq we may write

2 = log2sinxe, = = y(n) = log M(9),
n <9 n <9

2 = log 2 cos xn = A(n) = log u(2).
n<g n<g 2

27*

( 414 )

If we consider the quotient Jog M(qg):logg as an approximate
(but always too small) value of the number A(g) of prime numbers
less than g, to Kronucker’s result

= log 2 sin 1On

A
og Inca

we may add

q 2,
A|—)=— =& log 2 cos xo, .
é lo LES
79

Astronomy. — “Researches on the orbit of the periodic comet Holmes
and on the perturbations of its elliptic motion. IV.” By Dr.
H. J. Zwmrs. (Communicated by Prof. H. G. vAN DE SANDE

BAKHUYZEN).

At the meeting of the Academy on the 27 January of 1906, a com-
munication was made of my preliminary researches on the pertur-
bations of the comet Holmes, during the period of its invisibility
from January 1900 till January 1906, and also of an ephemeris of
its apparent places from the 1st of May till the 31st of December 1906.
This time again this computation led to its rediscovery. Owing to
its large distance from the earth and the resulting faintness of its
light, there seemed to be only a small chance for its observation
during the first months. This proved to be true, as not before the
30 of August of this year, the Leiden observatory received a
telegram, that the comet was found by prof. Max Wo r at the
observatory Koenigstuhl near Heidelberg, on a photograph taken in
the night of the 28% of August of a part of the heavens where
according to the ephemeris it ought to be found. The roughly
measured place

a=612 01 d= + 42° 28'

for 1352™1 local time, appeared to be in sufficient agreement with
the calculation.

Afterwards the place of the comet has been twice photographically
determined: on the 25 of September and on the 10% of October,
and each time prof. Wor was so kind, to communicate immediately
to me the places as they had been obtained, after carefully measur-
ing the plates. Although Wotr declared in a note to the observed

~~ 4

( 445 )

position of the 25t of September’) that the brightness had increased
sufficiently, to make the comet visible in a powerful telescope, till
now I did not hear, that any visual observation of the comet has
been made. The three Heidelberg plates are therefore the only material
that can be used for testing the elements and ephemeris given before.

I communicate here the results as I had the pleasure to receive
them from prof. Wo r.

1. “Den Kometen Holmes habe ich auf der Platte von 28 August
rechtwinklig an die 4 Sterne

A.G. Bonn 3456, 3462, 3472, 3493
angeschlossen, und die Messungen nach der Turner’schen Methode
reduziert. Ich finde fiir 1906.0:

a = 4h 7m 34884 d= + 42° 30' 59"9
fiir die Aufnahmezeit: 1906 Aug. 28, 13852™1 Kgst. Das dusserst
schwache zentrale Kernchen wurde dabei eingestellt. Die Messung
und Rechnung bezieht sich auf die mittleren Orte der 4 Sterne fiir

1906; sonst ist gar nichts angebracht.”
(Note of the 5 of September 1906).

2. “Ich habe Ihren Kometen nochmals am 25 aufgenommen und
finde ihn entschieden etwas heller. Den Ort nach Turner mit 3
Sternen (A.G. Bonn 3710, 3760, 3778) fand ich

1906 Sept. 25: 1254670 M.Z. Kegst.
1906.0 = 45 32™ 10802 di9u6.0 = + 47° 34' 546

Ich habe auch den letzten Ort (viz. of Aug. 28) mit nur 3 Sternen
nochmals gerechnet (weil ein Stern sehr ungiinstig war) und fand
fir 1906 August 28: 13452™1 Kest.:

@1906.0 — 45 7™ 35s00 Ji906.0 = _— 4 a pee

Ich bin nicht sicher, ob diese Bestimmung aus 3 Sternen besser
ist als die erst mitgeteilte.”’

(Note of the 29" of September 1906).

3. “Herr Dr. Koprr hat gestern den Ort einer Aufnahme vom
10 Okt. 1906 des Kometen Holmes ausgemessen.....

1906 Okt. 10: 951™0 Kest.
1906.0 — 45 34m 48594 di906.0 = -+ 49° 54’ 59”2
piemne. AG. snean at90. 3168, 3777 ..... Der Komet war

1) Astron. Nachr., N°, 4123, S. 302.

( 416 )

diesmal schon recht schwach, wahrnehmbar schwacher als im Sep-
tember. Die Messung ist deshalb auch wohl etwas unsicherer.”’
(Note of the 13 of October 1906).

Concerning the observation on the 28 of. August I preferred the
position obtained from 3 reference stars.

For the reduction to the apparent place, I used as before in the
ephemeris the constants of the Nautical Almanac, where the short
period terms are omitted. Assuming for the parallax of the sun
880, I find for the Heidelberg Observatory the following constants:

A= — 0b34m5 458
tg yp! = 0.06404
A= 9.58267
D = 0.82425

which are used for the computation of the parallax of the comet.
The following table gives an account of the reduced observations.

a eA Baa:

| Red. on app. pl. | Parallax Apparent geoc. place.

Pig CG ce OMS Se Z

Ax | Aé Aa | Ae a | é

" " h | ° ! "

s | s m s |
1 | + 1.888 | —8.55 | — 0.191 | 44.24) 4 7 36.697 | 442 3050.99
2| + 2.929 | —8.57 | — 0.217 | 40.92 | 4 32 12.732 | 447 3446.95

3 | + 3.593 | —7.51 | — 0.298 | 42.35 | 4 34 52.235 | +49 5454.04

I used for comparison with the ephemeris my original computations,
which contained in @ as well as in dé one decimal place more than
the published values. The computed places and their comparison with
the observed positions, are given in the following table.

T +.B An,

| Comp. apparent place Observ.—Comp. |

Aberration- | - = we

Local time ihe.

a | é a | é
n

rc P| "

| d hm s | ed
Aug. 28.553602 0.013211 |4 7 29.753) +423024.28 | 46.94 | 426.7
| Sept. 25.507699 .012005 | 4 32 4.255) +473429.94 | +8.48 | 417.0

| Oct. 10.351449 011462 | 4 34 43.017] +4954 43.02 | 49.92 | +41.0
|

( 417)

Together with the ephemeris I communicated a table containing
the variations of the right ascension and the declination by a variation
of the perihelion passage of + 4 or — 4 days. In comparing the
above given values O—C with the numbers of that table, it is evident
that by a small negative variation of the perihelion passage, the
agreement between observation and computation may be nearly attained,
at least in a. The deviations in d cannot be used so well for that
purpose, as the variations of d, resulting from a variation of 7’, are
always much smaller than those of «, and this is especially the case
in the period during which these observations are made. Yet we
may conclude from the table for A7’=—— 4 days that the positive
errors in d will not entirely disappear by a variation of 7’.

By means of a rough interpolation I derived from the 3 differences
O—C in right ascension the following corrections for the time of
perihelion passage :

Observ. of Aug. 28: AJ’ = — 0.0900 day

3 Ge Sept: BS — 0.0916 _,,
rere. ocr 20 : — 0.0896 _,,
In the average AT — — 0.0904 day, which at the rate of a mean

daily motion of 517"448 corresponds to an increase of the mean
anomalies of 468.

As a first step to correct the adopted elements of the orbit, I
therefore computed the 3 places, in the supposition of an increase
of the mean anomalies: 1° by 40", 2° by 50". I interpolated the following
sun’s co-ordinates (with reference to the mean equinox of 1906.0)
from the Naut. Almanac.

7 A B i. I.
| SS aia
| | |
Aug. 28.540391 | — 0.9134887 | + 0.3947635 | ++ 0.1712510 |
| Sept. 25.495694 | — 1.0018399 | — 0.0318699 | — 0.0138250 |

Oct. 10.339987 | — 0.9565810 | — 0.2616405 | — 0.1135029 |

For the reduction to the apparent places I added to the mean @
of the comet: F+ 4 sin (G + a) ig J, to its mean J: g cos (G + @).
The following table contains the computed apparent places in the
two suppositions.

( 418 )

oA LS Ty:

ia aM —=-} 40" | aM = +50"
Io, | elect.

+ é a é

|
hm s eer hms | 7 0
4 4 7 35.758 | + 42 3034.72 | 4 1 37.266 + 42 3037.38

+ 47 3431.46 | 432 13.248 | + 47 3431.85

bo
&
ee)
bo
—
—
iS
oO

—

iS
ww
Ss

3] 434 51.050 | 4+ 49 5442.90 | 53.060 | + 49 5444.99

Ke gl it alana

|
|
|

A sufficient control is obtained here by comparing the values for
A M= 0" (ephemeris), A M = - 40" and A M= - 50".

In comparing with the observed apparent places we obtain the
following differences O — C:

fA Bi ive

AM =-+ 40" AM =-50"

| Se hoe

" | '

Ss Ss
4 | + 0.939 | + 16.27 | — 0.569 | + 13.64
2 | + 1.281 | + 15.49 | — 0.516 | + 15.10
3 | + 4.185 | + 11.84 | — 0.825 | + 12:05

4 ed

By means of interpolation between the values of Aa we find as
resulting value for A A/ + 46"412, leaving the following errors :

we | ae | Aa
: atts
4 |—0.03 | 4+ 14.7
| 2 | + 0.43 | 4 45.2 |
3 ia ad aoe

From this follows that by a variation of M alone, the differences
O—C in @ can be reduced to very small quantities, but this is
not the case with the differences in d. It could be seen beforehand

( 419 )

that no further improvement could be expected from alterations in =,
y or uw; at the end I will add a few words on these elements. So
we must try to bring it about by variations in the position of the plane
of the orbit, viz. of 7and sR, and for this reason I determined the relation
between those elements and the computed places of the comet. As
from the two suppositions A 1/=-+ 50" seems to be nearer to the
truth, I computed the apparent places of the comet: for AM = + 50"

=-+10" and Asj§=O and also for AM=—-+ 50" Ai=0
AS = — 10". Probably a somewhat larger value of AS
had been more convenient. The following table gives the variations
of « and d in the two cases.

fat «VI,

Se ee ee
oe. é Aa | Ad
:
" "
4 | — 0.149 | + 10.00 | + 0-040 | + 1.96
2 | — 0.108 | + 41.95 | + 0.067 | + 0.83
3 | — 0.141 | + 12.88 | + 0.080 + 0.50 |

The numbers from the tables V and VI give the following values
of the differential quotients of @ and d with respect to M,zand y,
which will be used as coefficients in the equations of condition.

Aug. 28 Sept. 25 Oct. 10
0a
os 4. 0.1508 +. 0.1797 +. 0.2010
qa 0.266 0.039 0.021
oa +0. a9: 48)
0a
ae — 0.0149 — 0.0108 —— Orel
od
os + 1.000 + 1.195 + 1.288
0a
—— — 0.0040 — 0.0067 — 0.0080
Odd
od
—- 0.126 — 0.083 — 0 056

( 420 )

For a the second of time and for the others quantities the second
of arc have been adopted as unities. I multiplied the equations of

condition for «@ by 15cosd, and instead of A sb I “introduced —

a unknown quantity.

Equations of condition.

a. From the Right ascensions :

ASb
0.22202 AM + 9.21681, Ai + 9.64568, —
0.25966 ,, -- 9.08853, ,, + 9.88118, ,, = 0.71776,

0.28811 ,, -+ 9.038023, ,, + 9.88800, ,, = 0.90136,

6. From the Declinations :

On uoren

ASsb
9.42488 AM + 0.00000 Az + 0.10037,

= 1.13386

10
8.59106. ,, = 0.07737 ..,--. 9.91908, _,, == 117898
8.32222, ,, + 0.10992 ,, + 9.74819, ,, = 1.08099

The coefficients are written logarithmically ; the second members
are taken from column 4 and 5 of table V, and therefore to AM, .
found from these equations, the correction + 50" has still to be applied.

From the above equations of condition we derive in the ordinary
way the following normal equations :

Adv
+ 9.9278 AM — 0.39596 Ai — 3.8260 7 eee 31.495
— 0.39596 ,, + 4.1875 ,, — 2.7484 ,, = + 49.637
— 3.8260 ,, — 2.7484 ,, + 3.8423 ,, = — 238.951

These equations are much simpler if we introduce besides AM,
only one of the two unknown quantities. If we try e.g. to represent
the observations only through variations of MM and 7 we have not
only AS&=O but the third equation falls out entirely.

1. Solution for ASo=0.

The results are:

AM=— 2" 7042
Ai =4+11.74
and the remaining errors :
1. Aa=+ 09014 Ad = + 2"59
2. = + 0.097 4,18
3. = — 0.151 — 3.13

( 421 )

2. Solution for Ai=0O.
In this ease we find:

AM =—_ 9"0461
AQ = — 2'32"41
and for the remaining errors:
1 Aa=-+03185 Ad= — 3"18
2, + 0.089 4+ 2.80
3. — 0.226 + 3. 32

3. Solution with 3 unknown quantities:
The results are:

AM =— __5"3045
(a a a Sy
AQ = — 1 2.90

and according to the equations of condition there remain the following
differences Obs.—Comp.

1. Aa= + 09088 Ad= —0"23
2. 4+ 0.095 4.1.34
3 — 0.181 ip

As we see the solution with Ad& =O and that with Ai—0
satisfy the observations fairly well, the first one somewhat better,
especially in right ascension. Stili we cannot deny that in the values
Obs.—Comp. of d in both solutions, there exists a systematic varia-
tion. On account of that I prefer for the present the solution with
3 unknown quantities, where such a systematic variation doesnot
appear. I therefore take the following elements as the most probable
for the return in 1906:

Epoch 1906 January 16.0 M.T. Greenw.
M, = 1266456"838
= 351°47'36"838

ue = 517"447665

loga = 0.5574268
T = 1906 March 14.09401
@ = 24°20'25"55
e = 0.4121574
~ == 20°49' 062
x = 346 231.63 } 1906.0
Se = 331 4437.85 |

( 422 )

Yet it is evident that the accuracy of these elements is not equal to
the accuracy of those I could derive for previous returns of the comet.
In the first place the observations include only a period of 43 days, in
which the heliocentric motion of the comet with its large perihelion
distance was not even 12°. Secondly three observations with their
inevitable errors are in general only sufficient to obtain a mere
approximate idea of the orbit. We must admire the ability and
accuracy of the Heidelberg astronomers, who, from measurements on a
short focal photographic plate taken of a still wholly invisible nebula,
could deduce the position of the comet with an accuracy that could
be compared to that of micrometer measurements of objects several
hundred times brighter. Still we must bear in mind that the rejection
of only one of the 4 reference stars on the plate of the 28 of August,
had an influence of 0816 in @ and 1"6 in declination, or of 239
in are of a great circle.

As a test to my calculations, I derived the 3 places finally by
direct computation from the obtained elements.

Heliocentric aequatorial co-ordinates :
a — 19.993 7648.63] sin (v + 77°37'28"36)
y = [9.876 2140.59] sin (v — 20 58 46.82)
z = [9.832 7020.56] sin(v — 1 4646.76)

The following table contains the computed apparent places of the
comet and the differences Obs.—Comp.

dpa oe a 8

NO. | a e | Au | Aé |
| | 1

Omni elt i " |

{ho “ai 3s s
4 |4 7 36.602 | + 42 3051.32 || + 0.095 | — 0.33
4 32 12.633 | + 47 3445.69 || + 0.099 | + 1.26 K

434 52.42 | 4+ 49 5455.19 || — 0.177 | — 1.15
| |

The agreement between these differences found directly, and the
quantities obtained by substitution in the equations of condition forms
a sufficient control on the whole computation.

The elements p, a and g.

The elements from which the ephemeris for 1906 has been derived
are those given in “Systeme VII” p. 78 of my Deuxiéme Mémoire,
reduced to 1906 by applying the perturbations, arising from the
action of Jupiter. The mean error of the obtained value for pu is so

( 423)

small, that although not absolutely impossible, it is hardly probable
that the correction obtained for the mean anomaly should have been
caused totally or for the greater part by an error in wu. Taking the
obtained AM for the 25 of Sept. we get:

44" 6955
2662.50

and thus the real error of uw should be 67 times the mean one.
Adopting this correction of u, the mean anomalies for the 28t of
August and the 10% of October would be only 0" 469 smaller and
0" 249 greater than the adopted ones.

It is more probable that the correction of JM arises from neglected
perturbations of that element by Saturn. This perturbation is given

by the formula
t t
dM du
M = dt —— dt’.
A f ap ok f dt
to to

Even if instead of the sum of the values each term was known
separately it would be equally impossible to conclude from the value

d
of the double integral, the final value of f — dt, or the correction

Au=+ — + 0" 016787

t
of w for 1906. Observations during a much longer period can only
decide in this case.

Something like this holds for 2 and g. During the short period of
the observations, we may even substitute for a part of the correction
AM corresponding variations of a and g. If we keep to the plane
of the orbit, the apparent place, except for small variations in the
radius-vector (of little influence near the opposition), depends wholly
on the longitude in the orbit, or on

l=a-+».
So we can apply small variations to the elements without varying
perceptibly the computed positions, if only
Al=Az+Av=0
or
ize —— — Av.

This relation provides us with the means to throw a part of the
correction found for M on 2 or on y or on both together. In the
first case we have to satisfy the equation

Ov -

BB — —— A M.
0M

( 424 )

We can derive the values sg directly from the comparison of
the two former computations with A M= - 40" and AM=-+ 50".
And so I find for the three dates of the observations:

AM = — 0.506 Aa
— 0.549 Aa
— 0.573 Az
If we keep a constant and want to substitute a part of the correc-
tion of M/ by a variation of gy, we must satisfy the relation
Av = 0
or
AM= (5 Ag.
oy

v const.

0M :
I derived the values of (5) by computing from the three values
f /v const.

of v, with a varied excentricity, the corresponding values of the
mean anomaly. Hence I got for the three observations:
AM = — 1.040 Ag
— 1.186 Ag
— 1.260 Ag
Although the coefficients as well those of Aa as of Ag show a
small variation in the influence of the corrections of the elements
on the three positions, practically this influence differs too little from
that of a constant variation of J to allow a determination of
4M, 4g and Aa separately from the three observations.

Leiden, November 1906.

Mathematics. “On the locus of the pairs of common points and
the envelope of the common chords of the curves of three
pencils.” (1st part). By Dr. F. Scaun. (Communicated by
Prof. P. H. ScHovure).

1. Given three pencils (C,), (Cs), (C1) of plane curves of degree
r, 8, t. To find the locus L of the pairs of points through which
passes a curve of each of those pencils.

Let P and J” be the points of such a pair. When determining
the locus we shall notice but those points P and P’ which are for
each couple of pencils movable points of intersection (i.e. points not
necessarily coinciding with the basepoints), a distinction to be made
only when the pencils have common basepoints. The locus L arrived

( 425 )

at in this way we shall call the locus proper, to distinguish it from
the total locus to be arrived at by allowing one of the points P and
P’ to be a fixed point of intersection of two of the pencils.

Suppose the pencils (C,) and (C,) show a fixed points of inter-
section and that this number amounts to # for the pencils (C;) and
(C,) and to y for the pencils (C,) and (C;).

The degree n of ZL is determined from its points of intersection
with an arbitrary straight line 7. On / we take an arbitrary point Q,, and
through Q,,; we let a C,. and a C; pass, which cut each other besides
in the basepoints and in Q,. still in rs —y—1 points. Through
each of these points we let a curve C; pass. These rs — y — 1 curves
C, eut / in t(rs — y—1) points Q,, which we make to correspond
to the point Q,,. To find reversely how many points Q,; correspond
to a given point Q of 7 we take on / an arbitrary point Q, through
which we allow a C, to pass cutting the C; through Q; in rt — 86
points differing from the basepoints. Through each of those points
we allow a C, to pass, of which the points of intersection with /
shali be called Q,. To a point Q, now correspond s(r¢— 8) points
Q, and to a point Q, correspond r (st—e) points Q,. The 27st—ar.—8s
coincidences Q,Q, are the ¢ points of intersection of / with the C, passing
through Q; and the points Q,, corresponding to Q,, whose number
therefore amounts to 2 rst — ar — Bs —t.

So between the points Q,, and Q, of / we have a (rst—yt—t,
2 rst — ar — Bs —?)-correspondence. The 3 rst — ar — Bs — yt — 2t
coincidences are the points of intersection of / with Z and the points
_ of intersection of 7 with the curve of contact of the pencils (C,) and
(Cs), i.e. the locus of the points of contact of the curves C, and C,
touching each other. If there are two systems of curves (u,, »,) and
(u,, v,)*), the order of that curve of contact is

#,?, + #,P, + H,U, *)-

1) A system of curves (,, v) is a simply infinite system of curves, of which
» pass through an arbitrarily given point and » touch an arbitrarily given straight line.

*) This order is found by counting the points of intersection with an arbitrary
line 7. To this end we consider the envelope of the tangents of the curves of the
system (yj, +) in its points of intersection with /; this envelope is of class 4) +1,
the tangents of that envelope passing through an arbitrary point Q of / being
the tangents in @Q to the ~, curves of the system through Q and the line /
counting », times. In like manner does the system (yg, v2) give an envelope of class
Bo vg. The (j++) (4g-+- ve) common tangents of both envelopes are the
line 7 counting »». times and yyy9+ p,%9-+ po», other lines whose points of
intersection with / indicate the points of intersection of 7 with the curve of contact.
For a deduction with the aid of the symbolism of conditions see Scuusert, “Kalkiil
der abzdhlenden Geometrie’”, p. 51—52.

( 426 )

If we take for ithe systems the two pencils (C,) and (C;) then
u, =u, = 1 and (as ensues immediately from the principle of corre-
spondence) », = 2(r—1), », = 2 (s—1). So the order of the curve of
contact is

Qn + 2s—B.

For the number of points of intersection of 7 with Z remains
3rst— ar —Bs— yt—2t—(2r+ 2s—3) = 38(rst+- 1)—2(r4 s+-t) —(ar+fs-+ yt).

So we find:

The locus L of the pairs consisting of two movable pomts by which
a curve of each of the pencils is possible is of order

n—=3(rst + 1)— 2(r +s +2) — (ar + Bs + yt);
here a is the number of fixed pots of intersection of the pencils
(C;) and (C), B that of the pencils (Ci) and (C,) and y that of
(C,) and (C3).

2. Whilst the preceding considerations remain accurate when of the
basepoints of one and the same pencil some coincide, we shall suppose
in the following that the pencils (C,), (C,) and (C;) have respectively
r?, s? and ¢ different basepoints, so that we can only allow the
basepoints of one pencil to coincide in part with those of an other
pencil. Then a@ is the number of common basepoints of the pencils
(C,) and (C;) (which can however also belong to (C,)), ete. If the
pencils have no common basepoints («= 8—=y= 0), the order of
the locus becomes

8(rst + 1) — 2 (r+s-+2).

This is also in the case of common basepoints the order of the
total locus as long as that is definite, i.e. as long as there are no
basepoints common to the three pencils. If there is such a point, this
furnishes together with an entirely arbitrary point a pair of points PP
through which a curve of each of the pencils is possible; of this
pair of points however only one is movable. The locus proper however
is still definite then.

A basepoint of the pencil (C,) only we call A,, a common base-
point of the pencils (C,) and (C;) which is not a basepoint of the pencil
(C,) we call A, and a common basepoint of the three pencils we
call A, If d is the number of points A, then the number of
points A, amounts to a = aW— d, that of the points A,,to p =B—d
and that of the points A,, to y' = y— d, whilst the number of points
A, is equal to r? — p' — y' — d, ete. By introduction of a’, @’, y' and
J the order n of this locus proper becomes

( 427 )
naz3(rst+ W—2(r e+ )—(a'rt hist yQ—dr +549.

From this we see that the order of the locus proper is lowered by 7
en account of a common basepoint A,,. If there are no points
A;s:(d = 0) one can easily account for that lowering of order
by noticing that from the total locus the (C. passing through
A, separates itself, as not belonging to the locus proper. The point
A,, furnishes namely together with an arbitrary point of that Ca pair of
points satisfying the question ; of which points bowever only the latter is
movable '). Farthermore we see that a point A,.; diminishes the order
of L by r+s-+47, a fact one cannot account for by separation, the
total locus becoming indefinite *).

3. The locus proper L has in the basepoints of the three pencils
multiple points, the multiplicities of which are easy to determine.

A basepoint A, of the pencil (C,) only is an (st — a —1)-fold
point of Z. In fact, the curves C; and C; passing through A, have,
A, and. the basepoints excepted, still ss:—«—1 points of intersection
each of which combined with A, furnishes a pair of points satisfying
the question. The tangents in A, to the curves C’, passing through
the st—-a—1 mentioned points of intersection are the tangents of
J, in the multiple point.

To determine the multiplicity of a point A, we remark that to
obtain a pair of points satisfying the question and of which one of the
movable points coincides with Ay, it is necessary for C, to pass
through Ay (by which it is determined), whilst C, and C; which
always pass through A, must present a movable point of intersection
in As, thus must touch each other in A,. The question now rises:
How often do two curves C, and C; touching each other in Aj, in-
tersect each other again on the curve C, passing through Ay? To
answer this question we introduce an arbitrary C, intersecting the
above mentioned, C, in rs — y — 1 points differing from the basepoints.
Through each of these points we allow a C;, to pass which gives
rise to a correspondence between the curves C, and C; (so likewise
between its tangents in A,,) where rs— y—1 curves C; correspond
to a C, and rt— B—1 curves C, to a (:. Thus for the curves C,
and (; touching each other in A, it happens (7s ++ rt — 8B — y — 2)

1) If Ase counts for ¢ fixed points of intersection of the curves C, and Cy, the
C, passmg through Aw separates itself < times by which the degree of L is lowered
by &Y. ;

*) If As: counts for = fixed points of intersection of C; and C, for € fixed points
of intersection of C, and C; and for » fixed points of intersection of C, and C;, then
Ay dimimishes the order of L by er+¢s +t; this holds for a point As. too,
but then we must regard ¢ and , as being zero.

28
Proceedings Royal Acad. Amsterdain. Vol. [X

( 428 )

times that C, and C. have besides A,, another movable point of inter-
section, being at the same time movable point of intersection of C; and
C.. Here is included the case in whicb this second point of inter-
section coincides with A,,, thus where the curves C, and C, touch
C, in Ay; then only one movable point of intersection of C, and
C, still coincides with A,, whilst there need be no other movable
point of intersection lying on C;,, so that in this way we get no
pair of points furnishing a branch of ZL passing through Ay. So
the point Ay is an (rs + rt— B— y — 3)-fold point of L.

To determine the multiplicity of a point A, we have to consider
how many times three curves C,, C; and C; touching each other
in A, pass once more through a same point. To this end we con-
sider an arbitrary C, and the C, which touches this C, in A,s:.
Through each of the 7s — y—41 points of intersection of these C, and
C,, differing from the basepoints, we allow a C; to pass. Then the
question arises how many times this C; touches C, and Cs in A,st.
Let us call /,, the common tangent in A,,, of C. and Cs and J; the
tangent of C, in that point. To J/,, correspond rs—y—1 lines
/, To find reversely how many lines J/., correspond to an arbi-
trary line /, we consider an arbitrary C;, intersecting the C deter-
mined by / in rt—8 points differmg from the basepoints. Through
each of those points of intersection we imagine a C,. If J, and /
are the tangents in A, of C, and C, then rt—@ lines /, corre-
spond to /, and st—a lines /, to /, The rt-+ st—a—B rays of
coincidence indicate the lines /,, corresponding to i; to those rays
of coincidence however belongs the line /; itself, which must
not be counted, so that rt-+-st—a—f—t1 lines /,, corresponding to
l, vemain. So between the lines 7, and /; exists an (7s—y—1, rt +
+ st—a—pB —1)-correspondence.

The required lines /,,, are indicated by the st--tr--rs—(a-+-p+-y)—-2
rays of coincidence of this correspondence of which however three
must not be counted. When namely the contact in A, of C. and
C, becomes a contact of the second order one of the 7s—y—41 points
of intersection differing in general from the basepoints of C, and C,
coincides with A,s, namely in the direction of /.,. The C, passing
through that point of intersection will touch J/,, in A,s; in other
words /, coincides with /.,. As however the curves C. and C,, but
not the curves C. and C;, neither the curves C,, C; have in A,., a
contact of the second order we do not find in this way a pair of
points satisfying the question. Now it happens three times with two
pencils 0 curves with a common basepoint, between which a pro-
jective correspondence has been in such a way arranged that the

( 429 )

curves must touch each other in that basepoint, that this is a contact
of the second order, so that from the number of rays of coincidence
three must be subtracted to find that of the lines J... wanted.
From this ensues that the multiplicity of the point A,., amounts to
st + tr ee eek 1) 9.

So we find:

A basepoint of the pencil (C,) only is a

(st — a — l)-

fold point of the locus proper L. A common basepoint of the
pencils (C3) and (C,) which is not basepoint of (C,) is a

eat 8 7 —3)-
fold point of L and a common basepoint of the three pencils is a
(st + tr + rs — a — B— y — 35)-
fold point of L*).

4. With the help of the preceding the points of intersection of
£ with an arbitrary curve of one of ke pencils, e.g. a C;, are
easy to indicate. These are:

1. The r?> —8B—y-+d points A, counting together for

(r? meee) 1)

points of intersection.
2. The 8—d@ points A, counting together for
(8 — d) (sr + st —a—y— 3)
points of intersection.
3. The y — d points A,,, giving
(y — d)(@ +s —a——83)
points of intersection.
4. The d points A,., giving together

1) If there are no points A;s (3 =0) and therefore the total locus is not inde-
finite, we can also ask after the multiplicities of the points A, and Ay as
points of the total locus. Now the improper part of the locus consists of z curves
C, , B curves C; and y curves (; . Of these pass through a point 4, the @ curves
C, and through a point As, the B curves C;, the y curves C; and one of the
curves C,. From this ensues:

A point A, is an (st — 1), a point Ay an (rs+rt — 2)-fold point of the
total locus.

So the multiplicity of A, as a point of the total locus is not changed by the
coincidence of the basepoints, whilst the multiplicity of As: is equal to the sum
of the multiplicities which this point would have if it were only basepoint of the
pencil (Cs) or only basepoint of the pencil (C.).

28*

( 4380 )
dst + trp + rs —a—B—y—5d)

points of intersection.

5. The movable points of intersection of L with C, ; these are
those points of intersection which displace themselves ed we
choose another C’.. These are found as the pairs of common points
of the simply infinite linear systems of pointgroups intersect on C,
by the pencils (C,) and (C;).. The number of these are found from
the following theorem:

If there are on a curve of genus p two simply enfinite linear
systems of pointgroups consisting of a and b points, the number of
common pairs of points of those systems ts

(a — 1)(b— 1) — p.

In our case a=rs—y, b=rt—f8 and (as C, is am arbitrary curve
of the pencil (C,)) p= 4 (r—1) (v7—2). For the number of pairs of
common points we therefore find

(rs — 7 Dt 8 1) aie Die
and for the number of movable points of intersection of L and C,:
2(rs — y — 1) (rt — B — 1) — (r — 1) (vr — 2).
So the total number of points of intersection is:
r(drst +- 3 — 2r — 2s — 2t — ar — Ps — yt),
in accordance with the formula we have found for the order of L.

5. The pairs of points PP’ through which a curve of each, of
the pencils is possible determine on Z an involutory (1,1)-correspon-
dence; in the following we shall indicate P and P’ as corresponding
points of L.

If P falls into a doublepoint of Z differing from the base-
points, then in general two different points P' and P” will correspond
to P according to our regarding P as point of the one or of the other
branch of Z passing through P. The curves of the pencils passing
through P now have two more common points P’ and P", so that
we get a triplet of points ? P' P", through which a curve of each
of the pencils is possible.

It may however also happen that the points P' and P" coincide.
In that case correspond to the two branches through P two branches
through P’, so that P' is likewise doublepoint of Z. The curves of
the pencils passing through P have now but one other common
point P’, but now the particularity arises that P or P’ can be
displaced in two ways such that the other common point is retained.
So PP' is then to be regarded as a double corresponding pair of
points.

( 431 )

If reversely we have a triplet of points PP'P" lying on curves
of each of the pencils, then P is a doublepoint of ZL, for P' as well
as P" corresponds to P, and so it must be possible to displace P
in such a way that the corresponding point describes a branch passing
through P' and in such a way that a branch passing through P"
is described. The curve Z has thus two branches’? 1 and P2 passing
through P to which the branches P'l and P'2 correspond. Through
the point P’ (which is of course likewise doublepoint of L”as~well
as’ P") a second branch P'3 passes and through P" a second branch
P"3, which branches ‘correspond mutually. If a point Q describes
the ‘branch P1 the curves C,, C;, C, passing through Q have’a
second common point describing the branch /P’1, whilst a third
common point P” appears and again disappears when Q passes the
point P. This third common point displaces itself (along the branch
P"2) when Q describes the other branch passing through P, whilst
then the common point coinciding with /’ appears and disappears.

Triplets of points PP'P", and therefore doublepoints of ZL
differing from ‘the basepoints, there will be as a triplet of points
depends on 6 parameters and it is a 6-fold condition that a curve of
each of the pencils must pass through it. So we have:

The curve L has doublepoits, differing from the basepoints of the
pencils, belonging in triplets together and forming the triplets of points
through which a curve of each of the pencils is possible. To one or
other branch through a doublepoint of such a triplet corresponds a
branch through the second resp. the third doublepoint of this triplet.
Moreover L can however have pairs of doublepoints indicating the
double corresponding pairs of points. To the two branches through
the doublepomt of such a pair correspond the branches through the
other doublepoint of the parr.

6. The number of coincidences of the correspondence between P
and P’ can be determined as follows. The points ? and P’ coincide
if the curves C,, C, and C, passing through P have in P the same
tangent. Then P must lie on the curve of contact /,, of the pencils
(C;) and (C,) as well as on the curve of contact R,, of (C,) and (C,).
The number of points of intersection of those curves of contact
which are of order 27 + 2s — 3 resp. 27 + 2¢— 3 amounts to

(2r + 26 —:3) (2r + 2t — 3).

Some of these points of intersection however do not lie on the
third curve of contact R,, , and so they must not be counted. The curve
R,, namely passes once through a basepoint A, or A, and three
times through a common basepoint A,; or A,:; in fact in a point of

( 432 )

R,, two movable points of intersection of C, and C;, coincide so that
the point A,, as a point of the curve of contact is found when C,
and C, show in A,, a contact of the second orderwhich takes place
three times. Further Ff, passes through the doublepoints of the curves
C, and ©,, of which the number for the pencil (C.) amounts to
3(r—1)? and for the pencil (C,) to 3(s—1)’, which follows imme-
diately from the order of the discriminant.

Each of the 7? — p’ — y'— d points A, is a simple point of inter-
section of R,, and R&,, (simple, the tangents in A, to A,, and Ry,
being the tangents of the curves C, and C;, passing through A,,
differing thus in general), but no point of Ry. Each of the a’ points
Ay is a double point of intersection of A,, and R,,, as those curves
of contact in Ay have a simple point with the same tangent, namely
that of the C, passing through A,, ; these points are also points of Rg ,
namely threefold ones. Each of the # points A, is threefold point
of intersection of #,, and AR, (it being simple point of #,, and
threefold point of #,;) and lies at the same time on Ry; the same
holds for the y' points A,;. Each of the d points A, which are common
basepoints of the three pencils is 9-fold point of intersection of R,.
and R,,, being threefold point of each of those curves ; moreover it
‘is threefold point of Ry. Finally the 3(— 1)? doublepoints of the
pencil (C,) are simple points of intersection of #,; and A, , but not
points of Ry; of the curves C., C, and C; passing through such a
doublepoint C. has an improper contact with Cs and with C,, without
however C, and C, touching each other.

From this we see that the curves of contact R,, and R, have
r— p—y'—d4+ 3(r— 1)? = 47? — 6r + 3 — ' — y'— Gd
points of intersection which are not points of Ry, and so do not

furnish coinciding points P, P'. Moreover f&,, and R&R, have
2a’ + 3p + 3y' + 9d
points of intersection coinciding with the common basepoints, which
do fall on Ry, but which do not give any coinciding points P and
P', as for this it is necessary that of three curves C,, C, and C
passing through the same point each pair shows two movable points
of intersection coinciding with that point. So for the number of coin-
ciding points P and P' remains :
(2r + 2s — 8) (Qe + 26 — 8) —= (4 Geo 8 — By ee
— (2a + 38’ + 3y' + 9d) =
= A(st + tr + 7s) — 6(r +84+4H4+6—2(¢4+ 84+ y' 4+ 40).
So we find:
It happens
A(st + tr + rs) — Or +s +2) 4+6—2%et+e+y4)

( 433 )

times that the two points P and P' through which a curve of each
of the pencils is possible coincide.

7. With the help of this result the class of the envelope of the
lines connecting P and P' can easily be determined. To this end
we have to count how many lines PP’ pass through an arbitrary
point SS. We find this number by regarding the correspondence
between the rays SP and SP’, which we call / and /. This is an
involutory (7, 7)-correspondence where 7 represents the order of the
locus L of the points P and P'; for on an arbitrary ray / (or 7)
lie n points P (or P'), to each of which one point P' (or P) cor-
responds. So there are 2m rays of coincidence which can be furnished
either on account of PP’ passing through S or of Pand P’ coinciding.

So for the number of rays of coincidence where PP’ passes
through S we find:

2 {8(rst + 1) — 2 (r + 84 t) — (ar + Bs + yt)} — f4(st + tr + rs) —
— 6r+s64+4+4+6—2(a4+ 8+ y4 d)} = Grst — 4 (st + tr + rs) +
4+ 2(r+s- t) — 2a(r — 1) — 28 (s — 1) — 2y (¢ — 1) 4 29.

These rays of coincidence however coincide in pairs. For if the
line connecting the corresponding points P, and P,’ passes through
S, then to P,P,’ regarded as line / correspond 7 lines /', two of
which coincide with P,P,', for if point P of / is taken in P, or in
P,' the corresponding point P’ lies in P,' resp. P,. Likewise to P,P,
regarded as line / correspond n lines /, of which also two coincide
with P,P,', from which ensues that P,P,’ is a double ray of
coincidence’). So to find the number of the lines PP' passing
through S, thus the class of the envelope, the above found number
must still be divided by 2, so that we get:

1) One can easily convince oneself of the accuracy of this conclusion by a
representation of the correspondence between the rays SP and SP’. To this
end we regard the parameters of the lines SP and SP’ as rectangular Cartesian
coordinates x and y of a point which is the representation of those two lines.
The curve of representation (which is symmetrical with respect to the line y=
on account of the correspondence being involutory) indicates by its points of
intersection with the line y=-z the rays of coincidence. If B is the point of
representation of the rays J and /’ coinciding in P,P,’, the curve of representation
is cut in two coinciding points B by a line parallel to the y-axis as well as by
a line parallel to the x-axis, on account of P,P,' regarded as / or /' corresponding
twice to itself regarded as J’ resp. /. So B is doublepoirt of the curve of repre-

sentation, so that the line y=. furnishes two points of intersection coinciding
with B.

( 434 )

The envelope of the lines connecting pairs of points, through which
a curve of each of the pencils is possible, is of class

3 rst — 2 (st-Lir+rs) + (r+s+t) — a(r—1)— B(s—1)-y-l) + d=
— 3 rst — 2(st+tr+rs) + (rts+t—e'(r—1)— B(s—-)— 7 €—-)—
— S(r+s-+t—A4).

8. If the pencils have no common basepoints then the class of
the envelope is 3 rst — 2 (st + tr + rs) + (r+ 5-2). By a common
basepoint A, of the pencils (C) and (C;) that class is lowered with
r7—1. This is because point Ag has separated itself from the
envelope r—1 times. In fact, the curve C, passing through A, has
separated itself from the locus of the points ? and /’. If we take
P arbitrarily on this C,, the corresponding point ?’ coincides with
Ay. So an arbitrary line passing through A, is to be regarded
(r—1) times as a line connecting P and P”’, as any of the r—1
points of intersection with C;, differing from Ay may be chosen for P.

If the three pencils have a common basepoint A,; the total envelope
of PP’ remains definite (in contrast to the total locus of P and
P’. It is true P can be taken quite arbitrarily, but then
coincides with a point A,., so that the line P/’ passes through that
point A,., and therefore is not quite arbitrary. As the class of the
envelope proper is lowered by the point A, with r+ s + ¢— 4 iit
follows, that A,s: separates itself (r+s+t—4) times from the
envelope. As one of the points of the pair becomes entirely indefinite,
that multiplicity is not easy to explain, as far as I can see.

Physics. — “On a new empiric spectral formula.” By KE. E.
MocenporFr. (Communicated by Prof. P. Zeeman).

3y the fundamental investigations of Kayser and Runer and those
of RypprrG the existence of spectral series was proved. The formulae
of these physicists, however, give in general too great deviations for
the first lines of a series. | have tried to improve the formula given
by RypBeEre:

Particularly noteworthy in RypsBere’s formula is the universal
constant NV,. From Barmer’s formula, which is included as a special,
case in Ryppera’s formula, follows for hydrogen for the observation:

corrected to vacuo N, = 109675. > i aitiz

( 435 )

Assuming for a moment that the .V, was also variable for the
different series, I have calculated the constants A, a@ and .V, from three
of the best observed curves. For NV, the following values were found :

Principal Series Lithium — 109996
Z “A Natrium 107178

cs Z | Potassium | 105638

; » | Rubidium 104723

03 = Caesium 104665

1st associated series) Hydrogen 109704
¥5 5 Helium 1097038

to Natrium 110262

5 63 Potassium | 109081

9 $5 Silver 107162

a a Magnesium | 108695
a3 - Zine 107489
9 A Oxygen 110660
Second _,, Natrium 107819
= a Magnesium | 105247

s és | Calcium 103702

» A Zine 105399

is pe | Aluminium | 105721

These values have been calculated from wave frequencies not
corrected to vacuo.

As appears from these values NV, is not absolutely constant. As
Kayser *) found in another way, we see, however, that relatively

1) Kayser, Handbuch IIL. p. 553.

( 436 )

NV, changes little from element to element '). The supposition lies at
hand, that a constant of nature will occur in the rational formula.
For the first associated series of Aluminium calculation gives a con-
siderable deviation. Calculating from the first terms of this series we
find N,—207620 calculating from the middle lines N,—138082,
and from the lines with smaller 2 V, = 125048.

The first asssociated series of aluminium behaves therefore quite
abnormally.

In Rypsere’s formula another function than (m—- a)—? must be
used to get a better harmony, specially with the first terms of a series.

In my thesis for the doctorate, which will shortly appear, I have
examined the formula:

109675

eo SRE
(m +a +5)

in which m represents the wave frequency reduced to vacuo, A, a
and 6 are constants which are to be determined, m passes through
the series of the positive integers, starting with m=1. In most
eases with this formula a good agreement is obtained, also with the
first lines of a series. The associated series converge pretty well to
the same limit, while also the law of RypBERG—ScuustTEr is satisfied
in those cases where besides associated series, also a principal series
is observed.

A spectral formula has also been proposed by Rutz ?).

In my thesis for the doctorate I have adduced some objections to
the formula of Ritz, as it gives rise to highly improbable combinations
of lines. Moreover for the metals of the 2°¢ column of MENDELEJEFF’s
system his views are not at all in harmony with observation.

In the following tables the observed wavelength in A. E. is given
under 2,, the limit of error of observation under F, the deviation
according io the formula proposed by me under A, the deviation
according to the formula of Kayser and Runer under A. K. R. The
mark * on the right above a wavelength indicates that these lines
were used as a basis for the calculation of the constants A, a and 6.

The constants are calculated from the wave frequencies reduced
to vacuo *).

(hs

1) The * in Kayser and Runee’s formula varies within considerably wider hmits
than the VV) of Ryppere’s formula.

*) Ann. d. Phys. Bd. 12, 1903, p. 264. W. Rivz, Zur Theorie der Serienspectren.

8) Where it was possible, | have always taken these values from the “Index of
Spectra” from MarsHatt Warts.

( 487 )
Lithium.

Principal series: A = 4348013; «= + 0,95182 ; b= + 0,00722
1st ass. series : A = 28581,8 ; a=-+1,998774; 6 = — 0,000822
ands =: A= 285818 5 a= + 1,59872 5 6 = — 0,00821
ga) 5 A= 285818 5 a= + 1,95085 5 b= + 0,00404

The associated series converge here evidently to one limit.

The difference of wave frequency between the limits of principal
and associated series is 43480,13—21581,8 = 14898 33. The wave
frequency of the 1t line of the principal series is 14902,7. So the
formula satisfies the law of RypBrre-ScuustTER pretty well.

m dy E * kK VN a
:

1 | 67082 * | 0,20) 0 | + 408
2 | 3232,7" | 003 | 0 0

3 | 9741,39 | 003! — 006 | 0

4 | 62,60" 1003]. 0 0

5 | 947513 | 010 | — 0,2 | — 02

6 | 495,55 | 0,40 | — 0,18 | — 001
7 | 9304,54 | 0,20} — 043 | + 0,30

| ?
}

Ser eeers oe AMS (a Rell O02} 0,75
9 | 93594 LD.) ? | +047 |:4 418

FIRST ASSOCIATED SERIES.

m a | abla ake. Ke R-
|

a | 6103,77" 0,03 0 0
9 4602,37* | 0,10 Gmc 63.0
3 413944 | 0,20 | — 0,11 0
4 | 3915,20* 0,20 n= 0.20
5 3794,9 | 5,00} -—- 0,09 | — 0,35
6 3718.9 5,00 | — 1,94 | — 2,95
7 36706 5,00 | — 1,06 | — 4,44

( 438 )

SECOND ASSOCIATED SERIES.

nn nr ncn nnn es enn
|

m dw F A: Sp AGREES
4 | 8127,0° S 0,30 | 0 | — 65
2 | 497244 0,10 | — 0,13 | 0
3 4973,44* 0,20 | 0 | 0
4 3985,94 0,20 | + 0,22 | 0
5 3838,30 3,00 | + 2,40 | — 02

THIRD ASSOCIATED SERIES.

m dw | Eat A AAT
— :

4 | 6240,3*° S 040) Oe he

2 | 4636,3* S | 040/ 0 Sop

3 | M482 S 400 | 46. 4s

4 | 3928 EH dc eS

The capitals after the wavelengths denote the observers: L. D.
Liveinc and Dewar; S. Saunpers and E. H. Exner and HAscnek.
Where no further indication is given, the observation has been made
by Kayser and Runeér.

Natrium.

Principal series (the lines of the doublets with greatest 2)

A = 41447,09; a—=1,147615; b= — 0,031484
Principal series (lines of the doublets with smallest 2)
A = 41445,20; a= 1,148883; 4b = — 0,031908.

For the calculation of the limit of the associated series RyDBERG-
ScuustEr’s law has been used. With a view to the constant differences
of wave frequencies of the doublets of the associated series, I have
only carried out the calculation for the components with small
wavelength.

For the 1% ass. series A = 244911; a—1,98259: 6 = + 0,00639
For the 2"™¢ ass. series A = 24491,1; a= 1,65160; 4 = — 0,01056

( 439 )

PRINCIPAL SERIES.

re Oo HO

m iw F A Ae Ke
|
1 | 5896,16" a 0 |+478
1 | 3890,19° _ 6, £4.86
9 | 3303,07° 0,03 0 0
2 | 330247" + 0,03 0 0
3 | 2852.91 (0,05 | — 0,14 0
3 | 9852.91 1005 | —006 | 0
4 | £680,46" 0,10 0 0
4 | 9680,46" oso | -0 0
2593,98 (040 | +003 | + 003
9593,08 010 |. — 0,02 | + 0,09
2543.85 L. D. | 040 | — 0,06 | + 0,10
| |
2543.85 L. D. | 040 | — 014 | + 0,24
9512.93 L. D. | 0,20 | + 003 | + 0,50
2512,93 L. D. | 0,20 | — 0,0 + 0,60
FIRST ASSOCIATED SERIES.
= | = ae
818433" L. | 02 0 | 0
5682,90 0,15 0,04 | 0
4979,30° 020} 0 | 0
4665,20 0,50} — 013 | + 0,52
4494,30 | 400 | — 0,28 | + 0,50
4390,70 L. D. | 2 | +098 | + 4,30
4395,70 L. D | 2 | +400 | + 4,76

|
|
{

SECOND ASSOCIATED SERIES.

11404 2 +400 | + 100,
. 6154,62* 0,10 Gove 26
5149.19* O10 0 0
4748,36 | O45 + 0,12 0
4549.75 | obo + 0,65 + 1,39
4420.0 L.D.| 2 +002 4,55
4343,70 r | + 2,00 | — 1,36

( 440 )
Zine.

For this element I have calculated the formulae of the 1s* and
2d associated series for the components with the greatest wave-
length of the triplets.

The limits are determined for the two series separately, for the
first associated series the calculation gave 42876,25 and for the
second associated series the limit appeared to be 42876,70. A very
sood agreement. ,

The formula gives as 1*' line of the 1st associated series of Zinc the line
8024.05, which has not been observed. The 8'" line of the first associated
series 2409,22 has not been observed either. As 9 jine of this series
2393,93 was calculated, which is in remarkably good harmony with
the intense line 2393,88. As yet this line had not yet been fitted
in the series. The great intensity of a curve in the root of the series
is certainly strange; an investigation of the magnetic splitting might
decide whether it is correct to range this line under the first associated
series.

The formula for the 1st associated series is:

109675

n = 42876,25 —

0,007085 \?
ASN OOO NS Ss ee
m
and for the 224 associated series:
mee 109675
nm == 42876,70 —
We 0,058916\?
m + 1,286822 —
m
FIRST ASSOCIATED SERIES.
m dw | Neat A ie Keeht
| | |
|
1 | eae pte ae cas
2 | | 334513" — | 0,03 | 0 os 0@8
3 | © 2801,00° | 0,03 0 +. 0,08
4 | 2608,65* | 0,05 One as ope
5 | — 2816,00 0,20} + 0,04 | — 041
6 263,41 0,20 | — 0,14 | — 039
7 2430,74 0,30) + 0,22 | + 9,00
8 = Es | 2 —
9 2393,88 005 | -=e0rel

( 441 )

SECOND ASSOCIATED SERIES.

m dw Bah Soke A ee
4 4810,71* | 0,03 | Oh ta6
| |
ae 3072,19* | 0,05 | oy | > 16:00
3 Q712,60* | 0,05; 0 | + 0,02
4 9567,99* | 0,10 | + 0,14 | — 0,01
2493.67 . | 015 | + 042 | — 0,04
6 219,76 =| (0,95 | — 04 | — 0,20
Thallium.
The formula for the 18* associated series is:
109675
nm = 41466,4 —
0 ,00366
pia Ne emis
mm
for the satellites :
109675
(m Br 68956425 2 =
m
and for the second associated series:
109675
n = 41466,4 — TEES -G 07 10557
(m eet 9651 6: — ee)
m

The limit has been calculated from three lines of the 1st associated
series; only two more lines were required of the satellites and of the 24
associated series. So in this spectrum all the constantshavebeen calculated
from 7 lines and 31 lines are very well represented by the formula.

FIRST ASSOCIATED SERIES.

m | = | F A A. K. R.
1 3519,39° | 0,03 0 a
2 2918,43 0,03 | — 004 | —
3 9709,33* 0,03 0 =
4 2609,08 003!) =. 004 | —
5 2552,62* 0,10 0 ae
6 9517,50 040} — v6 | — 0,34
7 9494,00 0,10 | — 003 | — 0,19
8 QAT7,58 0,10 | — 0,09 | + 0,06
9 | 9465,54 0,20 | —0,A7 | + 0,2%
10 | 2456,53 0,20 | — 015 | + 0,47
1 4957 0,30 | — 0,7 | + 068
12 2444,00 0,30 — 0,28 | + 0,79
43 2439,58 0,30 | — 0,24 | + 0,95

( 442 )

SATELLITES.
| is So ee eee Pye
‘ |
1 3509.58 003} 90 | 4 002
2 292163 | 0,03} +006 | — 007
3 ono77* | 009} Oo | +4 0,43
4 260986 | 0,03 | — 0,03 | — 0,02
5 | 955307 | 0,10) — 005 | — ofe

| |

SECOND ASSOCIATED SERIES.

m hw i A hee ie ke da
|
1 5350,65* | 0,03 | 0 |= 468
2 | sea9gs- | 003; 0 Pe 2,7
3 20627 ~—s«|-:0,05 | — 0,05 | — 365
4 2665,67 | 0,05 | — 4,32 | — 4,69
5 2585,68 | 0,05 | — 0,16 | + 0,01
6 | 953897 .| 040] — 047 | + 004
7 | 2508,03 015 | — 044 es 0,01
8 | 248757 | 0,20 | — 0,06 | + 0,08
9 | 249765. | 020 | — 0,34 | — 0,21
10 246201 | 030} — 020 | — 0,3
44 | 9453.87 | 0,30 | — 0A7 | + 007
12 | Q4AT 59 | 0,30 | — 0,05 | + 0,22
13, | 2442.94 = | 030 | — 0,37 — 001

I shall just add a few words on the spectrum of Aluminium.
None of the formulae given as yet represents the first associated series of
this element at all satisfactorily; nor is a satisfactory result attained
with my formula. In the beginning of this paper I have pointed out,
that very deviating values for NM, were calculated from three of the
1st lines of the series.

The formula runs :

109675 .
1,038060\?

n = 48287,9 —
(m + 0.89436 + ——_—_—
m

( 443 )
The constants have been calculated from the lines 4, 5 and 6.

ALUMINIUM. FIRST ASSOCIATED SERIES.

m dy F | A AC Kees
| | bi cated)
1 31 82.27 0,03 | —268,82 | +3848
D) 9568,08 003! + 346 | + 535
3 2367,16 003) + 252 | + 64
4 | 9963.83" 0,10 0 + 0,03
5 | 9204,73" 0,10 0 + 0,7
6 2168,87* 0,10 0 = O13
7 145,48 0,20/ + 006 | — 0,31
8 129,52 0,20; + o44 | — 01
9 2118.58 oy | — 098 | 4 o4

The agreement with the first lines (1,2 and 3), leaves much to be
desired. The value of the constant > is here 1,03806, greater than
the value of @ in that formula; this does not occur with any of the
other series.

With 4 constants, so with:

109675
a

b ey
(m+o4+-4+35)

a better result is most likely reached. When the constants / and here
probably also the c, are not small with respect to a, then the influ-
ence of those constants is very great, particularly for small values
of m. The deviation for the first line of the above series (3082,27),
however, is so great, that I doubt if this is really the first line of
this series.

The behaviour of this Aluminium series is certainly peculiar, and
a further investigation is desirable.

For the way in which the constants in the formula were calculated,
and for the spectra of Potassium, Rubidium and Calcium, of Magne-
sium, Calcium, Cadmium and of Helium and Oxygen, I refer to my
thesis for the doctorate, which will shortly be published.

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 444 )

Astronomy. — “Mutual occultations and eclipses of the satellites
of Jupiter in 1908.” By Prof. J. A. C. OupeMans.

SECOND PART. — ECLIPSES.

(Communicated in the meeting of October 27, 1906).

From occultations to eclipses there is but one step.

Between the two phenomena there is this difference that, as has
been communicated on p. 305, the occultations have been observed
more than once, but that of the eclipses of one satellite by another
we have but one, incomplete account given in a private letter of
Mr. Stantey Witu1AMs dated 7 December 1905. In his letter to us
he writes: “With regard to the heliocentric conjunctions there does
“seem to be one observation of the rare phenomenon of the eclipse
“of a satellite in the shadow of another one on record. It occurred
“on the 14 August 1891 and was observed by Mr. J. Comas at
“Vatis in Spain and by the writer at Hove. Mr. Comas’ observation
“was published in the Frenca periodical L’Astronomie, 1891, p. 397
“(read 398) 1). The following is an account of my observation. No
“particulars of this have hitherto been published.”

« «4891 Aug. 14. 6'/, inch reflector, power 225. Definition good,
“but interruptions from cloud. Satellite I. transitted on the $. Equa-
“torial belt, (N. component). Jmmediately on its entering the dise
«<it became lost to view. At 11°49™ a minute dark spot was seen
“ «about in the position which the satellite should have then occupied.
«<The shadows of satellites I. and Il. were confounded together at
“this time, there seeming to be one very large, slightly oval, black
“spot. At 11°59™ the two shadows were seen neatly separated,
“<«thus, @® . The preceding shadow must be that of II., the follow-
‘ing and much smaller one that of I.. At 12'10™ satellite I. was
“certainly visible as a dark spot, much smaller than the shadow
“““of either satellite. It had moved with respect to the shoulder of
“<«<the Red Spot Hollow, so that there could be no doubt of its
“<identity. It is on the north band of the north (south) equatorial
“ «belt 2). Satellite I {this should evidently be II.] shines brightly
““on the dise near the limb. Definition good, but much thin cloud
oS eapout.-)

n
“

“The foregoing is an almost literal transcript from my observation
“book. I take it that when satellite I. entered on the disc of Jupiter,
“it was already partly eclipsed by the shadow of IL, so that it
“became lost to view immediately, instead of shining, as usual, for

tee

Tpowiee

“
iy

Pla
J. A. C. OUDEMANS. “Mutual occultations and eclipses of the satellites of Jupiter
in 1908.” Second part: Eclipses.

=
s
N

Scale — : ——, On this scale the sun’s diameter is 0.24 meter and its
30 168 000 000

distance 25.783 m.

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 445 )

“some time as a brillant disc. Also that the minute dark spot seen
“at 11549™ was produced by the portion of the shadow of II., then
“projected on I. Also that the small size of the following shadow
“spot at 11'59™ was due to a part only of the shadow of U. being
“projected on the disc of Jupiter, the other part of this shadow
“having been intercepted by satellite I. 3)

— — — “But combining Mr. Comas’ observation with my own,
“there can be no doubt but that satellite I. was actually partially
“eclipsed by the shadow of II. on the night of August 14, 1891.
“So far as I am aware, this is the only indubitable instance of one
“satellite being eclipsed by the shadow of another.”.....

“PS. The above times are Greenwich mean times. The Nautical
“Almanac time for the transit ingress of satellite I. is 11533™.” 4).

Before proceeding to the computation of epochs of such heliocentric
conjunctions we have investigated to what extent generally eclipses
of one satellite by the shadow of another are possible. That they
may occur is proved by the shadows of the satellites on Jupiter
itself. The question however is: 1s* whether the shadows of the
foremost satellite reaches that of the more distant one in every helio-
centric conjunction and 2°¢ whether the occurrence of total eclipses
is possible in any case. In order to find an answer to these questions
we assume that the orbits all lie in a single plane which, being
prolonged, passes through the centre of the sun. We further imagine
a line in the plane of the orbits starting from the sun and passing
Jupiter at a distance equal to its radius, the distance from the centre
thus being equal to its diameter (see Plate I). This line cuts the
orbits of the four satellites each in two points. Beginning with the
point nearest the sun we shall call these points g, @, ¢, a, Baie
and h. For clearness, sake the figure is given below (Plate [).

Now suppose that I is placed either at a or at 4. In both cases
the other satellites will be involved in its shadow cone as soon as
they come: II, at d, Hl,at f and IV; at 4.

The points of intersection with the orbit of II are c and d. If
II, is at ¢ then I,, may be eclipsed in a but also I; in 6; Illy at f
and IV; at h.

But if I, is in d then only II, and IV; can be eclipsed, the former
at f and the latter at h.

The points of intersection with the orbit of III are e and /. If
III is at e there is the possibility of an eclipse for’ Ey, at<, 1, at a,
I, at 6, ly at d and IV; at 4. If on the other hand it is in F there
is such a possibility only for IV; at /.

It is evident that IV can only cause the eclipse of another satel-

29%

( 446 )

lite if it is at the position g, one of the other three satellites being
then at one of the points of intersection already mentioned.

Each of the satellites might thus produce six different eclipses ;
if however we compute the radii of the umbra for the positions of
the other satellites we are led to a negative value in some of the
cases. This means of course that the vertex of the cone of the
umbra does not reach the other satellite.

If for the radii of the satellites we adopt the values mentioned in
the first part of this communication, diminished however by the
amount of the irradiation, it appears that a total eclipse is only pos-
sible in two cases. III, may cause a total eclipse of I, and In; I+
may nearly produce such an eclipse of Il, If the shadow does not
reach the other satellite then an inhabitant of the latter would see
an annular eclipse of the Sun.

This case presents itself

for the shadow of I, in respect to IV;.
ULE ee yk TR and, Ve
pape ib eee i ebay RAS
re) IV, » > > Il; and Ill.
In the fifteen remaining cases there may be a partial eclipse.
It need hardly be said that this case can only present itself if, at

the time of heliocentric conjunction, the difference of the heliocentric
latitudes (y’—y), is smaller than the sum of the radi. In computing

however the occultations observed by Messrs FautH and NIJLAND it

appeared that this difference in latitude, according to the tables of
DamolsEAu, is sometimes slightly greater. The latitudes found by these
tables are therefore not entirely trustworthy. For this reason we in-
cluded all the heliocentric conjunctions between 1 April and 20 May
1908 (both dates inclusive).

The preparation for the computation, vz the drawing of the orbits
of the satellites is the same as for the computation of the geocentric
conjunctions (see 1st part). First however the epochs of the helio-
centric superior conjunctions must be derived from the epochs of
the yeocentric superior conjunctions taken from the Nautical Almanac
by the aid of the hourly motions of the satellites and of the angle
G, i.e. the angle Earth—Jupiter—Sun. Furthermore, the jovicentric
mean longitudes should be corrected for their equations and pertur-
bations and diminished by S. i.e. the heliocentric longitude of Jupiter,
instead of by S—G which is its geocentric longitude.

Of the arguments N°.3 need not be computed; for this argument
only serves, combined with 1, tor the computation of the jovicentric

ii li,

-”

( 447 )

latitude of the Earth, which need not be known in the present case.
The number of columns in our tables will thus be found to be
diminished by one for each of the satellites.

Our results are contained in the annexed table. Between 1 April
and 20 May we found 81 heliocentric conjunctions; the last column
but one, (y—y’), shows that in a very great number of the cases an
eclipse is possible.

(1) The account of Mr. José Comas is as follows:

Ombres de deux satellites de Jupiter et eclipse. — Dans la nuit
du 14 aofit, j'ai observé un phenomene bien rare: la coincidence
partielle, sur Jupiter, des ombres de ses deux premiers satellites, et
par suite l’éclipse de Soleil pour le satellite | produit par le satellite II.

A 11 (temps de Barcelone)') lVombre du satellite IL est entrée
sur la planete. Pres du bord, elle n’était pas noire, mais dun gris
rougedtre. Comme limage était fort agitée, j'ai cessé d’observer,
mais je suis retourné a Jlobservation vers 11°37™ pour observer
Yimmersion du premier satellite, qui a eu lieu a 11°42™ (erossis-
sement 100 fois; lunette de 4 pouces). Jai été surpris de voir
disparaitre Io?) a son entrée sur le disque, ne se détachant pas
en blanc, quoiqu’il se projetat sur la bande foncée équatoriale
australe.

A 11"52™, avec des images plus tranquilles et un grossissement
de 160, je remarquai que l’ombre completement noire que lon
voyait était allongée dans une direction un peu inelinée vers la
droite, relativement a l’axe de Jupiter. La phase maxima de I’éclipse
du satellite I était déja passée de quelques minutes. A 11556™ je
pris le petit dessin que j’ai l’honneur de vous adresser; les deux
ombres se touchaient encore’). Aussitdt elles se séparerent et,
quoique je n’aie pas pu noter lVinstant du dernier contact, je crois
étre assez pres de la vérité, en disant qu’il s’est effectué vers 11°58".

L’empiétement d’une ombre sur l’autre pourrait étre de la troisieme

1) Barcelone is 2°10' East of Greenwich; mean time at Barcelone is therefore
8m 40s later than of Greenwich.

2) Since a few years the Nautical Almanac mentions the names of the Satel-
lites of Jupiter proposed by Srwon Marius: Io, Europa, Ganymedes and Callisto.

3) This drawing shows, as seen in an inverting telescope, the right hand
(following) part of the well know Red spot in the Southern Hemisphere of Jupiter.
Below it, at some distance, a dark band and still further two dark shadows each
4 mm. in diameter, which are not yet separated, The common chord is 2,5 mm,
in length; the total length of the two shadows together 7,2mm, The line connecting
the centres makes an angle of 40° with the vertical. Meanwhile the motion of the
two shadows must have been nearly horizontal.

( 448 )

partie du diametre. Dans cette supposition la distance minima des
centres des deux ombres a di avoir lieu vers 11°47™ et le premier
contact vers 11»37™. Le premier satellite pénétra dans le disque de
la planéte a 11°42™, comme j'ai dit plus haut, done l’éclipse a com-
mencé quand le satellite se projetait encore dans lespace, cing
minutes avant |’immersion.

L’invisibilité de lombre d'Europe sur lo peut s’expliquer par la
mauvaise qualite des images. Toutefois, la penombre et lombre du
II satellite ont été suffisantes pour diminuer notablement l’éclat du

premier.

(2) The meaning evidently is that, as seen in an mverting tele-
scope the dark spot seemed to be situated on the North band of the
North belt, but that in reality it was on the South band of the South
belt. It is well known that the so-called Hed spot is there situated.

(8) The author does not refer here to the visibility of a shadow
of II on I. This may be explained, in my opinion, by irradiation and
diffraction.

(4) According to the tables of Damoisgav, second part, the time of
the heliocentric conjunction of the two satellites is 28°45™ civil time
Paris = 11"36™ Greenwich. In the Nautical Almanac of 1891 we
find the following data for 14 August:

II Shadow. Ingress 10"51™ M. T. Grw.

I = ne A GI cer k ee
I Transit gee | eS eae
II a Piper Vee bags: ee ae Ree
I Shadow. Egress 1318 ,, ,, ,,
II ‘ 5S 43 eee
I Transit ryan de ee ee
ll z ion EEO oe pee

If from the 1st, 294, 5t and 6 line we compute the time at
which the shadows must coincide we get 11"31™. This result differs
by 5™ from that found just now. We have to consider, however,
that the two satellites went the same way, and that their relative
mofion in five minutes, consequently also that of their shadows, was
very minute.

Mr. STanLEY WILLIAMS seems not to have perceived a shadow
before 11°49" M. T. Greenwich; Mr. Comas already saw an oblong
shadow at 11°43™20: M. T. Greenwich. For the rest Mr. STaniey

( 449 )

Wituams makes the shadow of II larger than that of I whereas
in the estimation of Mr. Comas they were equal. It seems hardly
doubtful but the English observer must be right.

(5) In 1901 Sex repeatedly measured the diameters of the satellites
of Jupiter at the 26 inch telescope of Washington. He made use of
the filar micrometer but took a special care to eliminate the syste-
matic errors peculiar to this instrument ( Vid. Astron. Nachr. N°. 3764,
21 Jan. 1902. The communication of Sex is dated 19 Oct. 1901).

During the months May—August (both inclusive) of the year 1901
he measured the diameters in the night. He was then much troubled
by the undulation of the limbs caused by the unsteadiness of the
‘air. Afterwards in the months of September and October of the same
year he observed a little before and a little after sunset. Artificial
illumination was then not needed; and the satellites appeared as
quiet discs. Moreover the field and the satellites were coloured greenish
yellow by a screen filled with protochloride of copper and picric acid.
Tne results for the diameters turned out to be smaller in every case
than those formerly found. The difference was attributed to irradiation.

The results, reduced to the mean distance of Jupiter to the sun
(5,2028), are as follows.

Difference, attributed

Satellite At night In daytime Fie acti ie
I 1",077 + 0”018 0"834 + 0,006 0"243 + 0"019
II 0 976 + 0,043 0,747 + 0 ,007 0 229 + 0 0435
Ill 1 604 + 0,038 4 ,265 + 0 ,009 0,339 + 0,039
FY 1 44 + 0,018 1 169 + 0,006 0 372 + 0,019

It is remarkable that the brightest satellite, II1, shows also the
strongest irradiation. If however we consider the difference insuffi-
ciently established, and if therefore we combine the several results
obtained for the irradiation, duly taking into account the weights
corresponding to the probable errors, we get

Irradiation = 0",264 + 0,012.

This is the irradiation for the whole diameter and we thus get
0",132 for each of the limbs. This number however holds only for
the telescope at Washington for which, owing to its great aperture,
the diffraction must be exceedingly small.

( 450 )

It seems worth while to call attention to the differences between
the diameters found by the same observer in 1900 and 1901.

| 1900 | 1901 - 1901—1900
1 | ove72 + 07098 | 0",834 + 0"006 + 07162
1 | 0,624 + 0,078 0,747 + 0,007 + 0,124
Il | 1,361 + 0,103 | 1,265 + 0,009 | — 0,096
Iv | 1,277 + 0,083 | 1,169 + 0,006 | — 0,108

Stone, at Oxford, once told me that Arry, in a conversation on

the determination of declinations at the meridian circle, remarked to.

him: “I assure you, SToNE, a second is a very small thing’.

If we consider the differences just adduced between the results
obtained by a single observer in two consecutive years we are led
to conclude that, for micrometer observations, even now ‘a tenth of

a second is an exceedingly small thing”.

Appendix. Jn how far are the tables of Damotsnav still reliable ?

In the first part of this paper, pages 319 and 521, we explained
why we felt ourselves justified in using the tables of Damoisravu for
these computations in advance. We may now add that we also
investigated the differences of the eclipses, as observed in some recent
years at different observatories, from these tables, or rather from
the epochs given by the Nautical Almanac. In these investigations
we have been assisted by Mr. Kress, amanuensis at the Observatory
of Utrecht, who has carefully searched some volumes of the Astrono-
mische Nachrichten and of the Monthly Notices for the time of
“disappearance and reappearance’ of each satellite. He has further
combined these times, reduced them to the meridian of Greenwich,
and has then compared them with the data of the Nautical Almanac.
In order to simplify, we requested him to note only the observation
of the last light seen at disappearance and the first light at reappear-
ance'). We intended to extend our investigation from 1894 to 1905

1) DeLamBre in the introduction to his tables, does not state explicitly the
precise instant to which his tables refer but from some passages we may conclude
that he also means the instant as here defined. So for instance on page LIII
where he says: “Les demi-dureées ont été un peu diminuées, pour les rapprocher
des observations qu’on a faites depuis la decouverte des lunettes achromatiques’’.

That Laptace also takes it for granted that such is his real meaning, appears

from Ch. VIII, 8th book of the Mécanique Céleste.

9 gh eel

( 451 )

or 1906, but after having completed som> four years there seemed
reason to think that there was hardly need for further information.
The general result arrived at was, that the tables were still sufti-
ciently accurate for our purpose, which was no other than to prepare
astronomers for the observation of the mutual occultations and eclipses
of the satellites.

Now that the work is finished we will not suppress its results
though it cannot at all claim to be complete. It never was our
intention to make it so, and the journals appearing in France, in
America ete. have not been searched.

The following observatories have contributed to our investigation.

Aperture of the telescopes

in m.m.

cecamiee nt = ty , te LOD E70; 254, 714!
Uc 5) i Oh ee i er a 260

JE le oe it ee 5 ta 150
demaeCWWINKEBR YS =< < i. . 162

Palitax (GEEDHIEL) ocs.. 2°. 230

Ls DUE ee SI Ria Ae ee 162
GChrsitatien 64, tes —_. as. ~5.-j. 74, 190
Marwnere, oie, Yo. a. of 66, 84, 84, 96,244
OD. ea rr 2 161

Windsor (Tebbutt) near Adelaide 203

Lyon (a single observation) 2

At Greenwich, Christiania and Kasan the eclipses have been often
observed by two or more astronomers using telescopes of different
aperture. In such cases we have only taken into account the instant
observed by means of the telescope of largest aperture. As a rule
the observer at this telescope could follow the satellite longer at
“disappearance” and he would pick it up earlier at “reappearance”.
There are however a few exceptions to the rule.

For the eclipses observed during the period of a single opposition
of Jupiter the corrections to the data of the Nautical Almanac in
no case showed a regular progression. They fluctuated on both sides
of the mean in such a way that there could be no objection to
adopting their arithmetical mean, a proceeding which still would be
perfectly justified, even if there had been a regu/arly increasing or
decreasing progression. No further attention was paid to the diffe-
rences in the aperture of the telescopes. If these apertures exceed
a certain amount, for instance 150 mm. we find, theoretically as

( 452 )

well as practically that the differences due to the varying apertures
are very small.
The results arrived at are as follows:

Corrections to the epochs given in the Nautical Almanac for

the eclipses of Jupiter’s satellites.

Oppo- | ee \Num-| Mean ES Num-}| Mean Mean
| Corr. N.A. Corr. N.A. 3(D+-R)
sition. Disapp. ber. error. Reapp. ber. | error. error.
E>
i
1894/95 | -+ 37s | a | +14 | — 18s | + 4s |-+ 98.5! + 7s
1895/96 | + 30 | 9 8 | 0 | 32 | +145 | 45
1897 — 19° | 2 | 18 | — 5 12 6 | — 12 9°
4s98 | 441 one = es a, 6 [ 4
II.
9418 | — 78 2 +32 0 7 +11s | — 39s | +17s
1894/95 | + 52 4 225 | — 42 15 Fla b 12
1895/96 | + 73 6 18 — 4 19 6° | + 34 10
1897 — 72 3 26 + 11 10 9 | — 30 14
1898 — 36 5 20 — 15 9 95 | — 26 41
Ill.
1894 +4151s 3 +228 | —24%s 3 +38 | — 45s | +25s
1895 +101 4 419 —127 4 33 | — 13 11°
1895/96 | + 87 9 13 — 50 9 22 | + 19 Ei
1897 +181 4 | 19 + 37 law 22 | +109 148
1898 +266 eae ae ee ae | 1 66 | +4138 34
1899 +361 3 2 | 49% ee: 33 | +118 20
iV.
1895 | a oa et —17m9s| 2 | +138s |
uF a) = aa
| | |
1895/96 + 3 49 | 10 Se fn 7 | +22 Poe 16 | +17s
iM | = 0 2) Say Oh ee ee 1 | 60 | 437 4M

( 453 )

Average mean error of a single observation.

Disappearane Reappearance | Mean _ | Delambre *) Introd. p. LIV
I | + Bs A 203 | Sie 2255 1785
II 45 29 37
7 ea F 88,5
= "3 be (795 rejecting the observations
IV 80 60 70 deviating more than 3 mi-
; nutes).

According to these numbers the complaints about the increased
inaccuracy of the tables of DamorsEau seem rather exaggerated, at least
fur the first and second satellites.

Taking into account the mean errors contained in the last column
we get the most probable correction at the epoch 1894—98

for I + §:,0 with a mean error of + 28,6
similarly ,, II — 3 8 ee eas bee aa ge
Both corrections can hardly be vouched for.

For III the case stands otherwise. It is true, the subtractive cor-
rection at the reappearances as well as the additive one at the dis-
appearances may be attributable to the use of more powerful tele-
scopes; still there seems to be a progression in the numbers of the
last column but one, which calls for a more exhaustive investigation.

In regard to IV, we found great corrections for the year 1895.
After some years in which this satellite had not been eclipsed, owing
to the fact that at the opposition it passed to the north of the shadow
cone of Jupiter, there began a new period of eclipses in this year.
In such a case the satellite travels high above the plane of the
orbit of Jupiter, and describes only a small chord in the shadow.
The consequence is that any small error in the latitude appears
strongly magnified in the duration of the eclipse. The observations
of Mr. Winker at Jena and of the observer at the observatory at
Uccle near Brussels, of 8 March 1895 are very suggestive in this
regard. The corrections were found to be:

Jena. Brussels. Mean
at disappearance -+ 19™48s + 21™58s + 20™53s
at reappearance —19 36 —18 33 —19 45

which shows that it is not the mean longitude of this satellite which
is mainly in error.

*) DELAMBRE gives mean differences; we have multiplied his numbers by 11/4
in order to get mean errors.

( 454 )

The explanation of these extravagant differences must rather be
sought, either in a correction needed by the longitude of the node
of the satellite’s orbit or in the adopted flattening of Jupiter. It is
also possible that for suchlike eclipses the diminution of light is
very slow.

For the rest, according to the Nautical Almanac, this eclipse
would be the fourth after the long period in which no eclipse of
this satellite occurred. The data, on pages 450, 452, 454 are as
follows:

1895
17 Jany. D. 15 36™16s M. T.Gr., R. 2" 8™17s, duration 32™ 15

Devic 4, 19°26 12.~—5, 53 3° See ee af: 1610 46
MO oe 14:5, dd 24), CO ityig eign, Ae 4 Bs 5 1 34 57
poMarch;, °°% D4) ako 2s ee oa ko = 1 54 14.

Only, according to Scott-Hansen, who, on the North-Polar expedition
of NAaNsEN, was in charge of the astronomical observations, the
satellite has not been eclipsed at all on the 17 of January °).

On the 2"¢ February 1895 too an eclipse of IV was not observed;
(I cannot now call to mind where I saw this negative observation).
On the 19% February, however, an observer at Greenwich, using the
Sheepbanks equatorial, aperture 120 mm., got a correction of +- 23™30s,
for the disappearance of IV. This agrees quite well with the preceding
results, obtained at Uccle and at Jena on 8 of March.

If we adopt the mean result of the observations at Brussels and
at Jena, the duration of the eclipse on that day was

144] 4s — 3957s,5 = 11141655,

The number might be of some use for the correction of the ele-
ments of IV.
The difference here found cannot be attributed to a too small value

of the adopted flattening, for DamorsEavu’s value

exceeds already

that found by direct measurement by most observers. Taking into
account however the results obtained by De SrrrEr, as communicated
at the meeting of the Section (Proceedings Vol. VIII p. 777), it
appears that the longitude of the ascending node of the 4* satellite
must be increased by about + 10°, whereas for the inclination on

1) The Norwegian North Polar Expedition 1894 — 1896. Scientific Results, edited
by Friptsor Nansen. Vi. Astronomical Observations, arranged and reduced under
the supervision of H. Gretmuypen, p. XXIV.

( 455 )

the fixed plane is found the value = 0°,2504 — 15' 2"4, which exceeds
DamorsgEav’s inclination only by somewhat less than a minute.

The remaining eclipses of IV in 1895 and the two following years
do not show any extraordinary divergencies.

Now, as in 1908 the eclipses of the satellites will be nearly central,
as may be gathered from the drawings in the Nautical Almanac
accompanying the table of these phenomena, there is no need to
fear that such great divergencies will occur for IV in that year.

Our result therefore is that the Nautical Almanac, which is based
on the tables of Damotszav (taking into account only a few necessary
corrections), may be considered sufficient for preparing ourselves for
the coming observations. The only exception would be for an early
eclipse of IV after a period in which it is not eclipsed at all.

Utrecht, 23 November 1906.

( 456

ES U

:
LTS.

Mutual heliocentric conjunctions of the satellites in April and May 1908.

A.A. = Ann Arbor ; Fl. = Flagstaff ; H.K. = Hong Kong; La Pl. = La Plata; P. = Perth ; Tac. = Tacubaja ;
1a, == Tokio ; ve; wlan Wi: = Waar

Visible at

Kas., Taschk., Madras, HK., Pertt

| Lick, Fl., Tac., AA., Harvard.

Grw., Pulk., Kas., Taschk., Rio.

| a — near |
sr &
> ap =>
Mean time | | Zo ——-)
No.| TS a Ss ar | BS a= | y—y’
at Greenwich | 2= | a= | | se ss
ae |e — eo
1/4 April4h 8m) 1; | Is 45°70 | —0r30 | —0r955} —or04
Soe e483. 1:17 «| Tie | +3,21 | —0,46 | —0,20 | +0,04
|
3 | 3» 445 |Iy | Uh —2,49 | +0,08 | +0,10 | —0,02
4/3 > 95 My | | 44,30 | —0,105] —0,09 | —o,015|
513 » 440 | IVy | Mh —6,19° | 10,40 | 40,32 | +008
6 | 3 » 169 | IV; | Is —4, 03 | +0,30 | +0,49 | +041
7/4 » 1652 |IVy | Tl, | +46,03' | 0.20] —0.97| +007
8/4 > 17% |Iy |e | 45,75 | 0,31 | 0,05] —o06 |
9/5 » 1956 |Iy |i. | 9,94 | 40,54 | 40,30 | 0,94
| |
10/6 » 2012 | Iy|Ie | +43,61 | —0,485] 0,94 | +0,095)
44/6 » 2258 |My | 1. | +1,37 | —0,10 | —0,09 | —0,01
12/8 » 631 |Iee |Te | -+45,82 | -0,31| 0,95 | —0,06
13/9 » 252 | Ty Bue +3, 85° | —0,18 | —0,24 | 10,06
14/10 » 728 | Wy | Te | —2,055 | 40,06 | 40,095) —0,03°
4510 » 12 4 |My | Ip | 441, 4 | —0,09 | —0,08 | —0,01
| b
16 44 » | | | Il 1-7, 87 |
4542 | | Iv | —0,36 | —0,48 | +012
114 »\smallest | IV+7, 96 | |
distance |
47 » 19 43 Tee. | IIn | +5, 88 | —O0, 314 —0), 26 | —0,055 |
| }
18 is > 20% |Iee. |1Ve | 45,99 | —0,32 | —0,38 | 10,06
19 a >» 93.33 | Illy | Ine. —9, 4 | 40,54 | 10, 45 | +0,09
20 te » 357 | Il; | IV, | —7,35 | 40.42 | 10,97 | 0,15
a1 ie » 2322 | ly |In | +49,64 | 0,16 | —0,49 | +0,03
| '
22 ie > 444 |My | in | 441,41 | 0,09 —0, 06 | —0,03
}
Paes g) aiog aay | Ti +5,93 | —0,32 | 0,96 | —0,06
%46 » 9344 |Iy | I. +4, 45° | 0,93 | 0.96 | +0,03
25 17 » 10 41 | Ily | I, | 4,61 | +0,03 | +0,08 | —0,05
96 17 » 1417 | iy eis | +0,98 | —0,09 | —0,06 | —0,03
b] ?
97 ie 22 1-03 AA | +5,97 | —0,32 | —0,26 | —0,06

2. Kas., Taschk., Madr., HK., Perth, Te

Grw.,Pulk.,Kas.,Taschk.,La Pl.,Ric
Grw., Pulk., Kasan, La Pl., Rio.
Lick, Fl., Tac., AA., Harv., La Pl
Lick, Fl., Tac., AA., Harvard.
Lick, Fl., Tac., AA., Harvard.
Wi., We., Lick, Fl., Tac., AA.
Wi., We., Lick, Fl., Tac., AA.
Perth, Tokio, Wi., We. |
Bresl., Pulk., Kas., Taschk., Madras
Wi., We., Lick, Fl.

Grw., Pulk., Kas., Taschk., Ma
Grw., Pulk., Kas., Harv., La PL.,Ri

Lick, Fl., Tac., AA.,Harv.,La PL. Ric

Wi., We., Lick, Fl., Tac., AA:
Wi., We., Lick, FI,

Perth, HK., Tokio, Wi., We.
Kasan, Taschk., Madr., HK.
HK., Perth, Tokio, Wi., We.
HK., Perth, Tokio, Wi.

HK., Pe., To., Wi. We.
Grw., Pulk., Kasan, La PL, Ri
Grw., Fl.,Tac., AA.,Harv.,La PI.,Ri
Perth, To., Wi., We.

Mean time

at Greenwich

Eclipsed
satellite

49 April 5hi5m| IV;

49 » 2257
2 » 3144
A » 14
2 » 1 46
med 2-52
2A » 323
42 19

22 a 45 413
17 56

22 » 9 44
22.» 41 27
A, 2M
ao > «OM
aed. 7 5
28 » 4 28%
a 2 5 36
(|28 » 13 29
i298 » 16 18

-“' diminishes

29
29

Be 0. 45
meas 57

4 May 17 5

1
3

on en a ct ce

» 18 43
» 3 16
¥ ft7

» 748
Be G47
»| 142

»\ 47 57
»{ 24 39

Tee.

Ill ;
Il;
Ill ¢
Ill ;
greatest

Ill

Kclipsing
satellite

IV»

++et++

2,33
0,845
6,62

8,01 and 7.84

& 88

+ 14,84

—
+
+

+
+
+
II

6,04
5,02
6,03
9,62
4,57
0,58
644

+7,31

III +7,80

absolute value, reaches its minimum 0,49 at the time assigned and then
increases again. So there is no eclipse.

>
+

+
-—

0,27
6,05
0,74
0,44
6,06
9,61
0,31
0,10

4,63

4-6,64-1-6,85

8,43

Eclipsed
satellite y

—0,088
— 0,39
— 0.32
et (ite
— 0,66
ie 0,32

— 0,33
+ 0,54
— 0,08
— 0,03
— 0,36
= 098

|
|

Eclipsing
satellite y! .

-L O85
+ 0,43
+ 045
— 0,13

— 0,12
— 012

— 0r05 |

+ 0,05
+ 0,10
+ 0,01
+ 0,07
+ 0,07

— 0,044] — 0,044

- 0,44
— 0,40
— 0,44
2083
— 0,26

— 0,255] — 0,28

— 0,25

+ 0,45
— 0,03

+ 0,05
+. 0,08
+. 0,06
+. 0,17
— 0,06
+L 0,025
— 0,08
+ 0,09
+ 0,04

— 0,065] ++ 0,035

— 0,36
— 0,49

— 0,02
— 033 2 0,26
50.0. |= 062
— 0,07 |— 0,02
— 0,325} — 0,24
ib 0,54 |-+ 0,45
|_ 0,07 | — 002
ie 0,09 | — 0,01
/— 0,26 | — 0,26
= 038 |— 0,36
— 0,42 |-— 0,49

0,00
+ 0,06

— 0,07
+ 0,61
— 0,05
— 0,085
+ 0,09
|— 0,05
— 0,08

0,00

+ 0,02

| -+ 0,07

Visible at

Kasan, Taschk., Madras.

HK., Perth, Tokio, Wi., We.
Taschk., Madras, HK., Perth, To.
Madras, HK., P., Tokio.

Madras, HK., P., Tokio.

Taschk., Madr., HK., P., Tokio.
Grw., Pulk., Harv., La Pl., Rio.
Lick, Fl., Tac., A.A., Harv., La PI:
We.; Lick. -Fl., Tac., A.A. Har,
Grw., Pulk, Kasan, La PI., Rio.
Grw., Pulk., (Kasan), La Pl., Rio.
Taschk., Madr., HK., P., To.
HE. 2:; Te. Wr

Bresl., Pulk., Kasan, Taschk.
Kasan, Taschk., Madr., HK.
Kasan, Taschk., Madras.

Grw., Tac., AA., Harv., La Pl., Rio.

|} — 0,08 |-+ 0,08 | Taschk., Madr., HK.

Tac., AA., Harv., La Pl., Rio.
| Lick, Fl., Tac., AA., Harv., La PI.
Lick, Fl., Tac., AA.

| Taschk., Madr., HK., P., Tokio.
Grw., Pulkowa.

| Grw., Pulk., Kasan, Tascbk.

| Grw., Pulk., Kasan, (Taschk ).

Tac., A.A., Harv., La Pl., Rio.

We., Lick, Fl., Tac., AA.

To., Wi., We., Lick.

( 458 ) -

j =far | A 2

Mean time Lie ? ae | $ 2 a

No. |ee|s2| c=" | 23 | 22
at Greenwich | 25 gS = = | 32
pests 2 Ce i

53 16 » 16h38m| Je. | IL, | +4 603 | — 0915] — 094
aw | ee i 7 LAV |, Ca — 3:7 + 0,20 la 0,18
55 | 7 Mei 6 37 5) IV, | In <= 248 ee 0,19 | -++ 0,20
56 |8 >» 7 4 |IVy | Ih | + 690 |— 0,38 |— 0,99
57 |8 » 205% | Uy | In + 018 | — 0,06 — 0,00
Baws 99 93-95.-| 1, | His. |) — 908 Jee eons
pos 529 | toe, |. .| = 5,36. |= Ose ost
mote 6 0 |i, | Ie + 599. |— 032 |— 0,38
61 10 » 1641 | te. | = @en dee Obs |S oan
poem 9-7 0\13 | ee ats — 5,80 |-+ 0,32 |-++ 032
63 41 » 1596 | Wy | Wwe | — 9,46 | + 0,95 |+ 0,43
pe o>. 954 | Wty |, + 044 |— 603 | + 0,01
2s 10.0 cigaaee 1 08) 1038 049
66 12 » 104 | My | iy | + 0,58 |— 0,10 l— 043
67 |18 » 338 |My | Uy | + 932 |— 0,55 |— 045
6g 13 » 19 31 lh, Il, | 42-58) | = 10 33 |= 095
69 |14 429. | In ih) + 0,0. |—.0,05 |+ 0,02
70 |14 657 | Wie ave at A Agia 10-8 | 0,75
Bn 43 26 | Tees | The — 598 |+ 0,32 | + 0,33
72, \15 Gos Lp: ) Te + 559 |— 032 |— 0,29
73 5 2259 | Wy | IV¢ | — 048 |— 0,06 |— 0,02
7h 45 Nes lly ig — 0,09 |— 0,06 |-+ 0,02
75 |16 1159 | Ty IVn — 5,69 |+ 0,30 ie 0,24.
16 17 9 4 In In +. 5,75 | —-0,31 }— 023
77 47 11°22 | 1s IV, — 14,82 |+ 0,80 |+ 0,70
7g ty 46°9 ft |The '| 4 4,92! | 20,088} = 002
79 (48 » 342 | Twe)} Un — 6,00 |+ 0,32 |+ 0,32
80 18 » 2036 | Illy | ly 8,935 | + 0,51 | 0,41
g1 [200 » 8 8 | Iy|Uy | + 957 |— 056 |— O45

y—y'

+ 01025
+ 0,02
|— 0,04
|— 0,09 |
|— 0,06 |
es 0,01
— 0,02
+ 0,06
— 0,05
0,00
018
0,04

0,50

+ 003
— 0,10

0,08
0,07

0412 |

001

0,03 |
0,04

0,08

0,54
0,08
0,10

0,065

0,00
0,10 |

O44 |

Visible at

Lick., Fl., Tac., AA., Harvard.
Bres]. Pulk., Kasan, Taschk. Ma‘
Bresl. Pulk., Kasan, Taschk ,Mad
Bresl., Pulk., Kasan, Taschk., Mad
Wi., Wellington.

Perth, Tokio, Windsor.

Kasan, Taschk., Madras.

Lick., Fl., Tac., AA., Harvard.
HK., P., Tokio.

Lick., Fl., Tac,, AA., Harvard.
Grw., Pulk., Kasan, La Pl., Rie,
Grw., Pulk., Kasan, La Pl., Riv,
Grw., Pulk., La Pl., Rio.
Taschk,, Madr., HK.

We., Lick., Fl.

Taschk., Madras.

Grw., Pulk. Rio.

| Tac., AA., Harv., La Pl, Rio.

Grw., Pulk., Kasan, La Pl, Rio.

| Perth, To., Wi.

Perth, To, Wi.

Grw., La PI, Rio,

Grw., Pulk., Kasan, La PI., Rie.
Grw., Pulk., La Pl, Rio:

Lick, Fl., Tac., AA., Harvard.
Taschk., Madras, HK,

Wi., Wellington.

Grw., Pulk., Kasan,

P= oie.

( 459 )

Physics. — “Contribution to the knowledge of the w-surface of
vAN DER Waats. XI. A gas that sinks in a liquid.” By Prof.
H. KamertincH Onnes. Communication N°. 96 from the
Physical Laboratory of Leiden.

If we have an ideal gas and an incompressible liquid without
vapour tension, in which the gas does not dissolve, the gas will gather
above the liquid under the action of gravity, if the pressure is suffi-
ciently low, whereas the compressed gas will sink in the liquid if
the pressure is made high enough.

I have observed a phenomenon approaching to this fictitious case
in an experiment which roughly came to this, that helium gas was
compressed more and more above liquid hydrogen till it sank in the
liquid hydrogen. Roughly, for so simple a case as was premised is
not to be realized. Every experiment in which a gas is compressed
above a liquid, is practically an application of the theory of binary
mixtures of vAN DER Waats. In such an experiment the compressi-
bility of the liquid phase and the solubility of gas and liquid inter
se may not be neglected, as generally the pressure will even have
to be increased considerably before the density of the gas-phase
becomes comparable with that of the liquid phase.

If the theory of vAN per Waats is applied to suchlike experiments,
the question lies at hand whether in the neighbourhood of the plait-
point phenomena where gas and liquid approach each other so closely
that of the ordinary gas and liquid state they have retained nothing
but the name, perhaps on account of a higher proportion of the
substance with greater molecular weight’) the phase, which must be
called the gas phase, may become specifically heavier than the phase,
which must be called the liquid phase. On closer investigation 1!
appears however, to be due to relations between the physical proper-
ties and the chemical constitution (so also the molecular weight) of
substances, that a liquid phase floating on a gas phase has not been
observed even in this favourable region.

I was the more struck with an irregularity which I came across
when experimenting with helium and hydrogen in a closed metal
vessel, as I thought that I could explain it by the above mentioned no!
yet observed phenomenon, and so the conviction took hold of me,
that at — 253° and at a pressure of 60 atmospheres the gaslike phase
which chiefly consists of helium, sinks in the liquid phase which
chiefly consists of hydrogen.

1) The limiting case is that in the y-surface construed with the unity of weight
the projection of the nodal chord on the xv-plane runs parallel to the line v ==)

30
Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 460 )

In order to ascertain myself of this I compressed hy means of the
mercury compressor described in Communication N°. 54 a mixture
of about one part of helium and 6 parts of hydrogen in a glass tube,
which had a capillary inflow tube at the top, and a capillary
outlet tube at the bottom, and which was merged in liquid hydrogen.

Up to 49 atmospheres the liquid hydrogen was seen to deposit
from the gas mixture, bounded by a distinct hollow meniscus against
the helium. At 49 atmospheres the helium, or properly speaking
the gas phase consisting chiefly of helium, went down just as water
through oil, and remained on the bottom as a large drop. With
further compression to 60 atmospheres and decrease of pressure to
32 atmospheres the volume of the bubble appeared to follow the
change of the pressure as that of a gas. At 32 atmospheres the
bubble rose again. By changing the pressure the bubble was made
to rise and descend at pleasure.

The closer investigation of these phenomena in connection with
the isotherms of helium and the y-surfaces of H, and He is an
extensive work, so that in anticipation of the results which most
likely will be definitely drawn up only much later, I feel justified
in confining myself to this sketchy communication.

One remark may be added now. It appears that the 6 of helium
must be small, from which follows again that @ must have an
exceedingly small value, because the critical temperature, if it exists,
must lie very low. In this direction points also a single determination
of the plaitpoint of a mixture of helium and hydrogen which I have
already made. Whether @ has really a positive value, whether it is
zero, or whether (what is also conceivable) @ is negative, will have
to be decided by the determination of the isotherms of helium.

(December 21, 1906).

J. A. C. OUDEMANS.
in 1908.”

N. B.
the penumbra.
Scale 1

- 314 250 000.

lly in a. 42

lily in

[In in ¢.

ly in 7.

In in a. If in b,

IIIn in e.

Iln in ¢. In in a. lf in 6.

Totaal. Totaal.

—— ee
Vein A.

IVy in A.

IlIn in e. IIln in ec. In in a.

ly in 7.

@ :

Plate

“Mutual occultations and eclipses of the satellites of Jupiter
Second part: eclipses.

The continuous circles show the contour of the satellites, the dotted circles represe

]2m 3 | heloce
lf in 6.
ee A
lly in ad lyin 7. IVF it
@ ;
Illy in a.
——_ —

IVrin A,

lly in 7
lif in @. Vf in A. Vf in /
@
\x E \
IVn in &
If in 6. IIy in a. lily in
a yas
« Deter]
cy <

Proveedings Royal Acad. Amsterdam. Vol.

\ e N a ‘
‘ . and
‘ i cy ° t'e
1! a ‘
i
j 7 :
' i
' af :
* ‘ |
ry j i’ a
d j : \
‘ = by —— 7 .
* ‘ ‘ ‘
. , ,

IX.

*
%
.
iL S
=
5
‘
¢

af Om eh!

~4
4
J
~

Litre

KONINKLIJKE AKADEMIE
VAN | WETENSCHAPPEN
:- TE AMSTERDAM -:-

PROCEEDINGS OF THE
SECTION OF SCIENCES

VOLUME Ix
( — 28> PART — )

JOHANNES MULLER :—: AMSTERDAM
JULY 1907

(Translated from: Verslagen van de Gewone Vergaderingen der Wis- en Natuurkundige 5
Afdeeling van 29 December 1906 tot 26 April 1907. Dl. XV.) "]

: apres

ae nae tay!

~ * .. 4: ie
es

» »
» »
a >
» »

OoN T. PAN T S.

eedings of the Meeting of December 29 1906

January 26 1907 . .

> > February 23 »

» » March 30 3 -
> » April 26 >

Page
461

513

599

683

799

wee ahd een
Pi gal ot
Pea) ee ee ae ee
. ‘ 1
Ade | ald a
< 7 eT i oly
Ne

ome

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday December 29, 1906.

Co _____—_—_—_—_——

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 29 December 1906, DI]. XY).

a aes IN TL SS.

Max Weser: “On the fresh-water fish-fauna of New-Guinea”, p. 462.

N. H. Cowen: “On Lupeol’”. (Communicated by Prof. P. van Rompurcn), p. 466.

N. H. Cowen: “On « and f-amyrin from bresk”. (Communicated by Prof. P. van Rompcurcny
p. 471.

F. M. Jaecer: “On substances which possess more than one stable liquid state. and on the
phenomena observed in anisotropous liquids”. (Communicated by Prof. A. P. N. Francuimonr),
p- 472.

F. M. Jarcer: “On irreversible phase-transitions in substances which may exhibit more than
one liquid condition”. (Communicated by Prof. A. P. N. Francuimonrt), p. 483.

O. Postma: “Some additional remarks on the quantity H and Maxwerv’s distribution of
velocities”. (Communicated by Prof. H. A. Lorentz), p. 492.

H. Kameriincu Onnes and W. H. Kersom: “Contributions to the knowledge of the y-surface
of vAN DER Waats. XII. On the gas phase sinking in the liquid phase for binary mixtures”,
p. 501. (With one plate).

W. H. Kersom: “Contribution to the knowledge of the ¢-surface of van DER Waats. XIII.
On the conditions for the sinking and again rising of the gas phase in the liquid phase for
binary mixtures”. (Communicated by Prof. H. Kameritincu OnneEs), p. 508.

Erratum, p. 511.

31
Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 462 )

Zoology. — “On the fresh-water fish-fauna of New Guinea”. By
Prof. Max Wersrr.

(Communicated in the meeting of November 24, 1906).

In the year 1877 there appeared a ‘‘Quatrieme mémoire sur la
faune ichthyologique de la Nouvelle-Guinée”’, written by P. J.
BLEEKER and containing 841 species. These species are exclusively
marine and brackish-water fishes and shew clearly, as might be
expected, that the littoral fish-fauna of New Guinea belongs to the
great Indo-Pacific fauna which extends from the East coast of Africa
to the islands of the Western Pacific.

The same result is arrived at from the lists published by W.
Macieay in 1876 and 1882, which treat of the fishes of the South
coast of New Guinea and Torres Straits. But none of these lists
accomplished what BLEEKER desired, namely, to give some insight
into the nature of the fresh-water fish-fauna of New Guinea. The
information which Breeker desired was partly supphed by certain
communications, published by W. Macreay, E. P. Ramsay, J. Dov-
eLas Ocinpy, A. Perucia and G. BovLEneer, about fishes caught in
the Strickland, Goldie and Paumomu rivers, and in a number of
rivulets all situated in the south-eastern part of the island. The number
of fishes mentioned amount to about 30, but so long as the fish-fauna
of German and Dutch New Guinea remained unknown, it was
impossible to give a complete idea of the ichthyological fauna of this
big island.

This was the more to be regretted inasmuch as fresh-water fishes
are of very great assistance in solving zoo-geographical problems.
In using them for this purpose we should however keep well in
mind the following points.

If in regions, at present separated by the sea, identical or closely
allied fresh-water forms are found, to which the sea affords an insur-
mountable barrier, one may freely draw the conclusion that these
regions were formerly either directly or indirectly connected. Among
the fresh-water fishes there are however whole categories which
cannot be used as factors in such an argument or only with great
caution. These are the migratory fishes and those that can live also
in brackish water and indeed even in sea-water.

The so-called law of E. von Martens states that from the Poles
to the Equator the number of brackish water animals increases.
This is also true for fishes and especially for those of the Indo-
Australian Archipelago, and in a very remarkable degree for those
of the islands east of Borneo and Java. The great Sunda Islands

( 463 )

in consequence of their former connection with the continent of
Asia possess a fish-fauna of which the most important elements, both
as regards quality and quantity, had no chance of further distribution
in. an eastern direction. The rivers of the eastern islands of the
Archipelago were therefore almost devoid of fishes, and offered a
good place of abode for such forms as, though denizens of the sea
or of brackish water, possessed sufficient capacity for accommodating
themselves to a life in fresh-water. The competition of those Asiatic
forms (Cyprinidae, Mastacembelidae, Ophiocephalidae, Labyrinthici
ete.), originally better fitted for a fresh-water life, failing, everything
was in favour of the immigrants from the sea. The river-fishes of
Celebes favour this view, as also does all that we know about the
fishes of Ternate, Ambon; Halmahera, etc.

We observe the same phenomenon in the fresh waters of Australia.
These however contain also indigenous forms, partly very old, partly
younger forms; the latter were obviously, at least in part, marine
immigrants, which have accommodated themselves so entirely to a
fresh-water life as to adopt the characters of fresh-water fishes.

The fauna of Australia enjoy at present a general and vivid
interest — are there not even people who believe that the cradle
of mankind stood there? A remarkable point of interest in the
study of its fauna is the question how long Australia has been
isolated from other parts of the globe. New Guinea plays a pro-
minent role in answering this question.

It is therefore a welcome fact that the Dutch New Guinea Expe-
dition of 1903 under the direction of Prof. A. Wichmayn has brought
home, besides other treasures, a large collection of fishes from diffe-
rent lakes and many rivers and rivulets, giving us a good insight
into the fresh-water fauna of the northern part of the island. It was
of great help to me, while studying this collection, that I was able
to make use of the fishes collected in the brackish water at the
mouth of the Merauke river, by Dr. Kocu the medical man of the
Royal Geographical Society's Expedition to South New Guinea. The
results of this investigation will be published elsewhere, but some
more general conclusions may be mentioned here.

When we reckon up all the fishes known up to the present date
from the lakes, rivers, and rivulets of New Guinea, we find that
their number amounts to more than 100 species, but only about 40
of these were found exclusively in fresh-water.

A careful examination shews further that the latter species, with
a few exceptions, are either known from brackish or sea water at
other places, or that their nearest relatives may be found in brac-

31*

( 464 )

kish or sea water. New Guinea shews clearly the fact that immi-
gration from the sea or from brackish water has played and perhaps
still plays a predominant part in the populating of its rivers.

Let us now return to the point at issue: namely, that the marine
fish-fauna of New Guinea forms part of the great Indo-Pacific fish-
fauna and particularly of that of the Indo-Australian Archipelago.
Keeping this in mind one might be inclined to draw the conclusion
that there is not much to be learned from the fauna of the rivers
of New Guinea concerning the history of this island. Such a con-
clusion however would be erroneous, for it is clear that the very fishes
which are characteristic of the fresh-water of New Guinea belong:

1. to genera which outside New Guinea are known only from
Australia (Pseudomugil, Rhombatractus, Melanotaenia, Eumeda) ;

2. or to genera nearly related to exclusively Australian genera.
Lambertia for instance is nearly related to Eumeda; Glossolepis to
Rhombatractus and the three new species of Apogon are closely
allied to Australian ones. Finally the species of Hemipimelodus
from New (Guinea form a special group, distinct from those of the
neighbouring Indian Archipelago. Everything that gives to the
fresh-water fish-fauna of New Guinea a character different from that
of the Indian Archipelago is at the same time characteristic of
Australia. Twelve of its species belonging to the genera Pseudo-
mugil, Rhombatractus, Melanotaenia, Glossolepis, belong to the family
or subfamily of the Melanotaenidae, only known from Australia.
I do not hesitate therefore to maintain that the river-fishes of New
Guinea belong to two groups:

1. A fluvio-marine group, which is Indo-Australian or, if one prefers,
Indo-Pacific and which may also be met with, for instance, in Ambon
or Celebes. To this category belongs also Rhiacichthys (Platyptera) novae-
guineae Blgr. discovered by Pratt in mountain rivers of the Owen
Stanley Range four thousand feet high. Boulenger speaks of the disco-
very of a fish of the genus Rhiacichthys ‘“‘so admirably adapted to life
in mountain torrents’ as highly interesting. He tells us that the closely
allied Rhiacichthys asper C. V. is known from Bantam, Celebes and
Luzon. This is likely to create the impression that Rhiacichthys novae-
guineae does not belong to this category, but is a species whose nearest
relative is confined to rivers in regions occupied by the Asiatic fauna.
Rhiacichthys asper however, differing but little from Rhiacichthys novae-
guineae, was also found by BLEEker in Sumatra and, what is far
more interesting, it occurs, according to Giinther, also in Wanderer
Bay on the island of Guadaleanar in the Solomon Islands — in
“fresh-water. At all events it is thus found close to the sea. This

( 465

is also true for a specimen which I described from Ambon and still
more so for a specimen that I caught near Balangnipa in the lower
part of the Tangka, close to its mouth in the gulf of Boni. The
water was here already brackish and ran slowly. Rhiacichtys has
therefore a very wide distribution, it does not fear brackish water,
and its presence in New Guinea loses therewith much of its importance.

2. The second group, the characteristic element, is Australian.
This last group requires further explanation as to its origin. In the
present state of things, now that New Guinea is separated from
Australia by Torres Straits, these offer a barrier impassable to those
fishes which I called characteristic. Some species of Rhombatractus
and Melanotaenia may it is true, descend to the mouth of the river
and be able to endure even slightly brackish water, but none of the
24 recorded species is known from the sea. The barrier can therefore
not be bridged by the group of islands in the Torres Straits. They
are too poorly supplied with fresh-water and far too strictly coral
islands, even when we leave out of consideration the fact that they
are separated from each other, from New Guinea and from Australia
by broad tracts of sea with a high salt percentage and strong tidal
currents. The simultaneous presence of these characteristic forms in
New Guinea and in Australia cannot be explained otherwise than by
the existence of a more solid and extensive connection in former
ages. This connection must have been so far back in the past that,
to take an instance, the representatives of the abovenamed Melano-
taeniideae had time to separate themselves specifically. And this
actually happened; for among the 12 species of Melanotaeniidae
already known from New Guinea and among the 12 species described
from tropical or sub-tropical Australia not one is common to the two
regions, although the differences between some species are very
small. On the other hand therefore it cannot have been so very
long ago from a geological point of view that this connection between
Australia and New Guinea existed. How long a time may have
elapsed since that period is at present a matter of hypothesis. But
if zoo-geographical and more particularly ichthyological experience
may venture an opinion, | should seek the period of this connection
not earlier than in the pliocene, and the breaking up of it in the
pleistocene. Other zoological observations may perhaps be in favour
of this supposition.

It will be a long time yet before the last word is spoken on this
question. We may express the hope that the new expedition to Dutch
Southern New Guinea under the guidance of Dr. H. A. Lorentz, which
intends to investigate especially its big rivers, will bring us further light,

( 466 )-

Chemistry. — “On Lupeol’'). By Dr. N. H. Conny. (Communi-
cated by Prof. P. vy. Rompuren).
(Communicated in the meeting of November 24, 1906).

Notwithstanding the many and_ beautiful researches of several
chemists, the structure of cholesterol, which is important also from
a physiological pomt of view, is far from being known. Therefore,
Prof. van Rompcrcn invited me to investigate a substance closely
connected with the same, namely lupeol, a phytosterol. For the
phytosterols may be included with the cholesterols in one common
group “the cholesterollic substances”. The original intention was to
study the alstol found by Sack *) in ‘“bresk’*). From the “bresk”
investigated by me, alstol, alstonol and isoalstonol could not be iso-
lated, although Sack claims to have found them in the same, but
I obtained a@- and B-amyrin and lupeol. It appeared afterwards that
Sack’s alstol is not a chemical individual.

Lupeol was first found by Likrernik*) in the skins of lupin seeds;
afterwards Sack *) met with it in the bark of Roucheria Griffithiana,
whilst van Rompuren and van DER LinpEN*) demonstrated its presence
as a cinnamate in the resin of Palaquinm calophyllum. Finally,
VAN RompurGu proved that Tscuircu’s’) crystal-albane simply consisted
of lupeol cinnamate. The lupeol was prepared from ‘“bresk” by
extracting the same first with boiling aleohol. On cooling, a white
mass was deposited which, without any further purification was
saponified with alcoholic potassium hydroxide. The saponified product
was then benzoylated with benzoyl chloride and pyridine and the
reaction product treated repeatedly with acetone by heating just to
boiling on the waterbath and then filtering off without delay.

Finally, a lupeol benzoate was left, which after repeated recrystal-
lisation from acetone, consisted of fine, flat needles; m.p. 265°—266°,
(corr. 273°—274°).

Found C 83.71—83.81 Calculated for C,,H,,0, 84.07
H 10.41—10.36 10.03

These, like all subsequent combustions, were made with lead
chromate.

[«]p = + 60°,75 in chloroform.

1) For a more elaborate description see Dissertation N. H. Coney, 1906, Utrecht.

) Sack. Diss. 1901, Géttingen.

8) Bresk or djetulung is the dried milky juice of some varieties of Dyera.

*) Ztschr. f. physiol. Chem, 15. 415 (1891).

6) Sack l.c.

6) Ber. 37. 3440 (1904).
7) Arch. der Pharm. 241. 653 (1903),

( 467 )

By saponification of lupeol benzoate with alcoholic potassium
hydroxide and recrystallisation from alcohol or acetone, the lupeol
was obtained in the form of fine, long needles m.p. 211°, (corr. 215°).

Found: C 84.62 84.65 84.40 84.50 Calculated for C,, H., O 84.85
H 11.78 11.93 11.82 12.02 11.49
faJp = + 27°,2 in chloroform.

In the first place it seemed to me of importance to ascertain
whether double bonds occur in lupeol. Therefore, a solution of
lupeol in carbon disulphide was treated with a solution of bromine
in the same solvent. Hydrogen bromide was evolved. By recrystal-
lising the reaction product from methyl alcohol, needles containing
1 mol. of the latter are formed. The melting point of this substance,
dried at 100°, was 184°, (corr. 185°).

Found: | i >be Sey V VI VII eale. for C,, H,, OBr.

C 72.14 72.30 71.90
H 10.26 10.07 CARIUS LIEBIG 9.55
Br 44.48 14.50 15.40 15.07 14.67 15.45

fe]p = + 3°,8 in chloroform.

Most probably, « monosubstitution product had formed and I now
tried to obtain an additive product of the benzoate. When dissolved
in a mixture of glacial acetic acid and carbon disulphide and then
treated with a solution of bromine in glacial acetic acid, it yielded,
after spontaneous evaporation of the carbon disulphide, beautiful
leaflets. On extracting this product with boiling acetone a less easily
soluble substance was left, which proved to bea monobromide. After
repeated recrystallisation from aethyl acetate, I obtained fine, thick
crystals which when melting were decomposed. Placed in the bath
at 240° it melted at 2439.

Found I I TEE cu kV aves VIE VIII IX xX
C 72.62 72.90 72.58 72.46 72.59

H 8.85 8.88 8.72 9.09 8.84 Carivs LIEBIG
ee Se
Br 13.14 13.04 12.97 13.40 13.01

Calculated C_,H,;O, Br, C=73.38, H=8.61, Br = 12.87.
[a]p = + 44°,9 in chloroform.

The bromine atom is contained in the lupeol nucleus, because on
saponification an alcohol containing bromine, and benzoicacid are formed.

The more readily soluble portion crystallises from acetone in
beautiful leaflets. It is also a monobromide but could not with
certainty be characterised as a chemical individual.

One of the means to trace the structure of a substance is the
gradual destruction by oxidation.

( 468)

The lupeol was, therefore, oxidised with the Kimianr mixture’),
Lupeol dissolved in benzene was shaken with a weighed quantity
of the oxidising liquid, 6 atoms of oxygen caleulated for 1 mol. of
lupeol. Titrations of the oxidising liquid with potassium iodide and
sodium thiosulphate showed, that after six hours one atom of oxygen
had been consumed and as the amount of chromic acid did not
diminish any further, this one atom had been taken up quantitatively.
The oxidation product, which crystallised from alcohol in beautiful,
thick needles, melted at 169° (corr. 170°) and proved to be a ketone,
to which I gave the name of dupeon.

Found C 84.95 84.91 85.07 84.76 Cale. for C,, H,,0 85.24
MW 41.64 11.81 11.62 11.68 1%a3 11.09

[@|p = + 63°,1 in chloroform.

Dr. Jagcer was kind enough to examine the crystalform of the
lupeon. It belongs to the rhombo-bipyramidal class. A complete
deseription will appear elsewhere.

With hydroxylamine an oxime of the lupeon was obtained, which
is but little soluble in alcohol.

Reerystallised from ethyl acetate, it forms white, soft, light needles,
which are decomposed when melting. Placed in the bath at 278°,
they melt at 278°,5.

Found C 81.98 Cale. for C,,H,, NOH 82.41
H 11.44 with lead chromate 10.94
N 3.08 3.11

[@|\p = + 20°,5 in chloroform.

sromine dissolved in glacial acetic acid added to a solution of
lupeon in the same solvent gave hydrogen bromide and a dibromide,
which was deposited from the acid. Recrystallised from a mixture
of benzene and glacial acetic acid it consisted of beautiful, hard
needles, which were decomposed when melting. Placed in the bath at
253° the melting point was 254’.
Found J II Il IV Y. VI Vil Vib tea

H 813 8.26 8:05. S06 Carivs LIEBIG
ee TE ES
sr 926.88 26.91 27.08 26.85 27.35 27.23

Cale. for C,, H,, O Br,, C = 62.58, H = 7.80, Br = 26.90.

[a|p = + 21°,4 in chloroform.

When dissolved in ether, lupeon gave with hydrogen cyanide
under the influence of a trace of ammonia a cyanobydrin, which

!) Ber, 34, 3564 (1901).

( 469 )

after some time deposited in the form of beautiful, thick needles.
This substance is decomposed at a higher temperature and also on
melting. Placed in the bath at 192°, it melts at 194°. By collecting
the hydrogen cyanide liberated on heating in aqueous potassium
hydroxide and then titrating with silver nitrate I determined the
nitrogen content.

Found: I II bit PV V ME VIL “eale; for C,H, ON
C 82.63 82.76 82.86
Bm 14.20°11.26 copper oxide lead chromate _ titrated 10.66
N So aoe 3.00. OL 20 3.03

One mol. of cyanohydrin gave, with one mol. of ethyl alcohol
and one mol. of hydrogen chloride, a substance, which, when placed
in the bath at 230’, melted at 235°; as shown by a combustion,
this was not, however, the expected ethyl ester of the corresponding
acid. This substance has not been investigated further.

Lupeol benzoate treated in the same manner as lupeol with the
KiILIANI mixture was not affected. Lupeon dissolved in benzene and
stirred with the mixture for four hours at 40° also remained unaltered.

By the action of chromic anhydride on lupeon at a higher tem-
perature, acid products were formed, which could not be obtained
in a crystalline state.

The neutral oxidation product of lupeol with potassium perman-
ganate and sulphuric acid consisted of a mixture, which could be
separated only with extreme difficulty. Excepting lupeon no well-
defined substance could be isolated from it. As Senkowsk1') had
obtained phthalic acid from cholic acid by oxidation with alkaline
permanganate, | treated 23 grams of lupeol in the same manner,
but it suffered complete destruction. This fact does, therefore, not
favour the idea of a benzene nucleus in lupeol.

By the oxidation of an acetic acid solution of lupeol acetate with
chromic acid, I obtained a product which, on analysis, gave figures
which agree satisfactorily with the calculated values for C,, H,, Q,.

Placed in the bath at 285° it melted at 295° to a dark brown mass.

In alcoholic solution this substance did not turn blue litmus red,
not even on diluting with water, but still it could be titrated very
readily with alcoholic potassium hydroxide, phenolphtalein being used as
an indicator. Assuming that one mol. consumes one mol. of KOH the
titrations pointed to a molecular weight of 521 and 524, the formula
C,, H,, O, representing 512,5.

Found: © 77.59 77.23 76.87 77.24 calculat. for C,, H,, O, 77.28
H 10:75-10.49° 10.09 10.79 10.23
~ 1) Monatsh. f. Chem. 17. 1 (1896).

( 470 )

On saponification with alcoholic potassium hydroxide a substance
was obtained which erystallised from ether in needles. Placed in the
bath at 260°, the melting point was 263-—265°. In regard to litmus
this substance behaves like the unsaponified product, but it may be
again titrated with alcoholic potassium hydroxide and phenolphthalein.
From these titrations the molecular weight was found to be 452
and 461; the formula C,, H,, O, represents 470,5.

Found: C 78.42 78.61 calculated for C,, H,, O, 79.08
H-41.07 71208 LOVE

The potassium compound of this substance is soluble, with diffi-
eulty, in alcohol, and erystallises from this in needles.

On treating either the saponified or the unsaponified oxidation
product the same compound was obtained, which seems to be a
diacetylated substance. The results of the combustions, however, were
not very concordant, but I have not been able to account for this.
Found: C 75.39 74.71 75.67 74.96 74.47 caleul. for C,, H,, O, 75.75

H 10.12 10.16 10.51 10.24 9.81

By boiling with excess of alcoholic potassium hydroxide and
titrating with alcoholic sulphuric acid the molecular weight was
found to be 549, assuming that the molecule contains two acetyl
groups. The formula C,, H,, O,; represents 554.5.

It is desirable to investigate more closely these oxidation products,
which are so important in the study of lupeol, before trying to
explain their formation.

Lupeol is not reduced by metallic sodium and boiling amy] alcohol;
whereas lupeon is reduced by sodium and ethyl alcohol to lupeol.
Therefore, if lupeon should possess a double bond, this is sure not
to be in @$-position in regard to the carbonyl group.

Neither lupeol, nor lupeol acetate dissolved in boiling avetone are
acted upon by potassium permanganate. This behaviour does not
agree with the theory of a double bond, but the presence of the
latter in lupeol and lupeon could be satisfactorily demonstrated by
means of Hitst’s iodine reagent. On the other hand the oxidation
product C,, H,, O, no longer seemed to contain the double bond.
On the strength of various combustions and bromine determinations,
particularly of dibromolupeon, I consider C,, H,, O to be the most
likely formula for lupeol. The formula C,, H,, O given by Likmrntk *)
and Sack *) is certainly not correct.

Utrecht, Org. Chem. Lab. University.

1) Likrernik |. c.
2) Sack l. c.

at AY. }

Chemistry. — “On a- and 3-amyrin from bresk?*). By Dr. N.H.
Conen. (Communicated by Prof. Van Romevren).

(Communicated in the meeting of November 24, 1906),

Communications as to pP-amyrin, which is present as acetate in
“bresk’’ or “djelutung’” have already been presented (These Proc.
1905, p. 544). Since then, I have prepared also B-amyrin cinnamate.
This erystallises from acetone in small needles, which melt at 236.°5
(corr. 241°).

In addition to @-amyrin and lupeol another substance was obtained
from “bresk”’, which proved to be identical with the e-amyrin found
by VESTERBERG.

This substance crystallises from alcohol in long, slender needles ;
m.p. 185° (corr. 186°). VesrerBerG gives the melting point as 181—
181°,5.

Found: C 84.22 84.30 calculated for C,,H,,0 84.43
H 14.91 12.02 11.82

These, like all subsequent combustions have been made with lead
chromate.
[¢]p=+82°,6 in chloroform; in benzene was found [«] p—-+88°,2.*).

For the purpose of characterisation, different esters were prepared
from @-amyrin.

a-Amyrin acetate was obtained by heating with acetic anhydride
and sodium acetate. Recrystallised from alcohol it forms needle-
shaped leaflets; m.p. 220—221°, (corr. 224—225°). VeEsTERBERG gives
the melting point as 221°.

Found: C 81.85 82.27 81.79, caleulated for C,,H,,O, 81.98
H 11.384 11.40 11.33 11.19

|@|p = + 75°,8 in chloroform.

a-Amyrin benzoate was obtained with the aid of benzoyl chloride
and pyridine. From acetone it crystallised in long, prismatic needles ;
m.p. 192°, (corr. 195°). According to VESTERBERG it melts at 192°.

a-Amyrin cinnamate, which has not yet been described was obtained
like the benzoate. When recrystallised repeatedly from acetone it
forms small hard needles which melt at 176,5—177°, (corr. 178°).

Utrecht. Org. Chem. Lab. Univ.

1) For a more elaborate description see, Diss. N. H. Conen. 1906, Utrecht,
*) VesTERBERG found in benzene [z]n = + 91°,6.

( 472).

Chemistry. — “On substances, which possess more than one stable
liquid state, and on the phenomena observed in anisotropous
liquids.” By Dr. F. M. Jagerr. (Communicated by Prof.
I’ RANCHIMONT).

§ 1. The compounds now investigated belong to the series of fatty
cholesterol-esters, which were the subject of a recent communication *).
They are intended to supplement the number of the synthetic esters,
studied previously and include: Cholesterol-Heptylate, Nonylate,
Laurate, Myristate, Palmitate and Stearate. The Palmitic ester, as
is well known, is also important from a physiological point of view,
as it occurs constantly in blood-serum accompanied by the Oleate
m.p. (43° C.) *).

I have prepared these compounds by melting together equal parts
by weight of pure cholesterol and fatty acid, and purifying by frac-
tional crystallisation from mixtures of ether and alcohol, or ethyl
acetate and ether. The details will be published later on in a more
elaborate paper in the “Recueil”. The substances were regarded as
pure, when their characteristic temperature-limits and the typical
transformations occurring therein, remained the same in every parti-
cular, even after another recrystallisation, whilst also the solid phase,
when examined microscopically, did not appear to contain any
heterogenous components.

Most of these esters were obtained in the form of very flexible,
tabular crystals of great lustre and resembling fish-scales; some of
them, such as the heptylate and the /aurate, erystallise in long, hard
needles.

The investigation showed, that most of these esters of the higher
fatty acids possess three stable liqued phases. Whereas, in the first
terms of the series one at least of these anisotropous phases was
labile in regard to the isotropous fusion, all three are now stable
under the existing circumstances, although sometimes definite, irre-
versible transitions may still occur. It is a remarkable fact, that the
stearate again exhibits an analogy with the lower terms, as it appears
that only labile liquid-anisotropous phases may occur, or else none
at. all. A relation and similarity between the initial and final terms

1) F. M. Jaeger, These Proc. 1906; Rec. d. Trav. d. Chim. d. Pays-Bas, T.
XXIV, p. 334—351.

2) K. Hirruie, Z. f. physiol. Chem. 21. 331. (1895); The blood serums of:
man, horse, ox, sheep, hog and dog were investigated.

( 473 )

of the homologous series is plainly visible here. In what follows
there will be described, tirstly, the thermic, and then the microscopic
behaviour of these substances.

§ 2. The Thermometric Behaviour of these Substances.

Cholesterol- Laurate exhibits the following phenomena: The isotropous
fusion £ of this substance has still, at 100° the consistency of gly-
cerol, and gradually thickens on cooling. At 87°.8 C. (=4#,) there
suddenly occurs a peculiar violet and green opalescence of the phase,
which commencing at the surface, soon embraces the whole phase.
The still transparent thin- jelly-like mass quite resembles a coagulating
colloida! solution; the opalescence is analogous to that often noticed
in the separation of two liquid layers.

As the cooling proceeds, the opalescence colours disappear and the
mass gradually becomes less transparent and also more liquid. It is
then even thinner than the isotropous fusion Z. This doubly-refracting
liquid A now solidifies at 82°.2 C. (=7,) to a crystalline mass S,
accompanied by a distinct heat effect.

If, however we start with the solid phase S and subject the same
to fusion, the behaviour is apparently quite different. The substance
softens and yields after some time a thick doubly-refracting mass, which
will prove to be identical with the phase A. On heating further the
viscosity decreases, and at about 86° it becomes very slight. There
is, however, no sign of opalescence this time. The turbid mass may
be heated to over 90°, without becoming clear and now and then A
seems as if solid particles are floating in the liquid phase. At 90°.6 C.
(= t,) everything passes into the isotropous fused mass /. The micros-
copical investigation shows, that between A and Z another stable,
less powerfully refracting liquid phase 6 is now traversed, and that,
owing to retardation occurring, the phase S may be kept for afew
moments adjacent to L, when A and B& have already disappeared.

This is therefore, a case where a substance may be heated a few
degrees above its actual melting point without melting.

It should, however, be observed that the order of the temperatures
is here quite irreconcilable with the phenomena considered possible
up to the present, with homogenous substances; the temperature of
90°.6, at which these crystals disappear in contact with ZL finds no
place in the p-t-diagram of Fig. 1. Such a position of the said
temperatures might be possible, when the system could be regarded
as containing two components, for instance, if there was question
of tautomeric forms which are transformed into each other with
finite velocities. I think it highly probable that in all these substances,

( 474)

“phenomena of retardation” play a great role; moreover the enormous
undereooling which the phase A can undergo without transformation,
proves this satisfactorily in the majority of these esters.

The different behaviour of the laurate on melting and on cooling
the fused mass is so characteristic, that no doubt can be entertained as
to the irreversibility of each series of transformations. Fuller details
will be given below in the micro-physical investigation.

§ 3. Cholesterol-Nonylate forms at 90° an isotropous fused mass of
the consistency of paraffin oil; on cooling to 89°5 a stable, greyish,
doubly-refracting liquid appears which, gradually thickening, passes
into a second strongly doubly-refracting liquid phase A, — which trans-
formation is accompanied with a brilliant display of colours. All three
liquids are, however, quite stable within each specific temperature-
traject. On melting, as well as on cooling the substance, they succeed
each other in the proper order.

The viscous, strongly doubly-refracting, liquid phase A now
becomes more viscous on cooling, and is finally transformed into a
horny, transparent mass which exhibits no trace of crystallisation.
Even after some hours, the often still very tenacious mass has not
got crystallised. In the case of this substance it is therefore impos-
sible to give the solidifying point or the exact temperature at which
the heated mass begins to melt. The reason of this is, that the doubly
refracting liquid A can be undercooled enormously and passes gra-
dually into the solid condition without crystallising.

As the micro-physical research has shown, a spherolite-formation
occurs afterwards suddenly in the mass, which ultimately leads to the
complete crystallisation of the substance.

The velocity, with which such spherolites are formed appeared in
some cases not to exceed 0.0000385—0.000070 m.m. per second !

§ 4. Cholesterol-Myristate, at 80°, is still an isotropous, paraffin
oil-like liquid. On cooling, it gradually becomes viscous; at about
82.°6 the glycerol-like phase then turns, with violet-blue opalescence,
into a thick, strongly doubly-refracting mass A which, gradually
assuming a thicker consistency, is finally converted into a horny
mass, without any indication of a definite solidifying point. In this
respect the substance is quite analogous to the previous one. On the
other hand, on being melted, it behaves more like the laurate, in so
far as it is converted into a double-refracting liquid 4, before passing
completely into “. The transition temperature cannot be determined
sharply, but I estimate it at about 80°.

( 475 )

§ 5. Cholesterol-Palmitate at 80° is a clear, isotropous liquid
as thick. as simple syrup. On cooling, the isotropous phase is con-
verted at 80° with green opalescence into a fairly clear, transparent,
doubly-refracting jelly A, which rapidly assumes a thinner consistency,
and becomes at the same time more turbid, and finally solidifies at
77.°2, with a perceptibte caloric effect, to a crystalline mass S. In
this case also, a doubly-refracting phase B appears to be traversed
when the mass is being melted, before the occurrence of the isotropous
fusion Z; I estimate the transition temperature at about 78°.

§ 6. With Cholesterol-Stearate, 1 did not succeed in demonstrating
the occurrence of a doubly refracting liquid. The isotropous, thick-
fluid fusion solidifies at 81° to well-formed crystals S.

§ 7. Cholesterol-Heptylate exhibits, in undercooled fusion only, one
doubly-refracting liquid phase which is labile in regard to the solid
phase SS. The compound behaves, thermically, analogously to the
caprylate. The temperature of solidification is at 110.°5, the tran-
sition-temperature of the labile doubly-refracting phase lies a little
lower.

Of Cholesterol-Arachate, 1 could only obtain an impure product
on which no further communications will be made. The ester could
not be purified properly as it is not soluble to any extent in the
ordinary solvents. The crude substance obtained does not seem to
exhibit any anisotropous liquid phases.

§ 8. Micro-physical behaviour of these substances. If a
little of the pure solid cholesterol-laurate is melted on an object
glass to an isotropous, clear liquid Z, and the same is allowed to
cool very slowly, there is formed, usually, a very strongly doubly-
refracting, liquid phase, gleaming with lucid interference colours. It
consists of large, globular drops, which exhibit the black axial cross
and, on alternate heating and cooling, readily amalgamate to a syrupy,
_highty coloured, but mainly yellowish-white liquid. This phase will
be called A in future. On cooling, it gradually thickens, until no
more movement of the mass is noticed, which continues to exhibit
a granular structure. Around this mass an isotropous border liquid
is found. At first I felt inclined to look upon this tenaceous, isotropous
mass, which is visibly different from the fusion L, as a distinct
phase differing from the fusion A. But on using a covering glass
and pressing the same with a pair of pincers, or by stirring with
a very thin platinum wire, I found that this border liquid is only

( 476 )

“‘pseudo-isotropous’ (LEHMANN) and is, in reality, not different from
A; only, the optical axes of the liquid crystals are all directed
perpendicularly to the glass-surface. The other cholesterol-esters also
exhibit this phenomenon. On further cooling, this phase A crystal-
lises like the pseudo-isotropous border to a similar mostly spherolitie
erystalline mass _S.

Between the spherolites one often sees currents of the pseudo-
isotropous border liquid.

If now the entire mass is allowed to solidify to S, and then again
is melted carefully, it is at once transformed into the liquid A, recog-
nisable by its high interference colours and its slow currents. Then,
there appears suddenly a new, greyish liquid 4, consisting of smaller
individuals with a less powerful double refraction, which after a
short time is replaced suddenly by the isotropous fusion ZL. If Z is
now cooled again, it is A which appears at once and not the phase B.

Only a very feeble, greyish flash of light, lasting only for a
moment, points to a rapid passing of the phase 6; it cannot, however,
be completely realised now. On further cooling, S is formed sud-
denly, sometimes in plate-like crystals. When once crystallisation has
set in, S will not melt when the mass is heated, as might have been
expected, but actually increase in size of the crystals occurs, and
the velocity of crystallisation is now many times increased. It must
be remarked that the growing flat needles of |S drive before them,
at their borders, the liquid phase A amid violent currents. If the
heating is now continued a little longer, we may notice sometimes,
that whilst the little plates of S remain partly in existence, A passes
first into the grey phase 4, which then is converted into the isotropous
mass L. We then have adjacent to Z the solid phase S, which
therefore, may be heated above its melting point, before disappearing
finally into the isotropous fusion ZL.

All this shows, that the dawrate possesses three stable liquid phases
and also that the isotropous fusion being coled, B is always passed
over, but is realised when the solid phase is heated. All this is repre-
sented in the annexed p-é-diagram; the arrows, therefore, indicate
the order of the phases traversed on melting and on cooling. The
phase A in its quasi-immovable period may be kept a long time
solid at the temperature of the room, and may be considerably under-
cooled before it passes into S. Notwithstanding its apparently solid
appearance in that undercooled condition, A is still a tenaceous,
thick liquid, as I could prove hy stirring the mass with a thin
platinum wire.

The point (f,) agrees with the opalescence which occurs when

( 477 )

Fig. 1. Schematic p-t-diagram for

Cholesterol-Laurate.

the isotropous fusion is cooled; this indicates, therefore, the moment
where the stable phase B is replaced by the as yet still less stable
phase A, which will soon afterwards be the more stable one;
a fact which may be perhaps important in the future for the
explanation of the analogous phenomena observed in the separa-
tion of two liquid layers and the coagulation of a colloidal solution.

Indeed, the transition at (f,) presents quite the aspect of a gela-
tinising colloidal solution. The temperature of this transition point
may be determined, but not sharply, at 87°.8. The temperature at
which, when the solid substance melts, the liquid may be still
kept turbid, owing probably to the presence of the meta-stable plate-
like crystals of S, was determined at 90°.6; the solidifying tem-
perature (¢,) lies at 82°.2.

32
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 478 )

That the border liquid, obtained by cooling the isotropous fusion Z,
differed from JZ itself, could be demonstrated in more way than one.
By heating and cooling we may get so far that, apparently, nothing
more of A is visible, but that we have only the border liquid, which
on cooling, crystallises immediately to S. Occasionally, the doubly-
refracting individuals of A turn up in the mass for a moment to
disappear again immediately. However, that isotropous liquid thus
obtained is nothing else but A itself, when owing to the temperature
variations, all individuals have, like magnets, placed themselves
parallel with their (optical) axes and the whole has, consequently,
become pseudo-isotropous. This same phenomenon also occurs with
the other esters, for instance very beautifully with the nonylate and
the myristate. The difference between these pseudo-isotropous phases
and the isotropous ftsed masses “4 of these substances, is shown by
the fact that the pseudo-isomorphous mass of A, and also the doubly-
refracting portion of the same has a very thick-fluid consistency ; the
isotropous fusion £ of the laurate has a consistency more like
that of glycerol.

As regards the solid phase and its transformation into the liquid condi-
tion, it cannot be proved in this case that there exists a continuous
transformation between the last solid partic lesand the first anisotropous
ones. From the velocity, with which~the diverse phases usually
make room for each other in the microscopic examination, one would
feel inclined to believe just the opposite. The thermic observation
of the transformation, which generally exhibit only insignificant
caloric effects, would, however, make the observer feel more inclined
to look upon the matter as an uninterrupted concatenation of more
or less stable intermediate conditions, which I have observed pre-
viously with cholesterol-cinmamylate. A somewhat considerable heat
effect occurs in some cases in the crystallisation of the solid phase
only; in all other phases the exact transition temperature cannot be
determined accurately by the thermic method.

§ 9. Cholesterol-Nonylate exhibits microscopically the following
phenomena:

Starting from the crystallised substance, this was fused first on an
object glass to an isotropous liquid Z. On cooling a greyish doubly-
refracting liquid phase appears, which, at a lower temperature,
makes room for a very tenaceous, strongly doubly-refracting, mostly
vellowish-white phase, 4. This phase A is often surrounded by an
isotropous border; if pressure is applied to the covering glass or if
fhe mass is stirred with a very thin platinum wire, this isotropous

( 479 )

liquid appears to be identical with A, and to be pseudo-isotropous
by homoeotropism only. The optical axes of the doubly-refracting
modification A again place themselves perpendicularly on the surface
of the covering glass. On continued cooling A becomes increasingly
thicker; at last a movement in the mass can be seen only on stirring.
After a longer time there are formed from numerous centres in this
tenacious mass thin, radiated spherolites, whose velocity of growth
is but very small. When a number of these spherolites have formed
and the mass is then heated carefully, the spherolites do not melt,
but actually decrease owing to the greater crystallisation-velocity.
Soon afterwards —- however, they melt, on further heating, to the
doubly-refracting phase A, where the circumferences of the spherolites
and the black axial crosses are preserved for some time; so that the
whole much resembles a liquid mozaic. Subsequently the phase 6b
reappears and afterwards the isotropous fusion L. The whole series
of phases is traversed in a reversible manner; the liquid phase A,
however may be so much undercooled, that a proper melting or
solidifying point of the substance cannot be given. In larger quantities
of the substance, the crystallisation does not set in till after some
hours, and the substance turns first to a horny mass, which always
remains doubly-refracting to finally exhibit local, white spots, from
which the spherolite-formation slowly spreads through the entire
mass. One would feel inclined to call this transformation of liquid-
anisotropous into crystallised substance a continuous one, if it were

only possible to observe, even for a moment, the intermediate con-

ditions in that transition. As the matter cannot be settled by direct
experiment, the transition must be put down, provisionally as a
discontinuous one.

In this case also, and the same applies to the other cholesterol-
esters as well, the spherolite-structure of the solid phase is of great
importance for this entire transformation of undercooled, anisotropous-
liquid condition into the solid one. At the end of this communication
I will allude briefly to a few cases from which the particular signi-
ficance of the spherolite-structure in the transitions between aniso-
tropous-liquid and anisotropous solid phases is shown also plainly in
a different manner.

§ 10. Cholesterol-Palmitate behaves in quite an analogous manner :
I observed one solid phase and three liquid conditions A, B and L:
as in the case of the laurate, B is generally observed only on
warming. The succession of the liquid and solid phases takes place,
however comparatively rapidly, so that a real solidifying point may

32*

( 480 )

be observed, which has also been proved by the thermic research.
The solid phase crystallises in broad flat needles, when fused and
then solidified in conglomerated spherolites. On melting, the thick,
doubly-refracting liquid A is mostly orientated in regard to the
previous solid spherolites.

§ 11. Cholesterol-Stearate could not be obtained in a doubly-
refracting liquid form: the isotropous fusion always crystallises
immediately amid rapid, rotating movements, to small needles, which
often consist of a conglomeration of rosettes. It is possible that labile
anisotropous phases are formed, owing to strong undercooling or by
addition of some admixture *).

§ 12. Cholesterol- Myristiate \ends itself splendidly to the experiment.
It behaves mainly in the same manner as the /aurate; the phase B
can only be observed on heating, but not on cooling the isotropous
fusion 4. Most brilliant is the formation of large, globular crystal-
drops of the modification A, also the colour-zone which precedes the
formation of A from LZ, on cooling. This phase A also exhibits the
phenomenon of pseudo-isotropism in a particularly distinct form. On
the other hand, an important difference between this compound and
the laurate is the much smaller velocity with which, on cooling,
the spherolites S are formed from A; in this respect the compound
exhibits more similarity with the nonylate. Sometimes it may be
observed readily how in the phase A, which consists of an enormous
number of linked, globular crystal-drops, which all exhibit the black
cross of the spherolite crystals, centrifugal current-lines are developed
from a number of points in the mass, along which the crystal-drops
range themselves. After the Japse of some time those doubly-refracting
globules are seen to disappear, while the current-lines have now
become rays of the spherolite. Here again, the question arises whether
the transformation of the doubly-refracting liquid globules, which are
orientated along the current-lines, into the true spherolite form, does
not take place continuously, and whether we do not speak of a
sudden transformation merely because we are not able to observe
the stadia traversed in this transformation.

The liquid globules of the phase <A themselves exhibit much
similarity with a kind of liquid spherolites; a few times, | have
even been able to observe such “liquid spherolites” of greater dimens-

') Prof. Leamany informed me recently that the stearate possesses mdeed two
labile, anisotropous liquid phases.

( 481 )

ions, which rapidly solidified to solid spherolite crystals. In the case
of this compound also, one feels convinced that there must exist a
very intimate relation between the spherolite-formation of a substance
and its power of forming anisotropous liquid phases: on the nature
of this relation, | hope to make a communication later on.

It may, however, be observed, provisionably that in all trans-
formations: liquid SS solid, where serious “phenomena of retardation”
may occur, the undercooling, or superfusion, for instance is generally
abrogated amid a differentiation of the phase into spherolites. All
the cholesterol-derivatives, mentioned in this paper, exhibit this sphe-
rolite-formation. In the case of «-phytosterol-propionate, | have been
able to show, that a complex of a large number of doubly-refracting
microscopic spherolites may imitate the optical peculiarities of the
liquid phases in process of separation and of the colloidal opalescence.
This might lead to the strengthening of the previous conception of
the colloidal solidification as a separation-phenomenon of labile liquids.

§ 13. Cholesterol-Heptylate contains only /adile liquid anisotropous
phases. It exhibits great similarity with the caprylate described previ-
ously: I have only a few times been able to obtain one single thick-
fluid phase A from the undercooled isotropous fusion LZ. The solid
phase erystallises rapidly and in beautiful flat needles, which exhibit
high interference colours. On warming, the substance readily migrates
towards the colder parts of the object glass.

§ 14. In conelusion, I will communicate a few more points as
regards some phenomena, which prove plainly the signiticance of
the spherolite structure for with these questions.

Some time ago, I published a research on the fatty esters from
Phytosterol from Calabar-fat and stated how they all are wont to
erystallise in the spherolite-form from their cooled, isotropous fused
mass, while anisotropous liquid phases are not observed therein, with
the exception of the normal valerate which possesses a thick-tluid
anisotropous modification, and exhibits the phenomenon of the chang-
eable melting point, which again becomes normal on long keeping:
a fact also observed in the case of a few fatty glycerol-esters. Since
then, Wiypats has proved that the phytosterol, extracted from Calabar
fat is a mixture of two isomorphous phytosterols, which cannot be
separated by crystallisation. Being engaged in preparing the pure
fatty esters from the principal of those two phytosterols, namely
the a-compound (m.p. 136°), I discovered that the fused propionate
of a-phytosterol (m.p. 108°), when cooled rapidly in cold water,

( 482 )

exhibited the most brilliant interference-colours, which is also the case
with the cholesterol esters (acetate for instance), which possess labile
anisotropous liquid phases. The thought naturally at once occurred,
to attribute these phenomena to the appearance of liquid crystals in
the now pure a-phytosterol-ester. A similar behaviour was also
shown by perfectly pure a-phytosterol-acetate, but with a much less
display of colours. It was, however, a remarkable fact, that a-phyto-
sterol-propionate even after complete solidification sti/l retained those
colours for an indefinite length of time, particularly at those sides of
the testtube, where the layer of the substance was thinner and had
cooled rapidly.

The microscopic investigation now showed that these two sub-
stances exhibit extremely rapidly disappearing anisotropous liquid phases
or, more probably, none at all‘); but that the said colour-phenomenon
is caused by a very peculiar spherolite-structure.

In what follows, I have given the description of the solidifying
phenomena of the «-propionate, and also a figure representing the
typical structure of the fused and then cooled compound, such as is
present at the coloured sides of the tube.

If a little of the solid substance is fused
on a slide to an isotropous liquid the fol-
lowing will be noticed on cooling. The
mass solidifies completely to spherolites,
namely to a conglomeration of circular,
concentrically grouped figures, which appear
connected with a series of girdles. When
three spherolites meet, they are joined by
means of straight lines which inclose angles
of about 120°.

The mass is slightly doubly-refracting
and of a greyish colour; the rings and girdles are light greyish on
a darker back-ground. Each spherolite exhibits besides a concentric
structure, the black cross, but generally very faint. The whole
resembles a drawing of polished malachite from the Oeral, or of

some polished agates.

') Whereas the phytosterol-esters from Calabar fat which, of course, contain
a definite amount of the B-homologue, exhibit no liquid crystals, the pure z-esters
commencing with the butyrate [or perhaps the propionate] did show this pheno-
menon. This discovery is a powerful argument against the remarks often made
in regard to the cholesterol-esters, that the remarkable phenomena described are
attributable to an admixture of homologous cholesterols. Foreign admixtures
prevent as a rule these phenomena altogether; in any case they are rather spoiled
than improved.

( 483 )

The walls of the test-tube or the object-glass, which exhibit the
said colour-phenomena, have that same structure, but with this diffe-
rence, that the globular, concentrically deposited spherolites have
much smaller dimensions and lie much closer together. Each little
spherolite has also a cross; this however, is not dark, but coloured
with yellow and violet arms. The spherolite is also coloured in the
alternate circle-quadrants.

This ensemble of small, coloured spherolites is the cause of the
said brilliant colour-phenomena; they are quite analogous to those
which are wont to appear in the case of liquid crystals and remain
in existence for an indefinite period. Each of them exhibits one or
generally two luminous points in the centre; they exhibit a strong
circular polarisation and are left-handed. The whole appears between
crossed nicols as a splendid variegated mozaic of coloured cellular
parts. The size of each individual is 0.5—1 micron.

The acetate also exhibits something similar, but the spherolites are
built more radial and the whole ts not at all so distinct.

I hope to contribute more particulars as to these remarkable
phytosterol-compounds shortly. I have mentioned them here merely
to show the importance of this structure-form for the optical pheno-
mena, observed in the anisotropous phases.

Zaandam, 14 November 1906.

Chemistry. — “On irreversible phase-transitions in substances which
may exhibit more than one liquid condition.’ By Dr. F. M.
JAEGER. (Communicated by Prof. FRaNcHionr).

(Communicated in the meeting of November 24, 1906).

§ 1. The fatty esters of a-Phytosterol from Calabar-fat, which the
Phytosterol mostly occurring in the vegetable kingdom, and which has
also been isolated from rye and wheat under the name of “sitosterol”,
exhibit very remarkable properties in more than one respect.

In my previous communication, I alluded briefly to the colour
phenomena and the spherolite-structure in the propionate and the
acetate. In the latter | could not observe anisotropous liquid phases;
n the former a doubly-refracting phase is discernible just before
melting, but it lasts too short a time to allow the accurate measure-
ment of the temperature-traject.

With the following four terms of the series, however, these pheno-
mena are more distinct, and occur under conditions so favourable as

( 484 )

could hardly be realised up to the present in the other known sub-
stances. They also exhibit enormous phenomena of retardation in their
diverse transitions and often a typical irreversibility thereof, of
which I will now communicate some particulars.

§ 2. Thermometrical behaviour of the fatty a-phytosterol-
esters.

A. a-Phytosterol-n.-Butyrate, on very slowly raising the temperature,
melts at 89°.5 to a turbid, doubly-refracting liquid A, which at first
is very viscous but rapidly becomes thinner and is converted, at~
90°.6, into a clear isotropous fusion ZL of the consistency of glycerol.

On cooling the same carefully, the thermometer falls gradually ~
while the isotropous liquid thickens more and more but remains
quite clear. At 80° the whole mass crystallises all at once to small
crystals S with so great a caloric effect that the thermometer goes
up to 85°. There is no question now of anisotropous liquid phases
at all. These two experiments may be repeated at will but always
with the same result. As to the nature of the turbid phase, compare
“micro-physical behaviour”.

[If the isotropous fusion is suddenly cooled in cold water, a bluish-
grey coloration appears and a soft, doubly-refracting mass is obtained,
whieh does not become crystalline until after a very long time.

Bb. a-Phytosterol-Isobutyrate, when treated in the same manner,
melts at 101°.4 to a glycerol-like, turbid, doubly-refracting liquid A,
which gradually assumes the consistency of paraffin-oil and is con-
verted at about 108°.2, apparently continuously, into a clear fusion L.

If this is cooled, it certainly becomes gradually thicker but it
still remains quite clear and isotropous.

At 80°.4 it becomes turbid and doubly-refracting ; this phase is
identical with A, and it has the consistency of glycerol; at 73° it
has become as thick as butter, and at 66° the thermometer can be
moved only with difficulty, whilst it may now be drawn into sticky,
doubly-refracting threads. At 65° the thermometer suddenly goes up
to 68°.8 and the mass crystallises in long, delicate needles ‘5S.

On rapid cooling of the fused mass, this is converted into a turbid,
greasy looking, doubly-refracting mass, which crystallises but very
slowly ; no colour-phenomena occur.

C. a-Phytosterol-n.- Valerate melts, when in the crystallised condi-
tion, at an wncertain temperature. At about 48°, the substance com-
mences to soften visibly, at 54° its consistency is that of thick butter,
at 80° it is somewhat thinner, at 85° it is actually liquid, but still
turbid and doubly-refracting. A// these transformations proceed quite

( 485 .)

continuously. At about 97.°5 the liqnid is clear and isotropous; it
has then the thickness of paraffin-oil.

If, however, the isotropous fused mass is cooled, the isotropous
varaftin-oil-like liquid remains clear to about 87.°3, when a turbid
doubly-refracting phase is formed. This, on further cooling, gradually
becomes more viscous; at 80° it is as thick as butter, at 66° it can
hardly be stirred, and may be drawn into threads. It may be cooled
to the temperature of the room without solidifying. It remains in
this condition for hours, but after 24 hours it has again become
crystalline. The substance, therefore, has no determinable melting
or solidifying point.

D. «-Phytosterol-Isovalerate behaves quite analogously to the n-
valerate. Neither a definite melting point, nor a solidifying point can
be observed. The mass softens at about 45°, is anisotropous thick-
fluid at 65°, and becomes clear and isotropous at 81°.

On cooling to 78.°1, a beginning of turbidity is noticed, the liquid
gradually becomes thicker and is converted at an uncertain temperature
into a tenacious sticky, doubly-refracting mass, which after 24 hours
has again solidified to a crystalline mass.

§ 3. The thermometrical behaviour of these remarkable substances
is represented in the annexed schematic p-/-diagram, for the case
of the n-butyrate and isobutyrate. The typical irreversibility of these
phenomena is thus seen at once. Moreover in the case of the two
valerates, the whole behaviour can be described only as a real,
gradual transformation, solid = liquid with an intermediate realisation
of an indefinite number of optically-anisotropous liquids.

)4. The micro-physical behaviour of the fatty «-phytosterol
esters. Perhaps, there are no substances known, which exhibit under
the microscope the characteristic phenomena of anisotropous liquids
in so beautiful and singular a manner as these esters; in this respect
the isobutyrate and the valerate excel in particular. In the normal
butyrate, the traject, where the liquid crystals are capable of exis-
tence iS rather too small. For this reason, although the behaviour of
the four substances differs in details, I will describe more particularly
the behaviour of the n-valerate and as to the others, I will state
occasionally in what respect they differ from the valerate. In conse-
quence of the totally different circumstances which the microscopic
method involves, nothing more is seen of the thermically observed
peculiar irreversibility and even progression of the transformations.
For the study of the nature of the diverse phase-transformations, the

( 486 )

Fig. 1. Schematic p-t-diagram for »-Phytosteryl-lsobutyrate.

thermometric method is certainly preferable to the microscopic one,
because in the latter, the delicate changes in temperature cannot be
controlled so surely as in the first method. For this reason, the phase-

( 487 )
transformations, when observed microscopically, convey the impression
of being more sudden than in the thermic observation.

Still, the microscope completes the task of the thermometer in a
manner not to be undervalued, at it gives an insight into the structure
of the diverse phases and allows one to demonstrate their difference
or their identity.

§ 5. If a little of the beautifully crystallised n-valerate is carefully
melted on an object glass, the substance, at a definite temperature
changes, apparently suddenly, into an aggregate of an enormous
number of globular, very large and strongly doubly-refracting liquid-
drops, which all exhibit the black cross of the spherolites *) but can
flow really all the same. This condition may be rendered permanent
for a long time at will. But they may also amalgamate afterwards
to larger, plate-like, highly coloured liquid individuals, somewhat
resembling sharply limited crystals. These are frequently multiplets
of liquid drops; the demarcations between the separate individuals
vary constantly by changes in temperature.

The isotropous border of the mass is very striking. By pressure
or by moving the covering glass, also by the sliding currents which
we can induce herein by changes in temperature, it may be readily
shown that this isotropous border, owing to a parallel orientation
of the liquid individuals, is only psendo-isotropous and really identical
with the rest of the phase. Sometimes one may succeed even in
communicating this pseudo-isotropous aspect to the entire mass *) by

') We can, however, often observe a slanting projection of the optical symmetry
axis, which gives the same impression as if we look perpendicularly to one of
the optical axes of a biaxial crystal, or on a monoxial crystal cut obliquely to
the optical axis. We observe at the same time coloured rings which exhibit an
elliptic form. It is very remarkable that, when the phase has become very viscous
on cooling, these ellipsoidal drops, provided with rings and slanting but mutually
parallel-directed axes may he kept for a long time in an apparently immobile conditicn
in the midst of the pseudo-isotropous or double-refracting liquid. They place them-
selves mutually like little ellipsoidal magnets.

However, | could observe, that ihese drops are often not quite ellipsoidal, but
that they are sharply broken a little at the one side, just there, where the optical
axis is slanting. By turning the object-table, the axial point turns in the same
direction as the table, while the black line or cross is preserved. (Added in the
English translation Januari 1907).

*) The anisotropous-liquid phase has, in the case of the two valerates, an extra-
ordinary tendency to place itself in this pseudo-isotropous condition. We can

- observe this, because the border of the drop often moves inward with widening

of the isotropous-looking line. It is also remarkable to see how the flowing crystals
when meeting an air bubble arrange themselyes close together, normally on the
border thereof.

( 488 )

often repeated warming followed by rapid cooling. This substance
is about the best known example of this phenomenon.

§ 6. If now we go on heating very cautiously, the larger flowing
crystals and also the smaller drops situated between them are seen
to move about rapidly; the larger individuals, which consist mostly
of twins or quadrupleis, are split up into a multitude of globular
drops and these, together with the smaller ones, disappear at a definite
temperature entirely in the isotropous liquid, which is now isotropous
in reality. Tbe globules of the liquid rotate to the right and the left
under distortion of the mass, as may be observed from the spiral-
shaped transformation of the black cross. Sometimes, before the mass
becomes isotropous we may notice a temporary aggrandisement of
the plate-like flowing crystals at the expense of the smaller interjacent
globules; a result of the momentarily increased crystallisation-velocity
due to heating.

§ 7. On cooling the isotropous fusion this is first differentiated
into an infinite number of the double-refracting liquid globulus, which
here and there amalgamate to the more plate-like flowing crystals.
On further cooling, these latter individuals remain in existence
notwithstanding the undercooling, while the little globules in the
meanwhile unite to the same kind of plate-like individuals. This aggre-
gate, brilliant in higher interference colours becomes in course of time
thicker and thicker in consistency while the aggregation, owing
to an apparent splitting, becomes more and more finely granu-
lated. But even after the lapse of some hours, the phase remains
anisotropous-liquid as may be easily proved by shifting the mass and
by the pseudo-isotropous border, which commences to exhibit delicate,
double refracting current-lines. In the end when the pseudo-isotropous
liquid has passed like the remainder into the same, almost completely
immobile aggregation of doubly-refracting individuals, it is, gradually,
transformed after a very long time into an aggregate of plates and
spherolite-like masses, which possess a strong double refraction.

§ 8. If, after the lapse of some hours, the partially or completely
solidified mass is melted cautiously, we sometimes succeed, in the
‘ase of the two valerates, in keeping the crystals of the phase S
(therefore the solid crystals) for a few minutes near the isotropous
fusion L at a temperature above the highest transition point. This
phenomenon is, therefore, again quite homologous to that first observed
by me with cholesterol-laurate and which might be described as a

( 489 )
heating of a solid substance S above its melting point without fusion
taking place. For the present, at least according to existing ideas, this
behaviour can only be explained by assuming the presence ofa two-
-component-system with tautomeric transformations subject to a strong
retardation.

When the isotropous fusion 1 which has scarcely cooled to afew
doubly-refracting drops is melted cautiously, we may observe some-
times that where a moment before the strongly luminous, yellowish-
white globules were visible, there are now present greyish globules
showing the black cross, which gradually decrease in size and also
darken, to disappear finally as (isotropous ¥) little globules in the
isotropous fusion’). This phenomenon, in connection with those of
erystallised ferric chloride to be described later, and with similar
phenomena observed with the cholesterol esters appears to me to
have great significance for the theory of the formation of liquid crystals.

§ 9. Finally, there is something to be observed as to the separation
of «-Phytosterol-valerate from organie solvents. The substance may
be obtained from ethyl acetate + a little alcohol in beautiful, hard,
well-formed little crystals. If, however, the saturated co/d solution
in ethyl acetate is mixed with much acetone (in which the substance
is but sparingly soluble) the liquid suddenly becomes a milky-white
emulsion which deposits the compound not as a fine powder, but in
the form of a doubly-refracting, very thick and very sticky liquid.

| have repeated this precipitation in a hollow object glass under
the microscope. The emulsion consists of a very great number of
doubly-refracting, globular liquid-globules, which are either moving
about rapidly in the liquid, or, when united to larger masses, are
quite identical with the ordinary anisotropous phase A, when this
is cooled to the temperature of the room. These little globules all
exhibit the cross of the spherolites, and the doubly-refracting liquids.
They soon become solid and then form small needles and spherolitic
aggregations. It may be easily proved by stirring that the globules
deposited first are liquid; moreover, the doubly-refracting masses
often communicate with each other by means of very narrow, doubly-
refracting currents, while they often exhibit the phenomena of pseudo-
isotropism.

Therefore, we have evidently obtained here the liquid-anisotropous

') Before that happens, we may sometimes see here the globules becoming
enlarged to multiplets by amalgamation there larger ones being changed into
smaller ones, sometimes here one disappearing in the liquid while very close by
new individuals appear.

( 490 )

phase A from a solution by rapid precipitation at the temperature of
the room, and that in isolated drops! A few other phytosterol esters
exhibit analogous phenomena which | will describe later on in a
more elaborate communication on these substances.

§ 10. A very remarkable fact in the w-valerate, the iso-valerate
and the zsobutyrate, is the differentiation of the isotropous fusion into
a large number of globular, doubly-refracting liquid drops of con-
siderable dimensions, which like the circles of fat on soup float
alongside and over each other and often unite to multiplets, whose
separate parts are still recognisable. Wreathed aggregations of the
liquid globules are also observed occasionally. In most cases the
separate liquid globules exhibit the black cross and the four lummous
quadrants grouped centrically. They are, however, also seen to roll
about frequently, so that the projection of the optical symmetry axis
now takes place excentrically. Owing to the enormous size of the
individuals and the low temperature-limits, these esters lend them-
selves to the study of these phenomena certainly VorLANDER’S p-
azoxy benzoic-ethy lester.

If the temperature of the mass, when totally differentiated into

liquid globules — and the isobutyrate is particularly adapted for this
differentiation — is slightly raised, the liquid globules are often seen

to disappear suddenly just after they have enlarged their limits as it
were by an expansion. It is like a soap-bubble bursting by over
blowing.

§ 11. Finally, I wish to observe that the thermical transitions
just described and particularly those of the two valerates, can only
be interpreted by assuming a quite continuous progressive change.
For all these gradual transformations. either on melting or on soli-
difying, a measurable time is required and nowhere is to be found
any indication of a sudden leap. An exception is, however, afforded
by the sudden crystallisation of the two butyrates.

§ 12. As regards the differentiation of the fusion Z into an
aggregate of anisotropous liquid globules, I will now make a com-
munication as to an experiment upon the erystallising of ferric
chloride heaahydrate, which substance exhibits something similar, and
which, like most undercooled fusions and like many compounds
which exhibit liquid crystals, crystallises in typical spherolites.

If we melt the compound Fe, Cl, + 12 H,O cautiously in a little
tube, taking care that no water escapes, and a drop of this brownish-

( 491 )

red fusion is put on an object glass, it may be lett for hours at the
temperature of the room without a trace of crystallisation being
noticed. The liquid is now greatly undercooled and exists in a state
of metastable equilibrium. For all that, it has the same chemical
composition as the solid phase from which it was formed.

On prolonged exposure, smal/ liquid globules appear locally in the
fairly viscous mass, probably owing to local cooling, or by a spon-
taneous evaporation of water at those
points. These liquid globules are quite
isotropous and are surrounded by a
delicate aureole having an index of
refraction different from that of the
rest of the liquid (fig. 38a). The ob-
servation shows that, optically, they
are, practically, no denser than the
liquid, and from the fact that they
afterwards become, im their entirety, a

Fig. 3a. spherolite of the hexahydrate, we must
conclude that their chemical eomposi-
tion does not differ from that of the
fused mass.

These globules of liquid are con-
verted gradually into doubly-refracting
masses whose section is that of a
regular hexangle with rounded off
angles: individual crystals are not yet
visible in the doubly-refracting mass
and the luminous zone around still
appears to exist (fig. 34). Fig. 30.

Here and there, hexangular, sharply
limited, very small plate-shaped crystals
are also seen to form in the liquid
without previous formation of liquid
globules*). In the end, the doubly-
refracting hexangular mass gets gra-
dually limited by more irregular sides,
while a greater differentiation of the
mass into light and dark portions
points to a crystallisation process com-
Fig. 3c. mencing and progressing slowly.

1) These may, however, be formed perhaps owing to the presence of traces of
sal ammoniac,

(497 }

Finally, we can observe a spherolite of the hexahydrate with a
radial structure which now grows centrifugally to the large well-
known semi-spheroidal spherolites of ferric chloride (fig. 8c).

§ 13. This experiment proves that the abrogation of the metastable
condition, or at all events of a liquid condition which is possible
under the influence of phenomena of retardation may happen owing to
the formation of spherolites which are preceded by the differentiation
of the fusion into an aggregate of liquid globules. True, the latter
are here isotropous in contrast with the phytosterol esters just
described, but the anisotropism of the latter liquids may be caused
also by factors which are of secondary importance for the apparently
existing connection between: metastability of liquid conditions, their
abrogation by spherolite formation and the possible appearance of
liquid globules as an intermediate phenomenon. I will just call atten-
tion to the fact that if we set aside a solution to crystallise with
addition of a substance which retards the crystallisation, this will
commence with the separation of originally isotropous liquid globules,
so-called globulites, which BrHRrENpDs and VoGELsANG commenced to
study long time ago.

All this leads to the presumption that the formation of the aniso-
tropous liquid phases as aggregates of doubly-refracting liquid globules
may have its origin in a kind of phenomena of retardation, the nature
of which is still unknown to us at the present. Before long, I hope
to revert again to this question.

Zaandam, 24 Nov. 1906.

Physics. — “Some additional remarks on the quantity Hand MAXWEL1’s
distribution of velocities.’ By Dr. O. Postwa. (Communicated
by Prof. H. A. Lorenz}.

§ 1. In these proceedings of Jan. 27% 1906 occur some remarks
by me under the title of: “Some remark§ on the quantity A in
BoLtTzMANN’s Vorlesungen iiber Gastheorie’’.

My intention is now to add something to these remarks, more
particularly in connection with Grpps’ book on Statistical Mechanics a
and a paper by Dr. C. H. Wind: “Zur Gastheorie” ae

In my above-mentioned paper I specially criticised the proofs given

1) J. Wittarp Gipes ‘Elementary Principles in Statistical Mechanies’’, New-York,
1902.

*) Wien. Sitzungsber. Bd. 106, p. 21, Jan. 1897.

( 493 )

by Borrzmann and Jeans that Maxwei’s distribution of velocities in
a gas should give the most probable state, and demonstrated that they
wrongly assume an equality of the probabilities a priori that the point
of velocity of an arbitrary molecuie would fall into an arbitrary
element of the space.

The question, however, may be raised whether it would not be
possible to interpret the analysis given by BorrzMann and JEANS in
a somewhat different way, so that avoiding the incorrect fundamental
assumption, the result could all the same be retained. And then this
proves really to be the case. When the most probable distribution
of velocities is sought from the ensemble of equally possible combi-
nations of velocities with equal total energy, we make only use
of the fact that the different combinations of velocities are equally
possible, how they have got to be so is after all of no consequence.
Or else, it had not been necessary to occupy ourselves with the
separate velocities of the molecules and make an assumption as to them.

This way of looking upon the matter is of exactly the same nature
as that constantly followed by Gisss in his above-mentioned work.
Gipgs treats in his book all the time instead of a definite system,
an ensemble of systems of the same nature and determined mostly
by the same number of general coordinates and momenta (p,... pn;
G1 +++Qn), Which he follows in their general course. Such an ensemble
will best illustrate the behaviour of a system (e.g. a gas-mass), of
which only a few data are known and of which the others can assume
all kinds of values. He calls such an ensemble micro-canonical when
all systems, belonging to it, have an energy lying between / and
E+ dE and for the rest the systems are uniformly distributed over
all possibilities of phase or uniformly distributed over the whole
extension-in-phase the energy of which lies between 7 and E+ dL.
When the energy of a gas-mass is given (naturally only up to a
certain degree of accuracy) we should have reason according to Grpps
to study the microcanonical ensemble determined by this energy, and
to consider the gas-mass as taken at random from such an ensemble.
The -extension-in-phase considered is thought to be determined by

for. ... dda; but in the case of a gas-mass with simple equal
molecules this is proportional to
sf dx, dy, dz,... din dyn dzn, dx, dy, dz, ... dity dyn Un;

so that we may say that every combination of velocities and con-
figuration is of equally frequent occurrence in the ensemble.
It is now easy to see that when the energy is purely kinetic- the
33
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 494 )

same cases occur in such an ensemble, with regard to the distribu-
tion of velocities, as are considered as equally possible cases by
Bo.tzMaNn and Jeans. The difference in the way of treatment of
Gipps on one side, and that of Botrzmann and Jeans on the other
consists besides in the fact that the one occupies himself with separate
velocities and the other not, in this that Gress treats the configuration
and the distribution of velocities at the same time (both belong to the
idea phase), whereas Jzans treats the latter separately, and Bo.tz-
MANN does not occupy bimself with the configuration in this connection.

Every phase of BoLtzMann (combination of velocities) corresponds
with as many phases of Gisss (combination of velocities and con-
figuration) as the molecules can be placed in different ways with
that special combination of velocities. This number being the same
for every combination of velocities according to the independence of
the distribution of velocities and configuration following from the
fundamental assumption, it will be of no consequence, comparing
the different combinations of velocities inter se, whether we also take
the configuration of the molecules into account or not. So when
seeking the most probable distribution of velocities (that, with which
the most combinations of velocities coincide), we must arrive at the
same result whether we follow Gisss or BoLTZMANN.

It is obvious that the phases of the microcanonical ensemble meant
here are what Grsss calls the specific phases. Gisss distinguishes
namely between specific and generic phases: in the former we con-
sider as different cases those where we find at the same place and
with the same velocity, other, even though quite equal molecules,
in the latter we do not. In other words: in the former we consider
also the individual molecules, in the second only the number of the
molecules. So we may now say that in such a microcanonical
ensemble the most probable distribution of velocities and that which
will also occur in the great majority of cases (compare JEANs’
analysis discussed in the first paper) will be that of Maxwe t.
When therefore an arbitrary mass of gas in stationary state may be
considered as taken at random out of such a microcanonic ensemble,
Maxwel1’s distribution of velocities or one closely resembling it will
most probably occur in it. In this way a derivation of the law
has been obtained to which the original objection no longer applies,
though, of course, the assumption of the mivrocanonical ensemble
remains somewhat arbitrary ').

‘) With the more general assumption of a canonical ensemble Maxwett’s law is
derived by Lorentz; “Abhandlungen tiber Theoretische Physik”, Lpzg. 1906 I,
p. 295,

( 495 )

Finally the question might be raised, when we want to consider
the separate velocities, whether it is possible to arrive at the
equally possible combinations under discussion on another suppo-
sition a priori about the chances of every value for the velocity than
the one indicated by BoLTzMAnn and Jeans. The supposition must of
eourse be such, that the chance is independent of the direction of
the velocity, so that the chance of a velocity c, at which the point
of velocity ‘falls into a certain element of volume didyd5, may be repre-
sented by f ()dddi,d=. When we moreover assume that the probabilities
for the different molecules are independent of each other, the probability
of a certain combination of velocities is proportional tof (¢,) 7 (¢,)--- 7 (¢),
and this must remain the same when the kinetic energy ZL, or
because the molecules are assumed to be equal, 2c? remains the
same. For every change of c, and «| into cz, and c, so that
cy, + c?; — ce, + c7, must f(cc)-f()=—/ (cr) -f(c). This is an
equation which frequently occurs in the theory of gases, trom which
follows /(c)= ae. As a special case follows from this: /(c) =a,
i. e. the assumption of BoLtzMANN and Jeans, that the probability a
priori would be equal for every value of the velocity.

§ 2. In the second place | wish to make some remarks in con-
nection with the proof that BoLTzMann gives in his ‘“Gastheory”’,
that for an ‘“‘ungeordnetes’” gas with simple suppositions on the nature
of the molecules in the stationary state Maxweri1’s distribution of
velocities is found. Dr. C. H. Winp shows in his above-mentioned
paper that in this Bottzmann makes a mistake in the calculation of
the number of collisions of opposite kind. Bottzmann, namely, assumes,
that when molecules whose points of velocity lie in an element
of volume dw, collide with others whose points of velocity lie in
dw,, so that after the collision the former points lie in dw’ and the
latter in dw,’, now the elements of volume dw and dw’, dw, and dw,’
would be equal, so that now dw' dw’, = dw dw,. He further assumes
that when molecules collide whose points of velocity lie in dw’
and dw,', they will be found in dw and dw, after the collision.
These last collisions he calls collisions of opposite kind. Wisp now
shows that this assumption is untrue; dw is not =dw', dw, not
=do',, nor even dwdw, = dw'dw,', except when the masses of the
two colliding molecules are equal’).

Further the points of velocity of colliding molecules which lay
in dw' and dw,', do not always get to dw and dw, after collision,

1) I point out here that even then it is not universally true, but only when the
elements of volume dw and dw; have the shape of rectangular prisms or cylindres
whose side or axis has the direction of the normal of collision.

33*

( 496 )

so that another definition is necessary for collisions of opposite
kind, viz. such for which the points of velocity get in dw and da,
after the collision. Wixp proves further that the number of collisions
of opposite kind is all the same represented by the expression which
BoLTzMaNn had found for it.

It is ‘then easy to change (what Wuyp does not do) the proof
given by Borttzmann in § 5 of his “Gastheory”, that Maxwell's
distribution of velocities is the only one possible, in such a way that
it is perfectly correct. But the error in question makes itself felt all
through Boirzmany’s book. Already with the proof of the H-theorem
given in more analytical form in a footnote to § 5 we have some
difficulty in getting rid of this error.

We meet the same thing when the molecules are treated as centres
of force, and when they are treated as compound molecules. At the
appearance of the second volume of his work, Boitzmann had taken
notice of Wunp’s views, but the inaccurate definition for collisions
of opposite kind has been retained *).

In connection with this error, made by BoLTzMANN in a geometrical
treatment of the phenomena of collision, is another error of more
analytical nature, so that also JrANs, who treats the matter more
analytically, gives a derivation which in my opinion is not altogether
correct. ‘Though preferring the geometrical method, BOLTZMANN repeat-
edly refers to the other’). The method would then consist in this,
that the components of the velocities after the collision §'7/'$'s',7/',9,
are expressed by (/(§$S,7,5,) and then by means of Jacosr’s func-
tional determinant d&'dy'de'd§',dy',ds',_ is expressed in d§dydSd&,dy,d6, .
We find then that here this determinant is = 1 and so

deidyd§idé dy! ,d8', = d&dydéd§,dy,d8, or dw'do', = dwdw,.
The number of collisions of opposite kind = /'f",dw'de' oy cos ddddt
according to BortzmManxn, and so also = /'F",dwdw,o*g cos Idadt. In
this the mistake is made, however, that dé&dy/d¢'d&',dy/,d0', the
volume in the space of 6 dimensions that would correspond with the
volume didydSd3,dy,d5, before the collision, is thought as bounded
by planes such as §&'—c, which is not the case. JEANS too equates

the products of the differentials, in which according to him, dg’... do,
being arbitrary, the d§...d$ must be chosen in such a way,
that the values of §..., caleulated by the aid of the functions

§'— f(§...6,) ete. fall within the limits fixed by d§’ etc *). This,
however, is impossible.

1) Cf. § 78, 2nd paragraph.

*) Cf. among others volume I, p. 25 and 27.

8) Cf. *The dynamical Theory of Gases” p. 18.

( 497 )

In my opinion the correct principle that the calculation of the
extension occupied by the combinations of the points of velocity
after the collision when that before the collision is known and vice
versa, would come to the same thing as a transition to other vari-
ables in an integration, has not been applied in exactly the correct
way. The property in question says that in an integral with transi-
tion from the variables §'7’S'§',7',5', to §7$5,7,5, the product of the
differentials d§'dy/d8id3',dy',d3', may be replaced by asuSsi0S)

A(55§,7,5,)
dzdyd$ds,dy,d8,, if we integrate every time with respect to the corre-
sponding regions, but these expressions are not equal for all that.
The first expression may be said to represent the elementary volume
in the space of 6 dimensions, bounded with regard to §)...¢,, the

second the elementary volume bounded with regard to —...S, *).
We have a simple example when in the space of three dimen-

sions we replace frteana, which e.g. represents the weight of a

body, by { pr? sin 8drd9dp, which represents the same thing, without

du dy dz having to be equal to 7? sin & dr dd dg.
So we have here:
1 I 1(§' 5 Q ) | |
dg’ dy! dS’ dg’, dy, a5 = — d5 dy do ds, dn, a,
ae —26,).|
which two expresssions represent the “extension”’ in the space of 6 dimen-

sions after the collision. That before the collision is {uaa dgds,dy,d,,

so that, when the determinant = 1, the extension remains un-
changed by the collision. This proves really to be the case, as
JEANS shows. We may, however also consider this property as a
special case of the theorem of Liovuvinin, and derive it from this *).
This theorem says, that with an ensemble of identical, mutually inde-
pendent, mechanic systems, to which Hamiron’s equations of motion

apply, [eedn= dP,...dQn, when p,..dn represent the coordinates

and momenta of the systems at an arbitrary point of time, P,... Qn
those at the beginning. Grsps calls this law: the principle of conser-
vation of extension-in-phase, which extension we must now think
extended over a space of 22 dimensions. When now ihe two collid-
ing molecules are considered as a system which does not experience
any influence of other systems, and it is assumed that during the

1) Cf. Lorentz, l.c. Abhandlung VII.
*) As Borrzmann cursorily remarks: volume II p. 225,

( 498 )

collisions forees act which only depend on the place of the particles

and not on the velocities, we may apply the formula f op ee

| dp,...dg, to an ensemble of such pairs of molecules, the former

representing the extension-in-phase after, the latter that before the
collision. In the case discussed by Bottzmann the masses of these
molecules are m and m, so that we get:

' ! ’ “! ! ' ' ' 9 tod | ! al)
fae dy' dz' m® d&! dy! dS da’, dy’, dz, m,* d3', dy’, dS, =

= fae dy dz m* dg dy d§ dea! di' dz' m,* dg dy dq.

As we may consider the coordinates during the collision as inva-
riable, it follows from this that:

jas dy! dq ds’, dy, de, == ds dy Tks ds, dy, dS.
. .

§ 3. However as has been referred to above, we may, without
assuming anything about the mechanism of the collision, prove the
property by means of the formulae for the final velocities with
elastic collision, making use of the functional determinant. Another
method is followed by Wuyp in his above-mentioned paper (the
second proof) and by Borrzmann (vol. I p. 225 and 226); this
method differs in so far from the preceding one, that the changing
of the variables takes place by parts (by means of the components
of velocity of the centre of gravity), which simplifies the caleu-
lation’). A third more geometrical method is given by Wunp in his
first proof. This last method seems best adapted to me to convey
an idea of the significance of the principle of conservation of exten-
sion-in-phase in this special case. I shall, however, make free to
apply a modification which seems an abridgment to me, by also
making use of the functional determinant. So it might now also be
called a somewhat modified first method.

In the first place I will call attention to the fact that with these
phenomena of collision it is necessary to compare infinitely small
volumes; if we, therefore, want to use ms formula:

|
jee d§' dr} d3' dg’, dy, ds, = (Fe ig eu! ds dy d§ dS, dy, ds,
eth I

1) It seems to me that in this proof aya does not abide by what he
himself has observed before (§ 27 and § 28, vol. Il), viz. that the equality
of the differential products means that they may be substituted for each other in
‘integrals. The beginning of § 77 and the assumption of du dv dw, and dUdV
dW, as reciprocal elements of yolume, is, in my opinion, inconsistent with this

( 499 )

we must take infinitesimals of the 2°" order. We can, however, also
proceed in a somewhat different way. For how is the above formula
derived? By making use of the fact, that with a volume d§ dy ds dé,
dy, dS, in the region of the §.. $, corresponds a volume

ia@.. &)|
es ~| d5 dy d5 dz, dy, dS
id ae | bo} ] -) 1 Ny “il

in the region of the &'...%,, or also that the first mentioned exten-
sion, occupied by the representing points in the space of 6 dimensions
before the collision, will give rise to the second extension after the
collision. We can, therefore very well compare these expressions
inter se, without integration, if only the second expression is not
interchanged with ds’ dy d3' dz’, di’, dS',, i. e. the volume element obtained
by dividing the extension after the collision in another way.

We now suppose the points of velocity before the collision to be
situated in two cylindres, the axes of which are parallel to the
normal of collision. The bases of the cylindres are dOdQ, and the
heights dd and dd,. The extension occupied by the combinations of
the points of velocity is evidently equal to the product of the con-
tents of the cylindres : dOdO, dddd,. In case of collision the compo-
nents of the velocities perpendicular to the normal remain unchanged,
so the points of velocity are shifted in the cylindres in the direction
of the axis, so that d becomes J’, and d, becomes d',. Between
md + m, (2d,—d)

m+ m,

these quantities exist the relations: dd’ =
m,d, + m(2d—Jd,)
m + mM,

molecules (i.e. the same relations as between the normal initial and
final velocities with elastic collision.

If we now wish to calculate the extension after impact we may
make use of the fact that dO and dO, have not changed, so that
we need only examine what happens to dddd, or what extension in

the region of the d’d’, corresponds to the extension dddd, in the
region of the dd,.

and d', =

, When m and m, denote the masses of the colliding

; ae aw
According to the above this is: | \

: (aay dddd, , and as it follows

from the formulae for d' and Jd’, that the absolute value of the
determinant = 1, the extensions before and after impact are equal.

The extension after the collision is, however, not equal to the
product of the cylindres in which the points of velocity will be found
after the collision. This will be easily seen with the aid of the
geometrical representation given by Winp. The extension before
impact may be thought as the product of the extension in the space

( 500 )

of four dimensions dV0dO, and the extension dddd,, which we may —
imagine as a rectangle in the region of the Jd,, when we project them
as two mutually normal coordinates in a plane.

Every point in the rectangle represents therefore a number of
combinations of velocities with equal Jd and d,. The sides of the
rectangle with equations d—=c and d, —c,, correspond in the region
of the dd’, with the right lines md’ + m, (2d, — d)=(m-+ m,)e
and m,d', + m (2d — d',) = (m+ m,) ¢,, so that from the combina-
tions represented by points within the rectangle after the impact
others follow represented -by points within an oblique parallelogram.

The formula ——» = 1 expresses that the two figures have the

d(d6,) :
same area. Now the extension after impact is equal to this paralel-

( 501 )

logram XX dOdO, or the product of the two cylindres in which points
of velocity were found before impact. The product of the cylindres,
in Which points of velocity are found after impact is equal to the
product of dOdO, and the area of the rectangle with sides parallel
to the axes O'd’ and O'd', deseribed round the parallellogram under
investigation. In this rectangle lie a number of points which have
no corresponding points in the first rectangle. Only when m—= m,
rectangle and paralellogram coincide.

Collisions of opposite kind, now, are such for which the combina-
tions of velocity before impact are represented by points of the
paralellogram in the plane d'Od', and after impact by points of the
rectangle in the plane d0d,.

Physics. — “Contributions to the knowledge of the wesurface of
VAN DER Waats. XII. On the gas phase sinking in the liquid
phase for binary mixtures.” By Prof. H. KamErninan OnNus
and Dr. W. H. Krrsom. Communication N°. 96 from ithe
Physical Laboratory at Leiden. ;

§ Ll. Introduction. In what follows we have examined the equi-
librium of the gas phase with the liquid phase for binary systems,
with which the sinking of the gas phase in the liquid-phase may
occur.

It lies to hand to treat this problem by the aid of yw (free
energy)-surfaces for the unity of mass of the mixture (VAN DER WAALS,
Continuitaét I] p. 27) for different temperatures construed on the
coordinates 7 (volume of the unity of mass of the mixture) and «
(quantity of mass of the second component contained in the unity
of mass of the mixture).

As VAN DER WAALS (loc. cit.) has already observed, the laws refer-
ring to the stability and the coexistence of the phases are the same
for these y-surfaces as for the more generally used ~w-surfaces for
the molecular quantity: in particular also the coexisting phases are
indicated by the points of contact of the y-surface with a plane
which rolls with double contact over the plait in the y-surface. In
what follows we have chiefly to consider the projections of the con-
nodal curve and of the connodal tangent-chords on the zv-plane.

More particular cases as the occurrence of minimum or maximum
critical teinperature or minimum or maximum pressure of coexistence
we shall leave out of account; we shall further confine ourselves
to the case that retrograde condensation of the first kind occurs.
Moreover we shall restrict ourselves to temperatures, at which the

( 502 )

appearance of the longitudinal plait does not cause any irregularity’).
The component with the higher critical temperature (7;) is chosen
as first component; its critical temperature is, accordingly, denoted
by 7;,. The special case that 7;,—0, is that of a gas without
cohesion with molecules having a certain extension. The investigation
of the w-surfaces becomes simpler for this case. For the present it
seems probable to us that helium. still possesses some degree of
cohesion. We will, however, in a following communication compare
the case of a gas without cohesion with what the observations yield
concerning mixtures with He.

§ 2. Barotropic pressure and barotropic concentration. We shall call
» and « of the gas phase v, and z,, of the liquid phase v, and ay.
At a temperature 7’ a little below 7;,, we shall always have
vg>vi1. For then the plait extends only little on the w-surface (see
fig. 1), the plaitpoint is near the top of the connodal enrve, which
is turned to e=—1, and all the projections of the connodal tangent-
chords deviate little in direction from the v-axis, the angle with
Lg—tl ; -

/_" inereases regularly if we go from « =0
Vg—V
along the connodal curve to the plaitpoint, but it has but a small value,
when 7; —T is small. Only when we take for 7’ a value a certain
amount lower than 7%, the plait extends sufficiently on the w-surface

x 4

to allow that vy =v, and 6=-<.

the v-axis, 6, = arc tg

If at a suitable temperature 7’ we have substances as mentioned
at the beginning, as e.g. helium and hydrogen at the boiling-point
of hydrogen, we shall find the projection of a connodal tangent-
chord denoting the equilibrium considered in the zv projection of the
gas-liquid-plait on the y-surface for 7’; to reach it we shall have
to ascend from «=O along the connodal curve up toa certain value
of the pressure of coexistence p, before @, which itself is zero for

Pt -
«==0, can become >. A pressure of coexistence p= po, under

which v, = at the temperature 7, we call a barotropic pressure
Jor that temperature, the corresponding concentrations of liquid and gas
phase the barotropic concentration of the liquid and of the gas phase
at that pressure and that temperature. For when v,—v, with increasing
pressure of coexistence p passes through zero at p= po, we find
in equilibria with pressures of coexistence above and below the value
p, the phases to have changed positions under the influence of gravity.

1) This will be treated in a following communication.

( 503 )

In order to examine how a barotropic tangent-chord first makes
its appearance on the plait on decrease of 7’, we point out that with

extension of the plait from 7%, at first Fe remains positive all over
v

the liquid branch of the connodal curve, so that at tirst we have to
look for the greatest value of @ at the plaitpoint, where we shall
denote its value by 6p).

When, however, on decrease of 7’ the plait extends over the y-
surface, this need not continue to be the case, and we may find
dO 3 ae .

a alternately positive and negative. This is immediately seen when

v
we notice that this must always be the case when the plait extends
all over the w-surface.
If with decrease of 7’ the maximum value of 6 more and more in-
creases, and 7’ has fallen so low, that the maximum of 6 somewhere
big : eh
in the plait has just ascended to >, then at this 7’ the condition for the

barotropic equilibrium v, =v, will be satisfied just for the corre-
sponding tangent-chord, and only for this tangent-chord. The higher
barotropic limiting temperature is then reached. On further decrease
of temperature the barotropic tangent-chord will then split into two
parallel barotropic tangent-chords, the higher and the dower tangent-
chord, which at first continue to. diverge with further falling tempe-
rature, so that the higher barotropic tangent-chord may even vanish
from the plait through a barotropic plaitpoint, and then, at a lower
temperature, make its appearance again through a barotropic plaitpoint’).

At still lower temperature it follows from the broadening of the
plait in the direction of the v-axis, which at sufficiently low tempe-
rature renders the occurrence of a barotropic tangent-chord impossible,
that the maximum of 6 falls again, and the barotropic tangent-chords
draw again nearer to each other. At == the tangent-chords

coincide again, and the lower barotropic limiting temperature is reached.
At lower temperatures vy, = 7, is no longer to be realized, and v,
is always > w.

Figs. 2, 3 and 4 represent different cases schematically. In the
spacial diagram of the y-surfaces for different temperatures the
barotropic tangent-chords supplemented with the portions of the con-

‘) The latter supposes that 7;,/Z', is not very great; in accepting the contrary
we would come in conflict with the supposition that the longitudinal plait dves
not become of influence Moreover we preliminarily leave out of account the case
that both barotropic tangent-chords follow one another in disappearing or appearing
through a barotropic plaitpoint. [Added in the translation].

( 504 )

nodal curves between the lower and the higher tangent-chords form
together a closed surface, which bounds the barotropic region.
If on the other hand G@nax = 4,i remains, till it has reached or

w
exceeded the value —, and if not asecond maximum value ford > 5

4,
occurs on the plait, a barotropic plaitpoint will oceur at the higher
barotropic limiting temperature, whereas at lower temperature a
single barotropic tangent-chord on the plait indicates the equilibrium
with v,= vi. With decreasing temperature this barotropic tangent-
chord will at first move along the plait starting from the plaitpoint,
but at lower temperatures it will return, and finally (the occurrence
of a longitudinal plait being left out of consideration) it will disappear
from the plait through a barotropic plaitpoint at the lower barotropic
limiting temperature. In this case the barotropic region is bounded
on the side towards which the plait extends by barotropic tangent-
chords, on the other side by the portions of the connodal curve which
are cut off.

It follows from the above that — when the occurrence of barotropic
tangent-chords on the y-surfaces for a definite pair of substances is
attended by the occurrence of barotropic plaitpoints — if 7, > Toyis
(higher baratropic plaitpoint temperature) or Ts << Topix lower barotropic
plaitpoint temperature there always exists at the same time a higher
and a lower barotropic tangent-chord; if Ti. > Ts > Tsp there
exists only one barotropic tangent-chord.

The nature of the barotropic phenomenon for He and H, may
serve for arriving at an estimation of the critical temperature of He.
According to the investigation of one of us (K. See Comm. N°. 96 ¢.)
it is probable that the appearance of a single barotropic tangent-
chord for He—H, at the temperature of boiling hydrogen would
point to 7ye< about 2°, whereas on the other hand when Tye
is higher, a higher and a lower barotropic tangent-chord is to be
expected. Further that, as was already observed in Comm. No. 96 a.
(Noy. °06) a barotropic tangent-chord can only appear in the gas-
liquid-plait when very unusual relations are satisfied between the
properties of the mixed substances, which for the present will most
likely only be observed for He and H,.

Whether it is possible that more than one barotropic region occurs,
and whether one or more barotropic tangent-chords can move from
the plaitpoint past the critical point of contact, is still to be examined.
Also whether it is possible that the lower barotropic limiting tem-
perature descends lower than 7%,, so that fig. 5 might be realised.
With regard to these questions too it is only of practical importance to

(505 )

know in how far the properties of He and H, create that possibility.

§ 3. Barotropic phenomena at the compression of a mixture of definite
concentration. What will take place in this case is easy to be derived from
the foregoing survey of the different equilibria which are possible at a
same temperature. For the further discussion we have to trace the
isomignic line, the line of equal concentration (@ = const.) for this
mixture, and to examine the section with the connodal curve, the
successive chords, and finally again with the connodal curve.

In the deseription of the barotropic phenomena we shall confine
ourselves to the more complicated case, that at the 7’ considered
both a higher and a lower barotropic tangent-chord occur, after
which it will be easy to survey the phenomena when only one
barotropic tangent-chord appears.

To distinguish the different cases we must divide the liquid branch
of the connodal curve at 7’ into an infra- (7 =0 to «= 2y;7, lower
barotropic concentration of the lhquid phase at T), inter- (xy;7 to xys7)
and supra-(@=tpsr to «= x, )-barotropic part, and the gasbranch
into the three corresponding pieces falling within and on either side
of the region between the two barotropic tangent-chords (the lower
b;r and the higher 6,7) at that temperature.

Whether the phenomena of retrograde condensation attend those
of the barotropic change of phase or not depends on this: whether
both barotropic concentrations of the gas phases fall below the plait-
point concentration or not.

Let us restrict ourselves in this description to the case that this
complication does not present itself or let us only consider mixtures
for which «<< #,;. On compression the first liquid accumulates in the
lower part of the tube for «<< &giT and for rR > «> wys7, and in the
higher part for 2j;7 >“ > xyi7. On further compression, when
Lypbit > tisT, Change of phase will take place once for mixtures of
the concentration 2, so that LgbsT > L > Lghi7 OF LysT > L > LT;
it will take place twice ‘for- mixtures of the concentration ., so
that aynr > ¢ > ausr. So the last remains of the gasphase will
vanish above for «< wy7r and for Lp > & > Xps7, and below for
tsT > & > xpir. If it is possible that over a certain range of
temperature the barotropic tangent-chords get so far apart that
LbsT > XgviT, Change of phase will again take place once for these tem-
peratures for mixtures of the concentration z, so that 2.57 > @ > Lusr
Or LynT > L > Uw -

This description will, of course, only be applicable to He and H,
when the suppositions mentioned prove to be satisfied.

(506 5

$4. Disturbances by capillary action. As is always tacitly assumed
in the application of the w-surface when the reverse is not expressly
stated, the curvature of the surfaces of separation of the phases is put
zero in the foregoing discussion. If the curvature may not be ne-
elected, e.g. at the compression of a mixture in a narrow tube,
then, when the barotropic pressure is exceeded, the phase which has
thus become heavier, will only sink through the lighter phase under it,
when the equilibrium has become labile taking the capillary energy of
the surface of separation into account. For this it is required that 4,

a ae :
has become larger than =i to an amount of AG cays which will depend

on the capillary energy of the surface of separation and the diameter
of the tube in which the experiment is made. Thus capillarity causes
a retardation of the appearance of the barotropic phenomenon: both
with increase and with decrease of pressure the barotropic tangent-
chord must be exceeded by increase or decrease of pressure to a
certain amount, before the two phases interchange positions. In this
way the difference of pressure mentioned in Comm. N°. 967, (Nov.
1906, p. 460) between the sinking of the gas phase chiefly consisting
of helium and its rising again at expansion (49 and 32 atms.) is
e.g. to be explained by the aid of the following suppositions which
are admissible for a first estimation.

1. that at — 258° and 32, resp. 49 atms. He is in corresponding
state with H, at 150° and 160, resp. 245 atm., in agreement with the

3
assumptions Mf. 0.2 = rie vi according to the ratio of the molecular

refractive powers, 7:7. —=1°.5 (according to O1szewskt < 1°.7); if
the gas phase consisted only of He (molec. weight 4), the density
at the temperature and pressures mentioned would be 0.062, resp.
0.081, and if moreover the liquid phase had the same density with
the two pressures, 46,,, would have to correspond to a difference
of density of + 0.01; owing to the fact that the two last mentioned
suppositions are not satisfied, the difference of density will be smaller ;

2. that the capillary energy of the surface of separation between
the phases coexisting at the above temperature and pressures is not
many times smaller (or greater) than that of liquid hydrogen at that
temperature in equilibrium with ifs saturated vapour, and that the latter
may be derived from that of nitrogen *) by the aid of the principle
of corresponding states. The gas bubble will then in a tube like
that in which the experiment described in Comm. N°. 96a was made
(int. diam. 8 mm.) only sink through the liquid or rise again, when

1) Baty and Donnan, Trans. Ghem. Soc. 81 (1902) p- 907,

‘Y] JOA “wepaojsury "peoy ekoy sS8urpo0 904

= IC

ea?

‘g ‘S14

<Dy, |

cay

‘y Sl

esvydses oy} TO ‘TIX

_somyxrur Areurq toy osvyd piubiy oq} Ul suryuts
‘Iq pue SAaNNO HONITYANVH “H JOtd

‘s[eUM op WA JO ooVJANS-T OY} JO ESpo[MOUY Of} OF moTyNgI.AyM09,, "(MOSMAN “H “M

( 507 )

the difference of the radii of curvature of the tops of the bounding
menisci exceeds that between 3 and 5 mm.

T
At those temperatures for which maz << > + 4 Oca), the phenome-

non of the phase which is uppermost at low pressure, sinking and
rising again does not make its appearance in consequence of gravity
alone. If this condition is satisfied for mixtures of a definite pair
of substances for every temperature between the lower and the
higher barotropic limiting temperature, the phenomenon could only
be realised for these mixtures by the aid of a suitable stirrer.

§ 5. Remarks on further experiments with helum and hydrogen.

a. In the experiments mentioned in Comm. N°. 967 the gas phase
proved to remain below on compression to the highest pressure which
the apparatus will allow. When we repeat these experiments at a
higher temperature (which may e. g. be obtained by boiling the
hydrogen of the bath under higher pressure')) it is to be expected
that the barotropie pressure will first rise, as in the beginning starting
from — 253° the gas phase will continue to expand more strongly
than the liquid phase. At higher temperature the liquid phase begins
to expand more strongly than the gas phase, but the mutual solu-
bility plays already such an important role then that a definite expectation
cannot be expressed, unless this, that on account of the retreating
of the plait and the impossibility of the barotropic tangent-chord to
reach the side of the hydrogen, the higher barotropic limiting tem-
perature may be pretty soon reached. Also in connection with the
estimation, which may be made from this concerning 7.7, it will
be of importance to investigate whether with a suitable concentration and
at a suitable higher temperature we may observe the liquid phase sinking
after ut had first risen. That the phenomena at higher temperature,
if the glass tube used should prove strong enough to bear the pres-
sure, should be prevented by capillary action, is not probable, as
capillarity together with the differences of density decreases at higher
temperature; moreover in spite of capillarity the phenomena might
be realised by the aid of a suitable stirrer.

6. With decrease of temperature the limit is soon reached at
which we meet with the solid phase. The question rises whether
then the phenomenon: the solid phase, (the solid hydrogen) floating

on the gas phase (chiefly the as yet still gaslike helium), might not
be realised.

1) Or by using the vapour from boiling hydrogen in a separate vessel [added
in the translation].

(508 )

Physics. — “Contributions to the knowledge of the w-surface of
van per Waats. XII. On the conditions for the sinking and
again rising of the gas phase in the hquid phase for binary
mixtures,’ by Dr. W.H. Kresom. Communication N°. 96c from
the Physical Laboratory at Leiden. (Communicated by Prof.

H. KAMERLINGH ONNES).

§ 1. Introduction. As has been observed in Communication N°. 966
(See the preceding paper) (cf. Comm. N°. 96a, Nov. ’06, p. 459,
note 1) it lies to hand, to take as point of issue the y-surface for
the unity of mass of the mixture considered, in the investigation
according to vAN per Waats’ theory of binary mixtures, of the sinking
and subsequent rising of the gas phase in the liquid phase, i.e. the
barotropic phenomenon. Two coexisting phases of equal density are
joined on this y-surface by a tangent-chord whose projection on the
v,v-plane') is parallel to the z-axis. It has already been observed
in Comm. N°. 964, that with decrease of temperature starting from
the critical temperature of the first (least volatile) component such
a barotropic tangent-chord may make its appearance in two ways:

a. by the angle of inclination of the tangent to the plait in the
plaitpoint, 4,;, reaching the value of = at a certain temperature 7%,,;

and by its exceeding this value at lower temperature.
b. by @ showing a maximum and a minimum on the plait at a
certain temperature, and by this maximum reaching or exceeding

wu : : ; :
the value of 5. Also in this latter case one of the two barotropic

tangents-chords which then appear, might reach the plaitpoint at

1 4

lower temperature, and thus become 4,7 = =

: : 7v ; :
In both cases in which 6,; => at a certain temperature it should

“/

be expected apart from complications as e.g. a longitudinal plait

etc. (cf. Comm. N°. 966, p. 502), the description of which will be
ie i Leela 3 ;
given later on, that 4): becomes again = — for mixtures of the same

substances at a lower temperature.
In the first part of this paper the conditions are discussed on which

a plaitpoint with 4,. = —, barotropic plaitpoint, occurs on the y-sur-

DOS |

1) Cf. for the meaning of x and v Comm. N?. 96d.

(509 )

face, whereas in the second part the conditions are treated for the
appearance of a barotropic tangent-chord on the plait.

A. On the conditions for the occurrence of a barotrojic-plaitpoint.

§ 2. In a barotropic plaitpoint the isobar, which in a plaitpoint
always touches the plait, must run parallel to the v-axis. This gives

the condition :
0?»
ae SS Oe Ee
i , =

Moreover a section v =v, of the y-surface with the limiting
position of the tangent-chord must then have a contact of the 3"¢
order. This furnishes:

07yp Ow ;
== =e — . . . . . 1 d c
= 1 as es (1 } and c)

The two conditions (1 6) and (1c) follow also by applying ( a)
to the general relations for the plaitpoint of Comm. N°. 75 (Dec.
1901) p. 294.

The same may be obtained from the property of the barotropic

shows & maximum or minimum'), so that the substitution-curve

;

Ow
points on the connodal curve that there ( |) along the connodal

Ow ae
i) = const. (see for the substitution curves on the y-surface for the
U

molecular quantity Comm. N°. 597), touches the connodal line in
these points.

§ 3. For a first investigation we shall use van DER WAALS’ equation
of state:

sa Sek ec onan 6 Se)

with an a, and 6, not depending on v and 7’ for a definite «.
In this:

1) This property for coexistence with 7; —7, is analogous to the property that

( \= — p along the connodal line is a minimum or a maximum for coexisting

) along an isome-

phases with 2, =. In the same way the mean value of (
©
tric line v =v, which joins two barotropic phases on the J-surface, is equal to

)
the value of &

a“

) for these phases.

34
Proceedings Royal Acad. Amsterdam. Vol. VIII.

( 510 )

he BE (lay Ee, a) oe + ee tae
if R, = R/M,, R, = R/M,, F representing the molecular gas-constant,
M, and M, representing the molecular weights; we put for this first
investigation a,, = Va,,4,,, so that
Wa, (i —2)'- \/0,, 20 2 ee
and when we put for the molecular volumes 26,947 = 6)131 + boo,
the relation for }: given by VAN DER WAALS Contin. II p. 27, reduces to:
bz = 0, (1—2) =~ bag att ig Go eee, er
We get then (van per Waats Contin. I p- 28):
pee ade 6,) — = + TR, (1—z)ln(l—a) + Ryelna} . (4)
§ 4. Taking equations (8) into consideration, and putting
b..- eee = 1+

v—b Z

we get by the conditions (1)
1-F2{ 2 eis 2 v—b da
ea Ei Stas aay» pee es — :; oe re
ef |) 2 | eee w
1+< hs eas 1—z ra lia
hk, 5 ieee ct ce R, ee | “u— = = Pi dai (65)
1+-z 2 | 1 1l—<z aoe d
Rp EN yp | ee eee
ae git ne eae eae Eee + Bg peat fo eee

These equations are sufficient to calculate the data for a barotropic
plaitpoint 2,1, Vopi, 7,, for a definite pair of substances. Eliminating
T from (6a) and (64), we get, taking (6c) into consideration and
putting :

(a, + V2,,)/Ve. =e) = 2) eee

2 4
b (v—b) ir -+ mae (ute) +0 ie _ = = 0, 8)

while elimination of v from this equation and (5), putting:
(6,,+5,,)/ Gg—=),,) = 8. ee eo

i =

eo
Ae + |\«- = fata t fut =a] botan+t=0. ©

Bese ee equation (et (6c): 25), may be found for given #,/f,,
wand vy, after which x»,;, vy, and Th,i, as well as po, follow easily.

§ 5. That a barotropic plaitpoint exists on the liquid-gas-plait
with the assumed suppositions (2), (34), (3c) and with suitable

( 511 )

values of the constants, appears as follows: for 2=0.5, R,/R,=°/,,
b,,/b,, =/,, (6c) yields: u = — 1.957, after which (9) yields:
w= — 1.176, so that a,,/a,, = 0.00653. Thus we find fora mixture

of two substances with M, = 2 M,, v,,='*/, vz, (so that the ratio of
the molecular critical volumes is ‘/,), 7), = 0.052 f ae barotropic

plaitpoint for 1 — 0.26 Vk; » 1 ALE) igs » Phyl = 4.8 Pky -
(To he continued).

ERRATUM.

In the Proceedings of the meeting of September 29, 1906.

p. 209, line 15 from the bottom: for § 10 read § 9.
p. 210, Table I, line 5 and 4 from the bottom, for: 5 July read
6 July.
ed from the bottom, for 3 March ’65
read 3 March ‘O06.

(January 24, 1907).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday January 26, 1907.

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 26 Janunri 1907, Dl. XV).

CO een nN TT s.

A. F. Hotteman and G. L. Vorrman: “a- and f-thiophenic acid’, p. 514.

J. D. vAN per Waats: “A remark on the theory of the ¥-surface for binary mixtures”, p. 524.

W. A. Wistnorr: “The rule of Nerer in the four-dimensional space”. (Communicated by
Prof. P. H. Scuoure), p. 529.

P. H. Scnovre: “The locus of the cusps of a threefold infinite linear system of plane cubics
with six basepoints”, p. 534.

W. J. H. Morr: “An investigation of some ultra-red metallic spectra”. (Communicated by
Prof. W. H. Junius), p. 544.

F. Scuun: “On~the locus of the pairs of common points and the envelope of the common
chords of the curves of three pencils”, (2nd Communication. Application to pencils of conics).
(Communicated by Prof. P. H. Scuoure), p. 548.

F. Scuun: “The locus of the pairs of common points of four pencils of surfaces”. (Commu-
nicated by Prof. P. H. Scouts), p. 555.

C. H. Winn, A. F. H. Datuuisen and W. E. Rincer: ‘“Current-measurements at various
depths in the North-Sea”, (!st Communication), p. 566. (With one plate).

F. Scuun: *The locus of the pairs of commen points of n-++1 pencils of (n—1) dimensional
varieties in a space of n dimensions”. (Communicated by Prof. P. H. Scuoure), p. 573.

H. G. van bE SanpE Bakuuyzen: “On the astronomical refractions corresponding to a dis-
tribution of the temperature in the atmosphere derived from balloon ascents’, p. 578.

H. ZwaaRDEMAKER: “An investigation on the quantitative relation between vagus stimulation
and cardiac action, an account of an experimental investigation of Mr. P. WoLTERSON”, p. 590.

Erratum, p. 598.

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 514 )

Chemistry. — “«- and 8-thiphenic acid.” By Prof. A. F. HoLLEMAN
and Dr. G. L. VoERMAN.

(Communicated in the meeting of December 29, 1907).

A very remarkable research on these acids was published in 1886
by V. Meyer, who discovered the same. In the main it amounts to
this, that in addition to the theoretically possible monocarboxylic
acids of thiophen

z|CO,H |CO:H
EEA
5 S
Mp. 126°.2 Mp. 138°.4

a third isomer was obtained, called a-thiophenie acid m.p. 117.5—118°,
which, however, in its derivatives such as the amide, the phenylurea
derivative of the amide, the amidoxime and the thienone (C,H,S),CO,
so completely resembled the corresponding derivatives of the a-acid
that they would have been declared identical, were it not that from
the a-derivates the a-acid m.p. 118° was regenerated, whilst the
a-derivates yielded the a-acid m.p. 126°. V. Mryrr expresses himself
as follows: “Die Vergleichung der a- und a-Sdure ergab immer von
“Neuem das merkwiirdige Resultat, dass die beiden Sauren wirklich
“in ihren Eigenschaften durchaus verschieden sind, und dass die Ver-
“schiedenheiten sich als constante, durch keinerlei Reinigung oder
“Umwandlungen zu entfernende Eigenschaften erwiesen; dass aber
“alle Derivate der beiden Saduren in ihren physikalischen Eigenschaften
“absolut zusammenfallen und fiir identisch (im gewoéhnlichen Sinne)
“erklart werden miissten, wenn sie nicht die Eigenschaft besassen,
“dass jedes aus der a-Sdure dargestellte Derivat bei der Riickfiihrung
“auch wieder «-Siure, jedes a-Derivat dagegen a-Siure lieferte.”

As the a-acid had also been obtained by oxidation of tar-thiotolene
which is a mixture of a- and }-thiotolene {2—3 methylthiophen |],
V. Meyer suspected that this a-acid might be after all a mixture of
a- and #-acid, and he really succeeded, by oxidation of a mixture
of the two thiotolenes in definite proportion, or by slow erystallisa-
tion of a mixture of «- and B-acid from cold water, in obtaining an
acid which agreed in every respect with the a-acid. This was no
doubt an important step forward, but the behaviour of such a
mixture and also of the derivatives obtained therefrom still remained
a very remarkable one.

1) A, 236, 200; also V. Meyer, die Thiophengruppe p. 188—207.

( 515 )

Notwithstanding all this, nobody, since this elaborate research of
V. Meyer, has been engaged during the last 20 years in the study
of these acids, although it might have been expected from the deve-
lopment of the phase-rule that the latter might possibly give us a
closer insight into the phenomena described above.

The probable cause of all this is that these acids are not readily
accessible, and that those engaged in researches connected with the
phase-rule have not ventured to prepare the same. When Dr. Vorrman,
at my request, undertook the closer study of these acids we had, first
of all, to find a better process for the preparation of these substances.

In the case of the a-acid we have indeed succeeded in a very
satisfactory manner. We have also worked out another and improved
method for the preparation of the B-acid; but it is still unsatisfactory
owing to the small yield. Therefore, we have been obliged to restrict
ourselves, provisionally, to the study of the acids themselves; the
derivatives will be taken in hand when more material has been
obtained.

Preparation of a-thiophenie acid. By V. Meyer and his pupils, this
acid was best obtained by oxidation of propiothienon C,H,S.COC,H,,
because the oxidation of the much more readily accessible acetothienon
C,H,S.COCH, yielded a mixture of «-thiophenic acid and thienyl-
glyoxylic acid, which it was rather troublesome to separate. We have
succeeded in converting acetothienon almost quantitatively into a-
thiophenic acid, being guided by the following considerations. If
we oxidise a methyl ketone, experience has taught that the methyl
group very readily changes to carboxyl thus forming a glyoxylic
acid: R.CO.CH,—>R.CO.CO,H. If, however, we attempt to go further
and obtain the corresponding carboxylic acid: R.CO.CO,H—R.CO,H
a difficulty is experienced and the oxidising mixture then also attacks
the group R so that the yield of the carboxylic acid becomes
generally unsatisfactory. Now some time ago, I found a method for
converting acids R.CO.CO,H quantitatively into R.CO,H; this is
rendered possible by the application of hydrogen peroxide which
causes a ready resolution according to the scheme:

R.CO60,8 =,
HO OH = R-COOH + CO, + H,0

This method has led to the desired result in this case. The
oxidation of acetothienon is, therefore, done in two stages, first the
formation of thienylglyoxylic acid which is subsequently oxidised to
-thiophenic acid. The practical application of these processes was
as follows: ;

35*

( 516 )

Acetothienon was prepared from thiophen according to the method of FriepeL
and Crarrs, and a very good yield was obtained. The thiophen was prepared by
ourselves by distillation of sodium succinate with P,S;. 11.5 grams of ketone and
12 grams of sodium hydroxide were introduced into a litre of water, and to this
was added slowly, at the ordinary temperature, a solution of 42 grams of potas-
sium permanganate dissolved in a litre of water. After each addition the pink
colour was allowed to change to green before addition of a fresh portion.

After all the permanganate had been added the liquid was allowed to remain
overnight; the solution was heated gently on the waterbath until the green colour
had disappeared, then filtered off from the manganese dioxide, and concentrated to
250 ce. Without isolation of the thienylglyoxylic acid, beforehand the liquid, after being
nearly neutralised with hydrochloric acid, is mixed with 9 grams of 30°/9 hy-
_ drogen peroxide previously diluted with its own volume of water. The whole is
set aside for a few hours, and afterwards heated for a few moments on a water-
bath. On acidification the liquid the greater part of the z-thiophenic acid formed is
precipitated in a pure condition; a further small quantity may be recovered from
the mother-liquor by extraction with ether. By recrystallisation from water and
distillation in vacuo, the acid may be obtained pure and quite free from thienyl-
glyoxylic acid. The yield amounts to about 9 grams.

The solution of a-thiopbenic acid saturated at 24°.9 contains 0.75 °/,.

Preparation of @-thiophenic acid. V. Meyer has effected this by
oxidising with potassium permanganate in very dilute, cold solution.
The yield of B-acid was however very poor, in fact only about
5—8°/, of the thiotolene employed. After trying various modifications
of this direct oxidation process without arriving at a better result
we decided to follow an indirect way by first chlorinating the side
chain, then preparing the aldehyde from the thienalchloride and
finally oxidising the former to the acid:

C,H,S . CH,-> C,H,S . CHCl, C,H,S . CHO > C,H,S . COOH.

Bearing in mind van per Laay’s research") on the bromination
of toluene where it was shown that in presence of PCI, the sub-
stitution in the side chain is accelerated, this substance was added
in the chlorination of §-thiotolene. The above mentioned processes
all proceeded very smoothly, but unfortunately an acid rich in
chlorine was finally obtained as, apparently, the chlorination had
also extended to the nucleus. This certainly could be freed from
chlorine by treatment with sodium-amalgam but a large proportion
of the p-thiophenic acid was lost thereby so that the yield did, finally,
not exceed 10°/, of the thiotolene employed.

We add a few particulars as to the modus operandi followed.

G-thiotolene was prepared by ourselves by distilling sodium pyrotartrate with
phosphorus trisulphide. The chlorination took place in direct sunlight in the presence
of 10°/, of PCI;. The reaction product is boiled in a reflux apparatus with water
and calcium carbonate. The aldehyde is distilled in steam and purified over the

1) These Proc. Oct. 1905).

bisulphite compound. From 10 grams of thiotolene about 5 grams of the aldehyde
are obtained. Of this, 3 grams are oxidised with 500 cc. of water containing
3.2 grams of potassium permanganate and !.3. gram of 80"/) potassium hydroxide ;
afler standing over night the liquid is filtered from the manganese dioxide, con-
centrated and acidified when about 3 grams of thiophenic acid are precipitated.
The dechlorination of this product with sodium amalgam in dilute aqueous solution
takes 15—20 days during which a large portion of the acid gets lost.

As regards the solubility of @-thiophenic acid at 25° it was found
that the saturated solution contains 0.45 °/, of acid.

Melting point line of mixtures of the two acids. According to the
present views of the phase rule it was natural to suppose that the
impossibility of separating these~ acids by crystallisation is due to
the fact that they yield mixed crystals. In fact by determining the
melting point line, Dr. Vorrman has succeeded in demonstrating
with certainty that they give an interrupted mixing series. The
initial solidifying points may be observed very sharply but the final
solidifying points can only be determined within 0°.5.

A list of the initial and final solidifymg points is appended; and
in the annexed curve these figures are represented graphically.

( 518 )

SOLIDIFYING POINTS OF @ AND £-THIOPHENIC ACID AND THEIR MIXTURES.

Wy a % # | 4st solidifying prez 2nd solidifying point
400 0 196°9
99.01 0.99 125.4 | 493°9 494°
98.25 4.75 195.4 |
96 .56 3.44 194.6 | 123.4—193.6
94.3 5.7 124.4 121.0
93.60 6.40 193.4
90.60 9.40 422.2
88.20 41.80 121.3 + 118
85.82 14.98 | 120.3 + 116.6
85.0 15.0 120.4
79.45 20.55 417.7 443.5114
77.45 92.55 417.2 112 112.6
15.3 4.7 116.3 110.8—111 .2
74.60 95 40 116.0 + 110.6
69.45 30.55 114.3 440 —144
66.20 33.80 113.3 110.5—110.8
63.35 36.65 | 412.5 410.8
59.70 40.30 411.5 110.5
58.0 42.0 111.0 410.7
55.0 45.0 112.6 110.8
00.85 49.A5 115.0 410.7
42.50 57.50 | 119.6 id.
38.9 61.4 121.2 111.2
33.60 66.40 124.0 416.5—117.5
93.80 76.20 498.2 193194 (+193.5)
14.0 86.0 132.6 129.5—129.8
54 94.6 136.3 134 —134.3
0 | 400 138.4

It appears that the series of mixed erystals is interrupted on one

side at 25°

/, B-acid, and on the other side at 61°), ;
/, B-acid at a temperature of 111°.

is a eutectic point at 42.5

and that there

(519 )

The erystallographical investigation of these acids and their mixtures
kindly carried out by Dr. JAncrr leads to exactly the same result.
Dr. Janger reports as follows :

Of the two isomeric compounds p-Thiophenic acid erystallises the
most readily in sharply defined, small erystailine plates.

Whether obtained by crystallisation from solvents or by fusion
and subsequent cooling, the compound exhibits the microscopical
appearance of the subjoined figure. The crystals are monoclino-pris-

4]
}

aa
, AY) Vg
“GU, op Z&

Fig. 2. Microscopcial aspect of z+ and @-Thiophenic acid.

matic, and combinations of the form: {001}, very predominant,
{110} and {100}; the angle of inclination ¢ deviates considerably
from 90°, so that the smaller individuals often exhibit rhomboidal form
owing to simultaneous development of {110} and {001}. Often the
plates are so thin that only a single parallelogrammatic circumference
can be observed with a very slight stunting of the sharp angle which
was determined at 42°—48°, by {100}.

In addition, small rectangular plates occur which, as the investi-
gation shows, are formed with {100} as predominant form, and
therefore show prolongation along the /-axis. Although representing
apparently a second form they are, however, quite identical with
the parallelogrammatic phase.

The optical axial plane is parallel { 010} and falls along the longest
diagonal of the parallelograms or perpendicular to the longitudinal
direction of the needle-shaped individuals. In convergent light one
hyperbole with rings is visible at the border of the field of vision.

( 520 )-

Very feeble inclined dispersion with @ > v; double refraction nega-
tive. The longitudinal axis of the parallelograms and the shortest
dimension (breadth) of the more needle-shaped individuals are the
directions of a smaller optical elasticity.

a-Thiophenic acid crystallises from solvents or from the fused
mass in long more or less broad needle-shaped individuals, which
usually exhibit only a rudimentary limitation and cannot therefore
be properly determined morphologically. Although the optical proper-
ties seem to point to a monoclinic symmetry one might also feel
inclined to conclude to a triclinic symmetry on account of the form
limitations occurring here and there. The extinction of the needles
is, however apparently orientated perpendicularly to their longitu-
dinal direction. The smaller optical elasticity axis coincides with the
longitudinal direction of the needles. The optical axial plane is
orientated perpendicularly to the longitude of the needles; in con-
vergent light a single very characteristically coloured hyperbole is
visible at about ?/, of the diameter of the fields of vision.

Knormously strong dispersion with @ < v; the sign of the double
refraction around the correlated bissectrix is positive.

The two isomers are, therefore, readily distinguished microscopic-
ally by the following properties :

B-Thiophenic acid a-Thiophenic acid
Parallellogrammic limitation, or Long, very slender needles,
short rods of rectangular form. mostly with rudimentary limita-
Very high interference colours. | tion.
One optical axis with elliptical Grey, or unconspicuous colours.
rings, very weak dispersion @ < v. One coloured hyperbole ; very

Negative double refraction. strong dispersion: @ < v.
Monoclinic symmetry ; angle of Positive double refraction.
the parallelograms 42°—43°. Triclinie or monoclinie sym-
Optical axial plane for the paral- | metry.
lelograms // to the longest dia- The optical axial plane is per-
gonal for the needles perpendicular | pendicular to the longitudinal
to the longitudinal direction. direction of the needles. The latter

The largest elasticity-axis is | comeides with the shortest elas-
parallel to the longitudinal direc- | ticity-axis of the crystals.
tion of the needles or to the
shortest diagonal of the paralle-
lograms. |

(521 )

On account of the evaporation of the two substances when melting,
one is obliged always to use a covering glass under the crystallisation-
microscope.

On mixing the two isomers I have noticed the following on melting
and subsequently cooling the mixtures.

a. Mixture containing 61,6°/, of B-acid yields exclusively mived
crystals of the @-form; formation of minute traces of a-erystals is
not improbable.

4. Mixture containing 42 °/, of p-acid yields chiefly mixed crystals of
the a-form; at the edges of the fused mass, however, are found also
very small, slightly coloured parallelograms of the p-form. Negative
mixed erystals in the a-form (see d@) were not noticed; only positive
ones with 9 < v.

c. Mixture containing 35,5°/, of 8-acid behaves on solidification like 0.

d. Mixture containing 22,5°/, of B-acid, only yields mixed crystals
of the a-form both positive and negative doubly-refracting but with
o<v like the a«-acid itself.

e. Mixture containing 86°/, of a-acid only gives mixed crystals
of the a-type with strong dispersion @ < v and a positive double
refraction.

Dr. JAEGER comes to the following conclusion :

“There exists here an isodimorphous mixing series with hiatus. This
extends from a f-acid concentration > 22,5°/, to mixtures containing
61—62 °/, of the @compound. The mixed crystals of the a-type
become on addition of the negative B-compound less strongly positive
optically and in the immediate vicinity of the hiatus even negative ;
they, however still retain the strong dispersion with @ << v, which
is so characteristic for the pure a-compound.

On the other hand, the mixed crystals ef the p-type have at all
concentrations of 62—100°/, a negative double refraction and a
very feeble inclined dispersion.”

V. Mryer states in his treatise that in the oxidation of mixtures
of the two thiotolenes, he has obtained various other mixtures of
a- and 8-thiophenie acid, and that these showed no sign of separation
into their components when subjected to fractional crystallisation.

Dr. VorrMan, however, cannot confirm this observation. When he
recrystallised a mixture of 85.3°/, a-acid and 14.7°/, of p-acid (solidi-
fying point 120°.3) from hot water, the solidifying point increased
to 121 .6 which corresponds with a mixture of 89°/, «- and 11°/, of
B-acid. As, however, V. Meyer does not state the temperature at

(oa)

which he earried out his fractional crystallisations, it is possible that
this was not the same as in VoERMAN’s experiment and this might
account for the difference.

Dr. VorRMAN has finally been engaged in the determination of the
conductivity power of the two acids and their mixtures in the hope
of obtaining indications of a combination of the two acids when in
solution. The observations are as follows :

CONDUCTIVITY POWER OF a-THIOPHENIC ACID AT 25°,

v p a 100 &
25 | 32.44 | 0.085 | 0.0314
50 45.37 0.118 0.0319
100 - 62.49 0.163 0.0319
200 | 85.06 0.222 0.0318 Mop = 382.7
400 113.87 0.298 0.0315
800 149.44 0.390 0.0312
[1600 189.34 0.495 0.0303]

Average 0.0316

In this table, 7 represents the volume in which 1 mol. is dissolved,
w the molecular conductivity power, «@ the degree of dissociation,
100 4 the dissociation constant according to Ostwatp X 100.

The conductivity power has been determined, previously, by
Ostwatp (Ph. Ch. 38, 384) who found for 100% 0,0302, and by
Baber, (Ph. Ch. 6, 313) who found for 100% 0,0329.

CONDUCTIVITY POWER OF g-THIOPHENIC ACID AT 25°.

|
|

p | | x | 100 &
50 23.20 | 0.061 pusher 00783
100 32.32 0.084 | 0.00779
200 44.90 0.417 | 0.007795. - == seeen
400 62.06 0.162 0.00784
800 84.66 0.2914 (), 00785
[1600 144.17 0,298 0.00793]

Average 0.00783

( 523 )

CONDUCTIVITY POWER OF MIXTURES OF

a+ §@THIOPHENIC ACID AT

44 0/, ‘
thiophenic acid
89/% «

v pe g

50 | 13.96 | 0.413 |
100 Seb. 485 |
200 80.94 0.21
400 108.05 0.282
800 141.29 0.309 |
1600 180.75 0.572 |

33 .339/9
thiophenic acid

66 .66/p
7] b- a
33 .333 31.79 0.083
66.666 4h 28 0.416
123.333 | 60.86 0.159
266.666 82.33 0.215
533.333 409.83 | 0.287
1066. 666 AB 3 Ne 02375
50%/, 2
thiophenic acid
50%/, 4
D ia a
50 35.12 0.092
400 48 52 0.127
200 66.04 0.472
400 88.58 0.234
800 447.45 0.306
4600 ADA .7 0.396
70.4°/, 2
eniopnene acid
99.69), x
PY - a
100 42 O04 0.442 |
900 58.63 | 0.153
400 79:06 | 0.207 |
800 405.02 | 0.274
1600 198.90 | 0.361

25°.

400 &

(0). 0288
0.0286
0.0284
0.0278
0.0270
0.0264

100 &
0.0226
0.0227
0.0226
0.0221
0.0217
0.0210

100 &

0.0185
0.0184
0.0180
0.0174
0.0170
0.0163

100 &

0.0143
0.0139
0.0135
0.0130
0.0128

( 524 )

The conductivity of the pure ~-acid has been determined previously

by Loven (Ph. Ch. 19, 458) who found :
100 £ = 0,0078.
which agrees well with the value found by myself.

The influence of the position of the sulphur atom in regard to
the carboxyl group is very marked.

From these observations it appears that the acids in aequeous
solutions exert but very little influence on their mutual conductivity
power, as the conductivity power of the mixtures agrees fairly well
with the calculated result. A condensation of their molecules in such
a solution cannot therefore be supposed to take place.

Physics. — “A remark on the theory of the w-surface for binary
mixtures.” By Prof. J. D. VAN DER WaAats.

(Communicated in the meeting of December 29, 1906).

KAMERLINGH QOnNzs’ startling experiment, in which a gas was
obtained that sinks in a liquid, has drawn the attention more closely
to the direction of the tangent in the plaitpoint of a binary mixture.
Leaving the further particulars required for the realisation of sucha
mixture to the investigations of KaMERLINGH OnnEs and his collaborators,
I will make a remark of general significance, in close connection
with this experiment.

In my Théorie moléculaire and more fully in Cont. II I have
examined the condition, on which the tangent in the plaitpoint runs
parallel to the v-axis, or in other words () =o. The problems

related to this may be reduced to 3. All three refer to the inter-

dp dp
section of the two curves | — ]|=0O and {| — |= 0.
dv) .7 dz) yT

As first problem I should like to regard the principal one, namely

2 2
yw wy ‘
that where re 0 and ae 0. The point considered lies, there-
v v& ov
‘ : 07)
fore, on the spinodal curve, and at the same time the curve = ——
v
3
has for constant value of 2 two equal values for v and so also 8 =@,
v

Then the point considered is the critical point of the mixture
taken as homogeneous. The value of 7’ is that of 7), for such
a mixture and the value of x is then found, when the approximate
equation of state with % constant is applied, from :

which value ?7/, becomes =
volume is relinquished.

/,, when the independence of 4 of the

2 2
cuts the line —— — 0 still in two

Ox Ov Ov?

points. One point is that above mentioned, the second lies at smaller

x and larger v. So nearer the component with the smallest value of +.
With increase of temperature the two points of intersection draw

nearer to each other, and as second problem we may put: to exa-

mine the circumstances under which the two points of intersection

of these curves coincide. The three equations from which this cir-

Op : Ow
Ov ~=——d Ow Ov

which expresses that these curves touch, viz. :

dp \? dp d*y
(53 dz) dx? Ov v3

0p ches 0?p 0?p
dcdv ) Ox? du?

Above the temperature at which these circumstances are fulfilled,
O7y Ow
= (0 and
Ov? dxdv
cation in the course of the isobars, viz. that there is one that
intersects itself, has disappeared.

The third problem is more or less isolated, but yet I should like

In this ease the line

=0 and a third

cumstance is determined, are then:

=O do not intersect any longer, and the compli-

to treat it in this connection: viz. that for which the line eS 0
7
: Of Oty

-has a double point, and so at the same time Fe = Oand 5 aor 0.
v xzov

If there is a minimum 7’, for mixtures taken us homogeneous, such
a point is really a double point. If there should be a maximum 77,
it is an isolated point. We find then again »v = 7,, 7’= 7, and the
value of x is that for which 7). has a minimum or maximum value.
Let us call the three values of « obtained for those three problems
a; , #, and a,, then :

ay <a <a,
fe ee fe e i, :

( 526)

Now there are three more problems, and to this [ will call atten-
tion in this note, which may be considered as the analogues to the
three above-mentioned ones.

If in the above problems we substitute the quantity « for v and

d*y ee dy
vice versa, so that —— changes into ——,and=—— remains unchanged
Ov? Ow? Oxvdv
2 2
then the intersection of the curves ——- —0O and = 0 will give
0a? Ox0v

rise to three problems, which are of as much importance for the
theory of the binary mixtures as the three above-mentioned problems,

| | 3 O° ory
which relate to the intersection of —~— = 0 and ——— = 0.
Ov dvdv
*~
In the first place the points at which the two curves 5,2 Vand
ve

0*yp ,
nay intersect will belong to the spinodal curve, as appears

&OvV
F 07 0? 07w \?

‘om = :

Ox? Ov? dxdv

In the second place these points of intersection will have the same
dw
significance for the course of the curves (5") = g = constant, as
wv
WA és OB Oy ;
the points of intersection — ~ = 0Oand.=~-=—0O have for the course
Ov? adv
dw 5 tet
of the curves ei —= — p=constant. The first point of inter-
v r

section will be a double-point for the qg lines, whereas the other
point of intersection will present itself as an isolated point, the centre
of detached closed portions of the g lines.

In the third place there will be a limiting temperature for the

: w font mre
existence of the locus 3 = Q. With increasing value of 7’ this curve
v

i — 0 with

contracts to an isolated point, just as is the case with
v

maximum 7, or as the curve — —O has a double point with

Ov?

minimum 7%.

In the fourth place there is a temperature at which the curves
07yp d*y eh
> = 0 and = yn) =0( only touch, and the two points of intersection

have, accordingly, coincided.
And finally, and this is the most important case, there is a tem-

—

(527 )

perature at which the intersection of these curves takes place in

such a way that at one of the points of intersection a tangent may
2

} dv
be drawn to = (), for which — ().
Ox? dx
To determine these circumstances we have the three equations
07 02y O*yp :
au 0, =) and —- =O and this problem proves to be the
0x? dxdv 0a*
07 yp Oy
analogue of that mentioned above, for which iae =a), arn =0 and
] vav
0° dv da dv
= 0. If there was =o, now — => or — =
Ov? da dv da

i a: |
So if the 3 equations = — 0 and —-—0 admit of a
Ow? d2dv dx°

solution, the circumstances may be realised in which at the plaitpoint
a tangent may be drawn /) «z-axis. Neglecting the variability of 4
with v we find for the three equations:

b\?
Wea iva hs
mp MRT da) da

== ie De I aT
On? vl—e (v —b)? v ( )
db\?*
Ree | —
0*w MRT(1 — 2.) °F
a 4 =e enc
02? Head = fae (v—b)°
db da
DORE a
Op dix da ;
O.x0v ig (v—b)? v io (5)

If.we puta = A + 2Bue+ Co’ and b = b, + «v8 = b, + w (b, — 4,)

we get the equation:

Va ob — c F re) ea 4(B + Cw) (Cb,—BB)
(23245 2a 7° BS Ce Ca BO =e)

If B=a,,—a, should be small in comparison with a.--a,—2a,,)v=Cr,

h
we get « equal to */, by approximation, at least if — is aiso small.

Then real values are found both for x and for 7 and v; only
this value of 7’ can lie below the melting point in many cases, and
consequently it cannot be observed.

However, I shall not enter into a further discussion. I will only

point out, that for suitable values of 7’ the curve a
Uv

a closed curve, which contracts with increasing value of 7, and
may contract into a point.

= 0 represents

; dv ia: : :
In the problem, for which — =o at the plaitpoint, this case is
at

dv . ate :
the transition for the cases where ote positive or negative. In the
Xe

dv dy: Ne
same way in the problem for which — = 0 in the plaitpoint, this
Ak

dv = .
is a transition case between — positive or negative. So the cases
Ae

may also exist for which on the side of the small volumes, the
2 00: ee
quantity aa the plaitpoint may have reversed sign.

Ax
d°y d*y
= 0 and — = 0,
dx0v Ou?
it appears that it is required for the realisation of the case, that when

db As da da a
— is positive, also — and —~ are positive, and that the calculated
aL ak Ak

temperature must lie above 7). of the first component when we want
to apply the result to the coexistence of gas and liquid phases.

At the top we have the limiting case of two coexisting phases.
If the tangent is // w-axis, the molecular volume is equal and the
density will be proportional to m,(1—«) + m,a.

Put

_m, (1—2) + mex ad m, (1 = eee __(m,—m,) (#'—a)

— and d = nd d'—d—=——____—_—_
v v v

When we examine the shape of the curves

When (m,—m,) and (v’— x) have the same sign, d’—d is positive.
As «’—wv is negative when the first component has the smallest size
of molecule, m,—m, must also be negative, which is satisfied for
helium and hydrogen.

We can, in general, represent the limiting density of a substance

m : .
by a and then the law would hold: When the most volatile substance
}
has the greatest limiting density, the gas phase can be specifically
heavier than the liquid phase. For Helium the limiting density is
probably equal to that of the heavy metals. From the supposition
that it is formed by splitting off from heavy metals this follows
already with a certain degree of probability.

1) On further investigation it has appeared to me that a point that satisfies the

07yp 07 O*~p

sae Se one 0, and ar = 0, possesses the analytical character of

a plaitpoint, but at least in many cases, does not behave practically as such. I
hope to show this before long. (Added in the English translation).

equations

( 529 )

Mathematics. — “The rule of Neper in the four dimensional space.”
By Dr. W. A. Wytsorr). (Communicated by Prof. P. H.
SCHOUTE.

(Communicated in the meeting of December 29, 1906).
1. The wellknown ‘rule of Neper”
as follows:

If we regard as elements of a spherical triangle A, A, A,, rectan-
gular in A,, the two oblique angles A, and A, the hypothenuse a,
and the complements of the two other sides 4a — a, and } a—a,’)
we can apply to each formula generally holding for the rectangular
spherical triangle the cyclic transformation

can in principle be formulated

(A, 4+@7—a,, a, 42—a,, A,)

without its ceasing to hold.

Fig. 1.

We prove this rule by prolonging the sides A, A, and A, A, which
(Fig. 1) for convenience’sake we shall imagine as < } 2, through the
vertex A, = A’, with segments A’, A’, and A’, A’, so that A, 4’, =
A, A’, =i. The spherical triangle A’, A’, A’, then proves to be
again rectangular, namely in A’,, whilst furthermore between the
elements of both spherical triangles the following relations prove to
exist :

tele
a
Co
o
Pay
~

2
ON eee: Fe

From this is evident that the above mentioned cyclic transformation
can be applied to the elements of each rectangular spherical triangle
without their ceasing to be the elements of a possible rectangular

Al i '
i = SE es G Oks ine —— Oe x 6
!

+ — a, =A)

1) These are the complements of what Neper himself calls the “quinque
circulares partes’? of the rectangular spherical triangle. See N. L. W. A. GRAvELAAR,
Joun Napier’s werken, Verh. K. A. v. W., First section, vol. VI, N°., 6, page 49.

Proceedings Royal Acad. Amsterdam, Vol. LX,

(530 ) -

spherical triangle, from which further the rule of Neprer immediately
follows.

The train of thoughts followed here will be found back entirely
in the following.

2. A hyperspherical tetrahedron I shall call doublerectangular, if
two opposite edges stand each normal to one of the faces.

Let us suppose .the letters A,, A,, A, and A, at the vertices of
the tetahedron in such a manner that A, A, is perpendicular to the
face A, A, A, and A, A, perpendicular to the face A, A, A.

To make the tetrahedron doublerectangular it is necessary and
sufficient for the angles of position on the edges A, A,, 4, A, and
AAS 40 -be right: *)

0, = 24 =f S—eas
from which then ensues:
A, = A= ee
A

Ang = G&,°

Asn53

21 —

If we do not count the rectangular elements and if we count
those which are equal only once the doublerectangular hy perspherical
tetrahedron has 15 elements, namely @,,, G15; G41 Go3> a4 349 ras
Woes ens: As oy: Ay Se yea gee ee ee

12° 13)

3. We now form, starting from a doublerectangular hyperspherieal
tetrahedron A,A,A,A, of which we think the edges all <42, a second
hyperspherical tetrahedron (Fig. 2) by prolonging the edges meeting
in A, = A’, through this vertex, namely the edge A, A, with a seg-
ment A’, A’,, the edge A, A, with A’, A’, and the edge A, A, with
i A'

By very simple geometrical considerations we find that the tetra-
hedron A’, A’, A’, A’, is again doublerectangular, that namely 4’, 4’,
is perpendicular to A’, A’, A’, and A’, A’, perpendicular to 4’,4’, 4’ ;
furthermore it is evident that the following relations exist between

= ; Mie po Ali tg eee
4? sO that Jae A’, == batt A. —— vale AN —— 4 nN.

1) The signs used here | have derived from Prof. Dr. P. H. Scuoure, Mehr-
dimensionale Geometrie, 1st vol., page 267, Sammlung Scuupert XXXV, Leipzig,
G. J. Géscuen, 1902.

So I understand

by Gy, the edge A, Ap;

by «2 the angle of position formed by the faces lying opposite the vertices
A, and Ag, i.e. the angle of position on the edge A, A,;

by Aje the facial angle having A, as vertex and lying in the face opposite Ag,
i.e. the angle As A, Ay.

(531 )

Fig. 2.

the elements of the two tetrahedra

1 ee
gUW—4,, = 4,
! ——
G44 — 47 —4,, »
1
=U — Ay, = As, ;
' —-
G34 = &o53 3
! —
@o3 — 41.
' os °
GREG Be
' —— ! ee
ee Ay e (fa—A,,)+4,=47 ;
' ee |
ah pe 4 e,
'
er ee a ae ee
'
a te a hoe
(eis eee . ! ) —
AoA » 1. €. Se eee a — IE ’
y \ Se 1 : L ! 1 —— 2 A
A, + Ay, =} 2"), 1. €. (¢272—A,,) + a7—A,)=432 ;
! — ]
a Se Gy 5s
a _— gin.
$a —a,,=—A,, ,
! oe
Ag adit

4. So if we regard instead of a,,, a,,, A,,, 4,, and a,, their
complements as elements of the tetrahedron, then the elements of the

1) In giving the proof of this we must remember that A', A's and A, A; lie on
a sphere and therefore cut each other in a point P, just as A’, A’, and A, A; cut
each other in a point Q.
o6*

( 532 )

doublerectangular hyperspherical tetrahedron can be arranged in
three cycles, two of 6 and one of 3 elements, namely

1. (BO — ayy s Og) $ — Myo Begs Gar Gs):

2. (3 JS ro 22 ’ Ad > A139 Tas AG ; 4 Jt. — A,,),

3. (A,, 3 u— Os3 1 A,;);
so that it is possible to allow the elements of each cycle to undergo
all simultaneously a cyclic transformation, if only afterwards those
of the second cycle are replaced by their complements,*) without
these elements ceasing to be the elements of a possible doublerect-
angular hyperspherical tetrahedron.

These same simultaneous transformations may thus be applied to

each formula holding in general for elements of the doublerectan-
gular hyperspherical tetrahedron.

5. If we again apply the construction described in § 1 to the
newly formed spherical triangle, ete. we find a closed range of five
spherical triangles of which the hypothenusae form a_ spherical
pentagon.

The sides of these five spherical triangles are parts of five great
circles on the sphere, namely the circles part of which is formed by
the three sides of the original spherical triangle and the two polar
circles of the vertices of its oblique angles. These five great circles
form, however, another second similar range of five spherical triangles,
namely that of the opposite triangles of the former range.

6. We can likewise deduce in a manner indicated in § 3 out of
a doublerectangular hyperspherical tetrahedron a range of such
tetrahedra of which the faces all belong to six spheres, namely the
spheres part of which is formed by the faces of the original tetra-
hedron and the polar spheres of the points A, and A,.

Let us call B, the polar sphere of A,, 6, that of A,, 5, the
sphere A, A, A,, B, the sphere A, A, A,, B, the sphere A, A, A,
and B, the sphere A, A, A,.

Each of these spheres divides the hypersphere into two halves of
which I shall designate the one to which the original tetrahedron —

1) If we write the second ‘cycle

L ee 1 a=, 2
(2% Ai,, 3m A,,

1 ,, 4% —a,,, Ay A,,)
or
(A
then no replacement of the elements by their complements is necessary, but the
cycle has lost its symmetry with respect to the tetrahedron.

/ er ea 1 / <=
14? Ay,» 4m Gigs Cea tx—A,,, a A,,);

(533 )

belongs by +, the other by —. The following list then indicates on
which side of each of the six spheres the successive tetrahedra I,
II, etc. are situated and by which they are limited. For the non-
limiting spheres the sign has been placed in brackets.

B, | By | Be 19 B, B,
eee yo) +f te | ele +
eee. ey = | |
ashe eG) |B]
Bere (are 1 er) | =
Vv uf zy = + Coy ees
NE ee) z a3 + ele
A pat ed eg = = = —
FT Sh) Ne Oy Ss ea es a ee a
Pera eee Pe | =|
Sha | eS | | OO +t
eee seem). ef co
ee ere | |
eG ee tl | +

It is clear that the tetrahedra I and VII are opposite to each
other, likewise II and VIII, III and IX, ete., whilst the tetrahedron
I again follows tetrahedron XII.

Thus the whole range consists of 12 tetrahedra which are two by
two opposite to each other, in contrast to what we found in the three-
dimensional space, where fvo ranges of spherical triangles are formed
_ of which one contains the triangles opposite to those of the other.

7. Between the volumes of each pair of tetrahedra belonging to
the range exists a simple relation.
If we call V7 the volume of the first tetrahedron then the relation:
dVyj= 3a,, da,, + 44a,,da,, + 44,, da,,

holds for each variation of the tetrahedron remaining doublerectan-
gular (thus @,,,@,, and a@,, not changing).
Likewise

dV — 3 (32 = a, 4) da,, == 3 ($7 =< As) da,, =; (4x <5 at; 5) da,, -

( 534 )

So therefore
Vit j= ina, + i 2a,, —}4,, (4a — a,,) + constant.
The constant is found by putting @,, equal to @,, = @,, = 4, = 2%,
in which case V; takes up the sixteenth part of the whole hyper-
sphere, i.e. | 2?, whilst V7z becomes = 0.

The constant then proves to be —- } 2’, hence
ie i ee AO, eo, = 4 a,, (4% — @y,)-
Likewise we find.
7; : ; : F
Vil ae Vin = 8 n* = i 7 (37 — ,,) + ; os a (3% zs a, 4) ($2 = a5),
Vin + Viy=-;2 +4 %4,, +120 ($n —a,,) — $4,, (3% —- a,,), ete.
Every time the sum of the volumes of two successive tetrahedra
can be expressed by means of four successive elements of the first
cycle mentioned in § 4. We deduce easily from this:

whe

Vy — Vyi1 = $ 44485, — $4,, (9% -— Oy 5)s
whilst in like manner we can find Vy — Viy, Viz— Vy, ete.
Further we find
Vi + Viv = 44,,4,, — 44;, (4% — @,,) —
and in like manner Vz + J'y, ete.

If we remember that the tetrahedra I and VII are alike with
respect to their elements and volumes, I and VIII also, ete. and that
with respect to the volumes we have to deal with only a closed
range of six terms we see that of each arbitrary pair always either
the sum or the difference of the volumes can be expressed in a
simple manner.

rhe

a1, (420 = ats)

Mathematics. — “Vhe locus of the cusps of a threefold infinite
linear system of plane cubics with six basepoints.” By Prof.
P. H. ScHOUTE.

In the generally known representation of a cubic -surface S* on
a plane @ to the plane sections of S* correspond the cubies through
six points in @; here to the parabolic curve s’* of S* answers the
locus C™? of the cusps of the linear system of those cubies. The
principal aim of this short study is to deduce from wellknown
properties of s'* properties of c'? and reversely.

1. If a plane rotates around a right line / of S* the points of
intersection of that line / with the completing conic describe on /
an involution, the double points of which are called the asymptotic
points or /. According to the condition of reality of these asymptotic

(535 )

points the 27 right lines of S*, supposed to be real, are to be
divided into two groups, into a group of 12 lines with imaginary
asymptotic points, the lines

2 3 4 5 6
ie a he B.D, SB,
of a doublesix and into a group of 15 lines ¢,,,¢,,,...,c¢,, with

real asymptotic .points. If to the six basepoints A’; of the linear
system of the cubics the six lines a correspond and this case
we shall in the following continually have in view — then to the

six lines 6; correspond the six conics } through all the
basepoints except A’; and to the fifteen lines cz correspond the

5 Z : 2 ~ z 2
connecting lines cz, —=(A’;, A’,), whilst to the systems of conics (a;)

in planes through 4a;, (bj) in planes through 6;, (cy) in planes
through cy correspond successively the pencils of thg curves of the
linear system with A’; as doublepoint, the lines (4';) through A’;

and the conies (cz) through the four basepoints differing from
A; and A;,. The situation of the six points A’; is then such that
each of the fifteen lines cc’, is touched in real points by two conics

of the pencils (cz), whilst on the other hand the points of contact
of the tangents out of the points A’; to the conics b; are imaginary

. . . . . 2 . .
so that each point A’; lies within the conic 6; with the same index.

2. As a matter of fact all real points of a line / of S* are hyper-
bolic points of this surface with the exception of the two asymptotic
points of this line showing a parabolic character; whilst each of
these asymptotic points is point of contact of 7 with a conic lying
on S*, / touches in both points the parabolic curve S**. If we apply
this to each of the six lines a;, imaged in the points A’; and if
we consider that to a definite point P of a; corresponds the point
P’ lying infinitely close to A’; connected with A’; by a line of
detinite direction (Versl., vol. I, pag. 143) we find immediately :

“The six basepoints A’; of the linear system are fourfold points
of the curve c™ of a particular character, consisting of the combi-
nation of two real cusps with conjugate imaginary cuspidal tangents,
the cuspidal tangents of the curves out of the system with
a cusp in A’;”.

The twelve points of intersection of the line cz. with ct? consist
of the isolated points A';, A’, counting four times and the real points

( 586 )

of contact with two conics out of the pencil (cy) counting two
times. Likewise do the 24 points of intersection of the conic

b; with c'? consist of the five basepoints differing from A’; counting
four times and the imaginary points of contact with the tangents
through A’; counting two times.

3. From the investigations of F. Kiem and H. G. Zevtuen dating
from 1873 and 1875 it has become evident that the surface S* with
27 real right lines has ten openings and the parabolic curve s*‘? has
ten oval branches. In connection with this we find:

“The locus c'? has ten oval branches.”

We ask which situation of the six basepoints A’; corresponds to
the particular case of the ‘“‘surface of diagonals” of CLEBscH, in which
the ten oval branches of the curve s'* have contracted to isolated
points. In this case the fifteen lines with real asymptotic points, i.e.
in our case the lines cj, pass ten times three by three through a
point; this is satisfied by the six points consisting of the five vertices
of a regular pentagon and the centre of the circumscribed circle.

Fig. 1.

What is more, each six points having the indicated situation can be
brought by central projection to this more regular shape. The ten
meeting-points of the triplets, of lines then form the vertices of two
regular pentagons (fig. 1). The curve c' corresponding to these six
basepoints then consists of merely isolated points, namely of fourfold

( 587 )

points. in the six basepoints and twofold points in the ten meeting
points of the triplets of lines.

The remark that the curve c'? belonging to the six basepoints of
fig. 1 has the line c’,, as axis of symmetry and transforms itself
into itself when rotating 72° around A’,, enables us to deduce in a
simple way its equation with respect to a rectangular system of coor-
dinates with A’, as origin and c’,, as x-axis. The forms which pass
into themselves by the indicated rotation are

= 27 + 77, P= a’ — 102*y? + day*, Q= 5aty — 10a7y? + y'.

If we pay attention to the axis of symmetry and to the identity
P? + Q? =v" the indicated equation can be written in the form
o* + ap* + bo* + co” + do” + P(e + fo" + g9* + ho*) + P*(t+ko*) = 0,
so that we have to determine only the ten coefficients a,b,..,k. If
now the common distance of the points A’,, A’,,.., A’; to A’, is

unity, then
3—e\" 8 +)
= (« — 5 j («—1)* (« — = ‘) =O,

where e stands for 5, represents the twelve points of intersection
of the curve with the z-axis. By performing the multiplication this
passes into
w* (#® + 227 — 72° — 6a§ + 2024 — 6e* — 727 +22 41)=— 0.
From this follows
a= i, b= -20, -¢-- t= — 7, d+k=—1,
= Wee: f= =, g= — 6, h =22:

So the equation
@ — 19° +209* — 79-9" +-2P (I — 39" — 3e*+0') — Q? (i +407) = 0
is determined, with the exception of the coefficients 7 and /: still
unknown. Now the parallel displacement of the system of coordinates
to A’, as origin furnishes a new equation, of which the form
(4— 1—h) y? + 2 (12 —4i—dk)awy? + a4 + (54 —281—45h)a7y? 4 (54-414 8h)y'
represents, after multiplication by 25, the terms of a lower order
than five. The new origin being a fourfold point of c'? and the
terms with y* and ay? having thus to vanish, we find

: i=8 , k=—4,
on account 6f which the indicated form passes into
(a? + 5y7)?.

The correctness of this result is evident from the following. Just
as the two tangents in the old origin counting two times are represented
by 2* + y* =0,and therefore coincide with the tangents out of 4’,
to the conic through the other basepoints, so x? + 57? = 0 represents

( 538 )

for the new origin A’, the pair of tangents out of A’, to the conic through
the other basepoints. Or, if one likes, just as 7? + 7? is with the exception
of a numerical factor, the fourth transformation (‘‘Ueberschiebung’’) of the
first member of the equation Q—O of the lines connecting A’, to
the remaining basepoints, so #* + 5y° represents, likewise with the
exception of a numerical factor, the fourth transformation of the first
member of the equation ee = 0, which indicates with respect to the
new origin A’, the five lines connecting A’, to the remaining base-
points.

Finally the equation of c’*? is
o*(o°—7e° + 2009*— 79° + 1) + 2P(9*—39* —30* + 1) +4Q7(97—2)=0, (1)
or entirely in polar coordinates (@, g)
(o"-2)g%cosp=(e? + Ve"49" + )H(@-'V(@"-de"+ Ne"). Q)

It is easy to show that this curve admits of no real points differing
from the six basepoints A’; and tbe ten points of intersection of the
triplets of connecting lines. If tor brevity we write (2) in the form

Lcstig=M+ YN,

then we tind
—Dsintbp=(M+N—L)+2MYN .. . (3)
and ;
M? +N — L? = 2 (9?—1)? (20? —1)(0* —60° + 1404+ 20’?—1)
(M?+ N—L?*)? ae 4M? N == 40° (e?—1)’ (v?-— 2)? (o*—7o?+1)’
If now we moreover notice that N is negative and therefore

(4)

1
cos 5g complex when go? lies between — and 1, the following is

immediately evident :
a. The first member of the second equation (4) tends to zero,

when 0? assumes one of the values 0, 1, 2, = (7 + 382); it is positive

for all other values of 9?.
6. If VN is real and g? differs from unity the second member
of the first equation (4) is positive; for the equation
0° — 60° + 140% + 207 —1=0

Rela 1
has, as is evident when the roots 9? are diminished by 1 —, besides

ae 1
one real negative root only one real positive one between — and 1.
vo

c. If @° differs from 0, 1, 2, ey (7 + 3e) the second member of

( 539 )

(3) is positive when A is positive, and therefore g is imaginary.
d. Neither does 9? = 2 give a real value for g; for substitution

o
in (1) furnishes for cos 5g the result rua

e. So we find only the real points :

o—, wantemie, -~.. . . A’,,

eee eee ee. rs A, Aly A’,, A’, A’,

oa cosdp—=—1. . . the ten points of intersection

of the fifteen connecting lines three by three.

4. We now consider a second case, in which the position of the
six basepoints is likewise a very particular one, where namely these
points form the vertices of a complete quadilateral. Through these
six points not one genuine cubic with a cusp passes. For the three
pairs of opposite vertices (4,, A,), (B,, B,), (C,, C,) of a complete
quadilateral (fig. 2) form on each curve of order three, containing

them, three pairs of conjugate points of the same system, and these
do not occur on the cubic with a cusp, because through each point
of such a curve only one tangent touching the curve elsewhere
can be drawn. In this special case the locus of the cusps has broken

(540)

up into the four sides of the quadrilateral each of those lines counted
three times. For it is-clear that an arbitrary point of the line A,B,C,
e.g., as a point of contact of this line with a conic passing through
A,, B,, C,, represents a cusp of the linear system of cubics. We can
even expect that each of the four sides must be taken into account
more than one time, because each of those points instead of being an
ordinary cusp is a point, where two continuing branches touch each
other. And finally the remark that the sides of the quadrilateral
divide the plane into four triangles e with elliptic and three quadran-
gles h with four hyperbolic points, so that they continue to form
the separation between those two domains, forces us to bring them
an odd number of times into account, namely three times because
we must arrive at a compound curve c’.

Some more particulars with respect to the domains e and h. The
nodal tangents of the cubic (fig. 3) passing through the three pairs
of points (A,, A,), (B,, B), (C,, C,) and having in P a node, are
the double rays of the involution of the pairs of lines connecting

‘Be

Fig. 3.

P with the three pairs of points mentioned, so also the tangents in
P to the two conics of the. tangential pencil with the sides of the
quadrilateral as basetangents, passing through P?; now, as these two
conics are real or conjugate imaginary according to P lying in one

—*.~

( 541 )

of the three quadrangles / or in one of the four triangles e, what
was assumed follows immediately.

To the case treated here of c*? broken up into four lines to be
counted three times corresponds the parabolic curve of the surface
S* with four nodes.

5. In the third place we consider still the special case of six
basepoints lying on a conic, in which the linear system of cubies
contains a net of curves degenerating into a conic and a right line;
in this net of degenerated curves the conic is ever and again the
conic c’ through the six basepoints and the right line is an arbitrary
right line of the plane.

This case can in a simple way be connected with a surface
S* with a node V. If we project this surface out of this node
O on a plane e@ not passing through this point, then the plane
sections of the surface project as cubics passing through the six
points of intersection of @ with the lines of the surface passing
through 0; because these six lines lie on a quadratic cone, the six
points of intersection with a@ lie on a conic. Besides, the sections
with planes through O project as right lines; therefore the completing
conic c*? must evidently be regarded as the image of the node 0.
Of course we must here again think that c’ corresponds point for
point to the points of UO? lying at infinite short distance from O*;
for c? is the section of @ with the cone of the tangents to S* in 0.

As c? with one of its tangents represents a curve of the linear
system, this conic belongs at least twice to the locus of the cusps.
Here too this locus of cusps improper with continuing branches must
be accounted for three times, so that the locus proper is a curve
ce’ of order six, touching c’* in the six basepoints.

Let us suppose that c’ is a circle and that the six basepoints on
that circle (fig. 4) form the vertices of a regular hexagon, then the
curve c* has the shape of a rosette with six leaves having the centre
O' of the circle and the points at infinite distance of the diameters
A,A,, A,A,, A,A, as isolated points. Of the ten ovals there are four
contracted to points, whilst the six remaining ones have joined into
the circle of the basepoints and the curve c’.

If we take point O' as origin and the line O'A, as z-axis of a
rectangular system of coordinates, then if O'A, is unity of length
we find for the equation of c°

4y? (y? — 3a)? + 9 (x? + y*)? — 9 (a? + y?) = 0.

It is evident from this equation that the curve c® can really stand

Fig. 4.

rotation of multiples of 60° round OQ’, for then x? + y? and y(y? — 32?
are transformed into themselves.
Out of the equation

3
sin 3@ = = ae Yi —?r

on polar coordinates it is evident that the curve c* (with the excep-
tion of its four isolated points) is included between the circles de-

1
scribed out of O' with the radii 1 and =e oe

If we pass from the locus of the cusps to the parabolic curve of
S* we must notice that the last curve has the node O of S* as
threefold point, because c? has separated itself three times from the
locus c’*, So this parabolic curve is an s* of order nine, a result
which will presently be arrived at in an other way.

We shall give — without wishing in the least to exhaust this
case of the six basepoints situated on a conie — some degenerations

of the remaining curve c* corresponding to some definite coincidences
of the basepoints.

a) The cases (2,2,2), (4,2), (6). If the six basepoints coincide
two by two in three points of the conic, then c* consists of the sides
of the triangle of the basepoints counted double, originating from
compound cubics with ‘a double line; there is not a locus proper.
In reality the case (2, 2,2) of a conie touching in three points cannot
occur for a genuine cubic with a cusp.

( 543 )

The cases (4,2) and (6) are to be regarded as included in the
preceding. By allowing two of the vertices or the three vertices of
the triangle just considered to coincide we find for case (4, 2) the
connecting line of the two basepoints counted four times and the
tangent to the conic in the basepoint of highest multiplicity counted
two times, for case (6) the tangent to the conic in the point counting
for six basepoints counted six times. That there can be no locus proper in
the last case ensues also from the fact, that a genuine cubic with
cusp allows of no sextactic point.

b) The case (3, 3). If the six basepoints coincide three by three in
two points of the conic, then c* consists of a part improper, the
connecting line of the two points counted four times, and a part
proper, a conic touching the conic of the basepoints in these points.
The new conic lies owtszde the conic of the basepoints.

c) The case (1,5). This case agrees in many respects with the
preceding. We find a part improper, the tangent in the point counting
for five basepoints drawn to the conic of the basepoints, and a part
proper, a conic touching the conic of the basepoints in these points.
The new conic lies mside the conic of the basepoints.

6. Of course it is possible to call forth by the curve c'? succes-
sively all the different special cases which can put in an appearance
by the parabolic curve s’* of the various surfaces S*. As this would
lead us here too far, we limit ourselves to a single remark, which
can eventually facilitate an analytic investigation of this idea.

- According to the general results with respect to a linear system of
curves c” obtained as early as 1879 by E. Caporanr the locus
c#@n—3) of the cusps of this system has in each r-fold basepoint of
the system a 4(27—1) fold point and besides 6(n—1)?—2>(3r?— 2r-+1)
nodes C. Each of those points C' is characterized by the property
that each curve of the system passing through this point is touched
in this point by a definite line c.

For the case under observation, 7 = 3 of the cubies, the number
of points C is represented by 24—6p, when p is the number of
basepoints.

If we wish to investigate analytically what peculiarity the locus
of the cusps shows in a basepoint of the system, or how a line

through three basepoints separates from it, then the result — and
this is the remark indicated — will be independent of the fact,

whether the remaining basepoints occur or not, if in the former case,
that some of these basepoints appear in a real or in an imaginary
condition, we assume that these points both with respect to each

( 544 )

other and to the former basepoints have not a particular position.

With the aid of this remark we can easily find the following
theorems, with which we conclude:

“Both cusps of which the fourfold point of the curve c,, coin-
ciding with a basepoint A; seems to consist and the two cusps of
the curves of the system showing in this point a cusp, coincide in
cuspidal tangents, but they turn their points to opposite sides.”

“If the three basepoints A’,, A’,, A’, lie on a right line /, the locus
proper of the cusps reduces itself to a enrve c’ touching the line /
in A’,, A’, A’; If the three remaining basepoints exist then the
points of intersection of / with the sides of the triangle having those
basepoints as vertices are points of c*”.

The last case answers to that of a surface S* with a double point;
the parabolic curve having in this doublepoint a threefold point,
because / separates itself three times from c**, is as has been found
above already a twisted curve of order nine.

Physics. — “An investigation of some ultra-red metallic spectra.”
By W. J. H. Moti. (Communicated by Prof. W. H. Juuivs).

(Communicated in the meeting of December 29, 1906).

Among the spectra of known elements those of the alkali-metals,
by their relatively simple structure, lend themselves particularly well
to an investigation of their ultra-red parts. Many observers have
consequently sought for emission lines of these metals in this region.

For the first part of the ultra-red spectrum the photographie plate
may be sensitised; especially LeamMann*) measured in this way
various lines with wave-lengths ranging to almost 1m. By means
of the bolometer Syow *) could advance to 1.5m.

For the further region, however, nothing was known about these
spectra. CoBLeNtz*), to be sure, was led by a series of observations
in this respect, to the conclusion that the alkali-metals emit no
specific radiation beyond 1.54, but I had reason to doubt the
validity of this conclusion.

In what follows I will briefly describe the method by which some
ultra-red spectra were investigated, and the lines thus found. In an

1) H. Lenwann. D.’s Ann. 5, 633, 1901.

2) B. W. Snow. W.’s Ann. 47, 208, 1892.

3) W. W. Costentz. Investigations of Infra-red Spectra. Carnegie Inst. Washing-
ton. 1905.

( 545 )

academical thesis, which will soon be published, further details
will be given.

For the investigation of the alkalies, the metallic salts were volatilised
in the are in the ordinary way. The very complicated band-spectrum,
emitted by the are when no metallic vapour is present, extends far
into the ultra-red. But this interferes in no way with the investi-
gation of the metals, since it is entirely superseded when the are con-
tains a sufficient quantity of metal. On the other hand the continuous
spectrum, emitted by the incandescent particles in the are, makes it
somewhat difficult to observe some feebler lines: besides, the radiation
of carbonic acid, the product of combustion of the carbons, (with a
maximum near 4,444) persists with almost unchanged intensity.

The image of the are is projected by a concave mirror on the
slit of a reflecting-spectrometer; the rays are analysed by a rock-
salt prism and part of the so formed spectrum falls on a linear
thermopile. This thermopile, like that of Rusens, is built up of iron
and constantan: all the dimensions were chosen smaller than in the
original pattern and a great sensitiveness was obtained. As well the
emitting slit as the thermopile are mounted in fixed positions; in
order to throw on this latter different parts of the spectrum in
succession, the prism can be rotated through small angles. A
WapwortH combination of prism and plane mirror maintains minimum-
deviation during rotation.

In chosing and designing the instruments, the desirability,was kept
in mind of replacing the very tiring reading of the galvanometer
and the simultaneous noting of the corresponding position of the
prism, by an automatical recording-device. [ had in mind _ the
splendid arrangement by which Lanenry has for years recorded the
intensity-curve of the ultra-red solar spectrum on a photographic
plate. That this method has not been followed for recording heat-
spectra instead of the time-absorbing visual observations, must be
ascribed in the first place to a very complicated mechanism being
required for obtaining complete correspondence between the linear
displacement of the photographic plate and the rotation of the spectro-
meter, and secondly to the difficulty of keeping the surrounding
temperature perfectly equal during the observations.

With very simple means I devised a method of recording, which
avoids these two difficulties, while yet it warrants a sure “corres-
pondence”, and yields accurate results also when changes in the
surrounding temperature cannot be prevented. For this purpose the
continuous recording has been replaced by the marking of a series

Proceedings Royal Acad. Amsterdam. Vol. IX.

(546)

of dots, while for the continuous rotation of the spectrometer an
intermittent one has been substituted. In this way for any recorded
radiation-intensity the corresponding position of the prism can be found,
not by measuring abscissae, but by counting dots. Since moreover not
only the deflections of the galvanometer but each time also the zero-
positions are recorded, it is possible to determine on the spectograms
the radiation-intensities also when during the observations the surround-
ing temperature, and consequently the zero-position, was variable.

The principal advantages of this method of observation over the
usual one are:

1. the absolute reliability of the observations,

2. the very short time required for a set of observations,

3. the accuracy with which interpolation is possible when the
zero-position shifts,

4. the non-existence of disturbances, caused by the proximity of
the observer,

5. the complete comparability of the different observations,

6. the possibility of estimating the probable error from the shape
of the zero-line.

The short time in which a set of observations is made, is of
importance when e.g. heat-sources are investigated which, like the
arc, show slow changes in radiation-intensity. A spectrum, ranging
from 0,7 to 6u was recorded with 200 displacements of the spectro-
meter in two hours.

In the spectrograms a spectral line is represented by 5 to 6 dots.
With one displacement of the spectrometer namely the line is
shifted over a distance amounting to */, of the breadth of the image
of the slit, or of the equal breadth of the thermopile. Hence the
same kind of radiation will strike the thermopile during five successive
displacements. From the mutual position of the dots, the place where
the radiation-intensity has its maximum may be accurately determined.
In order to derive from this the place occupied by the line in the
spectrum, it is sufficient to know one fixed point in the spectrum.
This fixed point was as a rule taken from a comparison spectrum,
for which the carbonic acid emission of a Bunsen flame was chosen,
the maximum of which, according to very accurate measurements of
Pascuen, lies at 4.403. Part of the flame spectrum was for this
purpose recorded simultaneously with the spectrum to be studied.

A simple caleulation then gives the refractive index for the
unknown ray. In order to derive from this the wave-length of the
line, a dispersion formula must be used. I became aware that the

—_——--. - —es

—

( 547 )

well-known dispersion curves of Laneiry and of Rvcprns show
considerable differences, and although at first sight LaNnGLry’s deter-
minations seem to be much preferable, yet on closer examination |
their excellence must be doubted, especially for the longer wave-
lengths. To prefer one of the dispersion curves to the other seems
to be at present a matter of arbitrary choice. So I have given
in the tables besides the observed refractive indices, the wave-
lengths, calculated from them as well by Laneiry’s as by Rusens’
formula. The refractive indices hold good for a temperature of 20°;
their determination is based on the index 1.54429 for the D-line,
a value, derived from very accurate determinations by LANGLEY.

The tables given below contain the lines of Na, K, Rb and Cs
(I have been unable to obtain reliable results with Li in the arc)
and of Hg. The results were derived from a large number of, spec-
trograms (10 to 12 for each metal). For the investigation of the
mercury spectrum a mercury are-lamp was devised, furnished with
a rock-salt window. The spectrum of mercury has been repeatedly
investigated as far as 10”; no measurable emission has been found
beyond 1.7m.

In the tables the first column gives the refractive index n of
rock-salt, the second and third the wave-length mw of the line,
according to the formulae of Lanciey and Rvsens, and the fourth
the approximate value / of the intensity.

For the lines of which the exact position was difficult to ascer-
tain, the refractive index is only given in four decimals.

SODIUM. POTASSIUM.
a |«(Langley)| 2 eee) | oe | n (Langley) + (Rubens) | Bea)
a: |
| 1.53529 | 0.819 o.sie | a0! | 4.53654 | 0.771 | 0.768 | 620
53062 | 1.44 4.43 [480] | 5025 | 0.97 | 0.96 | 10)
2 es ae te | 4 | | 5810 | 4.44 | 1.40 | 20
5286 1.44 a a 53030 | 1.48 4.17 | 320 |
5981 1.57 1.54 5 | | 59972! 1.95 | 1.2% | 200]
59711 | 1.85 1.80 | 05 | | 59803 | 1.53 1.50 | 95 |
| 59613 | 2.91 2.16 | | | bee | 2.28 | 2.48 5
| 59589 | 2.34 9.95 | 35 | 586 | 2.76 | 2.70 | 20
| 150455 | 2.90 2.84 20 | 52401 | 3.44 | 3.08 | 20
| 5234 | 3.42 | 3.36 5 | | 52963 | 3.73 3.67 | 5
4.52178 | 4.06 | 4.00 10 | 1 5284 | hos | 3.98 | 40 |
= 4 ee eee ee

RUBIDIUM. CAESIUM.
Wen. a at ae > 2 I | -
| n oan) (Rubens), J | | u v(Langley)|# (Rubens)) J |
| 4.53733 | 0.744 0.742 | 412 | | 4.53566 | 0.803 0.801 40
| 53624 | 0.782 0.779 | 450| | .53454 0.855 0.851 | 950
| 5859 0.795 0.792 | 300! | .53375 0.895 0.891 | 200
| 5332 | 0.93 0.92 10 5333 0.920 0.94 | 75
| 53909 | 1.01 1.00 | 35 53202 | 1.04 1.00 | 90
| 15309 | 444 4.40 10 52902 | 1.37 4.35 70
59912 | 1.35 | 1.33 | 200 59846 | 41.48 4.45 80
59830 | 41.49 4.54 180| | .5975 4.74 4.70 5
59597 | 2.98 2.92 90| | .5964 2.08 2.03 5
52477 | 2.80 2.73 95 | 5957 2.41 | 9.35 5
| 4.52186 | 4.03 3.97 40| | .59433 | 3.00 2.93 50
|
| 59345 | 3.54 3.45 30 | -
: 4.52203 | 397 3.91 40
MERCURY.
| a »(Langley)|# (Rubens)) /* —

| 4.53198 | 4.01 4.00 | 28
| .53076-| 4.43 144 8
52907 | 1.36 1.34 | 44
52828 | 1.52 | 1.49 5
4.52759 | 1.70 | 1.66 | 5 |

* The intensity of the green and yellow mercury lines has been put = 10.

Mathematics. — “On the locus of the pairs of common points and
the envelope of the common chords of the curves of three
pencils.” 24 part.: Application to pencils of conics. By Dr.
F. Scnuun. (Communicated by Prof. P. H. Scnovurr.)

(Communicated in the meeting of December 29, 1906).

9. If the pencils of curves are pencils of conics (r= s = t= 2)
then in the case of there being no common base-points the locus is
of order fifteen and the envelope of class six. In the following we |

—.

wish to treat the case more closely, that one of the pencils has two
points in common with each of the two others, where we shall
attain at results in another way, which will prove to agree to the
general ones and complete these in some parts.

Let ABCD, ABEF and CDGH be the three pencils of conics. On
one conic of the pencil ABCD the two other pencils describe two
quadratic involutions of which the connecting lines of the pairs of
points pass through a point A of HF, resp. a point 1 of GH. The
pair of common points PP’ of these two involutions is thus deter-
mined by the right line AL. If the conic ABCD describes the whole
pencil, A and Z describe projective series of points on //' and GH.
For, if we take A arbitrarily on EF’, the conic ACD is determined
by it, as it must pass through the second point of intersection of
CK with the conic ABEFC (as likewise through the second point
of intersection of D&A and the conic ABEFD); by the conic ABCD
the point / is unequivocally determined. Reversely to a point 4
of GH now corresponds one point AK. The projective series of
points are however in general not perspective; so the line KL or
PP’ envelops a conic N touching EF and GH.

Of that conic three other tangents are easy to construct, namely
by taking for the conic ABCD in succession each of the three
degenerations. If that conic is AB.CD then the movable points of
intersection with conies of the pencil ABEF lie on CD so that A
lies on CD, thus in the point of intersection A, of CD and EF;
likewise does Z coincide with the point of intersection L, of AB
and GH. The line A,Z, is thus tangent to N. The construction
becomes a little less simple if we take one of the other degenerations
eg. AC. BD. By cutting this by the degenerated conic AL. BF
of pencil ABEHF it is evident that A coincides with the point of
intersection of H/F with the line connecting the point of intersection
of AF and BD with the point of intersection of BF and AC; in
similar manner Z/ is found.

To the locus of the points P and P’ belongs the locus of the
points of intersection of the conics of the pencil ABCD with the
projectively related series of tangents AZ of the conic NV. This locus
(as is easily evident out of the points of intersection with an arbitrary
right line or with an arbitrary conie of the pencil ABCD) is of
order five with double points in A, B, C and D; further it passes
through £, F,G and H, as K coincides with H when the conic
ABCD passes through £, etc. If we take for the conic of the pencil
ABCD the degeneration AB.CD, then KL passes into A,Z, which
line cuts the conic AB. CD in the points A, and L,, which thus

( 550 )

lie on the locus of the points of intersection too. By taking the two
other degenerations we find four more points of C;. Altogether there
are 10 single and 4 double points by which C;, is determined.

If we take the degeneration AL. CD, the particularity occurs,
that the pair of points of the involution described by the pencil
ABEF can become indefinite on AB, if namely the conic ABEF
breaks up into AB. EF. By this the whole line AB (and of course
the line CD too) will belong to the locus proper of P and P’’).
To the part proper of the envelope of the lines PP’ the pairs of
points PP’ lying on AB or CD contribute nothing but the lines
AB and CD (which belong also to the part improper of the envelope,
the points A, b, C and PD), which does not give rise to a higher class.

So the locus proper of P and P’ consists of the lines AB and
CD and the curve C, and is thus in accordance to the general
results of order seven. The line 44(CD) intersects C, in the points
A and B (C€ and JD) to be counted double and in ZL, (K,). The
curve C, has three double points differing from the base-poimts (of which
E, F, G and H are single and A, B, C and D threefold points
of C,) namely K,, L, and the point of intersection T of AB and
CD. These form a triplet of double points belonging together of which
we spoke in §5. The conics of the three pencils passing through |
one of those double points, also pass through the two others; these
conies are AB. CD, AB. EF and CD.GH. To the branches 7A,
and JTL, of C, passing through 7’ correspond respectively the
branches A,7’ and L,7' passing through A, and L,, whilst the
branches of C, passing through A, and /, correspond mutually.

Summing up we find:

For the conics ABCD, ABEF and CDGH the locus proper of
the pairs of common points PP’ consists of the lines AB and CD
and a curve of order five, having in A, B, C and D double points
and in FE, F, G and H_ single points and further passing through
the point of intersection K, of CD and EF and the point of mter-
section L, of AB and GH. The envelope proper of the lines PP’
is a conic touching the lines HF, GH and K,L,.

10. If the points A, B,C, D,H and F lie on ascomep ie
latter then belongs to the locus, so that the C, breaks up into that
conic and a C, passing through A, B, C, D, G, H, K, and L£,. To
each conic of the pencil ABCD now belongs the same point A,
namely ,, as is immediately evident when we make the conic of

1) More generally: if two base-points of one pencil lie with two base-points of
another pencil on a right line, that line belongs to the locus proper.

(551 )

the pencil ABEF to pass through C and D. If we take ABCDEF
for the conic of the pencil ABCD, then & is indefinite on LF,
whilst point Z is to be found somewhere in 1, on GH. The corre-
spondence between the points A and Z is of such a kind that to a
point ZL differing from ZL, the same point A always corresponds,
namely K,, whilst when Z coincides with L, point A’ is arbitrary
on EF. So the conic N breaks up into the two points K, and L,.
The relation between the conics of the pencil ABCD and the tangents
KL or PP’ of N is of such a kind, that to the conic ABCDEF
every line through JZ, corresponds and that, for the rest, between
the conics ABCD and the lines through A, a projective relation
exists, in which to the conics ABCDEF, ABCDG, ABCDH and
the degenerated conic AB.CD respectively A,L,, A,G, K,H and
K,L, correspond. From this is also evident, that the curve C,
breaks up into the conic ABCDEF and a C, passing through
A, B, C, D, G, H, K, and L, and farther that C, passes through the

points of intersection of K,L, with the conic ABCDEF.

The double points of C, = AB.CD.ABCDEF. C, differing from
the base-points are K,, L,, 7 and the two points of intersection of
KL, with ABCDEF. The latter two doublepoints do not furnish a
triplet of points through which conics of the three pencils pass, but
two coinciding pairs of points; the branches through one doublepoint
correspond to the branches through the other and, it goes without
saying, in such a way that the branches belonging to (, corre-
spond mutually and likewise the branches belonging to the conic

ABCDEF.

11. If moreover the points A, B,C, D,G and H le on a conic,
C, breaks up into that conie and the line A,LZ, (L, then coincides
with Z,) so that the locus proper then consists of the conics ABCDEF
and ABCDGH and the lines AB, CD and K,L,. When conic ABCD
does not pass through /, and F neither through Gand H, the point
K coincides with AK, and‘ “ with Z,; so that the pair of points PP’
lying on that conic is always determined by the same line A, /,.
Hence A,Z, forms part of the locus. The C. has now seven double
points differing from the base-points, namely one triplet Ay, L,, 7;
and two pairs, the two points of intersection of AZ, with the conic
ABCDEF and those with the conic ABCDGH.

If the point A, coincides with Z, and therefore also with 7’, i.o.w.
it the four lines AB, CD, EF and GH pass through one point, on
each conie of the pencil! ABCD the two involutions coincide. 7he
locus proper then becomes indefinite. If we bring through an arbi-

552 )

trary point P a conic of each of the pencils, then those conics have
another second common point, namely the second point of intersection
of the line TP with the conic ABCDP. The envelope proper is then
still definite and consists of two coinciding points 7’.

12. If the points EF and G corneide, then if the conic of the
pencil ABCD passes through /7 the point A as well as the point
coincides with /. The series of points A and JF are perspective, the
lines AZ all pass through a selfsame point C’.

The conic VV breaks up into two points # and Ul. As Fy belongs
to the part improper of the envelope, the envelope proper now consists
only of point U. By taking for the conic of the pencil ABCD the
degeneration AL.CYD it is evident that U lies on the line K,Z,.
Another line AZ and by that the point U itself can be constructed
in the way indicated in § 9 by allowing the conic ABCD to break
up into AC’. BD or AD. BC.

Between the lines AZ or PP’ through U and the conies of the
pencil ABCD exists a projective correspondence, where to the conics
ABCDE, ABCDF, ABCDH and AB.CD respectively the lines
Uk, UF, UH and K,L, correspond. The locus of the points of
mlersection is a cubic through the points A,B, C, D, U, bh, F, H, K,
and £,, which is determined by these 10 points; the third points
of intersection of that curve with AC, AD, BC and BD are easy
to construct.

On the conic ABCDE the two involutions coincide, so that that
conic has separated from the (, of § 9 and has become improper.

The locus proper consists now of the lines AB and. CD and the
above-named C,, so it ws of order five. Differing from the base-
points the C, has three double points, A,, 2, and 7’ (the point of
intersection of AL and CD) forming a triplet.

If moreover the points A, B,C, D, i and F le on a conie, no
other particularity appears than the pomt U coinciding with K,. Of
the three points of intersection A,, 1, and U of A, 1, with the C,
the points A, and U now coincide, so that the C, touches the line
Kk, L, mm K,. In comparison with § 10 the particularity that appears
is this that the point Z, coincides with # whilst the pencil of rays
LL, has passed into the part improper of the envelope and the conic
ABCDEF into the part improper of the locus.

13. The case treated in the preceding paragraph is of course not
the only one in which the series of points AK and / are perspective,
the condition of that perspectivity being single, the condition of the

( 553 )

coincidence of H and G being double. The condition of perspectivity
can be found out of the condition that the point of intersection |
of LF and GH (as point A) corresponds to itself (as point L).
Now the conic ABCD belonging to V (as point A’) passes through
the second point of intersection W of CV with the conic ABLIU,
whilst C,W is a pair of points of the involution described on
the conic ABCDW by the pencil ALLF. If this pair of points
also belongs to the involution described on that same conic by the
pencil CDGH, the point ZL coincides evidently with V. So this
is the case when the cone of the pencil CDGH touching the conic
ABCDW in C passes through W. This condition for the perspec-
tivity of the series of points A and ZL (where of course it must be
possible to interchange C with D and likewise A’ resp. LF with
UD resp. GH) is evidently satisfied when / and G coincide.

If U is the centre of perspectivity, there exists between the rays
of the pencil U and the conics of the pencil ALC'D a projective
correspondence, where to the conics AACDL, ALCDF, ALCDG,
ABCDH, ALCDW and AB.CD correspond respectively the rays
UE, UF, UG, UH, UV and K, L,, whilst moreover to the conic
ABCDW all the rays of the pencil V correspond. So the C; of $9
breaks up into the conic ABCDW, still belonging to the part proper
of the locus, and a C, passing through the points A, 5, C, D,U, £,
F,G,H, kK, and JZ,, cutting the conic in A, 6, C and D and the
two points of intersection of UV with that conic.

The locus proper is thus a C. consisting of the lines A/ and CY,
_ the conic ALCDW and the C, before mentioned. This C, has five
double points differing from the base-points, namely, the triplet A,, L,, 7’
and the pair formed by the points of intersection of UV with the
conic ABCD IW.

The C, is determined by the ten points, A, B, C, D, E, F, G, H, K,
and £, so these ten points will have to lie on a C, if the above
condition for the perspectivity is satisfied, and reversely it is easy
to prove that when those ten points lie on a C, the series of
points are perspective. Suppose namely that the series of points were
not perspective. Then it would be possible by keeping the points
A, b, CU, D, HK, F and G to construct on the line GH (thus by
keeping the points A,, 4,, Vand |W) by means of the former condition
for perspectivity a point A/’ in such a manner that the series of
points A and ZL are perspective; H’ is then the second point of
intersection of VG with the conic through C, D, Gand W, touching the
conic ABCDIW in C. So now the ten points A, B, C, D, EL, F, G, Ay,
L, and H’ will lie on a C,, however already determined by the

(554).

nine former points‘) and thus the same as (C’, through the ten points
A, B,C, DE, F,G,K,, Ll, and H. The line VG would then however
have four points G, L,, H and H’ in common with this (,.

So we arrive at the following simple result:

If the ten points A, B,C, D, E, F,G, H, K, and L, lie on the
same cubic, the series of points K and L are perspective, whilst the
centre of perspectivity coincides with the third point of intersection U
of KL, with C,. The envelope proper breaks up into the point U
and the point of intersection V of EF and GH. The locus proper
consists of the lines AB and CD, the cubie just mentioned and the
conic through A, B,C, D and the two points, in which the right line
UV intersects moreover the C, besides in U.

If / and G coincide, we immediately see that the above condition
is satisfied. The point V lies then in point / so that one point of
intersection of UV with C, differing from UL’ becomes the point £;
the indicated conic is thus the conic ABCDE, which now however
belongs to the part improper of the locus.

14. If G coincides with E and H with F, then the series of
points A and ZL are connective with double points in Z and F.
The pair of points PP’ on an arbitrary conic of the pencil ABCD
is now continually described by the same line /F, thus belonging
to the locus proper. If the conic passes through / or F the two
involutions coincide, so that the conics ABCDE and ABCDF belong
to the locus; but to the part improper of it. Moreover the lines AB
and CD belong to the locus proper, so that the latter consists of the
three lines AB, CD and EF. An envelope proper is no more at
hand, the line connecting P and P’ coinciding with AB, CD or EF
when P and P’ differ from the base-points.

In comparison with § 12 the particularity appears that UV coincides
with /’, that the pencil of rays U’ passes into the part improper of the
envelope and that the C, breaks up into the conic ABCDF becoming
improper and the right line HF.

The case of the pencils of conics ABCD, ABEF and CDEF can
be profitably used to define with the help of the principle of the
permanency of the number the order of the locus of P and P”’ and
the class of the envelope of PP’ for the case of pencils of conies
lying arbitrarily with respect to each other. Starting from this simplest

') The C; is only then not determined by these nine points if two of those
points coincide in such a way that the connecting line is indefinite (e.g. G with
E or K, with 1). Then the ten points lie on a Cy, whilst it is easy to prove that
the correspondence between K and L is perspective.

ee Tee

( 555 )

case, it is easy to reason that ?/” coincides with AB, CD or EF
apd so the locus proper consists of these three lines and there is
no envelope proper. The part improper of the locus however consists
of six conics ABCDE, ABCDF, ABEFC, ABEFD, CDEFA
and CDEFB, the part improper of the envelope of the six points
A, B,C, D, E and F. The total locus is thus of order fifteen, the
total envelope of class six, so that for arbitrary position of the pencils
of conics this same holds for the locus proper and the envelope proper.
Sneek, Nov. 1906.

Mathematics. — “The locus of the pairs of common points of four
pencils of surfaces.” By Dr. F. Scuun. (Communicated by
Prof. P. H. Scnovure).

(Communicated in the meeting of December 29, 1906).

1. Given four pencils of surfaces (F;), (F;), (7) and (F,) respect-
ively of order 7,s,¢ and w. The base-curves of those pencils can
have common points or they can in part coincide, in consequence
of which of three arbitrary surfaces of the pencils (/,), (4%) and (/",)
the number of points of intersection differing from. the » base-curves
can become less than sfw; we call this number a, calling it 4 for
the pencils (/;), (/,) and (f,); ¢ for the pencils (/,) (7) and (F,)
and d for the pencils (/,), (/) and (f,). We now put the question:

What ts the order of the surface formed by the pairs of points
P and P’, through which a surface of each of the four pencils is
possible ?

If the points P and /” do not lie on the base-curves we call the
locus formed by those points the locus proper L on which of course
still eurves of points ? may le for which the corresponding point
P’ lies on one of the base-curves. If one triplet of pencils furnishes
at least several points of intersection which are situated for all sur-
faces of those pencils on one of the base-curves, then there is a
surface that does satisfy the question but in such a manner that if
we assume /? arbitrarily on this surface the point /” belonging to
it is to be found on one of the base-curves; this surface we call the
part improper of the loeus, whilst both surfaces together are called
the total locus.

2. To determine the order » of the locus proper Z we find the
points of intersection with an arbitrary right line /. On / we take

( 556 )

an arbitrary point Qy, and we bring through that poimt surfaces
F,, F,and F, of the pencils (F;), (4/1) and (/,). Through each of
the a—1 points of intersection of those surfaces not situated on the
base-curves of those surfaces we bring a surface F/. These a —1
surfaces F’. intersect the right line / together in (a —1)r points Q,,
which we make to correspond to the point Qi. The coincidences
of this correspondence are: 1st the points Q,.:, determining four
surfaces which intersect one another once more in a point not lying
on the base-curves, thus the 2 points of intersection with the surface
1, 2°4 the points of intersection with the surface R,,, belonging to
the pencils (/°), (4) and (/",), the locus of the points S determining
three surfaces whose tangential planes in |S pass through one line.

To find the number of coincidences we have to determine the
number of points Q,,, corresponding to. an arbitrary point Q, of 1.
To this end we take on / a point Q,, arbitrarily and bring through
it an F, and an ¥#,. Through each of the 4 points of intersection
of these surfaces with the surface / through Q, (not lying on the
base-curves) we bring an /’,, which 6 surfaces /’, intersect together
the line / in ds points Q, which we make to correspond to Q,,.
To find the number of points Q,, corresponding to an arbitrary
point Q, of 7 we take Q, arbitrarily on /, we bring through Q, an
/’,and through @, an F, and through each of the ¢ points of inter-
section of those surfaces with /. an /, which furnish ¢ surfaces
F, cutting / in ct points Q,; reversely to Q, belong du points Q,,
so that we find between the points Q, and Q, a (ct, dw)-correspond-
ence, of which the cf+du coincidences give the points Q,, belong-
ing to the point Qs. So between the points Q,, and Q, exists a
(bs, ct+-du)-correspondence, of which the coincidences consist of
the 7 points of intersection of / with the surface /’, through Q, and
of the points Q., corresponding to Q,; the number of these thus
amounts to bs + ct + du—~r.

So between the points Q,,, and Q. there is an (ar—r, bs--ct-+-du—?)-
correspondence with ar+-bs+ct-+du—-27 coincidences. To find out
of this the number of points Qs: Wwe must first determine the order
of the surface Py, .

This surface may be regarded as the surface of contact of the
surfaces of the pencil (/’,) with the movable curves of intersections
(|, of the surfaces of the pencils (/’,) and (/,)'). So the question is:

!) We shali call this surface the surface of contact of the three pencils meaning
by this that in a point of this “surface of contact” the surfaces of the pencils,

though not touching one another, admit of a common tangent.

3. To determine the order of the surface of contact of a twofold
infinite system of twisted curves and a singly infinite system of
surfaces.

- To this end we shall first suppose the two systems to be arbitrary.

To determine the order of the surface of contact we count its
points of intersection with an arbitrary right line /. To this end we
consider the envelope #, of the o* tangential planes of the curves
of the system in their points of intersection with / and the envelope
FE, of the ow’ tangential planes of the surfaces of the system in their
points of intersection with /.

The common tangential planes not passing through / of both
envelopes indicate by means of their points of intersection with /
the points of intersection of 7 with the surface of contact.

In order to find the class of the envelope £, (formed by the
tangential planes of a regulus with / as directrix) we determine
the class of the cone enveloped by the tangential planes passing
through an arbitrary point Q of /. If the system of curves is such
that g curves pass through an arbitrary point and w curves touch
a given plane in a point of a given right line, the tangential planes
of EL, through Q envelope the g tangents in Q of the curves of the
system through Q, and the line / counting w times; for each plane
through / is to be regarded y times as tangential plane, there being
y curves of the system cutting / and having a tangent situated in
this plane. The envelope EL, is thus of class g +w and has | as
w-fold line’).

To find the class of the envelope /, we determine the number of
its tangential planes through an arbitrary point Q of /. If now the
system has w surfaces through a given point and r surfaces touching
a given right line, the tangential planes of the envelope passing
through Q are the tangential planes in Q to the w surfaces passing
through Q and the tangential planes of the » surfaces touching /. So
the envelope E, is of class w+ yv with v tangential planes through 1.

Hence both envelopes have (g +) (u +r) common tangential planes.
Each of the » tangential planes of FE, passing through / is however
a w-fold tangential plane of /, and so it counts for w common
tangential planes. So for the number of common tangential planes
not passing through /, thus the number of points of intersection of /
with the surface of contact we find:

(p + y) (w+ v) — yr = gv + Yu-+ gu,
therefore :

1) The regulus as locus of points has however line / as ¢-fold line.

( 558 )

The surface of contact of a system (g, W) of w* twisted curves *)
and a system (u, v) of w' surfaces*) is of order gv + wu + gu*).

4. To determine the order of the surface of contact *) of the systems
“,,%,), (4,,v,) and (u,,¥,) each of oo’ surfaces, we regard the
system (y,w) of the curves of intersection of the systems (tu, , v,) and
(u,,1,). Of these curves of intersection 4,4, pass through a given point,
so p=un,. The w points, where the curves of intersection touch a
given plane in a point of a given right line, are the points of inter-
section of that given line with the curve of contact of the systems
(u,,¥,)°) and (u,,¥,) of plane curves, according to which the given
plane intersects the systems of surfaces (u,-,7,) and (u,,7,). This
curve of contact is of order u,v, + wir, + wu,, thus:

w = —,", =f (,P, Fi HU,

The surface of contact to be found is thus the surface of contact
of a system (uu, ,,r, + 4,r, + u,u,) Of a7? twisted curves and a
system (u,,97,) of a’ surfaces, so that we find:

The surface of contact of three systems (u, , ,), (4, , ¥,) and (uy , P3)
of w' surfaces is of order

HP, -E H3Hy)P, = Har =k: 21, UH, «
If the three systems are the pencils (F,), (Ff) and (/,) we have
= fo ee
Y, =Ae-1) °, vf =26=])). 2 es = 2e—2)t

So we find:

The surface of contact Fu, of the three pencils of surfaces (Fs),
(F,) and (F,) is of order

') System with curves through a given point and y curves cutting a given
line and touching in the point of intersection a given plane through that line.

2) System with w« surfaces through a given point and » surfaces touching a
given right line.

3) This result is also immediately deducible from the Scuusert formula

ap? =p. G+ p'g'e. p'ge + p® . p°ge
(Kalkiil der abziihlenden Geometrie, formula 13, page 292) for the number of
common elements with a point lying on a given line of a system &! of o% and
a system © of o*# right lines with a point on it. If we take for =’ the tangents
with point of contact of the system of curves (?, ~) and for © the tangents with
point of contact of the system of surfaces (u,v), then
Ps=? , POs Oy ie — verre ais

whilst ap? is the order of the surface of contact.

4) Locus of the points, where the surfaces of the three systems have a common
tangent.

5) System of c! curves of which 4, pass through a given point and », touch
a given right line.

1 4 f
=e

( 559 )

2(s +¢+ u— 2).

5. To return to the question which gave rise to the preceding
considerations we find for the number of points Q,.;, on the arbitrary
line 7, which are the points of intersection of / with the locus
proper L:

ar + bs + ct + du — 2r —2(s+t+u—2)=
= ar + bs tc + du—2(r+s+t+u)4 4.

So we find:

The locus L of the pairs consisting of tivo movable points common
to a surface out of each of the pencils (FP), FS, Fi) and (Fw
of orders 1, 8, t and u, and not lying on the base-curves, is a
surface of order

ar + bs + ct + du— 2(r+s+t+4+ u)4 4.

flere a is the number of points of intersection not necessarily
situated on the base-curves of the pencils (F), (F.) and (F,); 6 the
analogous number for the pencils (F,), (F:) and (F,), ete.

6. It the pencils have an arbitrary situation with respect to each
other, then a—=sfu, ete., so that then the order of the locus becomes

4(rstu + 1)—2(r+s+t+ nu).

That order is lowered when three of the base-curves have a common
point or two of the base-curves have a common part, which
lowering of the order can be explained by separation as long as
the total locus is definite, i.e. as long as the four base-curves have
no common point and no triplet of base-curves have a common part.
For, if A,s;, 1s a common point of four base-curves then the surfaces
of the four pencils passing through an entirely arbitrary point P
have another second point in common, namely 4A,.),; if By, is a
curve forming part of the base-curves 5,, 2, and B, of the pencils
(F.), (F.) and (Ff), then the surfaces of the pencils passing through
an arbitrary point P have moreover the points of intersection in
common of Bs, with the surface F, through P; so in both cases
the arbitrary point P belongs to the total locus.

If the basecurves B,, B, and B, have a common point A,,, then
on account of that point the number a is diminished by unity without
having any influence on /, c and d. The order of Z is thus lowered
by 7 on account of it, which is immediately explained by the fact
that the surface F, passing through Ag. separates itself from the
locus.

(560)

If the base-curves B, and B, have a curve 4,, in common of
which for convenience we suppose that it does not intersect the
base-curves B, and B,, this B,, has no influence on ¢ and d, whilst
a is lowered with sm and 6 with rm, where m represents the order
of the curve B,,; for, when F,, /, and F, are three arbitrary
surfaces always sm points of intersection he on &,,;. “The order
L is thus lowered with 2rsm by 6,,. This can be explained by
the fact, that the locus of the curves of intersection C,, of surfaces
FE. and F, passing through a selfsame point of Bu") separates itself
from the locus of P and P'. That the locus of those curves of inter-
section is really of order 27sm is easily evident from the points of
intersection with an arbitrary line 7. We can bring through an
arbitrary point Q, of / an Ff, cutting 4, in rm points; through
each of those points of intersection we bring an F,, which rm _ sur-
faces F, cut the right line 7 in rsm points Q;. To @, correspond
rsm points Q, and reversely. The 2rsm coincidences are the pomts
of intersection of 7 with the locus of the curves of intersection C..

7. The base-curves B,, B,, B: and 6, of the pencils are morefold.
curves of the surface L. If A, is a point of 4, but not of the
other base-curves, then A, is an (a — 1)-fold point of Z. For, the
surfaces F's, /; and F, through A, intersect one another in a —1
points, not lying on the base-curves, each of which points furnishes
together with A, a pair of points satisfying the question. Each point
of B, is thus an (a —1)-fold point, i.o. w. B, ts (a—1)-fold curve
of the surface L.

Let A,, be a point of intersection of the base-curves 4, and b,,
but not a point of 5, and 5,. An arbitrary point P of the curve of
intersection C}, of the surfaces /;, and F, through A,, furnishes now
together with A,, a pair of points PP’ satisfying the question pro-
perly, as A,, is for each triplet of pencils a movable point of inter-
section not lying on the base-curves. If we let P describe the curve
Ci,, then the tangent /,, in A,,; to the curve of intersection of the
surfaces /’, and F, through P describes the cone of contact of Z in
the conic point A,,. The tangents m, and m, in A,<s to B,.- ands
are (a—l)- resp. (6—1)-fold edges of the cone. This cone is cut
by the plane through m, and mg, only according to the line m,
counting («@—-1)-times and the line m, counting (6— 1)-times, as
another line /., lying in this plane would determine two surfaces

1) If Bm cuts the curve Bs in a point Astu, then the surface /- passing through
Asiu separates itself from the locus of the curves of intersection Crs .

( 561 )

F. and F, touching each other in A,;, whose curve of intersection,
however, does not cut the curve Cy. The tangential cone of L
in Ay, is thus of order a+ 6— 2").

Let At be a point of ‘a common part 5,; of the base-curves B,
and B,; but nota point of B, and b,. We get a pair of points PP’

3 e ~ Se : 1) > =
with a point 2” coinciding with A,, when the surfaces F. and F,

have in A‘. a common tangential plane V,,; and pass through a

selfsame point P of the curve of intersection (,,, of the surfaces /, and

= Te. : Y “

F, through A”. If we let P describe the curve C;, , then on account
£ a seca

of that between the planes V, and V,, touching in A;, the surfaces

F, and F, through P, a correspondence is arranged, where to JV’,

correspond /—1 planes V, and to V. correspond @—1 planes

V,. One of the a+4—2 planes of coincidences is the plane through

: a ‘ :
the tangents in A,, to B,; and C;,,; this plane furnishes no plane
V,;. The remaining a+ 6 —3 planes of coincidence are planes V,.

and indicate the tangential planes in AW to the surface L. So B,,

is an (a+b—8)-fold curve of L.

8. Let us then consider a common point A, of the base-curves
B,, B; and B,. We get a pair of points PP’ with a point P” coin-
ciding with A,s;, when the tangential planes in A,., to F,, Fy and F;
pass through one line /,,, and these surfaces intersect one another
again in a point P of the surface F, passing through A,.. There
are oo’ such lines /,,,, forming the tangential cone of Z in point
A,s The tangents m,,m, and m: in A,, to B,., B, and B; are
(a — 1)-, (6 —1)- and (c —1)-fold edges of that cone. So the plane
through m, and m; furnishes a + 4—2 lines of intersection with
the cone coinciding with m, and m,. Moreover c —2 other
lines /,.; lie in this plane. For, the surfaces /. and /, touching this
plane intersect #, in ¢ — 2 points not lying on the base-curves; the
surfaces /, through those points intersect the plane through mm,
and m, according to curves whose tangents in A,,; are the mentioned

1) The order of this cone can also be found out of the number of lines
of intersection with an arbitrary plane < through A;,;. If J, and J, are the lines
of intersection of < with the tangential planes in A;; to the surfaces F, and F; through
P, then to 7, correspond b—1 lines 7; and to /; correspond a—1 lines /,, so that
in the plane = lie a+ — 2 lines 1s.

38
Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 562 ).

lines J... So the tangential cone of L in A,s: is of order a+ 6 4+- c— 4").

: tc : (1 ‘i E ;
A point of intersection A;,, of 4B, with a common part By, of the
base-curves B, and £, is a conic point of L, the tangential cone of
which is formed as in the previous case by a’ lines /,.;. The tangents

m, and mgs, in vie to B, and By are (a —1) and (6+ ¢ — 3)-fold
edges of that cone. As no other lines /..; lie in the plane through

3 : s . : ly}: .
m, and mg, it is evident that the tangential cone of Lin AY is like-
wise of order a+6-+c—4 "*).

‘9 : .
Let A, be a point of a common part B,.; of the base-curves 6,,
B, and B,. The point P’ of the pair of points PP’ coincides with

(y . i 7 7 > 2 ;
A;., when the surfaces /,, /; and F; have in AS) the same tangen-
tial plane Vs, and cut one another in another point P of the surface

(2 5 ° 5 :
F, through Ay. If we now consider an #7; and an F's having in

(2) . ae . .
Aj; the same tangential plane VV, and if we consider through each
of the c—1 points of intersection of /,, Ff, and /, not lying on
the base-curves an /, of which we indicate the tangential plane in

(2)

Ax by V; then to V,; correspond ¢—-1 planes V; and to V; cor-
respond a+ 6—1 planes J’,; (as for given V; a (6, a)-correspondence
exists between V, and V, of which JV; is one of the planes of coin-
cidence). Among the a+ 4+ ¢—2 planes of coincidence V,; V;
there are however three which give no plane V,.;, namely the planes
V.., for which the corresponding surfaces /. and /, furnish with

F, three points of intersection coinciding with A;,s;. For this is neces-

7 A "9 - 2 -
sary that /, touches in A;,, the movable intersection of /, and
F.. Now the tangents of those intersections for all surfaces /,

; (2) ich

and /’, touching each other in A; form a cubic cone having for
: 5 2) ;

double edge the tangent mm, to 6s: im point AS *). This cone

is cut by the tangential plane in Ae) to /, according to three lines,
furnishing with #,s, planes V;,,; which are planes of coincidence

') This order can also be determined out of the number of lines /,s¢ in a plane
e passing through A,». In this plane we finda (¢ — 1, a+b — 2)-correspondence
between lines /,, and lines J, of which however the line of intersection of « with
the tangential plane in A, to Fy is a line of coincidence, but no line /,se.

2) This is immediately evident if we take for (F',) a pencil of planes and for (F's)
a pencil of quadratic surfaces all passing through the axis B, of the pencil of
planes. The cone under consideration then becomes the cone of the generatrices
of the quadratic surfaces passing through a given point of B,. We can easily
convince ourselves that the same result holds for arbitrary pencils of surfaces.

ee SS ee

—— hc

( 563 )

of V,, and V,, but not planes J’,,.. So there are a+4-+-c—5 planes

° > : : 4 ie
V,., which are the tangential planes of Z in the point A;x, i. 0. w.

Bras ts (a +b + ¢— 5)-fold curve of surface L.

9 We then consider a common point A,,;, of the four base-curves.
We get a pair of points PP’ with point /” coinciding with A,.,, when
F., F,, F, and Ff, have in A, a common tangent /,,, and all
pass once again through a selfsame point P. The o° lines /,,,, form
the tangential cone of LZ in A,.. To determine the number of lines
[sm in an arbitrary plane ¢ through A,.;, we take in this plane an
arbitrary line J. through A,s, and we bring through the d— 1
points of intersection (not lying on the base-curves) of the surfaces
F,, Fs and F, touching J... the surfaces /,, whose tangential
planes in A,s cut the plane ¢ according to lines, which we shall
call /,. To /., now correspond d— 1 lines /, and to /, correspond
a+b+e—2 lines /,.., as there exists between /,, and /, when /,
is given a (c,a-+)-correspondence, of which /, and the line of
intersection of ¢ with the plane through the tangents in Bi: ron (oa 3
and #B; are lines of coincidence, but not lines /. So there are
atbte+td— 83 lines of coincidence /,s, /, of which however three
are not lines /,.;,. The common tangents in A,,;, of the surfaces F,,
F, and F, possessing three points of intersection coinciding with
A,, and where therefore the intersection of two of those surfaces shows
a contact of order two to the third, form namely a cubic cone *) of
which the lines of intersection with ¢ are lines of coincidence but
not lines /,s4. So in ¢ lie a+6+ c+ d—6 lines (ys, i.o0. w. the
tangential cone of L in Ay;sy is of order a+b+ce+d—6").

1) This is again evident when taking for (/.) and (F,) pencils of planes with
coplanar axes B, and B; and for (F%) a pencil of quadratic surfaces passing through
a line containing the point of intersection S of B, and B,. The line of intersection
of the planes F. and F, shows only then a contact of order two to Fy when that
line of intersection lies entirely on F,, so that the cone under consideration becomes
again the cone of the generatrices of the quadratic surfaces passing through S.

2) That order can also be found out of the lines of intersection with the plane
V;s through the tangents m-, and m in A,su to B, and B;. Those lines of inter-
section are: the line m,, counting (a — 1)-times, the line m, counting (b — 1)-
times and c-+dW— 4 other lines. This last amount we find by drawing in plane
V;s an arbitrary line ; through A,s.. The surface Fy touching /; cuts the surfaces
F, and F, touching V;; in d—1 points (not lying on the base-curves) through
which points we bring surfaces F,, whose tangential planes in A,s/« cut the plane
V,s according to lines to be called 7,. Between the lines 7; and 7, we now have
a (d — 1, ce — 1)-correspondence of which the nodal tangents in A,s‘u of the inter-
section of the surfaces F, and F; touching V;s are lines of coincidence. The
remaining ¢ 4-d — 4 lines of coincidence are lines /rstu.

38*

( 564 )-

The preceding considerations hold invariably for a_ point ACS
lying on the base-curves 8, and £B, and the common part By, of
the base-curves 4, and 6&,,').

In a point of intersection ee of 6,, and A, the tangential cone
is likewise of order a+6-+c+d—BO as that cone has the tan-
genis m,; and m, to B,; and By, as (a + 6 — 3) and (ce + d—3)
fold edges, whilst in the plane through m,s and m,, no other right
lines J,s1, are lying.

A point of intersection A®) of B, and By is also a (a+b +c+d—6)
fold point of Z as m, and ms are (a —1)- and (6 + ¢-+ d—5)-
fold edges of the tangential cone and the only lines of intersection
of that cone with the plane through m, and mix.

If finally AM) is a point of a common part B,s,, of the four base-
curves, then the point P’ of the pair of points PP’ coincides with

(4 > 5 4)
A.;, when the surfaces F., F, F; and F,, have in A;., the same
tangential plane V7, and all pass through a same point P. Let us
now assume an arbitrary plane V,,, passing through the tangent

; (4) . 7 : : :
Mrstu iN Apsin to B,si,. The surfaces F,, F’; and F; touching this plane in
(4) ae ; BAG : °
A, eut one another in d—1 points P, through which we bring

surfaces F',, of which we call the tangential planes in AQ, V
Thus we obtain a correspondence, where to V;s: correspond d—1
planes V7, and reversely to V, correspond a+ 6+ ¢—1 planes
Vis; for when V, is given there is between V,, and V; a
(c,a +) correspondence, of which V, is plane of coincidence, but
not a plane V,.. So there are a+ 6+c¢c+d—2 planes of coin-
cidence V,s:Vu, of which however jive are not planes V;s,. These
are namely the tangential planes of the surfaces F,, /; and F; of

which one more point of intersection coincides with AS which

1) [t is also easy to see from the lines of intersection with the plane Vstu
through the tangents ms and mm to B; and By, that the tangential cone in

Ain is of order a+ b-+c-+-d—6. The line ms counts for 6 — 1 lines of inter-
section, the line mu for ¢ +d — 3. Further, the surfaces Fs, F, and F, touching

Vsm cat one another in @ — 2 points not lying on the base-curves; through those

: 5 (1)
points we bring surfaces F, whose tangential planes in A;stu cut the plane Voix
along to lines which lie on the tangential cone.

a
3 .

i
be
%

(565 )

occurs five times '). So there remain a + 6 + c+ d—7T planes Vio,

: P : : ; (a
which are the tangential planes of JZ in the point A,</,, so that

Brsu is a (a +b+¢+d—7) fold curve of L.

10. So we find:

Of the locus proper L of the pairs of points P and P’ the
base-curve B, of the pencil (f,) is (a—1)-fold curve, the common
part B,, of the base-curves B, and Bs is (a + b — 3)-fold curve,
the common part B,s: of the base-curves B,, Bs and B, is (a + 6 +-¢—5)
fold curve and the common part Byrsu of the four base-curves is
(at+b6+c+d—%)-fold curve. The points of intersection of the
base-curves are cone points of L, namely a point of intersection of
B, and B, is (a+ b—2)-fold pomt, a point of intersection of
B,, Bs and B, or of B, and By is (a+b+c¢ —A4)-fold point
and a point of intersection of By, B,, Bi and B, or of B,, Bs and
Bu, or of B,s and By or of B, and Bsy is (a+ 6 + c+d—6)-
foid point. *)

11. The base-curves of the pencils are not the only singular
curves of the surface ZL. There are namely o’ triplets of points
lying on a surface of each of the pencils. These triplets of points
form a double curve of L. If P, P', P" is such a triplet and if P41
and P2 are the sheets through P of the surface, then the sheets
P11 and P"2 correspond to them. Through P’ passes another sheet
P’3 and through P" a sheet P"3 which sheets correspond mutually.
The pair of points not lying on the base-curves is movable along the
sheets P1, P’1, along the sheets P2, P"2 and along the sheets
P’3, P"3; on the base-curve a third point then joins the pair.

Further there is still a finite number of quadruples of points,

1) The number five is found in the following way. The tangents of the movable

; : : : A) ;
intersections of surfaces Fs; and F; touching each other in ve form a cubic cone
having the tangent mse. to Brs as double line. Such an intersection shows to
the surface F a contact of order two when it touches the movable intersection

of F, and Fi, so if its tangent in At. lies on the cubic cone belonging to the
pencils (fF) and (fF). As this last cone has also mrstu as double edge, both cones
have 9—4=5 lines of intersection differing from mrs which connected with
Mrstu furnish the five planes under consideration.

*) If the total locus is not indefinite, i.o.w. if there is no point common to
the four base-curves then B, is a (stu — 1)-fold curve and B,; a (stu + rtu — 2)
fold curve of the total locus whilst a point of intersection of By and Bs is a
(stu + réu — 2)-fold point and a point of intersection of B,,B, and By or of
B, and By a (stu + rlu-+ rsu — 3)-fold point of it.

( 566 )

through which passes a surface out of each of the pencils. Through
the points P, 2”, P" and P" of such a quadruple pass three sheets
of the surface Z and three branches of the double curve. The 12
branches of the double curve through those four points we can call
Fi, P2. P3. P11, P’2, P'4, P'1, P"3, PF" Ps eee
a way that the triplet of points is movable along the branches
Pi, P1,P"1, along P2, P’2, P"2, along P3, P"3, P"3 and along
P’4, P"4, P'"4. If the sheet of ZL passing through P1 and P2 is
called P12, then the corresponding sheets (i.e. sheets along which
the pair of points not lying on the double curve is movable) are
iz ana P12, P13 and*7' 13, ete

Geophysics. — ‘‘Current-measurements at various depths im _ the
North Sea.’ (First communication). By Prof. C. H. Wino,
Ltt. A. F. H. Da.auisen and Dr. W. E. Rineer.

In the year 1904 accurate measurements of the currents in the
North Sea‘) were started by the naval lieutenant A.M. van Roosen-
DAAL, at the time detached to the ‘“Rijksinstituut voor het Onderzoek
der Zee’, having been proposed and guided by the Dutch delegates
to the International Council for the Study of the Sea.

By him four apparatus were put to the test, viz. 2 specimens of
the current-meter of PrErTERssoN ’), one of that of Nansen *) and one
of that of Exman ‘*), all destined to determine the direction and the
velocity of the current at every depth.

The experiments were partly made on the light-ship “Haaks’,
where Dr. J. P. VAN DER Stok, the Marine Superintendent of the

Kon. Nederl. Meteorologisch Instituut, also took part in them. Other —

experiments were made in the harbour of Nieuwediep and further,
from the research-steamer ‘“‘Wodan’, in the open North Sea at a
station (H2) of the Dutch seasonal cruises ‘), situated at Lat. 53°44’ N.
and Long. 4°28’ E.

) Cons. Perm. Intern. p. l’expl. de la mer, Publications de circonstance No. 26 :
A. M. van Roosenpaat und C. H. Winn, Priifung von Strommessern und Strom-
messungsversuche in der Nordsee. Copenhague, 1905.

*) Publ. de circ. No. 25,

ee tread ia’ ANG. (O4.

ae cease YANG. | sek

®) Quarterly cruises of the countries taking part in the international study of
the sea, along fixed routes, observations being made at definite points or “stations”.

( 567 )

The apparatus of NANseN appeared to be unfit for the measure-
ments on the North Sea; it was not calculated for the strong tidal
currents occurring there (e.g. 60—100 em/sec.), and also the putting
out of the apparatus in unfavourable weather was hardly possible
without doing harm to the instrument. In more quiet water, however,
it seems to be very useful.

The apparatuses of Perrersson and EkMAN appeared to be better
fit for the observations in the North Sea. Some improvements in the
construction were proposed, partly also put into practice, by Van Roo-
SENDAAL and Winp, by which the instruments have gained in fitness.
For a description of the construction cf the current-meters used, and
the experience made in using them, we may refer to the publications
mentioned. The following few words may be sufficient here.

It appeared that pretty large oscillations, e.g. 15° to both sides
round the longitudinal axis, did not yet render observation impossible.
In 32 out of nearly 200 observations by Van RooseNDAAL as much
as the figure 4 was noted for the motion of the sea, in 40 to 50
cases the oscillations amounted to 10 a 20° to either side, and yet
the accuracy and certainty of these measurements were ouly excep-
tionally insufficient.

In the parallel-observations with the apparatus of PerrErsson and
EkMAN the agreement in indicating the velocity appeared satisfactory.
In one series of 23 measurements e. g. the average difference amounted
to 4.8 cm/sec, whilst the smallest was 3.1, the greatest 6.3.

Nor did the indications of direction, as given by the two instru-
ments, show great differences. The observations with EkMAN’s appa-
ratus bear to some extent a check in themselves, as, by the construction
of the instrument, every observation includes a series of consecutive
readings at small intervals. In by far the greater part of the readings-
observations these separate did not considerably vary. In 128 cases
the direction of the current could be estimated from them :

To less than 10° in 105 cases,
10—20 15
20—30 2
30—40 0
* 40—50 2
more than 50° 4.

Compared with the probable direction, as derived from the instru-
ment of Ekman, that which was determined by means of Prerrerson’s
instrument deviated :

( 568 )

in 65 cases less than 10
37 1020
15 20—30
5 30—40
| 40—50 :
8 more than 50°.

Van RoosenpaaL and Winp took from the whole of observations
made at station H, the most probable values direction and velocity
of current at the various depths and represented them graphically. They
constructed for the different series of observations, each lasting 12 or
24 hours, in the first place central vector-diagrams, by drawing from
a fixed point the successively determined currents as radii-vectores and
connecting the terminal points by means of straight lines or ofa curve, and
in the second place progresswe vector-diagrams, by drawing the current-
vectors, this time interpolated for the successive full hours, one after and
attached to the other. In the first kind of diagrams the periodical
currents, and in the second the residual currents make themselves

most apparent.

The measurements were continued at the station H2 during all
the following seasonal cruises of the “Rijksinstituut”, first by Van
RoosEeNDAAL and afterwards by the naval lieutenant DaLuuisen, who
succeeded the former in his detachment. At the more recent measu-
rements the current-meter of EkMAN was always made use of.

The following table gives the dates of the series of observations
and the number of measurements °).

— SEES SaSES

| |
Number of)
) Depth ale

NO, Time. Measure- (M.) Observer.
| ments. : |
| |
1, vm 16 05 4412 _ ‘head :
from 16 Aug. ‘05 4.12 eee 56 5,20,35 EKMAN. | VAN ROOSENDAAL.
fy Se oe ee {
2. |from 7 Nov. 05 748 am.) 55 ete car : | Van RoosENDAAL
Wl” G6 Cg AZ. poms a and DALHUISEN
‘ 7 mA 79 | |
3. |from 7 Febr.’06 7.20 p.m.| 4g »»» > | DALHUISEN,
| till Seta a O.0d case |
4. |from 2 May ’06 6.35 am. KY pik : “
ee es GAT. > mite ia?

1) A more detailed description of these observations forms the contents of the
last issue of the “Publications de circonstance”’ No. 36,

(569 )

At these researches wind and weather were on the whole favour-
able; the wind was in a.few cases noted 7 at most, at which
force, however, the observations had to be put a stop to in Feb-
ruary 1906 *).

On the plate added, the new measurements are again represented
graphically in central and progressive vector-diagrams. Also the central
diagrams, have been constructed this time with the aid of values
interpolated for full hours, the directly measured values however,
having still been indicated by dots.

It is principally to give a full idea of the variability in direction
and velocity of the currents, that these diagrams of the new series
of observations have been reproduced fully here.

Comparing the values of the velocity near the surface and in the
depth, we see that in 3 out of the 4 cases they show a rather
distinct decrease at an increase of depth. Also at the former series
of observations at H2 (5—4 Aug., 8—9 Aug. and 2—3 Nov. 1905 °*),
also 8—9 Febr. 1905*)) the same result was arrived at.

Also differences of phase in the periodical currents are noticed in
most cases between the surface and the depth, though a distinct law
may not immediately be obvious here.

The striking difference in amplitude of the tidal currents during
the observations in August 1905 and February 1906 on the one
side and that of November 1905 and May 1906 on the other, is
certainly connected with the age of the tide, as it was with the first
nearly spring-tide (15’/, and 14 days after N.M.), with the last
nearer to dead neap (10 and O days after N.M.).

The small number of series of observations that can be disposed
of, does of course not allow at all to already think of a calculation
of tidal constants, nor to give a correct description of the average

variation of the currents. The unmistakable generai agreement,

however, between the different current-diagrams justifies sufficiently
an attempt to compose them. As no doubt moon-tide will have played

1) The reliability of the new observations is no doubt greater than that of the
former, if we take into consideration, that in August and November 1905 and in
February and May 1906 the Wodan lay moored, so that her motion was conside-
rably smaller than on the former occasions, when she had cast only one anchor.

It may still be mentioned that an experimental and theoretical investigation
was started about the influence of the movements of the ship upon the indications
of the current-meter, which, however, has not yet led to a satisfactory result.

2) Publ. de Cire. N°. 26.

3 5 yp Saal eee

(570 )

the principal part, we have thought best for this purpose to compose
for the successive full moon-hours the current-values as they follow
by interpolation from the different diagrams. The averages thus obtained
have been combined in new diagrams, which are represented on the
plate, in the last column of figures, and that by black curved lines.

In order to complete the matter and to allow comparisons, in the
same way average diagrams have been derived from the observations
made in the past year at H2 (see above) and represented in the same
figures on the plate by black-and-white curves.

The arrows drawn in these figures indicate: in the central dia-
grams the direction of the current at the moon’s transit, in the
progressive diagrams the total residual current during a half moon-day.

A comparison of the average current-diagrams for various depths
or also of the newer with the older ones might give rise to all
kinds of remarks. With a view to the small number of data, how-
ever, on which the diagrams are based, it would perhaps be incon-
siderate to mention all of them here. We therefore confine ourselves
to what follows.

Difference in Phase of the tide at different depths.

| August 1905 —May 1906 August—November 1904
IF “ain a iy = | ‘
20 M.—5 M. | 35 M.—20 M.| 20 M—70 M. | 30 M.—20 M.
ES
@ Transit 18° | ae | 0 | 3°
ore hour after » 24 | — 6 —13 | 14
2 29 ae uit | 5
3 20 5 3 6
4 25 —2 D 14
5 | 26 0 18 —5
6 25 | 8 | 3 22
5 » before » 49 | 17 | 0 17
4 25 | 15 —9 a
3 8 6 —3 25
2 M, 10 | — 6 24
i] 6 41 — 6 141
Average | 13°85' | doug), | = 4050' | 49045
|

The tidal curve shows not only at different depths, but also
in the older and newer observations, generally the same shape.
Its size, on the other hand, both in the older and more recent
observations, appears to be smaller near the bottom than near the
surface. Also its orientation and the situation of the point in it,
which relates to the moment of the moon’s transit, or, more generally,
the phase of the tidal current, seems to change in a definite sense
as the depth increases. This last relation may be specially illustrated
by the following table.

It appears from the table, that the tide is on the whole accelerated
in the depth, compared with higher layers: but the table also proves
that the phenomenon underlies varying influences, besides constant
causes, among which perhaps may be reckonned the shape of the
bottom of the sea and ihe rotation of the earth.

The residual current is by no means constant; at the new obser-
vations it has been much stronger than at the old; it shows consid-
erable fluctuations also, when the progressive diagrams of the different
days of observation are compared. At the new observations this
residual current was on an average stronger near the surface than
in deeper layers. This particular may perhaps be principally attributed
to the action of persisting winds, which at least on the observations of
August 1905 and May 1906 had a very marked influence, rendered
quite obvious by the special diagrams for these dates.

The figures for the residual current as deduced from the newer
observations are the following:

These results are worth comparing with the following table of
values for the year-average of the residual current at the Noord-
Hinder (Lat. 51°35'5N., Long. 2°37,E.), calculated by van DER STOK ”)
from current-estimations neai the surface during five consecutive years.

1) Average of depths of 1, 4, 5, 6, 10 M.; at a depth of 35 M. measurements
were made by Van Roosenpaat only in February 1905.

2) J. P. van pveR Sox, Etudes des Phénoménes de Marée sur les cétes néer-
landaises ; Kon. Ned. Met. Inst. No. 90, Il. p. 67, 1905.

(572 )

Year. | Direction. | Velocity.

a Ro ES SEE SS

1890 N 16° E 0,024 miles p. hour.
91 15 62
92 16 25
93 29 47
94 27 4]
Average | N 21° E Byrrmenre p. hour.

Here it appears that the average residual current, which — as we
mention in passing — has at this point quite another direction than
at H2, even from year to year does not at all-remain constant in
strength which may perhaps be an indication for differences in the
quantity of Atlantic water, entering through the English Channel
from year to year.

The question may be put, whether and how far the results attained
by the current-measurements described, deviate from what is known
from the charts, in general use, about the currents near.the station H2,
The subjoined table allows of a comparison with statements, bor-
rowed from a chart, published by the British Admiralty *), and shows

From the Charis. Observed.
Hour. | Direction. | TEN Direction. | (alma

5 before H.W.Dover| N 90° E 0,3—0.2 N Er | 0,3
4 110 0,5—0,3 115 | 0,4
3 135 | 09—0,6 | 147 0,4
9 160 ; 0604 189 03
| 180 0,3—0.2 297 O4

H. W. Dover — O 266 0.5
4 after H.W. Dover 260 0 °—0,2 | 280 05
9 300 06—0,4 | 296 | 0,6
3 300 | 4,0—0,7, «| 331 05
4 315 | 06—0,4 | 342 0.4
5 0 03—02 | 9 | 0,4
6 50 | — | 40° 0,4

1) Tidal Streams North Sea 1899.

——s;
han
i

(573 )

that the deviations for a part considerably exceed the limits of aceu-
rateness of the statements.
It should be observed that the charts refer to currents near the

surface, whereas the values of the table derived from our observations

refer to a depth of 5 M.

Finally we may mention that the observations at station H2 up
till now have been continued in the same way, that is to say, they
are still made every quarter of a year, as far as possible, during
24 hours. Moreover, owing to the kind co-operation of His
Excellency the Minister of Marine, a current-meter of PETTrrRsson has
been placed on the lightship ‘Noord-Hinder’, with which since
November 1906 daily, in so far as the state of the weather permits,
with intervals of three hours, measurements at various depths are
made by the ordinary staff of the lightship. The lists of observation
are forwarded to the ‘Rijksinstituut’’ and promise to yield important
material, especially for the inquiry into the way in which the tidal
and residual currents differ in layers of different depth.

Mathematics. — “The locus of the pairs of common points of
n+1 pencils of (n—1)-dimensional varieties in a space of
n dimensions.” By Dr. F. Scnvu.

(Communicated by Prof. P. H. Scuoure).

1. Let (V;)@=1,2,...,n+1) be n+ 1 pencils of (n — 1)-
dimensional varieties in the space of operation Sp” of 7 dimensions and
let 7; be the order of the varieties V;of the pencil ();). Let moreover
a; be the number of points of intersection of the m varieties
ee pe eee ee ee not of necessity lying.in
the base-varieties.

When considering the locus of pairs of points P,P’ through which
a variety of each of the pencils passes we have exclusively such
pairs in view of which neither of the two points lies of necessity
on a base-variety of one of the pencils and we call the locus thus
arrived at the /ocus proper L.

We determine the order of Z out of its points of intersection with
an arbitrary right line /. To this end we take on /an arbitrary point
Qio...» and we bring though it varieties V,,V,,V,,..,V,, having
a4+,—1 pomts of intersection not lying on Qjo.. , and the base-
varieties. Through each of those points we bring a JV,,4; and arrive
in this way at a,4;—1 varieties V,,41 intersecting together line /
IN (n41—1) 7,41 points Q,41.S0 to Qis..., correspond (4,41—1)rn4i
points Q,+1.

To find reversely how many points Qre. _.n correspond to Qn41
we take arbitrarily on / the points Q;41, Q42, Q4s,.-., Q,41 and
we bring through those points respectively a V4.3, ] Fda ia heen

+1. We now put the question how many points Qi23...; lie on
/ in such a way that the varieties mentioned V;+.1, Vi+-9,.-,Vn41
and the varieties J,,V,,..,Vi passing through Qjo3...; havea com-
mon point not lying on the base-varieties. For ¢< the answer is:
a,7, +a,r, f+... far.

To prove this we begin by noticing that the correctness is imme-
diately evident for ¢= L. If we now assume the correctness for ¢ = ),
we have only to show that the formula also holds for ¢= 7 + 1.
Given the points Q;+9, Qj+3 ania (Oa = To determine the number
of points Qios3...;41 Wwe take on / an arbitrary point Qi23...;, we
bring through it varieties V,,V,,...,Vj; and then through each of
the aj41 points of inter sacha (not lying on the base-varieties) of these
ey @,.<°, V3 and the varieties V;42, Vj+s,..., Vn+-1 resp. passing
through Q,42,Q;13,.--,Qe+1 we bring a variety V;+ ; these ajay
varieties V;4, cut / in a;4 1 rj41 points Q;11. So to Qios... ; corre-
spond wj417);41 points Q;+, and (according to the supposition that the
formula holds for ¢=)) reversely to Q;41 correspond a,r,--a,r,-
+ ...+a,r; points Qias...;- So there are a,r,;—-+ a,r,-+..-=p
+ ajrj+ aj4i7)41 coincidences Qios...; Qj41; these are the Aounte
Qi2...; +1 belonging to the given aie Q;4-0; Qj2-3;-2- 5 ae

in this way the correctness of the formula has been indicated for
b= — iE .

When asking after the number of points Qis..,, corresponding to
Q,4, we have +=, so that the formula furnishes a,7,-+ a,r,-+

+ ....+a,r, for it. This number must however still be diminished
by rnti, as each of the points of intersection of / with the V,z4;
passing through Q,.1 is a point of coincidence Qio3....,—1 Q, but

not one of the indicated points Qis...n.

So on / there exists between the points Qio,.., and Q,41 an
(Qn Tal — Tn 41, GP, + 4%, +... + ayn —7n+1) Correspondence.
The ar, + a,r, +... Gn417+1— 27n+1 coincidences are the
points of intersection of / with the locus Z to be found and the
points of intersection of ¢/ with the oe ae variety of

contact RVie..., of the pencils (V,), ( ,(V..); we understand
by that variety of contact the locus of te ean) oie ne rarieties
eb a ee Oe ee through them have a common tangent, so
where she (7 —1)-dimensional tangential spaces of those varieties

eut each other er {o a line.

( 575 )

2. To determine the order of PR Wy2..., we must observe that RVys___,
is the locus of the points of contact of the varieties Vj, with the
curves of intersection Cis.. ,—1 of the varieties V,, V,,...., Vin—y.
So the question has been reduced to that of the order of the variety
of contact of a system of o' (7% —1)-dimensional varieties and
system of o”—! eurves. That order can be determined out of the
points of intersection with an arbitrary line /.

In a point of intersection of / with a variety of the system we
bring the (2 — 1)-dimensional tangential space Sp"—! and in a point
of intersection of / with a curve of the system the «#"—? tangential
spaces Sp"—!'. If we act in the same way with all varieties and
eurves of both systems, then the tangential spaces of the varieties
furnish an 1-dimensional envelope FE, (i.e. a curve) of class w+ v (as
is evident out of its osculating spaces Sp”—! through an arbitrary
point of /) with v osculating spaces Sp"—' passing through 1; here
mw is the number of varieties of the system passing through an
arbitrary point, and py that of the varieties touching an arbitrary right
line. The tangential spaces of the curves in the points of intersection
with / have an (n —1)-dimensional envelope E, of class g + w with
/ as w-fold line, where g is the number of curves of the system
passing through an arbitrary point and wp that of the curves touching
an arbitrary space Sp”—! in a point of a given right line of that
space; for, if we bring through a point Q of / an arbitrary Sp"—2,
then each of the g eurves of the system passing through Q furnishes
a tangential space Sp"—' passing through this Sp"—? whilst the space
Sp"—' determined by / and Sp"—? (just as every other Sp'—! passing
through /) is wp times tangential space of the envelope.

Both envelopes have thus (f + v) (g + w) common tangential spaces
Sp'—'. Each of the » osculating spaces Sp"—! of /, passing through
i is a y-fold tangential space of E,, so it counts for » common
tangential spaces; so that ug + wap + ry common tangential spaces
not passing through / are left; these indicate by their points of
intersection with / the points of intersection of / with the variety of
contact, so we find:

The (n—\)-dimensional variety of contact of an o' system of
(n—1)-dimensional varieties of which wu pass through a given point
and v touch a given right line, and an x—1 system of curves of
which p pass through a given pomt and w touch a given space
Sp’! im a point of a given right line of that space, is of order

ap + vp + my.

3. With the aid of this result it is easy to determine the order

( 576 )

of the variety of contact (locus of the points with common tangent)
of » simple infinite systems (@,, 1,), (Ug: Pads +++ +s (Uns Pn) of (n—1)-
dimensional varieties.

This order is

Y YP Pn
1 a. hs !
ff, +. <> fin +—+...4 +n—l1}],
Uy, i, Un

as can be shown by complete induction. The formula holds for 7 = 2.

We assume the correctness of the formula for n =z and out of
this we must find the correctness for n=72-++ 1.

The variety of contact for 7+ 1 systems in Sp'+! is the variety
of contact of the system of varieties (4#,, »,) and the system of curves
formed by the intersections of the 7 remaining systems of varieties.
So we have:

k=, » 2S? eee

The points of contact of the curves of the system with a given
space Sp’ form the (¢—1)-dimensional variety of contact of the sections
of Sp’ with the systems (u,, »,), (Uz, 3), +--+ » (4i+1, Pi+1); these sections
are likewise systems (u,,,),-- ++» (iti, Pi-1), but of (7—1)-dimen-
sional varieties. The variety of contact mentioned is according to
supposition of order

tate tin ( PEt. EE ed),
U, Hs Mil |

The points of intersection of that variety of @ontact with a right
line 7 of Sp' being the points of / in which Sp' is touched by
curves of the system, we have:

Y= tat wigs (22 bE a i 1)
Cae (ip

Thus according to the formula py + rg + py the order of the
i-dimensional variety of contact of the 7-++1 systems of varieties
becomes

totes stot (2 Spr =...+4 rit 3)

My Hi+i
by which the correctness Sf the same formula for n =7-+ 1 has
been demonstrated. So we find:

For n a systems (Uys P1)> (las Padres +: 9 (ny Pn) of (n—1)-dimen-
sional varieties the locus of the points where the varieties of the
systems passing through it have a common tangent is an (n—1)-dimen-
sional variety (variety of contact) of order

a] Pe Py
ft, pls te 2. $e +n—1).
i; Bs Hn

( 577 )
If the systems are pencils, then
me ey Mg 2 (r;—1);
thus the order of the variety of contact RVy2_.., ts:

2(r7,t+7+..-+7)—n—1.

4. Returning to the correspondence between the points Qjs..,, and
Q,+4: we find for the number of coincidences which are points of
intersection of / with the demanded locus L/, i. e. for the order of L:

a,?v, ar Oy Mo ee | On Tat 1 2 (, Se tae ects ots 1) a

i=n+l
+n+1=> 2 {(a; — 2)7; 4+ 1}.
ei)

It is easy to see that a base-variety 6; of the pencil (V;)
is an (a; —1)-fold variety of LZ. The tangential spaces Sp"—! of L
in a point P of 4; are the tangential spaces in P of the varieties
V;, which are laid successively through one of the a@—1 points of
intersection (not lying on P and the base-varieties) of the varieties
eee Fy, Ve43,-.-; Vay passing through P.

So we find:

Given n-+1 pencils (V;)(¢=1,2,...,n +1) of (n —1)-dimen-
slonal varieties in the space of operation Sp". Let 7; be the order
of the varieties of the pencil (Vi) and a; the number of the points
of intersection (not lying on the base-varieties) of arbitrary varieties
mueereemee( .). (V.),:--,(Vi—i); (Vi+1),---,(Vn41)- The locus
proper of the paws of points lying on varieties of each of the pencils
is an (n — 1)-dimensional variety of order

rar
= {(a;— 2)7,4+1
i=1
having the (n — 2)-dimensional base-variety of pencil (Vj) as (a;—1)-
fold variety.

If n> 83, then also in the general case the base-varieties of the
different pencils will intersect each other. In like manner as we
have dealt with pencils of surfaces') we can also determine the
multiplicity of common points, curves etc. of base-varieties.

’
$>

Sneek, Jan. 1907.

1) See page 555.
39
Proceedings Royal Acad. Amsterdam. Vol. IX,

( 578 )

Astromony. — “On the astronomical refractions corresponding to
a distribution of the temperature in the atmosphere derived
From balloon ascents.’ Preliminary paper by H. G. van DE
SANDE BAKHUYZEN.

1. The various theories of the astronomical refraction in our
atmosphere consider the atmosphere as composed of an _ infinite
number of concentric spherical strata, each of uniform density, whose
centre is the centre of the earth and whose densities or temperatures
and refractive powers vary in a definite way.

The various relations between the temperature of the air and the
height above the surface of the earth, assumed in the existing theories,
are chosen so, that 1st they do not deviate too far from the suppo-
sitions on the distribution of the temperature in our atmosphere,
made at the time when the theory was established, 2™¢ that the
formula derived from this relation for the refraction in an infini-
tesimal thin layer at any altitude could be easily integrated.

At the time when the various theories were developed, only little
was known about the variations of the temperature for increasing
heights, and this litthke was derived from the results of a small
number of balloon ascents and from the observations at a few mountain-
stations. In the last decade, however, ascents of manned as well as
of unmanned balloons with self-registering instruments have greatly
increased in number, and our knowledge of the distribution of the
atmospheric temperature has widened considerably, and has become
much more accurate. Now I wish to investigate, whether by means
of the data obtained, we can derive a better theory of refraction, or
if it will be possible to correct the results of the existing theories.

2. The temperatures in our atmosphere at different heights have
been derived from the following publications :

I. Ergebnisse der Arbeiten am aéronautischen Observatorium Tegel
1900—1902, Band I, II and III.

II. Travaux de la station Franco-scandinave de sondages aériens a
Halde par Teisserene de Bord. 1902—1903.

ILI. Veréffentlichungen der internationalen Kommission fiir wissen-
schaftliche Luftschifffahrt.

From the last work I have only used the observations from
December 1900 till the end of 1908.

I wished to investigate the distribution of the temperature up to
ihe greatest heights, and therefore 1 used for my researches only
the balloon ascents which reached at least an elevation of 5000 meters;

and, following HercEseLi’s advice, I have used only the temperatures
observed during the ascents, as during the descents aqueous vapour
may condense on the instruments.

It is evident that for the determination of the refraction, as a cor-
rection to the results of the astronomical observations, we must
know the variations of the temperature at different heights with a
clear sky. For the temperatures, especially of the layers nearest to
the surface of the earth, will not be the same with cloudy and
uncloudy weather, as in the first case the radiation of the earth will
lower the temperature of those layers, and so cause an abnormal
distribution of temperature. It is even possible that in the lower
strata the temperature rises with increasing height, instead of lowering,
as is usual.

For this reason I have divided the balloon ascents into two groups,
1st those with a cloudy sky, 2™¢ those with a clear or a partly
clouded sky.

In working out the observations, I have supposed that for each
successive kilometer’s height the temperature varies proportionally
to the height, and after the example of meteorologists, I have deter-
mined the changes of temperature from kilometer to kilometer. For
this purpose, I have selected from the observations, made during each
ascent, the temperature-readings on those heights, which corresponded
as nearly as possible with a round number of kilometers, and I
have derived the variations of temperature per kilometer through
division.

The available differences of height were often less than a kilo-
meter, especially at the greatest elevations; in those cases I adopted
for the weight of the gradient a number proportional to the difference
of heights. Sometimes on the same day, at short, intervals several
ascents have been made at the same station, or at neighbouring
stations, from which the variations of temperature at the same heights
could be deduced. In these cases I have used the mean of the
results obtained, but I assumed for that mean result the same weight
as for a single observation, as the deviations of the daily results
from the normal distribution of temperature are only for a small
part due to the instrumental errors, and for the greater part to
meteorological influences.

3. The observations which I have used, were the following:

_ from publication I, 31 ascents of which 12 had been made in pairs

on the same day, so that 25 results were obtained ; from publication

HI, 38 ascents all on different days; and from publication II, 170
39*

(580)

ascents distributed over 119 different days; — I have disregarded the
observations marked as uncertain in this work. On the whole I have
obtained the results on 182 different days, of which 58 with unclouded

and 124 with clouded sky.

The temperature gradients for each month were derived from this
material, and to obtain a greater precision, I have combined them in
four groups, each of three successive months, December, January and
February (winter), March, April and May, (spring), June, July and

August, (summer), September, October, November, (autumn)..

TA BA ae

Variations of temperatures per kilometer.

(V.T. Variation of temperature per kilometer; 'N. Number of observations).

A. Clear sky.

Kil V.T. N. VT. |

o—-il4 fe] w | 3.6)

42.9 124.9) 50 Ie

2 3ll-s.2] 1 |-_ 49] as | 4a} as | ae) as = ah ss
g- 41s i 10 |- 5.8] 15 | 5.4 18 a 15 ||— 5.5] 58
4A— 5 |}— 5.3} 10 |I— 6.7] 14.3 |i— 5.9, 18 5.7) 14.9 ||— 5.9) 57.2
5— 6 ||— 5.6] 89 | 7.4] 13.6 — 6.0) 18 7.3] 138 |- 6.5] 543
af 5.8 Be 7.5 ap 6.6 17.3 6.7| 10.1 ||— 6.7| 48.1
7— 8||- 6.8) 7 || 7.8 10.8 ||— 7.5) 146 8.0) 8 ||— 7.5) 404
s—g|-_7.6 5 || 6.4 28 ||-- 7.4] 133 |- 8.4] 8s |t 7.3] 341
910 ||— 5.9} 4 | 4.4) 57 |/- 7.2) 13 6.9} 7 ||— 6.4 297
1011 ||— 3.81 29 ||- 2.8) 5 I-68] 104 6.4] 68 |I- 5.4] 25.1
M—12 |i— 6 2] 2 — 2.4 26 |i— 5.9) 52 2.0)>59 |= 335) 154
12-13 |i— 1.6 2 jH-2.0|; 1 —1.4) 2 1.0; 49 |i— 0.7) 99
3-14 47.0) 7 feaoes 4.0| 16 ||—0.8| 46
4415 | 2 0.7] 16 Bid) <a: i i
10 | | | 10,8 8 + 0.8} 1

APS

( 581 )
B. Cloudy sky.

| Winter. | Spring. Summer. | Autumn. | Mean.
Kil. | VT. | N. | vt.| N. |i vr] Nn |ivrl on. fl ver. | N.
| | |
o-ale tal a |e ssl a3 |_oa os — 3.9] 40 _ £sl to
4— 21l— 3.0] 27 5.6] 325 |l— 5.4] 24 | 3 i 49 ||— 4.3) 1235
ee eeneeee sels 54) 2¢ || 4.3] 40. | 4.5] 124
s—3ie5.8\ 2 |l-5.5| 2 |l-5.4| 238 |-5.8) 905 | 5.6) 1233
£— 5 6.8) 2) |= 6.7). 38 — 6.4 2), i= 6.4| 39 |-- 6.4) 122
5— 6 |l— 6.9} 26 |i— 6.7) 30.7. | 6 7 215 ||— 6.2) 365 || - 6.6) 114.7
6— 7 ||— 6.8 254 |} 6.7 2 |] 6.6) 177 |— 7.3 218 ||— 6.9) 95.9
7— 8 || 6.9| 197 | 7.9) 203 | 7.9) 168 || 5.9| 216 | o6.8| 784
s—9||_6.4| 142 | 6.01 162 | 7.9| 141 || 7.9| 13 || 6.9| 575
910 ||-. 6.2) 123 ||— 3.9| 129 ||- 8.4] 121 ||— 7.5] 114 |- 6.5) 487
40—14 |l- 5.4 94 |1-- 1.8] 96 || 5.9) 81 |l- 5.4] 85 ||— 4.5] 356
14—12 ||J— 2.5 76 |}4 1.0) 83 AI 5.1 |i— 1.9 68 |i— 1.2) 278
12-13 | 1.3] 5 |} 4.9 67 |1+0.2| 19 || 0.5] 41 JH 0.4) 177
1344 |l-_ 0.91 27 |—3.9| 1 47) pa ||— 0:8) SA
14—15 | 1.9) 19 |-- 3.2) 1 | 4+. 0.2} 29
1516 |I— 0.6) 1 |- 3.2) 05 = 4.5} $5
16—17 $04 0.8 | 4.0.4] 08

We may derive from these tables that the mean variation of
temperature with clear and with cloudy weather only differs in
the lower strata, but is nearly the same in the higher ones.

In order to deduce from the numbers in this table the temperatures
themselves from kilometer to kilometer, I have also derived from
the data the following mean temperatures at the surface of the earth:

clouded sky clear sky
Winter + 0°1 — 0°.9
Spring + 64 + 51
Summer +14 .4 + 14 .7
Autumn + 9.0 + 79

By means of these initial temperatures and the gradients of table I

( 582 )

C. Cloudy and uncloudy sky.

Winter. | Spring. | Summer. | Autumn. | Mean.
Kil. || V.T. | N. (Hann, V.T.| N. | Hann.| V.T. | N. 'Hann,|] v.7.| N. |Hann.| v.7.| N. | Hann.
as | |
o—al_Polsr | +8sl_Sslas | Sal Paleo | 24 55 |_Sull_ Psi ise |_Po
5 31 | —2.9/| 5.6) 475) —5.5]| 4.7) 42 | 5.6 55 | —4.1]}-4.3] 1815|-4.4
4.97/31 |-—5.0|—4.9} 48 5.44.8) 42 et 55 | —4.8-4.7| 182 |—5.0
5.71/37 | —5.8ll-5.6148 | —5.sl|_5.9 418 25 545) —5.8||-5.5| 181.3|—5.7
—6.4|37 | —6.7]|-6.7) 473| —6.7||6.0| 41 | —5.9|-6.0/539| —5.9l 6.9] 1792'6.3
—6.7/349 | —6.7||-6.8) 443 —7.3|-6 4/395) —6 5, 50.3| —6.8—6.6| 169 ae
—6.6| 33.4 | —6.7||-7.0| 37.7) —7.9||-6 635 | 7 1| 37.9) —7.4]|—6 9} 144 |—7.0
—6.9| 26.7 | —7.9||-~7.4] 31.1 ).—6.3||-7.4| 31.4] —7 29.6| —7.3||--7.3] 1188 |—7.4
—6.5) 19.2) —6.9//6.1/24 —6 |—7.6|27.4| —7 21 | —7.6|--7.4| 91.6 |—7.4
—6.2| 163) —6.4||-4.0| 186) —4.8 meas —6.9|—7.4 184) —6.6) 6.5) 78.4|—6.3
5.0) 123) —3.9}—2.0) 146 | —0.9} 6.4) 185, —5.0 | 15.3] —6.4||—4.9] 60.7|—4.0
—2.6| 96, 0.0 ||—0.2) 10. | +0.5)|—4.0) 103) —9.4 | 12.7) —2.7||—2.4| 435/—1.2
4.9] 7 4.3 7.1 0.5) 39 9 0.9, 276,
—0.9 27) 1.6] 2 | LA.0 2 3 —0 8} 97]
146-15)|41.9 19 3.9) 1 40.7] 1.6 1 | 06, 55
15—16|0.6 1 ia 05 10.8] 1 —0.6| 25
1617)-40.1 08. | | +o. 08
| |

which in a few cases have been slightly altered, I have derived the
following list of temperatures for clear weather from kilometer to

kilometer.

Although the adopted values for the temperature of the air above

13 kilometer are not very certain, yet the observations indicate that
at these heights the temperature decreases slowly with increasing
height. The refraction in those higher strata being only a small
part of the computed refraction, nearly '/,,, an error in the adopted
distribution of temperature will have only a slight influence on my
results.

1 must remark that almost all the observations have been made
during the day, generally in the morning. It is evident that the varia-
tion of temperature, especially near the surface of the earth, is not
the same during the day and during the night, but the number of

( 583 )

Ay E be, i:

Temperatures at heights from 0 to 16 kilometer for clear weather.

|
| Winter. — | Spring. | Summer. | Autumn. | Mean.
| T | j Tr
Height. Temp.; Diff. |/Temp.| Diff. age | Diff. |Temp.| Diff. ‘Temp. Diff.
| | |
° fe} ° O° °
Oe, sf ty A) 4414.7 + 7.9 4+ 6.4
1.2 | —3.6 —2.8 +0.6 —1.1
ees Sey | 2503 441.9 4+ 85) + 5.3
—4.2 | —5.4 =A3 —3.2 || 4.3
i490 — 3.9 7 6) 5.3 + 1.0
sas. | —4.9 v —44 ie —4.6 Bee —4 8
oa Se — 8.8] Pe ia.2 Oi ae
—5 4 so | —5.4 =e sot
& |—45.5 —14.6| See — 4.9 — aes
—5.8 ECW, | —5.9 | 63 —6.1
eS p=o4 3 aed —11.0| 45.4
60 ey —6.0 —6.9 | —6 4
6 |—27.3) —28.0) 144 —17.9 —21.8
) —6.2 |_6 9 —6.6 Peay ear eel
7 \—3355| —34.9} 2307 ~ 95.4 98.5
| <3 yen ges aes = 77 | Bar
8 |—40.3 ae —928 .0| —32.8 —35.8
| 78 | 6.9 |: 7.6 | 7.6 baa a
9 |—47.6| —49 4 —35.6 -40.4 | - 43.2
| —6.4 [54 0: —6.9 || —6.4
10° .|—54.0 —54.5 —49, 8 —A7 3 —49 6
| 4.9 | 95 =26/8 peas | 54
44 |—58.9] —57.0} —49.6 —53.4 —54.7
| —2.4 _—e | —4.0 20 = 93
42 |—61.0 —58.0) —53.6 —55.4 57.0
Le iat ALO =) | —1.0 || =4.0
13 | —62.0 —59.0 —54.6 —56 4| 58 0
—0.6 =0.6 | —0.6 | —0.6 || —O0.6
44 |—62.6 —59 6 55.2} —57.0 —58.6
| —0.4 | —0.4 | —0.4 | —0.4 —0.4
45 |—63.0 —60.0 55.6 —57.4| —59.0)
| —0.2 —0.2 =. Pee ece
46 |—63 2 —60.2 —55.8 |—57 6) 59.2
| l
observations was not great enough for a reliable determination of

this difference. Lastly I remark that the various balloon ascents have
been made from different stations, Halide (in Danemark), Berlin, Paris,
Strasbourg and Vienna and conseyuently the given values do not
hold for one definite place, but for the mean of the area enclosed
by those stations.

After I had derived the temperatures given in table II, I got notice
of two papers, treating of about the same subject, namely: J. Hany,
Ueber die Temperaturabnahme mit der Hohe bis zu 10 Km. nach
den Ergebnissen der internationalen Ballonaufstiege. Sitzungsberichite
der mathematisch-naturwissenschaftlichen Klasse der K. K. Akademie

der Wissenschaften Wien. Band 93, Abth. Ila, 8.

571; and S.

(584)

GRENANDER. Les gradients verticaux de la température dans les mmima
et les maxima barometriques. Arkiv for Matematik, Astronomi och
Fysik. Band 2. Hefte 1—2 Upsala, Stockholm.

Of the results which Hann has given, up to a height of 12 kil.,
I have taken the means of groups of 3 months, which are printed
in table I by the side of the values I had obtained ; the agreement of
the two results, which for the greater part have been deduced
from different observations, is very satisfactory.

GRENANDER in his paper chiefly considers the relation between the
changes of temperature and the barometer readings; his results cannot
therefore be compared with mine directly, but probably we are most
justified in comparing the variations of temperature at barometer
maxima, with those which I have computed for clear weather. For
great elevations, till nearly 16 kil., GrenanpEr also obtains with
increasing height a small decrease of temperature.

It is difficult to state with what degree of precision the tempe-
ratures of table II represent the mean values for the different seasons;
the deviations, especially at great heights, may perhaps amount to
some degrees, but certainly they represent the mean distribution of
temperature better than the values adopted in the various theories
of refraction, and we can therefore derive from them more accurate
values for the refraction.

4. It is hardly possible to represent the relation between the
temperatures in table II and the heights by a simple formula, and
to form a differential equation between the refraction, the zenith distance
and the density of the atmosphere at a given height, which can be
easily integrated.

Therefore I have followed another method to determine the refrac-
tion corresponding to the distribution of temperature I had assumed.

According to Rapav’s notations (Essai sur les refractions astrono-
miques. Annales de l’Observatoire de Paris. Mémoires Tome XIX), the
differential equation of the refraction, neglecting small quantities, is :

l
| (: _ RY — 3 so) dw
ds = a’ ——____—— 7 - i oes

Here is:

R radius of the earth for 45° latitude,
yr, radius of the earth for a given point,
h height above the surface of the earth,

( 585 )

rr, th,

uw, index of refraction at the surface of the earth,
| eee: # » » height A,

0, density of the air at the surface of the earth,

density at the height 4,

» temperature at the surface of the earth,

/, height of a column of air of uniform density at 45° latitude,
of a temperature 7¢,, which will be in equilibrium with the pressure
of one atmosphere, the gravity being the same at different heights.
According to Rerenavuit’s constants, we have /, = 7993 (1 + at,)
meter, if @ represents the coefficient of expansion of the air.

Between these quantities exist the following relations:

Q
t

Q
uw? =1-+4 2co (ce being a constant), o — 1 — —
Qo
0, . a R Rh
Clie ed a Sl ET eS = (2 = -——-
1+ 2co, sin 1 l, (7, +A),

To determine the value of ds at each height, we require a relation
between w and y or between @ and /, which can be obtained when
we assume that the temperature varies according to Ivory’s theory,
or that the temperature varies as represented in table II. For the
same given values of z and w, the two values of ds in formula (I)
can be computed by means of the first and by means of the second
supposition, and the differences of these two values of ds can be
found. By means of mechanical quadrature, we can then determine
the differences As of the refractions s according to Ivory’s theory
and according to table IT.

The relations between y and w may be found in the following
manner.

5. If in a given horizontal initial plane, at a distance 7, from the
centre of the earth, the pressure is p,, the temperature f, and the
density of the air 9,, and in another horizontal plane, 2 kil. above the
former, the pressure is p, the temperature f, the distance from the
centre of the earth 7, and the density of the air oe, then we have
(see RApDAv) :

) a) da ah R Ri 7 D Rh
l,d (“) = (“) eS cae d ( ) or "0 d (4) == is d (=) ’
Po Qo\r Qo% Ar RK \Po eo or

: fe) Rh
or, putting ~ = yj and

0, Gant”

Ra(2 == nays Sarat see Meee? (EL)

( 586 )
further is :
L

Pict ieee a (& at, = y = (1—B) 9, See (II)
Po aes 1l-+at
a (¢ Neans ted

ea ae
When dividing equation oye by (III), we get:

if we put

ay
Po dd dy
At —
p 1—vd 7
Po
From the two last equations follows:

=% ao + a—m) I. . @y)

According to Ivory’s theory a where 7 is a constant value
(RapAvu assumes 0,2); if we introduce this relation into the equation
(111) we obtain after integration :

yj = 0,4 w = 1,8420681 Br tog (1) ee
: R moe

By substituting (VY) in (1) we can therefore calculate for each
value of w the value of ds according to Ivory’s theory.

6. Now I proceed to determine the relation between @ and y
according to the temperature table II.

Of two horizontal planes, one above the other, the first is situated
n kil. (2 a whole number), the second n’ kil. (n’ = or <n-+ 1)
above the surface of the earth; their distances from the centre of
the earth are 7, and 7,, their temperatures ¢, and ¢, and the values
of y, y, and y,. The temperature between 2 and vn’ varies regularly
with the height and, to simplify the formulae, I suppose ¢, — ty

: * a (¢ —t, ye
proportional to 7, — y,, 80 that, if 3, = — Se

1+-at,,
R . :
= (Yt — Ue) SS Cn. SD eee ee

R
Hence follows — dye, d% and after substitution of dy in (IV)

and integration
Piast —9,) —Wg(l— ow). . ._. (VID

in which 1—o represents the ratio of the densities in the two
horizontal planes.

If we substitute »-+ 1 for n', we can find in table II the tem-
perature for the two planes and hence also %,; as y, and y,4) are
also known, we can derive from (VI) the value of ¢, and we can
deduce from (VII) the ratio of the densities in those planes. By put-
ting for mn successively O, 1, 2, ete. we can consiruct a table con-
taining the densities of the air, D,, D,, D, etc. at the height of
1, 2, 3, ete. kil. above the surface of the earth, the density at the
surface being unity.

It is easy to derive from this table the height of a layer of a
given density d. If d< D, and > D,+41, the layer must be situated
between » and n+ 1 kil., and we oniy want to know in which
manner, within this kil., the density varies with the height / above
the lower plane.

We may assume:

a — |(0—a,

Peel eo — J), 4.\,. hence a = — lg —_—.

a being known, we may determine for each value of d,/ and
also y. By substitution in (1) we find then for each value of o the
value of ds.

7. Now we are able to form the differences of ds after the theory
of Ivory and after the table of temperatures LI, for values of w which
increase with equal amounts, and then determine the whole diffe-
rence of the refraction for both cases.

For great values of < and small values of y and o the coefficients
of dw in (1) will become rather large, which derogates from the
precision of the results.

This will also be the case when the differences of the successive
values of w are large; small differences are therefore to be preferred,
but they render the computation longer.

Both these difficulties can be partly avoided if, according to
Rapav’s remark, we introduce [/w asa variable quantity instead of o ;
the value of ds thus becomes:

( 588 )

" Ge tiga

ame = Va ed — coz + y

or approximately :

(VID)

“w

‘Ve Var |

It is evident that for small values of @ the coefficient of dV@ in
(VIII) is smaller than that of dw in (I), and that also the refraction
in the lower strata will be found more accurately by means of the
formula (VIII) than by means of (I). For if we increase V@ in
formula (VIII) and w in formula (1) with equal quantities, beginning

with zero, we find, that, from wo =O to w= 0,2, the number of

values in the first case is twice as great as in the second case, hence
the integration by means of quadrature will give more accurate
results in the first case.

Therefore I bave used the formula (VIII) and computed the coef-
ficient of dV w for values of Wo, 0, 0,05, 0,10, 0,15... to 0,95.

The density of the air which corresponds to (/w = 0,95, occurs
at the height of about 18 kil. From the observations at my disposal
I could not deduce reliable values for the temperature at heights
above 16 kil.; yet it is probable that the gradients at those heights
are small and I have assumed the temperature at heights of 17 and
18 kil. to be equal to that at a height of 16 kilometers.

In this way I have determined by means of mechanic quadrature,
and an approximate computation of the refraction between V @ = 0.925
and w= 0.95, the differences As of the two values of the refraction
corresponding to Ivory’s theory and corresponding to the table of
temperature IL in the part of the atmosphere between the earth’s
surface and a layer at a height of about 18 kil. where /’o@ is 0.95.
I have worked out this comptuation for the zenith distances 85°
86°, 87°, 88°, 88°30’, 89°, 89°20’, 89°40' and 90°.

An investigation, made for the purpose, showed me that in for-

lo
mula (VIIT) the terms re (y — 3ew) in the numerator

in the denominator may be neglected for all zenith distances except
z= 90°; therefore | have taken them into account in the computation
of the horizontal refraction only.

The

( 589 )

The results which I have obtained for the differences -
4 s = Ivory—table of temperatures

are the following:

ee Wa Be TAT.

To test the computations, we may compare the mean of the values
of As for the four seasons, and the values of As in column 6
which have been computed, independently the former, for the mean
yearly temperatures, which are almost equal to the mean of the
temperatures in the four seasons. Only for z= 89°40' and z— 90°
do these values show deviations exceeding 0.1.

From table III follows, 1 that for a distribution of temperature,
as derived by me from observations, the refraction deviates percep-
tibly from that deduced from Ivory’s theory, 2 that the differences
in the refraction in the different seasons are about of the same order
as the deviations themselves. I want it to be distinctly understood,
1 that the adopted distribution of temperature above 13 kil. and
especially from 16 to 18 kil. is uncertain, and 2 that I have not
taken into account the refraction in the layers which are lying more
than 18 kil. above the surface of the earth, in other words those
layers where the density, as compared to that of the surface of the
earth, is less than 1 —0,95?, or less than 0.0975.

Refraction after Ivory — Refractions after the table of temperatures IT.
— ——$$_____
Pee | | | eee aan ae
Bestance Winter | Spring Summer | Autumn aie ePx mes ot gs
| Winter | Spring Summer Autumn
85° + 021 | + 0"78 + 066] + 0731 | + 049 | + 028 | — 0799 | — 0117 | + O18
86° + 0.13 | + 4.26) + 0.95) + 0.30 | + 0.66 | + 053 | — 0.60 | — 0.29 | + 0.36
87° — 0.47 | + 2.08] + 1.31] — 0.20 | + 0.66 | +143 | — 1.42 | — 0.65 | + 0.86
880 — 3 93 | + 3.10) + 0.95] — 3.29 | — 0.83 | +340] — 3.93 | — 4.78 | 4 2.48
38°30' | — 9.64 | + 3.06| — 0.67| — 8.51 | — 3.95 | + 5.69 | — 7.01 | — 3.98} + 4.56
9° | 7-93.69 | + 1.08) — 5.45] —91.45 | 12.31 | 444.38 | —13.39 | — 6.86 | + 8.84
89°20’ | —43.80 | — 3.17) —12 68) —38.77 | —24.51 | +19.29 | —24 34 | —11.83 | +-14.26
gg |4"94"97, 43.07) —95.95|—1' 4446) —97.74 | 434.93 | —34.67 | 99.49 | $23.42
oO" |—2 32.4 | —33.4 | —52.9 |—2 9.6 —1'30"9) 1'1"5 | —57.8 | —38.0 | +38.7

( 590 )

Physiology. — “An investigation on the quantitative relation be-
tween vagus stimulation and cardiac action, on account of
an experimental investigation of Mr. P. Wo.rterson’’ ’). By
Prof. H. ZwAARDEMAKER.

(Communicated in the meeting of December 29, 1906).

The experiments were performed on Emys orbicularis, whose right
nervus vagus was stimulated by means of condensator charges and
non-polarising electrodes of Donpers*), while auricle and ventricle
were recorded by the suspension method. The mica-condensators had
a capacity of 0,02, 0,2 and 1 microfarad, the voltage varied from
a fraction of a volt to 12 volts, occasionally even more. From this
the intensity of the stimulus was calculated in ergs (or in coulombs
by Hoorwee’s method). Only a part of this energy, passing through
the nerve, when it is charged, acts as a stimulus. What part this is
remains unknown, but it is supposed not to vary too much in the
same set of experiments. In the typical experiments a summation
took place of ten stimuli, succeeding each other in tempos of 7/,
second; in particular experiments single stimuli or other summations
were investigated. Of fatigue little evidence is found with our mode
of experimenting, rather a somewhat increased sensitiveness of the
vagus system towards the end of a set of experiments.

Stimulation of the right vagus produces in the tortoise in the first
place lengthening of the duration of a cardiac period *), in such a
way that in the second period, after a stimulus, starting during
the cardiac pause, the diastolic half of the period is considerably
retarded, while in some subsequent periods a decreasing retardation
of the diastolic part of the period is noticed.

Then stimulation of the vagus causes contraction to become feebler,
this phenomenon becoming gradually more distinct and reaching its
maximum some periods after stimulation. This decrease of contractile
power is primary, since it may also occur when any change in the
automatic action is absent (e.g. when the stimulus consists of one
condensator charge and when the left vagus is stimulated). Finally
vagus stimulation as a rule produces slackening of the tonus, rarely
tonic heightening. Changes in conductivity were only observed once.

1) For details we refer to the author’s academical thesis, which will be published
ere long.

*) Onderzoekingen Phys. Lab. Utrecht (3) Vol. I p. 4, Pl. I, fig. 1, 1872.

5) The duration of a cardiac period is reckoned from the foot-point of a sinusal
contraction or if this is not visible, of an auricular contraction, to the foot-point
of the next following sinusal resp. auricular contraction.

(591 )

The negative chronotropy holds good for sinus, auricle and ven-
tricle to the same extent, the negative inotropy exists exclusively
for the sinus and the auricle, is mostly positive for the ventricle, if
it is found; the tonotropy is met with in auricle and ventricle.

A latent stage of the phenomenon, measured by the time-difference
between vagus stimulation and vagus action, was always observed.
It is smallest for the inotropy; already the first period often shows
an enfeeblement of the contraction, which in the subsequent periods
increases still further. The latent stage of the chronotropy is greater,
for only in the second, sometimes in the third period, a retardation
is noticeable; on the other hand this phenomenon reaches its maximum
at once. Inotropy and tonotropy do not coincide. On the contrary,
the maxima of effect form the following series as to time: first
maximum of chronotropy, then maximum of tonotropy, finally
maximum of inotropy.

In regard to the sensitiveness for vagus stimuli, we remark that
for the ipotropy the “threshold value” lies below that for the chrono-
tropy and for this latter lower again than for the tonotropy. So we have:

Threshold value for inotropy < idem for chronotropy < idem
for tonotropy.

From the fact that dromotropy did not occur in our experiments,
one would infer that the threshold value of the dromotropy lies
higher still in the present case. _

Physiologists are generally convinced that the rhythmic processes
at the bottom of the cardiac pulsations, are based on chemical actions
in the cardiac muscle. Leaving apart the founder of the myogenic
theory TH. W. ENGELMANN, we mention some authoritative writers,
Fano and Borazzi in Ricuer’s Dictionnaire and Hormann in Nacet’s
Handbuch, who embrace this point of view °*).

Also experimental results may be adduced in support of this
theory. SNypDER *) showed that the frequency of the contractions with
respect to temperature follows exactly the law, formulated by van
‘t Horr and Arruenius for chemical reactions *) and experiments,
independently made by J. Guwin, entirely confirmed this. ‘) Whereas
the influence of temperature is considerable, that of pressure is very
small. This agrees with the small significance of external pressure

for so-called condensed systems, i. e. systems in which no gaseous
phases occur.

') Fano and Borazz, Ricuet’s Dict. de physiologie t. IV. p. 316.

*) Snyper, Univ. of California Publications I. p. 125. 1905.

5) E. Couen, Voordrachten. Biz. 236 1901.

‘) J. Gewin, Onderzoekingen Physiol. Lab. Utrecht (5). Dl. VII, p. 222.

(592 )

For the automatism it seems to me to be settled, that it must be
based on chemical processes.

For the remaining cardiae properties: conductivity, local sensitiveness
to stimuli, contractile power and tonicity the decision is more
difficult. The law of van ’r Horr-ArrRHENIUs concerning the relation
between reaction-velocity and temperature can only be applied if
the duration of the reaction is known. Now the velocity of condue-
tion, measured with this purpose, increases with temperature up to
a certain optimum‘) whereas correspondingly the duration of the
contractions is diminished’). The local excitability, however, has
not been studied yet from this point of view, while also for the
contractile power the time factor is still lacking. But the contraction
of a muscle and also that of the cardiac muscle is so universally
considered a truly chemical process, that the reader will not object
to classing it among chemical phenomena without further arguments.
As to the tonicity we are absolutely in the dark, although we know
that rise of temperature chiefly brings about a change, in which
the tonus is definitely abolished.

In preparing his thesis Mr. Worrrrson had chiefly to deal with :

1. changes in the automatism (chronotropy) ;

2. changes in the contractile power (inotropy).

Both these changes are purely chemical phenomena, as was shown
above.

For chemical processes the law of GuLDBERG and Waagu holds *),
and we may apply this law to the processes here dealt with. For
this purpose we shall have to give a nearer definition for our special
case of the conception ‘times of equal change’.

By “times of equal change” we mean the times in which a defi-
nite reaction has taken place between two accurately fixed and in
the corresponding cases analogous terminal points. The total duration
of a cardiac period is such a characteristic time element, the begin-
ning and end of which cannot be determined with the balance after
chemical analysis, but still are determined by biological characteris-
tics. The time between the beginning and the end of a cardiac
period may be looked upon as a time of equal change provided no

1) Tu. W. Eneetmann. Onderz. Physiol. Lab. Utrecht (3d series) III p. 98. Above
the optimum the conductive velocity diminishes again.

2) Hormann I. c. p. 247. Recently confirmed by V. E. Niersrrasz; vide acad. thesis,
Utrecht 1907, p. 145, fig. 22: a fall in temperature of 9° gave an increase of
the duration of the systole to the double value.

5) E. Conen. Ned. Tijdschr. v. Geneesk. 1901, Vol. I, p. 58. Cf. also ZwAARDEMAKER,
ibidem, 1906. Vol. IL. p. 868.

(593 )

inotropic changes occur’) and the mechanical resistance which the
heart has to overcome, has remained the same.

These premisses made, we may at once apply the fundamental
equation of GULDBERG and Waacr’s law ;

== kCn

Here & is a constant, the constant of the reactional velocity, C is
the quantity of the substance, taking part in the reaction, 7 is the
exponent, determining the so-called order of the reaction, while ¢
indicates the reactional velocity. About the exponent » nothing can
be stated a priori for the heart. Toxicological experiments, in which
the quantity of the reacting substance diminishes, might perhaps
teach us something in this respect; perhaps also experiments on
fatigue might give us some clue; at present, however, no data at
all are available. Whether there are intermediate reactions and in
what number, cannot be ascertained. Under these circumstances |
assume, quite arbitrarily, that the present case is the simplest and
that the exponent is unity. If later this assumption turns out to be
wrong, our calculations will still apply, mutatis mutandis, without
losing their meaning. In this simple case the formula runs:

g=—kC.

When the vagus is stimulated a very marked alteration of the
times of equal change is noticed. The reactional velocity of the
hypothetical chemical process, which les at the bottom of the auto-
matism, must consequently undergo a very considerable change.
Such a change cannot take place unless either / or C' are modified.
In the literature on the subject both views are put forth, but only
the conception that / changes, leads to a clear explanation without
further auxiliary hypotheses. It also fits in best with a recent paper
of Martin *), according to which vagus-inhibition is aseribed to the
action of A-ions and is counter-acted by rise of temperature. The
significance of the ions of the alkalies and alkaiine earths for the
cardiac muscle is indeed by no means fully explained, even after
the mumerous investigations of J. Lozs and his followers and critics
— they are regarded by some as the cause of the continually
excited condition of the cardiac muscle, as the stimulus for the
automatism *), by others as the condition, necessary for keeping the

') In the ventricle vagus-stimulation produces no inotropy.
*) Martin. Amer. Journal of Physiol. Vol. XI, p. 370, 1904 (Martin himself seems
to assume a compound of K-ions with C).
3) Wencxesacu. Die Arythmie etc. Eine physiol.-klinische Studie. Leipzig. 1904.
40
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 594 )

active substances in solution!) — they certainly do not enter into
simple chemical combination with the cardiac substance, by which this
latter would become unfit. If this latter were the case, the life-
prolonging influence of Rrycxr’s solution and the remarkable anta-
gonism of Na and X on one hand and Ca on the other, would be
entirely unexplainable.

By placing the principal weight on the hypothesis that the vagus
alters the constant of velocity, of reaction we were led to the applica-
tion of the formula for the catalytic acceleration of a chemical reac-
tion. The catalytic acceleration is here negative. The explanation of
the formula will be found in G. Brepie’s work. It runs:

a7 Sp ee
Wa

By application to our experiments, the normal duration of the period
being indicated by (¢,—7,) the altered one during the principal retardation
by (t, —?,), a relation became evident which appears to be constantly
found between the intensity of the vagus stimulus on one hand and
the retardation, indicated by 6 on the other. (An examination of the
curves, recorded by the heart would show that the retardation affects
principally the diastolic part of the process, but since for this part,
taken separately, the times of equal change cannot be sharply deter-
mined, our calculations enclose the whole process).

When the vagus stimulus increases the retardation increases also
very gradually, until a definite degree is reached; from this moment
the reactional velocity of the hypothetical process of the automatism
remains the same, independent of any rise in the intensity of the
stimulus. Only by increase of the duration of the vagus stimulus, a
new retardation may be produced, which is pretty well proportional
to the extension of the duration of the stimulus. For a warmed
heart all this holds without any alteration.

1) H. J, Hameurcer. Osmotischer Druck und Jonenlehre. Bd. III, p. 127.

(595 )

Exp. 8, VI. 1906. Emys orbicularis. Right nervus vagus stimulated
on non-poiarising electrodes with charging currents. Capac. of the
condensator 1 microfarad. Number of stimuli LO (2 per second). Between
the series of stimuli pauses of 4 minutes; external temperature 18° C.

Initial § Total

Micro- : :
| Ergs 8
coulomb | | retard. in °/, | retard. in °/,
0.80 | 3.20 | 13 23 0.0392
0.82 3.36 | 92 143 0.1662
0.84 | 3.53 95 133 —0.1694
0.86 | 3.69 | 282 347 0.2555
Gee |r: 3.87- | 320) |} BB —0.2716
0.9 | 405 | 32 | 364 0.2635
0.92 | 4.23 329 360 —0.2648
0.94 4.49 346 64 | —0.9765
0.96 4.61 337 366 —0.9575
0.98 4.80 337 398 —0.2667
1.00 5.00 | 343 | 398 —0.2679
1.04 5 AA 333 394 —0.2570
1.08 5.83 346 40 —0.2765
1.12 6.27 333 367 —0.2661
1.20 7.20 | 330 322 —0.2480
1.28 8.19 | 346 373 0.2592
1.36 9.95 336 370 0.9575
1.52 11.50 343 374 0.2679
1.68 14.41 360 421 —0.2790
1.84 16.93 340 377 0.2673
3.68 67.74 S74 | 405 —0.2723
5.52 | 159.35 a.0O| 8 —0.2798
7.36 | 270.85 371 AMY —0 2723
9.20 | 493.90 357 377 —0.2702
41.04 | 609.40 333 347 —0.2661
0.80 3.20 330 343 | —0.2654
|

( 596 )

Exp. 15, VI. 1906. Emys orbicularis, Right nervus vagus. Non-
polarising electrodes. Charging currents. Capac. 0.2 microfarad. Number
of stimuli 10; (2 per second). Resting pauses between the series of
stimuli 2 minutes. Experimental animal in 0.6°/, NaCl solution,
heated to 28° C.

0.48 | 5.76 | 13 | 139 | 294 | —0.0785 | —0.3889
0.496 | 6.45 | 20) | 99 | S456, | 0: Sue
0.504 | 6.35 | 20 439 | 905 | 0.4114 | —0.3889
0.52 | 6.76 | 26 | 452 | 994 | —0.4404 | —0.4035
0.552 | 7.61 | 26 | 152 | 994 | —0.1404 | —0.4035
0.6146 | 9.48 | 26 | 152 | 218 | —9.1404 | —0.4035
0.74% | 13.83 | 98 | 499 | 970 | —0.1587 | —0.4762
1.446 | 31.43 24 157 | 270 | —0.1261 | —0.4579
5.58 [455.65 | 28 ely) | 984 | —0.4587 | —0.4579

Two particulars deserve notice:

1. that the greatest retardation falls not in the second but in the
third period. }

2. With stimulation with 7,61, 9,48, 13,83 ergs turbulent motions
occur in the ventricle, followed by the post-undulatory pause, namely
in the first systole after the preliminary retardation.

The relation brought to light in both these cases might be explained
by assuming with Laneiny that the vagus fibres do not reach the
heart directly, but first pass a station of the intra-cardial ganglia.
If this be the case the stimulated condition of the prae-ganglionic
fibres will only be communicated to the post-ganglionic by contact
in the ganglion cells. But then the quantitative coercion of WEBER’s
law holds for these ganglion cells and a relation as sketched above
is not astonishing. To this conception may be objected that probably
with stimulation of the post-ganglionic fibres (in the so-called n.
coronarius ') the same relation will be found in its principal features.
If on this point not only preliminary, but decisive experiments will
have been made, it will be found that the just-mentioned explanation

1) On the n. coronarius as a post-ganglionic nerve vide J. Gewin, |. c. 82.

(597 )

is untenable. Mr. WoLtrrson accordingly gives an alternating expla-
nation which, in my opinion possesses some probability, and which
agrees with Martin’s hypothesis on the nature of the vagus action.

Let us suppose that by the action of the vagus some catalytic
substance — say Martin’s A-ions — is produced in the receptive
substance of the cardiac muscle, then the above stated quantitative
relation will be explained, if we may assume that the substance,
produced by vagus action, is only to a limited extent soluble in the
medium. For with a small production of the catalyser this latter
will be dissolved and will increase the retardation, but when the
medium has become saturated with the catalyser, further secretion
is without effect. It must further be assumed that the newly formed
catalyser is at once removed from the substance by diffusion or is
deposited in the form of indifferent compound, for the vagus action
is known to cease after a short time. Only when the duration of
the stimulus is increased and catalytic substance is again and again
produced, the disappearance of the catalyser may be compensated
and the retardation may be lasting.

The second chemical process we meet in Mr. Wotrtrrson’s thesis,
that of the contractility, cannot be submitted to the above followed
treatment, since the time-factor is wanting. We tried to introduce
this latter by seeking the relation between the intensity of the vagus
stimulation and the duration of the inotropic action, but this latter
is not itself a chemical reaction, but only a modification of the
conditions under which periodically recurring reactions take place.
The negative inotropy may at the utmost be regarded as a diminution
of the quantity C in the formula g=C, which amounts to the
assumption that by vagus stimulation the quantity of the just men-
tioned substance, undergoing chemical change, is diminished. but
this also is uncertain, for in the chemical reaction of the automatism
C represents part of Laneiny’s receptive substance, which is different
from the contractile substance. So I prefer to keep the two chemism
apart and to consider the inotropy entirely by itself.

Placing ourselves on this point of view, we notice: 1. that with
feeble and increasing vagus stimuli the inotropic effect on the
sinus and auricle gradually increases with the intensity of the
stimulus, until a certain degree of inotropy has been reached,
after which it does not increase further for any intensity of the
stimulus; 2. that aa analogous relation holds good for the duration
of the inotropic effect; 3. that the pessimum of contractility is found
about the end of the first third or fourth part of the total duration,
for which the inotropy exists.

(598 )

Summarising we arrive at the following conclusions:

A. the chronotropy, produced by stimulation of the vagus, may
be reduced to a negatively catalytic action on a chemical process
which lies at the bottom of the pulsation.

B. the inotropy admits by analogy of a similar interpretation, but
it is impossible to prove this, since at present no times of equal
change can be determined here.

As secondary results we mention:

a. the existence of twofold negatively chronotropic fibres in the
right vagus of the tortoise.

b. a particularly great sensitiveness of the heart of the tortoise for
inotropy of the auricle by vagus stimulation, in such a degree that
a single condensator discharge may produce the stated modification
and that also with cumulative stimulation it appears sooner and
lasts longer than the chronotropy.

c. the occasional occurrence of spontaneous cardiac turbulence in
a warmed tortoise heart, immediately after a principal retardation
brought about by vagus stimulation.

BH RURGAT a i
In the Proceedings of the meeting of December 29, 1906.

p. 504, line 13 from the bottom: for 2 read 4
p. S11, line 5 from the top: for 0.052 read 0.104

(February 21, 1907).

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday February 23, 1907.

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 25 Februari 1907, Dl. XV).

GC, @ a, TF? et ST oe S.

A. P. N. Francumont: “Contribution to the knowledge of the action of absolute nitric acid
on heterocyclic compounds” p. 600.

F. A. H. Scurememaxkers: “On a tetracomponent system with two liquid phases”, p. 607.

J. BoésreKen : “On catalytic reactions connected with the transformation of yellow phosphorus
into the red modification”. (Communicated by Prof. A. F. Hotieman), p. 613.

J. D. vAN DER Waats: “Contribution to the theory of binary mixtures”, p. 621. (With one
plate).

Pr. Kounsramm: “On the shape of the three-phase line solid-liquid-vapour for a binary
mixture”. (Communicated by Prof. J. D. van DER WAALS), p. 639.

Ps. Kounsramm: “On metastable and unstable equilibria solid-fluid”. (Communicated by Prof.
J. D. vAN DER Waats), p 648. (With one plate).

W. H. Kersom: “Contributions to the knowledge of the 4-surface of van DER WaAats. XIII.
On the conditions for the sinking and again rising of a gas phase in the liquid phase for
binary mixtures”. (Continued). (Communicated by Prof. H. Kameriincu Onnes). p. 660.

H. Kameriincu Onnes and Miss T. C. Jorres: “Contributions to the knowledge of the
y-surface of vAN DER Waaxs. XIV. Graphical deduction of the results of KvENEN’s experiments
on mixtures of ethane and nitrous oxide’, p. 664. (With 4 plates).

P. Nrzuwenuuyse: “On the origin of pulmonary anthracosis”. (Communicated by Prof.
C. H. H. Spronck), p. 673.

41
Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 600.)

Chemistry. — “Contribution to the knowledye of the action of absolute
nitric acid on heterocyclic compounds.” By Prof. A. P. N.
FRANCHIMONT.

(Communicated in the meeting of January 26, 1907).

When searching about twenty years ago for the rules according
io which nitric acid’) acts on hydrogen compounds, not only on
those which contain the hydrogen in combination with carbon, but
also on those which contain it in combination with nitrogen, I
found that the hydrogen combined with nitrogen to the atomic group
NH, does not act on nitric acid, when, in cyclic compounds, this
group is placed between two groups of CO, but it does act if placed
therein between the group CO and a saturated hydrocarbon residue °),
and it may be added: not if placed therein between two saturated
hydrocarbon residues, although I have not mentioned this previously.

It is a peculiar fact that the hydrogen of the group NH does not
act on nitric acid if this group is placed between two similar groups
such as CQO, or saturated hydrocarbon residues, but it does act if
placed between two dissimilar ones; so that it might be thought
that a tautomeric form is essential for the reaction.

There are, therefore, in reality three rules, which, when considered
more closely, apply also to acyclic compounds and which, although
the cycle also exerts an influence, appear to spring mainly from the
nature of the substance in which the group NH is placed: viz.
secondary amine, amide or imide. In acyclic amides it was found
that not only the acyl group in particular, but also the alkyl group
exerts an influence on the reaction; we may, therefore, expect
something similar in the cyclic ones.

The first of the above rules was mainly deduced from the behaviour
of penita- and hexa-atomic cyclic urea derivatives, but was confirmed
also in the case of other compounds. For instance

CO—NH
CO—N H a
ie San Fe a
bee an |
CO—N H CO—NH
parabanic acid alloxan

') Namely the real (absolute) acid which may be obtained by distilling a mixture
of nitric acid 1.42 with twice its weight of sulphuric acid at a gentle heat under
reduced pressure (Recueil XVI. p. 386).

*) Which, however, weed not be the group CH, as stated wrongly by Harries
(Annalen 327. p. 358). The pages of the Recueil referred to by him contain
exactly the proof of the contrary. | have also never spoken of ‘héchst concen-
trirter Salpetersiiure’’ as he says, and of which he thinks he must ‘den Begriff
festlegen”, and for which he then recommends something which in many cases
cannot give a good result.

( 601 )

could be evaporated with nitric acid on a boiling waterbath without
suffering any decomposition,

CO—NH
CH,—CO | 3
and also \u The CH, CO
bi Ae
CH,—CO CO—NH
succinimide malonureide

gives a nitroderivative, but with the nitro-group attached to the
carbon; the two NH-groups do not act.

The second rule is also based mainly on the behaviour of penta-
and hexa-atomic cyclic urea derivatives. For instance

CH,—NH

a \co
Biee

CH,—NH

ethyleneureine

gave a dinitroderivative, which on boiling with water yielded carbon
dioxide and ethylenedinitramine. To this I may now add:

CH,—NH
| x
CH. =--_60
) ys
CH,— NH

Trimethyleneureine
of which I have stated recently with Dr. FrimpMann that it gives
directly a dinitroderwative, which on boiling with water yields carbon
dioxide and trimethylenedinitramine.

CH,—NH
; | eo and its methyl derivatives
CO—NH
Hydantoin
CH,—CH—NH (CH,), C —NH CH,—NH
ecg 00 eee eo |. 00
CO—NH CO—NH CO—N.CH,
a lactylurea acetonylurea 1 Nimethylhydantoin

gave mononitroderivatives, which on boiling with water were decom-
posed with evolution of 1 mol. of carbon dioxide and formation of
a nitramino-amide; for instance nitrohydantoin yields nitramino-
acetamide.

41*

( 602 )

CH,—NH
|
To this I may add the CH, CO recently investigated with Dr.
eats
CO —NH

hydro-uracil

(8 lactylurea)
FriepMANn, which yields, equally readily, a mononitroderivatiwe’),
which on boiling with water yields, in an analogous manner, carbon
dioxide and 6 nitraminopropionamide, from which we have prepared
8 nitraminopropionic acid, also its barium and silver salt.

This decomposition proves the position of the nitrogroup, and at
the same time these substances are all a confirmation of the first rule
because the group NH, which is placed between the two CO-groups,
has not taken part in the reaction.

CH,
CH,- CH, aa
Eee hae
CH, CO and Ge UC
am, | |
CH,—NH CH,.CH— NH
a Piperidone a methylpyrrolidone

gave with nitric acid N,O, presumably derived from a nitro-compound
unstable towards nitric acid at the ordinary temperature ; for it has
been shown that some nitramides are decomposed by nitric acid at
the ordinary temperature with evolution of nitrous oxide; whilst
others may be evaporated with this acid on a boiling waterbath
with impunity. ;

The rule was confirmed five years ago with cycles in which
oxygen takes part, for instance

vH
NH—CH /NCH,

Vg | |

GO | and” <" COM> Mars
. | i tee
0——ons O—CH,

u. célo tetrahydro-oxazole. wu. céto pentoxazolidine
gave on evaporation with nitric acid, mononitroderivatives, which on
\) Tare: stated about this substance (Ber. d. D. ch. G. 33 p. 3385) that it is

not affected even by prolonged boiling with concentrated nitric acid; evidently he
has not used absolute nitric acid

( 603 )

boiling with water were decomposed with formation of carbon
dioxide and a nitramino-alcohol.

CH,—CH,
The third rule is derived from the behaviour of CH, CH, which

CH,—NH
piperidine
yields with nitric acid a nitrate, but not directly a nitro-compound.
This, however, may be prepared from a number of piperidides, to
which we added recently the piperidides of suw/phuric and succinic acids,
or from the nitrate with acetic anhydride as found by Bampercer.
CH, —NH—CH,
I have noticed recently that | | behaves in the same
CH,—NH—CH,
piperazine.

manner.

The above cited new investigations and those which follow origi-
nated in a research by Mr. A. Donx. He had prepared for practice
CH,—NH—CO
| and we treated this with nitric acid. But even
CO— NH—CH,

_glycocol anhydride
on evaporation on a boiling waterbath it gave no evolution of
nitrous oxide, no nitroderivative, but a nitrate. I had expected this
CH,—NH—CH,
behaviour sooner from the unknown | | which is one of its
CO— NH—CO
iminodiacetic imide
isomeres, and in which one NH-group is placed between two CO-groups
and the other between two saturated hydrocarbon residues, but not
from glycocol anhydride in which each NH-group is placed between
CO and a hydrocarbon residue, and about whose structure no doubt
could be entertained. At most, we might suspect here a tautomer
which does not react with nitric acid, or in all other cases in which
nitric acid does act we might assume a tautomer and not here. *)

Mr. Donk’s nitrate, a very loose compound, appeared to be a mono-

nitrate, and on applying BamBercer’s method for amines (treatment

1) Harries l.c. suspects in 1 N methylhydantoin a tautomer CH;—N
| COH
CO —NCH,

which, however, yields with nitric acid the same nitromethylhydantoin.

( 604 )

of the nitrate with acetic anhydride) he obtained a mononitroderwatie,
of which he proved the structure by acting on it with methyl
alcoholic potassium hydroxide, which yielded a properly crystallised
acid, namely NO, NH CH, CO NH CH, CO, H.
nitraminoacetylaminoacetic acid

The reaction therefore took place as in all other cases where
NO, and CO are both linked to a nitrogen atom; by absorbing the
elements of water H and OH the group CO leaves the nitrogen
whilst NO, remains attached to it.

After the departure of Mr. Donk, who did not wish to prosecute
this matter, Dr. FrrepMann took it ap and obtained the dinztro-compound
from glycocol anhydride by treatment with excess of nitric acid and
acetic anhydride. By the action of ammonia on dinitroglycocol anhydride
nitroaminoacetamide was obtained, and by means of sodium hydroxide
nitraminoacetie acid was formed in such a quantity that the formation
of two molecules was no longer doubtful. The position of the two nitro-
groups on the nitrogen atoms has, therefore been sufficiently proved.

CH,-CH—NH—CO

| | when evaporated with nitric acid also
CO—NH—CH-CH,
Alanine-anhydride
gave a nitrate only, which on treatment with acetic anhydride
yielded a dinitroalanine anhydride.

These results, which formed a first deviation from the rule previously
laid down, incited to further research. For it was shown plainly
that besides the placing of the group NH between CO and a saturated
hydrocarbon residue, the other part of the molecule may also influence
the reaction in such a manner that a direct nitration is prevented,
even on warming, although nitro-compounds actually exist.

The question, therefore, arose as to the behaviour of those isomers
of glycocol anhydride, which possess the same atom-groups, but
arranged in another order.

There may be eleven cyclic compounds which consist of two
groups of NH, two groups of CO and two groups of CH,, of whom
however three only are described in the literature, namely :

CH,—NH—CO CH,—CH, CO CO—NH —CH,
| | and =!

CO —NH-—CH, NH— CO—NH CO—NH—CH,

glycocol anhydride hydro-uracil ethyleneovamide

The last one, however, only in an impure condition, as described
by Horrmann in 1872, and which we have not yet succeeded in
obtaining in a pure state.

This substance had a special importance. It has the two grcups

( 605 )

NH, also between CO and CH,, and, according to the rule, it ought
to yield readily a dinitro-derivative; either stable or unstable. Still
it might be that it was not attacked at all by absolute nitric acid,
for if we remember that diacetamide, although slowly, still evolves
N,O with nitric acid and, therefore, presumably forms an unstable
nitro-compound under those circumstances, and if we compare this
with the cyclic sueccinimide, which is not attacked at all even on
warming and which is connected with it in such a manner that if
contains two hydrogen atoms less, and thus causes the cyclic combi-
nation, one feels inclined to attribute to the cyclic combination the
prevention of the action of the nitric acid. We might also compare
ethyleneoxamide to dimethyloxamide which is readily nitrated, and
is related to it in the same manner as diacetamide to succinimide,
and if the cycle formation has the same effect here as it has in
the other case, ethyleneoxamide should not be attacked.
Preliminary experiments with the impure substance showed that
no stable dinitro-derivate app2ars to be formed; at most, one which
is at once decomposed by nitric acid, or it is not attacked at all.
A very slow evolution of N,O and CO, takes place, but this may
be due to the impurity.
Of the eleven possible isomers there are only two urea derivatives
namely hydro-uracil, which, as stated, conforms to the rule and gives
CH,—CO—CH,
a mononitroderivative. The second is | | . RiiGHEMeEr
NH —CO—NH
acetoneureme
thought in 1892 that he had obtained this substance by the action
of chloro-formic ester on diaminoacetone, but it was merely a surmise;
no analysis was made and the properties were not investigated ; and
from our investigations it is extremely doubtful whether he had this
substance in hand, for although we made the experiments in various
ways we could obtain nothing else but acetondiurethane, from which
a dinitro-derivative was readily obtained. A number of other methods
for preparing acetonureine from diaminoacetone were tried but always
without good result. In the meanwhile we are continuing our experi-
ments for, we attach great importance to this substance as a second
urea derivative, seeing that the first one conforms to the rule.
CO—CH,—CO
A fifth isomer would be | | which we have tried in
NH—CH,—NH
methylenemalonamide
vain to prepare from malonamide and formaldehyde. In this case
it is the group CH, of the malonic acid which appears to react

( 606 )

principally; but even with the amide of dimethylmalonie acid and
formaldehyde we have not arrived at the desired result. Methylene-
malonamide is of importance for this reason, that the CH,-group of
malonic acid might give a nitroderivative, whilst this may be equally
expected from the two NH-groups.

CH,—NH—CH,

A sixth isomer is the already quoted | | of which one
CO —NH—CO
iminodiaceticimide

might expect that it should yield with nitric acid only a nitrate,
but not a nitro-derivative.

On heating the diamide of iminodiacetie acid in vacuo, Mr. JONGKEES
obtained a substance which sublimes and has the composition of
the imide. This, however, does not behave as was expected, but
when evaporated with nitric acid, seems to give a nitro-derivative,
whose properties are, however, somewhat different from the usual
ones of nitramines or nitramides.

The last isomer of some significance for the problem under con-
sideration, for the preparation of which no experiments have, as yet,

CO—NH —CH,
been made, would be | | , in Which one NH-group between
CO—CH,—NH
CO and CH, renders probable a nitro-compound, whereas the second,
placed between two CH,, could only yield a nitrate.

The other four are derivatives of hydrazine, and are of no importance

for our problem, because the two NH-groups contained therein are
CH,—CO—NH
in a state of combination. One of those | | has been pre-
CH,—CO—NH
pared by Dr. FriepMann and, when it was brought in contact with
nitric acid a violent evolution of red vapours was noticed, evidently
caused by oxidation.

The details of these researches which of course, are being continued
will appear in the “Recueil des Travaux chimiques des Pays-Bas.”

But it is evident that the second rule will have to be altered,
namely in that sense that the direct nitration (if any) of the hetero-
cyclic compounds, which contain NH placed between CO and C,H,,
depends also on the manner in which the groups, between which
the group NH is placed, are combined; therefore it is the same as
has been noticed with acyclic compounds. In how far the eyele itself
plays a role has not yet been satisfactorily made out but we may
point, provisionally, to one peculiarity, namely, that the three com-
pounds which do not seem to conform to the previously established
rule contain the NH-groups in the para position in regard to each other,

( 607 )

Chemistry. — “Ona telrucoueponent system with two liquid phases.”
By Prof. F. A. H. ScHREINEMAKERs.

(Communicated in the meeting of January 26, 1907).

Although in the systems of three components with two and three
liquid phases there may occur many cases which have been predicted
by theory, but have not yet been realised by experiment, I have
still thought it would be as well to investigate a few systems with
four components to have a glance at this as yet quite unknown,
region.

I will now describe more fully a few of those systems built up
from the substances: water, ethyl alcohol, lithium sulphate and
ammonium sulphate.

We may represent the equilibria with the aid of a regular tetra-
hedron as in Fig. 1; the angular points represent the four components :

Fig. 1.

W = water, A = alcohol, Li = lithium sulphate, Am = Ammo-

( 608 )

nium sulphate. The side AW being invisible has been left out, also
the side Li Am.

Li,SO,.H,O and the double salt LiNH,SO, may also occur as
solid phases besides Li,SO, and (NH,),SO,. The first is represented
by a point Zon the side /iIV, the second by a point D, not
indicated, on the side Li Am.

The equilibria occurring at 6°5 are represented schematically
by Fig. 1. The solubilities of the (NH,),SO, and of the Li,SO,. H,O
in water are indicated by the points a and e; point c¢ indicates
the solubility in water of the double salt and must, therefore, be
situated on the line HD (the point D is on the side Li Am). As
Li,SO,, (NH,),S0, and LiNH,SO, are practically insoluble in alcohol,
their solubility may be represented practically by the point A.

The curve @A is the saturation line of the (NH,),SO,; it indicates
the aqueous-alcoholic solutions which are saturated with solid
(NH,),50, -

The aqueous-alcoholic solutions saturated with Li,SO, and Li,SO.H,O,
are represented by the curve eA which, however, must show a
discontinuity in the immediate vicinity of the point A, for the curve
consists of two branches, of which the one to the right indicates the
solutions saturated with Li,SO,.H,O and the one to the left those
saturated with anhydrous Li,SO,.

The equilibria in the ternary system: water, lithium sulphate and
ammonium sulphate are represented by the curves ab, bcd and de,
which are situated in the side plane of the tetrahedron. ad is the
saturation line of the ammonium sulphate, dcd that of the double
salt LINH,SO,, de that of Li,SO,.H,O. In my opinion, however,
this latter is not quite correct, for, according to several analyses,
Lithium sulphate seems to mix with the ammonium - sulphate,
although only to the extent of a few per cent, so that branch de
indicates solutions saturated with mixed crystals. As, however, I have
not accurately investigated this mixing, ' will continue to speak in
future of lithium sulphate monohydrate Li,SO, . H,O.

Let us now look at the equilibria in the quaternary system. The
surface Am or Aabb,h,b,A represents solutions saturated with solid
ammonium sulphate; surface D or Ab,k,b,bcdA represents the solutions
saturated with LiNH,SO,; the curve Ac of this surface has as pecial
significance, because it indicates the solubility of LiNH,SO, in aqueous-
alcoholic mixtures. The points of the surface D facing the curve Ac
represent solutions which, in relation to the double salt, contain an
excess of (NH,),5O,; the points behind this line show solutions
containing an excess of Li,SO,.

oo _

( 609 )

The curve Ac must therefore, be situated in the plane passing
through AW and the point D of the side Lz Am. The surface Li or
Ade indicates the liquid saturated with Li,SO, or Li,SO,H,O, or
with the above mentioned mixed crystals; it must, therefore, consist
of different parts which however, are not further indicated”in the
figure. At the temperature mentioned here (6°5) systems of two liquid
phases may occur also; in the figure these are represented by the
surface L,L, or 6,K,b,K, which we may call the binodal surface; this
binodal surface is divided by the line A,X, into two parts LZ, and L,
in such a manner that each point of L, is conjugated with a point
of L,. Two conjugated points indicate two solutions in equilibrium
with each other: with each solution of the surface L, a definite
solution of the surface 1, may be in equilibrium.

Instead of a critical point, such as occurs with ternary mixtures
at a constant temperature and pressure, a critical line is formed here,
represented by A,A,. Each point of this line represents, therefore,
a solution which is formed because in the system of two liquid
phases L, + ZL, the two liquid phases become identical. Let us now
look at the sections of the different surfaces: Ad then represents the
solutions saturated with LiNH,SO, as well as with Li,jSO,H,O; Ad,
and 6,5 indicate the liquids saturated with LiNH,SO, and (NH,),SO,.

The intersection of the binodal surface with the surface Am namely,
the curve 4,K,b, indicates the system: L, + L, + (NH,),SO, namely,
two liquid phases saturated with solid ammonium sulphate. With
each point of the curve 6,K, a point of 6,K, is conjugated. Each
liquid of 56,AK, may, therefore, be in equilibrium with a definite
liquid of 6,4, while both are saturated with solid (NH,),SO,.

The intersection of the binodal surface with the surface D, namely,
the curve 6,4,5, represents the solutions of the system L, +L, + Li
NH,SO,. With each liquid of 6,4, another one of 4,4, may, therefore,
be in equilibrium while both are saturated with solid Li NH,SO.,.

The points of intersection 6, and 6, of these two curves give the
system: L, + L, + (NH,),SO, + Li NH,SO,, namely two liquids both
saturated with ammonium sulphate and lithium ammonium sulphate
which may be in equilibrium with each other.

The points £, and &, have a special significance; both are critical
hquids which, however, are distinguished from the other critical
liquids of the critical curve £4, in that they are also saturated with
a solid substance: £, is saturated with ammonium sulphate and /,
with lithium ammonium sulphate.

-
It the temperature is raised the heterogeneous sphere is extended:

( 610 )

at about + 8° the point #, arrives on the side AWAm, so that
above this temperature a separation of water-alcohol-ammonium
sulphate may occur in the ternary system.

I have further closely investigated at 30° the equilibria oceurring
in this quaternary system; the results are represented by the schematic
figure 2.

The saturation surface Am which at 6°.5 still consists of a coherent
whole, now consists (experimentally) of two parts separated from
each other: this is because the binodal surface L,L, now terminates
on the side plane AW Am in the curve a,k,a,.

hi vi
ya
WA

Fig. 2.

Of the critical line /,4, the terminal point 4, represents a ternary
critical liquid; all other liquids of this line are quaternary critical
ones, of which /, is saturated with solid lithium ammonium sulphate.

The phenomenon of the existence of a second heterogeneous region
at this temperature was quite unexpected; it is represented in the
figure by the binodal surface L,'L,* or d,k,d,k, with the critical line
kk, 1 have not further investigated at what temperature this is
formed; it is sure to be present at about 18°,

( 611 )

The binodal surface L, ZL, intersects the saturation surfaces Am and D:
we have, therefore, one series of two liquid phases, saturated with
solid (NH,),SO,, and one series saturated with solid LiNH,SO,. The
binodal surface L,'Z,' intersects the two saturation surfaces D and Ly.
We have, therefore, one series of two liuid phases saturated with
LiNH,SO, (curve k,d, and k,d,), and one series saturated with
Li,SO,.H,O (curve £,d, and k,d,). By d, and d, are represented
two liquid phases which are in equilibrium with each other and
saturated with LiNH,SO, and Li,SO,.H,O. Of the series of the
critical liquids represented by the curve 4,4, 4, is saturated with
LiNH,SO, and &, with Li,SO, . H,O.

The curve Ac which indicates the liquids saturated with LiNH,SO,
without any excess of either of the components runs between the
two heterogeneous regions. From this it follows that this double
salt at 30° cannot give two liquid phases with water-aleohol mixtures.

We, therefore, have at 30° the following equilibria in the quaternary
system.

liquids saturated with

# (NH_),SO, , represented by the surface Am
2. LiNH,SO, , $4 Ona ae D
3. Li,SO,H,O, F ee Li
4. (NH,),SO, and LiNH,SO, ,, ,, the curves: 66, and b,A
5. Li,SO,H,O and LiNH,SO, ,, Me e dd, and d,A

system of two liquid phases :

6. in itself represented by the surface L,L,
ee, a ae y See Oa
8. saturated with (NH,),SO,, represented by the curves a,b, and a,b,
9. " 3) LANEHSO, ; 2 cate aay 2 CO RG OTe.
10. - 3; taNH,SO,, ¥ heme hy iin a ad ae RE Th
fa os aol Ee Os) -~,,. ue case a ey BHO ee

two liquid phases saturated with:
12. (NH,),SO, and LiNH,SO,, represented by the points: 6, and 4,
13. Li,SO,H,O and LiNH,SO,, is ete bh ~ te d, and d,

critical liquids :
14. one series represented by the curve KK,

15. 9 »? 2? 9 ? ” KK,

16. one critical liq. saturated with LiNH,SO,, represented by the point K,
17. 2? ” > ” > LiNH,SO,, > oe 2? » K,
18.

»” Li,SO,H,0, 2» »

a
Boat Gas Hees

( 612 )

On raising the temperature over 30° the two heterogeneous
regions gradually approach each other and finally unite; at what
temperature this happens has not been determined, but from the
experiments it is shown that this is already the case below 40°;
I have also not been able to determine whether this point of con-
fluence is situated in front or behind the curve Ac, or perhaps
accidentally on the same.

I have closely investigated the equilibria occurring at 50° and
represented the same by figure 3; any further explanation is super-

Fig. 3.

fluous. I must, however, say something as to the points ¢, and c,,
namely the intersecting points of the curve Ac with the saturating
curve of the two liquid phases: 6,d, and 6,d,. At first sight we
might think that these two liquids may be in equilibrium with each
other. That possibility, of course, exists. Suppose we take a water-
alcohol mixture of such composition that two liquid phases occur on
saturating with LiNH,SO,. Both liquids will now contain Li,SO,

( 613 )

and (NH,),5O, and it is evident that two cases may occur. It may
be that the two liquids contain the two components in the same propor-
tion as they occur in the double salt; it is then as if the double
salt dissolves in both liquids without decomposition. If this is the
case the liquids c, and ec, will be in equilibrium with each other.

The second possibility is that one of the liquids has in regard to
the double salt an excess of Li,SO, and the other, therefore, an
excess of (NH,),SO,; in this case, c, and c, cannot be in equilibrium
with each other. The experiment now shows such to be the case.
When I saturated a water-alcohol mixture with LiNH,SO, at 50°,
the alcoholic layer contained a small excess of Li,SO, and the aqueous
layer a small excess of (NH,),SO,. From this it follows that the
conjugation line does not coincide with the surface DA JV but intersects
it; the part to the right of the line must be situated in.front of the
plane and the left part behind it. The alcoholic solution c, of the
double salt cannot, therefore, be in equilibrium with the aqueous
solution c, of this double salt, but may be so with a solution con-
taining an excess of (NH,),SO,.

Chemistry. “On catalytic reactions connected with the transformation
of yellow phosphorus into the red modification.” By Dr. J.

B6OESEKEN. (Communicated by Prof. A. F. HoLieman).

(Communicated in the meeting of January 26, 1907).

From the researches of Hirrorr (Pogg. Ann. 126 pag. 193) -
Lemos (Ann. Ch. Ph. [4] 24. 129) Troosr and Havrerevrnie (Ann.
Ch. Ph. [5] 2 pag. 153), R. Scuenck (B. Ch. G. 1902 p. 351 and
1903 p. 970) and the treatises of Naumann (B. Ch. G. 187 2p. 646),
Scoaum (Lieb. Ann. 1898. 300 p. 221), Weescuemrr and Kavrier
(Cent. Blatt 1901 I p. 1035) and Roozesoom (Das heterogene Gleich-
gewicht I p. 171 and 177) it appears highly probable that red phos-
phorus is a polymer of the yellow variety, which polymerism is,
however, restricted exclusively to the liquid and the solid conditions:
the vapour (below 1000°) always consists of the monomer P,,.

From the above considerations it moreover follows that the yellow
phosphorus is metastable at all temperatures below the melting point
of the red phosphorus (630°); it may, therefore, be expected that
it will endeavour to pass into the red variety below 630°.

(614)

Although there are many instances where a similar transformation,
as with phosphorus at a low temperature, proceeds exceedingly slowly,
the velocity in this case is certainly strikingly small. Even at 200°,
when the metastable substance possesses a considerable vapour tension,
it is still immeasurably small eveh though red phosphorus may be
present. *) This extraordinary slowness, notwithstanding the considerable
heat quantities liberated during the transformation, and the complete
alteration of properties caused thereby, have a long time since esta-
blished the conviction that the two modifications of phosphorus are
each other’s polymers and that the red one has a much more com-
plex molecule than the yellow one, but the real cause of that slowness
is not elucidated thereby.

As regards the question fow this condensation takes place,
SCHENCK (I.c.) was the first to endeavour to answer this experimentally.
On boiling yellow phosphorus with an excess of PBr,, he succeeded
in changing it to the red modification at 172° with measurable
velocity; and from his first investigations he concluded that the
order of this reaction was a bimolecular one:

2P,— P,.

This was meant to represent the first phase, for ScHENcK pointed
out that red phosphorus had no doubt a higher molecular weight
than P,, which subsequent condensation should then take place with
great velocity; in other words he arrived at the rather improbable
result that the condensation of P, to P, would take place much
more rapidly than that of the simple P, molecules to P,.

At a repetition of these measurements with one of his pupils
(E. Buck), they came indeed to the conclusion that the reaction is
monomolecular (B. Ch. G. 1903 p. 5208). He remarks ‘Daraus
geht mit Sicherheit hervor, dass die Reaction der Umwandlung des
weissen Phosphors in rothen monomolekular verlauft.”

He, however, adds ‘Daraus kénnte man den Schluss ziehen, dass die
Molekular-gewichte des weissen und rothen Phosphors identisch sind.”

It strikes me that ScneNck arrives here at a less happy conclusion.
From the occurrence of a mono-molecular reaction we need not
necessarily come to the conclusion that the entire process proceeds
in this manner.

1) Roozesoom (l.c.) compares this to the retardation of the crystallisation of
strongly undercooled fusions as 200° is more than 400° below the melting point
of red phosphorus: I am, however, of opinion that this view is untenable on
account of the relatively high temperature, and particularly the very great mobility
of the yellow phosphorus (Roozegoom l.c. p. 89). The cause of the phenomena
must be looked for elsewhere.

( 615 )

On the contrary as in so many other chemical transformations,
we must assume that the measurements executed only apply to a
subdivision of the reaction, namely to that with the smallest velocity.

In this case it is only natural to suppose that the velocity deter-
minations of ScHEenck and Buck apply to the decomposition of the
P, molecule’) into more simple fragments (P, of P), then at once
condense to the red modification so that we may represent the
whole process in this manner for instance:

Paiyetowe = are. et. Oe OG,
nP = be bee mae ¥.)

in which the reaction velocity of (2) is very much larger than that of (1).

(We might also suppose, as a primary reaction the transformation
of the metastable phosphorus into a lJabile P,; this, however, I do
not think so probable because, in the determination of the vapour
density above 1000°, a splitting has been indeed observed).

It cannot be a matter of surprise that this decomposition velocity
at 200°, (without catalyst) will still be extremely small, Jooking at
the great stability of P, in the state of vapour; and if this decom-
position, as I suppose, must precede the condensation, the separation
of the red pbophorus at that temperature will proceed at least equally
slowly.

There is also nothing very improbable in the very rapid transfor-
mation of the dissociated P, or P into red phosphorus.

The fact that the allotropic transformation takes place particularly
under the influence of sunlight is certainly not in conflict with the
idea of a primary splitting, as we know that the actinic rays accelerate
the decompositions (such as of HJ, AgBr, C,J,, etc.).

I wish also to point out that a primary splitting is also accepted
in other monomolecular reactions, such as in the decomposition of
AsH, (van “Tt Horr’s Vorlesungen), of CO (Scnenck B. Ch. G. 1903
p. 1231 and Smits and Worrr. (These Proc. 1902 p. 417). *)

The monomolecular splitting of C,J, into C and C,J, ScuEnk and

1) Although the size of the molecule of the liquid yellow phosphorus is not
known with certainty, the identity with that of the vapour is however very probable;
for the rest it does not affect the argument.

*) | omit purposely the beautiful researches of M. Bopvensrein, although for the
union of S and H. he also arrives at the conclusion that a primary splitting of
the S; molecule precedes the union with H,, because we are dealing here with
heterogeneous systems in which solubility velocities play an important réle. It is not
impossible, that in all cases in which amorphous substances separate we are
dealing with such solubility velocities.

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 616 )

SirzenporFr B. 1905, p. 3459, may be interpreted in the simplest
manner by the succession of the reactions :

CA OS ae a
nC Gn and 2 CJ,—> CJ, °".. 2 2 2a
Il.
The measurements of ScHENcK and Buck have been made at the

boiling point of PBr,. As this is situated at 172°, it appears that the
solvent exerts a considerable accelerating influence on the transfor-
mation, as pure yellow phosphorus at 200° remains practically
unaltered.

The solvent, therefore, acts catalytically ; a still more powerful
influence has AICI,. If this is brought together with phosphorus in
vacuum tubes, the transformation takes place even below 100°,

The catalyst is at once covered with a layer of pale red phos-
phorus, which it is rather difficult to remove by shaking, so that it
is necessary to add now and then a fresh quantity of AICI, The
action proceeds much more regularly if benzene (and particularly
PCI,) is added as a solvent. At the boiling point of this, the trans-
formation is completed after a few hours (respectively, minutes) ;
the product is ScHENCK’s searlet-red phosphorus but much contaminated
with benzene and condensation products, which are retained with
great obstinacy.

In connection with the explanation in part I. I believe that the
observations of Scuenck and of myself throw some light on catalytic
actions in general.

For it is very probable that in this allotropie transformation a
splitting occurs first; we notice that the transformation, consequently
the splitting, is accelerated by PBr, or AICI,. Will this not occur generally
in catalysis? As a dissociation precedes most reactions it is probable
that this question must be answered in the affirmative. (I wish,
however, to lay stress on the fact, that in answering this question
we do not penetrate into the real nature of catalysis. The reason why
the dissociation acceleration occurs, whether this is connected with
a temporary combination of the catalyst with the active molecules,
or whether the catalyst removes the cause which impedes the
dissociation, remains unexplained and need not be discussed here
any furtber.)

As far as I have been able to ascertain, this conception is not
antagonistic to the facts observed; in fact a number of cases are
known where a catalyst causes directly a splitting or considerably
accelerates the same.

( 617 )

Platinum, for instance, powerfully accelerates the decomposition of
hydrogen peroxide, ozone, nitric acid, hydrazine ete.

Aluminium chloride causes a direct splitting of the homologues of
benzene, of the very stable polyhalogen derivatives, of aromatic
ethers, of sulphuryl chloride, etc. The number of these decomposi-
tions is so considerable that, in other cases where we cannot prove
a direct dissociation by the catalyst, we may still argue that it takes
place primarily, or rather that an already present but exceedingly
small dissociation is accelerated in such a manner that a system
attains the stable condition of equilibrium much sooner than without
the catalyst.

The great evolution of heat in the process

HCCI, + 3 C,H, + (AICI,) = (C,H,), CH + 3 HCl + (AICI,)
points to the fact that the system to the right is more stable than
that to the left. I attribute its slow progress when no AICI, is used
to the small dissociation velocity of chloroform : the catalyst accelerates
this dissociation so that the stable condition of equilibrium is attained
in a short time. This reaction gets continuously more violent (the
temperature being kept constant). This phenomenon may be readily
explained if we bear in mind that the reaction proceeds in different
stages (C,H, CHCI,, CHC] (C,H,), and CH(C,H,), are formed in suc-
cession) and that the chlorinated intermediate products are decomposed
much more readily than CHCl,.

If sulphur is boiled with benzene and aluminium chloride we obtain
almost exclusively (C,H,), 5, (C,H,), 5, and H,S. Without the catalyst
hardly any action takes place because the dissociation of S, in benzene
solution at 80° is negliglible: (if sulphur is boiled with toluene H,S and
condensation products are formed without AICI, being present) the alumi-
nium chloride accelerates the reaction 5,—>4S,, and consequently
the formation of the condensation products. This explanation is
therefore quite the same as that given for the reaction of P, with
benzene and aluminium chloride; the sole difference is that in the
latter the second stage of the reaction consists exclusively in the
condensation of P, to red phosphorus, a condensation to which
sulphur does not seem to be liable to the same extent, so that the
dissociated sulphur forms with benzene the above products.

I consider the formation of a compound of the catalyst with one
of the reacting substances of importance for the taking place of the
reaction in so far only that one phase can be formed ; otherwise it
rather obstructs the reaction, because the catalyst becomes to a
certain extent paralysed. One of the most powerful catalysts, platinum,
is actually characterised because it does nut (or at least with great

42*

( 618 )

difficulty) unite with the reacting molecules, but forms a kind of
solid solution. Carbon tetrachloride which forms no compound with
aluminium chloride is certainly attacked by benzene in presence of
that catalyst not less easily than benzoyl chloride which does form
an additive product; whilst also the chlorine atom in the acid chloride
is certainly not less “mobile” than that of CCl,.

GusTavsoN imagines that the formation of compounds, such as
C,H, (C,H,), Al,Cl, is necessary for the action of C,H,Cl on benzene ;
these were separated from the bottom liquid layer which forms
during the action of C,H,Cl on benzene and aluminium chloride ;
if, however, the formation of this layer is prevented as much as
possible, the yield of ethylated benzene improves. Therefore I do
not call its formation necessary. That it may act favourably perhaps
is because the catalyst and also the two reacting molecules are
soluble in the same, thus allowing them to react on each other in
concentrated solutions.

As has been observed above, there is something unsatisfactory
in assuming intermediate reactions in order to explain catalytic
phenomena. I will try to explain this matter more clearly.

As is known, we may express the reaction velocity of a condition

impelling force

change by the ratio: in which the impelling

resistance
force for that change in condition possesses a definite value which
a catalyst cannot alter in the least; the resistance, however, is
dependent on influences for the greater part unknown. Therefore,
the resistance must be lessened by the catalyst and the question to
be solved is: “On what does this decrease in resistance depend?”

If we suppose that intermediate reactions take place we divide
the process into a series of others of which each one considered by
itself is propelled by a force less impelling than the total change;
the resistance of each of those division processes must, therefore,
be much less, and the question then becomes: How is it that those
intermediate reactions proceed much more rapidly than the main
reaction? which is in fact nothing else but a circumlocution of the
first question: how is it that the catalyst decreases the original
resistance? Therefore, by assuming intermediate products, we have
not been much enlightened, on the contrary we have made the
problem more intricate, because, instead of having to account for a
single increase of velocity, we have to look for that of at least two.

I call to mind the theory of OstwaLp who supposes each process
to be a succession of condition changes, which will be all possible
if they occur with potential diminution. If, however, the first of those

( 619 )

changes can commence only with absorption of free energy, the
process will not take place unless a catalyst is added; this, therefore,
Opens another road . . . Now, in my opinion too much attention is
paid to the milestones on that road and too little to the opening itself.

This is chiefly caused by the fact that we know so little of the
so-called ,,passive resistances’, for instance we cannot give a satisfactory
explanation of the fact that iodine acts much more rapidly at low tempe-
ratures on metals than does oxygen, although the potential decline is much
smaller. Still, 1 think that we must look for this mainly in the
ready dissociation of the iodine molecuie, always supposing: that
atoms react more rapidly than molecules, a supposition, moreover
nearly a century old.

If this should be so, the action of a catalyst must be sought for
in the increase of this dissociation.

Now, we know of a number of reactions where the catalyst forms
undoubtedly a compound with one of the reacting molecules,
which additive product then reacts with the second molecule to form
the final product, with liberation of the catalyst, but even in. such
a case, which is called by many “pseudo-catalysis” (Wacner, Z. Phys.
Ch. 28 p. 48), I do not consider the formation of this compound as some-
thing essential without which the acceleration would not take place.

I certainly do not consider the formation of such an additive
product as being without any significance, as it is an indication
that the catalyst can exercise a particular influence on one of the
molecules ; the real increase of velocity is, in my opinion, due more
to that influence than to the formation of the additive product, and
in view of what precedes this, that influence consists presumably
of an increase of the dissociation (and through this of the active mass).

It is, of course, obvious that a catalyst will act all the more
energetically when the additive products are more labile. I have
already mentioned platinum and now point also to the H-ions
with which the formation of additive products, for instance when
accelerating saponification, is far from probable. As a very lucid
example, I mention the different catalytic influence which iodine
and AICI, exert on the transformation of yellow into red phosphorus.

From the researches of Brodie (Ann. de Ch. Ph. 1853 p- 592)
which I have found fully confirmed, a small quantity of iodine
can convert a large quantity of yellow phosphorus very rapidly
into red phosphorus at 140°. (As in many other cases, there is a limit
because the catalyst is precipitated by the colloidal phosphorus formed.

The velocity at the ordinary temperature is very small but becomes
plainly perceptible at 80°. We are undoubtedly dealing here with a

( 620 )

case where the catalyst combines with the phosphorus to P,I, ;
this substance commences at 80° to dissociate measurably [so that
its vapour density can only be determined at a low temperature
(Troost CR 95 293)] with separation of red phosphorus. We may,
therefore give here a fairly positive answer to the question: How is
it that the second division process proceeds more rapidly than the
original ? Because P,I, dissociates much more rapidly than P,.

But this is after all but a lucky circumstance, the real cause must
be sought in the fact that in order to obtain P,I, the P, molecule
must be dissociated to begin with. With AICI, I have not been able
to find an additive product, only some indications that, besides the
allotropie transformation, a trace of PCI, is formed (even with per-
fectly dry substances the manometer, after a few hours’ heating to
100°, showed a slight increase of the vapour pressure).

The fact that the red phosphorus formed has in a high degree
ihe property of coprecipitating the catalyst might perhaps indicate the
possibility of a compound being formed between yellow phosphorus
and AICl,; from the above it follows that there is a possibility of
a certain reciprocal influence*) but I attribute this coprecipitation to
the colloid properties of the red phosphorus, which, when obtained
from solvents and also under the influence of rays of light, carries
with it a certain quantity.

But even if an additive product is fc the existence of this substance
is no more the cause of the acceleration than it is in the case of P,I,.

On the contrary, I consider the formation of a compound of the
catalyst to be a case of “poisoning’, caused by one of the reacting
molecules, just as arsenic and prussic acid are poisons for platinum,
because in combining with it, they prevent the entrance of O, and
H, (respectively 5O,); just as ether is a poison for AICI,, because it
unites with it to a firm compound, which does not decompose until
over 100’, the temperature at which the catalyst again recovers itself.

Now, I cannot deny that we have not advanced much further
with this dissociation theory (which is also not absolutely novel) for
the question is now: How is it that a catalyst accelerates the
dissociation? But my object was to point out that the formation
(and eventually the admitting of the formation) of intermediate pro-
ducts can certainly never lead to an explanation of the catalytic

phenomena.
© Chem. Lab. University, Groningen.

1) I have also found a similar reciprocal influence in the action of C,H; Br on
AIC], in which C,H;Cl and AlBrs are formed; it undoubtedly points to a disso-
ciation.

( 621 )

Physics. — “Contribution to the theory of binary mixtures.’ By
Prof. J. D. vAN Der WAALS.

The theory of binary mixtures, as developed in the ‘Théorie
moléculaire’, has given rise to numerous experimental and theoretical
investigations, which have undoubtedly greatly contributed to obtain
a clearer insight into the phenomena which present themselves for
the mixtures. Still, many questions have remained unanswered, and
among them very important ones. Among these still unanswered
questions I count that bearing on a classification of the different
groups of w-surfaces. For some binary systems the plait of the
y-surface has a simple shape. For others it is complex, or there
exists a second plait. And nobody has as yet succeeded in pointing
out the cause for those different forms, not even in bringing them in
connection with other properties of the special groups of mixtures.
It is true that in theory the equation of the spinodal curve which
bounds the plait, has been given, and when this is known with perfect
accuracy, it must be possible to analysis to make the classification.
But the equation appears to be very complicated, and it is, especially
for small volumes, only correct by approximation, on account of
our imperfect knowledge of the equation of state. Led by this consi-
deration I have tried to find a method of treatment of the theory
which is easier to follow than the analytical one, and which, as the
result proved, enables us to point out a cause for the different shape
of the plaits, and which in general throws new light upon other
already more or less known phenomena.

Theory teaches that for coexisting phases at given temperature

d dt d dw
three quantities viz. — ie F ae and w—v a —xz| —
dv },7 \ dz) 7 dv ) eT dz )yT

must be equal. The first of these quantities is the pressure, which
we represent by p; the second is tue difference of the molecular
potentials or M,u,—M,u,, which we shall by analogy represent
by gq. The third of these quantities is the molecular potential of the
first component, which we shall represent by M, u,. Now the points
for equal value of p lie on a curve which is continuously trans-
formed with change of the value of p, so that, if we think all the
p-curves to be drawn, the whole v,z-diagram is taken up by them.
In the same way the points for given value of g lie on a curve
which continuously changes its shape with change of the value of q;
and again when all the g-lines have been drawn, the whole »v,.-
diagram is taken up. Both the p-lines and the g-lines have the
property, that through a given point only one p-line, or only one

( 622 )

g-line can be drawn. One single p-line, however, intersects an infinite
number of jines of the g-system, and every q-line an infinite number
of lines of the p-system. One and the same p-line intersects a given
g-line even in several points. However, it will, of course, be neces-
sary, that if two points indicate coexisting phases, both the p-line
and the g-line which passes through the first point, passes also
through the second point. If we choose a p-line for two coexisting
phases, not every arbitrarily chosen value for a q-line will satisfy the
condition of coexistence in its intersections with the p-line, because
a third condition must be satisfied, viz. that M, u, must have the
same value. The result comes to this: when all the p-lines and all
the g-lines have been drawn and provided with their indices there
is one more rule required to determine the points which belong
together as indicating coexisting points. So in the following pages
I shall have to show, when this method for the determination of
coexisting phases is followed: 1. What the shape of the p-lines ts,
and how this shape depends on the choice of the components.
2. What the shape of the gq-lines is, and how this shape depends
on the choice of the components. 3. What rule exists to find the
pair or pairs of points representing coexisting phases from the infinite
number of pairs of points which have the same value of g, when p
has been given — or when on the other hand the value of q is
chosen beforehand, to find the value of p required for coexistence.

But for the determination of the shape of the spinodal curve the
application of the rule in question is not necessary. For this the
drawing of the p- and the g-lines suffices. There is viz. a point of
the spinodal curve wherever a p-line touches a qg-line. We have viz.

Py (dv d*y Gy (dv d*y dv
fr = ——_— === OSandir =- = 0 for| —
one dv? ( :) = dvudx ae dadv \ da /q + dx? ss: da},

ay aw
dxdv dv da?
the value — - and for | — the value —-—, and so we may
d’y du Jy dy
dv? dadv

write the equation of the spinodal curve:

dv dv
da raee da *

So if we are able to derive from the properties of the components
of a mixture what the course of the p- and of the g-lines is, we
can derive much, if not everything, about the shape of the spinodal
curve. And even when the course of these lines can only be predicted
qualitatively, and the quantitatively accurate knowledge is wanting,

— q y ae

( 623 )

the qnantitatively accurate shape of the spinodal eurve will, indeed,
not be known, but yet in large traits the reasons may be stated,
why in many cases the shape of the plait is so simple as we are
used to consider as the normal course, whereas in other cases the
plait is more complex, and there are even eases that there is a
second plait.

Particularly with regard to the p-lines, it is possible to forecast
the course of these lines from the properties of the components.
With regard to the g-lines this is not possible to the same extent,
but if there is some uncertainty about them, we shall generally have
to choose between but few possibilities.

THE COURSE OF THE /p-LINES

In fact the most eel Peres of the course of the p-lines
were already published by me in “Ternary Systems” and only
little need be added to enable us to determine this course in any

; : E dw
given case of two arbitrarily chosen components. As p= — a
Uy

, it is required for indicating the course of

dp
these p-lines to know the course of the curves 7 =O and
rT

av
d
() er
dx).

The former curve has a continuous liquid branch, and a continuous
gas branch, at least when 7’ lies below every possible 7%, when we
denote by 7, the critical temperature for every mixture taken as
homogeneous that occurs in the diagram. If there should be a minimum
value of 7), for certain value of 2, and 7 is higher than this mini-

mum 7%, the curve (2) = 0 has split up into two separate curves.
Zhi dere &

In either of them the gas and the liquid branch have joined at a

value of v=v,. In this case a tangent may then be traced // to

: d
the v-axis to each of these two parts of the curve (=) ==, Gp.
Ly fe

(ap re
The second curve {— ]} =O is one which has two asymptotes,
Se

and which may be roughly compared to one half of a hyperbola.
The shape of this curve derived from the equation of state follows
from the equation:

( 624 )

_, db da

MR1 ae =e
Lv ; az ae 0 ‘

(v—b)? v*

If we now always take as second component that with the greatest

db . were
is always positive, it appears from the given

value of 6, so that

dp :
=O cannot possess points for these

equation that the curve (
Ce) yT

da
values of «, for which 7q, is negative. Only at that value of x for
Av
5 da ; Hee 4 :
which SS) this possibility begins, but then only if 7 = 0. If
av

da :
T has a definite value a must be positive, for points of this curve
av

to be possible. For v=o, = must be == MRT— And the value
of wv which satisfies this equation, indicates one asymptote of the
discussed curve by a line // to the v-axis. If this asymptote has
been drawn, we may think the mixtures with decreasing critical
temperature to be placed on its left side. And on the right the
mixtures with increasing critical temperature do not yet immediately
follow. For a separation between the mixtures with decreasing and
ae only when MRT = =
dx b
7), would immediately ascend again on the right of this asymptote;
but then 7’ would have to be chosen so high, that it was 7’/, 7},
and for the present at least we shall choose 7’ far below that limit.

That the line ec, where ¢ has the value which follows from

da el Sie ; : ;
— = MRI az? 1S an asymptote, is seen when we think the equation
wv

; : da a
those with increasing 77, a must be ==
a“

av
da
d 3 dix
of the curve :. = 0 written as follows: ed ae : > ie
da) 7 (v —b)? db
Mia =
dx

1 aa ; : v ;
the value of = becomes larger from left to right, ; must increase
av

v—

° v + .
from left to right, or = decrease. For the value of 2, following

da pe MM at ile 048) 5 ; ie
from = MRI ry infinite; for larger values of 2, ms decreases
AL av )

( 625 )

v da
more and more, and as 3 can never become = 1, because

ie
cannot become infinite, the curve v=b is the second asymptote.
So if « is made to increase more and more, also beyond the values
which for a given pair of components are possible in order to
examine the circumstances which may occur with all possible systems

: db . ae
for which with positive value of a increasing value of 7). is always
ax §

dp :
found, a minimum volume must occur on the curve (Z) =O; So
a) yT
d'p
for this point 3/=9
dix? oT

Now that we have described in general outlines the two curves
which control the course of the p-lines, we shall have to show in

what way they do so.

From.
dp
5 dx
dx )yT “@),
follows that to a p-line a tangent may be drawn // 2-axis when it
dp
passes through the curve {| — }, and a tangent y-axis, when it
&/yT

dp
passes through the curve (2) . But though these are important
I] 2T

properties they would be inadequate for a determination of the course
of the isobars, if not in general outlines the shape of one of these

dj :
lines could be given. The line (z) = 0 viz. intersects the line
Lv vT

d ;
(? ‘= = 0 in two points, and it is these two points which are of
pee Y

fundamental significance for the course of the p-lines. The point
of intersection with the liquid branch is viz. for a definite p-line a
double point, the second point of intersection being such an isolated
point that it may be considered as a p-curve that has contracted to
a single point. The surface p=/(#,v) is namely convex-concave
in the neighhourhood of the first mentioned point. Seen from below
a section // v-axis is convex, and a section // x-axis is concave. A
plane, parallel to the v,«-plane touching the p-surface intersects,
therefore, this surface in two real lines, according to which p has

( 626 )

the same value. But for the second point of intersection the two
sections are concave seen from below — and there are no real lines
of intersection. This second point isa real point of maximum pressure.
With all these properties, and also with those mentioned before or

; da ae
still to be mentioned, —— is assumed to be positive. *)

Now the curve p=constant passing through the first point of
: ; ’ dp dp ;
intersection which the curves =O and |—]=0 have in

dv) oT U/l

common, is the isobar whose shape we can give, which shape
at the same time is decisive for all those following, either for
larger or smaller value of p. In the adjoined figure 1 its course is
represented. Coming from the left it retains its direction to the

dp
right also in the point of intersection with the curve i =e
Ax vo

the convex side all the time turned to the «-axis till it is directed
straight downward in the point where it meets the vapour

branch of the curve ($) =. There it has a tangent // v-axis, and
Of 27

from there it has turned its concave side to the v-axis. When it

dp . Oe eae : ;
meets the curve =) —0, | —) is equal to O for this as for all
ae ]}yT pP

dv de
nitely large, and pursuing its course, it passes for the second time
through the double point, and further moves to the right, always
passing to smaller values of v, till it has again a tangent // to the

dp dv
isobars. Passing again through the curve (2 : is again infi-
mye: p

: - : ip :
axis of 2, when it meets the curve (? = () once more, after which
> v

it proceeds to larger value of v. It is clear that in the path it describes
from the double point till it passes through this point for the second
time, it has passed round the point we have called the second point

d.
1) That the characters of the two points of intersection of the curve (Z)=0
iv vT

dp ;
with the curve bel =O are different appears among others from this that when
Gv

ie «rT
these points of intersection coincide as is the case when these curves touch each
_ dp d'p d*p \* :
other, the quantity —- —— — {| ——— ]=0. The character of the points of inter-
dv* dz? da dv

section depends on this quantity being positive or negative.

af. 4 7

( 627 )

of intersection with the curve (2) = 0, and where maximum pres-
Pe ON

sure is found. In fig. 1 some more isobars have now been drawn

besides this one. We obtain the course of the isobars for lower value

of p by drawing a curve starting from the left at higher value of

v, bearing in mind that two p-lines of different value of p can

never intersect, because the p is univalent for given value of «

dp
and v. Such an isobar cuts the curve (7) =Q on the left of the
av / rT

lv
isobar with the double point in two points, where a =o, then
f

a x )

passes through the curve (Z)=0 in a point where é a4 \)
dz oT da pT
and has then also on the right of the said isobar again two points
of intersection with the curve (7) = 0, in which points of intersection
ys
j dv
again = = 00.
An isobar of somewhat higher value of p has split up into two
isolated branches. One of them starts on the right at somewhat smaller
value of v; further this branch follows the course of the isobar with
the loop, but must not cut it. Arrived in the neighbourhood of the
double point it is always obliged to remain at small volumes; there

d dv
it meets the curve (2) = 0, and it has ( 6 From this point
j L v Pp

ag
it proceeds to smaller volumes, till anew meeting-point with the same
curve causes this branch again to turn to larger volumes. but the
second branch of this isobar of higher value of ; is entirely inclosed

~ within the loop of the loop-isobar. Such a branch forms a closed

curve surrounding the point which we have called the second point

dp dp ee
of intersection of the curves 7 = 0 and re =~ Sen a
v ne Lv v

d,
closed branch passes twice through & ) = 0, and also twice through
az},

dp dv : 2
— |=0, and has again in the first cases { — ] = 0, in the second
dv }, dx},

: : é dv
points of intersection | — }= oa

dz /,

With ascending value of p the detached portion of the p-line

contracts more and more, till it has contracted to a single point. So

at still higher value of p only one single branch of the p-line remains.

( 628 )

A similar remark must be made for the curves of lower value of p.
The smallest value of p for gas volumes is of course p=0O; but
this limit does not exist for the minimum pressure of the mixtures
with given value of z. For this we know that also values of p may
occur which are strongly negative. For values of p which are negative,
the p-line has again divided into two disjointed portions, viz. a
portion lying on the left in the diagram, which is restricted to
volumes somewhat larger and somewhat smaller than that of the
P ¥ dp
liquid branch of the curve (5

) = 0, and a similar portion lying
x

on the right in the diagram.
Also on the locus of the points of inflection of the isobars the
given diagram can throw light. So it is evident in the first place,

) = 0 starting

that between the two branches of the curve (
v

from the double point, both on the left and on the right a connected

ee d*v dp\.
series of points is found where =. If the curve =i
p un

dx? dv

itself should possess a double point, which is the case when 7’ has
exactly the value of 7), minimum, this locus of the points of inflec-
tion of the p-lines passes through this double point, and when the

curve (2) =O has split up into two separate portions, as is the
av /x

case for still higher value of 7’, then those points of the two portions

dv
where —= belong to this locus. It is also apparent from the
at

diagram that two more series of points start from the double point,
one on the right and one on the left, as locus of the points of
inflection, and that these run to smaller volumes.

An isobar with somewhat larger value of p than that of the loop-

shaped isobar has a tangent // to the w-axis where it passes through
at

Ip : : Cake
the curve (32) = 0. On the right and on the left of that point it
v

turns its concave side to the a-axis, whereas at larger distance it
must again turn its convex side to it on both sides. So there start

d*v
from the double point four branches on which (=) == 0 tas
Bie

also easy to see that the branch which moves to the right towards
dp
smaller volumes, must pass through that point of the curve | — }] =0
v v

where the tangent is // z-axis. For an isobar which passes through

( 629 )

d ee :
the curve (2) = 0 on the left of this point, turns its concave side
a

v
to the z-axis, but when it passes for the second time through the
said curve on the right of the point, its convex side. Hence an isobar
where these two intersections have coincided, has its point of inflec-
tion in the point itself. If we wish to divide all the v,7-diagram into
d?v

regions where (4 -) is either positive or negative, it must be
& a
: p

eh dp

borne in mind that also the two branches of line {— ) = 0 them-
av yz

selves form the boundaries for these regions, because on that line

dv P.
rie a. [e @) .

In’ all. ‘this

is supposed to be positive. For on the contrary

; : E .
the course of the line (2 = 0, to which we could now assign
Ax v
an existence on the right of the asymptote which is given by

db da
MRT ee would be directed to the left of this asymptote,
&

a
Ma . 5 : g
when = should be negative, so if 2a,, could be >a, + a,. Foras
da
VU 3 dx v da
( ;) —= ———.,, the value of ; decreases only, when — increases.
ae av
; MRT—
dx

d
If we put a= A+ 2 Be + Cr’, and so —_ 2(B-+ Cz), it appears

: ; , da
that with C negative « must decrease in order to make P increase.
L

For the points of this line p would then possess a minimum for given

d at x -
value of v, and so = would be positive. From this follows then that
ay

v ny
have interchanged réles. The point of intersection with the smallest
volume represents then a real minimum of p, and will have the

same significance for the course of the p-lines as the second point

. ' : a ae : d
the two points of intersection of this line with the curve (3) = 0)

of intersection has, when

is positive. And the point of intersection

( 630 )

with the smallest volume has now become double point. I have,
however, omitted the drawing of this case 1. because most likely
the case does not really oceur, and 2. because the drawing may
easily be found by reversing the preceding one. There are e.g. with

the solution of salts in water cases which on a cursory examination
2

present some resemblance with the assumption negative, but

Ak
which yet are brought about by influences perfectly different from
. aa
the fact of a negative value for Fe
i
2 ; da
Such a diagram for the case ee negative, though, would quite

fall in with the right side of fig. 1. As in the given figure 7%,
increases with « on the right side, and there is a maximum value
2
of 7; on the supposition > fig. 1 might be still extended to the right
till such a maximum 7% was reached. But then we should also have
to suppose that a value of zx could exist or rather a mixture for
da
which at a certain value of & the quantity qa? reverses its sign.
Every region of fig. 1 of certain width which is taken parallel to
the v-axis can now be cut out for a, + a, — 2a,, positive, to denote
the course of the isobars. Regions on the left side indicate the course
of the isobars for mixtures for which with increasing value of 6 the
critical temperature decreases — regions on the right side for mixtures
for which with increasing value of 6 the critical temperature increases —
the middle region with the complicated course of the isobars when
there is a minimum 7. The left region would be compressed to an

: da
exceedingly small one if we wished to exclude the case —— negative

or — =0. We do so when putting a,, = Va,a,. On such a suppo-
ax

sition a minimum 7;¥ is still possible, but the left region must then
have an exceedingly narrow width. There is, however, no reasonable
ground for the supposition a,a, = a,,?. There would be, if the quantity
a for the different substances depended only on the molecular weights,
and so a= em* held for constant value of «. If the attraction, just
as with Nrwron’s attraction, is made to depend on the mass of the
molecules, and so if we put a, = &,m,*, and also a, = &,m,’, it appears
that ¢, and & are not equal. If we now put a,, = Va,a,, we put
a,, = m,m,/e,e,. What reasonable ground is there now for the sup-

Prof. J. D. VAN DER WAALS. Contribution to the theory of binary mixtures.

| SK Xx Oe x KX x X| Xx XK KK PR HH AAV

Fig 1.

Proceedings Royal Acad, Amsterdam. Vol. IX.

( 631 )

position that if there is a specific factor ¢, for the mutual attraction
of the molecules of the first kind of which we do not know with
what property of these molecules it is in connection, and if there
is also a perfectly different factor ¢, for the mutual attraction of the
second substance, we must not represent the specific factor for the
attraction of the different molecules inter se by é,,, but by a8.
It is true that this supposition renders the calculations simpler ; I had
already drawn attention to this in my Théorie Moléculaire (Cont. I,
p. 45). But whether the calculations are somewhat more or some-
what less easy does not seem a sufficient ground, after all, to intro-
duce a supposition which involves that naturally a great number of
possible cases, among others also for the course of the spinodal line,
are excluded. If we put all possibilities for the value of a,,, then

a = 2 & a, 7 ass =
— can also be — 0, viz. for = ———.. We need not go so far
dz l—x a,—a,,

however, to give sufficient width also to the left region.

THE COURSE OF THE q-LINES.

d
The value of a =q is found from the value of p:
az v

SS MRTE = +{(2) dv
: ars —- Vira) ee
3
For z=O this expression is negatively infinite, for 2=1 it is
positively infinite, so that we have g,=—o and q,=+o.

But it follows also from the equation of state that for all values

ao d .
of x the value ot (2) dy is also positively infinite for the line v=6.
& fy

v
It is true that for such small volumes the equation of state
(a a .
P= > ~ Gy 38 not accurate when 6 is not made to depend on
= Vv

v, and the quasi association in the liquid state is left out of account,

a
dp + ie
and that the conclusion: = dv is infinitely large for v equal to the
Lv)» :
v
limiting volume, calls for further consideration before we may accept
this as an incontestable truth. But it seems to me that simple con-

siderations lead to this conclusion. For the limiting volume p is
infinitely great, and if 4 increases with x, (Z is infinitely large
& v

43
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 632 )

a
of higher order, whereas f (=) dv can again diminish the order of
a he
v
infinity by a unit, because the factor of dv has this higher order of
infinity only for an infinitesimal value of dv. But still the thesis

@o
remains true that { (2) dv is infinitely great for v= 6.
a

So there is strong asymmetry in the shape of the qg-lines. Whereas
qg==— holds for =O and every value of v > 6,, g= + @ holds
all over the line of the limiting volumes, and for all volumes on the
line z—1 which are larger than 6,. We derive immediately from
this, that all the g-lines without exception start from the point « =0
and v= 6,. In this point the value of q is indefinite, as also follows
from the value of g as it is given by the approximate equation of
state, viz.:

db dv

di dit

oS ee OE ae
1—az vo—b v

It also follows from the approximate equation of state that at
their starting point all the g-lines touch the line v= 6, of course

with the exception of the line g=—o. For we derive from
dw\ _
(3 —
(25) (2) B=
dedv) \dx),' da?
or
dy
dv dx?
(2 - Py
dadv

a, ee $ ,
For —— the approximate equation of state yields:

lz?
d*b ab\**
MRT = UT | ee
ay MRT dz* dx dx®

dx* ~~ @(1—2) v—b (v—b)? Temas

d? d d
We already found the value of 74 =(2) bove. For (=)

we find therefore:

RN i

( 633 )

db\? da
MAT) = Dee
MRT MRT d?b a (=) da?

dv a(1— a) v—b dx? | (v—b)? v
dig MRT db dal

; : . : d ll
put v=b, we find for the starting point of the g-lines (Z) samp

s Coe
at least if we can prove that ———— is equal to zero for «=O and
v

v=b,. To show this, we put =), + Pe + ye

* and so: 0 — b=

v—b
= (v — b,) — «8 — yx’, and then we find for (v — b) —— the value:
L

v—b, ‘
(v — b) Se Sal as :

Vv —
The term

is indefinite, but nevertheless the given value

multiplied by » —6 is really equal to zero. This result, too, is still
to be subjected to further consideration, because it has been obtained
by the aid of the equation of state, which is only known by approxi-
mation. And then I must confess that I cannot give a conclusive
proof for this thesis. But I have thought that I could accept it with
great certainty, because in all such cases where a whole group
of curves starts from one vertex of an angle, e.g. for the lines
of distillation of a ternary system, I have found this thesis confirmed
that then they all touch one side of the angle. Only in very
exceptional cases the thesis is not valid.

Moreover, the theses which I shall give for the further course of
the g-lines, are independent of the initial direction of these lines.
Only, the q-lines themselves present a more natural course when
their initial direction is the indicated one than in the opposite case.

dv
From the value given above for (Z) follows that they have a
av q

d
tangent // v-axis, when (2 = 0, and a tangent // z-axis, when
v

ay
daz?

= 0. Hence they have a very simple shape in a region where

d d? :
the lines (2) = 0 and—~=0 do not oceur. Starting from the
de}; dx?

point c=0O and v=), they always move to the right and towards
43*

( 634 )

dvu\ . me dp
larger volume, and 77m Pe always positive. Therefore ah and as
In
q v

2
. “W nn : :
will presently appear, ga 00° always positive in such a region. As
aX

v becomes greater the value of g approaches to MRT / , and

for very large value of v the q-lines may be considered as lines
parallel to the v-axis, for which the distribution over the region
from «—0 to e=1 is symmetrical. The lines for which q is
negative, extend therefore from «=O to x=3 and for =} the
value of g=0. It will only appear later on that yet in their course
probably two points of inflection always occur for small volumes,
a fact to which my attention was first drawn by a remark of Dr.
KounstamMM,. who had concluded to the presence of such points of
inflection in the g-lines from perfectly different phenomena.

dad.
But as soon as the line S = 0 is present (the case that also
av v

dy : :

aS may be =O will be discussed later on), a new particularity
&

makes its appearance in the course of the q-lines. A q-line, viz.,
which cuts this locus, has a tangent //v-axis in its point of inter-
section, and reverses its course in so far that further it does not

proceed to higher value of z, but runs back to smaller value —

dv eds ae
so that (> , which is always positive in the beginning, is hence-
q
forth negative. From that point where they intersect the line

daz Fy

& = 0 and where (=) may be considered negatively infinite,
v q )
this quantity becomes smaller negative. Still for v= 2, the g-line
must again run parallel v-axis. So there must again be a point of
inflection in the course of the q-line. In fig. 2 this course of the
g-lines has been represented, both in the former case when they do
: dp
not intersect the curve (2), and when they do so. In the latter
case they have already proceeded to a higher value of x in their
course than that they end in. They end asymptotical to a line v=<2¢,
and at much smaller volume they also pass through a point # = a.
The point at which with smaller volume they have the same value
of « as that with which they end, lies on a locus which has a

_ (ap
shape presenting great resemblance with the line (3) = 0. The value
ty»

( 635 )
X-axis.
'
'
{
T
{
Den Boe yore Pa
, VO ee
Py ane
7 P
‘
f / i
Peet
' /
' /
'
‘ n =C
‘ ’
‘ /
¢ ag
ps) ‘
x. ,
un '

ree wenn. .

Fig. 2.

for the points of this locus may be derived in the following way.

If we write p= MRT {(1—xa) log A—a) 4 « log x} +f pe

d
then (=). Sl iL),
az }»

At infinite volume the value of ¢= MRT 1
—wWL

as we saw above.

The locus under consideration must therefore be determined by

<2)
d
S@ dv = 0. Hence on the line «= the final value, a point must

( 636 )

Vd
be found such that, proceeding along that same vine, {(%)de = 0
e v

So from this follows immediately 1. that the points of the said locus

é d : d
restrict themselves to those values of z in which the curve —e
AH v

occurs, 2. that the points must be found with smaller volumes than

d
those of () =0(Q. For such points with smaller volume is viz.
Lv Dv

d aS :
= positive, and for points with greater volume negative —
us) y

however when the volume may be considered as a gas volume this
negative value has an exceedingly small amount. And even without

fee , )
drawing up the equation () dv = O, we conclude that the said locus
ax v
dp ia
has the same a-asymptote as ae = 0 itself, and is further to be
dz},

found at smaller volumes. Hence it will also have a point where
its tangent runs // a-axis. There is even a whole series of loci to be
given of more or less importance for our theory, which have a

{dp 7
course analogous to that of (2 —( and Be dv — 0.
dz r az),

v °

; d :
The latter is obtained from (1) by integration with respect to v;
aay;

all the differential quotients with respect to v of the same function

dp
a) put equal to O have an analogous course — thus
dz},

wdv

which is a locus of great importance for our theory. That it has the same
dp ; : ; :

x asymptote as i. ) = 0 itself, and that all its other points are to
oh ag) he

be found at higher value of v, follows immediately from the follow-

, : , . dp
ing consideration. For a point of the line (=) = 0 the value of
WU oT

dp ;
(2 =0. For points of the same w and smaller v this value is
ax

positive — but for points with larger v negative. For v = o this negative
value has, however, again returned to 0. So there must have been
a& maximum negative value for a certain volume larger than that

( 637 )

: 2 : ? ; d*p
for which this value = 0. These are the points for which Penal
aAtaAV

For smaller value of the volume

is therefore negative — on the
adxvxav iy

other hand positive for larger volumes. The approximate equation
of state yields for the loci mentioned and for following loci these
equations :

db da
d d: d
enn for oP dv = 0
v—b v dx
v
db da
dix d. d;
eee SS as for *) — 0
(v —6b)? v? dz)»
db da
Saat dz d*p
aE EE ope (eS 0
(v—b)’ v* dadv

And so forth.
But let us now return after this digression to the description of
the shape of the q-lines. Whenever a g-line passes through the locus

dp Tie a eee

iG dv =0, the asymptote to which it will draw near at infinite
Xv

volume is known by the value of « for that point of intersection. For

the present it does, indeed, pursue its course towards higher value of 2,

but when its meets the locus (2) = 0, it has the highest value of
uv) y

x, and a tangent //v-axis. From there it runs back to smaller value
or

And this would conclude the discussion of the complications in
the shape of the g-lines, if in many cases for values of 7’ at which
the solid state has not yet made its appearance, there did not exist
another locus, which can strongly modify the shape of the gq-lines,
and as we shall see later on, so strongly that three-pbase-pressure
may be the consequence of it.

2 2

The quantities and aa occur in the equation of the spinodal
v av
curve in the same way. It may be already derived from this that
: d* dy
the existence of the loci —~ —0 and = 0 will have the same

dv* ait?

significance for the determination of the course of the spinodal line.

( 638 )

That as yet our attention has almost exclusively been directed to
ay
dv*
given binary mixture furnishes points for the latter locus for values

of 7 below 7). for that mixture, whereas the conditions for the
2

—0 is due to the fact that we know with certainty that a

: W : :
existence of a locus —-=—0 are not known — and it might be

x
suspected that this remained confined to temperatures so low that
the solid state would have set in, and so the complications which
would be caused -by this, could not be observed. That such a sup-
position is not quite unfounded may still be safely concluded from

the behaviour of many mixtures, which quite answer to the consi-
2

: : : Ue
derations in which the curve a is left out of account. But that
Hi

the behaviour of mixtures for which more complicated phenomena

occur, cannot be accounted for but by taking into consideration that
a

=a can be =O, seems also beyond doubt to me.
2

The approximate equation of state gives for this quantity the

following value:
db? dh da
MRT | — ——
@y MRT ( ) ie ae
dx? x(l1—x)° (v—b)? rae eee:

which I shall still somewhat simplify by assuming that 6 depends
3

linearly on w, and so Tie 0. We can easily derive from this form
&

ee ee

that if qr can be =O, this will be the case in a closed curve. At

F : : aw .
the boundaries of the v,7-diagram aaa 18 certainly positive. For «= 0
Ax ;
and «= 1 even infinitely great. Also for v6. And ‘for »=@
MRT
z (1—2)
ryy r ar lr. 2 . *
4 MRT. That, if only 7 is taken low enough, it can be negative,
2

d?a

at least if oa is positive, is also obvious. At exceedingly low value
dz :

it reduces to , the minimum value of which is equal to

of 7 it can take up a pretty large part of the v,7-diagram, which
must especially be sought in the region of the small volumes. With
rise of temperature this locus contracts, and at a certain maximum
temperature for its existence, it reduces to a single point. So it is
no longer found above a certain temperature.

(To be continued).

( 639 )

Physics. “On the shape of the three-phase-line solid-liquid-vap Ab
for a binary mixture.” By Dr. Pa. Kounstamm. (Communicated
by Prof. J. D. van per Waats.)

Already for a considerable time I have been engaged in arranging

Prof. Van per Waats’ thermodynamic lectures, and having arrived
at the discussion of the three phase line solid-liquid-vapour, and the
metastable and unstable equilibria solid-fluid which are in connection
with it, I have formed on some points a different opinion from that laid
down in the literature known to me on this subject. It does not
seem unprofitable to me to shortly discuss the points of deviation in
this and the following communication.

The first concerns the shape of the three phase line solid-liquid-
vapour when the solid substance is one of the components, viz. the
least volatile one. We find given for this that this line must always
possess a pressure maximum’), and that it must also possess a
temperature maximum’) when the solid substance, — as is usual,
— melts with expansion of volume. The latter remark is the
generalisation of a supposition, advanced by Van per Waats*) with
respect to the line for ether and anthraquinone. These consider-
ations, however, hold only for definite assumptions on the extent
of the difference of volatility of the two components. This appears
immediately from the differential equation of the three phase line
given by VAN DER WAALS‘) :

Ly
# No—Ns — — (HI—7Ns)
Pp Z|

7 So a ae Pe ee ee ee ae
Vy — Vs — —(ViI—?,)
|

in which 4, candy denote resp. entropy, concentration and volume
of the coexisting phases, the index v, / and s denoting that resp.
the vapour, liquid and solid phase is meant. .r, does not oceur,
because we assume, that the solid phase is the first component itself
so z, 0. The pressure maximum will now occur in the line when
the numerator, the temperature maximum when the denominator can
become zero. Now Nyv—Ys > hi—. and v,—v, > vi—v,; the two
cases are therefore only possible when z, >.;, i.e. when the vapour
is richer in the component which does not form the solid phase,

') Baxnuis Roozesoom. Die heterogenen Gleichgewichte II. p. 331.

*) Suits. These Proc. VIII, p. 196; Zeitsch. phys. Ch. LIV, p. 498.
3) These Proc. VI, p. 243.

*) Verslag Kon. Akademie V, p. 490.

( 640 )

(for in the equation is put 7,0) than the liquid. Or in other
words, as we said above, the points sought can only present them-
selves in the three phase line with the least volatile component as
solid substance’). However, whether those points awi// occur, depends

Ly ;
on the value which lim (=) will get. If this value may be put

Bi —

= infinite, we get for <= 0:
dp Nl — Ys
AP De:

so equal to the slope of the melting-point curve. So we must have
both pressure and temperature maximum, at least when the solid
substance expands on melting. ‘This was the purport of the above
cited remark of vAN DER WAALS about ether and anthraquinone; if

however ( ‘) may not be put infinite, this conclusion is no longer

vl i)

valid ; it then depends on the value which:

Ly
0): 0 5 (v) = Vs)
el

assumes for «=O whether there exists a temperature maximum

or not; if the difference in volatility, so =, should not be so large,
|

that this expression becomes negative at the limit, the maximum

does not occur, even when vs > 7.

The question whether such a maximum will occur in many systems,
cannot be answered with certainty for the present. For this many
data would be required, which we have not at our disposal as yet ;
it is, however, possible to show the probability that only in very
extreme cases the volability of the components will be so diversified,
that a temperature maximum is to be expected. For this maximum
to be just present, viz. in the triple point of the solid component, it
is evidently required that :

Ly VI—Vs

ae
&] Vy—Vs
Now the first datum we should want, would be the variation of
volume during melting. It seems, however, that only a few data have
been collected for this; I have found some in WINKELMANN’s *‘Hand-

1) If has of course been tacitly assumed here, that there is no maximum vapour
pressure; in that case the points in question could be found in both three phase

lines,

( 641 )

buch” '), and in Baxknuts Roozesoom’?); LANDoLT’s and BérnsTEIN’s
tables do not give anything on this subject. The values indicated at
the places mentioned confirm that the percentage of these expansions
is not very considerable, which was a priori to be expected ; they amount
for the highest cases to little more than 10°/, and for most substances
they are considerably lower. So if we take 10°/, as basis, we
shall find for by far the majority of the cases a too great, so for
our proof a too unfavourable value. If we introduce this value, we
get as condition (neglecting v by the side of »,):

&1 Vy

So we must now try and get a rough estimation of the relation
between liquid and vapour volume in the triple point. If at the
triple point the vapour tension was of the order of an atmosphere,
this ratio would be about of the order of magnitude 1000. Now,
however, the vapour tension is always very considerably lower ;
almost for every substance the melting point lies very considerably
below the boiling point. If we now assume that the triple point
lies at about */, 7%, we find the order of the vapour tension from
the well known formula :

Ti.
— l, ae <i Bde. 1).
Pk 1

With -~=7 and .=—1'/, 7,. this gives og ===) OP ge aoa’
If we put p, at 100 atms.’), p, becomes of the order of 0.1 atm.
So we may safely say that in general v,/v, will be smaller than
0.0001. For a temperature maximum it is, therefore, necessary, that
at least :

Ly Ly ~
—=—10* or l. nat — = 11.5.
L| 2

Now according to a formula which has been repeatedly derived
by VAN pbeR Waats‘), for low temperatures (a condition which in
this case is certainly fulfilled) the equation :

L| TSE 1) ad; 1 db

Ty. dz b da’

4) Il p. 612 Qnd p. 775.

Py Ae. a Bsteo:

3) In the table of Lanpott and BérystEIN only two substances occur, ammoniac
and water which have a higher p;; the majority by far is considerably lower,
particularly that of the little volatile substances which we have in view.

*) See e.g. These Proc. VII, p. 159.

( 642 )

Iz,
holds, or for the limit, where ==:
Shy
FF aT, toa
l SS a 1 RoI eer ae, . [ . . « 2
“9 Ly m li dx b dex (2)

It is clear that everything will depend on the first term here,
because the second would not amount to more than —1 in the
utmost case, i.e. when the 6 of the other component would be zero.
Moreover it might even be possible that the second term was positive,
it would hence decrease the value of the second member.

The greatest difficulty for our calculation lies now in our igno-
rance of with the variability of 7; with 2, or more strictly in
this that for this variability not one fixed ruie is to be given, because
in every special case it will depend on the special properties of the
mixture in question, viz. on the quantity a@,,, a quantity which does
not admit of being expressed’) in the characteristic quantities of the
components, at least for the present. It is, therefore, certainly not
permissible to try and derive results for all kinds of systems. But
it is only our purpose to determine the course of 7}, for those cases,
in which the components differ exceedingly much in volatility, and
for those cases it is perhaps not too inaccurate a supposition to assume
for the present that the line which represents 7; as function of a,
does not deviate too much from a straight one.*) On this supposi-
Ty,—Ty, f i aac

tion then we may write —~——— for ———. As now —=1]4, as

Tr, Te dex m
Shy

ke —TL x,

we already supposed, must not descend considerably below

1) The equation of GALITZINE-BERTHELOT d= dy, which I rejected as general
rule already on a former occasion on account of the properties of the mixture
ether-choroform (These Proc. IV, p. 159), can certainly not be accepted as such.
Not only is it easy to mention other examples which are incompatible with this
rule (see e.g. Quint, Thesis for the doctorate p. 44; Gerrits, Thesis for the
doctorate, p. 68); but besides, — and perhaps this must be considered as a still more
serious objection — by assuming this equation we wilfully break up the unity of
the isopiestic figure (v. p. Waats, Proc. of this meeting p. 627) by pronouncing its
middle region on the left of the asymptote to be impossible, whereas the left and right
regions are considered as real. For if @,=Va,dq itis never possible that da/dx = 0
for whatever system; and this takes exactly place in the middle region. 1 had
overlooked this in the paper mentioned; Prof. vAN DER WAALS has since
drawn my attention to it. The already mentioned system of Quint gives an

a
= 0; dg is there smaller than even the
ax

example of the occurrence of this case

smallest of the two a’s.
2) Cf. VAN DER .WAALS, These Proc. VIII, p. 272.

( 643 )

0.9, that log. = may not become smaller than the required value
11.5, or in other words, for the maximum in temperature to be
reached, the critical temperature of one component must be about
ten times as high as that of the other. A system, in which hydrogen
occurs, will most likely show the temperature maximum when the
other component has its critical point above 0° C., but already when
the more volatile component is nitrogen or oxygen, we shall be more
restricted in the choice of the other component. For then the latter
must have its critical point at about 1000° C. resp. 1250° ©. If
ether were the more volatile component, this temperature would
almost amount to 4500° C.

This conclusion is hardly affected when we put the temperature
of the melting point not at */,, but at '/, of the critical temperature,
as it really is for a number of substances whose critical temperature
and melting temperature are known. It is true that this consi-

: cs ce
derably increases the second member of equation (2), and so —, but
; LI

‘ sini Ue : :
in the same ratio — increases too, so that the quotient remains about
Vv! ;
unchanged. This is most easily seen when the condition on whicha
temperature Maximum occurs, is written :
&l Vy

v
oe Usk or lag. ete + log. vy — log. vy < log. 0.1.
e Xy

Vy vl
&|
Now for Jog. — we may introduce the value from the equation’):

Tes

i Brel Fy 73 ah, 1 dpi.
0d. a — — :
I if T dz pe da

Seas av] vy

and write for /og. v, :

MRT
log. vy = log.

= log. MRT +15 — 1) — log. Pk
c
so that the condition becomes:
dT},

ff T).dx

So an increase of 7’ will only affect the first term and the term
logy. MRT, and the logarithmic change of the latter will certainly
amount to less than the change of the former. This now increases
when 7 becomes smaller, hence when at 7’='/, 7), the inequality

aes os — log. py — log. v1 — f + log. MRT < log. 0.1.
Pk de

1) These Proc. VII, p. 559.

( 644 )

is not satisfied, this will certainly not be the case for 7’= */, Ty.

Still, it would be too hazardous to assert that it has now been
incontestably proved that e.g. for the system ether-anthraquinone no
temperature maximum can occur. For we have had to make use
of the supposition that 77, depends linearly on 2, and though this
supposition may possess some degree of probability for critical tem-
peratures that differ much, it is just with substances which — as
ether and anthraquinone-lie’ closer together, that there is some ground
for expecting a deviation from the straight line. Only very few ex-
perimental data are at our disposal. As such may e.g. be used
the determinations on the increase of the plaitpoint temperature by
addition of little volatile substances, made by Smits, CENTNERSZWER
and bBicuner. For by means of the formula given by vAN DER WAALS’)

dT dT; 49 ( dT), lap.)
Tde, Tydze 45 (Tydz 7 pda

(3)

in which we need only pay regard to the principal terms (those

ryy

sae Nees . ale ;
with 7;), we may calculate the value of from those directly
v

T pd:
measured. If we now calculate by the aid of the thus found
; dT; » vie § aa : a :
value of and the supposition of rectilinearity, 7%; i.e. the

Ty.dx
value of 7; for the admixed substance, we find the data collected in
the following table. (P. 645).

From this appears that the values calculated in this way at least
for some substances, and particularly for anthraquinone according to
the determination by Smits, are not inconsiderably lower than double
the melting point temperature. It may, therefore, be considered highly
probable that these lines are convex seen from below, and so the

ry.

absolute value of a will be larger than might be expected from
the supposition of rectilinearity. With our imperfect knowledge of
the further course of the plaitpoint line, and hence a fortiori of the
line for 7; an estimation as to this will, naturally, remain very
uncertain; but yet it seems to me that something about this may
be ascertained in the following way. We have on the side of the

ether:
dT}. ae da db
Tyde)y=0 \ade bdxJz—o

1) These Proc. VII, p. 272 and 296.

( 645 )

| Second First | T. cal-| Double the

Observer melting-point

Orthochloronitro benz. 5 “ | 760 610
BiicHNER’s values have been borrowed from his thesis for the doctorate |
| (Amsterdam 1905); those of CENTNERSZWER from a table by vAN Laar
(These Proc. VIII, p. 151); that of Smits has been calculated from his
determination: plaitpoint at 203° and 2 —G.015, (These Proc. VII, p.179).

and so when introducing for a the quadratic and for 4 the linear

function:
aT, _ 20,,— 20, b,— bp (i)
Ty.dz/,—0

Now it will not be too hazardous an estimation, when — keeping
in view that the formula for ether is C,H,,O and for anthraquinone
C,,H,O, —, we put the size of the anthraquinone molecule at about
two or three times that of the ether molecule; so 6, = 2d, a 36,.

AT}.
If we introduce this value and the value of P - , calculated by the
de

aid of equation (3), into equation (4), we obtain a value for a,,.
Assuming that the value of 7) for anthraquinone is 2 560°=1120°,
we can find an a, from the ratio of the critical temperatures ot
ether and anthraquinone, and the a for ether; and with these quan-

on the anthraquinone side

ae k
tities we ean finall leulate tl
y calculate the Tide

(=) iba 2a, — 24,5 b, —6,
Tydz ).—} iat a, b

from:

( 646 )

OME ie ee aT
Starting from 6, = 26, we find in this way ( ‘) == G65

k Hi == ||

with 6, = 2.5: 0.65 and with 6,—306,: 0.64. The error which we
committed in our choice of 4,, will, therefore, bring about no con-
siderable modification in the result; it would, indeed, be considerably
modified if the critical point of anthraquinone should prove to lie
considerably higher than 1120°. This is not in contradiction with
our former remark that it is of little importance whether the reduced
temperature is ‘/, or '/, at the triple point; for this we started from
the supposition of the linear dependence, whereas here we have
abandoned this supposition, and calculate this dependence from the
experimental data. So according to the course of reasoning followed
here the a,, is given by the experiment, and the smaller value of
m would now result in a higher value of a, at given 6,, 6, and a,,.
If our estimation may be considered as not too inaccurate, we may
conclude that the deviation from rectilinearity does increase the value of

reach the critical value 0.9. (The value derived from the supposition
of rectilinearity is 0,58).

Though the foregoing calculations teach us hardly anything
positive, they fix first of all our attention on the great desirability
of more data concerning the values of the quantities a and 6 of very
little volatile substances ; for it appears again that the whole behaviour
of all the systems in which such substances appear, is controlled by
these quantities, and it would exactly be of great importance for
the theory of mixtures, if its results could be tested by such cases
where the properties of the two components differ strongly. It is true
that it will not be easy to determine the critical point of such sub-
stances in the usual way, but we should have gained already much
if we could obtain an estimation of the critical temperature by
caleulation of the a and 6 from the deviations from the law of
Boye in rarefied gas state, so still some hundreds of degrees below
the critical point.

And further I think that after the foregoing I may be allowed to
draw this conclusion, that the appearance of a temperature maximum
in the three phase -line, far from being the general case, will be
confined to mixtures of very exceptional nature.

dT, : : :
( ) but by no means in the degree which would be required to

Much more frequently than a temperature maximum will a pres-
sure maximum occur. It appears from equation (1) that this will
always be the case, when the expression:

( 647 )

x
(ny — Ns) — — (qi — ms)
2]

may become negative. Now it is true that we cannot properly say

that 7,— 7, is a heat of sublimation and 7; — 4; a latent heat of

melting, because the 7’s do not refer to the same concentration, but

we may say that 4,— 7s is of the order of magnitude of a heat of

sublimation, 4; — 4, of the order of a latent heat of melting. Or in

other words 4, — 4, will be about 7 or 8 times 4 — 7s. So in all
By

cases where & <7 the pressure maximum in the three phase
®! }x—0

line will also fail. Here too the necessary data are wanting to

ascertain whether there are many systems for which the - at the
triple point will descend to this amount. For, determinations of vapour
tension or direct determinations of the required ratio have been
nearly always carried out at considerably higher temperature’), and
for the calculation by the aid of the just used formula the necessary
data fail here too; besides, it would be doubtful whether the
formula would be accurate enough, now that we have to deal with
such small amounts. But — quite apart from the existence of mix-
tures with minimum vapour pressure — the existence of a system
like ether-chloroform’?) where on the chloroform side 2, becomes
almost equal to «,, already proves, that such systems exist.

In any case to the scheme for the possible course of the two three
phase lines in a binary system plotted by Bakxuris Roozesoom in
Fig. 108 of Vol. I of his “Heterogene Gleichgewichte”’, must be
added types VII and VIII, characterized by a succession of sections,

1) Particularly when we notice that the ratio of a» and xi would have to be
calculated from the formula:

il dp Vy—t]| 1 dp By— aX]
= ———— or —- — = —_____
pdx, 2#,(1—2,) p da, x|{(1—a))

and the value obtained will, therefore, strongly vary in consequence of a change
of temperature of some ten degrees, which have generally an enormous influence
per cent on the pressure in the neighbourhood of the triple point.

®) KoHNSTAMM and van Da.rsEn, These Proc. IV, p. 159. BAKHUIS RoozEBoom
(lc. I p. 41) deems it probable that also systems of gases with water and of
water with many salts will show a similar shape. However, for such systems
whose three phase line for the least volatile substance shows a pressure maximum,
at least at temperatures that do not lie too far from the triple point, the shape
of the p,a-line will have to deviate considerably from the line drawn there in
Figs. 15 and 19, because from that shape would follow x. =v.

da
Proceedings Royal Acad. Amsterdam. Vol. [X.

( 648 )

denoted by 1.7.4.5 and1.7.8.5 in Roozrsoom’s nomenclature ’).
Type VII (see Fig. 1), is therefore distinguished from HI in this that

Fig. 1. Fig. 2.

Fig. 3: Fig. 4.

section 3 disappears; our Fig. 3 (lacking with RoozmBoom) takes
its place. Type VHI (see Fig. 2) is distinguished from type V in this
that instead of section 6 the section indicated in Fig. 4 appears
between 8 and 5.

Physics. — “On metastable and unstable equilibria solid-fluid.”
By Dr. Pu. Kouysramm. (Communicated by Prof. J. D. van
DER WAALS.)

In a preceding communication *) I discussed a point on which I
could not agree with the existing literature on the equilibria solid-
fluid. A second point which will prove to be allied to the preceding
one, concerns the course of the curves which are to indicate the

1) Loc. cit. p. 392.
2) Proceeding of this meeting, p. 639.

( 649 )

metastable and unstable equilibria solid-fluid in the 7’, .x-figures drawn
up by vaAN DER Waats’), and the v,2- and p, a-figures drawn up
by Smits?). VAN DER WaaLs himself has already pointed out a defect
in those figures *), viz. that the spinodal curve falls here within the
connodal one, whereas in reality it falls far outside it at low tem-
peratures; but it is not this that I have in view.

Let us first take the p, x-figures. According to them the complica-
tion which the binodal curve solid-fluid shows for temperatures below
the triple-point, will disappear in this sense that at the triple point
a new complication makes its appearance with three phase pressure,
horizontal and vertical tangent, that then these two complications
together give rise to the existence of a detached closed branch which
contracts more and more, and at last disappears as isolated point.
It is clear that in this way it is supposed that the complication can
only disappear above the triple point, and not in the triple point
itself, or in other words, that when the triple point is passed, always
another three phase pressure is added to the existing one, and that
these two more or less high, but always above the triple point
pressure and the triple point temperature concur and disappear. Or
expressed in another way still, it has been supposed in these figures
that there is always found a temperature maximum in the three
phase line. In the light of the considerations of our preceding com-
munication this supposition is by no means legitimate. But apart
from this there rise serious objections against these views. First of
all, if these views are held, it is impossible to see what the shape of
the binodal curve solid-fluid must be when the solid substance is the
more volatile component. Moreover all through the succession of the
p, #-figures the binodal curve solid-fluid has only one point in common
with the axis 7 =O. Now it is, however, known, that for the com-
ponents themselves, so for the concentrations 7—=0O and «= 1 the
p,f-diagram of fig. 1 holds (see the plate), i.e. at the triple point
temperature there’ exists by the side of the triple point pressure C
a second pressure of equilibrium solid-fluid (viz. of an unstable
phase) C’, and above and below the triple point temperature these
exist even two such pressures, one of which indicates metastable
equilibrium, the other unstable equilibrium. But then the binodal curve
solid-fluid for the mixture will not have to cut the axis of the
component which becomes solid, once, but three times. And _ finally
the p,.«-figures of Smits and the 7, 2-figures of van pER WAALS

1) These Proc. VIII p. 193.

2) These Proc. VIII p. 196.

S) loc; -cikerp: “95.

44%

(650 )

cannot be made to harmonize with the v, z-figures plotted by the
former; for in these threefold intersection of the binodal with the
rim does really occur before the detachment takes place (Compare
in fig. 6 of the said paper by Smits the line fed with /,c,¢,¢,'ef1's
between this a v,.a-line must necessarily be found intersecting the
rim in three points). Now that attention has once been drawn to
these unstable and metastable equilibria, it seems desirable to remove
these discrepancies.

For this purpose the best thing is to start from the », z-figure.
The general equation of coexistence of phases in the variables », x
and 7’ becomes in this case, if we now consider phase 2 as
solid phase, 1 as fluid phase’):

ab et Op O° d*y
9 * Sige dz Vs — “aw 2a Ss x
(vs vf) Ove? dvy | Ove Ou sh | nF (w af) laa dws Oar? das ain

4+ (*%\arT=0
T eS

so that we get for constant temperature :

d°y d7y
PME pds
de Ow 0?

Soy DD Pe ei
PMA ies LE Ria Gale.

In what follows we shall denote the numerator and the denomi-
nator of this fraction by .V and D. The geometrical meaning of
D has already been given by VAN DER WaAAts in his first paper
on these subjects*): the locus D= 0 is the locus of the points of
contact on the tangents drawn from the point for the solid substance
to the isobars. It is easily shown that the locus NV = O is the locus
obtained by putting the q-lines i.e. the lines ot Cin this instead
of the p-lines. So a double point or an isolated point, as they are
assumed by Smits, can only occur where the loci V = 0and D=O
intersect. As in such a point, as appears from the geometric meaning,
the p- and the q-lines have the same tangent, and accordingly touch,
such a point must also lie on the spinodal line’). In perfect agreement

1) Cont.. Il p. 104.
2) These Proc. VI p. 233.
5) For from the equation of the spinodal curve
0? O?y
0° yp 07» ee J 0 Ovdw Our?
— —— — | —— } = 0 or = ——
Ov? Ow? Oxdv 07yp 07

Ov? Ovdx

651 )

with this we easily obtain for the case that , is not O or 1, the
course of the loci mentioned indicated in fig.2. The dotted line
denotes the concentration of the solid. phase P; the lines A Q Band
CQ D are the two branches of the spinodal curve, the two other lines
eee Op

joining A with B and C with D the branches of eee When

v x
now 2z,—0 becomes, it is evident that at this rim the line D—O

ie er
must pass through the point where = 0, and this point coincides,

as is known, with the spinodal curve at the rim. The conclusion seems
obvious that the points Q and Q’, the points of intersection of the
spinodal curve and D) =O have shifted towards the rim, and that,
accordingly, the points of detachment and contraction from figs. 2—8 of
Prof. Smits (loc. cit.) would have to lie at the rim. However, this
conclusion would not be correct. For the inference that where the
spinodal curve and ) = 0 intersect, on account of the geometrical
meaning of D=0O and V=—0O, the latter must also intersect, does

2 ay

¥
Ov?

not hold good at the rim. This is in connection with becoming

07»
zero and becoming infinite. If we introduce the value MRT/z,

0a?
which the last quantity for «=O gets, then N assumes the value:
O°y ' MRT _ (0p aaah
FH 5) Sard. 1 = ther) |)
Ov02 On (vs te di ae 4 (vf Us)

and in general this expression will by no means be equal to zero

)
in the points where Ee = 0, as already appears from the simple
U

consideration that there can
vf—Us

quantity which depends exclusively on the properties of the pure

Op . mh : oe
component and 5, ) in the maximum and minimum points of its
& v

isotherm, because this latter quantity will also have to depend on
the properties of the second component. So the points Q and Q’
will certainly not lie at the rim, and in the points where D=0O

dp O7y ow 0g
e Ovda = 0? @) dv
aa ep oY (dg =-( ),

follows:

( 652 )

cuts the rim the binodal curve will simply have a tangent parallel to
the v-axis.

The shape which the different figures will assume, will now depend
entirely on the fact whether such points Q and Q will also exist
when the solid substance is one of the components and if so, where
they lie. The best and most general way of solving these questions
would be a full consideration of the different forms which the g-lines
may present. As however the solution of the special question we
are dealing with does not call for such a discussion, I believed to
be justified in preferring another briefer mode of reasoning. For this
purpose I point out first of all that it is easy to see that at least
in a special case such a point must exist also now. Let us imagine
a plait, the plaitpoint of which has shifted so far to the side of the
small volumes, that the tangent to the plait in the plaitpoint points
towards the point indicating the solid state’). The plait touching the
isobar in the plaitpoint, the plaitpoint lies evidently on the line
D=O0 in this case?). But the plaitpoint lies also on the spinodal line,

2 0?yp
so the point @ lies here in the plaitpoint, as neither ) : = 0; TOF
v
07) f
ag We may conclude from this that in such like cases, so
Hi

those cases where the plaitpoint has been displaced still somewhat
further or somewhat less far to the side of the small volumes, and
perhaps in general when the difference in volatility between the two
components is great, a branch of NW =O will pass through the figure,
and that it will most likely have a point of intersection with the line
D=O0. A closer investigation of this supposition can, of course, only
be given by the calculation.

1) The above was written before Prof. ONNES’ remarkable experiment (These
Proc. VII p 459) had called attention to “barotropic” plaitpoints. Now that the
investigations started by this experiment have furnished the proof that plaitpoints
can exist, in which the tangent runs // a-axis, the existence of plaitpoints as
assumed in the text, in which the slope of the tangent need not even be so very
smali, has, of course been a fortiori proved.

2) We may cursorily remark that it is therefore not correct to say in general
that the line D=O runs round the plait in the sense which vAN DER WAALS
(These Proc. VIIL p. 361) evidently attaches to this expression, i.e. that the
point of intersection of the line D=O with the binodal and spinodal curves would
lie on either side of the plaitpoint. For if the plaitpoint should have moved still
a little further to the side of the small volumes, the two points of intersection of
D=O with binodal and spinodal curves lie evidently on the vapour branch of
these lines (the part of these lines between the plaitpoint and the point with the
largest volume on the 2-axis).

i
a

( 653 )
For this we shall start with the case that / increases and a decreases
. . . ‘yy Op . os
with increasing «, so that 7}, decreases strongly, and 7, ) 18 positive
Ly v

everywhere; and for the present we confine ourselves only to the
solidification of the least volatile component, so 7,— 0. Let us write
the value which V gets at the rim by the aid of the value derived for

0
Ss from the equation of state, in the form:
eo

MRT db “4/4,

(us — vs) =

; (v—b)? dx v?
It is clear that this value will become negative for v =o, on

the contrary positive for v=/"'); so there will always have to be

a point on the axis «=O, where N—0O. The value which NV

assumes for «—1, is:
dp ' MRT
Oz ), cin tie t— >

and this expression will, accordingly, be negative for «= 1 for all
possible liquid volumes, and even negative infinite. From this follows
that from the point of intersection of WV —O with the axis «—0,
the locus N= 0 will run to smaller volumes. Now whether V=0
and DO will intersect in our figure depends on the place where
N =O cuts the axis x= 0. In this we may distinguish three cases:

1. The point of intersection of NV =O and the axis lies at smaller
volume than the points where D—O cuts the axis. Then no inter-
section of VN =0 and D=O will take place; the points Q and Q’
lie quite outside the axes e—1 and «—0O;

2. The point of intersection VW =O with the axis lies between
the points of intersection of DO with it. Then the point of
detachment does fall inside the figure, but not the point of contraction ;

3. The point of intersection of V —O and the axis lies at larger

208, of ee Ae |

') If we should object to putting »—b, yet assuming that v-> vs , we shall
in any case have to grant that there is nothing incompatible in the assumption
that at a certain high pressure the volume in the solid state can be smaller than
that in the liquid state, and that yet a great increase of pressure may be required
to keep the substance in the same volume after we have replaced some of the

molecules by much larger ones (so & )=2.
zy
*) As said, in every point of the line N —O the q-line passing through it, is
directed to the point indicating the solid substance. Every q-line for infinite yolume
being // v-axis, and terminating in the point v = 8, it follows from the existence
of the line N =O that every q-line cutting this locus, must at least possess one
point of inflection.

( 654 )

volume. Then both the point of detachment and the point of contraction
fall in the figure.

The consequences for the change of the v,«-projection of the binodal
curve with variation of temperature will probably be clear from the
figures 3—5 without further elucidation in these three cases. With regard
to the frequency with which the three cases occur, it is evident that
the last case will occur only rarely, with exceptionally high values of
i or in general of (32). This case would be altogether
impossible, if we had to take the temperature of the triple point,
and the volume which the saturated vapour then has, into account,
for this amounts certainly to some thousands of times 6, and hence
there will probably never be any question of an intersection of
NO with the branch D=O holding for the large volumes at
the triple point temperature. But for our case we have not to reckon
with this temperature, but with the highest temperature at which the
binodal curve solid-fluid has still three points in common with the axis
x=0, and this is evidently the temperature of point A in fig. 1.
This, now, can probably lie very considerably above the triple point,
and moreover as we observed before — not the volume of the
saturated vapour, but the much smaller one of the maximum of the
isotherm must be introduced here. If we e.g. put the temperature
of A so high that the maximum point of the isotherm lies at
a volume 4), the expression will already become positive with
= 3b,, or 6, = 4b, and v, near 6, (da/dz is negative). So, the
case of 3 is, indeed, possible, but it also appears thai it will oceur
only in exceptional cases *).

With none of these three cases do the 7’, .x- and p, 2-figures construed
by vAN perk WaaLs and Sits, agree. They agree in so far with that
mentioned under 3, that the point of detachment and the point of
contraction are assumed to fall within the figure. But it is at the
same time clear from the ve-figures, that a complication must begin

') It appears from what has been said here that the figures 6—9 are meant
quite schematically, for though we have drawn several binodals solid-fluid which
hold for different temperatures, we have left the loci NV =0 and D= 0 unchanged.
This has, of course, been done to save space, for else we could not have repre-
sented much more than one temperature in each figure without rendering the
figures indistinct. But after what has been said it is clear that also the points
Q@ and @' move, and that it might e.g. very well happen that at lower tempera-
tures the point Q' is not yet present in the figure, and that it makes its appearance
only at higher temperatures. The following figures, too, are meant schematically,
and serve only to elucidate the properties mentioned in the text.

( 655 )

in the p,a- and 7, .-figures far below the triple point, viz. already
at the temperature B of fig. 5, 1.e. the temperature, at which in
fig. 9 the new branch of the binodal curve (on the left side) makes
its appearance in the figure. Let us first consider the p, 2-lines.

At the temperature mentioned (7) a new branch begins to form
at the same height as the spinodal line, so far below the point of
the stable coexistence. In the p, a-figure the point where this appears,
is, in opposition with the v, x-figure, indeed a point where the tangent
is indefinite ; for the equation:

ay dy ap \?
N . 1 — Ws ae aay, SSS SSS SS —_——- -—- Ia rs
L ap (w vy) Oar? fe | € /

holds for the former figure; the factor of dv, is zero on the spinodal
line and the factor of dp on the line D=O, which both pass through
the point considered ; oP is there indefinite. The new branch extends
more and more (fig. 6); its maximum continues to lie on the spino-
dal curve, and the point with the vertical tangent on the line D=0.
When the temperature of detachment in the », x-figure (7',) has
been reached, the old branch and the new one unite (fig. 7), and
separate again as figure 8 represents. At the triple point temperature
(7) the middle one and the topmost one of the three points of inter-
section with the axis coincide (in the final point of the double line
vapour-liquid) (fig. 9); afterwards they exchange places. At still
higher temperature the downmost point of intersection with the axis
and that which has now become the middle one coincide; at this
place there is again a point with indefinite tangent (7’,, the tempe-
rature A of fig. 1) (fig. 10); at still higher temperature the binodal curve
solid-fluid has got quite detached from the axis, and its downmost
branch forms a closed curve, which contracts more and more, and
at last disappears at the temperature of the isolated point of fig. 5.
Here it is evidently essential that 7, lies above 7,, and 7’, above
T,, according to the significance which they have in fig. 1; also
T,, the point at which the detached branch disappears from the figure,
must lie above 7’, the triple point, because in the triple point the
binodal curve solid-liquid must still have two points in common with
the rim (a little above it even three). But it is not essential that 7,
lies between 7’, and 7,; 7, might just as well lie above 7,. Then we
get the succession: fig. 6, fig. 9a (triple point), fig. 10a. If now 7, lies
below 7’, there is confluence and section, and we get after fig. 10a
fig. 11, and then Smrrs’ figs. 4 and 5 (loc. cit); if 7, lies also above
7’, first the two lowest points of intersection of the binodal curve so!id-

( 656 )

liquid with the rim join, then they are detached from the rim, and
we get, therefore, in this case, but only above T',, so above tempe-
rature A of fig. 1, the continuous line drawn by Smits fig. 3 (Loe.
cit.), which then passes into figs 4 and 5 (loc. cit.).

The case mentioned under 2 that the point of contraction falls
outside the figure may after all, be derived from the foregoing by
putting 7’,, the temperature at which the detached branch disappears
from the figure, below ‘7’, the temperature at which it detaches
itself from the rim. In our figures it has only this influence that the
loop of figs. 9 and 10a cannot detach itself from the rim, as in
fig. 10, and disappear as isolated point; but this loop contracts more
and more at the rim and disappears there. In this case, too, 7’, can
lie above 7’,, but of course, not above 7’. If 7, lies under 7), we
have the succession 6, 7, 8, 9 and disappearance of the loop in the
rim; if 7, lies above 7’, then: 6, 9a, 10a, 11, and disappearance
of the loop in the rim.

The above case mentioned under 1, when also the point of detach-
ment falls outside the v, x-figure, may be considered as the case that
T, lies below 7,, and 7’, above 7. We have then the succession,
the upper portion of fig. 6 (viz. without the downmost loop), figs.
12, 8, 9, after which the loop merges in the rim. Now in all the
eases mentioned, except in the second subdivision of the case under
3 (so 7’, above 7’,), we meet still with two possibilities. Up to now
we have assumed for those cases, that the triple point témperature
is the highest temperature at which the two binodal curves intersect in
the stable region, and that they have got detached above it. It is now,
however, possible, that also in these cases the two binodals intersect
twice at the triple point and above it. Then fig. 96, is put every-
where for fig. 9, and then this is changed into fig. 11. .

We get then the following summary :

Case under 1.
Upper portion of 6, 12, 8, 9, disappearance of the loop in the rim

%3 fF 5 Oy, 12,18 4 ood $4 ;,
Case under 2.

Os Oune,

49 13 99° ‘So 99
G09d, 1Gg7at re 35°» 0, Sw gg ee
62070, 98,296; ad a ya 52

Case under 3.
Gai ee oe e0, disappearance of the loop in the fig.
6, 9a, 10a, 11, 4and5 Smits __,;, Pe ae a
6, 9a, 10a, 3, 4 andSSmits _,, Nom er ae.
6, 7, 8, 96, 11,4 and 5 Smits se, sd a-aeel bag: 5d, eee

( 657 )

The greatest chance to only one intersection with the binodal curve
liquid-vapour presents, of course, as is best seen from the v, v-figure,
the case under 1, more particularly when in this case the line V = 0
cuts the axis at such small volumes, that it has no longer any point
in commen not only with the spinodal curve, but even with the binodal
curve of the transverse plait. Only » ith a very exceptional course
of the binodal curve of the transverse plait double intersection could
take place in this case. On the other hand it will be highly probable
that always when the line V=—O cuts the binodal curve of the
transverse plait (which will always have to take place in the cases under
2 and 3), also double intersection of the two binodals will be found.

This shows at the same time the connection of this investigation
with that of the preceding communication. For it appears now that
the shape of the p,c-lines holding for 1 with single intersection is,
after all, by far the most frequently occurring, i.e. in almost all
cases where no temperature maximum occurs in the three phase line;
for in this case the triple point temperature is the highest tempera-
ture for which a three phase coexistence exists.

For a complete survey I have also indicated in figs 183—16, how
the binodal curve for the other solid phase gets detached from the trans-
verse plait. This is only possible in one way, because here there
cannot be intersection of the lines D=0O and N=0O. For «=1
for this binodal curve, and so the expression for 7’ at the rim becomes :

Op MRT
) OS Aaa

so always positive for both rims. The line N =O would therefore,
have to become a closed curve, which on account of the shape of
the g-lines may be considered as excluded ').

In the 7)2-lines double intersection will, of course, always occur
above the triple point when the three phase line has a maximum
pressure. For the rest nothing of interest is to be said of the 7\z-
lines ; they have the same general course as the p,a-lines given here,
provided the figures are made to turn 180° round the z-axis, or in
other words, provided a negative 7-axis is made of the p-axis. Then
the points with vertical tangent lie here, of course, on the line
W.=0, instead of on the locus D=0; only at the rim they coin-

1) At least as long as the complications, which are in connection with the
s
2
existence of a locus

= 0, do not present themselves. (See v. p. wAats, Proc.

of this meeting p. 637). I shall perhaps revert later on to the changes which are
to be made in what precedes in consequence of this.

( 658 )

cide. If the pressure maximum of the three phase line should be
found at higher pressure than the point A of fig. 1, we must, of
course, have the case mentioned under 3, i.e. the point of contrac-
tion must lie within the figure.
Ra has been assumed in the above that throughout the region
P

(3°) = positive, and that a decreases with increasing 6. The case
aL v

that @ inereases with increasing 6 does not present any new points
of view. If we have a system where a strongly increases, so that

Op
the critical temperature rises with 6 and (3 is negative, the ex-

Ox
pression
Op\ | MRT
ponds (ae) me
is evidently always negative for #, = 0. And this is obvious,

because this axis is now also that of the more volatile component ;
on the other hand the reversal of sign may now take place with
the other axis. What happened on the left just now, will now take
place on the right, and vice versa. It is only worthy of notice that
now the line N=—0O, if it exists, must intersect the axis 2—1 in
two points. For the expression

: MRT db da/dz

2). aera

where db/dv and da/dx are positive, becomes positive for v=6 and
v=o. From this follows that besides the just mentioned cases,
another possibility may be found, i.e. that the point of contraction
does fall within the figure, but not the point of detachment. For
the p,a- and 7’, x-figures it makes only this difference that a loop
formed in the way of fig. 12, (which always disappeared in the
rim in the other cases) may now also disappear like the loop of
fig. 10 in a point within the figure. It is further clear, that in this
case the point of contraction will much sooner fall within the figure

+ MRT

; s E 7 3 db
than in the preceding case. For according to formula (1) — must

dx
have an excessively high value for the expression to be able
still to become positive with a volume v=1006. If, however,

da/dx = 2a, — 2a,,=1.8a,'), then:
1) With the values for a and b of Lanpotr and Boérnstetn’s table 82 we find

b; a:
about 12 for the highest value of a about 250 for that of —; if hydrogen is

b a
l 1
excluded, the values become resp. 8 and 40. So, whereas with exclusion of
; Ogi dg—Q,
hydrogen, pairs with a ratio ——— > 7 cannot oceur, can reach the
1 Y

value 39.

(659 )

da/dx mai bool 18 (= od ?)
2 2

v v a—

da/da:

vo?

and so (v—vs) becomes of the order of magnitude 1.8 {J/R7' —

— p(v—v;)}. With this volume and the low temperature holding
here the latter term is certainly a small fraction of J/R7, also
MRT db

metip tg 3°50 that the expression becomes negative.
C= x

Ow
region, call for a further discussion, for it does not present any new

Op da.
points of view. If = becomes zero in consequence of —- first being
uv), da

negative then positive (minimum critical temperature), we shall have
on either side what in the first case took place on the left side
(fig. 6—12); if da/dr is first positive then negative (minimum of
vapour pressure) we have on either side what happens on the right

side in figs. 13 —16.

Op
Nor does the case that ( may become zero in the examined

Nor does, in view of the foregoing, the occurrence of cases in which
the plaitpoint curve meets the three phase line, offer any difficulty.
It is only clear, that the two points where this meeting takes place,
must lie below the point of detachment (double point of the binodal
curve solid-fluid) both in pressure and in temperature. For when detach-
ment has taken place, and so the binodal curve has split up into two
branches, it seems no longer possible, when the v,x-figure constantly
contracts and hence (és/), has a negative value, that the three phase
pressure coincides with a plaitpoint pressure’). But nothing indeed
pleads against this conclusion. Only when we cling to the supposition
that the point of detachment must always lie at the rim we are
confronted by unsurmountable difficulties. For then the temperature
and pressure of the point of detachment coincide with those of
BG (fig. 1), and this point, lying considerably below the triple point,
lies certainly, at least in pressure, far below any plaitpoint.

In conclusion we may remark that the cases where 2, lies between
1 and QO, i.e. where the solid substance is a compound entirely or
partially dissociated in the fluid state, may be derived in all their details
from the v,a-figure (fig. 2) without any further difficulty. We get
then at low temperatures Smits’ diagrams in the figures 4—-7 in his

1) Compare the figures referring to this in van perk Waats, (These Proc. VI,
p. 237, VIII, p. 194 fig. (2) and Sarrs (These Proc. VI, p.491 and 495 and VIII,

p. 200 (fig. 10).

( 660 )

paper: Contibution to the knowledge of the p, x- and the p, 7-lines*,,
at least when we take the maxima of pressure very much higher
and the minima very much lower, so that on the left side the figure
intersects itself twice. The detachment of the two binodal curves then
takes place in a very intricate way by means ofa series of modifications,
which I shall, however, omit, with a view to the available space.
So, for this I must refer to the lectures which I am arranging for
publication as mentioned in the beginning of my preceding commu-
nication, though certainly some time will elapse before they see
the light.

Physics. — “Contributions to the knowledge of the w-surface of
Van per Waats. XIII. On the conditions for the sinking and
again rising of a gas phase in the liquid phase for binary
mixtures. (continued). By Dr. W. H. Kersom. Communication
N°. 96° from the Physical Laboratory at Leiden. (Communi-
cated by Prof. H. KamerLincn ONNESs).

(Communicated in the meeiing of January 26, 1907).

§ 6. Conditions for the occurrence of barotropic plaitpoints for
mixtures with M,=2M,, v,, = 8 vz. Now that it had appeared in
§ 5 (These Proc. p. 510), that there exists a barotropic plaitpoint )
on the assumptions mentioned there, first of all the occurrence of
barotropic plaitpoints with M,/1/, = 2, Uk, = */, Was subjected to
a closer investigation, partly also on account of the importance of
these considerations for mixtures of He and H,*). The barotropic
_plaitpoints given in table I respectively for the ratio of the critical

') These Proc. VIII, p. 200.

2) The proof that this barotropic plaitpoint really lies on the gas-liquid plait, is in
connection with the discussion of the longitudinal plait. In a following Comm.
by Prof. KamertincH Onnes and me on this latter subject, the treatment of
which was postponed for the present as stated in Comm. N®. 962, the proof
in question will be included.

3) To enable us to judge in how far this last assumption is in accordance with
what is known about mixtures of He and Hy, the followimg remark may follow
here, in the name of Prof. KaMertincH Onnes too, (cf. Comm. N° 965 § 4, Dee.
‘06 p. 506) on & (cf. van peR Waats, These Proc. Jan. ’07, p. 528) and a for
helium: It proved in the preliminary experiment described in Comm. N°. 96a that
on analysis the liquid phase contained at least (some He has evaporated from
it during the drawing off of the liquid phase) about 3 °/) He, the gas phase at
least (a very small quantity of liquid has been drawn along with the gas phase
being blown off) about 21°/, Hy (estimations of the corrections which for the
reasons mentioned ought to be applied to the results of the analysis make
it probable that they will not considerably influence the results derived here [added
in the English translation]), Let us put the density of liquid hydrogen boiling under

( 661 )

temperatures of the components given there are found in the way
explained in § 5.

atmospheric pressure according to Dewar (Roy. Institution Weekly Evening Meetings),
25 March '04) at 0.070, and let us derive the coefficient of compressibility according
to the principle of the corresponding states, e.g. from that of pentane at 20° G,,
then the density at 40 atms. is 0,072. If we calculate the increase of densily in
consequence of the solution of helium from van per WaAats’ equation of state for

; : 1
a binary mixture by putting 0./He = > 0./H2 for this correction term, we get

for the density of the liquid phase at the p and 7 mentioned if it contained
a 0} Her >. 0-077.

The gas phase will have the same density at about the p and 7’ mentioned (cf. Comm.
N°. 96a, Nov. 06, p. 460). The theoretical density (AvocapRo-Boy.e-Gay-Lussac) at
T=20° and p= 40 atms. = 0.0885. If we assume Van per Waats’ equation of state
with a, and } for constant x not dependent on v and 7, to hold for this gas phase,

: : be dx 1
it follows with the above given value of the density that er
v—br RT Vv
= 0.00042 (Kounstamm, LANDOLT-BORNsTEIN-MEYER-

For Agg = A, — 0 with a uu

HorreR’s Physik. Chem. Tabellen), and with v,,— 0.0021, putting x = 0.80 for the
gas phase, we should obtain: be 0.21 v,,= 0.00044. We should then, if

we may put B, 4p 2, | yy T Ogg gy) Bet Ono gg = 9.00083 = 8's Dy CO g¢= 0.00088,

12M

Kounstamm |. c.). If we wish to assume positive values for a. and gq (cf. Gomm.

N°. 96a, p. 460), we should have to put 6,,),>%/s5,14, for T= 20°; if we

assumed that the gas phase contained 15°/) He we should derive from the above

mentioned experiment for positive values of a, and a2: 5,,,,> 0.31 0,,

These results harmonize very well with what may be derived about byyy.

at O° C.; the ratio of the refracting powers (RAYLEIGH) gives: oso 0.31 Ow

while the ratio of the coefficients of viscosity and also that of the coefficients of
the conduction of heat lead to a greater value for D yyy, (about 1/2 4 y/4,)-

If we take bg544/0), 44 = V2, we should obtain from the above given considerations
(putting @oy—=Vayiy 42)? Mui = Vis, 80 that Typy_ = about 0.35°.

This renders a value for the critical temperature of He <0.5° probable.

This conclusion would not hold if bz3f fer «0.8 were considerably greater
than follows from the hypothesis that bz@ varies linearly with x. This however
is according io the experiments of Kuevey, Keesom and Brinkman on mixtures of
CH;Cl — CO, and CO, — 03, not to be expected. The experiments of VeRSCHAFFELT
on mixtures of CO,—H, would admit the possibility, but give no indication for
the probability of it. [Added in the English translation].

So though probably 42/:,, for mixtures of He and H, is larger, yet we shall
here retain the supposition made in § 5 on 22/s,,, with which the calculations
were started, because the accurate amount is not yet known to us, and we only
wish to give here an example for discussion; moreover the course of the y-sur-
face will not be considerably modified by this difference in any essential respect.

( 662)

TABLE I. |

Barotropic plaitpoints at M,/M, = */,, ve, /vi, = */s-

D/T Gene| Aa JA a tl sen es opt Pry
0.002 06 0.3957 | 1.00 ! 4.805 |
0.0210 | 0.65 0.3481 0.934 4.772
4a | 9/8 1/3 444/484 | 576/124
0.0604 | 0.7 0.9048 | 0.867 | 4.758
0.1044 | 0.75 0 2636 0.300 | 4.780

| 0.1472 | 08 0.2934 0.726 | 4.844

| 0.1842 0.85 0.1833 | 0.638 4.800

0.2106 0.9 | 0.4424 | 0.581 | 4.567 |
0.2176 | 0.925 | 0.41212 0.444 | 4.202 |
0.2199 | 0.94 | 0.4081 0.387 3.751
0.82 | 0.95 0.0991 0.343 3.282 |
0.2148 | 0.96 | 0.0854 0.266 2.107 |
0.2106 | 097 | 0.072 | 0.204 |! 0687
0 2040 0985 | 0.0644 0.130 =. OT oy

| 0.1996 | 0.99 | 0.0585 | 0.078 — 5AM |

| 0.1960 | 0.99 | 0.0518 | 0.033 | — 9.793

| 0.1956 | 0.996 | 0.0495 | 0.028 | 10.86 |

| 0.1964 | 0.9975 | 0.0478 0.019 —12.086

| 4/4 | { | eae tat 0 | ay

For so far as the assumed suppositions hold, the barotropic plait-
points given in the table have only physical significance if 7;,, does
not become so low that solid phases make their influence felt (ef.
Comm. N°. 964, § 5 4), and if moreover the portion of the p-surface
in the neighbourhood of the plait-point is not covered by a portion
of the derived surface indicating more stable equilibria (as e.g. will
be the case for negative pressures). In how far the indicated baro-
tropic plaitpoints will belong to the gas-liquid-plait will be more
fully treated in a following communication (cf. footnote 2, p. 660).

In the first place it follows from table I that with the assumed

wa” .

( 663 )

ratios of the molecular weights and of the critical volumes, baro-
tropic plaitpoints prove only possible (quite apart from the question
whether they are physically realisable, and whether they belong to

1
the gas-liquid plait) for 7;,/7%,, Son

The barotropic plaitpoint for the ratio 7%,/7).,—= 0.0002 is a
plaitpoint for a mixture, one component of which is a gas almost
without cohesion (Comm. No. 964 § 1). A further consideration of
it would lead us again to the region of the longitudinal plait.

The conditions relating to barotropic plaitpoints for ,,; near 1
furnish a contribution to the knowledge of van per WAALS’ w-surface
for binary mixtures with a small proportion of one of the components ').
We find for 7;,/7%, near */,, putting «,;—=1— §:

my Haase 3h
Ter, = 7 | 5 b+ Te a TE

20 29
== bg a ea
are 16

It is seen from the series of the ratios 7;,/7;, in table I, that
in this a maximum and a minimum occur, respectively for about
T/T, = 0.219 and 0.196. From the formulae derived for Lbpl
near 1, a minimum and a maximum for uw is found, and hence for
Ty.,/Ti,, respectively at x); = 0.9968 and 0.969. That the latter is
in reality found at %,;=0.94 is due to following terms in the
development.

Bor 77/77, < 0.196. or. 0.219 < T,,/T;, < 0.25 one barotropic
plaitpoint is found, for 0.196 < 7T;,/T,, << 0.219 three. In connec-
tion with Comm. No. 966 § 2 (Dec. ’06 p. 502, ef. also this Comm.
§ 1, Dee. ’06, p. 508) it follows also from this that for the mixtures
considered here at lower temperature the longitudinal plait makes
its influence felt.

The experiment described in Comm. N°. 96a proved that for
mixtures of He and H, at — 253°, ie. about 7= 0.65 Tin,, a
barotropic tangent chord is found on the w-surface. If at that
temperature only one barotropic tangent chord occurs, this will point
to this (Comm. N°. 966 p. 504) that for the mixtures of these sub-
stances 1'%,,); > 0.65 Tin, and therefore according to this table
Tre <. 0.18 Tiy,, while the found considerable difference in con-
centration between the gas and the liquid phase (see Footnote 3,

1) Cf. Comm. N?. 75 (Dec. ’01), .N*. 79 (April ’02), N° 81 (Oct. ’02), Suppl.
N°. 6 (May, June ’03).
45
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 664 )-

p. 660) indicates, that 7%,;, would have to lie still pretty much
higher, and therefore 74, pretty much lower (probably < about 4°) ’).
Of this result we availed ourselves in the treatment of the estimation
of the critical temperature of He in Comm. N°. 960.

(To be continued).

Physics. — “Contributions to the knowledge of the w-surface of VAN
peR Waals. XIV. Graphical deduction of the results of
KUENEN’S experiments on miatures of ethane and nitrous ovde.”
Supplement 14 to the Communications from the Physical
Laboratory of Leiden. By Prof. H. KameriincH Onnes and
Miss T. C. JoLuxzs.

(Communicated in the meeting of Januari 26, 1907).

§ 1. Introduction. In what follows we have endeavoured to
derive quantitatively by first approximation the behaviour of the
mixtures of N,O and C,H, (mixtures of the II type’)), which has
become known through KvuENEN’s experiments *), by the aid of VAN DER
Waa.s’ free-energy surface. The w-surfaces construed for this purpose
(see plate I) are the counterparts of those construed in Comm. N°. 59
(These Proc. Sept. 1900) and Comm. N°. 64*) for the derivation
of the results of KugNEN’s and HartMan’s experiments on mixtures
of CO, and CH,Cl (mixtures of the I type). In the graphical treat-
ment’) of our problem we have chiefly followed the method given
in Comm. N°. 59, where the critical temperature and pressure of
some mixtures were borrowed from KurNEN’s determinations, and
then the results of another group of experiments — those referring
to the conditions of coexistence of two phases at a certain tempera-
ture — were deduced by the aid of van DER WAALS’ theory.

KUuENEN’S results for N,O and C,H, are principally laid down in

1) If 4x/p,, is taken larger than 1/, (Cf. Feotnote 3 p. 660) this supposition too
makes the upmost limit for 7’,,, on the said supposition smaller. This is seen
when we compare with table I that we obtain | i =0.679 for %:2/4,, = 1/4
with 2, /-p, = 0.15.

*) Hartman, Leiden Comm. Suppl. no. 3, p. 11.

3) Kuenen, Leiden Comm. no. 16, Phil. Mag. 40, p. 173, 1895, cf. also Kamer-
LiNGH Onnes and Zaxrzewski, Leiden Comm. Suppl. no. 8. (These Proc. Sept. 1904).

It is remarkable that the possibility of this case was foreseen by vAN DER WAALS,
Contin. Il, p. 49 [added in the English translation].

4) Arch. Néerl. Serie II, Tome V, p. 636.

®) Only graphical solutions for definite cases are here possible. (Cf. Suppl. 8,
These Proc. Sept. 1904. § 1).

( 665 )

four figures '); one of them gives the critical quantities from which
we shall start in our deduction, and the border curves for mixtures
of different concentration; the three others, which represent the xv-
projection of the connode with the connodal or tangent chords at
the temperatures 20° C, 25°C, 26° C, show the contraction of the
transverse plait with rise of temperature, and finally its splitting up
into two plaits.

We have thought that we could obtain a better comparison of
observation and calculation, when representing the observations by
the xv-figures for 5° C., 20°C. and 26°C., and the p7-figure, instead
of by the wv-figures at the before mentioned temperatures and the
pT-tigure.

§ 2. Basis of the calculation. Law of the corresponding states
and reduced equation of state. We start (cf. Supp]. N°. 8, These
Proce. Sept. 1904 § 1) from the supposition that the law of the
corresponding states — at least within the region of the observations
— holds as well for C,H, and N,O as for their mixtures. As reduced
equation of state we chose equation V.s.1 of Comm. N°. 74%) p. 12
For a region of reduced temperature and pressure which incloses
the region which corresponds to that of the observations under in-
vestigation, this equation is as closely as possible adjusted to CQ,,
which in thermical properties has much in common with N,O, and
there is no reason to suppose that this will not be the case with C,H,.

In the application we are, however, confronted by this difficulty,
that V. s. 1 deviates most strongly from the observations on CO,
exactly in the neighbourhood of the critical state. (Cf. Comm. N°. 74
and later Kresom, Comm. N°. 88). If from V. s. 1°) the point is
derived, for which

oY 0?

ae =—— (and — ths
we find ¢ 1.010595, 4» = 1.0407.10-*, »—1,06566. Kexrsom’s
observations, Comm. N°. 88, give for the critical volume, when it
is sought by the application of the law of the rectilinear diameter,
for CO,

vy — 0.00418 and for 2 = = 1.0027.10-3. . . (1)
k

1) Where it was necessary, Kuenen’s figures have been rectified in accordance
with the results of observation given by him.

2) Arch. Néerl. Vol. Jubil. Bosscha. Serie II, tome 6, 1901.

5) In the calculations 7 is put 273°,04 for the freezing point of water, because
V. s. 1 was calculated with this value.

45%

( 666 )

We find then:

t, = 1.010595 instead of 4% =1
b, = 1.0379 S. a) ea!
pe = £.065606~ ~:.;, oes,

The isotherms from which V. s. 1 has been derived by the com-
putation of the virial coefficients 3, © ete. (See comm. N°. 71, These
Proc. June 1901), indicate therefore, by means of interpolation
according to this mode of calculation, a critical state, which, drawn
in the pv-diagram, has shifted with respect to that which was found
by immediate observation; the critical temperature according to V.
s. 1 is namely t,7% when 7}, is the observed critical temperature.

So are also the values found for Ymaz., Vig. ANd Yyap, at t by the
application of Maxwe.u’s criterion, different from those which we
should find when dividing Paz. by px, Viig. ANd Vyay. by vz. The
deviations are of the same order as the deviations of the substances
inter se, when they are compared by the law of the corresponding
states. At t—0,9 they are about zero, but they increase as we
approach the critical state, so that the deviations agree with a gradual
transformation of the net of isotherms. The following table gives a
survey of the deviations in the corresponding values.

Column A refers to V.s.1 and implicitly to pz, 7% of CO,

» BS > thesobseryatious » CQ
” 6 ” ” ” ey) ”> N,0.
4 | F | c
| |
| t p p t p
{

1.0106 1.066 | |

| 1 1. 1.
| 9.0% 0.826 || 0.975 | 0.84 || 0.975 | 0.854
| 0.950 | 0.695 | 0.950 | 0.709 |} 0.950 | 0.720
| 0.925 | 0.589 || 0.925 | 0.599 |
| 0.900 | 0.490 || 0.900 | 0.494 | 0.900 0.486 |

In the neighbourhood of the critical temperature the phenomena
are governed by the difference of the temperature of observation and
the critical temperature, 7’— 7); for this reason we have chosen
for the detailed model of 26° such a temperature 7” for the com-

( 667 )

parison of the observations at that temperature 7’ with the result of
computation that
~— 1 = tT, —- T’.

At the general survey for 26° we have applied to t= T; a cor-
rection = At, so that At—0.01066, when 6=10(t—0,9) for
0<@4<1, whereas At—O for all other values of 6. The correction
Attot was accompanied by a correction Av to v and Ap to yp, so
that Ap —0.06576 and Av = 0.0379 6, which together represent a
regular increase of the corrections from t= 0,9 to the critical state.
For the detailed model of the y-surface for 26°, on which only the
part from «=—0,35 to 2=0,65 was represented, we used every where
the same correction viz. A t=0.0106, Ap = 0,0657, Av = 0,0379.

§ 3. Critical quantities for the mixtures. Kunnen has determined
the plaitpoints 7 plc» Pyle» Vple,» for some mixtures with the molecular
concentration v Tix, Pix, Ure, the critical points of contact, and
Tix, Pkx> Vix the eritical states of the mixtures taken as homogeneous
differ so little’) from these values, that this difference may be dis-
regarded for our purpose, and so they are also known for the
mixtures investigated by KUENEN.

‘yy

+ 1 » |
ok |
{ qieece 5} ee
a i IL i | ~ | ee
|
} < |
> |
| | | |
i — ——_ —_—+— —-— —+— —S |
| 1 it ex
e BIO, 23 oss o76 ° wo PT) 7] ole a)
ee Fig. 3
oud —— =: =~
|
rob =
}

i
i

1) This has been fully treated by van per Waats, Cont. II, § 11.

( 668 )

The critical quantities for the other mixtures were found by gra-
phical interpolation. Fig. 1 gives p,., fig. 2 Ti. as function of 2; in
fig. 3 v,, has been calculated from pz, and 7%;, by the aid of 2 (see
formula (1)); the vz, observed by KueENEN have been indicated there.

By carrying out the construction for the connode by the aid of the
P, and y — # ES —v se curves (see Comm. N°. 59a')), which the

& & v

models for those different temperatures yield, we may derive 742 — vpi.2,
Pkx — Ppl.c in first approximation (see Comm. N°. 59a). Applying
these corrections we should then have to repeat the calculation from
the beginning, to obtain more accurate values for vr, Pez. We have
confined ourselves to a first approximation in all our constructions,
as also a further correction of the equation of state V.s. 1., which
can cancel the deviations mentioned in § 2, has not been applied
and we were the more justified in this, as these latter deviations
are larger than those we have now in view.

§ 4. Construction of the w-surfaces.

From the equation of state V.s.1 we find immediately the reduced
w,7 curves, from which are then deduced the ordinary w-curves accor-
ding to Suppl. N°. 8,§ 4; or the ordinary virial coefficients, whic hare
then used for the calculation of yw according to Comm. N°. 59.

For the construction (cf. Comm. 59) use was made of:

Bie SD. wy, =wt 0107+ 250
ees ww, —~w+ 01024 36,50
» 2° w,,=—w+ 0,242 4+ 57,3 2,

while a suitable constant was subtracted from every y. Here v is
expressed in the theoretical normal volume, just as in the diagrams.
From the ~,7 curves (ef. fig. 1, pl. II) the y,.7 curves (see fig. 2,
pl. Il) and the pr curves were graphically derived. The models for
w were construed on a scale 5 times larger than the diagrams on
pl. Il, pl. II and pl. IV.

§ 5. Determination of the coexisting phases.
Applied was both the construction by rolling a glass plate on the

1) In giving the figure 3 in Comm. N°. 59a for this consuuction it was stated

that this figure was very imperfect. It appears now that the loop ought to contain
two cusps. We found out the error by the aid of the general properties of the
substitution-curves treated by van peR Waats (Comp. Proceedings of this meeting).
This error shows the more how necessary it is that graphical solutions are
controlled by such general properties as van per WaAats is now publishing,
| Added in the English translation].

AEG aly

ee ae

( 669 )

model, which yields the connode and the tangent chords, and the
simplified construction in the plane given in Suppl. N°. 8, § 7, to
which a small correction was applied. After viz., a provisional
connode, that of the mixtures taken as homogeneous, has been found
by tracing curves of double contact to the y,7-curves, and by
determining conjugate points 4 on the gas branch at some points a

ae : Ow

on the liquid branch of that connode, so that every time — is the
Lv

same for the two conjugate points, the lines which join every two

of these points a and 4, are produced outside the provisional connode,
till they cut the isobars which pass through a, in points ¢,
which together represent the required gasbranch of the connode; ¢
and a are then considered as conjugate points. In the w-surface at
5°C. the two constructions yielded fairly well corresponding results,
both with regard to the chords and to the connode itself, as appears
from pl. Il fig. 4, where — — — — denotes the connode and the
connodal tangent-chords found-by rolling a glass-plate on the model,
.__ those found by means of the just mentioned construction.
That the simplified construction, which was more particularly plotted
for equilibria far below the critical temperature (see Suppl. N°. 8, § 7)
still leads to our end, is probably due to the fact, that we have
here to deal with a mixture of the II type.

With the y-surface for 20° the slight depth of the plait rendered
it necessary, to considerably diminish the longitudinal scale for the
v-coordinate of the model. This compression (ef. pl. I, fig. 2) rendered
the plait sufficiently clear to determine the connode and the place
of the connodal tangent chords by rolling a glass plate. By means
of the simplified construction the connode was still to be obtained,
but the determination of the tangent-chords became uncertain.

With the y-surface of 26° the depth of the plait (here split into
two) becomes so exceedingly slight, that it does not appear but with
a computation with 7 decimals, and even then it manifests itself
almost quite in the two last decimals. Hence it is not possible to
model a y-surface (we mean a surface derived from the w-surface,
on which the coexisiing phases are still to be found by rolling a
plane}, on which this plait is visible, nor is it of any avail to confine
ourselves to a small part of the surface, because the curvature of the w,7-
curves is very strong exactly there where something important might be
shown. The determination of the connode and the connodal tangent
chords by construction according to §8 of Comm. N°. 59a, which
can always be carried out provided enough decimals are worked
with, remained still uncertain up to 7 decimals, so that we have

( 6T0>))

not pursued it any further. Thus the represented part of the y-surface
for 26° from 20.35 to 70,65 and from v—=0.0038 to v—0.0070,
has been given by us chiefly to demonstrate how exceedingly small
the influences must be on which a plait depends, and how much
care is required to determine a plait experimentally which is not at
all to be seen on the surface. The curves drawn on the surface,
which relate to the plait, were found by indirect ways, partly by
construction, partly by calculation. To facilitate a comparison of the
models inter se the region of « and v, on which the model for 20°
and that for 26° extends, has been indicated on the model for 5’,
on the model for 20° that for 26°.

§ 6. Further remarks on the different models and drawings obtained
by construction.

a. The w'-surface for 5°. The model, pl. I, fig. 1, and the drawings
pl. I, figs. 1, 2 and 3 show curves of equal concentration, w',7,
equal volume and equal pressure, the connode and the connodal

Ow’

tangent chords. AB en Saat some pressures are represented
by negative slopes on the stable part of the y'-surface, in consequence
of which the character of this w'-surface does not in this respect imme-
diately express that of the y-surface, where all the slopes are positive.
A connodal tangent-chord, near the concentration with maximum
pressure, almost touches the y-line. With the concentration of
maximum pressure this would be just the case. Just as the connodal
tangent chords the isobars are traced in the projection on the « v-plane
(Pl. II, fig. 8) in full lines, the connode is denoted by —_.- se

For the isobars') we may note several peculiarities, to which
vAN peR Waats has drawn attention in his theory of ternary
systems ”). The isobar which touches the connode on the liquid and
vapour side, belongs to the pressure p = 36,6, which is found for
the mixture which when behaving as a_ simple substance should
have a maximum coexistence pressure. The pressure curve 2
determines the transition between the continuous isobars (taking
the region outside the drawing into consideration) and __ those
split up into two branches. The parts of the continuous isobars
which point to P, have each a point of inflection on either

O*7y 0?yp

side: of the top.. The shape of the curves —— = 0 and———=@
Ovda Ov?

1) Cf. the sketch by Harrman, Leiden Comm., Suppl n°. 3, pl. Il, fig. 5.
2) These Proc. March 1902, p. 540.

( 671 )

is as has been indicated by vay perk Waats'). The points of intersec-
tion of these two curves are the centre Q of the isobars and the
double point of the pressure curve 2, P.

—b. The w'-surface for 20°. Fig. 1, pl. Ill denotes the y,7-curves
and the connode. Fig 2, pl. Ili the w,7-curves and the connode.
Fig. 3 gives the projection on the « v-plane of the connode, of
ihe tangent chords and of some isobars. The connode is denoted by
__.__._. Pl. I fig. 2 gives a representation of the model.

c. The w-surface for 26°. Fig. 1 pl. IV gives the y,7-curves, fig. 2
pl. IV gives the critical states, A, and A, the isobars and the con-
nodes for the mixtures which are taken as homogeneous, and whose
gas branch as well as whose liquid branch is almost a straight line.
Though in the calculations (see § 2) the plaitpoint «7, and the criti-
cal point of the homogeneous mixture 27;, have been considered
as coinciding, a distance has now been given between these points
which has been fixed by estimation®). The dotted parabola has been
taken from VeERSCHAFFELT’s calculation, Suppl. N°. 7, p. 7, though
properly speaking it holds only for the case that the maximum
pressure falls in P, or P,; the produced connode denotes the probable
course of this part by approximation. Pl. I, fig. 3 gives a repre-
sentation of the model. All this refers to a small region of v and v;
fig. 3 pl. IV, however, indicates by —_.__.__._. the connode according
to the construction for the mixtures taken as homogeneous all over
the width of the y'-surface. The square drawn denotes the extension
of the just treated part of the y’-surface.

d. The contraction and the subsequent splitting up of the plait
appears from fig. 4 Pl. III, where the xv-projections of the connode
and some connodal tangent chords of the three models have been
drawn on the same scale after the wv-figures for 5°, 20° and 26°
mentioned under abe.

1) Prof. van peR Waats was so kind as to draw our attention to a property
which might also have been represented in the figure, when also the curve for
Ow
ir had been drawn, viz. that the minimum volume in the vapour branch,
v 0x

3
and the maximum volume in the liquid branch lie on the curve i a = 0 which
v7 0a

2
has a course similar to that of the curve sg =o more particularly it has the
same asymptotes, and it deviates from it only in this, that with greater density
the curve passes over larger volumes.
2) Here the representation of the plait must come into conflict with the theory
or with the simplification introduced at the basis of the calculation. With a view
to the illustration of the theory by figures the latter has been chosen.

( 672 )

§7 Comparison of the construction with the observation. On the
whole this is very satisfactory, taking the degree of approximation
into consideration.

a. In pl. Il fig. 4 the diagram for the plait at 5° indicating
KvENEN’s observations, has been drawn in full lines. The figure
contains at the same time that obtained by construction. The single
observations have been denoted by [-| (see § 6a). Besides the con-
struction with the model indicated by — —— W— and by \.-/, also
the simplified constructions in the plane indicated by —_.—.— and
by ©, the outermost of which refers to the less simplified con-
struction, represent the character and also the numerical values satis-
factorily.

b. In pl. Il, fig. 3 the figure representing KUENEN’s observations

for the plait of 20°, has been indicated by —— ——. The figure
contains at the same time the __._.., obtained by construction on

the model (see $64). The correspondence at «= 0.3 is the worst,
which is no doubt in connection with this, that here we have already
vot very near the critical temperature, and that strictly speaking,
different values should be assigned to 7” (see § 2) for all v, and
corresponding Av and Ap should have been taken into consideration.

c. In pl. IV fig. 3 the figure representing KUENEN’s observations,
have been indicated by. full lines; the figure contains also the figure
derived in § 6c denoted by —..— curves.

d. Plate IV fig. 4 and 5 may serve for a comparison of KUENEN’Ss
pT-figure (fig. 4) with that derived by construction (fig. 5). In
accordance with the remark on 7” in §2, we have proceeded for
26° as follows :

For 5° and 20° the values of p and 7’ have simply been taken
from the construction with the model, mentioned under a and /.
Then we marked «) the p’s and 7”s, obtained by multiplying KuUENEN’s
pi and Ti. by py, and t, (see § 2); @) for the different values ofa the
values of 7” and of 6 for the temperature of 26° have been calculated,
and then 4¢ and Ay determined by the aid of this 6 according to
§2; the values of p and 7’ corrected in this way have been denoted
by +--+ + and joined by .—.—-.— eurves with the points men-
tioned under a). The full and the dotted curves give the corrected
values. Between the parts where we started from the critical
temperature, and the p7-lines derived fromthe models of 5° and
20° a space has been left open.

( 673 )

Pathology. — “On the Origin of Pulmonary Anthracosis.’ By
P. Niruwenuvse. (From the Pathological Institute in Utrecht).
(Communicated by Prof. C. H. H. Spronck.)

(Communicated in the meeting of January 26, 1907).

As is known, von Breurine and CaLMerre oppose the doctrine accord-
ing to which the pulmonary tuberculosis among mankind proceeds
in most cases from inhalation or aspiration of tuberclebacilli. They
presume the tractus intestinalis to be the porte d’entree of the virus.

In connection with this new hypothesis VANSTEENBERGHE and GRiskz')
have made some experiments at the end of 1905 in CaLmerrr’s
laboratory about the origin of lung-anthracosis.

They mixed the food of full-grown cavies with soot, Indian ink
or carmine and made the animals eat a large quantity of this. After
24 hours already they found resp. black and red spots in the lungs
especially in the upperlobes and along the edge of the underlobes.

VANSTEENBERGHE and Geisez concluded from these results that
the fine parts, taken up in the intestines, pass through the mesenteric
glands and thoracic duct and after having reached the blood in this
way, they are caught by the lungs.

According to their conclusion the carbon particles suspended in the
atmosphere would not be inhaled, but swallowed, thus reaching the lungs
via the intestines. The theory of the intestinal origin of the pulmo-
nary anthracosis was propounded half a century ago by VILLARET*);
it had however met with little success, and after the careful resear-
ches made by Arnonp*) on the inhalation of fine particles it was
totaliy forgotten.

Whereas VANSTEENBERGHE and Griskz tried to defend the theory of
ViLLaRET, after having made new experiments and no less a person
than von Benrinc doubted the exactness of the generally assumed
opinion, no one will be surprised that criticism soon followed.

Whilst I was working in the laboratory of Prof. Spronck, to
whom | offer my thanks for his continual interest in this research,
repeating the experiments of VANSTEENBERGHE and GrisEz, several
treatises appeared on this subject. First of all Ascnorr *) advanced

1) Annales de I’Institut Pasteur, 1905, p. 787.

2) Vitaret: Cas rare d’anthracosis, Paris, ~1862. ref. in Schmidt's med. Jahrb.
1862, Bd. 116.

3) Arnotp: Untersuchungen ueber Staubinhalation und Staubmetastase, Leipzig, 1885

4) Sitzungsber. der Gesellschaft zur Bef. der Ges. Naturwissenschaft, Marburg,
13 Juni, 1906.

( 674 )

the opinion that there must have been technical mistakes in the
experiments of VANSTEENBERGHE and Griskz; some time afterwards
he was enabled to convince himself of the incorrectness of their
opinion by his own experiments ').

Mirongsco *) after bringing fine particles into the stomach of rabbits,
was not able to recover them in the lungs.

In August 1906 VANSTEENBERGHE and SONNEVILLE*) described a
new series of experiments which confirmed the results of VANSTEEN-
BERGHE and GRisbz.

Fine particles which were brought into the mouth with a catheter
were already to be recognised in the lungs after a lapse of 5 or 6
hours.

Soon afterwards the opinion of VANSTEENBERGHE and GRISEZ was
opposed by two authors: ScHuLze*) in a temporary publication con-
cluded that the pulmonary anthracosis could not proceed from the
resorbing of fine particles from the intestines and Prof. SpRoNcK com-
municated shortly afterwards at the 5 International Conference on
Tuberculosis the results of some of the following experiments, which
were adverse to the results, gained by VANSTRENBERGHE and GRiISEZ.

In a more extensive treatise ScHuLzE*®) demonstrated further how
substances are lightly aspirated into the lungs either by administering
them with the catheter or by ordinary eating. A rabbit however,
had received within two months the total quantity of 200 grams of
vermillion through a gastrotomy, yet no trace of vermillion was to
be found in the lungs.

On the other hand some investigators took the part of VANSTEEN-
BERGHE and Grisez: Petit’) brought carbon particles into the stomach
of six children who were in an advanced state of tuberculosis or
athrepsy and after a post-mortem examination he found pigment in
the lungs in three of them and Hermann‘), on the authority of
experiments, esteemed an intestinal origin of the lung-anthracosis
possible, but compared with the inhalation-anthracosis of very inferior
_ significance.

Afterwards the results of VANSTEENBERGHE and Griskz were empha-

1) Braver’s Beitriige zur Klinik der Tuberculose, 1906, Bd VI, Heft 2.

2) Compt. rend. de la Soc. de Biol. 1906, T. 61, N°, 27.

3) Presse médicale, 11 Aout 1906.

4) Miinchener Med. Wochenschr. 1906, N° 35.

5) Zeitschrift fiir Tuberculose, October 1906,

6) Presse médicale, 13 Octobre 1906.

7) Bulletin de ’ Académie royale de médecine de Belgique, Séance du 27 Octobre 1906.
La Semaine médicale, 1906, N° 44.

tically contradicted from various sides. (Conn'), REMLINGER*), Basser’),
Kiss et Losstrern‘), Brrrzkn®)).

Some of the above mentioned considered the normal anthracosis
in test-animals as a source of mistakes, which VansTeENBercup and
Griskz had not taken into account whereas others described the
aspiration also as a source, which might give rise to wrong con-
clusions. :

Meanwhile VANSTEENBERGHE and GRiskz, supported by CaLmetrr °)
maintained their opinion. They explain the negative results of their
opponents in the following manner: some allowed too much time to
pass between the introducing of carbon particles into the stomach and
the killing of the test-animals, because after 48 hours the pigment
would have almost completely disappeared from the lungs; others
used rabbits or too young cavies as test-animals, in which the fine
particles are almost wholly retained by the mesenteric glands.

With a view to this last remark I wish to publish the following
experiments, because I have taken into account the age of test-
animals as well as the time which passed between the introduction
of the fine particles and the killing of the animals.

To me it also appeared that the physiological anthracosis is a
factor which must be considered, for among all my test-animals,
cavies as well as rabbits, black pigment was found in the lungs.

Among some animals this spontaneous anthracosis was rather
decided, with others very minute. As a rule there was much less
pigment in the lungs of my rabbits than in those of the cavies.

The physiological anthracosis impedes as a matter of course the
experimenting with black substances. Besides carmine, vermillion
and ultramarine, I have also used Indian ink and soot, because after
microscopic investigation it appeared that the first mentioned matter,
even after being intensively rubbed in a mortar, was not as fine as
the particles of carbon of the last mentioned.

In order to control the experiments of VANSTEENBERGHE and GRISEZ

1) Berliner Klin. Wochenschr. 1906, N° 44 und 45.

*) La Semaine médicale, 1906, N° 45.

5) La Semaine médicale, 1906, N? 47.

4) Bulletin médical du 21 Novembre 1906.

La Semaine médicale, 1906, N° 48.

5) Virchow’s Archiv, Bd. 187, Heft 1.

6) Compt. rend. des séances de l’Académie de Sciences, T. 143. p. 866.
Compt. rend. de Ja Soc. de Biol. T. 61, p. 548.

La Semaine médicale, 1906, N’. 50.

(676) |

the test-animals were killed already 5—48 hours after administering
the forementioned substances.

Some cavies (experiment n°. 1—5) had eaten bread, mixed with
soot, Indian ink or carmine. After the dissection of the animals, the
lungs showed only the ordinary physiological anthracosis, but car-
mine was to be seen neither in the lungs nor in the bronchial glands.

One of these animals (experiment n°. 4) had evidently aspirated soot,
for in many bronchi and corresponding alveolars, foodparticles and
soot were distinctly seen in large quantities.

Also after introducing various matters with the catheter into the
stomach of rabbits (experiment n°. 6—10), aspiration was observed once
(experiment n°. 10), whereas among other animals only the normal
pigmentation was present.

In order to prevent aspiration with certainty, tracheotomy was
performed with three rabbits and after that a suspension of carmine
was brought into the stomach with the catheter (experiment n°. 11
-—13); for the same purpose among some cavies I injected coloured
particles into the distal part of the cesophagus which was cut through
and then bound up (experiment n°. 14—18). After dissecting no traces
of coloured particles were to be found neither in the lungs nor in the
bronchial glands.

Further with different cavies the fine particles were directly brought
into the intestines after laparotomy (experiment n°. 19—35). Neither
was then any of the coloured matter to be found in the lung-tissue
nor in the bronchial glands, whereas everywhere else nothing was
to be seen except normal anthracosis in varying intensity.

Among some experiments I noticed that coloured particles which
were injected directly into the intestines, were later on to be found
also in the stomach, in the oesophagus and in the pharynx, sometimes
in large. quantities (experiment n°. 21, 22, 29 and 30). In the phlegm
of the trachea the coloured particles could be distinctly seen some-
times with the use of the microscope (experiment n°. 21 and 29),
whilst once (experiment n°. 29) the easily recognisable ultramarin-
grains were to be seen even in the phlegm of the chief bronchi. It
is quite probable that the animals in agony had aspirated these sub-
stances from the pharynx, for, according to NENNIGER*) e. g. bacteria
too are often aspirated from the pharynx in agony.

The question is now, how came the matter from the pharynx
into the intestines. Was it by a motion of the fine particles in a

1) Zeitschr. f. Hygiene u. Infectionskrankheiten, Bd. 38.

( 677 )

proximal direction, as e. g. GrirzNner') describes this for fine particles
in the intestines and as Kast’) has also shown for the oesophagus,
or, had the animals eaten their own faeces > *)

In order to solve this question, -four cavies were carefully wrapped
in a bandage, after ultramarine had been brought into the intestines
so that eating the faeces was quite impossible (experiment n°. 32—35).
It now appeared that the ultramarine had come some way proximal
from the place of injection, but in the oesophagus, in the pharynx and
in the chief bronchi no ultramarine was discernible.

From this I suppose that the ultramarine had simply come into
the pharynx owing to the eating of faeces and not through a proximal
motion of the fine particles ‘*).

From my experiments I conclude that the pulmonary anthracosis
does not originate through taking up fine particles from the intestines.
It may be acceptable a priori, that fine particles can be taken up
in the intestinal mucous membrane and can get into the lungs along
ductus thoracicus and right heart, but this phenomenon is with
regard to the pulmonary anthracosis of not so much importance, as
VANSTEENBERGHE and Grisez have supposed. Evidently these investi-
eators have given sufficient attention neither to the physiological
anthracosis of the test-animals, nor to the aspiration of the coloured
particles which cannot be quite prevented, not even, as is mentioned
above, by direct injecting the matters into the intestines.

If the physiological antbracosis originated by taking up carbon
particles from the intestines, not only the mesenterial glands but
also the marrow and the milt had to contain much carbon pigment,
because firstly it cannot be understood how carbon parts should
pass the mesenterial glands without leaving distinct traces of their
passing behind them and on the other hand there is no possible
reason why the carbon particles to a great extent should not pass
through the capillaries of the lungs and deposit in the marrow and
the milt.

*) Archiv. f. d. Ges. Physiol. (Pfliiger). Bd. 71.
2) Berliner Klin. Wochenschr. 1906, N° 28.

5) When starving cavies and rabbits usually eat their own faeces, it also often
occurs when they have sufficient food.

Swirskt: Archiv f. exper. Path. und Pharm. 1902, Bd. 48.

4) UrFENHEIMER, after injecting a suspension of prodigiosusbacilli into the rectum
of rabbits, ncticed a motion of the bacilli in a proximal direction; they ascended

up to the pharynx and from thence they were sometimes aspirated into the lungs.
Deutsche Med. Wochenschr. 1906, N°. 46,

( 678 )
Description of the Experiments.

1. Cavy 650 grams.

First 24 hours without food, then for 24 hours exclusively dough
and soot, then killed.

Results: Macroscop. intestines much soot, lungs grey with small
black spots, especially in the upper lobes, bronchial glands distinctly
pigmented.

Microscop. In the interstitial spaces of the lung-tissue are many
cells with black pigment especially under the pleura. A very small
quantity of it is also found in the alveolars and in the bronchi. The
bronchial glands contain a great many cells with black pigment.

2. Cavy 200 grams.

For 48 hours exclusively dough and soot, then killed.

Results: Macroscop. intestines much soot, lungs and_ bronchial
glands pale; microscop. lungs and bronchial glands few cells with
black pigment.

3. Cavy 760 grams.

First 24 hours without food, then 5 cem. of Indian ink in dough,
killed after 24 hours.

Results: as in experiment 1.

4. Cavy 350 grams.

First 24 hours without food, then for 48 hours exclusively dough
and soot, then killed.

Results: Macroscop. intestines much soot, lungs many black spots
and points, bronchial glands pale; microscop. there are foodparticles
mixed with soot in many bronchi and alveolars. For a part the soot
has already been enclosed in cells, many cells have already penetrated
into the interstitial spaces. No pigment is to be seen in the bron-
chial glands (so in this experiment the coal was aspirated during
life; not in agony).

5. Cavy 400 grams.

First 24 hours without food, then 0,5 grams of carmine in dough;
killed after 48 hours.

Results: Except in the intestines no carmine can be found.

6. Rabbit 1.75 K.G.

For three days 100 mer. of soot is brought into the stomach by
means of a catheter; killed after 24 hours.

Results: Macroscop. lungs and bronchial glands pale; microscop.
few cells with black pigment are to be seen in the interstitial spaces
of the pulmonary tissue.

( 679 )

7, Rabbit 2 K.G.

For three days totally 2,9 grams of soot is brought into the stomach
with the catheter; killed after 24 hours.

Results: as in experiment 6.

8. Rabbit 2 K.G.

A suspension of 2 grams of carmine in water is brought into the
stomach with a catheter; killed after 48 hours.

Results: Except in the intestines, carmine is not to be found.

9. Rabbit 2.75 K.G.

50 grams of charcoalpowder, suspended in water, is brought into
the stomach with the catheter; killed after 24 hours.

Results: as in experiment 6.

10. Rabbit 3 K.G.

A suspension of 40 grams of charcoalpowder is brought into the
stomach with the eatheter; killed after 24 hours.

Results: Macroscop. lungs show black spots especially after dissect-
ing them; the bronchial glands are faintly pigmented; microscop.
fine carbon particles and also coarser carbon pieces can be seen
in many alveolars. Carbon can be shown neither in the larger
bronchi, nor in the trachea; the bronchial glands show some
pigment-cells.

The presence of the coarser carbon parts in the alveolars made
the diagnose “aspiration” very easy.

11. Rabbit 5 K.G.

After tracheotomy 9 grams of carmine is brought into the stomach
with the catheter. About 18 hours afterwards the animal chokes, as
phlegm has gathered in the canule.

12. Rabbit 4.25 K.G.

After tracheotomy 8 grams of carmine is brought into the stomach
with the catheter; the animal is killed atter 24 hours.

13. Rabbit 3.5 K.G.

After tracheotomy 8 grams of carmine is brought into the stomach
with the catheter.

Killed after 48 hours.

Results of the experiments 11, 12 and 13: Except in the intestines
I could find nowhere carmine in the body at the microscopical inves-
tigation; in the lungs and bronchial glands black pigment is present.

14. Cavy 400 grams.

The oesophagus was freeprepared and cut through. Through the
lower part 5 gram of vermillion was brought into the stomach.

Then the lower part of the oesophagus was bound up whereas the

46
Proceedings Reyal Acad. Amsterdam. Vol. IX.

( 680 )

upper part was fastened with its opening in the wound of the skin.

Killed after 5 hours.

15. Cavy 720 grams.

10 grams of vermillion were injected as in experiment 14.

Killed after 6 hours.

16. Cavy 720 grams.

7 cem. of a suspension of vermillion in gum arabic was injected
as in experiment 14.

Killed after 5 hours.

17. Cavy 400 grams.

2 grams of carmine were injected as in experiment 14.

Killed after 5 hours.

18. Cavy 860 grams.

4+ grams of carmine were injected as in experiment 14; killed after
6 hours. ;

19. Cavy 790 grams.

After laparotomy 4 grams of vermillion (in suspension) were
brought into a twist of the intestines; killed after 18 hours.

20. Cavy 620 grams.

6 cem. suspension of vermillion in gum arabic was _ brought
into the small intestin as in experiment 19; killed after 19 hours.

21. Cavy 750 grams.

10 ccm. suspension of vermilliion in gum arabic was brought
into the small intestin as in exp. 19; killed with chloroform after
18 hours.

22. Cavy 610 grams.

5 cem. suspension of vermillion in gum arabic was brought into
the colon as in exp. 19; killed after 18 hours, by abruptly decapi-
tating in order to prevent vomiting in agony.

tesults of the experiments 14—22: In the lung-tissue and in
the bronchial glands no vermillion resp. carmine was to be found.

At experiment 21 the vermillion could also be shown in the
stomach, in the oesophagus and in the pharynx while some grains
could be shown in the phlegm of the trachea. At experiment 22
vermillion could also be found in the stomach, oesophagus and
pharynx whereas in the trachea no vermillion was to be seen. (At
the experiment 19 and 20 stomach, pharynx etc. were not investigated).

23. Cavy 700 grams.

After laparotomy 5 ccm. of Indian ink is brought into the small
intestines.

Killed after-18 hours.

tesults: as in experiment 1.

( 681 )

24. Cavy 880 grams.

After laparotomy 5cem. of Indian ink is brought into the coecum.

Killed after 18 hours.

Results as in experiment 1 (the pigmentation is somewhat less
intensive).

25. Cavy 750 grams.

After laparotomy 5ceem. of Indian ink is brought into the small
intestines.

Killed after 18 hours.

Kesults: macroscop. Lungs and bronchial glands pale microscop.
few pigmentcells.

26. Cavy 700 grams.

After laparotomy 5cem. of Indian ink is brought into the coecum.

Killed after 18 hours.

Results as in experiment 1. (here the pigmentation is more intensive.)

27. Cavy 730 grams.

After laparotomy 5cem. of Indian ink is brought into the small
intestine.

Killed after 18 hours.

Results as in experiment 25; one of the mesenteric glands con-
tains carbon parts which are also to be seen microscopically.

28. Cavy 750 grams.

After laparotomy 5 cem. of Indian ink is brought into the colon
at 20 em. distance of the anus.

Killed after 18 hours.

Results as in experiment 1.

29. Cavy 650 grams.

After laparotomy 4 ccm. of a suspension of ultramarine in 0.9 °/,
NaCl is brought into the small intestines.

Killed with chloroform after 18 hours.

Results: The ultramarine is in the intestines, in the stomach, in
the oesophagus and in the pharynx, while some grains can be
traced in the phlegm of the trachea, and in that of the chief broncii.

The pulmonary tissue and the bronchial glands are free of ultra-
marine.

30. Cavy 850 grams.

After laparotomy 4 ccm. of a suspension of ultramarine in 0.9 °/,
NaCl is brought into the small intestine.

Killed after 17 hours with chloroform.

Results: as in experiment 29; in the phlegm of the trachea and
in that of the bronchi however no ultramarine was to be found.

( 682 )

31. Cavy 820 grams.

4 ccm. of ultramarine is administered as in experiment 30.

Killed after 16*/, hours.

Results: No ultramarine can be found, except in the intestines.

32. Cavy 360 grams.

+ ccm. of ultramarine is brought into the intestines as in expe-
riment 30.

After this the animal is carefully wrapped up so that it can get
no faeces into its mouth and cannot lick itself.

After 6 hours the animal is decapitated abruptly in order to pre-
vent vomiting in agony.

Results: the ultramarine is in the small and in the large intestines,
also somewhat proximal from the place of injection.

In the stomach, in the oesophagus, in the pharynx and in the
phlegm of the chief bronchi no ultramarine can be traced.

33. Cavy 750 grams.

Treated as in experiment 32.

Killed after 16 hours.

vesults: as im experiment 32; here some grains of ultramarine
are in the stomach.

34. Cavy 475 grams.

Treated as in experiment 32.

Killed after 12 hours.

Results: as in experiment 32.

35. Cavy 540 grams.

Treated as in experiment 32.

Killed after 6 hours.

results: as in experiment 32.

(March 28, 1907).

tee,
—— *
x
~ 4 =

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM.

PROCEEDINGS OF THE MEETING
of Saturday March 30, 1907.

DOG

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Zaterdag 30 Maart 1907, Dl. XV).

CONTENTS.

J. P. van per Stok: “The treatment of wind-observations”, p. 684.

F. M. Jarcer: “On the anisotropous liquid phases of the butyric ester of dihydrocholesterol,
and on the question as to the necessary presence of an ethylene double bond for the occurrence
of these phenomena”. (Communicated by Prof. A. P. N. Francurmonn), p. 701.

P. van RompurGuH and A. D. MatrENBRECHER: “On the action of bases, ammonia and amines
on s. trinitrophenyl-methylnitramine’’, p. 704.

W. H. Jviivs: “Wave-lengths of formerly observed emission and absorption bands in the
infra-red spectrnm”, p. 706.

C. H. Wixp: “A hypothesis relating to the origin of Rontgen-rays”, p. 714.

J. H. Meersure: “On the motion of a metal wire through a piece of ice”. (Communicated
by Prof. H. A. Lorentz). p. 718.

J. D. van DER Waats: “Contribution to the theory of binary mixtures”, II, p. 727.

J. D. van DER Waats: “The shape of the empiric isotherm for the condensation of a binary
mixture”, p. 750.

H. Kamertisen Onyes and C. Braak: “Isotherms of diatomic gases and their binary mixtures.
VI. Isotherms of hydrogen between — 104° C. and — 217° C.”, p. 754. (With 2 plates).

H. Kameriincu Onnes and C. Braak: “On the measurement of very low temperatures,
XIV. Reduction of the readings of the hydrogen thermometer of constant volume to the absolute
scale”, p. 775. (With one plate).

H. Kameriincu Onxes and W. H. Kresom : “Contributions to the knowledge of the ¥-surface
of vaN DER Waats. XV. The case that one component is a gas without cohesion with mole—
cules that have extension. Limited miscibility of two gases”, p. 786. (With 2 plates).

47
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 684 )

Meteorology. — “The treatment of wind-observations.” By
Dr. J. P. vAN DER STOK.

(Communicated in the meeting of February 23, 1907).

1. When working out wind-observations we directly meet with
the difficulty that a method holding generally, in which the charac-
teristics of a wind distribution come to the fore in condensed
form, does not exist. The discussion held for many a year concerning
the desirability or not of an application of Lampert’s formula, i. e.
of the calculation of the vectorial mean of velocity or force has not
led to a definite result and the consequence is that for regions where
trade- and monsoon winds prevail the calculation of this mean can
be applied, not for higher latitudes, so that here we have to judge
by extensive tables of frequencies of direction and mean velocities,
independent of direction.

When working out the wind-observations made at Batavia | did not
hesitate to make an extensive use of this formula; the same method has
been followed in the atlas for the East Indian Archipelago; but in order
to give at least a notion of the value of the velocities annulling each
other here I have added to the resulting movement (called by Hann
windpath) a so-called factor of stability. If namely the wind were
perfectly stable, the vectorial mean would be equal to the mean
independent of the direction and the stability would amount to
100 °/,, which percentage becomes smaller and smaller according to
the direction of the wind becoming more variable. So here attention
is drawn to the fact, that a part of the observations is eliminated,
but it is not indicated what character this vanishing part has which
becomes chief in our regions.

In the climatological atlas lately published of British India the
same method is followed; in the “Klima Tabeller for Norge’ Monn
gives but the above mentioned tables without calculation of the
vectorial mean, which is, indeed, of slight importance for this climate.

The same uncertainty is found in the graphical representation of
a wind distribution by so-called windroses; almost everyone who
has been occupied in arranging books of prints has projected wind-
roses of his own; some of those roses, as e.g. in the “Vierteljahrs-
karte fiir die Nordsee und Ostsee’’ published by the “Deutsche
Seewarte”’, show only the frequencies of direction without velocities ;
in others, as e.g. those shown in the above atlas of the Easi Indies,
each direction is taken into account with the velocity belonging to it
as weight, so that mean velocities are represented. All these roses

( 685 )

furnish discontinuous quantities and change their aspect according
to their boundaries being taken differently.

In Bucuan’s general meteorological atlas no roses are projected,
only arrows indicating the most frequent direction without heeding
the force, and in the ‘“Segelhandbuch fiir den Atlantischen Ozean”
published by the “Deutsche Seewarte” for higher latitudes where the
wind is variable the use of wind-observations is entirely done away
with and arrows have been drawn in accordance with the course
of the mean isobars on account of the law of Buys BaLtot, where
a constant angle of 68° between gradient and direction of wind has
been assumed.

This short survey of the manner in which in the most recent
standard works this problem has been treated may show that indeed
there is as yet no question about a satisfactory solution, as has already
been observed.

The aim of this communication is to hit upon a general method
of operation and representation of an arbitrary wind distribution
in which to the variable part also justice is done, whilst the gra-
phical representation has a continuous course and shows at a glance
the five characteristic quantities which mark each wind distribution
and which may be, therefore, called the wind-constants.

The method proposed here is founded on the basis of the caleulus
of probability, but it is important to notice that it is not at all bound
to it; at the bottom it is the same which is generally applied in the
treatment of directed quantities: distribution of masses and forces in
mechanics, the theory of elasticity, the law of radiation and the
theory of errors in a plane.

2. A wind-observation can be represented by a point in a plane
such that the distance to an assumed origin is a measure for the
velocity of the wind (or force) and that the angle made by the
radius vector with the Y (North) axis counted from N. to E. indicates
the direction. If in this way all observations, V in number, are drawn
and if we think that to each point an equal mass is connected, then
in general the centre of gravity will not coincide with the origin
selected ; its situation may be determined by the quantities R, and u.
The distribution of the masses around the centre of gravity, is then
characterized by the lengths M and JA/’ of the two principal axes of
inertia and the angle 3 enclosed by the axes M and FY.

As is known the five constants by which such a system is charac-
terized can be calculated according to this purely mechanic notion
by determining the moments J/, and J/, with respect to the axes

47*

( 686 )

and the moments of inertia J/,? and M/,? and M,,, which furnish
the five equations necessary for the calculation of the unknown
quantities.

We arrive at quite the same equations when the distribution of
the winds according to direction and velocity is regarded as a system
of accidental, directed quantities in a plane. The centre of gravity
then represents according to size and direction the constant part of
the wind which is supposed to be connected with all observations
and of which, therefore, the probability is equal to unity; the axes of
inertia become principal axes of probability and the lengths M/ and
M’ are replaced by the reciprocal lengths h and h’, so that

1 1
pe Ce es
2M? 2M"

The sum of the masses is put equal to unity and for the proba-
bility that an observation lies between the limits R and R+ dk
of velocity and 6 and 6+ d6 as far as direction is concerned the
expression holds

| ie
— ¢ \*) Rp aidd. =) ee
ww

where:
F(R,A)=h? [RB cos (O@—B)—R, cos(a—B)]? + A? _Rsin(6 —B)—R, sin(a—8B)]?.(8)

In the language of the theory of errors \(&,,a@) would be the
so-called constant error, Jf and M' the greatest and smallest projections
of the mean errors. As observations of wind agree still less than
other meteorological quantities with the opinion held in the theory
of errors, where the constant part is regarded as the end of the
operation and the variable quantities as deviations, it is desirable
when applying the calculus of probability to quantities of this kind
to be entirely free of the terminology used in the theory of errors,
but which would be here without meaning and which would give
rise to misunderstanding.

The treatment must also differ somewhat from that of erroneous
quantities, it being if not impossible at least impractical to correct
all the observations for the constant part.

3. As examples of treatment two series of observations have been
selected from the treated material.

a. Observations of wind performed at Bergen (Norway) during
20 years, 1885—1904, three times daily at 8 A.M, 2 P.M. and 8 P.M.
The velocity (or force) of the wind is expressed in the so-called

( 687 )

half scale of Beavrorr \1—6) (Jahrbuch des Norwegischen Meteorol.
Instituts, Christiania).

6. Observations of wind performed at Falmouth (Channel) during
17 years, 1874—1886 and 1900—-1903; the observations made in the
years 1887—1899 are published in such a way as to be useless for
this investigation.

Observations have been used, made daily six times: at noon, + P.M.,
8 P.M., midnight, 4 A.M. and 8 A.M.; the velocity of wind is
expressed in English (statute) miles an hour (Hourly readings obtained
from the selfrecording instruments etc. London).

With respect to the force of the wind estimated at Bergen is to
be noticed that in this communication these scale-values are regarded
not as forces but as velocities, although in reality they are neither
one nor the other. According to a recent extensive investigation *)
the ratio of the Beaufort values to corresponding velocities can be
indicated by the following numbers

Beaufort velocity ratio Beavurort velocity ratio
meters a second meters a second

0 1.34 — 6 10.95 1.83
1 2.24 2.24 7 13.41 1.92
2 3.58 1.79 8: 16.09 2.01
3 4.92 1.64 9 19.67 2.19
4 6.71 1.68 10 23.69 2.34
5 8.72 1.74

As the various velocities do not appear in an equal number the
total mean out of these ratios would not give a fit factor of reduction
for mean Beravurort-values; so a certain weight must be assigned
to each separate ratio. For this the frequencies have been used of
the 36000 wind-velocities observed at Falmouth calculated for a
whole year; in this way has been found for the reduction-factor 1.83;
the English measure, miles an hour, can be reduced to m.a.s. and
Beaurort scale-values by means of multiplication respectively by

0.447 and 0.244.

1) The Beavrort scale of windforce.

Report of the Director of the Meteor. Office upon an Inquiry into the Relation
between the Estimates of Windforce according to Admiral Beaurort’s Seale and
the velocities recorded by Anemometers. London, 1906.

( 688 )

4. The calculation of the five charateristic constants of a wind
distribution amounts in one respect to the integration of (2), in
another respect to the means applied in this integration to a given
set of observations.

The integration of (2) takes place by the introduction of rectangular
coordinates :

2 hsm 0” 3. eee

where the element RdRdé is replaced by the element dzdy, whilst
the limits which were o and O for &, 2% and O for 6, now become
co and — o.

Then the expression (2) under the sign of the integral is multiplied

successively by
2, y, 2, y? and zy.
If we then put:
R, cos (a—B) = a, zx=ex'snB+ y' cos B,
R, sin (a—8) = 4, y =x cseB — y' sinB,
the variables 2 and y' can be separated and the integration can be

done; in this way we find for the determination of the five quantities
to be obtained the five equations :

M, =acosB—bsmB, M, =asinB+ beosB

2 aa
M,? ——— == + a’ cos? 8 + 6? sin? 8 — ab sin 2B

“ au — + a® sin® B 4+ 6? cos? 8 + absin2B +8)

1 ul
2M,, = = -- sa) sin 2B + (a? — b*) sin 28 + abcos 28

M,’ —

out of which, on account of (1)
M, =R,csa, M,=—R, sma \
M,? + My? — [\Wa)? + (My?) = M? + M” |
M,? — My? — [(Mz)? — (M,)"] = (M2 — ered
2Mry — 2M,M, = (M? — M") sin 28

(4)

( 689 )

TABLE I. Frequencies of the wind.
Bergen. June.

In half BeAurort scale-values.

5. In order to apply the formulae (4) to a given set of obser-
vations we must compose for each period, e.g. each month, in the
first place a table of frequency of direction and velocity, which can
be easily done. In Table I such a composition has been given as
an example.

Further out of this table have been calculated the products of
these frequencies / with the scale-values FR, the latter counted
double, so that the products have been expressed in the ordinary
Bravurort scale; finally these products have been once again multiplied
by the corresponding scale-values (/A*); in this simple way we
find the sums.

( 690 )

TABLE IT.

SSE 162 524 NNW 4336 5296
Som 4560 16888

The sums fR, multiplied respectively by cos@ and sin@ and
divided by 1800, immediately furnish the quantities J/, and M,;
the sums fR? must be multiplied successively by cos? 6, sin?@ and
sin @ cos 6.

It is easier to multiply the latter sums by cos 26 and sin 26; if
the total mean is S, we find:

M,? — fR’ cos? 6 =} 8+ 3 FR? cos 26
M,? = sR sin? 0 = 4S — 4 fR? cos 20
2Myy = fR sin 26.

So the whole operation greatly resembles the calculation of FourtEr
terms; indeed, also by the way of operation indicated here an
analysis of the movement of the air is obtained.

In the Tables III and IV we find the values of the wind-constants
calculated in this way; besides the five characteristic quantities we
find still given as quantities practically serviceable for various ends:

ad Me M*
M
M’ represent the half principal axes,

(R,’ and «’) the resultants of the squares of the velocities giving

an image of the mean flux of energy,

, the excentricity of the ellipse of which J/ and

V the mean velocity independent of the direction,

V* the mean square of the velocity independent of the direction,
i.€. a measure for the total energy; this quantity is according to (4)
analogous to the square of the mean error, not corrected for the
constant part, in the theory of errors,

N the number of used observations.

( 691 )

TABLE IIl@ Constants of the wind.
Bergen 1885—1904.

In BEAUFORT scale-values.

January 1.84 174° 0.873
February {5 172 0.858
March 1.16 174 0.872
April 0.40 169 0.859
May 0.68 169 0.879
June 4.07 171 0.891
July 0.93 168 0.885
August 0.73 168 0.904
September 0 97 174 0.876
October 1.10 174 0.857
November Te5t 474 0.880
December 1.78 174 O 866
Year 0.85 171 0.875

TABLE [IIb Constants of the wind.
Bergen 1885—1904.

In BEAUFORT scale-values.

January

February 1695
March 1860
April 1800
May 41860
June 1800
July 1860
August 1860
September 1800
October 5.413 185 2.16 7 AD419 1860

November ov ly 182 2.91 14.09 1800
December 9.30 483 3.06 | 15.50 1860
Year 4.42, 200 2.80 12.45 21915

( 692 )

TABLE [Va. Constants of the wind.

Falmouth. 1874—1886, 1900—1903.
In Eng. miles an hour.

January 15.20 13:93" 3h 1 ae
February 14.08 13.25 | 164
March 15.02 13.26 67 |
April 13.70 | 19.98 | 7 |
May 1202 | 41.52 40 |
June 11.39") 290-07" ioe
July 10.39 8.92 | 155
August 10.48 | 9.76 | 82
September 11.05 | 10.67 | 164
October 13.51 | = 13.08 | 81
November AS ik S103 o3
December 13.69 | 12.98 29
Year 4.44 | 239 12.60 | 12.43 | 96

TABLE IVb.
Falmouth.

Constants of the wind.

January

February 159.9 203 2675
March 88.8 241 2930
April 23.0 178 2879
May 04.4 | 241 3110
June 92.4 | 259 3015
July 122.2 252 3060
August 122.0 242 3154
September 83.7 224 3047
October 78.5 223 3154
November 114.2 237 3053
December 148.6 233 2888
Year 98 .2 229 35816

( 693 )

A closer discussion of the results arrived at in this way may for
shortness’ sake be left out; however, the observation is not super-
fluous that the two examples represent two types, a reason why
they were chosen. At Bergen the ellipse of the variable winds is
very constant of shape and the excentricity is very great; at Falmouth
the difference between JW and M’ is always very slight and the
differences found there are evidently to be regarded rather as accidental
arithmetical results than as facts, the angle § being subject to great
and irregular oscillations; evidently the ellipse approaches a circle, so
that in form (2) we may put h=/’. This leading to a considerable
simplification of the formula, these observations at Falmouth are
eminently fit for comparison of the results of calculation and obser-
vation, whilst also the fact that here real velocities have been
observed with well-verified instruments, makes this series very
favourable.

6. The expression (2) shows: the probability that an observation
lies between the limits R and R+ dR, 6 and6é+dé6; the same
expression without the element RdRdé indicates: the specific proba-
bility of a wind (&,6) i.e. the probability with respect to the
unity of surface when one imagines this surface to be small. If we
put for simplification :

h® +h? = 2p, h® —h* = 29, B* (p — qeos 2 (a — 8) =u
(p—qeos2(9@—P)) =v, 8? = R,? (p* + g? — 2pq cos 2 (a —8))

: Sf
s cos (6 —g)=Aa ees Fn (ee
p cos a — g cos (a — 2 B)

then (2) takes the form :
tes 0

pee ee ade. eB)

If here we put:

a a er (5)
then it follows out of the above formulated definition that the specific
probability of all observations lying on the circumference of the
excentric ellipse (6) is the same and equal to:

p?—q?
yee Le.
rs 4
The probability that the velocity of the wind does not surpass the

value #, expressed by (6) in function of 6, in other words the number
of observations which are to lie within the area of the ellipse, is

( 694 )

found by integrating (5), first with respect ‘to R between the limits
R, and O, then with respect to 6 between 2 and 0.

For the simple case R, = 0, so also w=O and 2—0O, the
first integration gives immediately

VP Sta lee

x j 2yv
and as
Qr
V p?—q (2- ;
2x Viste aac
7)

the probability to be found becomes simply :
Lee SOUL te ne

and the number of observations lying inside the circumference of
the ellipse (6) :

N(1—e- ),

This amount remaining the same whether we regard the ellipse
(6) from the excentric origin or from the centre, i.e. for ik. =,
if with the integration the limits are changed correspondingly, the
expression (7) must also be accurate when R&, is not equal to zero
and must thus hold in general.

Indeed, an other simplification, namely g = 0 (which is applicable
to the results for Falmouth) leads to a set of definite integrals, which
can be evaluated and which confirm this conclusion.

Amongst the series of ellipses represented by (6) two are
remarkable; if we assign to c the value 0.5, then on account of
(1) the half axes of the ellipse become equal to the greatest and
smallest projections J/ and M’ of the mean velocities, so that the
ellipse (6) then represents what we might call the specific or typical
windellipse, thus a kind of windrose, in which the characteristic
qualities of the wind-distribution under consideration inmediately
become conspicuous.

The radius vector £#,, drawn to an arbitrary point in the cireum-
ference is given in the direction determined by that choice by the
equation :

2R?nv—4RnA+2u—1=—0.
The probability that a velocity does not surpass this value is:

1 — e— 2 = 0.89847.

( 695 )

So among a_ thousand observations there will be 393 lying
inside this typical ellipse whilst the specific probability of each of
the velocities Ry» is:

0.6065

In the given diagram such a typical
windellipse is represented for Bergen in
the month of June by the dotted line.
the vector OC represents here the constant
part (R,, a), the half axes are equal to M
and M’, and the angle NOM=8; one
millimeter corresponds to */,, BEAUFORT scale-
value or to °*/,, x 1.83 = 0.275 meter a
second.

If necessary this diagram might be am-
plified with two circles, one of a radius

Vie Mm",

representing the mean monthiy wind velocity corrected for the
constant part, the other described with radius

Vi? = M+ (iy + (,)’, |
which is according to (4) a measure for the mean total velocity,
corresponding to the square root of the quantity V? of the tables
III and IV.

An other remarkable ellipse which might be called the probable
windellipse is obtained by requiring half of the observations to lie
within its dominion; we have then to determine c in such a way that

te, te 056932,
so that the axes of this ellipse are
26 2 0.6326 = 1.177

times longer than those of the typical windellipse ; the number 0.8326
is a quantity known in the theory of errors in the plane.

7. The frequency of the windvelocities, setting aside the direction,
cannot be represented in a finite form; we can arrive at a form
serviceable for comparison with the observation by writing (5) thus:

PT poe PRE Rp) 2B

RdRd6,. . . (8)
ww

( 696 )

by developing the last exponential factor and then by expressing the
powers and products of cosines in cosines of multiples.
It is clear that when integrating (8) with respect to 6 from 22 to
0 only those terms are left which are independent of @ and which
appear with the common factor 2.
The expression to be found for the probability that a velocity lies
between the limits R and R-+ dR then becomes :
2V p—q .e-*.¢ PP (lta +a hk...) Rdk, > ey
where :
Le 8,
a, = q,/27 + 9487/2! cos 2(¢—B) + s*/(2!)’,
a, = 978/27 4 ge*/B! cos 2 (p—B) + 8°/(3!)*.
For Falmouth, where as was noticed above qg can be put equal
to nought these coefficients become simply :

g2n ni 2
an — aa
[(z)

sph, , p=phs. PSPs. P]«, APR; oe

and farther

In practice it will frequently be only necessary to calculate a few
of these coefficients; if we put:
q/P — &,
the integration of (9) between the limits m and O leads to the
expression :

yi Care ~ \
: fm Mer 417 Seat 5 fc
(ee) ee eee
p . x
pmie—pm fa, . ala, ape (11)
7 ae Aa? oe
p?mie—pm* Ila
as Se eee ee
2! p
As for m= o this expression must become equal to unity, we have:
ba a, ua 2/a, 7 er
5 eae ee

or, for the case ¢ =O, (11) becomes :

( 697 )

1—e— pm?
pmie—em |
l—e-+
ae? Foo. age
p?mse—pm* a, |
ce a AT 1 — e—*# — —¢-* | ete.
2! P

from which is immediately evident that in many cases the three first
terms are sufficient, so that then the calculation of the coefficients
can be entirely avoided, or at most only a, must be taken into
account; for generally mw is small, so that already

Pee

will be a small quantity. If g is not small the calculation becomes
rather tedious.

8. To find expressions for the quantities VY and V*, the mean
velocity and the mean square of the velocity independent of the sign,
we have to multiply (9) successively by R and FR? and to integrate
between the limits «2 and O which, with the well known fundamental
equation, leads to the expressions :

a, 24a, 24.60
=A(3 4 ee Spe

= V5 by oc 11 Bay | 8.5.7.0, )
te Pp ears Gey | dS (13)

=a fa 4 AGa, | 46.805 )
ve oa war Peay (apy

= V1l—? e.

9. For the calculation of the frequency of the directions independent
of the velocity we have first to integrate (5) with respect to & between
the limits «© and O and then with respect to 6 between the desired
limits 6; the mean velocity as function of the direction is found by
the application of the same operation to (5) after multiplication by
FR. It is then easy to give to a frequency-formula found in this way
the form of a Fourier series. For brevity we treat here only the
case that g =O and the angle- limits are a to 0.

By putting

( 698 )

we get (5) reduced to the form:

a
2

et daly —ve A
Lat das ele “(e +2) a0 . eee
es 4 Yv

‘
—A}y

If g=0, so that the formulae (10) hold good, we then find for
the desired frequencies in the two easterly quadrants
RV p sina

Be a 2 ee (15
Wn t SiR Seine )

From this formula it is evident in what way and in what degree
the asymmetry of the distribution is dependent of &,, @ and p.

10. The application of the given criteria has been made for
Falmouth and the four seasons: _

Winter: December, January, February, number 8384,

p = 0.00258, q = 0.00004
i= pee a@ = 222°8'

Spring: March, April, May, number 8949,

p = 0.00298, q = 0.00028

R, = 2.21, a = 250°25'
Summer; June, July, August, number 9229,

p = 0.00485, gq = 0.00029

R, = 5.60, a —= 251922!

Autumn: September, October, November, number 9254,
= 0.00313, q = 0.00004
R, = 3.80, a = 239°16'

For each series the number of observations is reduced to 10.000
and everywhere we have put g=0O, the calculated values are
accordingly accurate as far as the fourth decimal.

In Table V we have compared the observed frequencies of wind-
velocities independent of direction with those caleulated according
to formula (12), from which it is evident that the differences havea
clearly systematic course. Just as is the case with all series of errors
the number of the observed small velocities is larger than would agree
with the normal distribution. The differences together amount in summer
to about 10°/,, in winter to 15°/,.

( 699 )

In the caleulation of the frequencies of the directions independent of
the velocity, the observations regarded as calms — and to these are
reckoned in the English records all velocities less than 4 miles an
hcur — have been distributed proportionally to the frequencies of
direction; furtheron the frequencies North and South are assigned
for one half to the eastern and western quadrants.

As is evident from the following table also in this comparison
systematic differences appear; in all seasons the observed frequencies
in the western quadrant are greater than the calculated ones, so that
an increase of the constant part A, to which this uneven distribution
can be attributed, would improve the correspondence.

TABLE VI.
Frequencies of winddirections at Falmouth
for 10.900 observations.

| |

, | Observed. |Calculated | Difference

E. quadr. || 3709 | 40c6 | — 297
Winter
$ 6291 | 5994 + 297
E. quadr. 4037 | 4354 ~ 4
Spring |
- 5063 | 5649 4+ 214
. E. quadr. || 2619 | 3009 | — 300
Summer
-. 7381 6991 + 390
E. quadr 34353 3980 — 527
Autumn
F 6547 6020 | + 527

45
Proceedings Royal Acad. Amsterdam. Vol. LX.

( 700 )

TABLE V. Frequencies of windvelocities at Falmouth.
For 10.000 observations.

S Pee ENG SUMMER

AUTUMN

WoDN. TER

Observ. |Calculat.| Difference] Observ. | Calculat.| Difference} Observ. |Calculat.| Difference} Observ. |Calculat.| Difference

Miles an hour
(As:
Ab— 95

95—145
145—195
495—245
245-9015
995—345
345—395
395— 4.45
h45—495

495—5A5

700 | 477-| +983 | 756 | 5e9 | +497 | 94 | s10 | +495 | 936 | 588 | + 348
1871 | 1482 | +389 | 2073 | 1759 | + 314 | 2610 | 9336 | + 974 | 2196 | 1779 | +377
1853 | 2026 | —173 | 2120 | 9979 | — 450 | 9538 | 9798 | — 190 | 2144 | 2804 | — 160
1701 | 2030 | —329 | 1875 | 2120 | — 245 | 1868 | aisa | — 965 | 1918 | 2110 | — 292
1466 | 1650 | — 484 | 1355 | 4558 | — 203 | 4164 | 1995 | —- 61 | 1984 1564 | — 283

967 | 119 | —158 | 917 | 939 | — 99 | sas | 5% | + 40 | 702 | 920 | — 128

B80 jc 668 ef ek a os | as. | 4 20 ond ame oe AB Paso |, ded) 0G

369 | 331 | + 38 | 250 | 901 | + 4 8) a | + 3 | a8 | 179 | + 79

199 | 144 | + 55 | 144 72 | + 42 20 9° | + 4 | 16 6 | ++ 50

( 701 )

Chemistry. — “On the anisotropous liquid phases of the butyric
ester of dihydrocholesterol, and on the question as to the
necessary presence of an ethylene double bond for the occur-
rence of these phenomena’. By Dr. F. M. Jarerr. (Communi-
cated by Prof. A. P. N. Francuront).

(Communicated in the meeting of February 23, 1907).

§ 1. In order to explain the behaviour of substances which are
wont to exhibit double-refracting liquid phases, some investigators
have started the hypothesis that, in this kind of organic substances,
it might be a question of systems formed of two components, and
of equilibrium phenomena between tautomeric and isomeric modifica-
tions, which would be converted into each other with finite velocity,

Although it is difficult to understand how such a supposition,
which is easy to propound, but very difficult to prove, could explain
the numerous well ascertained facts of the regular optical anisotropism
of these phases, it might explain, however, at least to some extent,
the peculiar irreversible transitions of phases, which I found more
particularly with the esters of cholesterol and a-phytosterol, and also
the hindrance phenomena noticed on that occasion’).

Such a supposition, however, is perhaps of some importance for
the interpretation of the brilliant colour phenomena which accompany
the phase-transitions in the cholesterol esters. For a mixture, or an
emulsion of substances, whose indices of refraction differ very little,
but whose dispersions differ much, might, like CHRIsTHIANSEN’s mono-
chromes, cause a similar display of colours.

§ 2. There is more than one cause for tautomerism (or isomerism)
in the case of these cholesterol esters, for all the esters, as well as
cholesterol itself, possess an asymmetric carbon atom, and in solution
they all polarise to the left.

Consequently, a racemisation during the esterification is by no
means excluded, and we might, therefore, have a mixture of the
optical antipodes. Cholesterol, moreover, possesses an ethylene double
bond, so that we may also expect an isomerism in the sense of
fumaric and maleic acids.

§ 3. As many other compounds (in fact most organic substances
which are wont to exhibit these phenomena of doubly refracting

1) F. M. Jagger, These Proc. 1906 p. 472 and 483 (29 December).
48*

( 702 )

liquid phases) possess such ethylene double bonds, one might indeed
imagine that the presence thereof in the molecule is of great
importance for the occurrence of the said phenomena, if not the
conditio sine qua non, as the structure of the azoxy-compounds is
not yet firmly established and because it may be assumed that they
contain, perhaps, similar double bonds between NV and O.

Moreover, the cholesterol esters all contain three liquid phases, so
that this peculiar complication might perhaps also be connected with
the possibility of very intricated isomerism-phenomena of those
substances.

§ 4. In order to answer these questions, I asked Prof. Dr. C. NruBErG
of Berlin to furnish me with a specimen of his synthetic Dihydro-
cholesterol, to which request this savant most willingly acceded.

I wish to thank Prof. NeuBErG once more for his kindness.

In this Dihydrocholesterol the ethylene double bond has disappeared
owing to the addition of two atoms of hydrogen, and the malenoid
and fumaroid isomerism is therefore, a priori excluded.

§ 5. I have prepared from this alcohol the acetic and the normal
butyric esters, by means of the pure acid-anhydrides, and have
examined the same as to their phase transitions. The acetic ester
will be described elsewhere later on; here the butyric ester only
will be discussed.

As a highly important result | may mention that the colour pheno-
mena on melting and the occurrence of three liquid modifications in
the normal butyrate remain unaltered as before, but that the irre-
versibility of the phase-transitions is shown in a manner just the
reverse as in the case of most of the cholesterol esters, e.g. the laurate.

Whereas of the two doubly-refracting liquid phases of the last
named substance, one is always passed over on cooling, whilst both
are found on melting the solid substance, this is just the reverse
in the ease of the dihydrocholesterol-n-butyrate.

6. The solid phase S consists of an aggregate of very thin,
colourless, and clear transparent laminae in which the plane of polari-
sation makes an obtuse angle with the sides of demarcation and
exhibit in convergent polarised light a hyperbole with very strong
colour dispersion @ > v.

On heating, this phase S passes into a doubly-refracting liquid 5,
consisting of very small, feebly doubly-refracting individuals, which
in turn passes at a higher temperature into the isotropous fusion L.

( 703 )

Of colour phenomena during one of these transitions, absolutely
nothing is noticed.

If, however, we start from the phase / and allow the same to
cool, we first notice the doubly-refracting phase £, which on further
cooling, amid violent sudden currents of the mass, passes into a
much more strongly doubly-refracting liquid A, which on continued
cooling crystallises suddenly, also amid very violent currents, to an
aggregate of flat needles, glittering in vivid interference colours. These
in turn, rapidly assume a spherolite structure so that the solid phase
S itself appears to be also dimorphous und monotropous, as the flat
needles are not reobtained on warming the spherolitic mass. The
transformation of A into these needles, during cooling, is accompanied
with the most vivid display of colours. Under the microscope these
may be recognised by the dark-green colour of the background of
the field of vision; with the naked eye, however, with incident
light, that colour-display commences with a brilliant violet gradually
turning into blue and finally into a radiating green when the
mass crystallises. [| have never noticed red or yellow colours with
incident light. These phenomena return in the same order when the
experiment is repeated.

That the phase A really exhibits the behaviour of a stable phase

Fig. 1.
Schematic p-(-diagram for Dihydro-cholesterol-n-butyrate.

( 704 )

is also shown by the fact that, the colour having become blue
or green on cooling, turns again violet on warming, sé long as
the solid phase S has not yet been attained. The phase is, therefore,
realisable at a change of temperature in fvo directions.

§ 7. As I had but very little of the substance at my disposal,
the thermometric determinations could only be studied in capillary
tubes with the aid of a magnifying glass.

At 82.°1 the phase S melts to a doubly-refracting phase B which
becomes clear at 86.°4 and passes into Z. On cooling this isotropous
fusion, it first passes properly into 4 at 86°.4, but at 84° into the
more strongly doubly-refracting phase A, which may be undercooled
many tens of degrees, and with retention of its violet colour, before
passing into the solid phase S.

Want of material prevented my determining the true solidifying
point of S by inoculation; I estimate it at about 80°.

Thus the positive proof has been given that the remarkable colour
phenomena accompanying the melting the cholesterol esters cannot be
attributed to the presence of an ethylene double bound; also that
an eventual presence of fumaroid and maleinoid isomers cannot be
considered as the cause of the occurrence of the three liquids.

Zaandam, 15 Febr. 1907.

Chemistry. — “On the action of bases, ammonia and amines on
s. trinitrophenyl-methylnitramine.” By Prof. P. vax RomBurGH

and Dr. A. D. MAURENBRECHER.

(Communicated in the meeting of February 23, 1907).

s.-Trinitrophenyl-methylnitramine, as has been known for a long
time, is decomposed at the ordinary temperature by ammonia in
alcoholic solution, or on warming, by an aqueous solution of potas-
sium hydroxide, or carbonate, in the first case with formation of
picramide, in the second (with evolution of monomethylamine) of
picric acid. One of us who formerly studied the reaction with bases
concluded, from the occurrence of the amine and the formation of
nitric acid which was also observed, that the methyInitramine which
might be expected according to the equation :

( 705 )
CaN O aN-CH, -— KOH = CH, (NO,), OK + HN CH,
|
VO: NO,

might have become decomposed *).

From the reaction of methylamine on tetranitropheny|l-methyInitra-
mine and on trinitromethylamidomethyInitramidobenzene he after-
wards concluded’) that, probably, there had been formed methyl-
nitramine, meanwhile discovered by FRANcHIMONT and KLOBBIE *).

The amount of amine formed by the decomposition of trinitro-
phenyl-methylnitramine by alkalis is considerably smaller than might
be expected from theory; the possibility, therefore, exists that the
reaction proceeds indeed mainly in the above indicated sense.

We have, therefore, taken up the problem again in the hope that
by suitable modifications in the reaction, we might get at a process
for the preparation of methylnitramine which would have the advan-
tage of yielding this costly substance from a cheap, easily accessible
material. We were not disappointed in our expectations.

If trinitrophenyl-methyInitramine, which is the final product of the
nitration of dimethylaniline and melts at 127°, is boiled with a 10°/,
solution of potassium carbonate a brownish-red solution is obtained,
which on cooling gives an abundant deposit of potassium picrate.
If after filtration the liquid is acidified with sulphuric acid and again
filtered off from the picric acid precipitated and then agitated with
ether, the latter yields on evaporation crystals, which after purifica-
tion, melt at 38°, and are identical with methylnitramine, as was
proved by comparing the compound with a specimen kindly presented
to us by Prof. Francuimont. The yield, however, was very small.

If the finely powdered nitramine, m. p. 127°, is treated with
20 °/, methylalcoholic ammonia this becomes intensely red, the mass
geis warm and after a few hours the reaction is complete, and a
large amount of picramide has formed which is removed by filtration.
The alcoholic solution is distilled in vacuo, the residue treated with
dilute sulphuric acid and, after removal -of a yellowish byeproduct by
filtration, the liquid is agitated with ether. On evaporation of the
ether, crystals of methylnitramine were obtained. In this reaction
also, the yield was not large, amounting to only 15 °/, of the theoretical
- quantity. With ethyl-aleoholic ammonia a similar result was obtained,
whereas an experiment in which ammonia was passed into a solution

1) Rec. d. Trav. chim. d. Pays-Bas, II. (1883) p. 115.
2) Ib. VIII (1889) p. 281.
3) Ib. VII (1888) p. 354.

( 706 )

of the nitramine in benzene gave results which were still less
favourable. |

One of us had noticed previously that among the aromatic amines
which generally react on an alcoholic solution of the nitramine
quite as readily as on picry! chloride, p-toluidine in particular gives
a beautifully crystallised p-toluylpicramide m. p. 166°*) whilst the
alcoholic solution contains only comparatively few, not very dark
coloured byeproducts. In an experiment in which 35 grams of the
nitramine were heated on the waterbath with an equal weight of
p-toluidine and 100 c.c. of 96°/, alcohol, a fairly violent reaction set
in after some time. The heating was continued for 5 hours and,
after the picramide derivative had been removed by filtration, the
alcohol was distilled off and the residue extracted with dilute sulphuric
acid. The liquid filtered off from the toluidine sulphate was shaken
with ether. On evaporation of the ether a still yellow coloured liquid
product was left which on being inoculated with a crystal of methyl-
nitramine became crystalline and after having stood for some
time over sulphuric acid weighed 7 grams. On pressing between
filter paper light yellow crystals were obtained which after being
sublimed in vacuo (a treatment which methylnitramine stands very
well) melted at 38°. On mixing the same with a preparation con-
sisting of pure nitramine the melting point was not affected.

p-Toluidine appears, therefore, to be a suitable means for readily
procuring in a short time methylnitramine from s-trinitrophenyl-
nitramine.

We are continuing our investigations with different amines and
also with other nitrated aromatic nitramines, and will state the
results more elaborately in the “Recueil”.

Org. Chem. Lab. of the University Utrecht.

Physics. — “Wave-lengths of formerly observed emission and ab-
sorption bands in the infra-red spectrum.” By Prof. W. H.
JULIUS.

If in the infra-red spectrum, as formed by means of a rock-salt
prism, the positions of emission or absorption bands have been care-
fully determined, the corresponding wave-lengths still are uncertain.
by an amount which, in a considerable part of the spectrum, is
greater than the probable error of those determinations, because the

1) We now obtained this substance in two modifications, one coloured dark red
and the other orange.

( 707 )

dispersion curve of rock-salt is not yet known with sufficient exactness.

Mr. W. J. H. Moun') has lately compared with each other the
dispersion curves that have been calculated according to Kerre.er’s
formula with two sets of constants, one given by Rvusens’), the
other by Lanenny*). Laneiry’s results held for a temperature
of 20°; the numbers given by Ruspens were corrected by Mr. Mou.
so as to apply to the same temperature. While coinciding in the
visible spectrum, the two dispersion curves appeared to diverge very
sensibly in the entire infra-red region, the wave-lengths correspond-
ing to given indices of refraction being smaller with Rupes’ than with
LANGLEY’s constants. At 2=1,54 e.g. the difference amounts to
0,028 w; it increases unto 0,062 u (at 234) and then decreases to
0.032 « (at 2— 8,54). If, on the other hand, the indices of refrac-
tion, which according to LaneLey’s and according to Rugens’ formula
belong to rays of given wave-lengths, be compared with each other,
the difference appears to be rather constant between 2=44.4 and
4=8,3u, namely 1,5 units of the 4% decimal of the index, and to
increase from © to 1.5 similar units in the region between 0.6 u
and 4 u.

The apparatus, nowadays available for the investigation of the
infra-red, admit of determining the position of sharp maxima or
minima of radiation with an accuracy, going a good deal farther
than 1,5 units of the 4 decimal of the index.

When between 1887 and 1891 I investigated several infra-red
emission and absorption spectra, our knowledge of the dispersion of
rock-salt was restrained to the outcome of LANGLey’s first determi-
nations *), which extended only as far as 5,3u. As a great part of
my work bore upon longer waves, I published my results in the
form given by direct observation, viz, as galvanometer deflections
and corresponding angles of minimum deviation, reduced to the
temperature 10°. The refracting angle of the prism being also recorded,
the indices of refraction of rock-salt for waves, corresponding to the
observed maxima, were thus implicitly given.

In order to obtain a rough estimate of the wave-lengths, I had
extended LanGLey’s dispersion curve in a straight line, though under
strict reservation. The wave-lengths as read on this lengthened

1) W.J.H. Mott, Onderzoek van ultra-roode spectra. Dissertation, Utrecht, 1907.

*) H. Rupens, Wied. Ann. 60, 724; 61, 224; 1897. Cf. also Kayser, Handbuch
der Spectroscopie I, 371, 1900.

*) S. P. Lanetey, Ann. Astroph. Obs of the Smiths. Inst. I. 1900.

+) S. P. Lanetry, Phil. Mag., Aug. 1886.

( 708 )

curve, to which I myself assigned little weight’), have found their
way to some text-books *), where they unfortunately appear as the
results of my investigation, with the incidental remark that they are
incorrect, as founded on a false extrapolation. It is clear, however,
that this incorrectness has nothing to do with the accuracy with
which the position of the bands in the prismatic spectrum has been
determined. Now I have reason to believe, that the spectrometric and
heat-measuring apparatus used in that research were not less valid
than those employed by many later observers of infra-red spectra
(Donat, Pucctanti, Ik1L.E, CoBLENTZ, Nicuots, RuBENs and ASCHKINASS
and others), so that the results still retain their value as a first
contribution to our knowledge of the examined spectra.

I therefore thought it suitable to republish the principal results
obtained at that time *), but now to mention the indices of refraction
for the maxima of emission and absorption, as following directly
from my observations, and to add the wave-lengths, as derived from
the more recent dispersion curves of Rupens and of LANGLey.

The positions in the infra-red were determined in my work with
respect to the place of the D-lines of a Bunsrn flame coloured with
chloride of sodium. But the latter were too faint to be observed with
the bolometer; and the transition from the visual observation of the
D-lines to the bolometric observation of infra-red radiations caused
an uncertainty in the determination of the relative positions, which
was still increased through the necessity of displacing the bolometer
along the optical axis of the rock-salt lens according to its different
focus for visible and invisible rays. It was chiefly in the part of

1) Cf. ,,Bolometrisch onderzoek van absorptie-spectra”, Verhandelingen der Kon.
Akad. v. W. te Amsterdam, Vol I, N°. 1, p. 8 (1892), or the German translation
in: Verhandl. des Vereins zur Beférderung des Gewerbfleisses, 1893, p. 235, where
[ have clearly stated that | considered the extrapolation of Lanetey’s dispersion
curve as quite uncertain, and that in the tables the direct data of observation
(angles of minimum deviation) were given, because | did not like to have my
results inseparably connected with a possible incorrectness of the dispersion curve.
The passage in question seems not to have been noticed by W. W. Costentz, for
in his excellent work ‘Investigations of Infra-red Spectra”, published by the
Carnegie Inst. of Washington, 1905, he says on p. 135, after alluding to LanGiey’s
extrapolation of the dispersion curve in a straight line: ‘Jutrus, with apparently
less hesitation, has applied this extrapolation to his work”.

2) Winxetmann, Handbuch der Physik; Kayser, Handbuch der Spectroscopie ;
Cuwotson, Lehrbuch der Physik.

3) Recherches bolométriques dans le spectre infra-rouge. Arch. neérl. 22, p.
310—383 (1888).

Die Licht- und Warmestrahlung verbrannter Gase, Berlin, Simion. 1890.

Bolometrisch onderzoek van absorptiespectra, 1. c.

( 709 )

the investigation, described on p. 69 of “Die Licht und Wiarme-
strahlung verbrannter Gase” that many pains were taken to reduce
this source of error. There the CO,-maximum of the Bunsen flame
was found at minimum deviation 38°54'20", the refracting angle of
the prism. being 59°53'20" and the temperature 10°. From this
follows n = 1,52103. Had the temperature been 20°, then the devia-
tion would have been found smaller by 1/50", giving for the index
of refraction: 2 = 1.52069.

If we suppose this value to be exact, then the angles of minimum
deviation given in my first paper in Arch. neerl. 22, and on pages
47—68 of “Die Licht- und Warmestrahlung” are too small by
nearly 3', owing to an instrumental error. In “bBolometrisch onder-
zoek van absorptiespectra” the deviation of the CO,-maximum has
been found 388°52'40" instead of 38°54'20"; 1’ of this difference
results from the fact that the refracting angle of the prism, then in
use, was smaller than that of the other one by 1’; only the
remaining 40" were owing to an instrumental error.

I have now applied the corrections resulting from this re-exami-
nation, and calculated the indices of refraction for 20°, the tempe-
rature to which the dispersion curves as compared by Mr. Mou
also refer. In finding the wave-lengths corresponding to the indices,
advantage has been taken of elaborate tables, prepared by Dr. Moun
for a research of his own, and which he was kind enough to put
at my disposal.

( 710 )

Indices of refraction Wave-lengths according to

Emission-spectrum : :
for the maxima eS Se Intensity !)
of: | (Temp. 20) of RUBENS | of LANGLEY
| |
aos | ;
BuNSEN-flame ... .- - | 4.5268 £206 1.953 | 0.5
| 4.5947 HO 2 769 2.831 | bee
| 4.52069 CO, | 4.410 | 4.462 | 10
Flame ofcarbon monoxyde 1.52445 CO, , 2.883 | 2.947 | |
or of cyanogen | 1.52069 CO, | 4.40 | 4.462 | 10
Hydrogen flame... . 1.5247 H,O | 2.77 2.83 | 10
1.5176 | 544A | 5.46 2
Luminous gasflame. . 1.5270 c | 1.84 4.89 2)
1.52947 H.O | 2.77 | 2.83 2)
1255907, COs | 4.41 4.46 2)
|
Hydrog. burning inchlor.) 1.5226 HCI | 3.68 3.74
Flame of sulfur... . | 1.5093. - SO, 7.49 7.59
Flame of carb. disulphide) 1.5247 2.77 2.83 1
4 208 — Gs 4.44 4.46 10
4.5125 COS(?) 6.76 6.80 3—0 2)
1.5093 - ..SO, 7.49 7 52 2—3 »)
Absorption-spectrum
of:
CC (giamond) 2 =... 2 4.5238 3.18 3.24
1.5202 4.58 4.63
10
1.5183 5.20 5 25
1.5088 ete. 3) 7.59 7.62 10
PUES igen eta Se ao 1.5287 1.44 1.43 4
4.5265 2.04 2.06 4
4.5236 3.25 3.31 ie
4.5194 4.85 4.90
1.5146 6.24 6.28

1) In each spectrum the intensity of the highest maximum is indicated by 10.
The letter s following an intensity figure means, that the band is rather sharp.

*) The relative intensity of these bands varies much with the place in the flame.

*) The addition “ete.” behind an index of refraction indicates, that the band marks
the beginning of an extensive region of strong absorption.

ecast)

aay
Indices ofrefraction Wave-lenghts according to
the dispersion curve

Absorption-spectrum

for the maxima | Intensity
2h | (Temp. 20°) | of RuBENS of LANGLEY |
|
aS ct: >t: +s eee | 4.5203 4.55 4.60 6 6s
| 1.5129 HA G67 6.71 10
2 5 si eee 1.5219 3.96 4.02 I
| 1.5163 5.78 5.82 1
1.5090 | 7.55 7.58 7 s
4.5049 | 8.36 8.39 |
4.5020 / 8.90 8.93 1
1.4992 eae | «9 Ak 8 s
1.4942 10.28 10.31 10
Ee x ee , 1.5221 3.88 3.04 05
1.5082 mie cy 775. | 10 s
1.5030 8.73 Rae 4 0.5
44944 10 25 10 28 10
Bae 8 1.5172 5.53 5.57 6 s
1.5154 6.03 6.07 {
1.5058 8.19 99 10 s
1.5014 9.02 05 | 10
4.4974 3 9.16 "h” ~46
2 16 Sa ere 1.5934 | 3.34 3.40 0.5
1.5173 fess 50 5.54 | S
1.5058 8.19 Boa | 2
1.5014 9.02 | 9.05 | 10
1.4974 9.73 9.76 3
areliee y/.) sacl GRO e 4.5137 6.47 ZA 9s
1.5058 8.19 2 | 10
4.4942 10.28 10.34 | 5 Ss
GAL... sige | 4.5234 3.34 3.40 3
1.5131 6.62 | 6.66 | s
1.5445 6.99 | 7.08 |
1.5058 s19 | 8.02 | 40
4.4980 9.62 9.65 7

( 712 )

|
Indices of refraction Wave-lengths according to

Absorption-spectrum | the dispersion curve

_ for the maxima Intensity
ole | (Temp. 20°) | of RUBENS | of LANGLEY
ot: SG Re itis | 1 5935 | -g.a0 |. 336° [ssa
| 4.5007. | 4.40 | 4.45 3s
eromees. | = 5276 +1) - eae As
1 5116 2 (0.01 > Vesna 3
| i503 =—ti«d(L; 89
4.5024 | ae | 8.86 10
‘402 | 9. | a
£5 5 ge Bee aoe P5259 |" 2.25 sae | 1
1.5236 9°95. eee 4s
| 1.5214 4.96 | ps 1
| 1.5173 | 5.50 5.54 6 s
1.5128 | - 660>< || Seige 10 s
1.5107 Sele 7.21 3s
4.5088 | ~ 7.5: |" Seg
| 1.5060 8.45 8.18 2
| 1.5039 8.56 8.59 | 7
(och eh a a ei ae 1.5259 2.95 > aay een
| 1.5230 3.51 3.57 s
1.5154 6.03 6.07 1
1.5118 | = 6.98 6.96 10 s')
1.5097 7.40 7.43 6
1 5068 7.99 8.02 4
1 5032 8.69 8.72 6 s
| 1.4980 9.63 9.66 5
1.4942 40.298 |» 403) 5
AL: ne ie | 1.5259 2.25 | 2.31 4
| 1.5229 3.56 | 3.62 10
1 5194 | 485 | 4.90 G45
| 1.5145 ete. | 6.97 | 6.31 10
CHO 2 eae. 1.5959 2.25 ce: aa |
1 5229 3.56 3.62 | ° 40
1.5183 5 20 5.25 | s
4.5154 6.03 6.07 2
1.5126 ete, 6.74 6.78 | 10

') Sharply limited only toward the smaller wave-lengths.

(e113)

Absorption-spectrum

of:
COT oa.
C,H,OH (normal)... .
CTOMs(iso) <>. +
S20) |
(C,H;),0 Ee oe ae
(Cs 2S) 5 (CPi

ties of refraction Wave-lengths according to

: the di sion curve
for the maxima dispersion

(Temp. 20°)

~_ =
—

—

.5230
.0152
5126

5230
.0162
9126
.9230
.5192
D154

.9126

5230

etc.

Intensity
| of RuBENS of LANGLEY |
3.51 3.57 10
6.09 | AB 3
4 | 78 10
iy 3.57 10
5.81 Bete | 3
6.74 Geleiee chal 40
3.51 3.57 10
4.92 4.97 1
6.03 | 6.07 9
Ga fh 1@.78 10
3.51 3.57 10
4.92 4.97 | 1
6.08 6.07 | 3
6.74 | 6.78 10
0.953 | 0.958 | 1
4.30 | 4.32 1
2.25 2.34 2
3.51 3.57 8 s
5.20 ayer 3 a)
5.76 5.80 1
6.97 7.04 10
Zeb) | ae 4
TED hot Bas a1. Vwi
8.44 8.47 10
2.38 2.44 1
3.54 251i 5
3.88 3.94 3
5.20 5.25 {
5.84 5.85 4
6.99 7.03 10
7.88 7.91 40
9.95 9.28 8
410.39 10.42 9

(712 )

Physics. — “A hypothesis relating to the origin of RONTGEN-rays.’
By Prof. C. H. Winn.

W. Wirn') has measured the energy of RONTGEN-rays, converted
into heat in a bolometer or in a thermo-element, and has compared it

with that of the cathode-rays, likewise converted — with exception
of the small fraction transformed in energy of R.-rays — into heat

in the anti-cathode. He finds for the proportion of the total quantities
of energy of the two kinds of rays

Br 918. W28 2).

Ey,

Supposing that the R.-rays are the radiation of energy, emitted
by cathode-ray electrons being brought to rest, and that this stoppage
may be considered as a continually decreasing motion, he proceeds
with the aid of the theory of M. ABranam to deduce the duration
of the stoppage and from it the thickness of the R-waves. For the
latter he finds

A=: 11S) 0S em:

Results of the same order of magnitude have afterwards been
attained by Epna Carter*) in an investigation, also made at the
laboratory directed by Wirn.

These results do not very well agree with the values, derived by
HaGa and myself for the wave-length of R.-rays from diffraction-
experiments :

A= 270:to 12 =, 19 em)
and
4= 160 , 120 , 50 . 10—%em. 5 E

If the R.-rays have to be considered as disturbances in ether of the

single pulse character assumed by Wren in accordance with the

3.7
current conception, the same numbers must be divided by ipo
2

2°) in order to represent the corresponding values of the thickness
of the pulse-waves, which consequently become

f= 110 .t6 5 i. 1O=) “om:

8, = 64, 48, 20 .°10—10 em.

1) W. Wien. Wittyers Festschrift, Leipzig, 1905; Ann. d. Ph. 18, p. 991, 1905.

*) L. c. p. 996. The number is doubled here, on account of the remark made
regarding it on page 1000.

3) E. Carrer. Ann. d. Ph. 21, p. 955, 1906.

*) H. Haga and C. H. Winn. These Proc. I. p. 426.

6) Id. Ibid. V. p. 254.

*) G. H. Winn. Physik. Zschr. 2, p. 96. Fussnote 2), 1901.

( 715 )

Wirn’s experiments would have led to results more in keeping
with the diffraction experiments, if the values found for the energy
of the R.-rays had been 20 to 100 times smaller. The difference is
too great to ascribe it to errors of observation. We must rather
think of fundamental errors in the method of observation or of a
viciousness in our conceptions concerning the mechanism of the
phenomena.

As for the method of observation Wren himself pointed out?)
the possibility that the quantity of heat, generated in the bolometer
or in the thermo-element, should not be to its full amount converted
energy of R.-rays, but partly also —. perhaps even for the greater
part — converted atom-energy, liberated by a, say, catalytic action
of the R.-rays.

J. D. v. p. Waats Jr.?) suggests the additional idea that the
electrons are not generally stopped at once by a simple uniform
decrease of velocity, but will mostly, by their interacting with the
particles of the anti-cathode, before being brought to rest move for
some time amidst the latter in rapidly changing directions with great
velocities, sending out a new R.-pulse at every change of motion.
Starting from this idea we could, indeed, expect from each electron
a much greater contribution to the energy of radiation than in
the theory accepted by Wren and find the results of Wien’s energy-
measurements in better agreement with those of the diffraction-
experiments.

Nevertheless it seems to me that by the side of this another idea
deserves our attention, which might be more in keeping with the
properties of cathode-rays as far as known. It would be this, that
not simply the cathode-ray electrons, but in combination with these
the atoms of the anti-cathode are the principal centres of emission
of R.-rays.

It should be imagined, that the electrons, arriving at the anticathode
with their immense velocities, are not, generally, thrown into an other
direction by the atoms, but will for the greater part pass straight through
them, and even, in doing so, will mostly not suffer any persisting
decrease of velocity. This idea is by no means a new one. It has
been worked out by LrNnarp’*), who sees in it the best explanation
for the laws of absorption of the cathode-rays. In very few cases only
it will happen that an electron, when piercing an atom, gets imprisoned

1) W. Wien. Drudes Ann. d. Ph. 18, p. 1005, 1905; cf. also E. Caper. Ann.
d.-Ph. 21) p: 957, 1906.
2) J. D. v. vp. Waats Jr. Ann. d. Ph. 22. p. 603. 1907.
3) P. Lenanp. Drudes Ann. d. Ph. 12, p. 734, 1903.
49
Proceedings Royal Acad. Amsterdam. Vol. IX,

( 716 )

or changes its direction considerably ') in a centre of exceedingly
strong electromagnetic action; in the great majority of cases it will,
by the abundance of vacant space in the interior of the atom’), fly
across it without experiencing a considerable decrease of velocity.
In this way the greater part of the electrons will pierce thousands
or tens of thousands of atoms before being stopped, and we find
easily explained the great penetrating power of the cathode-rays,
which may still in appreciable quantity pass through a layer of
aluminium 10°) thick or a layer of atmospheric air, some em thick *).
If we consider the values given by the diffraction-experiments
for the order of magnitude of the thickness of R.-waves as correc,

it follows from Wuren’s experiments — apart from a_ possible
catalytic action of the R.-rays — that the radiation of the cathode-ray

corpuscles, by the simple fact of their stoppage, could account only for

1 1
something like ao ind of the whole energy of the R.-rays. Conse-

quently for by far the greater part this energy must, if LENARD’s
views may be accepted, have a different origin. What this can be,
is obvious. The atoms namely will by no means remain undisturbed
during the sudden passage of an electron. Themselves probably con-
sisting of negative and positive corpuscles, they will see their electro-
magnetic fields during the passage altogether altered and at the same
time will no doubt send out a pulse or wave of disturbance *) into
the surrounding ether. About the character or shape of these pulses,
which moreover may vary from one case to an other, we can,
without making any more definite assumptions as to the structnre
of the atom, say little; but there is one important point, in which
all these pulses will be to a certain degree similar, viz. their duration.

1) Together with the expulsion of electrons originally belonging to the atom,
which will often occur at the same time, these changes of direction could very
well account for the diffusion of the cathode-rays according to LeNarp.

2) Lenarp calculates (Drudes Ann. d. Ph. 12, p. 739, 1903) that only 10—9 of
the volume of an atom is occupied by the “dynamids’”’, of which he considers it
to consist.

5) Lenarp. Wied. Ann. 51, p. 233, 1894.

4) Id., Ibid., p. 252.

5) Lenarp expresses himself (“Ueber Kathodenstrahlen”, Nobel-vorlesung, p. 37,
Leipzig 1906) as follows: *Das durchquerende Strahlenquant’” — the electron —
“wird vermége der abslossenden Kriifte, welche es auf die anderen, dem Atom
eigenen, negativen Quanten ausiibt, eine gewaltige Stérung innerhalb des Atoms
hervorbringen kénnen”, and then continues thus: ‘und als Folge dieser Stérung
kann ein dem Atom gehdériges Quant aus ihm hinausgeschleudert werden (sekundire
Kathodenstrahlung)”; but he does not speak of a radiation emitted by the atom.

f-7i7 |)

The latter will be, if @ represents the diameter of an atom and
v the velocity of the electron, which is piercing it, something like

a . . . .
(rather smaller than) —, causing the wave emitted to be of a thick-
v

a .

ness of something like (rather smaller than) c —, ¢ being the velo-
i

city of light in ether. By putting a—10-° and v= 10", we get

; . .
by this way for ¢ — 3.10—-§, a number which only slightly exceeds
_

the order of magnitude of the values of § (p. 714), derived from
diffraction experiments. It might therefore be possible, that the
waves of disturbance in question should be identical with the Rént-
gen rays.

As by this theory a single electron would disturb some thousands
or tens of thousands of atoms, every atom, being traversed by an

1
electron, need only send out something like we of the quantity of

energy emitted by an electron itself in its total stoppage, in order
to account for the relatively large amount of energy found by Wien
in the R-rays. That such proportions should exist, seems to me
. not impossible at all.

The views presented here as to. the origin of the R.-rays bestow
anew and great importance on the ‘wave-length’ of these rays,
as they intimately connect this measurable quantity with the
dimensions of the atoms. Whether there really exists such a close
connection, could perhaps be experimentally put to the test by
diffraction experiments with anticathodes made from different materials.
More generally it might be expected that experiments of this kind
would throw some new light upon the structure of atoms, and also
of molecules or molecule aggregates. In such experiments it would
certainly have a peculiar interest to use crystals as anticathodes, as
perhaps the regular structure of these bodies could manifest itself
both in rather sharply defined wave-lengths of the R.-rays emitted
by them as in a polarisation of these rays.

The question, whether R.-rays should or should not be expected
to show total or partial polarisation, may be treated on the basis
of the above hypothesis, as soon as this be supplemented by definite
suppositions about the structure of the atom.

The relation that, according to our views, should have to exist
between the wave-length of R.-rays and the velocity of the cathode-
rays, is of course liable to rather direct experimental verification.

49*

(718 )

Two further questions connected with those views and perhaps
liable also to be answered by way of experiment, are these:

1. whether the air molecules on the outside of the aluminium
window of Lenarp emit R.-rays in appreciable quantity ;

2. whether the y-rays of a radio-active substance, except by the
substance itself, are to a considerable extent emitted also by the
atoms of air in its neighbourhood on their being pierced by the
electrons constituting the §-rays.

Physics. — “On the motion of a metal wire through a piece of ice.”
By Dr. J. H. Mrersure. (Communicated by Prof. H. A. Lorentz).

(Communicated in the meeting of January 26, 1907).

During the last and the preceding winter I made some measurements
with a purpose of testing the formulae, expressing the velocity
of descent of a metal wire through a block of ice, which Mr. L. S.
OrnsTEIN had derived from the theory of regelation ').

') L. S. Ornsrei. These Proc. VIII, p. 653.

( 719 )

the legs A and PD of an iron frame, which, in order to secure
greater rigidity, had been cut from an iron plate. In the first measure-
ments the downward displacements of the wire were observed by
means of a small reading telescope, turning round a horizontal and
a vertical axis, and were determined on a measuring rod, mounted
at the side of the frame. The breadth of the ice-block was also read
on a horizontal measuring rod. In the later experiments a catheto-
meter was used, placed at my disposal by the professors of the
Technical University. at Delft. | wish to express here my sincere
thanks to these gentlemen, especially to Prof. pe Haas. The fall of
the wire was always derived from the change in the difference of
level between the top of the wire and the upper edge of a small
bubble, existing somewhere in the interior of the ice. Every ten
minutes or, when the descent was quicker, every five minutes, the
difference of level was measured in order to ascertain whether the
fall was regular. Each experiment lasted 20 to 40 minutes.

The ice used was artificial commercial ice. From a larger block
a clear smaller one was sawn out, in which some bubbles should be
present to serve as marks. The faces were melted flat by pressing
them against a metal plate, so that errors, caused by irregular
refraction, were avoided.

Heat conduction along the wire was prevented by hanging small
pieces of ice on the wire on both sides of the block. Still small
grooves were occasionally formed when the descent was slow.

The experiments were made with wires of steel, german silver
and silver. The thickness Of the wires was measured by means of
caliber compasses, giving results accurate to 0.01 mm. The thickness
was 0.5, 0.4 and 0.3 mm. Deviations from these numbers, amount-
ing to some hundredths of a millimetre, were occasionally found.

For the case, realised in my experiments, in which the two straight
ends of the wire make a certain angle 2a with each other, formula
(Illa) of Mr. Ornstern’s paper ') holds:

2aCP

d, sin a
in which v represents the velocity of descent of the wire, P the
total weight and d the breadth of the ice-block. C is a constant.
The value of this constant I calculated by formula (1) from the
values of v, found in my experiments.

The results are summarised in the following table:

( 720°)

Steel wire. Diameter 0.5 mM.
PB. C' average of (.
(in grammes).
455 0.0162
(a5 0.0151
1255 0.0172 ta
2160 0.0185 ee
2205 0.0169 \
5160 0.019
diameter 0.4 mM.
455 0.030
155 0.029 | 0.029
1255 0.029
diameter 0.3 mM.
DD 0.043
1255 0.042 | ri
German. silver. diameter 0.5 mM.
fers 0.0134
1255 0.0119
2150 0.0143 Dae
5160 0.0172
diameter 0.4 mM.
TDD 0.0196
$255 0.0204
2150 0.0208 | OO
5160 0.0255
diameter 0.3 mM.
355 0.0306 |
oa 0.0348 0.035
855 0.0393
Silver wire. diameter 0.5 mM.
ToS 0.0207
1255 0.0255 | oO”
diameter 0.4 mM.
536 0.0367
Bo 0.0384
1036 0.0392 | a
1255 0.0404

diameter 0.38 mM.
555 0.0347 |

755 0.0467

0.041

( 721 )

The quantity C is not expressed here in C.G.S. units, since the
dimensions have been taken in millimetres, the velocities in milli-
metres per minute and the forces in grammes. In order to reduce
them to C.G.S. units, the value of Chas to be multiplied by 170 > 10—.

The values given in the table are averages of several measurements.
In order to show the deyiations of different measurements, made with
the same weight, I give here an arbitrarily chosen set of separate
measurements.

German silver wire, diameter 0.4 mm.
Number of the

experiment e v d, 2a c averages.
8 1036 1.017 39.0 42° 0.0368
10 10386 141° 37.4 492° 0.0393 | a qaue
14 fie 2.98 SR SP ee ee
i 1036 1.73 29.3 30° 0.0393
87 1255 0.99 52.3 50° 0.0401 |
112 1255 1.09 51.27 53° 0.043 |' 0.0404
115 1255 0.756 66.46 53° 0.0387 |

The value of C is calculated by Mr. Ornstein in formula (I) of
his paper. He finds

pee aes Py ae) F
(=) EB k, — “Rr (hk, — *,) fe |
po dp}, ho ae ta)
ae zx R? WS;

Here &,, k, and &, are the coefficients of heat conductivity respec-

(2)

dt
tively of the wire, of water and the ice, (=) is the rise of the
P),

melting temperature by pressure, measured at the melting tempera-
ture, JV’ is the latent heat of melting ice, S,; the specific gravity of
ice, A the radius of the wire and d the thickness of the layer of
water. Now the value of C cannot be calculated by this formula,
since the quantity @ is unknown. But besides the equation (I)
Mr. OrnstEIN gives in his formula (II*) an expression, found by a
hydrodynamical reasoning, in which the quantity d likewise occurs.

This relation is‘):
Sa 2aP aye
v= — ———_— | — Le eee ry)
S; 12a pd, sna\R
ga ; Sw
) In Mr. Ornstery’s paper this formula is given without the factor = since

t
this latter has no perceptible influence.

& 722 )

Here S,. is the specific gravity of water at 0°, « the viscosity
coefficient. By equalising (1) and (3) we find:

C Sw 1 ax x
ae ee

d
and we should now have to eliminate p between (2) and (4). In order

to perform this elimination we simplify (2). We consider the form
d *.
in (2) between the brackets | | and keep in mind that R is very

small, that &, is very much greater than /, and that 4, may be
neglected with respect to the first term (which amounts to neglecting
the conduction of heat through the ice).

We may then write:

; k, Ex a/R (A, ia k,) aay ky k

hgh in ky Ee eee ee
; eae

(5 ) k,
dp k
P 0 1 == d/R ~

ot R? WSS;

Then we have

GS
|

If we put
d
(=) = — 74010 ss = 05101, Se ee,
dp ;
(’ becomes
C=—=aa 10—!! : hy
as . x me R? Sips
oe ze

In (4) we substitute
Sp = 1, S=0 9167, 2D

c= 1.600(4) 1 fae

Equalising the two values of C we have:

then

ee oY: 9.0 x 10501
BR) AR) a ae

( 793 )

d
From this equation Rp om easily be found by a tentative method,
L

when &,, &, and FR are given. In the different cases we find (in

CGS-units),

Steel wire k= 0.166 &,=0.0015 R=0.025 < — 0.10166
L
d

k, = 0.166 k= 0.0015 R=0.015 = = 0.00229
/

German silver wire k, = 0070 k, = 0.0015 R= 0.025 _ — 0.00128
|

k, = 0.070 &, = 0.0015 R—=0.020 “= 0.00149
1

k, = 0.070 k,— 0.0015 R—0.015 a= 0.00179
d

Silver wire b= 1.50 4, = 0.0015 R= 0.025 | = 0.00289
d

k= 150 &,= 0.0015 R=0.020 = = 0.00279

1
,=150 &=0.0015 R=0.015 s= 0.00818
Cis then found by substitution in (5). These values are given
below, together with the values found by experiment, but now
expressed in CGS-units.
Calculated Found
Steel wire KL = 0.025 fo) 1010 29. 36 1Q=10
fe == 0:020 POS 10-10 49 & 10—10
R= 0.015 192° 10-10 fac A0—9
German silver wire R= 0.025 34 & 10-10 24 X 10—20

R= 0,020 53x 10-10 837 x 10-10
R=0.015 9110-59 & 10-10
Silver wire R=0.025 218 10-10 46 x 10-10

Rk = 0.020 347 & 10—10 665< 10-10
f= 0.086 aloo 10-10 hE SS LOS

The agreement must be called bad for the silver wire, satisfactory
for the german silver wire. It may be called satisfactory, since
different circumstances may be mentioned which make us expect a
too small value. Leaving aside the great uncertainty in the values of
the heat conductivities of metals, to Which we cannot here ascribe
the bad agreement, since we do not know in which direction this

( 724 )

will influence the result’), the following causes may be mentioned.
1. The roughness of the wires. Already Mr. Ornstety pointed
this out. If the wire is not entirely smooth, the hydrodynamical
deductions are uncertain and hence also formula (3). In order to
ascertain the influence of this roughness I made some experiments
with a steel wire that had for a moment been scoured with fine
sand-paper in the direction of its length. Macroseopiecally no result
of this manipulation could be discovered on the wire. Yet the effect
proved considerable, for the following results were found :
5 &
, 455 0,009 |
steel wire, diameter 0,5 m.M. pe 0,0J1 » average 0.011 m.M.
1 2205 0,014
steel wire, diameter 0,3 m.M. 1255 0,028

So we find a diminution of about 40°/, in the value of C. After
having observed this influence I tried to obtain smooth wires, but
unsuccessfully ; all the wires that were used in the experiments showed
under the microscope numberless grooves in the direction of their
length and of a breadth that might be estimated at somewhat less
than 0.01 mm.

d
Since it is easily deduced from the calculated values of R that

the thickness of the layer of water increases with the size of the
radius of the wire and since the influence of the roughness of the
wire will be smaller with a greater thickness of this layer of water,
[ have still made some measurements with a thicker steel wire of
0,87 mm. diameter and heavier weights. The result was :
Yi GS C' (in C.G.S. units)
25200 0,00803 |
0,0081 13.8 Se 7O
25200 0,00822

5200 0 ,00667 13° x Ai
while calculation gives
l ,
k, —0,166, &,=0,0015, R=0,0435 = 0400120 C= 27.7 x 10-10
The agreement is now better indeed; the value found is half the

calculated one, while with the thinner steel wires it was slightly
more than a third.
') The values given by F. Kontrauscn (Lehrbuch der praktischen Physik 10

Auflage 1905), steel A =0.06 to 0.12 and silver &4=1.01, would give a much
better agreement.

( 725 )

2. In the deduction of formula (1) it was assumed that within
the layer of water the relation

holds. This relation, however, holds for a body at rest. Here, on
the other hand, we have to deal with a streaming liquid, in which
case the following formula holds:

ee. OF Sw Ot Ot
da?! dy? Es ("55 +°5)-

Here og is the specific heat of the liquid, w and v the velocity
components in the A- and Y-directions. If we use this formula we
take into account that the heat, conducted through the wire, does
not entirely serve for melting the ice, but that it is partly conveyed
upwards again by the streaming liquid. This also must result in
a diminution of the velocity of descent. Prof. Lorentz informed me,
however, that it can be shown that this influence must be regarded
as a quantity of the second order, so that the differences cannot be
explained in this way.

3. If the temperature in the interior of the block of ice is not
exactly O°, but lower, the velocity of descent will also become
smaller. But I observed no phenomena which point to a lower
temperature in the interior. Blocks of ice that had been kept for
24 hours in a space above 0° gave the same results as blocks that
had just been received. Moreover the wires as a rule went down
at a distance of only a few millimetres from one of the faces of
the block, and in some experiments they even came out of the block
by melting of that face. Yet in the last moment, before the wire
came Out, no acceleration of the descent was observed.

Nor does theory support such an explanation. Prof. Lorentz
informed me that when the surface of a ball of ice of 3 centimeters
diameter and at a temperature of — 2°, is raised to 0° and kept at
this temperature, it may be shown that in less than an hour the
temperature at the centre has risen to — 0.01°.

4. Another important influence on the velocity of descent is found
in the fact that it is possible that not all the ice, melting at the
lower side of the wire, freezes again exactly at the upper side, but
that this water perhaps flows off laterally. It is clear that this must
have a great influence since then the heat, necessary for the melting,
is furnished by conduction through the ice. Already J. THomson

( 726 )

BortomLEY ') showed that the lateral flow of water causes a great
retardation.

he experiments now showed that this lateral flow really exists.
For even when the ice was perfectly clear, in the places where the
wire had passed through it various small bubbles were observed.
Consequently not all the ice had been re-formed which had been there.

In this respect I also mention a curious change, found in the
values of C: these values rise with the weight. This is very con-
spicuous with the silver and german silver wires, but also with the
steel wires it exists, especially with the thick one of 0.87 mm.

Accordingly it was often seen that the bubbles on the path of
the wire were more numerous with small than with heavy weights.
This became particularly clear in experiments in which, during one
descent, first a heavy and then a small weight was used, With the
smaller weight more water flows off laterally.

I still made several experiments in which the wire was pulled
upwards through the ice, hoping to prevent this lateral flow. The
result was not the expected one, for bubbles also appeared and the
values, found for C, were even somewhat smaller than in the former
case. In regard to this question it would be desirable to investigate
the descent of a whole body, e.g. of an iron ball, through perfectly
clear ice.

In my opinion this lateral flow is the chief reason why theory
and observation disagree. It also explains why with the silver wires
larger differences were found than with the german silver and the
steel wires. For if the heat is only partly furnished by the freezing
process above the wire and if the rest has to be furnished by con-
duction through the ice, it seems to be of little consequence whether
the wire be a good conductor of heat.

5. Ice is a erystalline substance. This also may have its effect.
Perhaps the melting point is not the same at the different faces of
the erystals which the wire touches. Though this influence may exist,
we cannot say in which direction it would modify the result.

In order to find out whether such an influence makes itself felt,
I made the wire pass several times through the same block of ice
in three mutually perpendicular directions. But no perceptible diffe-
rence was found.

As the general result of the experiments I think we may state,
that they indicate that the regelation theory will be found capable
of explaining the phenomena not only qualitatively but also quanti-
tatively.

1) Pogg. Ann. 148, p. 492, 1871.

( 727 )
Physics. — “Contribution to the theory of binary mixtures, [1’’, by

Prof. J. D. van per Waats. (Continued, see p. 621).

Not to suspend too long the description of the course of the qg-lines

30s

in the case that the locus = 0 exists, we shall postpone the deter-

ak
mination of the temperature at which this locus has disappeared,
and the inquiry into the value of z and v for the point at which

it disappears — and proceed to indicate the modification in the
course of the qg-lines which is the consequence of its existence.
dw
dv i
From the value of -—— = W—-W—— follows that when a gq-line
dig ay
dadv

passes through the curve ae it has a direction parallel to the
&

Fig. 3.

( 728 )

z-axis in such a point of intersection. So a q-line meeting pies
aL

will be twice directed parallel to the z-axis, and have a shape as

Ed lig hes alilenst as, J i | ee
ins Ole eel Td) thee ay I east aS lone as the curve
P 2 = dedv dit :

does not occur. Such a shape may, therefore, be found for the q-lines,
in the case that the second component has a higher value of 6, and
lower value of 7). —, and such a shape will certainly present itself
in the case mentioned when the temperature is low enough.

Then there is a group of g-lines, for which maximum volume,
and minimum volume is found. The outmost line on one side of
this group of q-lines, viz. that for which g possesses the highest
value, is that for which maximum and minimum volume have coincided,

and which touches the curve a e O in the point, in which this curve
v

itself has the smallest volume. The other outmost line of this group

of q-lines, viz. that for which q possesses the smallest value, is that

for which again maximum and winimum volume have coincided, and

dp 2 bee

3 = 0, but in that point in which

this curve itself has its largest volume. So for these two points of
8

dy ’
contact the equation —— = 0 holds. These two points of contact are,
Ak

which also touches the curve

L?w 3

ae d Dw
therefore, found by examining where the curves eae 0 and = 0
Ax ae

intersect. This last locus appears to be independent of the temperature,

Ba
as we may put eS equal to 0. We find from the equation on p. 638
Ax

db\* db
1—2ea dz da? |
2 =

sis ure | 1a i.
da* x? (l—2)’? (v—b)* (v—6)?
d?b a:
If we neglect aa Oe find from "5 iaae
dx 3 1-20 _
es “Yr 2 #2? (1—
13

te au ‘ : Seyeke :
rhe locus sai = 0 occurs, therefore, only in the left side of the
Az

figure or for values of « below 4. The line « =%/, is an asymptote
for this curve, and only at infinite volume this value of wz is reached.

( 729 )

: Tha |b
And as for =O also r—h must be = 0, the curve Ta = 0 starts
ax
; : : : ah
from the same point from which all the q-lines start. If — should
aL
not be equal to 0, we have ground for putting this quantity positive

(Cont. II, p. 24), and we arrive at the same result for the initial

; d*p
point and the final point of the curve —— = 0.
Y
: Mp
So the points of the curve —~=0O, where tangents may be
Av

drawn to it parallel to the v-axis,
lie certainly at values of x smal-
ler than 4, and accordingly the
two outer ones of the group of
the g-lines with maximum and
‘, minimum volume have their hori-
on Ee zontal tangents also in the left
side of the figure. The gq-line
with the highest value of g at
lower value of « than that with
the lowest value. This is repre-

sented in fig. 4.
We notice at the same time
that the points in which a g-line

dw
~= 0, are

Fig. 4. touches the curve

points of inflection for such a q-line, just as this is the case with
2

the p-lines when a p-line touches the curve os From
(LU

follows

and

3 3, 391, 3 Jw
Py dv \ =) +2 (5 tee"
dadv Fee = dz datdv \ dx da?*

dw
In the points mentioned (> 6 because eee QO, and at the
ae Lk
q

( 730 )

é d’y a) d
same time —~ = 0. Hence ie = 0, which appears also immedia-
av v
q
tely from the figure.
mae _ ap
Within the curve —— = 0 every q-line that intersects it, has also
at

a point of inflection, because the latter must pass from minimum
volume to maximum volume in its course. So there is a continuous
series of points in which q-lines possess points of inflection between
the two points in which horizontal tangents may be drawn to
dw
da?
qg-lines must possess points of inflection on the left of the curve
Pw
dx*
side turned to the a-axis just afterit has left the starting point. If

=(. But there is also a continuous series of points in which the

= 0, so with smaller value of x. For every gq-line has its convex

d ;
it is to enter the curve a = 0 in horizontal direction and to pass
&

then to smaller volume, it must turn its concave side to the z-axis
in that point, and so it must have previously possessed a point of
inflection. Most probably the last-mentioned branch is somewhere
continuously joined to the first mentioned one. If so, there is a closed

av
curve in which —— = 0 — and then it may be expected that this
Av q

closed curve contracts with rise of 7’, and has disappeared above a
certain temperature. But these and other particulars may be left to
a later investigation.

We have now described the shape of the q-lines, 1. in the case

d? a
that neither an nor =
&

is equal to O, 2. in the case that the

3 2

d
= 0 exists, 3. in the case that the curve Y Ode fonmee
dadv dx?

It remains to examine the course of the g-lines when both curves

dw d*w
= c d
ae

curve

=( exist.

2 a
oe 0 it is only required that —
da? : da?

For the occurrence of the

be positive, as we shall always suppose, and that 7’ is below the

value of the temperature at which the curve Pring] 0 has contracted

to a single point. It may, therefore, occur with every binary system,
without our having to pay attention to the choice of the components.

€ 731 )

2 ad? dp }
But the occurrence of the curve =() = 0 is not always pos-
dadv dx),

sible, as we already showed in the discussion of the shape of the
isobars. If we consult fig. 1 (These Proc. IX p. 630) it appears

d :
that the curve (Z)=0 does not exist throughout the whole width
ae},

of the diagram of isobars.

With mixtures for which the course of the isobars is, as is the case in

f ; dp
the left side of the figure, the line (Z) = 0 does not exist at all.
aa] y

Only with mixtures for which the course of the isobars is represented
by the middle part of fig. 1 it exists and if the asymptote is found,
it can occur with all kinds of volames. Also with mixtures for which
the course of the isobars is represented by the right part of fig. 1,
it exists, but then only at very small volumes, and it possesses only

the branch which approaches the line v = 6 asymptotically.
. dp
Let us now consider a mixture such that the curve (3) =O is
wt),

d?
really present at such a temperature that also the curve a 0
v

exists; then we have still to distinguish between two cases, i. e.
1. when the two loci mentioned do not intersect, and 2. when they

dp
do intersect. If they do not intersect, and the curve (2) = 0 lies
dx),

dz?

on the right of = 0, then the q-line, after having had its maxi-

mum and minimum volume, will intersect the line (2) == Ohi
v

that point of intersection will have a tangent // v-axis; it will

further run back to smaller volume, just as this is the case with one

of the q-lines drawn in fig. 2. This may e.g. occur for mixtures cor-

responding to the left region of the diagram of isobars, when this

region is so wide that also the asymptote and a further part of the

dp at. *
curve (2) = 0 is found. If with non-intersection the relative position
v

of the two curves

d
_ O and (2) = 0 are reversed, this can
B v

probably not occur but for mixtures which correspond to a region
of the diagram of isobars which has been chosen far on _ the
right side. The course of the g-lines which then pass through

the curve —-=0, is represented in fig. 5. But when the two
; a“
50
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 732 )

Fig, 5.

dw du \? dv d*w
eee Se oe ==i0.
dadv? dx ]y dax*dv dx] q da*®

ced ee and (Z)=0
dx? dz),

intersect, which then necessarily
takes place in 2 points, the shape
of the q-lines is much more intri-
cate. Then numerator and deno-
minator are equal to zero in

curves

dv ez) da? q dv :

ae 3 = an ae is not
ae q
dx).

to be determined from this equa-

dv
tion. Then (3) must be deter-
daz :

mined from :

In the discussion of the shape of the p-lines we came across an

J2 2

dw d :
analogous case when the curves == 0 and —— = 0 intersect ;
av :

dado

and there we found that the two points of intersection had a different
character. For one point of intersection the p-line has two different

dv® dvdxz?

dy d*w Pipe ye
real directions, depending on the sign of : als (<4). If
LAU

this expression was negative, the
loop-isobar passed through that
point of intersection. In the
same way, when from the above
dv
quadratic equation for i
we write the condition on which
the roots are real, we find the
condition : |

dy dw dy \? é
da* dedv? \dadv Negative;

which may be immediately found
from the condition for the loop
of the loop-isobar, as require-
ment for the loop of the /oop-
qg-line, when we interchange «x
and v,

Fig, 6.

( 733 )

The g-line which passes through the first point where the above
condition is negative, has, therefore, a real double point, and runs round
the other point of intersection before passing through this double point

Baty

d
for the second time. In Fig. 6 the dotted closed curve

ees O has
; dp Py

been traced, and also the dotted curve 2) hema iP la 0. The
AL aAvadag

point of intersection lying on the left, is the double point. According
ay . : : : ;
to what was stated before, a is negative in this point, and the
¢ wv
3

quantity is positive, which is also to be deduced from what was

dadv

dp d*w
= — ——.. So the eri-
dadv dadv?

mentioned previously about the sign of

: Say dv . ’
terion by which the reality of the two directions of (Z) is tested is
Ake
q

satisfied in that point of intersection. In the second point of inter-
d® dw

~ Is positive, and
ae dxdv

section ; 18 also positive. It is true that it

dx’? dxdv’ dvdx*

appears in the drawing of the loop-q-line that there is no other

possibility but that it runs round the second point of intersection,

and 2 it appears, that just as we have mentioned in the analogous

case for the shape of the p-lines (Footnote p. 626), only when the
wy

== 0)

dy d® i 4
does not follow from this that Ue bed >(Z ") — but 1 it

two points of intersection coincide, so when the two curves

Cw Pw Bu ( Cw
and =

2
= 0 touch, the quantity is equal to
dadv | ’ ) |

dx* dadv? dvda?

2
az?
temperature, the loop-q-line will, of course, extend still much more
to the right, and the higher q-lines must be strongly compressed at

O. In the ease that

= 0 has greater dimensions, so at lower

d
the point where the curve (Z) = 0 cuts the second axis (the line
v

ak
== 8),
This loop-q-line determines the course of all the other g-lines.

: : ; a?
Thus in fig. 6 a somewhat higher gq-line passes through = ee 0,
av

in vertical direction just above the double point, rises then till it
50*

( 734.)

passes through this curve for the second time, reaches its highest
dp

! ) — (Qin vertical direction,
Vv

point, after which it meets the curve (2
Ak

and then pursues its course downward after having been directed
horizontally twice more.
It must then again approach asymptotically that value of a, at

1
which it intersected the line f = dv = 0 shortly after the beginning
a

of its course. This line has also been drawn in fig. 6. It is evident

= 0. In fig. 6 it has, accord-

that it may not intersect the curve
F &

ingly, remained restricted to smaller volumes than those of the curve
Pw ;
dx*

—0(. For the assumption of intersection involves that a q-line

d
could meet the locus {+ dv = 0 several times. As g= MRT a in
ae SH)
such a meeting point, it follows from this that only one value of

a can belong to given g. It deserves notice that in this way without

f dw
any calculation we can state this thesis: “The curves = 0 and
AX
ap 2 .
pee —( can never intersect.” According to the equation of state
aL

it would run like this: “The equations :
db? Pa db da
a tues ee eee
dx xv _ da

1
MRT \—
zw(1—a) (v—bd)?

can have no solution in common. Indeed, if v from the second equation
is expressed in « and 7’, and if this value is substituted in the first
equation, we get the following quadratic equation in MRT:

da
ao 1 1 (db fp 1 db dx?|

daz

1 dbda 1 wa l (dar
— 2 (MRT) Se es. — | 2") =
( ie da dx b. 2 a z b? (=)

A value of MRT, which must necessarily be positive to have

( 735 )

1 db da fe da

significance, requires — — — >— —.
0 Se b? dada~ 26 da?

From the foregoing remarks it

da Pr dp
is sufficiently clear that = must be positive to render the locus | dv
AX ;

; Ma ae iw

= 0 possible, and that —- must be positive to render
3 da? dx?
ble. The roots of the given quadratic equation, however, are then

: . 1 db da La i ae ;
imaginary, the square of eS 8E aa being necessarily smaller
1 db da

b? da da’

= 0 possi-

than the square of and the square of this being smaller

1 f/da\?
than the product of rz) and the factor of (MRT).
HH

But let us return to the description of the course of the remaining
g-lines. There is, of course, a highest qg-line, which only touches the

d
locus = = 0, directed horizontally in that point of contact, and
ve

a’
for which also — = 0 in that point. There is also a g-line which
aL

d* aS
touches the locus <7 = 0 in its downmost point, and which as a
ax

rule will be another than that which touches it in its highest point.
The q-lines of higher degree than the higher of these two have again
the simple course which we have traced in fig. 2 (p. 635) for that
mes dp
q-line which intersects the locus ) = 0. Only through their con-
dx} y
siderable widening all of them will more or less evince the influence
of the existence of the above described complication. The q-lines of
lower degree than the loop-q-line have split up into two parts, one
part lying on the left which shows the normal course of a q-line
’ d :
which cuts (4) = 0; and a detached ciosed part which remains
VU }ov

enclosed within the loop. Such a closed part runs round the second

4 : , : d d*
point of intersection which ( 4 — 0 and ; cs — 0 have in common,
xv

LB / y

nie 0, and

passes in its lowest and highest point through

through (2) = 0 in the point lying most to the right and most to
Ve

the left. With continued decrease of the degree of q this detached

( 736 )

part contracts, and disappears as isolated point. This takes place
before g has descended to negative infinite, so that q-lines of very
low degree have entirely resumed the simple course which such
: dp é
lines have when only the curve ()= 0 exists.

Also in this general case for the course of the q-lines we can

form an opinion about the locus of the points of inflection of these
q2,.

. . v .
curves, so of the points in which (=) = 0. We already mentioned
st q

dp a)

above that when the line (Z) = 0 exists at a certain distance from
ays:

it there must be points of inflection on the q-lines at larger volume.

dp . ue .

If also the asymptote of (7) =O should exist, also this series ot

adr),
points of inflection of the g-lines has evidently the same asymptote.
In fig. 6 this asymptote lies outside the figure, and so it is not
represented — but the remaining part is represented, modified, however,
in its shape by the existence of the double point. The said series
of points of inflection is now sooner to be considered as consisting
of two series which meet in the double point, and which have, therefore,
: ; : : dp
got into the immediate neighbourhood of the line (3) = 0 there.
B)y
So there comes a series from the left, which as it approaches the
dp

double point, draws nearer to (2) = 0, and from the double point
at} »

there goes a series to the right, which first remains within the space
dw

in which aa 9 is found, and which passes through the lowest
ak

point of this curve, but then moves further to the side of the second
dp
component at larger volume than that of the curve {| — ]=0O. The
; daJ,

double point of the q-loop-line is, therefore, also double point for the
locus of the points of inflection of the g-lines, and the continuation
of the two branches which we mentioned above, must be found

dp
above the curve (7) — 0. Accordingly, we have there a right branch,

aL
LS. ;
which rans within me —0, and passes through the highest point of
av

this curve, and a left braneh which from the double point runs to
the left of the loop-g-line, and probably merges into the preceding

Cts¢. )
branch. If this is the ease the outmost q-lines on the two sides, both

that lying very low and that lying very high, have no points of
inflection.

THE SPINODAL CURVE AND THE PLAITPOINTS.

The spinodal curve is the locus of the points in which a p- and

dy dry
’ ; dv dv dudv dx?
a g-line meet. In these -points — =— and so — =—
dx» dig iw dw
dv* dudv

aw Pw dw 2
= dv? dx? — \ dadx
points of contact, we shall have to trace the p and the q lines
together. As appears from fig. 1 p. 630 the shape of the p-lines
is very different according as a region is chosen lying on the
left side, or in the middle or on the right side; but the course of
the g-lines in the different regions is in so far independent of the
choice of the regions that g_. always represents the series of the
possible volumes of the first component, and g4. the series of the
possible volumes of the second component, and also the line of the
limiting volumes. As the shape of the p-lines can be so very different
we shall not be able to represent the shape of the spinodal line by
a single figure. Besides the course of the p-lines depends on the

. In order to judge about the existence of such

; dp @y
existence or non-existence of the curve ag = a8 = (, and the
av ave
course of the g-lines on the existence or non-existence of the curve
dy
rm = 0, and besides, and this holds for both, on the existence of

the curve = (. Hence if for all possible cases we would illustrate

dxdv
the course of the spinodal curve in details by figures, this examination
would become too lengthy. We shall, therefore, have to restrict
ourselves, and try to discuss at least the main points.

Let us for this purpose choose in the first place a region from
the left side of the general p-figure, and let us think the temperature
so low, so below (7%),, that there are still two isolated branches for

d
P —0 all over the width of the region.
av

the curve

("738")

In fig. 7 VT is thought higher
than the temperature at which
Pw
dz
below this temperature. In fig. 7
all the g-lines have the very simple
course which we previously in-
dicated for them, and the p-lines
the well-known course, with which

= 0 vanishes, and in fig. 8

dv : See
(=) is positive on the liquid side
P

of 2 = 0, and on the vapour side
v ‘
dp :
of a negative between the
Uv

two branches of this curve, the

dv ;
transition of from positive to
dz},

Fig. 7.

negative taking place through
infinitely large. The isobars p,, p, and p, have been indicated
in the figure, in which p, <p, < p,. Also a few gq-lines, 9g, < q,
and the points of contact of p, to q, and of p, to q,. Also on
the vapour side a point of contact of p, to g,. It is clear 1st
that every q-line yields two points for the spinodal curve, and
2nd that these points of contact lie outside the region in which
dp

a is positive. On the other hand we see that the distance from
Vv

dp
? —0 ean be nowhere very large.
av

_ Only by drawing very accurately it can be made evident that on
the vapour side the spinodal curve has always a somewhat larger

the spinodal curve to the curve

; dp
volume than the vapour branch of the curve ro In the four
av

dp :
points, in which eal intersects the sides, indeed, the spinodal
av

line coincides with this curve.

Fig. 75 has been drawn to give an insight into the cireumstances
at the plaitpoint. At 7’>(7;), the two branches of the curve
dp
--==0 have united at that value of 2, for which 7=(7;),. One

of the p-lines, namely that of the value p=(przx, touches in the

( 739 )

point at which the two branches have joined at a volume v = (v)z, and
has a point of inflection there. Two
parts of q-lines have been drawn
as touching the p-line. The two
points of contact (1) and (2)
are points of the spinodal curve,
and lie again outside the curve
dp ,
—=0. For a higher p-line these
dv

points will come closer together.
And the place where they coin-
cide is the plaitpomt. As in

int (1 d?v d?v
point (1) a ees im), 8 re-

: A d?y d?v
versely in point(2)( — ] >| — },
dx q dx? }y

Bu\ dv h 1 12
Fig. 7b. Ta Fe we laa (1) and (2)

have coincided, and this may be considered as the criterion for the
plaitpoint so that in such a point the two equations:

(=) a (=)
di), da Jq
ey se (=)
dx? p dx” }q

hold.

The following remark may not be superfluous. In point (2)

and

dv dv ;
— ] is not only smaller than {| — ]}, but even negative. In order to
dx* Jp dx* J q

find the plaitpoint, the point in which 2 points of contact for the g and

dv d*v
the p-lines coincide, and so i and a have the same value,
Lv av
p q

d*v es Nee , Soa
(= must first reverse its sign in the point (2) with increase of
av

dv
the value of p for the isobar before the equality with =) can
Ce gq
be obtained. And that, at least in this case, this reversal of sign
must take place with point (2) and not with point (1), appears from

Se d*v
the positive value of (5) So we arrive here at the already known
av #4 q

theses that in the plaitpoint the isobar surrounds the spinodal curve,
and also the binodal one.

Q
st
|
| eS
e| &
& iv
Nae
3

&
+
_

‘ —
sl
hie.
Sri oa
Slee
et
S

a
--
—
bo]

d*v
daz* etc.
A a, 7 oe p
and
F dv q 1=/ Pe da? + 1 d*v aa
= — || — Hs —— | —— AH == ) etc.
ce =) a 43 x} q : 1.2238 q 33

we find for a plaitpoint :
d’v d®v
mE fas ; da? ete.
dz P dz q

1
£25
So the p- and the g-lines meet and intersect in a plaitpoint, and
this is not always changed when a point should be a double plait-
point. We shall, namely, see later on that the criterion for a double
plaitpoint is sometimes as follows:

(i), =(@)

dz) p dx )q
(3) es
dz? } py dx” }g

Let us now proceed to the discussion of the case represented by
fig. 8. Here it is assumed that 7’ lies below the temperature at
Dy ahem: Beye
which =. vanishes, so that this locus exists, it being moreover
LL

dvy — dvg —

and

d,
supposed that it intersects the curve = 0. It appears from the
v

drawing that for the gq-lines for which maximum and minimum

volume occurs, two new points of contact with the p-lines are

necessarily found in the neighbourhood of the points of largest and
smallest volumes at least for so far as these points lie on the liquid

4 _ dp
side of —=— 0.
dv

So there is a group of g-lines on which 4 points of the spinodal
curve occur, and which will therefore intersect the spinodal curve
in 4 points. The two new points of contact lie on either side of
dw : :

FI —(, and these two new points of contact do not move far away
( L
from this curve, the two old points of contact not being far removed
ee
from — = 0.

If we raise the value of g, the two new points of contact draw

in its

nearer to each other. Thus e.g. the g-line which touches

dv d*»
highest point, and for which (=) =o and also ( ;) =0 in that
dx }y dix* ]y

ot
ae

ea ——

~~ T_T

( 741 )

point has also been drawn in the figure. Also this q-line may still
be touched by two different p-lines, which, however, have not been
represented in the drawing. For a still higher g-line these points
would coincide, and in consequence of the coincidence of two points

. , ee dv
of the spinodal curve a plaitpoint would then be formed. a) always

a2z~ v

2 *

. U ° bd . .
being positive, (3); which has been negative for a long time in
q

the point lying on the left side, must first reverse its sign before it
can coincide with the point lying on the right —— a remark analogous
to that which we made for the plaitpoint that we discussed above.

If on the other hand the value of g is made to descend, the point
of contact lying most to the left will move further and further from

Jats

uw dp
: = 0, and nearer and nearer to the curve ay till
AH v

the curve

for q-lines of very low degree, for which as we shall presently see,
the number of points of contact has again descended to two, the
whole bears the character of a point of contact lying on the liquid side.

But something special may be remarked about the two inner
points of contact of the four found on the above q-line. When
the g-line descends in degree, these points will approach each
other, and they will coincide on a certain g-line. Then we have

72, 2,
again a plaitpoint. In this case neither (2) nor & need reverse
eed ae
its sign because these quantities have always the same sign for each
of the two points of contact which have not yet coincided, i.e. in
this case the positive sign. but in this case, too, there is again besides
contact, also intersection of the p- and gq-lines. On the left of this
plaitpoint the g-line les at larger volumes, on the right on the other
hand at smaller volumes than the pz-line, the latter changing its
course soon after again from one going to the right into one going
to the left.

This plaitpoint, however, is not to be realised. With the two
plaitpoints discussed above all the p-line and all the g-line, at least in
the neighbourhood of that point, lie outside the spinodal curve, and so
in the stable region. In this case they lie within the unstable region.

Summarizing what has been said about fig. 8, we see that there is a
group of g-lines which cut the spinodal curve in four points. The
outside lines of this group pass through plaitpoints. That with the
highest value of g passes through the plaitpoint that is to be realised ;
that with the lowest value of g passes through the plaitpoint that
is not to be realised. All the g-lines lying outside this group intersect

(7742)

the spinodal curve only in two points. If, however, the temperature
chosen should lie above (7%), the g-lines of still higher degree than
of that, passing through the vapour- i plaitpoint, will no longer
cut the spinodal curve.

And finally one more remark on the spinodal curve, which may

: aby Pw
occur in the case of fig. 8. By making the line —- = O and
3 dv? da*
; dy ay ,
= 0 intersect, we have a region, in which both and is ne-
dv? dx?

gative. In such a region the product of these quantities is again

d*w \?
. If this should be the

dadv
case, it takes again place in a locus which forms a closed curve.
Within this region there is then again a portion of the spinodal
curve which is quite isolated from the spinodal curve considered.

With regard to the p- and g-lines this implies, that there both

dv dv
— al
da Ms diy

a portion of a spinodal curve encloses then a portion of the y-
surface which is concave-concave seen from below. If we consider
the points lying within the spinodal curve as representing unstable
equilibria, the points within this isolated portion of the spinodal
curve are a fortiori unstable. The presence of such a portion of a
spinodal curve not being conducive to the insight of the states which
are liable to realisation, we shall devote no more attention to them.

It appears from this description
and from the drawing (fig. 8)
that in this case the spinodal curve
has a more complicated course
than it would have if the curve
Pw
dx?
a portion on the liquid side
in which it is foreed towards
smaller volumes. There is, however,
no reason to speak here of a
longitudinal plait. We might speak
of a more or less complicated
plait here. But we shall only use
the name of longitudinal plait,
when we meet with a portion
Fig. 8. that is quite detached from the

positive, and it may become equal to (

is negative; and so that contact is not impossible. Such

= 0 did not exist. It has

( 743)

ordinary plait, which portion will then on the whole run in the
direction of the v-axis.

There remains an important question to be answered: “What
happens to the spinodal curve and to the plaitpoints with increase
of temperature ?”

At the temperature somewhat higher than (7%), there exist 3
plaitpoints in the diagram. 1. The realisable one on the side of the
liquid volumes. 2. The hidden plaitpoint also on the side of the
liquid volumes. 3. The realisable vapour-liquid plaitpoint. Let us
call them successively P,, P, and P,. Now there are two possibilities,
viz. 1. that with rise of the temperature P, and P, approach each
other and coincide, and the plait has resumed its simple shape before
P, disappears at Y7=(7},),; and 2. that with rise of 7 the points
P, and P, coincide and disappear, and also in that case the plait
has resumed a simple shape. In the latter case, however, the plait-
point is to be expected at very small volumes, and so also at very
high pressure. Then, too, all heterogeneous equilibria have disappeared

at T=(T;),. Perhaps there may be still a third possibility, viz.
2

when the locus =0O would disappear at a temperature higher

dz?

than (7%), Besides the plaitpoint P, another new plaitpoint would
then make its appearance at 7’—(7%,), on the side of the first com-

ponent. This would transform the plait into an entirely closed one,
2

and only above the temperature, at which ; —0 vanishes, all he-

ak
terogeneous equilibria would have disappeared.

Let us now briefly discuss these different possibilities. We shall
restrict ourselves to the description of what happens in those cases,
and at least for the present leave the question unsettled on what
properties of the two components it depends whether one thing or
another takes place. If P, and P, coincide, the portion of the locus
dw
da?

= 0 which we have drawn in fig. 8 for smaller volumes than

that of ae must have got entirely or almost entirely within

2
the region where
dv?

is negative in consequence of the rise of tem-

d
perature, or the whole locus ae

= 0 may have disappeared with

rise of 7’.
Now at P, in the previously given equation :

( 744 )

1 dv d*v le}
I — dv, = ak
dv, dr q 1.2.3 | dx Z dx’ :

the factor of dz* is negative, but at P, this factor is positive. If
the points P, and P, coincide, this factor —(). With coincidence of
these plaitpoints, called heterogeneous plaitpoints by Kortewse, besides

Iv lv Pe Pv hey) d®v
ge — fs and sac ee =) , also (=) = (= 4
dx }» dx) 4g dx? }» dx? } g dx* }»y dx* ]g

If P, and P, coincide, ing 0 has contracted with rise of tem-

av
aw : ay :
perature. Also = —Q contracts with rise of the temperature and
ae
is displaced as a whole, as I hope to demonstrate further. But the
wy

contraction of ais 0, whose top moves to the left, happens rela-
av

tively quicker, so that e.g. the top falls within the region in which

Pw ; : : dw
is negative. The existence of the point P, requires that
dz dx},
Ow "ap
is positive. The point P, lies on the right of = 0 and above ==.
dix? dv?
iw PA: Py
If the top of = —() lies within the curve —~ = 0, neither P, nor
els, 2

P, can exist any longer. Before this relative position of the two
curves they have, therefore, already disappeared in consequence of
their coinciding. Also in this case the coincidence of heterogeneous
plaitpoints holds. At P, the factor
of dzx*® was positive, and at P,
this factor is negative. In case of

d®v d®v
coincidence ( = . With
dx* p da*® q

further rise of 7’, however, the
2

_ vy ;
top of i will have to get
av

again outside the region where
ay . a?
— is negative. The curve ——- =—0,
dz? da

namely, cannot extend to «= 0,

d’?w
and the curve = = Vat T=(hy,
av

has its top at «=O. We draw
Fig. 9. from this the conclusion, that

( 745 )

with continued increase of temperature the curves ~=0O and
av

ae =O will no longer intersect, but will assume the position indicated

by fig. 9.

The spinodal line runs round the two curves, and so in conse-

' Suhel ee
quence of the presence of = 0 it is forced to remain at an
ae”

ay d

exceedingly large distance from the curve a = 0. The question

av~

may be vaised whether the spinodal curve cannot split up into two
"yp

- = 0, the other part

separated parts, one part enclosing the curve :
“Uv
2
passing round —— The answer must then be: probably not.
av ‘
: d® Pw : ;
In the points between the two curves = and qa are indeed, posi-
av AL

dx dv

tive, but still small, whereas does not at all occur in the figure,

and will, therefore, in general, be large. Now if the temperature at which

12 f2
he = 0 disappears, should lie above (7%),, - , — 0 shifts to the left,
av

dx?
till it leaves the figure, and the spinodal curve is closed at «= 0
and 7’=(7;,),, and the new plaitpoint makes its appearance, which
we mentioned above. From this moment we have a spinodal curve
with two realisable plaitpoints. The graphical representation of the
curvature of the p- and the g-lines is in this case very difficult,
because both groups of lines have only a slight curvature. If, howe-
ver, we keep to the rule, that the p- and the q-lines envelop the
spinodal curve at realisable plaitpoints, we conclude that the value of

dx? dx?

dv Go\ . EP aga sna
and is positive in P,, and negative in the other plait-
P q

point. When these points, called homogeneous plaitpoints by Kortnwse,

te av d*v : é
coincide, {| — }= = 0. Above the temperature at which this
daz* }y du? Jy

takes place, the p- and the g-lines have no longer any point of contact. In
2

: ; d
consequence of the disappearance of the locus se fy the course
av

of the p-lines has become chiefly from left to right, so in the direc-
tion of the z-axis. On account of the disappearance of the locus

( 746 )

Pw
dz?
least with a volume which is somewhat above the limiting volume,
they run chiefly in the direction of the v-axis.

Many of the results obtained about the course of the spinodal
curve, and about the place of the plaitpoints, at which we have
arrived in the foregoing discussion by examining the way in which
the p- and the g-lines may be brought into mutual contact, may
be tested by the differential equation of the spinodal line. This
will, of course, also be serviceable when we choose another region
than that discussed as yet.

From :
Py dw ps (= i. ;
"Boomutb hig dadv
we derive:

a fy dw dw Pw ay

—(Q the course of the g-lines has also been simplified, and at

a dv +-
daz? dv? dv? da?dv dadv dadv?

+\4 wp dy dy pete dw
dx? dadv? dx? dx*dv

dx +

at ——§-

+| d*7 yp dy dw iio ay dw
dx* dv? dy? dx? | dadv dadv

We arrive at the shape of the factor of d7’ by considering that from:

de = Tdy—pdv+qdz
follows :
dp = — 4 dT — pdv+ qdz
dw dw dy
so that| — |= — y and so ——~ => — { —— 5
Gae Per: dT dv? & ) ete

This very complicated differential equation may be reduced to a
simple shape.
Let us for this purpose first consider the factor of dv. By substi-

Mwy Mwy
dadv Py dv\ . dx dv
: , and | — } for —
dx),

tuting in it the quantity fou
Dy

dw da?
dv? dv?

this factor becomes :

Py \d*wp Py (dv ay
ae (2 + : d dx +
da dadv? \ dx p da?dv

f dw ‘
From p = — — we derive:
dv

( 747)

ow (a), ew
dv* \dzx/, dadv
Cy (dv dw (dv \? d* By dv ay
S| Fe 12 —
dv* \dx? /, dv* \dz/, dv?du\ dx p dardv

from which appears that we can write the factor of dv in the form of:
wv (dv
~ Gar) Ge);
We might proceed in a similar way with regard to the factor of

dz, but we can immediately find the shape of this factor by substi-
tuting the quantity « for v and gq for p in the factor of dv. We find then:

ae) (ae)

As long as we keep 7 constant, and this is necessary for the
course of a spinodal curve, the differential equation, therefore, may

be written :
ad? 2 Pa
aa! wa d« = 0.
dx? dv? q

d? 2 d?
Gar) (a),
dv? ) \dx*/),

; : az dz \* (d?v ;
By taking into account that s oon = oS ; we obtain

after some reductions which do not call for any explanation, the

simple equation :
d?v
dx: An, dz ==) ;
Pp

dz?

and

a a first fo we derive from this equation the thesis, that

dv dv
and must have the same sign, if{ — } and
da spin jee dix? P da?

have the same sign and vice versa. Thus on the vapour side in fig. 7

dv ;
(S a), 70 * have always reversed sign, and (Z) being
ax 1)

negative, (Z) is negative on the vapour branch of the spinodal
d spin
curve. Reversely the curvatures of the p and g-lines have the same

Li di
sign on the liquid side, and (2 ") = (Z) = positive. If, however,
de} in dz), =q
51

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 748 )

d*v ; :
(= should have been indeed negative there, as was accidentally
a“

represented in point 2 of the spinodal curve, the spinodal curve runs
towards smaller volumes with increasing value of. So if there occur

points with maximum or minimum volume on the spinodal curve,
dv dv rt :
—— |= 0 in those points. If on the other hand is infinitely
dx* }q da eee

large, which occurs in the case under consideration when the spinodal
2

d?v
curve is closed on the right side for 7’> (7%), then (Ss) must be
v
p

= 0, and so the p-line must have a point of inflection in such a
point, to which we had moreover already concluded before in another
way. A great number of other results may be derived from this
differential equation of the spinodal line. We shall, however, only

Suge dv dv
eall attention to what follows. In a plaitpoint =
spin p=

. Ee
For a plaitpoint it follows from this that ( Nee (a ae
da” py

: dv
If for a point of a spinodal curve ( ) is indefinite, both
Ww / spin

dv Pv
(=) and (=) must be equal to O. This takes place in two cases :
p q

da” da
1. in a case discussed above when the whole of the spinodal line is
reduced to one single point. 2. when a spinodal line splits up into
two branches, as is the case for mixtures for which also 7), minimum
is found. In the former case the disappearing point has the properties
of an isolated point, in the second case of a double point.

In the differential equation of the spinodal line the factor of dT
may be written:

= a (= Ty dy 5 TH Pw oe ’Ty yp
tf dz? Jy dv? dadv } 7 dadv dv* } Tz dx* \

and by putting e—y for 7 it may be reduced to:

1 rt We > tw We awd

eo I gia

D) eae
dv? dz? dadv dadv dx? dv?

1 dy | de dv ae (dv dé
Le +——(2) 45
T dv? (dv? \ da p=q adv \ da p=9 da?

ld’y .
The factor by which — —
\ y Lich Tap

or to

is to be multiplied, occurs for the

( 749 )

first time in formula (4) Verslag K. A. v. W. Mei 1895, and at the
close of that communication I have written this factor in the form:

/ 2
Ade Lda ae,~ay)
v dxy 2a dz a?
from which appears that in any case when a,a, >a,,?, this factor
is negative. Here, too, I shall assume this factor to be always negative,
but I may give a fuller discussion later on.

In consequence of these reductions the differential equation of the
spinodal curve may be written as follows:

Py (Pv Pie oid d*v 0
dv? da? ),, Tea Wiles:

From this equation follows inter alia this rule concerning the
displacement of the spinodal curve with increase of 7, that on the

2y . . . .
side where ( rd ee positive, the value of v with constant value of
&
P

x, increases, and the reverse. So the two branches of a spinodal
curve approach each other with increase of the temperature. But I
shall not enter into a discussion of the further particulars which
might occur when this formula is applied. By elimination of dv I
shall only derive the differential equation of the spinodal line when
we think it given Dy a relation between p, z and 7. We find then:

Gies)°= (wal (a ale nol ** [+ CG yet)

for a plaitpoint the factor of dz disappears, and we find back the
equation (4). Verslag K. A. v. W. Mei 1895, for the plaitpoint curve.
At constant temperature we tind for the spinodal curve :

=| &
z) a (Z v

(S
3
=)

(To be continued).

( 750 )

Physics. — “The shape of the empiric isotherm for the condensation
of a bmary mixture’. By Prof. J. D. vAN DER WaAALs.

Let us imagine a molecular quantity of a binary mixture with a
mass equal to m, (l1—2) + m,z, at given temperature in a volume,
so that part of it is in the liquid state, and the remaining part in
the vapour state. Let us put the fraction which is found in the
vapour state equal to y. The point that indicates the state of that
mixture, lies then on a nodal curve which rests on the binodal
curve. Let the end of the nodal line which rests on the liquid branch
be denoted by the index 1, and the other end by the index 2. Let
us represent the molecular volume of the end 1 by v,, and the
molecular volume of the other end by v,, then when v represents
the volume of the quantity which is in heterogeneous equilibrium:

v=r,(l—y)+ry
the constant quantity 2 being represented by :
w= x, (l—y) + 4, y-
From this we find:
dv = (v,—v,) dy + (l—y) dv, + y d,
and
0 = (a,—2,) dy + (1—y) dz, + y dz,.
By elimination of dy we obtain the equation:

Ete es Jay de, ty te,)| — (1—y) dv,—y do,.
&,—ek,
; dv dv :
Now in general dv =: ( — } dw + | — | dp. Let us apply this equa-
du}, dp )y

tion for the points 1 and 2 of the binodal curve, and let us take
the course going from v, to v,-+ dv, and from v, to v, + dv, on
the surface for the homogeneous phases. Then:

dv, dv,
dv, i di, +- =—=— dp,
da 1/p dp hom

and
dv, _[{(ad,
dv, =| — ] da, + | — } dp.
dx, P dp hom
a a ‘dv, dv, ;
[he quantities | —— ] and {| —— } must then be taken along an isobar.
dx, ), dx, /,

If we substitute the values of dv, and dv, in the equation for dv,
it becomes :

a lei
Wie. aan z:), | ale ) ss
Y aati — |< |

aoe i= ae |e. ae \

dv v,—?, dz, dv,
—()=a—-» a
dp het & C1) p 5 bin dp hom
+ ie 6 ‘Sr (9 |
G— & o/ p dp bin dp hom
as da
Now the factor of = , and we find:
= din &4—2, daz," ]»,T dp i
xe (C8 ( aia) u
dp het - pt dp bin dp hom
Pao ass) @
= Yy
¢ dx, 3 pst dp bin dp hom

If we consider the beginning of the condensation, and so y= 1,
the above equation becomes:

le ats). ()
dp cae dx,” Dok dp bin dp hom

in which we must put v, =v and v,=—vw. It appears from this
dv

eon. that never Be =— =) , and that there must
dp het dp hom

dv d
therefore be a leap in the value of ee of —— at the begin-

Ip dy
ning of the condensation, unless there should be cases in which

ao dey F ple. :
ae —} is equal to 0. The only case in which this is so, is
da? dp] vin

: ie : : dp da

in the critical point of contact. There |] =o and so |—]=0
aL } bin dp bin

But then there is properly speaking no longer condensation, and the

empiric isotherm has disappeared. We might think of a plaitpoint,
2

ag Pn: dp
because a Ee in it, but on the other hand = == 0 ane

da? aX / bin

ae dz\?
=o there. If the limiting value of ; _- or of
bin dx pT dp] vin
=
dx? },,.T
&)
dx ] yin

is sought, we find by differentiating numerator and deno-

( 752 )

minator twice with respect to a:

as as ass

dx” ) »T ig dx* pT se dx* ]»,T
2 a? E T- en

Ge ROE

dx } vin dx) yin \ dx? / vin dx* ] yin

ye ad? as dg
In a plaitpoint, besides (S) also (=) = (0; but e
dx* ],,T dx* )».T dx* )y,T

will have a value differing from 0, and so there is a Jeap in the

dv
value of ite in a plaitpoint too.

dv rh dv ;
As —|{— must always be positive, also = 5 will always
PF hom het

dv dp dp
be positive and larger than = =| or - >) (- Z) :
: = dp hom dv | hom 2 dv J het

At the beginning of the condensation the empiric curve will ascend
less steeply with decrease of volume than that for homogeneous
phases. |

] dv dp F
There are eases in which —(¢) == Gor Or (- zs —— re
dp het
2

; as ie ae
on the sides, so for z=0O and «—1. Then i is infinite, and
wv fs
Pp

MRT
z(1—z)

is represented by the principal term ( ). 2. if on the binodal

af. : dp .
curve — is infinite or ——O; this takes place for those mixtures
dp da
which behave as a simple substance.
If in equation (1) we put yO and v, =v and x, = « we could
derive the same conclusions for the end of the condensation.
: ; dp dp nee
The relation between — {| — and — {| — at the beginning
QU / het av Lhom
and at the end of the condensation, could be immediately derived
by applying the equation :

; dv ain dv ;

Co == — tp - = Ae

dp an a dx p

both for the surface of the homogeneous phases and for that of the

; dv

heterogeneous phases. If we then take into consideration that (=
v
p

Us—— 0
. : 2 1 .
on the heterogeneous surface is equal to ———, we find:
: ct, -~ &
2 1

s—?; dv dv dv
—— d« + | — dp =| — |de#+ ([{— dp,
&,— 2, dp het da p dp hom

and from this the former relation.

dv :
From the form for (- — ]} in general, so not only at the begin-
dp het
ning or at the end of the condensation, we see that the empiric
isotherin can have an element in which it has an horizontal direction

only when a nodal curve is intersected, at one or the other of

whose ends (=) is infinitely large. But as neither the sides nor
dp bin
the nodal curve which runs parallel to the v-axis can be intersected,
it would follow that the empiric isotherm can never run horizontally
in one of its elements. There are, however, cases which form
exceptions to this rule. First of all if we widen the idea empiric
isotherm, and understand by it the section of a surface // v-axis
with the derived surface of the y-surface, also in the case of a
complex plait. Then there are also nodal curves to points in which
the binodal curve passes through the spinodal, and where therefore

dx
. is oo. But as such equilibria are hidden equilibria, they cannot
PS bin

be realised in spite of this. Instead of this we have rectilinear inter-
section of the surface // v-axis with the three phase triangle, and in
; du\ See
this part | — } is, of course, again infinitely large. But secondly,
yp het
and this is a case which might, indeed, be realised, the binodal

d. v

curve has a point in which ( =o, when this point is a plait-
\ dp bin
point which with increasing or decreasing temperature will become
a hidden plaitpoint. This is a limiting form of the first mentioned
case, in which the three phase triangle was intersected. Then the
three phase triangle has contracted to a single line, and the above
mentioned straight dine has contracted to a single point. Then there
is, of course, a point of inflection of the empiric isotherm in that
point. With larger volumes it is curved negatively, with smaller
volumes positively.

Physics. — “Jsotherms of di-atomic gases and their binary miatures.
VI. Lsotherms of hydrogen between —104° C. and — 217° C.”
By Prof. H. Kamertincn Onnes and C. Braax. Comm. N°. 974
from the Physical Laboratory at Leiden.

(Communicated in the meeting of December 29th 1906).

§ 1. Introduction.

The investigation treated in this Communication forms part of the
investigation on the equation of state of hydrogen, which has been
in progress at Leiden for many years. *)

With that part of our measurements *) which we now deem fit for
publication, we have more directly carried on the work that H. H.
Francis HynpMAN had already done with one of us (K. O.) before
1904, so that, though all the observations, one for this, another for
that reason, but always for the purpose of reaching the desired
accuracy (which, we may add, was increased in the course of the
investigation) have been repeated, an important share of the final
success of the measurements is due to the said investigator.

The results obtained by us furnish data for applying the correction
of the readings of the hydrogen thermometer to the absolute scale
experimentally (see the following communication), and for determining
the deviation between the net of isotherms of hydrogen and that of
the mean reduced equation of state (see Comm. N°. 71, June 1901
and Comm. N°. 74, Arch. Néerl. 1901) *). The points determined in

1) In Comm. N’. 69 (March 1901), where the apparatus have been described which
were used in this investigation, the Communications referring to this subject, have
been mentioned. Since then the isotherm for 20°C. to 60 atms. was given in
Comm. N°. 70 (May and June 1901) with the accuracy of which the open standard-
manometer (Comm. N'. 44 Oct. 1898) and the closed standard manometers (Comm.
N°. 50 June 1899) admit, which investigation is carried on in Comm. N°, 78
(March 1902) for the isotherms of hydrogen at 20° C. and 0°C., which have been
determined with the apparatus that have also been used for this investigation.
The suitability of these apparatus for accurate determinations of isotherms has
been shown in Comm. N'. 78, and is confirmed by this Communication for low
temperatures. Several communications e.g. Nos, 83, 84, 94 and 94/, further 85
and 95%, finally Noes. 89, 93 and 95 are more or less in connection with this
investigation, the great importance of which, if accurately carried out, is demon-
strated in Suppl. N°. 9.

2) We soon hope to publish the results of measurements at higher pressures
and lower temperatures, and also those of supplementary determinations at lower
pressures. ‘.

) Definitive values for the virial coefficients B and C (§12 contains only provi-
sional values) from which the difference with those according to the reduced
equation may follow, are given in the following communication,

( 755 )

the net of isotherms are only few in number, but these few points
have been determined with particular care, so that, so to say, they
form normal places in the examined region of the equation of state,
with which without preliminary adjustment we may set about the
calculation of individual virial coefficients. Characteristic of them
is that every group of such normal places obtained by deter-
minations with the piezometer and manometer (see Comm. N°. 69
and 78) lies really on the same isotherm (that of about — 104°,
— 136°, — 183°, —195°, —205°, -—213° and —217°), and that on
these same isotherms every time a point has been obtained at small
density by a determination with the hydrogen thermometer (see
Comm. N°. 95¢ Oct. 1906). The great difficulty *) of this investigation
lies in obtaining the required constancy and stability of the low
temperatures. Accordingly the arrangement of reliable cryostats was
made a separate subject of investigation at Leiden. (cf. Comm.
Nes. 83 and 94).

This investigation comprises three series of piezometer-determina-
tions at densities respectively about 70, 160, and 300 times the
normal.’) Several of the observations mentioned here lie in the
neighbourhood of the curve of the minima of pv. They enable us
to determine the shape of this curve pretty accurately (see § 13).

We confine ourselves in this communication to our observations
themselves. A discussion of them, also in connection with the results
of other observers, will be given in a following communication.

§ 2. Survey of the apparatus used.

a. On Pl. I in fig. 1 we find a schematic *) representation of the
system of the apparatus for measurements and auxiliary arrangements,
excepted those which serve for keeping constant the temperature
in the cryostat. The compression apparatus A is the same as that
mentioned in Comm. N°. 84 (March ’03). For the meaning of the
system of tubes, cocks and other parts we may refer to Comm.
N°. 69 and N°. 84. The same figures have been used, except that
in this communication c, is used for the cock which admits the

1) Witkowski, whose important investigation on the expansion of hydrogen
(Bulletin de l’Académie des Sciences de Cracovie 1905) had already appeared
before the experiments mentioned in this communication had been completed,
already mentions this as an explanation for the fact, that he has dropped the
direct determination of isotherms at temperatures lower than — 147°.

*) The limits are chiefly given by the pressure under which the gas stands ;
they are about 20 and 60 atms.

3) The individual apparatus are represented on the same scale, the connections
schematically,

( 756 )

compressed air, and c, for the cock which shuts off the level glass.
Of the compression tube, provided with the system of cocks, mercury-
reservoir and level-glass belonging to it a front-view is given in
fig. 3 of Pl. I. The piezometer with the connections g, and g, has
been represented in detail in fig. 2 of Pl. I.

The arrangement of the cryostat 6 which has served for the deter-
minations mentioned in this communication, is described in Comm.
N°. 944.

For the description of the apparatus serving for keeping the tem-
perature in the cryostat constant, we may also refer to this last
communication. Fig. 4 of Pl. I may also serve for elucidating this
description for the special case that our piezometer is placed in the
cryostat.

The pressure is conveyed (see fig. 1, Pl. I) from the compression-
tube to the manometer along ¢,,, Cs, Cis. Cyy> Ci, and c,,. By closing
and opening c,, the differential-manometer /') may be shunted in
and out. By means of ihe cocks c, and c,, it may be shut off from
the rest of the pressure-conduit, when great differences in pressure
are brought about, or are to be feared. *)*). The apparatus are
placed in two rooms as has been indicated in the figure by a
dotted line. By closing one of the cocks c,, and c,, the two parts
may be treated as independent systems. This was done when the
manometer was compared with the open manometer connected at ¢,,.
The manometer C is the same as served for the investigations of
Comm. N°. 78. The reservoir D serves, if necessary, for eliminating
the injurious influence of small leaks, for which purpose it is placed
in ice. At c,, it can be coupled to the system. In the experiments
of this communication there was no need to use it *). The pressure
is exerted by compressed air, which enters through c, and c,, along
the drying tubes /'and G; and is regulated by blowing off along c,,.

The cocks ¢,,, ¢ sy, Cy, and c,, have analagous meaning to

?
18?

COR Gs Of, CF 6 ane -e

Cc c

20?

i a

1) This manometer, which was fcrmerly used with the open standard-mano-
meter, (see C fig. 1 Comm. N°. 44) and had now been removed to the piezo-
meter, was of great use for finding leaks.

2) A couple of mercury-receptacles, which served for receiving the mercury
that might overflow, have not been represented in the figure.

5) The system which we have so far described and which belongs to the piezo-
meter, is placed in one of the rooms of the laboratory, situated in the immediate
neighbourhood of the cryogen department. The remaining apparatus which chiefly
belong to the manometer are erected in the room with the standard-manometer.

*) The adjustment of trays of oil for the different couplings rendered the search
for leakages so easy, that an injurious leak needed never to remain,

( 757 )

All couplings of the conducting tubes in which air is to be kept
at constant pressure, have been placed (ef. the plate to Comm. N°. 944)
in trays filled with oil, according to what has been said in
Comm. N°. 94°.

6b. With regard to the means for keeping a constant temperature
in the cryostat, the system of pumps and auxiliary arrangements
for the regulation of the temperature, belonging to the circulation of
oxygen, has been represented in fig. 4 of Pl. I. For a description
we refer to Comm. N°. 94.

Some particulars about the ethylene circulation used for the deter-
minations of Series I, are to be found in Comm. N°. 94/ XIII § 1.

§ 3. The manometer.

The pressure measurements were performed by means of the closed
auxiliary manometer described in Comm. N°. 78°. As a comparison
of this manometer made in 1904 with the standard manometer A IV
(of Comm. N°. 78 §17), yielded an unsatisfactory result, and led us
to expect that the auxiliary manometer was no longer- reliable, it
was compared at four points with the open standard manometer, to
which the improvements mentioned in Comm. N°. 94° were applied °).
The results of this comparison have been combined in the subjoined
table.

Column C like column C' of table XVII of Comm. N°. 78° repre-
sents the reading of the pressure determined with the open mano-
meter (Comm. N°. 44). Every value is the mean of two observations.
Column F' gives the pressure read by means of our closed auxiliary
manometer. Each of the values has been obtained as a mean from
three observations. In the calculation the calibration derived in Comm.
N°. 78° has been used. In column G the difference of the columns
F and G is represented, column H contains this same difference
expressed in the numbers of column C’ as unity. The pressure given
by the auxiliary manometer appears to be too high for all pressures
observed. It was obvious to ascribe this to a too high value assumed
for the normal volume ”).

If we take the mean of the values in column H, we find 0.00087.
If we diminish the normal-volume and so also the pressures by
this part of the original amount, the differences represented in

1) In the investigations with this manometer of Comm. N’. 70 the total absence
of leaks was rare; here, however, it was easily brought about. Also the improved
coupling of the steel capillaries to the glass-capillaries of the open manometer by

platinizing proved satisfactory. (See Comm. N°. 94%),
”) In connection with this diminution of the normal volume see also Comin.

N. 95¢, § 11.

( 758 )

column A remain between the indications of the auxiliary manometer
and the open standard manometer. These differences, considerably
smaller than those in table XVII of Comm. N°. 78’, remain within
the limits of accuracy fixed for this investigation, and justify us to

1
estimate the mean error in the pressure measurements at = anil

In the following the pressures have been calculated with this new
value of the normal volume.

TABLE I. Manometer.

24,247 24.264 + 0.017 0.00070 — 0.09017
36.290 36.333 + 0.043 0.00120 + 0.00033
47 .960 48 .004 + 0.044 0.00092 + 0.00005
| v.08 | o.cor | + 0.000 60.022 60.061 + 0.039 0.00065 — 0.00023

§ 4. The piezometers and auxiliary apparatus.

The piezometer used in the first series for the observations at a
density 70 and the temperatures —104°, —136°, —183°, —195°, was
of about the same dimensions as that used for the observations of
Comm. N°. 78. In the subjoined table, just as in the corresponding
table II of that communication, the dimensions are given to facilitate
a survey of the amount and the influence of the many corrections.

| TABLE II. Data H,, Series I.

U, = 6-4110 em? B= sa

U,=0.0530 » A = 4.0% 10
¥,— 071 198: ~ >

Vs~=6.0174 »

U, has been determined from a calibration (able

V,=576.077 cm’,

vo ==) 0.722: > Mher.cm.

1) The values of 6 given here have been determined for the ordinary temperature
and those for lower temperatures have been put equal to them. We hope soon to
determine @ also for lower temperatures.

——wSTS — ~~ 7 Os

( 759 )

The stem 6, (see Pl. Il Comm. N°. 69), on which the volume
U, is read, was 30 em. long in order to enable us to determine
every time three points on the isotherm which did not lie too near
each other‘). In others of our piezometers it was taken still longer.

For the series II and III a piezometer of larger dimensions was used.

The necessity of the use of a larger gas-volume for determinations
at densities higher than 120 times the normal has already been men-
tioned in § 19 of Comm. N°. 84. The volumenometer described there
was not used, but just as in Series J the normal-volume was deter-
mined in the piezometer itself.

As in the preceding table the dimensions of the piezometer are
given here.

TABLE III. Data H,, Series III.

U, = 5.1583 cm’. pease 7 400
U,= 0.0382 » Bp 47° ..10
¥.= 0, 1-4 «2
V{= 10.9645 »
U,= (see preceding table)
= 2063. 30 cm».

oy =) ot OL per cm:

In Series II the piezometer-reservoir had a volume of 10.343 eM?,
but for the rest it had the same dimensions as in Table III.

To detect any escaping of gas during the measurements at high
pressure in consequence of leakages at the connections gy, and gq,
(see Pl. IT Comm. N°. 69), cylindric glass oil-trays were placed
_ round these couplings (see Pl. I fig. 2) which enabled us to discover
immediately even the slightest leakage; everywhere the oil-trays
rendered excellent services, but here they were of the greatest
importance for obtaining reliable results *).

1) They may serve, inter alia, to give us information about the curvature and
the inclination of the isotherms at the middle point.

2) Once the oil-trays near the couplings g, and gy rendered good services,
when before the determinations of Series II] gas escaped in consequence of the
nut gy being imperfectly screwed on. From a determination of the normal-volume
made immediately afterwards, it proved to have changed so much that the previous
determinations had to be rejected.

( 760 )

§5. The hydrogen.

The filling was accomplished for Series HI with all the improve-
ments described in Comm. N°. 94¢ § 2. For the first series the
purification by means of cooling in liquid air was not yet applied,
in the second series it was, but without application of high pressure.

§ 6. The temperatures.

The temperatures ¢, and ¢, respectively of the divided stem 0, and
the steel capillary g, (see Comm. N°. 69 Pl. Il) were determined
in the same way as in Comm. N°. 78 § 18. In series I three ther-
mometers were placed along the steel capillary, and one at the part
of the glass capillary 7, that remained outside the cryostat. The
refrigerating action of the cryostat proving to be very slight even
in the immediate neighbourhood, only three thermometers were used
in the following two series, two at the ends and one in the middle
of the steel capillary. The influence of an error of 1° C. in the
temperature of the capillary (comp. Comm. N°. 78 § 13) is only
_—— Of the total compressed volume at — 100° land ee at —200°.
4000 10000
For the temperature of the glass capillary we assumed here that
indicated by the thermometer at the end of the steel capillary.
This simplification is the more admissible as the temperature in
the cryostat is lower, and hence the volumes outside it contain less gas.

The temperature of the glass capillary in the eryostat has been
determined in the same way as was followed in the investigations
with the hydrogen thermometer mentioned in Comm. N°. 95%. As the
arrangement of the cryostat was the same in the two cases, and the
measuring-apparatus placed in it had almost the same form, there
was no objection to start from the previously found data for the
determination of the temperature of the capillary. (see Comm. N°, 95¢
§ 4). This method gives sufficient accuracy, as, reasoning ina similar
way to that followed in the said communication, we arrive at the
result, that an error of 50° in the temperature of the part of the stem
that is taken into consideration still gives a negligible error in the

1
final result, viz. less than :
5000

The temperature ¢, of the piezometer-reservoir was determined by
means of the resistance-thermometer, which (cf. Comm. N°. 95°) had
beforehand been compared with the hydrogen-thermometer.

They differ little from those at which the calibration of the resistance-
thermometer took place. Hence the reductions are simple and may
be effected with great accuracy.

The temperatures were not calculated directly from the resistance

( 761 )

formula of comm. N°. 95 § 6, but they were based on the separate
readings of the hydrogen-thermometer, because the latter must also
serve as points of the isotherms. From the above mentioned formula
— was determined, and by the aid of this factor the reduction was
dw 2

effected.

The difference of temperature for which the reduction was made
amounting to less than 0.°3, this method of calculation is perfectly
sufficient ; only for the temperature of — 135’.71, where the difference
amounts to 4°, another correction of 0.°O1 was required.

In the subjoined table’) the method of calculation has been repre-
sented for one determination of the temperature. The first column con-
tains the observed resistance JV’, in the following column JV’, represents
the resistance at which the resistance-thermometer has been compared
with the hydrogen-thermometer (of comm. N°. 95° Table I § 6), and
T represents the corresponding reading of the hydrogen-thermometer.
From the value = and W— W, follows now the temperature-correct-
ion At which is to be added to 7; in order to give f, on the hydrogen-
thermometer-scale.

! | > ae
, dt |
W | W, | 1 i | a | At | ty
17.295 | 17.290 | — 212°839 1.750 + 0-009 | — 212.89

§ 7. The measurements.

At the beginning and the end of every series the norimal-volume
was determined in the way described in Comm. No. 78 § 12, only
with this difference that in the series mentioned here every deter-
mination of the normal-volume was supplemented with a reading in
the U-tube }, and one of the barometer. In this way two determi-
nations were generally made before and after every series. The values

found before and after every series, differ nowhere more than 000°

The tables V and VI are analogous to the tables VII and VIII of
Comm. N°. 78. In the former the results are represented referring to
1) The difference of the numbers in this table with those of the Dutch text is

due to an improved calculation. The influence of this improvement enters also in
some numbers of the last part of this communication.

( 762 )

the determination of the normal-volume for series III, the latter com-
prises the three series.

TABLE V. Normal volume H,, Series III.

|
N°. Volume. | Pressure. | Prva | Mean. Mean. Difference.
i
1 | - 1955.78 75.546 | 1915.56 — 0.17
1945.58
2 | 1955.82 735.545 1945.60 1945 .73 — 0.13
9 1961.39 75.342 1945.87 1945.87 + 0.14
10 [1962.04 75.327 1946.13 4) + 0.40]
1 [1962.31 75.317 |

1946.14 | + 0.41]
1

TABLE VI. Normal volumes H,.

1) The two last determinations have been left out of account, though they show
but slight deviations, because on account of variations of temperature in the room
a certain cause could be assigned for the apparent rise of the normal-volume.

2) A determination of the normal volume was made in this series both before
and after, and also belween the determinations at high pressure. The value given
here is the mean from these three determinations.

( 765 )

As far as the pressure permitted, three points were chosen in
every series on every isotherm for the determinations of the isotherms
ascending with about equal differences of density, which offers
advantages for the calculation of the virial coefficients (see § 12).
The readings were adjusted at these points by bringing the mercury
at the bottom, in the middle and at the top of the divided stem 4,.
By way of control the two points in the middle and at the bottom
of the stem were for some determinations determined once more
with decreasing pressure.

For every determination we waited till both the temperature and
the pressure were constant, and we could assume that the equilibrium
of temperature and pressure had been established. This will be the
case when the meniscus in the divided stem is moving up and
down within the same narrow limits. The stability of the tem-
perature was ensured by a good regulation, and that of the pressure
was easily obtained and preserved by paying attention to the oil-
trays mentioned in § 2, which immediately betrayed the slightest
leakage ‘).

When the above mentioned constant state had set in, some readings
of the piezometer and the manometer were alternately made. If they
agreed, we proceeded to the next point.

With regard to the regulation of the temperature the measure-
ments took place under the same circumstances as the investigations
with the hydrogen thermometer described in Comm. N°. 95¢ (Oct.
1906). Besides the resistance-thermometer for regulation and deter-
mination of the temperature the thermo-element was also used here
by way of control for the determination of the temperature. The
indications of the resistance-thermometer, however, proving more
reliable than those of the thermo-element, only those of the latter
apparatus were used for the calculations. All possible care was
always taken that the temperatures at which determinations were
made, lay as close as possible to those which have been used for
the calibration of the resistance-thermometer to render the corrections
small, and the accuracy of the determination of the temperature as
great as possible.

The regulation of the pressure took place according to the indication
of the metal manometer J of Pl. I, fig. 1. If we passed but slowly
from one pressure to the other the thermal process in the reservoir
which attended it, was so slight, that the regulation and the measure-

) Formerly this often required a long and sometimes fruitless search (cf. e.g.
comm. N’. 70 p. 8).
52
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 764 )

ment of the temperature did not experience any perceptible disturbance,
on account of which the stability of the bath was the more ensured
throughout a whole determination of an isotherm.

§ 8. Calculation of the observations.

The calculation is made in the same way as described in Comm.
No, Os. 8:

For the ealeulation of the variability of the volume of the piezo-
meter-reservoir U7, at low temperatures we started from the qua-
dratie formula for &£ found in Comm. N °.95°§ 1, so that 4, =
Dare eam hs == 0.0272 << 10-6. No correction for glass-
expansion was required for the volume of the glass capillary U, nor
for that of the steel capillary U/,. For the reduction of the gasvolume
in the glass capillary U, to 0° we proceeded as follows: the volume
was divided into 5 parts Uoq, U2»), U2, Vog and U2,. Ura represents the
part in the liquid bath increased by 2 em. of the capillary above
the liquid, where the temperature may still be put equal to that
of the bath. Us, Us, and Usg form together the remaining part of
the capillary in the cryostat above the bath, Us, + Us. corresponds
to the volume w', of § 5 of Comm. N°. 95¢ and Uog to u",. Use
is the volume of the capillary that is outside the cryostat. For the
reduetion of the volumes U,,, Us, and Usa we started from the
same determinations of the temperature as in Comm. N°. 95¢. As on
account of the greater density at lower temperatures the mean tem-
perature found cannot directly be used for the reduction, each of the
above mentioned volumes is divided into 3 parts, the temperature of
each of these parts is derived, and from this the mean temperature

t
| a
is determined according to the formula ¢= —jzin which T represents
Soi
a

the absolute temperature. The coefficients of expansion, which are
required for the reduction, were determined by means of the general
development into series of Comm. N°. 71 with a_ slight modifi-
cation of the coefficients mentioned there. The results obtained in
this way do not give an appreciable difference with those which
were found when the reductions are made with the approximate
results for the determinations of the isotherms obtained in this way.

With regard to the corrections of the temperature of the volumes
(7, and U, we proceeded in a way similar to that of Comm. N°. 78. For
that of the reservoir at low temperature a somewhat different way

( 765 )

was followed. As _ practically the temperature for every individual
determination of the temperature might be considered as constant
(see § 1), a number of parts of isotherms could be immediately
obtained for every series separately. As they at the same time refer
to about the same densities, an accurate value may be derived for
| ae

dt
temperatures in the different series not differing more than 0.°2, the
results could be reduced to one and the same standard-temperature
in this way without the slightest difficulty. As standard-temperatures
were assumed the temperatures —103°.57 and —135°.71 of series
I, —182°.81 and --195°.27 of series II] and —204°.70, —212°.82

r é 1
| from the graphical representation of pv, on —. The
VA VA
1

and —217°.41 of series III. In the subjoined table the values of
Pra’ PvaAr :
=n ) are given, which served for the reduction of the
ae 1
{

remaining determinations of the isotherms to these standard tem-
peratures ¢;, which relate to the hydrogen-thermometer at constant
volume and 1100 mM. pressure at 0°.

ts, :
TABLE VII. H, («) Series II. |
5}
| a eae 195° .97 —904°.83 | 7 \—219°.98
Density. ay Gy ay
| —182°.81 | —195°.97 —2049 .83
150 0.004336 0.004406 0.004501
160 0.004390 0.004458 0.004540
| 170 0.004440 | 0.004513 0.004588
| 184 0.004508 | 0.004599 | 0.004667
} |

; d(pva
From these mean coefficients values for | could be derived
"4
for the different points of the respective isotherms, which for the
isotherm of —212°.82 have been given in Table XI of § 10.

§ 9. Survey of a determination.

As an instance of determinations of isotherms at low temperatures -
we give here one of the measurements from the 3"¢ series at a density
326 times the normal, in oxygen boiling under strongly reduced
pressure.

52*

( 766 )

TABLE VIII. H, Series III N° 7. Determination in oxygen at
about — 213° C. |

| BE.
Time 3.10—3.20 | A B | C | D | E F G | H K
| | |
Piézometer top | 5.8) 56.496 | — 213°/20°.76| 19°.5
rim | 4.8] 56.360 | — 188° 19°. 4) 2156
division n® 29 7.0} 56.864 | — 98° 49°:4
— 41°
Manometer | | 5.0 |] 93.97) 19.98
| 20.04
Piézometer top 5.9 56.493 |20°..76 19°. 4 |
rim | 4.8} 56.360 21.6
19223
| Manometer 5.0 || 93.97} 20.00
. 20.05
| Float 1.3

The columns of the table agree in the main with those of table
1X of Comm. N° 782. A and B& have the same meaning, C’ denotes
the temperature ¢, of the piezometer-reservoi in the bath, and the
temperatures f2, 2, and fq in the order of the parts Uoq Us, and
U2, (see § 8) of the glass stem in the cryostat above the bath. The
temperatures given in this column are to be considered as constant
throughout the determination, and have, therefore, been mentioned
only once. D gives the temperature ¢, of the waterbath round the
stem 6,, EH the temperatures ¢,', ¢,” and ¢,'" of the thermometers
placed along the steel capillary. The temperature of the part Uo,
that projects above the cryostat is put equal to ¢#,. The columns F,
G and H have the same meaning as in the above mentioned table.
In column K the indication of the float in the cryostat has been
given. All lengths are in ems.

These readings are corrected in the same way as was followed
for table X of Comm. N°. 78°. These corrected values are given in
the following table. The two readings of the piezometer have after-
wards been united to a mean after reduction.

( 767 )

TABLE IX. H,, Series III. Determination at about —213° C.
Corrected and recalculated data.

A B | C D E F G H |. K
re |
Manometer mean 145 .5)/93.97|49.95
Temperatures f¢,, t,, t, —213°/}20°.70)}18°.6
|
top —188°
t2¢ = gs°e
tod 1— 44°
28 .634)0.135 82.3
Piézometer
28.631} 0.432
1.9

Surface of the liquid |

The columns from A to H (# included), have the same meaning
as those of table X of Comm. N°. 78’. A denotes the position of
the liquid level above the boundary of the piezometer-reservoir and
the glass stem, derived from the indications of the float.

From this the following table is obtained, which gives the ecor-
rections of volume and pressure required for the calculation of pug .
It corresponds to table XI of Comm. N°. 78.

The volumes of the parts of the glass capillary with their correc-
tions have been separately given. Moreover the corrections w', and
h have been added, the former is a consequence of the packings
being pressed down at g, and q,, the latter accounts for the weight
of the compressed air in the connecting tube between the manometer
and the piezometer. The vertical distance of the levels of the mercury
is about 0.5 meters. Instead of the mean coefficient of expansion &,
the double term: £,-++4,¢ has been assumed (cf. § 8) for the
computation of the correction 7,.

Here we must point out that as standard temperature ¢, for the
reduction of the parts into which the glass capillary is divided,
-++ 20° has been assumed, so that the differences ¢,—1+, are not very small
here. The method of interpolation applied in Comm. N°. 78 for
small values of ¢,—/f, will be used as soon as we have tables
for dz», at our disposal. The values have been directly determined
here by calculation.

( 768 )

TABLE X, H,. Series III. Corrections and
final result.

Ue = 4.4904 em*. pm = 61.004 atm.
WL = 0.0003 » i = 0.434 »
wy = 0.0014 » h =—0.004 »
us = — 0.0104 »

uw“ = 0.0018 » ws = — 0.0005 cm?.
Uo, = 0.0018 » U2 = 0.0059»
Ua = 0.0020 » Unb. == 0.0049»
Uae 0.0025 » U2 = 0.0024 »
Uo = 0.0129 » a 0.0015 »
Use = 0.0190 » U2 = 0.0001 »
wi = — 0.0239 » w1 = 0.0012 »
(pva)t, = 0.18863 t, = — 212°.82
(pv4)__9490 99 = 0.18863 for p = 61.434 atms.

The total value of the correction of the stem appears to be very
small, so that we might apply the law of Gay—Lussac down to— 217°
without introducing appreciable errors.

§ 10. Values of pra.

The values of pv4 obtained in this way for the different determi-
nations have been represented for the isotherm of —212°.82 in the
following table. The values in the last columm refer to the redue-
tion to the standard-temperature ¢, (ef. the conclusion of § 8).

The values of this table have been obtained, as appears from
table IX, by calculation with the mean values of the separate read-
ings. The deviations in these separate readings which may be due
both to oscillations of the pressure and the temperature and to errors

in the readings themselves, amount nowhere to more than 5000"

The result found for about the same point at the beginning and
at the end of one determination of an isotherm are in very good
accordance, as moreover is to be seen by comparing observation

( 769 )

TABLE XI. A. Results for the isotherm of —212°.82.

j
} .
N°. fh P pra da di pra)
dt An
tvs

|
47 Sexes 30.591 0.19406 | 157.64 0.00458
\ 1 35.426 0.19134 185.145 | 0.00473
Series II ¢
Re 33.071 0.19264 | 171.68 0.00465 |
\ 20 30.554 0.19405 | 457.46 | 0.00458
— 242° 89 51.632 0.48767 975.12
Series III
61.434 | 0.18863 | 325.68

N°’. 17 and N°. 20 of table XI. The results are reduced to the

: _ | dpe.
same standard-temperature by means of the values of | |
a “U4

given in the last column.

In table XII the results obtained in this way are also given for
the remaining isotherms. Those belonging to series | are less certain
and will be repeated.

The results obtained at the beginning and at the end of a determination
of an isotherm at about the same density have been united to a mean.

For every temperature we have added to the results of the deter-
minations of isotherms those of the readings of the hydrogen-ther-
mometer to which the former are in direct relation.

The numbers do not agree with those of the preceding table,
because some determinations have been united to a mean, for which
reason they are indicated by { ).

The points of the hydrogen-thermometer have been obtained in
the following way. From ScHaLKwik’s determinations of isotherms
follows for 20° C.

pva = 1.07258 + 0.000667 d4 + 0.00000099 d 4’.

If we suppose the mean pressure-coefficient from O° to 20° not
to deviate appreciably from the value 0.0036627 between O° and 100°,
which is permissible on account of the insignificant deviations of
the indications of the hydrogen-thermometer of constant volume from
the absolute scale, it follows from this that:

(pv) 0°, 1100 mm. — 1.000275.

The value given in Comm. N°. 60 having been taken for the

pressure-coefficient of hydrogen for the calculation of the hydrogen

Lxaey

TABLE XII, 4. Values for (pva)t,-

NW”: | ts | Pp | pva da
Hy therm. (1) | — 103°.57 0.8°6 | 0.62082 1.444
(2) 32.985 | 0 . 63467 51.971
Series I (3) 39.659 | 0.63765 | 62.493
(4) 49.897 | 0.64274 | 77.632
H, therm. (1) | — 135°.74 0.727 | 0.50307 1.445
| (2) | 98.592 | 0.51064 | 55.991
Series I
(3) 33.437 | 0.51258 | 65.234
H, therm. (1) — 182°.8 | 0.479 0.23051 1.448
| (2) 46.572 | 0.32700 |142.42
Series II
(3) 55.293 | 0.32892 [168.46
H, therm. (1) | — 195°.27 0.443 | 0.2°486 1.449
(2) 40.599 | 0.27367 {4148.35
Series II { (2) | 45.484 | 0 27337 (166.36
(4) | 49.998 | 0.27343 1182.85
Hy therm. (1) | — 204°.70 | 0 363 | 0.25031 1.449
(2) | 35.487 | 0.23189 /153.03
Series II { (3) 38 640 0.23097 | 167.30
(4) 42.438 | 0.23010 [184.43
Series III (5) | 61.917 | 0.23009 |269.40
H, therm. (1) | —- 2129.82 | 0 320 | 0.22056 1.450
(2) | 30.689 | 0.19480 |157.64
Series II { (3) | 33.200 | 0.19339 [174.68
(4) 35.566 | 0.19210 /185.45
(5) 51.632 | 0.48767 |275.42
Series III | |
(6) 61.434 | 0.18863 |325 68
H, therm. (1) | — 2479.41 0.295 | 0 20375 1.450
(2) 46.419 | 0.16381 /|283.84
| Series III (3) 52.898 | 0.16336 |323 80
| (4) 58.971 | 0.16424 |359.04

(771 )

thermometer-temperatures, the value of pva at t, now follows from
the formula
(pva)r, = (pv), (1 + 0.0036627 ¢,).

§ 11. Probable error of a determination.
The mean error in the calibration of the large volume of the piezo-

; 1
meter may be estimated at + 4000° As to the volume into which the

gas is compressed during the measurements, the greater density
of the gas in the reservoir at low temperatures may be allowed

ae!
for by reckoning only with = of the amount of the volumes at the

temperature of the room. The errors in these volumes being predo-
minant with respect to those in the volume of the piezometer reser-
voir, the mean error for measurements below —180° with piezo-

1
meters of 5 c.M* may be put equal to + Sieh of the compressed

volume in accordance with the degree of accuracy as was calculated in
Comm. N°. 69, where for measurements at the ordinary temperature

the mean error is estimated at + for piezometers of 5 c.M’.

1000
1
For —100° the mean error will be + ——.
; 2000
The mean error of the determinations of the normal volume is

aa: that of the measurements of the pressure may also be

1
estimated at + ——.
3000
In the determination of the temperature there is no appreciable
error. The observations made for one point show that the mean error
due to variations of temperature and faulty readings of the position
: : 1
of the mercury in the stem, may be put smaller than + 5000"
3)
The mean error of the determination of temperature in the stem
1
remains below + ——_.
6000
The mean error caused by all these sources of errors together

1
amounts to = ae for piezometers of 5 ¢.M.* and not very low

ih Sli ;
temperature, to + 1700 for larger piezometers and very low tempe-
( .

Care y
ure. The different points on one and the same isotherm must
show smaller discrepancies inter se than corresponds with the
said mean error. The mean error namely, for a determination, apart
from the errors in the determination of the normal volume and the
calibration of the large volume is + from ile Oh 5
2000 2400
All this does not apply to the isotherms of — 103°.57 and
—135°.71. These belonging to series I are the earliest determina-
tions and for different reasons less accurate than the later ones.

§ 12. Provisional individual virial coefficients.

If the temperatures had not been given as readings on the hydrogen-
thermometer of constant volume at 1100 mm. pressure, but on the
absolute scale, the coefficients A4, 4 etc. calculated from the equation

Ba, Ca , DA se

puaAa— Ag + —-+ Shes . 2) See
vA vA VA VA vA

with the values of pv, from table XII, could be immediately com-
pared with those derived in Comm. N°. 71'). However, this is not
the case, because the latter relate to the absolute scale of tem-
perature. From the outset it has been our purpose to derive the
correction of the hydrogen scale on the absolute scale experimentally
from our measurements themselves. This might be attained by first
neglecting the correction, and by calculating provisional values
Al4, Bau, Ca ete. for each of the isotherms, which serve then for
finding provisional corrections for the hydrogen-thermometer; after
this the calculation is repeated with the corrected temperatures, etc.,
till further repetition would not bring about any change. A similar
treatment has been applied for the determination of the corrections
of the readings of the hydrogen-thermometer to the absolute scale,
where we purposed to draw through the observations for every isotherm
a curve, which does not only correspond as closely as possible to
the observations, but also to the general equation of state. In this §

1) We must call attention to the fact that in the calculations of Comm. N°. 71
we began by taking 273°.04 by first approximation for the absolute zero-point ;
we should find the correction to this from the results of the calculations of iso-
therms, and then proceed to a second approximation. We have still retained
273°.04 in VI. 1 Suppl. N®.8 and in VI. 2 Comm. N®. 92. Since then, however,
a set of coefficients VII. 1, which will be published in the following communi-
cation, have been calculated with the further approximation for the absolute tem-
perature, viz. the more accurate value 273°.09, and corrections have, moreover,
been applied in critical quantities ete.

(773 )

the method of least squares has been applied directly to the indivi-
dual isotherms, in order to obtain a formula which represents the
observations as accurately as possible.

The number of points on each isotherm not being large enough
for all six coefficients to be determined at once, definite values were
assumed for the last three values. #4 was put =O, and values
were calculated for D4 and Hy, from the sets of coefficients VII.1 °),
which was chosen instead of V of Comm. N°. 71. This assumption
means, that a definite course was prescribed for the isotherms at
higher densities, which corresponds as closely as possible to the
law of the corresponding states. The results of these calculations are
laid down in the subjoined table. D4 and /4 are the values assumed
for the calculation according to the above.

TABLE XIII. H,. Provisional virial coefficients.

ts | A's | 103. Bla » COL OU 10? “Dia NOW Es
— 103°.57 0.62048 0.24971 0.5584 | 0.9113 -— 0.648
— 135°.71 [0.50303 0.03234 1.7974 0.7028 — 0.408]
— 182°.81 0.33063 — 0.08384 0.4024 0.3809 — 0.088
— 195°.27 0.28503 — 0.13051 0.3565 0.2892 — 0.016 |
— 204°.70 0.25058 — 0.18030 0.3710 0.2166 | 0.034 |
— 212°.82 0.22090 ie 0.22433 0.3668 0.1544 0.066 |
— 217°.41 0.20410 — 0.25013 0.3715 | 0.1122 0.082 |

It appears from the table, that the coefficients of the same column
vary regularly with the temperature, except for — 135°.71, for which
we may account by taking into consideration that the two piezo-
meter-determinations which had to be used for the calculation, lie
so close together, that a slight difference in their relative situation
already produces a large difference in B’4 and C''4.

By the aid of the coefficients the values of pv, were determined
anew according to formula (1). The divergencies for every isotherm
between the assumed values of pva, W; and the F,; calculated with
A's, B'4 and C'4 (pv4=1 for 0’ and 760 mm.), where 7 indicates
the number of that observation in table XII, have been represented
in the subjoined table.

') For the calculation of Dt and #4 the uncorrected reading of the hydrogen
thermometer was used (see preceding note).

(774 )

TABLE XIV H,. Deviation from formula (1).

10° (Wi—Roi) in "/) of Roi
ts i=tli—s Slee Ol j=afims jaslins 14
—103°.57; —1 | 47 | —9 | +8 0.001 |0.011/0.015}0.005
—435°.71
—182°.31
—195°.27/ +4 | +1 | —4} +42 %.004}0.C04/0.014!0.007
—-204°.70} —1 | +9 0 —9 |} +1 0.004|0.036|0.000)|0.036/0.004

—212°.82; —2 | +5 | +6 | —2 | —17| +-10//0.007|0.022/0.027|0.009|0.077/0.045

—217°.41| 0 0 —3 |} +2 0.001 /0.000)0.014/0.010

The isotherm of — 212.°82 is best adapted to give an idea of the
accuracy of the mutual agreement on account of its larger number
of points. The agreement proves very satisfactory. The upper limit

1
of the mean error may be put at +——_.
) P 2000

§ 138. Minima of pv.
By means of the coefficients of table XIII the following minima
of the pv-curves were derived from the data of table XII.

TABLE XV. H,. Minima of pvg.

ts Pv dA P w—R,
— 182°.81 0.32630 102.24 63.36 — 0.08
— 195° .27 0.27338 174.45 47.69 + 0.50
— 204° .70 0.22935 227.17 52.10 — 0.75
— 212°.82 0.18780 285 .55 53.63 + 0.26
— 217° .41 0.16335 315.72 51.57 + 0.08

By means of the method of least squares the coefficients of a
parabola
| i te ae BP. (pva) = i ae (pv )
have been calculated from these data’). They are;

1) It is to be remarked that the less certain isotherms of — 104° and — 186°
are not used in this deduction,

P, = — 2.623
P,= 552.610
P, = — 1354.86

The differences W—Af, between the given values of p and those
calculated with these coefficients have been represented in the last
column of tabie XV. They amount to little more than } atmosphere.
The results given in the table have been reproduced in a diagram
on Pl. If*); the curve traced there is the calculated parabola.

It follows further from the values of the coefficients, that the
parabola cuts the ordinate p= 0 in two points, where pv, is respec-
tively 0.00480 and 0.40307, from which follows with the formula?)

(pea) 7 = 0.99939 {1 + 0.0036618 (7 — 273°.09)}
for the corresponding temperatures measured on the absolute scale,
T,=1°.3 T .— 110°.2. |

The top of the parabola lies at a pressure of 53.73 atms. the
value of pv, is here 0.20394, from which follows, in connection with
the value of (“x

dt
0.0053, for the absolute temperature of the isotherm which passes
through the top that

) determined from the isotherms, viz.
p =53.73

T = 63°.5. *)

Physics. — “On the measurement of very low temperatures. XIV.
Reduction of the readings of the hydrogen thermometer of
constant volume to the absolute scale.” By Prof. H. KAMERLINGH
Onnes and C. Braax. Communication N°. 97° from the Physical
Laboratory at Leiden.

(Communicated in the meeting of Jan. 26, 1907).

§ 1. Introduction.

As it is till now difficult to obtain pure helium, and_ very
easy to obtain pure hydrogen (c.f. Comm. N°’. 947, June 1906),
the scale of the normal hydrogen thermometer (that with constant
volume under a pressure of 1000 m.M. of mercury at 0°) is for the

1) The temperatures have been given in absolute degrees below zero. The
temperatures noted down on the plate undergo slight alterations on account of a
more accurate calculation of the corrections to the absolute scale. They become
—103°.54, —135°.67, —182°.75, —195°.20, —204°.62, —212°.73 and —217°.32.

2) This value of A ve has been calculated from ScHAtKwuk’s determinations of

isotherms (cf. the conclusion of § 10).
8) Im this the corrections to the absolute scale have been taken into account.

(776 )

present, just as when it (1896) was first mentioned as the basis of
the measurement of low temperatures at Leiden in the first com-
munication (N°. 27) on this subject, still the most suitable temperature-
scale to determine low temperatures down to —259° unequivocally
with numerical values, which come nearer to the absolute scale than
those on any other scale. It is therefore of great importance to
know the corrections with which we pass from the normal hydrogen-
scale to the absolute one.

As is known they may be calculated for a certain range of tem-
peratures, when the equation of state for this region of temperature
has been determined at about normal density. Up to now we had
to be satisfied for that calculation for the hydrogen thermometer
below O° with equations of state of hydrogen obtained in a theore-
tical way. BertHELotT') derives them by means of the law of the
corresponding states from experimentally determined data of other
substances in the same region of reduced temperature. CaLLENDAR *)
modifies VAN DER WAAITS’ equation of state so as to render it adapted
to represent the results of the experiments of JouLte—Ketvin for air
and nitrogen as well as those for hydrogen between 0° and 100°,
and supposes that a same form of equation holds also for hydrogen
outside this region. Chiefly this comes to the same thing as the
application of the Jaw of the corresponding states, aibeit to a limited
group of substances. Though such theoretic corrections as have been
given by BERTHELOT and CALLENDAR are a welcome expedient to help
us in default of other data’), yet an experimental determination of
these corrections remains necessary.

We have obtained them in this research by using the isotherms
of hydrogen between —104° C. and —217° C. given in Comm. N°. 97+.

1) Sur les thermométres a gaz, Travaux et Mémoires du Bureau International, T. XIII.
2) Phil. Mag. [6] 5, 1903.

8) Wrostewski’s determinations of isotherms at the boiling point of ethylene
and oxygen are not accurate enough for this purpose. In the results found for
the last temperature this is immediately apparent from the irregular situation of
the points on the isotherm. The values obtained at the boiling-point of ethylene
give more harmonious results. And yet a correction on the absolute scale would
follow from them which has the wrong sign, viz. — 0°.07.

At the temperature of liquid air Travers has determined the difference of the
hydrogen thermometer of constant volume and constant pressure, from which we
may also derive the corrections to the absolute scale for these temperatures. It is
obvious that this derivation cannot be very trustworthy.

Further it is now possible (see § | of Comm, N° 972) to derive data on the
expansion of hydrogen at low temperatures from the determinations of WirKowsk1;
they will be discussed in a following communication.

(777)

For the calculation of these corrections at a definite temperature
we might start from the individual virial coefficients in the development
into series of the equation of state (cf. Comm. N°. 71, 1901), which
we have derived in § 12 of Communication N°. 97¢. The results
obtained in this way show really a regular course‘), in spite of the
small number of points on the isotherms.

However, we wished first to adjust the results of the separate
isotherms by general formulae of temperature. Both in this case and
in general it is very difficult to succeed in this by application of
one of the equations of state drawn up in a finite form. Very
suitable for such a purpose is the general develo;ment inte series
(or more strictly speaking, development into a polynomial), which
has already been mentioned frequently. We chose for this the
form VII. 1 (cf. the footnote to § 12 of Comm. N°. 972). The
adjustment takes place by calculating for every isotherm modifications
in B and (, AB and AC, which we eall individual AZ and ray
with an approximate value of the correction to the absolute scale,
by then representing the values of AC by a general formula of the
temperature, and by computing new values for AB by successive
approximation in such a way that the value for the correction on
the absolute scale corresponds to the assumed value of 7. Finally
also the values of 4 were represented by a general formula of
the temperature. .

If we put the new values of 6 and C’ obtained by the aid of
these corrections, which special values we denote by VII. H,. 1 in
the polynomial of state, then this represents at the same time the
determinations of isotherms of Comm. N°. 70 at 20° very satisfactorily,
and those of Comm. N°. 78 at 0° and 20° by approximation.

By means of these general expressions the reductions on the absolute
scale have been carried out.

If 6 and C are known there is another way to derive the absolute
temperature from the observations with the hydrogen thermometer,
than by applying the corrections which lead from the hydrogen
scale to the absolute temperature scale. In the calculation of the
temperature from the observations we may namely take at once into
account, that the gas in the thermometer does not follow the law of
Boyie-Cuar.es, but that pressure and volume are connected in the
way, as is indicated by the development into series with the corrected
values of 6 and C. The formula which may serve for this purpose,
is given in § 5.

1) Only the isotherm of — 135°.71 gives a deviating result. (See the conclusion
of § 12 of the preceding communication).

( 778 )

§ 2. Reduction of the readings of the hydrogen thermometer of

constant volume to the absolute scale.

If v is the volume of the gas in the thermometer, expressed in
the theoretical normal volume, p the pressure in atmospheres, 7’the
absolute temperature, the equation of state for the thermometer gas
may be written in the form:

pr=4r(1 o

ics

Vv 0)

oe +) | (2)

Further we put:
¢ the temperature on the scaie of the hydrogen thermometer of
constant volume
and
a T0 C. = A.
¢ is determined by
(Pvt (Pr)
(pe), ep
where @, represents the mean pressure-coefficient between 0° and
400° for the thermometer with the specific volume v. This is given
by (PY), 00 — (pr),
Yo ane Ree
100 (pr),
If we represent the correction on the absolute scale by:
At=60—t,
we may write for this:

(T_T) ee Bhoo— ToB's ST, kod i ne i Ss a Le
3 100 v 100 v? v v?
SS es ee aa
44 T0000 — TP’ Dy 100— Fo“ (8)
100 v 100 v2

In agreement with what may be derived from the mean equation
of state VII. 1, it appears from our determinations, that the influence
of C’r is very slight, and down to — 217° does not amount
to more than 0°.0003, so that it has not to be taken into
account. Therefore in what follows will be put C’7=0, as is also
done by BrrtHEeLot but without proof.

For the absolute zero point the value 273°.09') is assumed, from

1) From Amaaat’s experiments with the development into series of Comm N®, 71
(cf. the note to § 12 of Comm. N°. 972) 1.26> 10—5 was found for the difference
between the pressure-coefficients of nitrogen at 1000 mm. pressure and 0 mm. pres-
sure, from which follows with Cuapputs’ pressure-coefficient for 1000 mM., i.e.
0.0036744 the value 0.0036618 for the limiting value at O mM. pressure, corresponding
to the absolute zero point — 273°.09. In the same way hydrogen gives for the
difference of the pressure-coetlicients at 1090 mM. and 0 mM. 2.1 &* 10—§, which
with the pressure-coefficient 0.0036629 given in Comm. N°. 60 (see XV) gives

(779 )

which follows 47—=0.0036618 7, Tyoc, =273°.09 and 7\o9°¢. =373°.09.
For the rednetion of the data given in Comm. N°. 97¢ to the

} vA
theoretical normal volume the value — — 0.99939 was taken, borrowed
Ei

ATO 7

from the determinations of isotherms of Comm. N°. 70 (ScHALKWuJKk).
The values of 5’, and 8B’,,, have been derived from the same
determinations of isotherms’) by the aid of the pressure-coefficient
0.0036629 (see XV at the end of this Communication), neglecting
the correction to the absolute scale for 20°. These values are: *)

B', = 9.000607 B49 = 0.000664

100

The values of b'y were found from the VII. H,.1 already

2
more fully discussed in § 1, which gives in a reduced form *)

] 1 1
10° B= + 173.247 ¢ — 462.956 — Mgt AS + 384.2458 — — 4.2530—
ts t

whereas VII. 1 gives:

| fume

1
10° 3} = 157.9500 t — 305.7713 — H2 22S — 97.5686 —— 4.2530

7
a

From this the values of Bb’; have been calculated for the standard
temperatures of the isotherms.

The subjoined table contains in the first column these standard
temperatures 7; measured on the scale of owr hydrogen thermometer,*)

the limiting value 0.0036608. The same value as was found above from nitrogen,
was derived by Berruetor (loc. cit.) from CHappuis’ results for nitrogen and those
for hydrogen obtained with a thermometer-reservoir of hard glass. In the same
paper he derives the value 273°.08 for the absolute zero-point for the case that
also the less concordant results found by Cuappuis for hydrogen with a platinum-
thermometer are taken into account. Afterwards (see Zeitschrift fiir Elektrochemie
N°. 34, 1904) the first mentioned value 273°.09 is again found by taking the
mean of the above values for nitrogen and hydrogen, and those which may be
derived by means of the experiments of Ketvin and Joute.

1) Compare the conclusion of §10 vf Comm. N°. 974,

*) The values found by Cuappuis are resp. 0.000579 and 0.000606.

Those of WirkowskI are 0.000616 and 0.000688.
Those derived in Comm. N°. 71 from the
observations of AMaGaT are 0.000669 and 0.000774.

3) According to Dewar, pkK=15 atms. and T7K=29° are used for the calcula-
tion, which also served for the derivation of VII. 1.

Further have been put 440=0.99939 and 44= Aao (1 + .0.0036618 2).

4) The slight differences with the value of table XII of Comm. N°. 97¢ are
due to a correction (see XV) in consequence of the application of the improved
pressure-coefficient 0.0036629 and the influence of the dead space on the deter-
minations of the temperature, which will be more fully discussed in the last part
of this communication.

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 780 )

in the second column the same temperatures measured on the absolute
scale. The two following columns contain the corresponding values
of the special 5’; and of the corrections to the absolute scale A ¢,
calculated according to formula (2) for a hydrogen-thermometer of
constant volume with 1090 mm. zero-point-pressure. The last column
gives the corrections for the normal hydrogen-thermometer.

The values for — 103°.56 and — 135°.70 are less certain than the
others (compare § 10 and § 11 of the preceding communication).

we
|
|

TABLE XVI. H,. Corrections to the absolute scale.

faa z
— 403°.56 | — 403°.54 | + 0.2892 | 0°.0214 | 0°.0196
_ 1350.70 | — 195°.67 | 0.2368 0°.0316 0.0290
_— 1920.80 | — 1899.75 | — 0.9327 09.0530 | 0°.0486 |
| — 195°.96 | — 195°.20 | — 0.4734 | 0°.0611 | 0°.0561° |
— 2049.69 | — 204°.62 | — 0.7244 | 0°.0683 0°.0697 |
— 212°.811)) — 2129.73 | — 1.0112 | 0°.0752 | 0°.0690 |

| — 217°.40 | — 2179.32 | — 1.2167 | 0°.0796 | 0° .0730 |

|
|

With very close approximation the results of the last column may
be represented by the formula:

At arc b ye ay 1 ah ti
= 4790 ~ “\i00)-* *\i00) “"\i00) rn
a = — 0.0143307
b = + 0.0066906
c= + 0.0049175
d= + 0.0027197

The greatest deviation is three units of the last decimal.
The formula gives the value Af—=O, both for t=-+ 100° and for
i= 0°, while At=-+0°.14 would follow from it for t= -— 273°.

where:

§ 3. Accuracy of the corrections.

The influences which may cause errors in the corrections, are of
two kinds.

1. Errors in the values of B'r.

2. Errors in the data which have been used in the further derivation.

\) The difference with Comm. N°. 974 remaining after the correction of the
preceding note is the consequence of an improvement applied in the calculation.

a ———

ail, .

( 781 )

The latter may be reduced to the error in B’, and the difference
of the pressure-coefficients used for the density =O and that at
0° and 1090 mM. If for the mean error in B', we compare the
valnes of 4', which may be derived from the data of Comm. Nes. 70
and 78 and from those of CHAppuIs, a mean error of + 0.000034
(about agreeing with the error per cent derived for the pv in § 411
of Comm. N°. 977) follows from their deviations inter se, which
corresponds with a mean error of + 0°.008 at — 100° and of
+ 0°.003 at — 200° for At.

We may further assume that the mean error in the pressure-
coefficients 0.0036618 and 0.0036629 amounts to one unit of the
last decimal for the first and to two units for the second, which
corresponds with a mean error in Az¢ of + 07.003 and + 0°.006 at
— 100° and of + 0°.005 and + 0°.011 at — 200°.

If we further put the mean error in 4'p equal to that of B, a
mean error in Af corresponds to this of + 0°.006 at — 100° and
of + 0°.002 at — 200°.

The total mean error in consequence of all these mean errors
together will amount to +£0°.012 for —100° and £0°.013 for —200°.

§ 4. Comparison of the results with those which have been theore-
tical'y derived.
Table XVII contains the corrections concerning the normal hydrogen

TABLE XVII. H,. Corrections to the absolute scale.

240° | | 02.0470 |} — 250° | 0°.1005 | |

OE? 0° .0925

At At
: Gore. according : | coe creda ee
values | Callendar | Berthelot | values
— 108°.56 | 0°.0196 0°.0017 — 10° | 0°.00021 | 0°.0015 |
== 135°.7) | 0°.0290 -]} 0°.0032 — 90° | 0°.00048 | | 0°.0034 |
= 182°.80 | 0°.0486 0? .0082 — 50° 0° .OO164 0°. 082 |
= 495° .96.| 0° 0561 0°.0108 — 100° 0? .0054 0° .008 : 0° .O187
— 204°.69 | 0?.0627 | 0°.0136 — 150° | 0°.0132 0°.0337 |
— 212°.81 0° .0690 0° .0168 — 200° 0°.0314 0°.96 | 0°.0593
—27°.40 0°.0730 , 02.0192 | — 240° } 0°.48 | |
l

( 782 )

thermometer. Besides the above mentioned values of A#, which were
directly found from the observation it contains the corrections deter-
mined according to the serial formula VII. 1 and those calculated
by CatLenpar and Berruetot. Moreover in the last column the
corrections, which may be calculated from the experimental values
adjusted with VII. H, according to formula (4) are given for a
comparison.

Besides the corrections derived from this investigation for the zero-
point-pressure of 1000 m.M., also the values found by BERTHELOT
and CALLENDAR are represented on the plate. The three curves have
been indicated by I, II and III in the above mentioned order. Also
II and III refer to a zero-point-pressure of 1000 m.M.

The values derived by CattenparR and Bertaenor by means of
the law of the corresponding states appear to deviate systematically
from the experimental ones. With regard to the corrections according
to VII. 1., in the derivation of which formula agreement in the
region of the equation of state (between O° and —217° for hydrogen)
treated here, was not aimed at, we may observe that a modification
is required for VII. 1 to give as good an agreement as possible also
in this region. In the first place this agreement would require that
for the calculation of VII. 1 those values were assumed for the
critical quantities of H, which follow from the data of Comm. N°. 97¢.
They are py=15 atms. and 7), = 48°. This value of 7; would
considerably increase the corrections given in table XVII according
to VEE ae

§ 5. Formula to derive the temperature directly from the obser-
vations with the gas thermometer of constant volume.

We suppose that the correction for the difference in pressure at
the mercury meniscus and the thermometer-reservoir in consequence
of the weight of the thermometer-gas is applied to Hy, and that
it is so small that it may be neglected for the small volumes.

The fundamental formula for the reduction is '):

B; C;
py =A (+ 24 -)
v v

which may also be written in the form :
(
pv=—Apr (1 ose pt fed r*) i 4 3] corn

We start from this latter formula. The equation for the gas-ther-
mometer (cf. formula (1) of § 5 of Comm. N°. 95°) becomes now :

1) Here v is expressed in the theoretical normal volume and hence AT=
= 1 + 0.0036618 §. We call the value for 0° C., at which 6=0, AZ. It is 1.

{785}

Hr ees oh fess a sat oe
A, (4B) Art OMH?) ' A (1+ BYP) |
rh =i 2: 7 1 TT T ) te ( T : 1h )
u " {
i im

Bir ie Baba t ult wu," u |

A, (1 +By) H,+ CY) H) — A,,(14+Be) H,)

This formula holds also for the carbonic acid thermometer up to
the number of decimals given by Cnappuis. In XV we shall further
discuss the deviation of the formula used by CHapputs.

With a sufficient degree of approximation the formula for the
determination of the temperature down to 0°.001 with a hydrogen
thermometer of 1100 m.M. zero point pressure and a dead space

ae! may be written in the simpler form:
Oo

A, (1+ BY) H,,) 1+.0.00366 1,

op Ee ae ae tl, 4

ae Us Us *
a 1-+. 0.00366 t,." ! 1-+.0.00366 ¢, | 1+0.00366 -|=

V ‘ ! " rs
aia ot Bot ut usu," +u, L : e (7)
ar ds Bes HT.) 1+.0.00366 «15

First an approximate value may be assumed for BY. With the
approximate value of the temperature found in this way a better

value of Br may be determined, and the correction term for the
expansion of glass calculated.
Thus we find Av, from which the value of @ follows through
Ay — AT

0.0036618

XV. Influence of the deviation from the law of BoyLE—Cuar rs
on the temperature, measured with the scale of the gas thermo-
meter of constant volume according to the observations with
this apparatus.

§ 1. When the formulae are drawn for the calculation of the
temperature on the scale of the gas thermometer of constant volume
the variation of pressure of the gas both in the thermometer-reservoir

( 784 )

and in the dead space has as yet (see e.g. Cuapputs) been generally
entered into the calculation. as if it took place at perfectly constant
density.

The error committed in this way, is so slight for the permanent
gases for small values of the dead space, that it manifests itself
only in the last of the decimals given by Cxappuis. For CHappurs’
carbonic acid thermometer, however, it attains an appreciable value
(the influence extends here to the last decimal but one), so that it
was of importance to examine in how far it is permissible to neglect
it. This appears when Cuappvis’ formula is more closely compared
with formula (6) of XIY.

The density not being constant, either in the thermometer-reservoir
nor in the dead space, on account of the fact that e.g. at low tem-
peratures gas passes from the dead space to the reservoir, and pv
as well as the pressure-coefficient varies with the density, four
approximations are applied in this treatment (two for reservoir and
two for dead space), all giving an error in the same direction.
(Adsorption is left out of account).

The errors caused by these approximations, are of the same order
of magnitude for the reservoir and the dead space, the first applying
to a large volume and a small difference of density, the second to
a small volume and a large difference of density. The correction
which is to be applied to the determination of temperature on
account of these errors, only amounts to — 0°.001 at — 100° for a
hydrogen-thermometer with 1000 mm. zero-point-pressure and a
dead space of 0.01 V,, to somewhat less for lower temperatures,
and so it may be neglected below O°.

Formula (6) differs from the preceding formula by one correc-
tion more, which is independent of the size of the dead space, and
which is the result of the variation of density in the reservoir caused
by the expansion of the glass. This error is of no importance for
the determination of the temperature by the hydrogen-thermometer,
but may exercise an appreciable influence in some cases. (ef. § 3).

The approximations mentioned have also an influence on the deter-
mination of the mean pressure-coefficient. The discussion, perfectly
analogous to that for the influence on the determination of the tem-
perature, gives -++ 0.QOOOO0O19 as correction for our thermometer,
which remains below the limit of accuracy given in Comm. N°. 60.
Hence the value 0.0036627 derived in Comm. N°. 60 for hydrogen
at 1090 mm. changes into the corrected value 0.0036629.

§ 2. We may pass from the temperatures derived in the way

Prof. H. KAMERLINGH ONNES and C. BRAAK. On the measurement of
very low temperatures. XIV. Reduction of the readings of the hnydro-
gen thermometer of constant volume to the absolute scale.

50" 5000 0200 250

Proceedings Royal Acad Amsterdam. Vol. IX.

te” Sh
ea a;
a’

v¢e

( 785 )

mentioned in Comm. N°. 95° to those on the normal hydrogen-ther-
mometer by availing ourselves of the subjoined table, in which the
corrections required for this have been given. These corrections give
an account of the variation in the assumed pressure-coefficient and
(with regard to the number of decimals given) of the influence of
the dead space.

TABLE XVIII. Corrections for the temperatures
calculated according to Comm. N®. 95¢
to those on the normal hydrogen scale.

}
er : = |

t | At | t at
iets ! | :
— 50° | + 0°.003 | — 200° | +4 00.016
| |
— 100° | + 0°.006 | — 220° | + 0°.019
— 150° | 402.010 | — 9500 | +4 0°.020
| i

By means of the fifth column of table XVI the corrections to the
absolute scale are found. Thus the tables XVI and XVIII enable us
to reduce the temperatures calculated according to Comm. N°. 95¢
and used in Comm. Nes 957, 95° and 95¢ both to the normal hydrogen
scale and to the absolute scale.

The temperatures ¢; occurring in Comm. N°. 97+, already corrected
in the first column of table XVI for the application of the corrected
pressure-coefficient 0.0036629 and the influence of the dead space,
are adjusted to the absolute scale by the corrections in the fourth
column of table XVI.

§ 3. The values found by CHappuis and Travers for the pressure-
coefficient of hydrogen (cf. the footnote to § 7 of Comm. N°. 95°)
are corrected to 0.00366266 and 0.00366297 (number of decimals
the same as given by thern).

For the fression-coefficient of carbonic acid found by CHAappuis
the correction is more considerable and amounts (because the dead
space is small here, the correction on account of the variation of
density caused by expansion of the glass is here about of the same
value as that on account of the variation of density by the dead
space) to — 0.25 >< 10-°, so that the value found by CHappuis *)
0.00372624 is corrected to 0.00372599.

1) Nouvelles Etudes, Travaux et Mémoires du Bureau International. T, XII, p. 48.

( 786 )

Physics. — “Contributions to the knowledge of the w-surface of
van per Waats. XV. The case that one component is a gas
without cohesion with molecules that have extension. Limited
miscibility of two gases.’ By Prof. H. Kamertinco Onnes and
Dr. W. H. Kezsom. Supplement N°. 15 to the communications
from the Physical Laboratory at Leiden.

(Communicated in the meeting of Februari 23, 1907).

§ 1. Introduction. In the Proceedings of Dec. ’06; p. 502.
(Comm. N°. 964) it was mentioned that the investigation of the
w-surface of binary mixtures in which the molecules of one compo-
nent have extension but do not exert any attraction, would be taken
in hand as a simpler case for a comparison with what the observations
yield concerning mixtures of He, whose molecules are almost without
cohesion. Before long we hope to give a fuller discussion of such a
w-surface '). In the meantime some results have already been obtained
in this investigation, which we shall give here.

Thus it has appeared, that at suitable temperatures, at least if the
suppositions concerning the applicability of vAN DER Waats’ equation
of state with a and 4 not depending on v and 7’ for constant 2,
mentioned in § 2 hold for these mixtures,*) two different phases
may be in equilibrium which must be both considered as gasphases.
Then the two substances which are the components of these mixtures,
are not miscible in all proportions even in the gas state. And if
certain conditions are fulfilled this may continue to be the case when
the one component is not perfectly without cohesion, but possesses
still some degree of cohesion, which, however, must be very slight.

From the considerations of van DER Waats, Contin. II p. 41 et sqq.
and p. 104, follows that the mixing of two substances in the fluid
state is brought about in consequence of the molecular motion
depending on the temperature 7’, and promoted by the mutual
attraction of the molecules of the two components determined by
the quantity a@,,, Whereas the attractions of the molecules of each
component inter se determined by a,, and a,,, Oppose the mixing.

1) Van Laan, These Proc. May ‘05, p. 38, cf. p. 39 footnote 1, treated the
projection of the plaitpoint curve on the v, x-plane for such a mixture, without,
however, further investigating the shape of the spinodal curve and of the plait.

2) The possibility of the occurrence of a longitudinal plait at temperatures above
the critical ones of both components was supposed by van perk WaAats in his
treatment of the influence of the longitudinal plait on critical phenomena. (Zittings-
versl. Kon. Akad. vy. Wetensch. Amst. Nov. 1894, p. 133), [Added in the English

translation |.

( 787 )

If the mutual attraction of the molecules of the two components a,,
is small compared with the attraction of the molecules ‘of one of the
components inter se, @,,, the appearance of complete miscibility will
be determined solely by the molecular motion, and then the tempe-
rature will have to be raised to an amount which, if some propor-
tions of the 4’s can occur then, may greatly exceed the critical
temperature of the least volatile component, 7’.,*'), and with it the
critical temperatures of all mixtures of these components. Thus from
the equation (a) of vAN DER Waars, Contin. II p. 43, follows
Tru = 1.6875 T;, for the critical temperature of complete miscibility
(vAN DER Waats l.c.) Tim, if a,,=a,,—=O0 and b,,=—b,, may be
put. At a lower temperature the two substances considered are only
partially miscible, whereas for such a temperature above 7’, there
may be coexistence of two phases which, as will be further explained
in § 3 and 4, are to be considered as gas phases.

Now it seems to follow from the nature of most of the substances
known to us, most likely from the structure of their atoms, that
b,, is also small, when a,, becomes very small; hence for a gas
without cohesion 6,, may not be put equal to 0,, of a gas with
cohesion, and as according to the equation cited of van per WaAAaLs
a small value of 4,, furthers the mixing greatly, the critical tem-
perature of complete miscibility cannot rise as high as was derived
just now. But though most likely the case mentioned just now as
example does not occur in nature, yet it is certainly conducive to
a better insight of what is to be expected for gases of exceedingly
shght cohesion.

§ 2. The shape of the spinodal curves and the form of the plait
on the w-surfauce for binary mixtures of which one component is a
gas with molecules with extension and without cohesion. In fig. 1
P].I the spinodal curves are represented for such a case. The figure
refers to the y-surface for the unity of weight of the mixtures, as
we hope to give a further discussion of such a w-surface (comp. $ 1),
also with a view to the treatment of the barotropic phenomena which
may occur for these mixtures *) in case of a suitable proportion of
the molecular volumes of the components, for which treatment the
use of the y-surface for the unity of weight readily suggests itself.
As was also mentioned in Comm. N°. 964, the conditions for

1) van DER Waatzs, in the paper cited p. 786 footnote[1], brought this in connec-
tion with the great amount of heat absorbed at the mixing of such substances.

[Added in the English translation].
2) Cf. Comm. N°. 96a (Nov. ’06), 966 (Dec. ’06) and 96¢ (Dec. 06, Febr. ’07).

( 788 )

coexistence may be studied by the aid of the y-surface for the unity
of weight in the same way as by the aid of that for the molecular
quantity; moreover it is easy to pass from the former to the latter,
which offers advantages for the treatment of many problems (ef. § 6)
if this is desired.

The equation of the spinodal curve on the y-surface for the unity
of weight of mixtures, for which Van per Waats’ equation of state
for binary mixtures with a and 4 not depending on v and 7’ for constant
a may be applied, and for which a,,=—Wa,,q,,, 6,,="/,(0,,1+4,.™)
(cf. Comm. No. 96c, Dec. 06, p. 510) may be put, *) runs:

R, R, To = 2 R, (l—2) fv a,, —},, Va? +2, elo a,, —5,,.V af.

Here R, and R, are the gas constants for the unity of weight of
the components concerned. For a,, =O this equation passes into:

hk, 22
4tw’* = (1—2z) [{3 w — (1-— 2)? 4+ : x(l — z)]
R OF a:

if we put Ty ==ti = =o. The roots of this equation in w have
1 “Ry

been determined by a graphical way for definite values of a and r.

The figure has been construed for mixtures for which #,/R, = */,,

b,,/b,, ='/, (ef. Comm. N°. 96c, Febr. ’07, p. 600, footnote 2).

With reference to Fig. 1 we point out that for 7< Tn (= 1.299 Ty)
and > 7, a spinodal curve closed on the side of the increasing v’s,
and together with it a similar plait, extends on the y-surface from
the side of the small v’s. At 7’— 7), this plait reaches the side or
the least volatile component. At lower 7’ the spinodal curve has
two distinct branches, and the plait runs in a slanting direction from
the line v= 4% to the side of the least volatile component.

Thus the investigation of mixtures with a gas without cohesion
calls attention to a plait that starts from the side of the small volumes,
and at lower temperature runs in an oblique direction to the side
of the figure, which plait can be distinguished from the transverse
and from the longitudinal plait.

The spinodal curve for t= 1.040 has a barotropic plaitpoint LP.
(see Fig. 1). For 1.299<1< 1.040 the angle with the v-axis of

I
the tangent to the plait in the plaitpoint*) 6,. >>, for 1.040<1r<1

is Bt <= The barotropic phenomena for such a plait will be further

1) The quantities @,, 22, G2, 01, Bo2, Pig, ete. relate to the unity of weight,

,%, dggu etc. to the molecular quantity.
*) Cf. Comm, N°. 960. .

( 789 )

discussed in a following communication ‘cf. N°. 96¢ Febr. 07, p. 660,
footnote 1).

In Fig. 2 the course of the plait has been schematically repre-
sented for a temperature between the barotropic plaitpoint temperature
and the critical temperature of the first component. The — - — -
curves denote the pressure curves, the —— — — curve the spinodal
curve, the continuous curve the connode. The straight line AP is
the tangent chord joining the coexisting phases A and 4, CVD is the
barotropic tangent chord (Comm. 964).

§ 3. Limited miscibility of two gases. For mixtures where as in
fig. 2 a plait giving rise to phases separated by a meniscus which
coexist in pairs, represented in the figure e.g. by A and 4, while
mixtures in intermediate concentrations are not stable, extends on
the y-surface from the side of the small v's at temperatures above
the critical temperature of the least volatile component, we shall
call not only the phase 46 a gas phase, for which it is a matter of
course, but also the other A; so the latter may be called a second
gas phase, and we may speak of equilibria between two gaseous
mictures at those temperatures. That there is every reason to do
so in the case treated in § 2 appears already from this, that the
reduced temperature of the phase A, calculated with the critical
temperature of the unsplit mixture with the concentration of A, is
so high that already through its whole character the phase must
immediately make the impression of a gas phase (so a second one).

The shape of the p-lines in fig. 2 shows further, how the two
coexisting gas phases may be obtained by isopiestic and isothermic
mixing, in which nothing would indicate a transition to the liquid
state, from the gas phases J/ and JN of the simple substances’).

We shall explain in the following § that it is really in accordance
with the distinction between gas state and liquid state for binary
mixtures in general, when we call A a second gas phase.

§ 4. Distinction between gas and liquid state for binary mixtures.
It is true that since the continuity of the gas and the liquid state
of aggregation has been ascertained, it may be said with a certain
degree of justice that it is no longer possible to draw the line between
the two states, but when in the definition of what is to be under-
stood by liquid and what by gas we wish properly to express
the difference and the continuity in the character of the hetevo-
geneous region and the homogeneous region and to preclude con-

1) Cf. footnote 1 p. 792.

( 790 )

clusions') which are irreconcilable with the most obvious conception
of phenomena, then the limits allowed for making this definition,
are very narrow.

Thus for a simple substance no other distinction will be possible
than by means of the isotherm of the critical temperature, and the
border curve (connodal curve on Gipss’ surface), which is divided
into two branches by the eritical state (plaitpoint of the connodal
curve), of which the branch with the larger volumes is to be defined
as gas branch, that with the smaller volumes as liquid branch?).
Liquid phases are only those which by isothermic expansion may
pass into such as lie on the liquid branch of the connodal curve,
and also the metastable*) phases lying between the connodal and the
spinodal curves, which may be brought on the liquid: branch of the
connodal curve by isothermic compression ‘).

For binary mixtures the consideration of the w-surface of VAN DER
Waats leads in many cases to definitions which are just as binding.

1) So Turesen’s definition, Z.S. fiir compr. und fl. Gase 1 (1897) p. 86,
according to which e.g. strongly compressed hydrogen at ordinary temperatures
would have to be called a liquid.

2) This is in harmony with the principle of continuity of phase along the
border curve according to which a change of the character of the phases on a
border curve can only occur in a critical point. For substances which at tempe-
ratures near the critical one, in states represented by points on, or in the vicinity
of that branch of the connodal curve on Grsss’s surface which connects the
liquid states at low temperatures with the plaitpoint, should be associated to mul-
tiple molecules of which the volume is greater than the volumes of the composing
molecules together, this principle would admit the possibility that on the liquid
branch of the border curve liquid phases should occur with greater volume than
the coexisting gasphase. Such simple substances would then show the barotropic
phenomenon, till now only found for binary mixtures. There is nothing known
that points im the direction, of making the existence of such simple substances
probable but there can be no more given a reason why it should be impossible.
{Added in the translation].

3) The metastable states have not been included in Botrzmann’s definition
Gastheorie Il, p. 45.

4) We do not accept the principle of the distinction of Leaman, Ann. d. Phys.
22 (1907) p. 474: “Erst die unterhalb der betrachteten Isotherme liegenden Kurven,
welche in ihrem S-f6rmigen Teil unter die Abszissenachse hinunterreichen, ent-
sprechen walrer (tropfbarer) Fliissigkeit, d. h. eimem Zustand, der negativen Druck
zu ertragen im stande is’, as depending on the meaning that the existence of
capillary surface tension in liquids which can form drops, would presuppose that
these liquids can bear external tensile forces, i.e. negative pressures without split-
ting up (cf. ibid p. 472 in the middle, and p. 475 at the top). [Added in the
translation ].

(791)

When discussing this we shall leave out of account the case of solid
states of aggregation and three phase equilibria.

In the first place gas states are ail the states on the y-surfaces
on which tliere are no plaits. As eriterion to divide states which
belong to the stable or metastable’) region of w-surfaces which show
plaits, into gas states and liquid states, analogy with the simple
substance indicates their relation with the connodal curves of those
plaits while for the metastable states the help of spinodal curves is
to be called in.

For this first of all the distinction between the two branches of
the connodal curve of a plait is required. For in the first place
we shall have to give the same name to each of the two branches
of a connodal curve separated by one or two plaitpoints throughout
its length *).

Now, on account of the existence of the barotropic phenomenon we
cannot simply call gas branch of the connodal curve that at which
one of the isopiestically connected states has the smallest density *).
It is therefore the question to indicate if possible on each branch a state
whose nature is already known through the definition holding for
simple substances or for those which behave as such when splitting
up into .two phases. In this different cases are to be distinguished.

For the case that the considered plait *) extends from one of the
side planes «=O or «=1 over the w-surface, follows from the
definition of gas phase and liquid phase of a single substance that
the branch of the connodal curve from the gas state of the pure
substance to the plaitpoint is to be called gas branch, and also that
the branch from the liquid phase of the simple substance to the
plaitpoint is to be called liquid branch. The gas branch and the
liquid branch of the spinodal curve may be distinguished in the
same way as those of the connodal curve.

Let us restrict ourselves for the present to the distinction of gas
and liquid in this case. In the first place we make use for this
purpose of the isomignic (Comm. N°. 96) compression and expansion.

1) It follows from the nature of the case that unstable states have not to be
considered here.

2) Cf. p. 790 footnote [2].

8) Even it if we wish to leave gravity out of account, and pay only attention to the
molecular volume of the phase, the barotropic phenomena have yet called attention
to the possibility that we may find the gas volume first larger and then smaller
than the liquid volume when passing along the same connodal curve.

4) The case of the two plaits at minimum crilical temperature is comprised
in this.

( 792 )

Every phase which cannot be brought on the connodal curve
through this operation, or if it can, comes on the gas branch, will
have to be called a gas phase, every phase which is made to lie on
the liquid branch through isomignic expansion is a liquid phase.
Besides the phases lying between the connodal and the spinodal
curve which isomignically may be brought on the liquid branch of
the connodal are metastable liquid phases.

Besides the isothermic and isomignic compression without splitting
there is another operation already mentioned in § 3, which may
help us to form an opinion about the similarity of different phases,
viz. the isopiestic and isothermic mixing.*) With regard to this
phases which have been obtained by _ isopiestic admixing without
splitting from phases of which it has been ascertained that they are
to be called liquid phases, must be called liquid phases until in
another way, (e.g. because no splitting takes place with isomigni¢
compression and expansion) they have been proved to have passed
into gas phases. ”).

Proceeding to the case that the plait from higher temperature
appears as a closed plait on the w-surface, as long as the plaitpoint
which first comes into contact with the side with decrease of tem-
perature, has not yet come into contact, and with decrease of tem-
perature the plait has not yet reached a mixture which on splitting
behaves as a simple substance, and for which the distinction in
liquid state and gas state is therefore fixed, we shall have to con-
sider that branch of the connodal curve on the side of this plait-
point, which passes into that of the gas phase at lower temperature,
as belonging to the ordinary gas phase, whereas the branch which
passes into the liquid branch at lower temperature may be looked
upon as a second gas phase, and we are the more justified in doing
so as the temperature should lie further above the critical tempera-

1) With the continuous isothermic and isopiestic mixing of two similar phases
a and b the case may present itself (divided plait in the case of minimum crit.
temp.), that an intermediate phase c of the other kind is obtained. So in general
we cannot conclude to the similarity of ¢ from the isothermic and isopiestic mixing
of similar @ and b.

2) This criterion is particularly of application to the retrograde condensation
2nd kind. For then phases on the connodal curve between the plaitpoint and the
critical point of contact are liquid phases, phases on the p-curve through the plait-
point and phases with the same « as the critical point of contact just the transi-
tions to gas phases. The phases within the triangle bounded by these two lines
and the connodal curve are also to be considered as liquid phases.

Here we abstract from the small uncertainties which would be caused in these
delinitions when capillarity ought to be taken into account, [Added in the translation].

tures of the unsplit mixtures belonging to the phases lying on them.

Whereas in the case, that at a temperature comparatively little
lower also the other side of the y-surface is reached by the origin-
ally closed plait, the difference of the second gas phase with a liquid
phase is still not very conspicuous, this may become very clear for
the case of § 2, to which we have now got at last, that viz. with
decreasing temperature a plait comes from the side 7» = 4, on the
w-surface, and the plait appears for the first time as longitudinal
plait. Now we may again call PSDF the branch of the first gas
phase, PACE the branch of the second gas phase. It will certainly
be obvious to speak of gas phases when a// the parts of the plait are
found above the critical temperatures of the unsplit mixtures, and
we shall decidedly have to speak of two gas phases, when the
second branch of the connodal curve is intersected all over its length
by isomignic lines on which beyond this plait no splitting up occurs,
or if it is at most touched by one of them in the point » = 4. For
then it is beyond doubt that the final point of that branch must be
called a gas phase.

Possibly also phases between the isomignic line of the critical
point of contact, the line v= 4, and the second gas branch belong
to the second gas phase.

§ 5. The surface of saturation for equilibria on the gas-qasplait.
In fig. 3, 4 and 5 the sections 7’— const. of the p, 7, .x-surface of
saturation for equilibria on the gas-gasplait have been schematically
drawn for a mixture in which one component is a gas without, or
almost without cohesion, in fig. 3 and + for temperatures higher
than the critical temperature of the first component, in fig. 5 for
this last temperature.

In these figures too the division of a gas phase into two gas
phases, and the transition of a part of the gas region into the liquid
region at 7’— T;, is clearly set forth. The — — — — curve is the
locus of the plaitpoints.

In a following communication, in which the properties of the
w-surface for such mixtures will be further discussed, 7’, z-sections
ete. will be drawn of this surface of saturation. At the same time
it will then have to appear in how far retrograde unmixing of a
phase into two other phases is to be expected.

That one of these phases may be called a second gas phase,
appears in § 4.

§ 6. On the conditions which must be fuljilled that limited mis-

( 794 )

cibility of two gases may be expected. Now that it has appeared that:
on the suppositions mentioned in § 2 for mixtures in which one
component is a gas without cohesion with molecules with extension,
limited miscibility might be expected in the gas state, the question
rises whether this phenomenon is also to be expected for mixtures
with a gas of feeble cohesion. As on the said suppositions no maxi-
mum critical temp. is to be expected, this will be the case when
Trem > Ty, is found.*) We have treated this question by the aid of
the w-surface for the molecular quantity (cf. § 2). We arrive then
at the equations developed by van per Waats Contin. II p. 48.
The condition that 7. > 7%, is:

vy (l—«y)
in which 4y and «(1—vr,,) follow from the equations given by
VAN DER Waats loc. cit. We find from this?) Tin > Ty,

ee ae
boo / bia — Vaam/ aim}? > 37 bo / 5? iat -

for booy/bw 2 , He asoy/any< 0.98

1 | 0.053
a 0.0037
V4 0.00023
uy 0.000015

It appears on investigation that only for few pairs of substances
the ratios of the as and }’s*) will be able to satisfy this condition.
The still unknown relations between a and / for a same substance,
to which we alluded in § 1, and from which ensues that in general
substances with small a also possess a small 4, and that as a rule
large } goes together with large a, seem to prevent this. H e, which
with a 4 which is still not very small compared with H, possesses a
very small a, so feeble cohesion, and H,O, which taking the value _
of a into consideration, has a comparatively small 6, so a molecule
of small volume, constitute exceptions to this general rule which are
favourable for the phenomenon treated here.

If for He — He boy /boum = iS and 92M / A\y\M ‘ee (Comm.
N°. 96c, Febr. ’07. p. 660 footnote 2), Tm << 7, must be expected
on the above suppositions. Also for helium-argon and helium-oxygen
e.g. the same thing must be expected. Most likely the ratios are

1) Whether limited miscibility in the gas state may also occur if Tim < Ty,
in certain cases and at suitable temperatures, will be discussed in § 7.

2) For baaar/ baw = Ys eg. we find also 7m > Ty, for 0.125 > azeu / Gum > 0.061.
These cases will be further discussed.

3) See e.g Konnstamm, LaAnpoLT-B6RNSTEIN-MEYERHOFFER’s Physik. Chem.
Tabellen. ;

more favourable for mixtures of helium and neon’) than for those o.
helium and hydrogen.

For mixtures of helium and water the ratios for the above assumed
ay. and by. are such that for them limited miscibility in the gas
state is to be expected, if the suppositions mentioned in § 2 are to
be applied.

The coefficients of viscosity and of conduction of heat (ef. Comm.
N°’. 96c, Febr. ’07 p. 660 footnote 2) admit a value of bg. which
is still somewhat though only little higher; this might render
it possible to realise the said phenomenon perhaps also for the other
pairs of substances mentioned, especially when we bear in mind
that its appearance is not excluded for 7’... < Ty, (ef. p. 794 footnote 1).

The experimental investigation of these mixtures has been taken
in hand in the Leiden Laboratory.

(Communicated in the meeting of March 30).

§ 7. The shape of the spinodal curves and of the plaits for the
case that the molecules of one component exert some, though still
feeble attraction. With very small value of the mutual attraction
a,, of the molecules of the two components, in connection with the
feeble attraction a,, of the molecuies of one component inter se, the
spinodal curve will with decreasing temperature extend more and
“more on the y-surface as in Pl. I fig. 1 from the side of the small
v's, come into contact with the line 2=O at 7’= 7;,,, and then
cross from the line vO to the side e—0O in two isolated branches ”).
We leave here out of account what takes place at lower temperatures
when the spinodal curve approaches and reaches the side «=1 too.

To examine what shape the spinodal curve can have with greater
attraction of the most volatile component, we shall avail ourselves
of the suppositions introduced in § 2 and also applied in § 6 con-

1) Cf. Ramsay and TRAVERS, Phil. Trans. A. 197 (1901) p. 47 for data con-
cerning refractive power and critical temperature of neon.

aT xpl :
2) Here = >0O fori x=0. We see here that VerscHAFFELT’s conclusion

(These Proc. March 1906 p. 751) concerning the maximum temperature in the
plaitpoint curve for mixtures, for which the component is indicated by a point
from the region OHK (see fig. 2) must be supplemented by the possibility that
he branch of the plaitpoint curve starting from the first component, goes to infinite
pressures.
54
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 796 )

cerning the equation of state and the quantities a,, Vrand: 6,5. 10
the net of spinodal curves for a given pair of substances 2 singular
points may then occur, belonging to the spinodal curves for different
temperatures. The values of « for these are determined by the
equation :

am 3b MV aaam 261MM aumasy — 522M V am

— — a ==
l-wM bi; Varo =F 2boom aiyimasem + 3622mMV aim

For very small a,,. we find from this two singular points with
x >1, so not belonging to that part of the w-surface which can
denote phases of mixtures. Of these two singular points that for
which the lowest signs hold, passes through infinity for increasing
@,,, and then approaches the line =O on the other side of the
u-surface. This line is reached for:
Wanulaimn =—|—1+V148bsubinj=m - @
With increasing a,,/a,, the singular point, which appears to be a
double point for this region, approaches the line v= 6, which line
is reached for:
Y avu/aiim = — (1—bsom By im) + W1—b6soy/biim-+ (b22m/b11m)?=m, (8)
In this we assume bsoy < bi)y”).
So if the mutual attraction of the molecules of the most volatile
component and those of the other in connection with the attractions

inter se attains a certain value — on the assumptions made for the
calculations for m=V asm/aiim—=m, — the spinodal curve for

T = T;, will no longer touch the side in A, (ef. fig. 1 Pl. 1), but it

}) In this first invesugation of what may be expected for mixtures of helium,
with a view of forming some opinion as to the conditions under which the ex-
periments for this purpose are to be made, we put (§ 2), biom= + (bum-+ 620m )
diem = V dima (ef. Comm. Suppl. No. 8, These Proc. Sept. ’04 p. 227) in
the calculations, no data concerning a@,, and 0b,, for those mixtures bein» available
as yet. Also Van per Waats (These Proc. Febr. ’07, p. 630) assumes that as a
rule diam < (dim d22.). It will be necessary for a complete survey concerning
the different possibilities to make also other suppositions about dom (cf. VAN DER
Waats l.c., Kounstamm ibid p. 642), at the same time taking care that @ and b
are not put independent of v and 7. at least not both (cf. Van per Waats, These
Proc. Sept. ’05 p. 289) and that they may only be put quadratic functions of x
by approximation.

If also for mixtures with very small a2, diem might be < V dum ow (ef.
Kounstamm |c.). the phenomena of limited miscibility under discussion might still
be sooner expected.

*) For bom > bum the other singular point comes from side «=1 on the
v-surface for a smaller value of @22m/diim. As probably this case does not present
itself for the pairs of substances with smal] @2/a,, known to us, we shall not discuss it.

( 797 )

will have a double point there, in which the two branches of the
spinodal curve intersect each other and the line =O at an angle.
In this case the critical temperature of the least volatile component
is not changed in first approximation by small quantities of admixtures.

With greater attraction of the most volatile component — on the
suppositions mentioned for m,< m<C m, — a spinodal curve on
the w-surface will have a double point. This will lie the nearer to
the side of the small ws, the more the attraction of the most volatile
component increases. With a certain value of the attraction — m=m, —
the spinodal curve reaches the line v = 4 with a double point, with
greater attraction the spinodal curve will proceed from 2 — 0 on the
w-surface with decreasing 7’, and touch the line vx=4 at T= 7;,,,.
On the suppositions mentioned for 22m/biia < *°/,, the contact with
the line »>=6 will here take place at temperatures > 7), , for
bam /biim >**/,, at 7’< 7;,, so that in the latter case the spinodal
curve comes first into contact with the line «— 1.

In the first ease (422y/biim<c''/,,) a plait will come from «= 0
and at lower 7, whereas for larger m a branch plait directed to the
side «1 may develop: if m< m, it will be united through an
homogeneous double plaitpoint (Kortewre, Archiv. Neer/. 24 (1891)),
with a plait coming from v= 6 to a plait that crosses from one side
to the other, if m—>>m, it wili pass into such a plait by contact
with v= b.

In the second case the plait which becomes from «=O will
again united with on2 coming from v = 4 for smaller mm; for larger im
a branch plait will have developed before this union takes place
or before the spinodal curve touches the line v = #. .

The shape of the spinodal curve for these cases with always
greater attraction of the most volatile component, where we shall
have to consider three phase equilibria, need not be discussed for
the present, as they do not belong to the case of a component with
feeble attraction *).

For some values of b22\1/4::y table I gives the values dzay/a);iq = 1",
calculated from the equations (2) and (3). If we compare with this
the values of @2y/aiim for which Ti, = Ty: (§ 6) we see that they
really lie between those calculated here.

The shape of the spinodal curves for a case, in which m,<m<m,,
has been represented on plate II, for the w-surface of the unity of
weight (cf. § 2), with the relations and data assumed in § 2,
except that a,, @,, = 0.00049 (or as2y/@i141 = 0.00196).

1) Cf. moreover Van Laan, Arch. Tevter (2) 10 (1906), These Proc. Sept. "06
p. 226.

( 798 )

TABLE I.

bso /O1uM ni | ise
Vo | 0.0014 | 0.0179
a | 000134 0.000527
|
ig | 0.000011 0.000022

The plait extending on the w-surface from v = 6 for a temperature
> Ti, will have to be considered as a gas-gasplait according to
§4 (ef. § 6). Also a similar plait for 7’< 77, if the connodal curve
is not touched by an isomignie line, and is nowhere cut by an
isomignie line which intersects the connodal curve of the plait coming
from z=0',. According to § 4 we shal! be justified in considering
also the plait lying on the side of the small v’s for 7;, >T7 >Ta,1
(temperature for which the double plaitpoint considered occurs) as

gas-gasplait, if the temperature is above the critical temperatures of

the unsplit mixtures for all parts of that plait. That there can be
some reason for doing so, appears when we calculate the reduced
temperature for the double plaitpoint for some cases, e.g. for the
ratios boom/biim and the m, belonging to it, mentioned in Table I.

Putting bs/bum =n the double plaitpoint temperature is cdleter-
mined by :

a 27 (n—m*)*
Pay ee re aR) as =m

and
2 m (n+ m)?

Dy = : ¢
dpl ky 3 (1-+-m)? m (2—m) — n (1—2m)

So for the case represented on Plate II we find :
dy = 0.587, Tayt/Te, = 9.966, Tayt/Tee = 2-17.
(To be continued).

1) Here it appears that a gas-gasplait can occur also if Tkm< Tk,, and for
temperatures T’< Tk, with Thm > Tk,, (cf. p. 794 note 1 and p. 794).

(April 25, 1907).

dM. KAMERLINGH ONNES and W. H. KEESOM. Contributions ‘o the knowledge of the
u-surface of VAN DER WAALS. XV. The case that one compoaent is a gas
without cohesion with molecules that have extension. Limited miscibility of
two gases. Plate

ae O25 Oe YF 1 TAF

|
fl |
F
y
a 1 7A 2
ax Oe Yar
ayy, : ~ a
é y
: gi
e |
ze
Fig. 3 Fig. 4 Pig. 3

Proceedings Royal Acad. Amsterdam. Vol. IX.

pay:

IL 9F*I1d

Wepla}suy “pVoy [ekoy ssuipaa001 dg

ox 8K

) clo re,

‘soseS omy Jo AQITIQIOSTUA poyTuT “Worse, xo eAey yey} soTN90]0UL
st quouodumos ouo yey} osvo ONL “AX ‘STIVVM UAC NVA J°

Y}iIM UOTSOyOO FNOUFIA ses 8
9 ‘WOSaay “H M PU? SANNO HONITHANV A “H

ooRjans-? oy} JO 9SpoTmMouy OY} OF SUOT}NGI1} U0

KONINKLIJKE AKADEMIE VAN WETENSCHAPPEN
TE AMSTERDAM,

PROCEEDINGS OF THE MEETING

of Friday April 26, 1907.

DOG

(Translated from: Verslag van de gewone vergadering der Wis- en Natuurkundige

Afdeeling van Vrijdag 26 April 1907, Dl. XV).

COU SS) as Ne S.-

J. Boeke: “Gastrulation and the covering of the yolk in the teleostean egg”. (Communicated
by Prof. A. A. W. Husrecur), p. 800. (With 2 plates).

F. M. Jarcer: “On the influence which irradiation exerts on the electrical conductivity of
Antimonite from Japan”. (Communicated by Prof. P. Zeeman), p. 809.

B. van Tricut: “On the influence of the fins upon the form of the trunk-myotome”. (Com-
municated by Prof. G. C. J. Vosmarr), p. 814. (With one plate).

L. J. J. Muskens: “Anatomical research about cerebellar connections” (8rd Communication).
(Communicated by Prof C. WiyxktEr), p. 819.

S. L. van Oss: “Equilibrium of systems of forces and rotations in Sp4’. (Communicated by
Prof. P. H. Scuoute), p. 820.

J. D. van DER Waats: “Contribution to the theory of binary mixtures”, III, p. 826.

C. Lety: “Velocities of the current in a open Panama-canal”, p. 849. (With 2 pl.).

A. A. W. Husrecut: “On the formation of red blood-corpuscles in the placenta of the flying
maki (Galeopithecus)”, p. 873.

Proceedings Royal Acad. Amsterdam. Vol. IX.

( 800 )

Zoology. -— “On gastrulation and the covering of the yolk in the
teleostean egg.” By Dr. J. Bouxs. (Communicated by Prof.
A. A. W. Husrecar).

(Communicated in the meeiing of January 26, 1907).

1. Generally the process of gastrulation in teleosts is described
by the greater part of the embryologists as a folding in of the margin
of the blastoderm and the forming, partly by this process of folding
and partly by delamination, of a mass of cells that contains the
elements both of the chorda and mesoderm and of the entoderm.
Only Waciaw Berent, M. v. Kowatewskr (in his paper of 1885),
F. B. Sumner and myself have described a more or less independent
origin of mesoderm and chorda on one side and the entoderm on
the other side. Sumyrr called the mass of cells lying at the posterior
end of the embryo, from which the entoderm originates, prostomal
thickening; I kept the same name for them and regarded these cells
as being derived from the periblast.

The large pelagic eggs of Muraenoids, which I could collect in
large quantities at Naples, offer an extraordinarily good object for
the study of these processes, much better than the eggs of Salmonides,
studied chiefly by French and German authors’). The formation of
chorda and mesodermic plates out of the folded portion of the blas-
toderm, and of the entoderm out of the “prostomal thickening’, the
mass of cells that lie at the hind-end of the embryo and are connected
with the superficial layer and with the periblast, is clearly to be
seen from the beginning of the formation of the embryo until the
closure of the yolk-blastopore (confirmed by Sumner in his paper of
1904) and after a renewed careful study of these eggs *) I can only
confirm entirely and in full the conclusions arrived at in my former
paper *) and the observations described there at some length.

But in accordance with the new and better definition of gastrula-

1) Neither Henxecuy, nor Kopscu or JasLtonowski, to take a few examples, did
see anything of these differentiations. Sumner gives however of Salvelinus very
clear figures and descriptions. (Arch. f. Entwickelungsmech. Bd 17. 1903).

2) During the last 2 or 3 years Muraenoid-eggs seemed to have disappeared
entirely from the Gulf of Naples. Now (summer 1906) I found them again in
sufficient quantities. When comparing the different eggs with each other, it seemed
to me that they belong to a still larger number of different species than I
concluded in my former paper (9), and that there must be distinguished at least
10 different species of Muraenoid eggs in the Gulf of Naples. Dr. Sanzo at Messina
came to the same conclusion.

5) Perrus Camper, Vol. 2, page 185—210 1902,

( 801 )

tion in vertebrates, given by Husrecur and Keiser and confirmed by
a number of other embryologists, this process in the teleostean egg
too must be revised and more sharply defined.

In my former paper I was led to divide the process of gas-
trulation into two “phases”, one by which the gut-entoderm is formed
and one by which chorda and mesoderm are differentiated. But now
I think the line must be drawn still sharper and the second phase must
be separated entirely from the process of gastrulation sensu strictiori.

According to the definition given by Husprecut gastrulation is a
process by which a gut-entoderm is differentiated from an ectodermic
layer, and thus the germ consists of two distinct layers. The process
of formation of chorda and mesodermic plates, which follows directly
on the process of gastrulation proper (notogenese Husrecur) is a
secondary complication of the process, characteristic of the vertebrate
embryo.

The most primitive mode of formation of the entoderm, according
to Husrncnt, is by delamination and not by invagination. But after
all it is chiefly the outcome, the formation of the two germ-layers,
that is of interest. As soon as these two layers are formed and may
be distinctly separated from each other, the process of gastrulation
is finished.

This is for example in amphioxus already the case at that stage
of development, in which the gastrula is cap-shaped, the two layers
(ectoderm and entoderm) are lying close against each other, the
segmentation-cavity has disappeared, but the blastopore still extends
over the entire breadth of the original blastula-vesicle. All the following
processes until the closure of the blastopore (‘Riickenmund” of Huprecut)
are notogenesis and lead to the formation of the back (chorda) and
of the mesodermic plates and to the closure of the gastrula-mouth.

When we study again the teleostean gastrulation-process from this
point of view, we come to the conclusion, that in this case the
process of gastrulation is ended as soon as the prostomal thickening
has been formed, viz. at the beginning of the covering of the yolk.
At that moment the “Anlage”’ of the entoderm is clearly differentiated,
and the ectodermal cells begin to invaginate to form the chorda and
mesodermic plates ; the concentration of the cells towards the median line
begins, the long and slender embryo is formed out of the broad and
short embryonic shield. The blastula-cavity, in the cases in which it is
developed, has disappeared as such ; all the following processes, the
longitudinal growth of the embryo, the covering of the yolk by the
blastoderm ring, the closure of the yolk blastopore, belong to the
notogenesis and we are no more entitled to reckon these processes

55%

( 802 )

to gastrulation proper as we are to do that of the covering of the
yolk by the entoderm in sauropsids. During this longitudinal growth
of the embryo new cells are produced by the prostomal thickening
and pushed inwards to form the entoderm, but this may not be
called gastrulation any more. The period of development, during which
the yolk is being covered by the blastodermring, differs much in
different embryos. In muraenoids at the time the yolk-blastopore is
closed the embryo possesses from 5 to 10 pairs of primitive segments;
the issuing larvae possess 58 to 75 segments. In salmonidae at the
closure of the yolk-blastopore of the 57 to 60 segments 18 to 28
are differentiated. The other organs too are developed to a greater
or lesser degree. To use the term gastrulation for the processes
during this whole period of development leads us into difficulties.

Tue first question we have to answer, when we study closer the
process of gastrulation in teleosts, is: at what time does the process
of gastrulation begin in the large meroblastic eggs?

Recently Bracnret') has called attention to a process, which he
ealls “clivage gastruléen’’, and which he describes for the eggs of
Rana fusca as the formation of a circular groove at the base of the
segmentation-cavity around the yolk-mass, before there is to be seen
a trace of a blastopore (Rusconic groove) at the outside of the egg:
“immédiatement?) avant que la gastrulation ne commence, la cavité
-de segmentation, sphérique ou a peu pres, occupe Vhémisphere
supérieur de l’oeuf (de Rana fusca).... Bient6t, sur tout le pourtour
du plancher de la cavité de segmentation, une fente se produit par
clivage; cette fente ’s enfonce entre les cellules de la zone marginale
et les divise en deux couches: l'une, superficielle, prolonge directement
la votite de la cavité de segmentation, mais est forme par des
cellules plus volumineuses et plus elaires qu’ au pole supérieur;
autre, profonde, fait corps avee les éléments du plancher. Cest ce
clivage, que j’ai appelé “clivage gastruleen’”’, c’est lui, qui caractérise
la premiere phase de la gastrulation, parce qu’il amene, en dessous
de Véquateur de Voeuf, la formation dun feuillet enveloppant et
d’une masse cellulaire enveloppée, d’un ectoblaste et d’un endoblaste.”
And farther on: ‘lorsque ce clivage est achevé, il est clair, qu’a
sa limite inférieure, lVectoblaste et Vendoblaste se continuent lun
dans l'autre, comme le faisaient antérieurement la voiite et le plancher
de la cavité de segmentation.”

This line of continuity Bracuer calls “blastopore virtuel”; after
a short time this virtual blastopore is converted into a real blastopore

1) Archives de Biologie Tome 19 1902 and Anatom. Anzeiger. Bd. 27 1905,
*) Anat. Anzeiger Bd. 27, p. 215,

( 803 )

by the formation of the groove that leads to the formation of the
archenteric cavity. This groove is formed by delamination; until
now there is no trace of invagination. This begins in what Bracuer
calls the second phase of the gastrulation process, which leads to the
formation of the archenteric cavity in its entire width, and is
synchronic with the process of notogenesis, of the formation of the
back of the embryo; “quand les levres blastoporales se soulévent,
quand de virtuelles elles deviennent réelles, c’est que le blastopore
va commencer a se fermer, c’est que le dos de lembryon va
commencer a se former” (I.c. 1902, p. 225).

Bracuet is right here. Also there, where be draws a sharp line
between the entirely embryogenic blastoporus of the holoblastic eggs
and the blastoporus of the meroblastic eggs with a large amount of
yolk, which is divided into two parts, an embryogenic blastoporus
and a yolk-blastoporus.

But when he reckons these processes, which occur in the selachian
and teleostean egg during the covering of the large mass of yolk
and the closure of the blastopore, still to gastrulation, when he ealls
the entire process of covering of the yolk ‘“clivage gastruléen’’, and
calls the whole blastoderm ring “blastopore virtuel’, he goes too far,
and forgets the significance of the phenomena, occurring at the end
of segmentation and during the formation of the periblast.

For the answer to the question, at what time does the gastru-
lation in the teleostean egg begin, bis analysis of the phenomena
of this process in the amphibian egg is extremely interesting.

The segmentation of the teleostean eggs is not regular during all
its phases. When we combine the very accurate observations of
KopscH on this account, we see, that in the segmenting blastoderm
at a definite moment, about that of the 10" division of the embryonic
cells, there occurs an important alteration.

Until the end of the 10 ¢ell-division (in Belone) the different
cells divide wholly synchronic; in Torpedo Rickert found synchro-
nism until the 9" division. By the tenth division the yolk-sac ento-
blast is formed (in Gobius, Crenilabrus, Belone), the two nuclei of
the marginal segments, resulting from this division, remaining in the
undivided protoplasm; where this does not occur at the tenth division
the deviation is very small (in Cristiceps argentatus it partly begins
at the 9 division, in Trutta fario at the eleventh division). Syn-
chronically with the differentiation of yolk-sac entoblast the super-
ficial layer (‘‘Deckschicht’’) is differentiated. At the end of the 10%
division all at once the blastoderm alters its form: it gets higher,
more hill-shaped and the diameter is lessened; the mass of cells

( 804 )

concentrates, the superficial layer is still more clearly visible as a
definite enveloping layer of cells. It is just the synchronic differen-
tiation of the superficial layer, which shuts off the blastoderm from the
surrounding medium and is the only way by which the developing
cells may get the oxygenium from the perivitelline fluid, on one side,
and of the periblast, by means of which the blastoderm is nourished
by the yolk, on the other side, which seems to me to be important ;
by this synchronic differentiation a new phase in the developmental
process is initiated, and the series of changes have begun that lead
to gastrulation.

Very soon the blastoderm-dise flattens, at first only because the
superficial layer contracts a little, and the blastoderm sinks a little
into the yolk-sphere (fig. 8) but after that because the blastodise
itself spreads out, flattens (fig. 9). The cells come closer together,
and soon the unilateral thickening that forms the first outwardly
recognisable beginning of the building of the embryo, becomes visible.

During these changes it is of no account whether a blastula-cavity
is formed, or not. As I have described elsewhere, in different murae-
noids during this stage a distinct blastula-cavity is formed, which may
be seen in the living egg. Afterwards follows the flattening of the
blastodise and the disappearance of the cavity as such. The closer
study of young stages of the eggs of muraena N°. 7’) showed me
however, that in these eggs no blastula-cavity is formed, and that
in this case the blastoderm, that takes just the same conical shape
as the hollow blastoderm in the other muraenoid eggs, remains solid
and is built up of a mass of loosely arranged cells. The further
development is the same as in the other series (c.f. fig. 1—3 on
plate 1).

This flattening of the blastodisc, following on the stage just described,
coinciding with the concentration of the cells of the blastoderm
towards the side where in later stages the embryo is formed, and
coming before the invagination (and partial delamination) of the
blastoderm cells, that leads to the formation of the chorda and the
mesodermic plates, is already a part of the gastrulation process and .
must be compared with the ‘“clivage gastruléen”’ of the amphibian ege.

Immediately on this “clivage gastruleen’”’ follows the formation of
the prostomal thickening (that is the ‘‘vlastopore réel” of BracHer),
there where the superficial layer or pavement layer is connected
with the periblast, out of the surperticial cells of the periblast *) (e-f.

1) Comp. Perrus Camper, Vol. Il p. 150.

2) Sumner (1. c. page 145) saw evidences for this mode of origin in the egg of
Salvelinus, but not in that of Noturus or Schilbeodes. On these two forms I| can-

( 805 )

fig. 4 5 and 6 on plate 1). It seems probable, that at least in some
cases entodermcells are formed by delamination from the periblast at
some distance from the surface in front of the prostomal thickening
(fig. 5e). So here, as in many vertebrates, the entoderm is formed
by delamination. At the moment of the differentiation of the pros-
tomal thickening (figs. 2, 4), there is still no trace of the invagination
of the mesodermeells, only a thickening of the mass of cells lying
just overhead of the cells of the prostomal thickening. Immediately
afterwards however a distinct differentiation of the mesoderm becomes
visible. At that stage the notogenesis begins and the gastrulation
process is finished. The prostomal thickening is the ventral lip of
the “blastopore réel” of the Amphibian egg. For the developmental
processes following on this stage I can contain myself with referring
to my former paper. That here only a small, not very prominent
tail-knob is formed and no far-reaching projecting tail-folds appear,
as in the selachean embryo, is caused by the relation of the pave-
ment-layer to the blastoderm and the periblast, which inflaence the
development of teleostean egg (‘‘développement massif” of HENNEGUY).

2. To determine the direction of growth of the blastodermring
during the covering of the yolk, I used in my former paper the
oil-drops in the yolk of the muraenoid eggs as a point of orientation,
on the contention that these oildrops maintain (in the muraenoid
egg) a constant position in the yolk. On this basis I constructed a
seheme of the mode of growth of the blastoderm in the yolk. *)
Both Sumner and Kopscu rejected this contention and the scheme,
SuMNER because of the fact, that by inverting the egg of Kundulus
heteroclitus in a compress, the oil-drops may be caused to rise
through the yolk and assume a position antipodal to their original
one, which shows, that here the oil-drop may not be regarded
as a constant point of orientation in the egg. In this SuMN»ER is
perfectly right. Not only in Fundulus, but in several marine pelagic
eggs too the oil-drops may be seen travelling through the yolk by
converting the egg or bringing the young larva (in some _ species)
in an abnormal position. In the muraenoid egg however the case
is entirely different. Here the structure of the periblast and of the

not judge, but I will only mention here, that the figures, drawn by the author, are
taken of much too late stages of development, to be convincing. And after all,
where the blastodermcells are so much alike, as is the case in most teleostean
eggs, one positive result im a favourable case as is offered in the muraenoid egg,
is more convincing than several negative results in less favourable eggs.

1) 1. ce. page 142.

( 806 )

yolk-mass, which I described at full length in my former paper,
completely checks the displacement of the oil-drops. This is to be
concluded already from the behaviour of the normal egg. 5o0 in the
egos of Muraena No. + a large number of rather large oil-drops are
lying at about equal distances from each other at the surface of the
yolk-mass. During the entire process of covering of the yolk, the
distance of these oil-drops remains the same, they maintain their
relative position absolutely, and only during the slight disfigurement
of the yolk-sphere, caused by the contraction of the blastodermring
during the circumgrowth of the yolk (fig. 4 on plate 2) the position
of the oil-drops is changed a little, only to become the same as
before, after the yolk has regained its spberical form. When these
oil-drops were lying loose in the yolk or in the periblast, they would
have crowded together at the upper pole of the egg, or at least their
relative position would have undergone a change during the covering
of the yolk. Only when the yolk-mass in the developing embryo
becomes pear-shaped and very much elongated (l.c. plate 2, fig. 6, 7),
the oildrops of course change their position. Even then, however,
they remain scattered through the yolk.

Experiments also show the constant fixed position of the oil-drops
in the muraenoid eggs. When we transfix the egg-capsule carefully
with a fine needle, it is possible to lift one of the oil-drops or a
small portion of the peripheral yolk out of the egg. The other oil-
drops retain their normal position, and in most cases such eggs
develop normally and give rise to normal embryos. When we operate
very carefully under a low-power dissecting-microscope, it is possible
to leave the oil-drop connected with the periblast by means of a thin
protoplasmatic thread. When we do this in a very early stage of
development, at the beginning of the gastrulation-process, we see
that this oil-drop, which surely may be regarded as a fixed point on
the surface of the egg, retains its position in relation to the other
oil-drops, until it is cut off from the periblast by the growing blas-
todermring. In fig. 2a, 26, 2c and 2d on plate 2 I have drawn
from life several stages of this process in an egg of Muraena No. 1.
During my stay at the Stazione Zoologica at Naples, in August and
September 1906, I performed several of these experiments with
different muraenoid eggs. They all led to the same result, and con-
firmed my former statements. And so I believe that my contention
was right and that the scheme I figured is a true representation of
the facts. Of course only in a general sense, for there are many
individual variations (so for example the case figured in fig. 3 on
plate 2). And after all, when we compare this scheme with that

( 807 )

given by Kopscu for the trout, we see that they do not differ so
very much, and that the displacement of the hinder end of the
embryo is almost the same. In the text of my former paper however
I expressed myself rather ambiguously, and brought my view into
a too close contact with that expressed by OxLiacurr. The tigures
however show that my scheme differs rather much from that of
OELLACHER.

But I differ from Kopscu in his supposition that the hea d-end
of the developing embryo is a fixed point on the periphery of the
egg. I find myself here quite in harmony with ScmNner, who draws
from the large series of his extremely careful and exact experiments
the conclusion, that ‘“‘the head end also grows, or at least moves,
forward, though to a much smaller extent” (1. ¢. page 115), and
says: “I regard it as highly probable (see Exp. 1, 3, 34, 35, 36
and Fig. 32) that the primary head end grows — or is pushed —
forward from an original position on the margin” (1. c. page 139).

From different experiments of the author we may draw _ the
conclusion, that in many cases this forward growth of the head-end
is rather extensive (exp. No. 6, 10, 11 (partly), 26, 35, fig. 10), and
experiment N°. 6 (table VIII) among others shows, that under circum-
stances the direction of growth may be entirely reversed, so that
the tail-knob of the embryo remains at the same place, and the
head-end bends round the surface of the yolk.

Kopscn too, in his paper: “Ueber die morphologische Bedeutung
des Keimhautrandes und die Embryobildung bei der Forelle’’, describes
an experiment with simular results in the trout.

So it is not unreasonable to suppose, that in the spherical egg of
the Muraenoids during the covering of the yolk the head end of
the embryo is moving forward, and to a certain extent follows the
growing blastodermring, which is the case chiefly during the later
stages of the covering of the yolkmass, as | showed in my scheme.
During the first stages of development it is chiefly the tail-end of
the embryo which travels backwards, (see the scheme in my former
paper and fig. 1—-5 or plate 2), and Kopscn is right to locate here
the centre of growth of the embryo.

The conclusion of Sumner, that for some time prior to the closure
of the blastopore, the ventral lip of the latter (former anterior
margin of the blastoderm) travels much faster than the dorsal lip
(l. e. p. 115) is quite in harmony with my results for the murae-
noid egg described in my former paper. *)

1) Petrus Camper, I. c. p. 196.

( 808 )

3. At the end of the covering of the yolk, at the closure of
the blastopore, Kuprrer’s vesicle is formed after the manner described
at length in my former paper. By Swarn and Bracuut?) and by
SUMNER the narrow passage connecting this vesicle with the exterior,
through the closing blastopore, is regarded as representing the neuren-
teric canal. I do not think they are in the right here. Kuprrer’s vesicle
is a ventral formation. Dorsally it is separated from the cells of the
medulla by the cells of the prostomal thickening and the pavement
layer. An open canalis neurentericus is not found even in these forms.
Kuprrer himself called the vesicle allantois. Husrecat followed him
in this. In my former paper I compared the vesicle with the allan-
tois of amniota on physiological grounds, and | think it is a very
good thought of Husrecut to take up the old name of Kuprrer and
compare the vesicle with the allantois on morphological grounds.

DESCRIPTION OF FIGURES ON PLATE 1 AND 2.
Plate 1.

Figg. 1—4. Median sections through eggs of Muraena N°. 1 on different stages
of gastrulation. In fig. 3 gastrulation is finished and notogenesis is begun. In fig. 2
the structure of the yolk is drawn. Enlargement 40 times. Fig. 4a, 5 and 6 give
median sections through the developing prostomal thickening and adjoining parts,
seen under a higher power.

Figg. 7—9. The flattening of the blastodise at the beginning of gastrulation in

—

eggs of Muraena N°. 7. Enlargement 4’) times.
Plate 2.

All the figures on this plate are drawn from life as accurately as possible.

Fig. la—le. Covering of the yolk in an egg of Muraena N.. 1.

Fig. 2a—2d. Covering of the yolk and closure of the blastopore in an egg of
Muraena N°. 1. By means of a fine needle one of the oil-drops is nearly severed
from the surface of the yolk, remaining connected with the periblast only by means
of a thin protoplasmatic thread. In fig. 2¢ this oil-drop is cut off from the surface
of the egg by the travelling blastodermring and is lying close against the egg-
capsule EK. In fig. 2d (closure of the blastopore) this oil-drop is no more drawn
in the figure.

Fig. 3. Unusually fargoing dislocation of the hinder end of an embryo during
the covering of the yoik. The head end lies approximately at the former centre
of the blastodise.

Fig. 4. Compression of the yolk-sphere by the growing blastodermring in an
egg of Muraena N’. 4. The oil-drops only temporarily changed their relative dis-
tances a little.

OD = oildrop.

pv = prostomal thickening
per = periblast.

bl = blastoderm.

D = pavement layer

e = entoderm

Leiden, 17 January 1907.
2) Archives de Biologie T. 20. 1904. page 601.

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( 809 )

Physics. — “On the influence which irradiation exerts on the electrical
conductivity of Artimonite from Japan”. By Dr. F. M. Janerr.
(Communicated by Prof. P. Zeeman).

(Communicated in the meeting of February 235, 1907).

§ 1. Having been occupied for a considerable time with the
determination of the specific electrical resistance in the three crys-
_tallographic main directions of the antemonite from Shikoku (Japan),
I had already found that with this substance, which belongs to the
very bad conductors, inexplicable irregularities presented themselves,
when the resistance was determined several times anew during a
long time, with identical electromotive force.

Generally the obtained deflection of the galvanometer first became
larger and larger, and decreased again in course of time, after which,
as I found, periodical increase and decrease sometimes followed. It
was impossible to detect any connection between tension, intensity
of current, and time.

As for rods of a length of some centimeters and a section of about
a quarter of a square centimeter, resistances were found in the different
directions lying between 500 and 20000 millions of Ohms, I first
thought of an impregnation of the electrical charge in the ill-con-
ducting material. On account of its opposed direction, however, an
eventual polarisation current would have to cause an apparent ‘crease
of the resistance, whereas experience generaliy showed a decrease of
the initial resistance.

§ 2. While I was trying to ascertain the cause of these deviations,
a sunbeam fell through an aperture of the curtain on the piece of
mirror-glass which closed the THomson-galvanometer, and was partially
reflected to the apparatus containing the piece of antimonite, cut
with its longitudinal direction parallel to the crystallographic J-axis,
The needle of the galvanometer deflected immediately towards that
side in which the total deflection was imcreased. At first I thought
that the heat of the sun penetrating the galvanometer on one side
had changed the cocoon thread so much as to cause a torsion. Some
moments later, however, when I happened to light a match in the
neighbourhood of the preparation, the mcrease of the already existing
deflection was repeated, and now in the same sense as before, and
at the same time stronger.

§ 3. So we have met here with a new phenomenon. Either the

( 810 )

radiation of light, or the heat must be the cause of the phenomenon.

1 then undertook the following set of experiments.

A rod of antimonite quite covered by paraffin, and cut parallel
to the J-axis, was shunted into the circuit of a dynamo, the tension
being kept at exactly 35 Volts by means of a resistance of incan-
descent lamps. When shunting in the THomsoy-galvanometer '), which
had been hung up in an antivibration apparatus of JuLius, and
which was so sensitive, that at a distance of mirror of two meters,
it still gave a deflection (double) of 26,5 mm. for a current of
0,000000006 Amperes, -— we obtained a constant, single deflection
of 10,7 cm. on the left of the zero point.

An incandescent lamp (of 110 Volts), placed at about 2 meters’
distance from the preparation, gave an icrease of this deflection of
+ m.m., i.e. 3,7 °), — agreeing in this case with a decrease of
resistance of about 53 millions of Ohms.

When the same lamp was placed at 1 meter’s distance it brought
about an increase of the deflection of 11 m.m.; at ‘/, meter’s distance
of about 20 m.m., and held near the rod for a short moment, of
more than 220 m.m., i.e. an increase of the conductivity of resp.
= 10°) a FST emt 206 >! *)

Then the lamp was removed, and after the deflection had resumed
about its original value, one of the curtains at the window was
drawn aside, so that the diffuse daylight (overcast sky) fell on the
apparatus. Instantly the deflection was increased by more than 4 m.m.
i.e. about 3,7°/,. Then a wooden box was placed over the apparatus,
and then removed. Every time the experiment was repeated the
constant deflections in the light were found from 3 to 8 m.m. larger
than those in the dark.

§ +. In the foregoing experiments only exceedingly little light
fell on the rod of antimonite, as it was quite covered by a coat of
paraffin *) about O,f em. thick, and so only the light penetrating
the half transparent coating could have any effect.

Then the experiment was repeated as follows.

A lamella‘) of antimonite was clasped between two much larger
copper plates, which two plates were well insulated. The condensator
(fig. 1) obtained in this way was suspended on silk threads. *)

1) The internal resistance of this instrument amounted to 6681 Ohms.

2) The resistance of the rod was diminished by an amount of more than 950
millions of Ohms in the latter case.

5) The purpose of these precautions will be explained later on in a paper
written in conjunction with Mr. Vas Nunes.

*) The antimonite splits perfectly // (010), so g b-axis.

{Sit }

The antimonite plate had a thickness of about 1 m.m. and a
section which may have amounted to about '/, ¢.m’.

Now if a source of light was placed at I, the remaining deflection
of about 1,8¢.m. obtained at a 10,5 Volts’ tension was only increased
by 2 m.m., i.e. by about 11°/,. If, however, the light was placed
at the same distance in I, the increase amounted to about 11,5 m.m.,

ie. 64°/..

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In the former case the plate A is viz. in the shadow of A,, and
so receives but very little light reflected by the walls; in the latter
case, however, the radiation is direct.

§5. Ifa thick plate of colourless plate-glass is placed between
the source of light in If and the apparatus, the remarkable fact
presents itself that the deflection is considerably increased. The
explanation of this phenomenon was obvious. For a copper bar,
heated to some hundreds of degrees, and brought near the apparatus,
immediately diminished the obtained deflection greatly. Hence —
and this is a most remarkable result — rise of temperature has an
influence directly opposed to that of radiation of light: it enlarges
the resistance instead of diminishing it, as rays of light do,

( 812 )

If the plate is again removed, the deflection decreases again to
the value it had before the plate was placed between ete. So this
fully proves that it is the radiation of light which influences the
conductivity of antimonite in so high a degree, and not the heat;
for the latter diminishes the conductivity, in contrast with the former.

§ 6. Finally glass plates of different colours were interposed between
the source of light and the rod.

It then appeared that the influence of the differently coloured
light was very different. The antimonite namely proved to be subjected
to hardly any change by green light; for red light the increase was
pretty large, for yellow light a little more, for green very small,
for violet light again stronger. In each of these cases the deflection
-appeared to have resumed its original value after removal of the
source of light’). With violet radiation | obtained an increase ot
conductivity which amounted to about 150 °/, of the original value ;
with white light with interposed glass plate one of about 250 °/,.

To get some insight into the guantitative action for a special case,
the following experiments were made. An ordinary electrical incan-
descent lamp was adjusted at 16 cms’. distance from an antimonite rod
covered with a coat of paraffin 1 cm. thick. First of all it was
ascertained that action of light by itself did not excite an electrical
current. It then appeared that the deflection of the galvanometer
was increased just as much irrespective of the direction of the
current. So the decrease of resistance is independent of the direction
of the current. By interposition of coloured glass plates I got a

rough estimation of the relative influence of the different colours —

of the spectrum. Thus I found:

White light, placed at 16ems’. distance, makes the conductivity rise to 200%, of its original value

Red ” ” 39.) a3 3039 ” ” ” ” 939 194% 5 45 ” ”
Orange ” ” shits 39 ” ” ” ” 999 153% 4, 45 ” .
Green ,, ” a | Oe | 7 ” ” ST 9 99 116% 4, 45 ” ”
Blue ,, ete Ss: oes 8 S55 a = + “ 95 5g A TOU Gee at ~

1) Not quite the original value. The substance shows hysteresis to a certain
amount, which, however, is smaller than for selenium. Already 20 a 40
minutes after the source of light had been removed, the original deflection was
sometimes found back. The mineral seems to be quite free from any admixture
of selenium, as a qualitative investigation taught me. Remarkable in a high degree
is the fact, that on melting the natural mineral, it obtains, when solidified as a
conglomerate of little crystals, a specific resistance, which is many thousand

times less than before, while at the same time it has lost its sensibility to light-

radiation quite. On heating the antimonite however, without melting, it conserves
. this property. Analysis has taught me, that there are present the elements: Sb, 5,
Ca, Ba, Sr, Si and, as Prof. Kury found, traces of Zn and Co; also SiQ,-crystals
are included. (Added in the English translation),

( 813 )

As heat-rays have only an exceedingly slight effect, and, as I
ascertained later on in conjunction with Mr. Vas Newnzs, also the
ultraviolet light emitted by cadmium poles causes only a small
increase of the conductivity, the dependence of wave-length and
decrease of resistance is evidently represented by a curve whose
minima lie in the ultrared, in the green and in the ultraviolet, and
whose maxima are situated in the red and in the blue part of the
spectrum *).

Later on when the determinations of the resistance of this sub-
stance will have led to favourable results, we shall make some closer
communication on the relation between thermal and electric motion
in this conductor.

§ 7. The phenomenon discovered here reminds strongly of that
observed for selenium’). It is, however, noteworthy, that though the
dependence of the increase of the conductivity on the radiation of
light, and even on the wave-length of the light manifests itself in a
perfectly analogous way to that for antimonite, yet the two differ in
some respects. First of all for the selenium polymorphous changes,
and the displacement of equilibrium attending them play an im-
portant part; then, however, the resistance always decreases here
with rise of temperature, so exactly the reverse of what happens in
my investigations, in which moreover there is no question whatever
of polymorphous changes, as far as is known. An analogy between
the two cases is to be found in the fact already discovered by
Apams*), that the resistance decreases with rising electromotive force,
also after continued action of it; such a deviation from the law of
Onm is also found for the antimonite.

On the other hand the behaviour of antimonite from Japan seems
to present a closer analogy with that of the crystallised te//urium ;

1) Though it is not intended as an explanation, I will yet call to mind that it
follows from Mi.ter’s investigation (N. Jahrb. f. Miner. u. s. w. Beil. Bd. 17, 187— 251)
on the optical constants of the antimonite from Japan, that the indices of refraction
nm, and wm, have their maximum values exactly for the green rays (between the
lines E and F) (viz. n) =5,47—5,53 and n, = 4,52—4,49), while also the double
refraction reaches its maximum value for these rays. The polarisation of the
reflected rays is right-elliptic (negative). However, on using polarized light, we could
not find any influence of the direction of vibration: the variation of the electric
resistance was in the two cases the same. (Added in the English translation).

*) G. Wiepemann, Die Lehre vy. d. Elektricitaét. (1882). I. p. 544—553).

3) Sate, Phil. Mag. [4]. 47. 216. (1874); Pogg. Ann. 150. 333; Chem. News. .
33. 1. (1876).

*) Apams, Phil. Trans. 157.; Pogg, Ann. 159. 621. (1876), Phil. Mag. [5]. 1.115

( 814 )

here, too, the resistance increases with heating, decreases with
exposure to. light ’).

In conjunction with Mr. Vas Noyes I hope shortly to publish also
some quantitative data on the phenomenon discovered by me, and
also on the behaviour of the melted and again solidified antimonite
and the analogous selenium compound. This investigation has been
made in the Physical Laboratory at Amsterdam.

Anatomy. — “On the influence of the fins upon the form of the
trunk-myotome”’. By B. van Tricnt. (Communicated by Prof.
G. C. J. Vosmanr). (From the Anatomical Institute at Leyden).

(Communicated in the meeting of March 30, 1907).

This research forms a direct sequel to Professor LANGELAAN’s work
“On the Form of the Trunk-myotome’, and is intended to show the
influence of the fins upon the form of the myotome. The method
which I followed, was based upon the chief result of the foregoing
research viz. that the differentiation of the myotome takes place in
a continuous manner by means of folding, and that it is possible to
follow the process of folding in dissecting the intermyotomal tissue.
Now the method of direct dissection proved to be restricted in its
application, so that it was necessary partly to apply a more indirect
one. This latter method rests upon the relation, which exists between
the form of the myotome and the structure of the transverse sections
of the animal.

Differentiation of the dorsal musculature.

From a rather large sample of Mustelus vulgaris the skin with
the underlying connective tissue was removed, so that the external
surface of the myotoms was laid bare (figure I). Then in the region
before the first dorsal fin the parts constituting one and the same
myotome were determined; the form of this myotome exhibited
about the same form as in Acanthias, only the lateral part of
the myotome proved to be displaced caudally; the breadth of
this displacement amounted to about half the breadth of the myo-
tome. This myotome was arbitrarily indicated by the number

1) | have made an arrangement with Mr. J. W. Gittay at Delft with regard to
the mounting of antimonite preparations, and the preparation of antimonite cells
for practical use.

( 815 )

1 and the following myotoms by subsequent numbers. After that,
transverse sections of the animal were made, of 1—2 em. thickness,
and the numbering transferred to these sections, so that the lamellae
belonging to one and the same myotome received the same number.
For the sake of an easy description, figure II gives a hemischematic
representation of the myotome, in which the peaks are indicated by
numbers, the lamellae by letters. In figure III which is the trans-
verse section, (indicated in figure I with an A) the peaks appear
as systems of concentric lamellae marked in accordance with the
marking in figure II]. If we now pass to the region of the first
dorsal fin (figure IV, section B of figure I and fig. V, section C of fig. I)
the image of the transverse section is changed, instead of being com-
posed by four peaks, the dorsal musculature only shows three peaks.
The peak, indicated as number 1, has disappeared and instead
of this peak we find the first lamellae of the dorsal fin. Now in all
subsequent sections this first peak does not reappear any more. By
the method of successsive numbering it was possible to determine
the first myotome losing its most dorsal peak (number 1). The
external surface of this myotome is blackened in figure I. From the
principle laid down in the beginning of this notice ensues, that the
myotome apparently losing its first peak, gives a muscular element
to the dorsal fin; this element is therefore also blackened in figure I.
It may be seen in figure I that the first myotome giving an element
to the fin lies a little caudally in respect to the front edge of this
fin. The number of myotomes giving a part to the first dorsal fin
may easily be determined, because these composing parts of the fin
are separated by intersegmental tissue; that we have really to deal
with intersegmental tissue follows from the fact that through these
lamellae bloodvessels and extremely fine nerve fibres reach the skin
(vid. v. Bisselick “On the Innervation of the Trunk-myotome’’). The
total number of muscular elements composing the fin, amounted to
34, so that the last myotome still giving an element to the first
dorsal fin already lies in the region of the second dorsal fin. From
the fact that the most dorsal peak (number 1) does not reappear
any more in the transverse sections, it follows that the next myotome
gives the first element to the second dorsal fin. The surface of this
myotome is also blackened in figure I to show its position in relation
to the front edge of the second dorsal fin. It is evident, that this
myotome occupies the same position in respect to the second dorsal
fin as the first myotome does in respect to the first dorsal fin. The
number of myotomes composing the second dorsal fin amounts to 30.
Upon the second dorsal fin follows the dorsal part of the caudal
56
Proceedings Royal Acad. Amsterdam. Vol. IX.

( 816 )

fin. In this fin the myotomes are pressed together so closely that a
direct counting of the number composing the fin is no more possible ;
by comparing the total number of vertebrae to the number of
myotomes composing the first and second dorsal fins, we find that
about 70 myotomes give an element to the dorsal part of the
caudal fin.

The results obtained by this indirect method are corroborated by
the result of the direct dissection. If we take a myotome giving a
muscular element upon the more anterior part of the dorsal fin and
begin the dissection with lamella 6 in the neighbourhood of the
second peak and proceed preparing caudally, we find lamella 6 being
rolled in, towards the mesial plane of the body, in the shallow
excavation in which the base of the fin rests, (tig. VI‘. Along this way
the muscular tissue becomes gradually atrophic and only a thin band
remains, consisting of the connective tissue which forms the frame-
work for the muscle fibres. In the neighbourhood of the sagittal
plane of the body this lamella is folded, in such a manner, that the
line of folding (figure VI L’ L") runs parallel to the sagittal axis of
the body. By this process of infolding the direction of the lamella }
is reversed, the infolded part proceeding cranially; this part of
lamella 6 passes into the dense sheath of connective tissue, which
is interposed between the dorsal musculature and the base of the
fin. As far as I can see this sheath of connective tissue is chiefly
built up by a large number of these lamellae, but they are so inextri-
eably united that I have not been able to follow lamella 6 in this
sheath. If starting from the fin, we prepare free one muscular ele-
ment of the fin, and this element is lifted up with enough precau-
tion, it may be seen, that from the base of such a fin-element as well
a thin lamella of connective tissue passes into that sheath of tissue in
which we could follow the reversed part of lamella 6. The direct
continuity however of both lamellae in the sheath of dense connec-
tive tissue, I have not been able to establish.

The muscular elements composing the fin (figure VI) are trian-
gular laminae; one side of the triangle is contiguous to the fin-rays
and the connective tissue which unites these rays in the mesial plane
of the body, the lateral side forms part of the lateral surface of
the fin, while the base is excavated and moulded upon the shallow
depression in the dorsal musculature. From the outside a septum of
intermyotomal tissue (s.i. figure VI) penetrates into the muscular
substance of the fin dividing this substance into a lateral (6) and a
mesial part (a). This septum inserts a little above the muscular
substance upon the fin-rays, and becomes thinner and thinner without

( 817 )

reaching the base of the fin. At the base therefore the lateral
and mesial parts of the musctlar substance are continuous and form
a peak (figure VI p. 1), lying quite near the mesial septum of the
body. This peak must therefore represent the peak which is lost in
the transverse section (figure IV) made at the level of the first dorsal
fin. The septum penetrating into the muscular substance of the
fin is therefore the intermyotomal septum stretched out between
lamellae a and / of figure II.

It ensues therefore from the combined observations, that the first
dorsal fin (and the same applies to the other dorsal fins) is differen-
tiated by a process of infolding similar to the differentiation between the
dorsal and the lateral and between-the lateral and the ventral mus-
culature. The line of infolding crosses lamella 6. In that part of the
lamella, which lies in the depth of the fold the muscular tissue is
atrophic. Proceeding from peak 2 caudally along lamella 4 the atrophy
of the musclefibres gradually increases, whilst on the other hand
proceeding from peak 1 caudally along lamella @ (as far as it lies upon
the fin) the atrophy of the muscular tissue is abrupt. The position of
peak 1 has not changed in respect to the mesial plane of the body,
only the lamellae have changed their direction. The superior cornu
(i. e. lamella a) is no longer directed cranially but turned upwards
and this is also the case with that part of lamella 4 that has passed
into the fin. In connection with this representation of the facts, I deter-
mined the direction of the muscle fibres in the fin; here they slope
downwards from the intermyotomal septum. Now if we imagine the
lamellae composing the fin restored to their original position, the
course of the fibres in lamella @ would be from mesial to lateral
and from caudal to cranial and this was actually the direction of
the muscle fibres in lamella @ in the region cranially of the first
dorsal fin.

Differentiation of the latero-ventral musculature.

The lateral musculature, as described by van Bissenick, shows a
peak directed caudally (peak 5, figure II and fig. VII) situated near the
second line of infoldmg L’'L’. Proceeding along the body a second
peak appears directed cranially. The first myotome showing this peak
(peak 6, fig. II and fig. VII), is the eleventh myotome following the first
myotome giving a muscular element to the first dorsal fin. The two
peaks lie near to each other in the neighbourhood of the second
fold. In consequence of the infolding of the myotome at that place,
they do not reach the surface of the body, being covered from the
outside by the ventral musculature. Meanwhile the ventral part of
the myotome undergoes a change in form, the first lamella belonging

56*

( 818 )

to the ventral musculature (lam. / figure II and fig. VII) becomes
shorter and the first peak of the ventral musculature directed crani-
ally (peak 7, figure II), more and more develops into a true peak.
Now by the disappearence of lamella 7 peak 6 and 7 approach each
other, remaining divided, however, by a thin lamella of connective
tissue penetrating into the second fold along the line L'L’ (fig. VIII
and IX). In consequence of the process of infolding peak 6 lies mesially
in respect to peak 7 which covers peak 6 from the outside. At the
level of myotome 15 (reckoned from the first myotome, giving an
muscular element to the first dorsal fin) the second: fold vanishes.
Together with the disappearence of the fold we notice. the vanishing
of the displacement of the lateral musculature in respect to the
ventral musculature, which was only a consequence of the process
of infolding, so that the two peaks (6 and 7) lie side by side in the
same transversal level of the body. At the place of disappearance
of the second fold the two peaks unite to a single peak directed
cranially. Together with the disappearance of lamella 7 we notice
the further development of lamella g.

At the same level where the second lateral fold disappears, we
find the appearance of the cartilagineous plate, uniting the two
basipterygii of the pelvic fins. With its front border, this plate folds
in lamella g (figure II and X) from the inside so that this lamella
covers the front edge of this plate; in this way the pelvie fin is
formed. The details of the formation of the pelvic fins I have not
yet investigated. By the formation of the pelvic fin peak 8 (fig. ID
and Jamellae g and / pass into the musculature of the fin, so that
in a transverse section through the animal, at the level of the pelvic
fin, the trunkmusculature is only composed by five peaks (viz. 2.
3. 4. 5. (6+ 7) of figure II). This structure of the transverse
sections does not change any more proceeding along the body
caudally (figure XI and XII).

The disappearance of the first fold, dividing the dorsal from the
lateral musculature, takes place in the same way as described for
the second fold, at the level of myotome 45 (reckoned from the
first myotome giving an element to the first dorsal fin); only the
case is more simple not being complicated by the presence of two peaks.

Finally I paid attention to the influence of the abdominal cavity
upon the form of the myotome. I found this influence to be very
restricted, as it only determines to some extent the dimensions of the
myotome, without producing any particular differentiation in its form,

In fine I wish to express my thank to Prof. LANGELAAN for his
aid and assistance in these researches,

( 819 )

Anatomy. — “Anatomical Research about cerebellar connections.”
(Third communication). By Dr. L. J. J. Muskens (Communi-
cated by Prof. C. Wrinxuer).

(Communicated in the meeting of March 30, 1907).
The ventral cerebello-thalamic bundle.

Whereas it is nowadays generally accepted, that the direction of
conduction in the superior crus cerebelli is cerebellofugal, there is
no uniformity of opinion attained yet by the authors regarding the
bundle, which is found degenerated in the predorsal region in the
pons after cerebellar Jesions. After PenLiizzi and vAN GEHUCHTEN,
Tuomas, OrEsTANO, CayaL and Lewanpowsky this bundle is built up
by fibres, which take them origin from the superior crus after it
has crossed the raphe in WERNEKINCK’s commissure, the direction of
conduction being rubro-fugal. Prosst however, describes this bundle
as the ventral cerebello-thalamic bundle, conducting nervous impulses
from the cerebellar basal nuclei upward towards the red nucleus.

The problem of this bundle really has to deal with two questions ;
1st. which are the two nerve centres, which are connected by means
of this bundle and 2°¢. what is the direction of conduction of impulses
in the same.

Cats appear to be more suitable for these experiments. In two
animals different parts of the cerebellar cortex, with the adjacent
part of the basal nuclei, were removed, except the flocculus. In these
animals there was hardly any degeneration at all in the ventral
cerebello-thalamic bundle, whereas in 3. other cats in which with
other parts also the flocculus was removed there was very extensive
degeneration of this bundle. That these fibres do not take their origin
from the cortical gray matter of the flocculus is proved by the fact,
that in another cat in which the cortex of the formatio vermicu-
lavis cerebelli was injured, no degeneration of the said bundle was
found.

In two cats (XXIII and LXI) a lesion was effected in the mid-
brain, by passing a curved knife in front of the lobus simplex
cerebelli in such a way, that the predorsal region on the right side
was cut, distally from the red nucleus. In none of these animals
any degeneration was found in the ventral bundle. If Caya’s sup-
position were correct, certainly a great many of the descending col-
laterals of the superior crus ought to have been found degenerated.

In one cat (LVIIJ) a longitudinal lesion- was effected in the teg-

( 820 )

mentum, the instrument (Prosst’s covered hook) passing through the
middle crus cerebelli. In this cat were found a certain number of fibres
degenerated, which passed through the regio reticularis of the side of the
lesion and then, crossing the raphe and running upward in the
predorsal region of the other side, took their way towards the red
nucleus. This experiment tends to show, that there are direct fibres,
coming from the basal cerebellar nuclei, which do not join the
superior crus, but follow the ventral course to arrive at the red
nucleus. Lrwanpowsky,s fibres O. P. (in fig. 66 and 37) are not to
be identified with these fibres on account of their entirely different
course.

In cat J.XII the anterior crus cerebelli was partially cut, and at
the same time an incision made into the middle crus. Also in this
animal there was found no degeneration on the distal side of the
lesion except the bundle of Monakow. Were the ventral bundle to
be regarded as being formed by descending collaterals of the anterior
crus this result could hardly be explained. A simular result was
obtained in cat LX VIL, where hemisection of the pons was effected.
Also here there was no degeneration on the distal side of the lesion.

Mathematics. — ‘“‘Lquilibrium of systems of forces and rotations
in Sp,.” By Dr. 5. L. van Oss. (Communicated by Prof. P. H.
SCHOUTE).

(Communicated in the meeting of March 30, 1907).

Referring to the following well-known properties :

a. The coordinates py and aj of a line p anda plane a satisfy the
five relations :
P; = pki Pim + Plj Pkm + PjkPim=9 » MEZA Ajm=0, ~. (I)

of which relations three are mutually imdependent.

4. The condition that a line p and a plane a intersect each
other is expressed by

2 py gO Oh Soe

c. The coordinates of the point of intersection X of two planes
x, a and that of Sp,3 through two lines p, p’ are:
ae = IL Tim tj Wem + jk Wn + Aim Wht + Ahm Wy + Wim 2 jE;
(= > pr p'jms ole be

t
we wish to draw the attention to the following properties :
If (7/) are ten arbitrary quantities: i.e. not satisfying the relations

( 821 )

S (jk) (lm) = 0, we shall continually be able to break up each of these
quantities into two parts (7)' and (7), so that =(A/)' (jm)'= (kl)" (jm)"=0.
i i

It is easy to see that this decomposition can be done in a@* ways.
For each decomposition holds good :
Sea tn) — — (I) (ym), . sw Sk. A)
for :
= (kl)! (jm)" = & (Kl) (jm) — (jm)'} = & (Kl) (jm),
t : 1

likewise :
> (kl) (jm) = & (i) (jm),

from which by addition :
= (Wy (jm)" = & (Hl) (jm).

Giving a geometrical interpretation we regard a homogeneous
system of 10 arbitrary quantities aj and aj as the coordinates of
a system a of c* lines, in pairs a system @ of w* planes, in pairs

«, a conjugated by the relations: a’, a" conjugated by the relations:

be Rig Gg ss (5) ag + «'g = ag... - (5)
All these lines le in one Sp, = All these planes pass through
having as coordinates: one point X, having as coordinates:
Si = 2 agi ajm. - - - (6) Z, = Zaki Aim... - (6)

c c

We now annul the homogeneousness of the p-, 2-, a- and a-co-
ordinates.

This causes those elements to assume vector-nature and makes
them interpretable respectively as force, as rotation, as dynam and
as double-rotation. The equations (5), (5’) determine the reduction
of the vectors a and e@ on the conjugate pairs of lines and of planes
of the systems a and @ under consideration and not yet partaking
vector-nature, whose structure now becomes revealed.

Il. In connection with the meaning given in 6 of the equation
= pi Xi; = 9 we interpret
2 ag pg... - te) ae gg ee Ot
as the condition that a line peuts as the condition that a plane x
a pair of conjugated planes of cuts a pair of conjugated lines
system @. of system a.
This gives us a very fair survey of the structure of the linear
complex of lines and planes. The reduction of the equation of the
complex of planes to its diametral space is now easy to do; likewise

( 822 )

the further reduction to the simplest form (4:7) =o (jm), assumed by
the equation when the edges £/ and jm, the planes ¢jym and cA of
the simplex of coordinates are conjugated elements of the systems
a or @.

Ill. If we assign to the elements p, a, a,a vector-nature, expres-
sions Say pj, Saya become of importance as virtual coefficients
(in Baxw’s theory of screws) and the disappearing of these coefficients
then gives the condition that the force p performs no work at a
displacement im consequence of a double rotation a, resp. that the
dynam «a performs no work at a rotation x.

So in Batu’s notation the equations (7), (7)’ give the condition of
reciprocity between force and double rotation, resp. between dynam
and rotation.

In like manner the equation

Sa eg = Oils Se (8)
which includes (7) and (7)’ and likewise (2), gives the condition of
reciprocity between the dynam a and the double rotation a.

IV. We shall now pass to the general equilibrium of forces and
rotations. It will be convenient to understand by p, 2, a, a vectors
unity and to indicate the intensity of these vectors by a factor.

It will be sufficient to limit ourselves to the equilibrium of forces,
leaving the treatment of the dual case to the reader.

In the first place we regard the case of n forces, n > 10 working
along lines given arbitrarily.

It goes without saying that for the equilibrium it is necessary and
sufficient that the intensities 4” satisfy the ten conditions:

Shp) 0) 5. Sn «ee

We can therefore in general bring arbitrary intensities along 2 — 10

vectors, those on the other ten then being determined by the above

equation (9).

In particular for n= 11 the theorem holds:

To vectors along eleven lines given arbitrarily belongs in general only
one distribution of ratios of intensity, so that the system on those
lines is in equilibrium.

The generality of the case is circumscribed by the requirement
that no ten lines can satisfy one and the same linear condition in

the form — 0, where the coefficients ¢, do not depend on

Ss oe. 1H?
— ij Pe
vy, in consequence of a well-known property of determinants tending
to zero,

( 823 )

So if there are among 7 lines at most 10 belonging to a linear
complex we can satisfy the equations (9) by choosing all intensities
except those belonging to these 10 equal to O and then (if not all
subdeterminants of order 9 tend to O) we shall be able to bring along
these last only one distribution of intensity differing from O in such
a way that the system of forces obtained in this manner is in
equilibrium.

We have thus at the same time arrived at the following theorems:

For the equilibrium of ten forces it is necessary that these belong
to one and at most to one linear complex. In this case always one
and not more than one distribution of intensity is possible.

If we continue the investigation of the equations (9) we then
obtain successively the conditions of equilibrium of 9, 8, 7, 6, 5
forces. We can express the result as follows:

In order to let n forces, 11 >n>4, admit only of one distri-
bution of intensity in equilibrium, it is necessary and sufficient for
them to be the common elements of exactly 21—n linear complexes.

In particular for n= 5 we find the condition that the forces must
belong to a system of associated lines of Sucre.

This has given us a connection with a former paper in which we
treated this case synthetically.

V. The condition that ten forces in equilibrium belong to one
complex follows almost immediately out of tbe interpretation of the
equation Yas pz = 0 as condition of reciprocity of force and double
rotation.

Let eg. ten forces be given in equilibrium; nine of these forces
chosen arbitrarily determine a complex, so also the double rotation
for which none of them can perform labour. The united system of ten
forces, as being in equilibrium doing no labour for no motion
whatever, it is necessary for the tenth force to be likewise reciprocal
with respect to the double rotation a, ie. this force belongs with the
former nine to the selfsame complex.

Equally simple is the deduction of the conditions of equilibrium
for nine forces.

For eight forces determine a simply-infinite pencil of complexes
whose conjugate double rotations @ + 4a’ are all reciprocal with
respect to these eight forces. So they must also be reciprocal with
respect to the ninth force in equilibrium with these, i. o. w. the
latter must belong to ali linear complexes to which the eight others
belong.

And so on.

( 824 )

VI. We shall now denote still, by means of a few words, in which
way we can arrive at an extension of the screw-theory of BaLL by
the application of the principle of exchange of space-element to the

equations = a;§;== 0.
l

By interpreting this equation either

1st. as condition of united position of a point X and an Sp, Zin Sp,,

2d. as condition of reciprocity (BaLL) of a dynam X and a double
rotation &,

we make a connection between the point- and Sp,-geometry in
Sp, on one hand and the geometry of dynams and double rotations
on the other hand.

To each theorem of the former corresponds a theorem of the
latter geometry. Nov the remarkable fact makes its appearance that
the fundamental theorems of the geometry of Sp, correspond to the
fundamental theorems of the theory of screws of Bat in Sps.

With this as basis we shall show, though it be but by means of
some few examples of a fundamental nature, that the principles of
a generalisation of the theory of screws are very easy to be arrived
at by transcription of the simplest properties of the point- and
Sp,-geometry in Sp, which examples can at the same time be of
service to explain the above observations on the theory of Bau in Sp,.

To avoid prolixity we introduce the following notation. We call:

dynamoid the system of lines whose conjugate pairs can serve
as bearers of a dynam. .

rotoid the system of planes whose conjugate pairs can serve as
bearers of a double rotation: So dynamoid and rotoid correspond
to dynam and double rotation as in the notation of BaLL “screw”
to dynam and helicoidal movement.

Let the following transcriptions be sufficient to explain the appli-
cation of the above principle.

oX: Point X bearing a mass Dynamoid X bearing a dynam
6. of intensity X.

o5: Sp, = with a density of Rotoid 3 bearing a_ double
mass 0. rotation of intensity o.

(X'X"): Right line, locus of the Pencil of dynamoids, locus of

centres of gravity of the bearers of the resultants of
variable masses in the two variable dynams on the dyna-
points Y' and X". moids X' and X".

(4'5"): Sp,-pencil. Pencil of Rotoids.
A right line has always A pencil of dynamoids always

( 825 )

a point in common with
an Sp.

An Sp, is determined by
nine points.

p spaces Sp, cut each
other according to Spo_p.

Ete. ete.

contains a dynamoid reciprocal to
a given rotoid.

A rotoid can always be deter-
mined lying reciprocal with re-
spect to nine dynamoids.

The dynamoids reciprocal to
the movements of a body with
p degrees of freedom form a
(9-p)-fold infinite pencil.

We shall now apply the above to the problem: “To decompose a
dynam according to fen given dynamoids’’, this problem being a tran-

scription of the following:

“To apply to ten given points a distribution of mass so that the
centre of gravity finds its place in a given point.”
We again put side by side the results.

To be defined successively :

a. An Sp, through nine of the
given points.

6b. The right line through the
remaining point and the centre of
gravity.

c. The point of intersection of
this right line with the Sp, found
in a.

d. The decomposition of the mass
in the centre of gravity according
to this point of intersection and
the 10% point named in 4, which is
possible, these three points being
collinear; gives at once the mass
to be applied in the last named
point.

The other must necessarily be-
come the centre of gravity of
the remaining nine points.

e. These treatments to be repeated
for the determination of mass in

the other points.
Zalt-Bommel, March 28, 1907.

The rotoid reciprocal to nine
of the given dynamoids.

The pencil of dynamoids through
the remaining dynamoid and the
bearer of the given dynam.

The dynamoid on_ this pencil
reciprocal with respect to the
rotoid found in a.

The decomposition of the given
dynam according to the dyna-
moid found in ¢ and the 10%
dynamoid named in 4, which is
possible, these three dynamoids
belonging to one pencil, gives
at once the intensity of the
dynam on the last mentioned
dynamoid.

The other must necessarily bear
the resultant of the dynam to
be applied to the remaining nine
dynamoids.

These treatments to be repeated
for the determination of intensity
on the other dynamoids.

( 826 )

Physics. — “Contribution to the theory of binary mixtures,’ ITT,
by Prof. J. D. vAN DER Waats.

Continued, see page 727.

We shall now proceed to describe the course of the spinodal
curve and the place of the plaitpoints when choosing regions of
fig. 1 which lie more to the right. But it has appeared from what
precedes that to decide what different cases may occur, we must

[2 2

: a |
know the relative position of the curves —_ = 0 and —~ = 0, to
da? dv
Pw
which now the curve = 0 is added; so the relative position at
avavu

different temperatures of the three curves which occur in the equation
of the spinodal curve.
Sats 2

The curves === O and
dv avtav

= 0 may be considered as sufficiently

known, and the knowledge of the relative position of these curves
with regard to each other enabled us already before to elucidate
sufficiently the critical phenomena of mixtures with minimum critical
temperature and though with regard to the relative position of
these lines some particularities are met with, which have not expressly
been set forth, I shall assume the properties of these lines to be

dw ; :
known. But the curve qa = is less known — and it has appeared
Av

from the foregoing remarks, that if we wish to understand the occur-

rence of complex plaits, the relative position of this curve with
2

respect to the curve = 0 must be known. If this line hes alto-

av
to speak of on the course of the spinodal line, but if it hes either
partially or entirely outside this region, the influence on the course
of the spinodal curve is great, and the existence of this curve accounts
for the complexity of the plait and gives rise to the phenomena of
non-miscibility. I have, therefore, thought it advisable to investigate

gether within the region where is negative, it has no influence

the properties of the curve

ee vee
~ = 9, before proceeding to the deserip-
AX

tion of the course of the spinodal curve also in other regions of
fig. 1. A perfectly exact investigation of this line would, of course,
require a perfectly accurate knowledge of the equation of state. But

( 827 )

the value assumed already before as an approximate equation of
this line:

(=) Pa
2 1 da la?
Ee a RT x | Ee steer
dx” u(l—v) — (v—by

will prove adapted to give an insight into the different possible

he Ae : Py dy
positions of this line with respect to —-=0O and —— = 0,
dv? dadv
yp
THE CURVE —— =
dx?
The differential equation of this curve:
as can Pw ’
gf ve —
aa dattien eo da*dT
may also be written in the following forms:
dw dy aly al
dx + —— dv — — = 0
da* da?dv dz*,7 T
or ;
dy Pap d? e—) aT
u dx +- = 4 dv — ( —=0
da? da? dv de7,7 I
or
d* Pw jk as A
eae ee ee aan ea
da* div? dv az* 7 I
or
ay dy iidaa dr
a+ dv — —_ =
dx* da*dv o dz? T

2al

F=0 can only be found for positive value of 7

The curve

when ms is positive. So we derive from the latter form that
ax

dx \ . = : : ; ; dy :
ap) 38 positive for the points for which eo negative, and the
= Ax

other way about.
dv

) is positive for the points for which

In the same way, that ( —
; ri

dw
dvda?

is negative and vice versa. The transition of the points for

( 828 )

‘eNO ; oe :
which ma 8 negative or positive, takes place in the points of the
ar
dp . .
curve —— =O with maximum or minimum volume or for which
ae
ly dw d*p

—0Q: and the transition of the points for which ——~ = —
dx* P dvdx? dx*», T

is negative or positive, takes place in the points with maximum or
or minimum value of 2. From all this follows that the curve
Py
da?
point for certain value of 7’= 7). It is now necessary for our
purpose to determine the value of 7/, and also the value «, and v,
of the point at which this locus vanishes. This means analytically
that we have to determine the values of 7, « and v, which satisfy:
Pw a dw See a Za a DR 52) ay

dx? da?* dvd? dx?

db\? da
dx oe da?
Saas

=O contracts with rise of 7, and has contracted to a single

or the equations:

db\?

1
MRT
| x(1—2) a (v—by?

LSTA AS —S——. 3 . *. e . (2)

227(1—2) (v—))

db \? aa
unr S@ ? 2 (3
anc MIe (ob) pa )

If (1) is divided by (8), we get a relation between 2 and 2,
which in connection with (2) may lead to the knowledge of 2,
and vg.

Then we get:

(o—b)>_ -=v}b=(—b) 428

aa 1 . db
u( —«(Z)

and as (v—b)? = db * 207(1—a)’
| : \de 1—2-2

b = a(1—2) eee ’

we find:

(4)

dh Te Ain oo ae

( 829 )

db
and putting b=b,+ 2 fa

ae 1 ( 2.n? 1—.r)? 1/.
b, ie n ey 17( r) (5)
db 1—27 a 1—2e |
dic

The Ist member of this last equation representing the ratio between
the size of the molecules of the first component and the difference of
the sizes of the two kinds of molecules, we see that x, depends
only on the ratio between the sizes of the molecules of the two
components.

If we take the two extreme cases 1*t that 6, may be put equal
to 0, 24 that 6, is equal to 6,, we find the two extreme values of 2,.

Bie \ ear ks)" en (1—22)* or 2a? — (1 ) (1-22)
— — ——— ies SY —# lh Lac), = —f& —awr
19.0 [ge a )

db
or «= '/, for 6,0. For the other extreme case Se we find
&

For some arbitrarily chosen values of 2, I have calculated the

corresponding values of

b,—),

: v—b
Ly wary 5 Yo (see p. 832)
7 hens Wir le 5) a See Os
0,4 i eo eel fA: AUN IS Sars Sd od 5 i, | ts.
A a he Sees ene ees." OLE
4G) oe + POS i: 0,457 . . . 0,186
O40 le x S06 ee tee S|. O54
(AS) us 2 3 SiO4 eee OO 2+. ies) OAVF
Ca in. 28 owe See re se . OORT
0,5 eae - ae a heey as | Far tt ee

If on the other hand the value of a, has been calculated by the

b
aid of the given value of re v, is determined by the aid of the
Ae:

dx
equation :
db (3 /2a(1—a)?
v— b= — Se
dx 1—2z
, ab ;
If —=0O, in which ease &g = '/,, this equation gives an indefinite

( 830 )

value for »v— 6. So it is better to express v— 6 in a value in

which 7A does not occur. We write by the aid of formula (4):
x
po
1-—2a
vai—wz) 1 ie
aor 12 1—2«

v—b 2

b <7 aS =) )
(1—22)?

v—b P ;
In the above table we find the values of aie calculated for arbi-

(ve —b)=b

trarily chosen values of a,. For values of x, differing very little

(Ui

b |
from '/, the value of approaches 2B~ (1 — 2a,)’.

The value of MRT, may be brought under the following form :

aa (1 oF 2a’)? Es
1 — | ——— _]3
Sener 4.v(1— «)
MRT, =-—« (1 — 2)- Say ir ys
b 1 it eae by
ER he
4x(1—.w)

é dw ;
So the temperature at which the locus ~ = ( vanishes depends
AX

in the first place on the value of ” at which it vanishes, and in

a ; k z
the second place on the quantity a As according to form. 5 &
We

ene +
, the factor of We may vary between 31

—

: 1
may lie between 7 and -

1
and . The value of that factor is therefore only determined by the
,
ratio between 6, and 6,. For 0, =O the value is 81° for.'b, ==";

this value is —. So the greater the difference in the size of the

molecules, the lower this factor, and the lower the temperature at
gp ee ; :
which oe = 0 has disappeared. And because the non-miscibility in
Ak
2
the liquid state is to a great extent due to the existence Of a 0,
Lv

( 831 )

d* .
molecules of the same size & being always thought equa will
ae

not so readily mix as those for which the size of one kind greatly
exceeds the size of the other kind, a property to which we might

have concluded without caleulations. But in the second place the
42

a : F .
quantity — = 2 (a, +4, — 2a,,) has great influence on the height of

va
this temperature, and indeed, in so high a degree that if = should
GAL
2

)
be = 0, the locus = 0 would already have disappeared at the
&

absolute zero point. Indeed, we might have seen from the very
Ma

beginning that this locus could never exist for ane negative. Every-
Lv

thing, on the other hand that diminishes a,,, makes 7% rise, and so

furthers non-mixing. In some limiting cases we may compare the

value of 7, with that of 7%, 1st in the case that into a given sub-

stance we should press a gas as 2"¢ component following the laws

of Boytr and Gay-Lussac perfectly. For such a gas we should have

to put 6, and a, equal to zero, and so probably also a,,. The value

1
of # of the formula for 7, is then equal to 3" The a for the mixtures

containing only one term then, and being equal to a,x*, —~ = 2a,.
av

The value of 6, for the mixture is then equal to 4,7. On these

suppositions MART, = and so 7, is equal to the critical tem-

a,
2b,
perature of the 2°¢ component. The value of 7). for every mixture
taken as homogeneous, is then equal to «(7%,),. Consequently
T, = 3(7T;),. For a value of T somewhat below (7%), the locus
dw
dv*

=O is restricted to a very narrow region on the side of the

ome
2d component, while - = 0 still exists, and may be compared to
Lv

a small circular figure whose centre is a point with the coordinates

«='/, and v=4,. The spinodal curve has then an equation which
may be written as follows :

MRT

Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 832 )
which equation shows that it consists of two straight lines, which
join the point =O and v =O with the points for which —? =0
for the second component. At temperatures which are not too far

dw
below (7%),, the locus = 0 lies, therefore, entirely outside the
le

curve = 0, and is then restricted to the left side of the figure.
av

2.4. As second limiting case we take 6, = d,, but a, differing from a,.

@aa(l—e« 1 2 A a ist
Then MRT, = fh Zh and because a = = MRT = ne
8 a,+a,-+2a,,

da? b
27 4h

whereas MRT, is equal to for a=='/,. Thenaleaee:

27
T, may be larger than 7, viz. when a (a, +a,—2a,,) >(a, +a, +2a,,)

23
or if 2a,,< 31 (a, + a,). But even if 7, should be < 7%, this implies

Py
by no means that shortly before its disappearance the locus SS = 0
v

lies in the region in which is negative. The previously calculated

dv?

v—b
values of gis show that this disappearance takes place at a very small

volume, which may be smaller, and in the limiting case will certainly

cee d
be smaller than the liquid volumes of the curve + =0. To ascer-
wv

{ oy ;
tain whether the disappearance of | ~ — 0 takes place in the region
&
; po EAB
in which a © negative, we may substitute the value of 7), 2 and
v ; j
dw ae
vy in the form for a and examine the conditions on which this
v
aw ; ; “ak, Aes
value of becomes negative. If we write for ——_—*— = y,°, then:
dv? 4x,l—a) ~

VRT, = Pax (1l1—ax) 1—yg wy—bg 24% %y 144g
wee ae eee ee See et a eh

and

(=) | MRT; 20
g

de Jy (vg—by)* ag os

dv?

( 833 )

tii

2a (v,—5,)? 6,
*) depends on MRT, —— (9 sl ” or on
9

dv 4 Vg Uq

d'a x (1 —zq) 1—y, 2a 4y,’ l—y,

dix by = =(l+y9)? by (L+- Yq)? 1+
4y9°
Lb yq—
For the discussion of this last quantity we first put the first
mentioned limiting case, in which a, and a,, may be neglected with

So the sign of (

or on (a, + a, — 2a,,) z, (14 — 2,) —a

respect to a,, and a=a,z* may be put; the value of 7, being =
3 2 A Yi o

pol

according to the table of p. 829. With this value this quantity becomes :
2
asa 3 -2—Fe|
3

so positive.

For the other limiting case for which y, = 0, it is also positive.
But for the intermediate cases, specially those for which a, + a,—da,,
is small with respect to a, and 4, and 4, are not equal, it will be

rp
dle an

negative, and shortly before its disappearance the locus a
ax
3

will le in the region in which

is negative, and the existence of
v

this locus will have little influence on the course of the spinodal
curve, and accordingly it will not give rise to a complex plait ’)
or rather to a spinodal curve which diverges greatly from the curve

d?
0.
dv?
er ee EAD
Let us now also examine where the point in which ==)
az

= : Py d
disappears, lies with respect to the curve = 0 or to | |=
adv da},

Let us substitute the value of MRT(,, x, and vy, in the expression for

(2) If this expression becomes positive, the point lies outside
&

the curve or rather at smaller volumes than those of the curve

d
(2 = 0 and the other way about. Then we find for:
w)y

1) I need hardly state expressly, that in this communication I no longer attribute
the cause of the complexity of the plaits exclusively to the abnormal behaviour
of the components, to which at one time | thought I had to ascribe it, in com
pliance with Lexretp. On the other hand it would be going too far to deny the
abnormality of the components any influence.

57*

( 834 )

db da
MRT, Bente 2
4 (uv, —b)? v?

d@ax(1—a,) 1—y, db 1 dx

dz? bs (1+-yq)*dx(vg—b)? v4?

b vg—b
and after the substitution of — = oa Ee ly and of (2 =

ap~ t=2 vy
da
ans the sign proves to depend on the expression:
(1-+y,)’
@a la eae
7 (1—2z,) — Pn 499 :
In the first limiting case in which gs ='26 = ee ee
; az? >” dx ad ee 2

1 : Bate
and & = = this expression = 0. Also in the second limiting case,

1 Sa
in which cs and y,=0. So in the limiting cases the curve

d?w a
4 = 0 intersects the curve -
v

up to the last moment in which
dadv

the latter disappears. Also in an intermediate case this quantity may
be zero, but the value of 2,, at which this takes place, depends on
d’a

da? (a,—a,,) — (4,,—4,)
— or on —
da (4,,—4) (1—2,) al kg (a,—a 13)
da
Pa
a,—a dix? A
If we write ~——" =1+4+A, then — = ————_ =
A,,—4, 2 da 1—a,+a,(1+A)
dx
4 ae ;
= ant We then derive the value of «, for this intermediate
—+— Ly .

case from the equation:
(1 — 2ag)'/s
[4a4(1—ay)]"

A _yf 1% =}
1+Ac, |. 162,7(1—a,)* |

A
——— (1 — 2x) = 4

( 835 )

For values of x, differing little from } we find approximately :

1 ea iy ia a
a 2 > TS\ 114A)

If for A we take the value 4, which must be considered large for

1 i |
molecules of about the same size, then 5 a would be = ae The
2 5
conclusion which we draw from all this as to the situation of the
2

d
point in which zs

da?
iw
dxdv

= 0 disappears, with respect to the curve

= 0, is the following. In most cases this point disappears within

dp d
the curve = = 0, and so in the region where js) is negative,
dz}, x

d
but this can also take place on the other side of (z) = 0, so at

a volume which is smaller than that of this curve.

a,+a,— 2a
That at positive value of A, so at positive value of Es ea
Ria t
A 5 (1—2z)!/s : :
= 4° ——,_ has always a root, appears immediately when
1+cA x*/3(1—a)"/ 7

we represent the two members of this equation graphically. The
first member, namely, represents then a branch of a hyperbola which
at x=O has a height above the axis of « equal to that of A, and

at «=1 a height equal to and which, therefore, proceeds

A
1+)’
continuously at a certain positive though decreasing height above the
x-axis. The second member represents a line which for c=0O has

a point infinitely far above, and for «1 a point infinitely far
ee: 1
below the z-axis. This line passes through the point t= 5: and on

the left and the right of this point the ordinates are equal, but with
reversed sign. So intersection will certainly take place, and for

: 1 L ;
positive 4 at a value of 2 For the case that —=0 disappears
wv

at smaller volume than that of the curve = = 0, the first member
Fr

must be larger than the second. As A is larger, the point of inter-
: ; : 1

section will be further removed from 2 = = and so the series of the

values of z for which the condition is that the first member be larger

( 836 )

than the second member, has increased. From this we conelude that
au

ae Q may disappear also for very different size of the molecules

( L

Pie ee eves ae

in the region for == positive, if A has a considerable size. But for
0 a

1 A
perfectly unequal size of the molecules (« =I rea would be
A

8 ’
Sid 408 34

> 1, which is not yet satisfied even at A =o.

Sere f . if ap ay
Fig. 6, in which the intersection of {— ]=0O and ;—=0 has
dx / da ,

been drawn in both points on the left of the point in which

dp “ie ad
(=) —() has the minimum volume, holds for this latter case. The
Taal

42
ae ea L. ae a ee
point in which oe OQ disappears, must viz., lie on the line
&
d*p : ‘ ae
re —Q(. As has already been mentioned before, this line passes
aL”

dp
through the point where (2) = 0 has its smallest volume, and as
v v

dv , af : <p in
is easy to calculate — is then always positive. If now in fig. 6
aL
Pap *P
the line —— = 0 contracts, and it must vanish on = = 0, then the
ave av

point in which it disappears, lies at smaller volume than that of
dp <

(=) =(. For the opposite case the two points of intersection must
td Lv »

therefore be drawn right of the point with minimum volume. Also

the intermediate case has now become clear. In this respect there
2

is an inaccuracy in the drawing of fig. 5. The line ee which
ax

has already almost quite contracted, must be expected there on the

right of the point in which (Z)=0 has the smallest volume. So
bys.

dp ; ae
ihe line (Z)=0 would have its minimum volume more to the

da
left in fig. 5. In fact, with rise of temperature all these lines are
subjected to displacements — however, not to such a degree that

the relative position is much changed by it.

( 837 )

All these remarks seem essential to me for the following reason:
we shall, namely, soon have to draw the relative position of the

d*p d*w : : ; :
curves —— — 0 and —— — 0, also in regions lying more to the right
da* dv? wets *

of fig. 1, in order to decide about the more or less complexity of
the plaits at the different temperatures. Then we shall have to make
assumptions as to this relative position, which otherwise might seem
quite unjustifiable. A great many more similar questions would even
have to be put and solved, before alle doubt as to the validity of
the assumptions would have been removed. And it remains the
question if for the present the imperfect knowledge of the equation
of state for small volumes does not prevent our ascertaining with
perfect certainty whether a phenomenon of mixing or non-mixing
is either normal or abnormal. So, before proceeding to the appli-
cations I shall subject only one more point to a closer investigation,
viz. the question whether in the critical point of a mixture taken

ae oie ;
as homogeneous, the quantity det 1S positive or negative, so the
ax

sign of the quantity:

db\? d*a
8a aI daz da?

27b |a(1—a) | 48?

or of

xv (1—2) Ab? Be a

da da\? ; :
As 2a age ae + 4 (a,a,—a,,"), we may also write for the
wv

last form:

db~? da’?
1 da: 9 \ dz 9 ast a a?
a(1—a)' 406? 16 a? 4 a

As a first special case we consider a substance mixed with a
perfect gas; then 6,=0, a4,=0 and a,,—0. Hence a=a, 2’,
b—b,«. With these values the above form becomes:

1 A ota
#(1—2) 2? a* (1—a)

is negative in the critical point for values of «< ?/,; for

; d
“== */, the curve 2 ss = 0 wil! pass through the critical point. But
L

( 838 )

for «> ?/, the two curves will lie
outside each other, as has been
drawn in fig. 10, and already
observed above. For all other cases in
which a and 4 cannot be equal to
zero, the value of the expression for
©=Q0 and «=1 will be positive
infinite. If it can become negative,
this will, accordingly, have to be
the case for two values of 2. Now
very different relations may exist

Aas ee 1 db
between ——-—} and — — Thus
a dz b dz
ida 4S See a
— — ="/, = > Jor the plane

a dz > 6 dx

circumstances of a mixture taken as
homogeneous. ') With these values the form reduces to a quantity
which will certainly be positive, as even if a, a, > a,,?, the value of
a, a, — te 4 1

can probably never be larger than — ———.,, the mini-
a . 9 «2 (1—2)

mum value of which is ah

= 1 db 1 db

‘ bdx a dx
da 1db_.1db 1B -teleal

value of = OF gage If 5 ney [Ys ire a negative value of the
form is possible. So for mixtures,

in which the components differ
greatly in the size of the mole-
cules, the case of fig. 11 occurs for
minimum 7%, and this minimum
value of 7). could not be rea-
lised. For mixtures, for which

1 da 1 db Laon; b
ops ; as may be ne-

, the sign of the form under discussion, depends on the

a dx

9 /1 da’!
glected with respect to ee ;
which is even perfectly allowable
in the limiting case, for which

2
b= 0, and

rig. 11. da

_~ Will be negative,

1, In all the above calculations the equation of state has been applied with

( 839 )

3 (1 da = a 1
4 \a du a(1 - 2) :

As minimum value for which this is the ease, we should then have:

when we put:

adzx,” 38
In all such cases, in which the critical circumstances of a mixture
2

Tos se ., ap
taken as homogeneous, fall in the region in which —— < 0, these

az
circumstances are not to be realised. Nor are they to be realised when
aw

da?

> 0, but then the spinodal curve passes at least at a small

distance round this point, and the plaitpoint circumstances are not
very different from these which would be the critical circumstances

21s

; tp Neh
with an homogeneous substance. If a < U0, a considerable difference
at

may be expected.

k
THE SPINODAL CURVE AND THE PLAITPOINTS WHEN tk IS POSITIVE.
( Lv

Let us now again proceed to the discussion of the course of the spinodal
eurve and the plaitpoints; but now in the case that with increasing

value of 6 the quantity 7% rises. Let 7%, be much higher than 77, ,

a, a Pw
and ee Now two eases are possible. The value of a. a AY be
) &

2 1

positive or negative in the critical points of every arbitrary mixture.

For «=O, and in general for very small value of «, iene a
av Se

is very large, —— is certainly positive, however large the value of
P = da

Ma fonace ae ee
eo may be, and also for values of «x differing little from 1.
av
da
dz” . Pw - Pi : By . ;
If — is small, rig positive in the critical points of all mixtures.
a ax

2
=

a
But for large values of —~: a there are two values of x, between
Wt

value of not depending on v. Hence in this equation we get the factor 2/. for

5
which, as we have already repeatedly observed, we should really substitute —.

( 840 )

d’w

Akt

which is negative in the critical points. If in this case we draw

C ' . ;
the line 7 =0 with a top, either at «=O, or at a small value
av

of w, the curve = 0, which is chiefly restricted to the left side

dx
4 ; ; dw

of the v,z-figure, lies partly outside the curve = = 0, and on the
‘

small volumes. If we now apply the reasoning of p. 737 etc. also

in this case, when we had the reversed state as far as =—(0 i

dv’
concerned, we couclude that for large values of a the spinodal curve
a EP
does not move for away from —— =O, but that it is forced back

av
: Py . :
to smaller volumes for those values of «2, where a is negative,
ax

Pw
and draws again very near to =
u

— 0 with very small values of z.

Naturally the course of the g-lines in connection with the course of
the p-lines must indicate this.

The course of the p-lines for
this case must be derived from
the right side of fig. 1, from which
appears at the same time that the

aw
curve = 0 occurs, but with
Lav
sensibly smaller volumes than
=u
those of ———0O. And the course
av

of the g-lines is then indicated by
fig. 5 or perhaps sometimes by
fig. 6. In fig. 12 a couple of p-
lines have been drawn and a q-
line which touches these p-lines,
which lines yield, therefore, points
for the spinodal curve. Here again
three plaitpoints are to be expected: 1s'. a realisable plaitpoint P,

a wey
above the curve — =0, 24, a hidden plaitpoint P, on the left of
at
dy dw : 2 : : E :
ae 0 and above a 0, and 3". the ordinary gas-liquid plait-
aL v

be.

( 841 )

point P,; on the left of = 0, but shifted to the side of the small

dv-
volumes. Now it is to be expected that the value of p in the first
mentioned plaitpoint is smaller than in the last mentioned. For 7%

strongly rising, the pressure strongly decreases when we pass along
ups : i,t

to the right and only if agree should strongly

x 5

the curve
dy?
43

project above —— = 0 we should enter the region of high pressures.

av
The hidden plaitpoit has, of course, far lower pressure than the
two others. The value of a for the first mentioned plaitpoint is larger
than that for the hidden plaitpoint. The gas-liquid plaitpoint has the
smallest value of «. Proceeding along the spinodal curve we get a
course of p, as has been previously drawn by me. (See These Proc.

dw
March 1905 p. 626). If 7’ is made to increase, + =0 contracts.
av
2

The top moves to the right, and reaches a position, in which a
Mv

is negative for the critical circumstances. But this means that the
gas-liquid plaitpoint and the hidden plaitpoint have coincided already
before. When they coincide we have again, as we observed p. 744

dv dv d*v dv a By +e i r
es — - s ‘J o.2 = Lae a cae — isis . : Md -
iZ p \da)q \da*}, \dx* ]q . dx}, \da*), =eY COE

ciding we have again a simple plait with a simple plaitpoint. But
the plaitpoint lies far more to the left than would be the case if the

2aTs

u ;
~ = 0 did not exist any longer, and it also has a much
ax

larger pressure. With further rise of 7’ nothing of importance is to
2

d
be expected. For neither the fact that — =0 lies quite outside

Lv
dw dw ,
ss, ==; nor that aay = 0 vanishes, gives rise to new phenomena,
Uv av

because this takes entirely place in the unstable region. If we now
draw either the value of the plaitpoint temperature or of the plait-
point pressure as function of z, and if we restrict ourselves to the
realisable quantities, so excluding the hidden ones, this line separates
into two detached parts. The right part begins at that value of z,
in which the plaitpoint P, possesses a pressure large enough to show
itself on the binodal curve of the plait whose plaitpoint is P,, and
passes then to «= 1. The left part begins at «= 0, and disappears
before P, and P, coincide, namely, when P, lies on the binodal
line, of which /P, is then the plaitpoint.

curve

( 842 )

That what we have called hidden plaitpoints, can never exhibit
themselves, requires no explanation. That what in general we have
called realisable plaitpoints, need not always show themselves, may
indeed be assumed as known from the former thermodynamic con-
siderations about the properties of the w-surface — but yet it calls
for further elucidation now that we examine the properties of stability
and of realisability by considering the relative position of the p- and
the q-lines. We shall, however, only be able to give this elucidation,
when by treating the rule to which I alluded in the beginning of
this communication, we have also indicated the construction of the
binodal curve.

To get a clear insight into the critical phenomena for the case
that for mixtures between two definite values of « the critical point

d*w

falls in the region in which is negative, we must again distinguish

two cases; viz.: 1. the case that already at 7 = 7%, the curve

d*wp ; dy , ;
aa partly projects above oa = 0, in which case already at 7= 7,
ak Uv

the two plaijpoints P, and P, are found, and 2. the case that at

daz v

fig. 13 the second case is
represented. Now if for values

1
of 7 >7;,, the top of =a)
v

- . . . a?
=Odoes fall within = O
t a

there must have been contact
of the two curves at a lower
7’, and intersection at a higher
T. As long as the two
curves do not yet touch, the
spinodal curve is little or
not transformed, and no other
plaitpoint is to be expected
as yet than the ordinary
gas-liquid plaitpoint which

dp
lies at smaller value of 2 than the top of = = 0. If the two curves
v

intersect, the plaitpvints P, and P, have appeared first as coinciding
heterogeneous plaitpoints, later as two separate ones. Naturally the

( 843 )

value of w for the two coinciding heterogeneous plaitpoints is larger
than the value of x for the plaitpoint ?,. With further rise of the

dw re Py
temperature, when ae 0 rises further above rite Q, the plait-
points P, and P, move further apart. /, moves towards larger
values of wz, and P, (the hidden plaitpoints) to smaller value of x.
And the two heterogeneous plaitpoints ?, and P, coinciding at still
higher value of 7’, there is a continuous series of values of ‘x from
«=O up to x=1, for which plaitpoints occur. For every value of
only one. I have drawn (These Proc. VII, p. 626) the transformation
of what I called there a principal plait and a branch plait. But
this transformation refers, properly speaking, more to the binodal
curve of such a complex plait than to the spinodal curve. If we
then draw aT as function of z, such a line has a maximum and
a minimum value, both lying above 7;,. The minimum value at
the origin of the double plaitpoint P, and P,, and the maximum
value at the disappearance of , and P, in consequence of their

coinciding. Also when /,; is drawn as function of «, we get a
- gl ; dp dp dp\ af dT,
similar curve. As in general — = — — :

de dri dT}, da da:

will be

dP yi
Ax

the value of P,; as function of 7); exhibits a more intricate form.
pl mn ( op =) dP, 3 :
ae , —— 1s determined by the proper-
c (Hh) + dv dT i Pee
dx? ] oT
a

dr
ties of the substance in the plaitpoint, e.g. by as
ax pl

d
ey if = 0, because (2) is equal to 0 in a plaitpoint. But
Lu) T

This quantity is

the same for double plaitpoint, and so = has two equal values
in such a double plaitpoint. The plaitpoint curve has therefore two
cusps in the case treated. The left branch extends from 7;, to the
temperature at which P, and P, coincide. The right branch begins
at 7).,, and runs then back to the temperature at which the double
plaitpoint P, and P, originates. The middle branch gives the hidden
plaitpoints. But here, too, we must again notice that not the whole
outside branches can actually be realised, the splitting up into three
phases when we draw near the cusps having a greater stability
then the homogeneous plaitpoint phase. These are the phenomena
observed by Kuenen for the mixtures of ethane and alcohols with
greater values of 6 than that of ethane. Perhaps the change of

( 844 )

db
(=) may already account for the fact that the peculiar feature of
Ae

this phenomenon disappears more and more when, retaining ethane,
we choose an alcohol with larger value of 6; so that the phenomena

point to the fact that a normal plaitpoint line might be expected if

; ae d?
we proceed in the series of the alcohols. As condition for au
av

being negative in the critical circumstances, we had:
db\? da
1 dix 9 dx? 0 937
Sepia see p.
x(1—~2) 4b? 8 a “< ( I )

For in general it is to be expected that this value cannot so
. db
easily be realised for large value of an than e.g. for almost equal
Ax
value of 6, and 6,. That the mixture of ethane with methy! alcohol
displayed quite different phenomena might already be expected on
account of the fact that we have then a case for which with in-
creasing value of 4 the value of 7). decreases. It is viz. almost sure

that 5 is smaller for methyl alcohol than for ethane.
2

d
i. for: 7 = 7, the curve E us =O should already partially project
&

d u
above
dv?

=(), this will bring about but little change in the pheno-

menon. Only the minimum value of 7), will descend below
T;, in the (7,,,«)-figure. In the same way the left cusp will have
to be drawn at lower value of 7’ than 7%, .

It is, therefore, required for the course of the plaitpoint pheno-
mena, that 7’, >> 7), and so according to the value of 7, (p. 830).

Ma
ee a (1—2) eda oie ay
b (1 ain yy)” 27 b,

b
In this inequality «, dependent on the value of es, lies between
1 1

1 1 1
Fr and re and y, between 7 and 0. Let us write:

ay 4- a, —

Th 2a,4 NM (124) O= Ys)
a, (1+ 49)

4 &
ae ay
b,—b,

3 1

( 845 )
1

1
By successively increasing « from a oe

, and deriving the cor-

b
responding values 7, and as from the table of p. 829, we can

2 1
a, + a, — 2a,,

calculate the value which must have at the least in

— 1 b
order to satisfy this inequality. If we put 2 = ee to which —— = 0
. )—

2 1
corresponds, we see that only a, =O might be put. If x is made

to increase, which implies that the ratio of the size of the molecules
at aah Sa 2a,,

approaches 1, the value of ——————-—— required to satisfy the in-
a,
. = . . . 1
equality, decreases. For the limiting case = >, 6,=6, and y,=0,
a, + a, — 2a,, 16

= must be > = to enable us to put 7, >7%,.
; 2

But this value must be larger for 6, smaller than 4,, and the
larger as the difference between 4, and /, increases. If this equality
is not satisfied, so if 7, < 7%,, we have a plaitpoint line of a per-
fectly normal shape. This is inter alia the case when for a low ratio
between 6, and 6,, also a not very high ratio between the critical
temperatures is found. First, however, we should have to know how

a,, depends on a, and a,, before for given ratio of 4, and 4, we

could indicate how large the ratio of - and “? would have to be to
1 2

justify us in expecting either the complicated or the simple shape of

the plaitpoint line. Moreover, I repeat that it should be considered

in how far numerical values occurring in the given equations, would

have to be replaced by others on account of the only approximate

validity of the equation of state.

From all this appears in how high a degree the properties of the
3

function influence the shape of the plait, and so also the miscibility

ce
or non-miscibility in the liquid state, and that the influence of the
properties of this function may be put on a level with that of the

3
function

We shall further demonstrate this by also examining
v
2

the case that the curve eM exists, and intersects the curve
Ai
dp

dadv =

( 846 )

Let us now take a region of fig. 1 such that the line ze = 0
occurs on it, and that this line has the position as drawn si fig. 6.
Then the liquid branch of = =0 lies on the right side of the region

=(. These two curves might

at larger volumes than those of

: ; dy

intersect on the left side. If now also the curve = = 0 “cecum
Lv

which will be the case if the temperature is low enough, and if

;
= 0 and — = 0, we have the shape
av

this curve intersects both

At v
of the g-lines as drawn in fig. 6, and there will again be formed a
complex plait, whose shape and properties we shall have to examine.

Strik-p-lijn = loop-p-line.
Fig. 14.

( 847 )

iw wp
That intersection may: take place of s =O with id ==" (>) ae

da dadv

been proved on pages 854 and 835.

We saw before that one gq-line may possess 2 or 4 points of
contact with p-lines, but now we have a case in which the number of
points of contact can rise to 6. In fig. 14 has beeu drawn: 1. the curve
d, dp ;
—=0 and = 0, 2. the loop-p-line, 3. a q-line to which horizontal
v v : i
tangents may be drawn in 4 points, and a vertical tangent in 1 point
and 4 portions of 6 p-lines touching the gq-line. The pressure in
point1 is much larger than in 2, rises then, has a maximum in 3, descends
again and reaches in 4 its lowest value. The gréatest pressure is
found in point 5, and in 6 the pressure has been drawn lower than
in 5, but it may be higher than in point 1. Consult fig. 1 for the
direction of the p-lines in the points of contact. These 6 points of
contact are again points of the spinodal curve. So on the right there
is again a portion of the spinodal curve which follows closely the
' yp : : ; :
line Fath in its course, also on the left a portion that does not

av
move far away from this line. But between these two portions the

spinodal curve must have been strongly forced back towards smaller
-]2

€
volumes to avoid the line Sas — 0.

dx
dp . lw
In the points where — =O intersects the curve fae 0 the
aL awe

dy 2
dy =)

spmodal curve touches this curve, because 72 must be ==

dx dw
: dv?
for the points of the spinodal curve, and so it must remain in the
2 2

region where dat is positive, except when =Q. It may then
a ;

AvavU
even be doubted if v > is found for all the points of the spinodal
curve.

Values of » <6 would mean that the left part and the right part
of the liquid branch of the spinodal line would remain separated
from each other; and this would imply for the miscibility or non-
miscibility of the components that at the temperatures for which this
is the case, even infinitely large pressure would: be insufficient to
bring about mixing. Already in my Théorie Moléculaire | raised this
problem, and I showed, that if 4 is a linear function of «, cases

58
Proceedings Royal Acad. Amsterdam. Vol. 1X.

( 848 )
are conceivable in which the spinodal line could intersect the line
.- ab vt
v = 4 twice, but that if = has positive value, as is really to be
a

expected, intersection will never take place. But if we acknowledge
again that the knowledge of the equation of state is insufficient for
very small volumes it follows that we had better not pronounce the
solution of this question too decisively.

If the spinodal line is closed on the side of the small volumes,
then a realisable plaitpoint will be found there, while there must
be a hidden plaitpoint in the neighbourhood of the points 2

2
=), as it now

d
and 3. If the temperature is raised, the line -
v

BP :
ey = 0, can contract to above pe :
dadv dv?

d?
before disappearing. If it has sufficiently ascended above > =e
v

also intersects the line

the spinodal line will get a point where it splits') up, at which 2
new plaitpoints (homogeneous ones) are formed. So at this splitting
dv dv ae!
= 0 and —— = 0. This furnishes an indication as to the
da?p da*,
place where this splitting point will lie. That the g-line below

point

dy f2 ;
aaa O must have a point of inflection, has been shown before
aALav

(p. 736), where we derived a series of points of inflection of the
2

d
q-lines passing through the point in which <r =0 has the greatest

volume. We have also previously (p. 628) derived a series of points

d,
of inflection of the p-lines starting from the point where —— 0
Ak

dp j Z
and ae A =O intersect, and passing through the point where — =0
i

(Db)
has the minimum volume. From this we conclude that the double

d, Tech
plaitpoint can only occur when the line — = 0 is intersected by
v

ae 0 pretty far to the left of the point with minimum volume,
ax

d.
and so not far to the right of the asymptote of the line - ==),
&

!) This splitting point I had already in view in my Théorie Moléculaire (Cont. II
p. 42 and 43) where I indicate the temperature at which the detached plait (longi-
tudinal plait) leaves the v,z-diagram, when it has not contracted to a single point.

( 849 )

I may remark in passing that Van per Ler’s observations for water
and phenol illustrate the case discussed here, and that through the
existence of a maximum pressure the properties of the vapour-liquid
binodal line give evidence either of the occurrence of the asymptote
of the line = = 0 in the v,#-diagram, or of its lying not far to the
left. So there are 4 plaitpoints after the appearance of this double
plaitpoint. So two serve as plaitpoints of the plait which is detaching
itself and they are both realisable according to our nomenclature
and when detachment has taken place, both can actually be realised.
They serve then as plaitpoints of what must properly be called a
longitudinal plait. The two other plaitpoints, viz. the hidden plaitpoint
which we placed in the neighbourhood of the points 2 and 3 above,
and the lowest of the newly formed plaitpoints then form a couple
of heterogeneous plaitpoints, which do not show themselves on the
binodal curve of the vapour-liquid plait and will soon coincide and
then disappear. From this moment the binodal lines of the two plaits
are quite separated and behave independently of each other. The
vapour-liquid plait is then simple and perfectly normal. But also the
longitudinal plait may then be considered as a normal one.

(To be continued.)

Waterstaat. — “Velocities of the current in an open Panama cana.”
By Dr. C. Lety.

(Communicated in the meeting of March 30, 1907).

§4. After an elaborate investigation the American Government has
resolved on the execution of a project of a Panamacanal at high
level, viz. at a height of 85 feet (25.9 M.) above the mean sea level.
It will have three flights of locks.

Against this project of the minority of the Board of Consulting
Engineers of 1905 there was a counterproject of the majority which
favoured a canal at sea-level or rather a canal with one pair of locks.
This canal would have been provided with one pair of locks in order
to separate the Atlantic Ocean from the Pacific, but for the rest it
would have been in open communication with these seas on both
sides of the locks.

As a matter of fact this canal would not have been an open canal,
therefore, like the Suez Canal, but a canal in which in most eases,

58*

( 850 )

if not in all, lockage would be necessary. A canal, therefore, which
probably would have resembled more closely to the lockcanal pro-
posed for Suez but not executed and strongly opposed, than to the
present open Suez-Canal.

The question therefore presents itself whether the Panama-Canal,
like the Suez-Canal might not have been made open and without
sluices.

The technical commission of the International Congress of Paris
in 1879 deemed a lock near the Panama-terminal an absolute neces-
sity, because it was supposed that, without it the tidal motion of the
Pacific would cause currents in the canal of a velocity of 2 to
2.50 M. per second ’).

On the other hand the Board of Consulting Engineers of 1905
rightly judged that the necessity of such a lock was not established
but, owing to lack of time, it was not able to investigate the matter *).

On page 56 of the report we find as follows:

“The question of the necessity of a tidal lock at the Panama end
“of the canal has been raised by engineers of repute, but the limited
“time available to the Board has not permitted the full consideration
“of this question which is desirable. It is probable that in the
“absence of a tidal lock the tidal currents during extreme spring
“oscillations would reach five miles per hour. ‘(2.24 M. per second)”
“While it might be possible to devise facilities which would permit
“ships of large size to enter or leave the canal during the existence

‘) This opinion clashed with that of the original projectors Messrs Wuse and
Recius. In a statement made by the latter at the meeting of the Technical
Commission of May 19, 1879 he explains that the inclination of the high and low
waterlines in the Panama-Canal will be about the same as on the Suez-Canal, as
a consequence whereof velocities of the current might be expected in the Panama-
Canal which would not exceed very appreciably those of the Suez-Canal. The latter,
as far as they are due solely to the tides, usually do not exceed 0.90 M. per
second; under the influence of wind they may increase to 1.30 or 1.35 M.

2) At the time of the meeting of the Consulting Board competent experts were
still of opinion that a lock at the Pacific-terminal would be necessary. Such appears
clearly from the letter of Mr. T. P. Saonts Chairman of the Isthmian Canal Commission
received by the Board at the beginning of its labours. In this letter occur the
following lines:

“A disadvantage which the two plans have in common is that the rapid develop-
“ments of naval architecture make it difficult to determine the proper dimensions
“of the lock chambers. It is to be considered, however, that up to the present
“time such developments has not been greatly hampered by deficient depth in the
“harbors of the world, and that development here after will have that obstruction
“to contend with. Moreover, it is not possible to dispense with locks entirely. Even
“with the sea-level canal a tide lock will be required at the Panama end”.

( 851 )

“of such currents, the Board has considered it advisable to contem-
“plate and estimate for twin tidal locks located near Sosa Hill
‘even though the period during which they would be needed would
“probably be confined to a part of each spring tide.”

It would require a special investigation, however, to know whether
in a canal provided with locks, those locks would have to be used
only during part of the spring tides.

For, the oscillations of the sea above and below the mean level
executed in a period of three hours are on an average + 1.23 M.
at neap tide and + 2.53 M. at spring time. This being so it seems
probable enough that, both in the interest of navigation and to
prevent eventual damages which might be caused by the closing of
the lockgates against a strong current, lockage of the ships would
be preferred to passing the lock with gates open. For, assuming the
total profile of the locks to be equal to the profile of the canal,
observations made in the Suez-Canal justify us in evaluating the
velocity of the current at 0.70 to 0.90 M. at mean neap tide and
at 1.00 to 1.50 M. at mean spring tide.

At all events, each time after the gates having been closed the
passing of the lock with gates open would not be possible before
the sea had again reached its mean level. As a consequence, at
each tide requiring the closing of the gates, the period during which
passing of the lock with open gates would be possible, would be
less than three of the six hours included between two returns of
the sea to its mean level.

Howsoever this be and leaving out of consideration the question
to what degree a lock in a sea level canal will be an obstacle to
navigation, it appears at all events that the necessity af such a lock
has remained an unsolved question when in 1905 the projects of a
Panamacanal were examined. The cause thereof lies in the uncer-
tainty about the velocity of the currents which will occur in an
open canal, particularly as a consequence of the tidal motion of the
Pacific.

In addition to the motion caused by the tides, great velocities of
the current may occur in a sea level-canal, with or without tidal
lock, at the time of high floods of the Chagres and other rivers, if
the water of these rivers must be carried off by the canal. In
contradistinction to the project of 1879 such would have been the
case in the sea-level canal according to the project of the Board of
Consulting Engineers.

The Board comes to the conclusion that in a sea level canal
with tidal lock currents will thus be caused reaching a maximum

( 852 )

velocity of 1.148 M. per second. (2.64 miles per hour). The Board
is of opinion that such a velocity will be no hindrance to navigation.

These same velocities will occur in an open canal as well as in
a sea-level canal with tidal lock, at least if in both cases the water
of the rivers must be carried off by the canal. They occur very
rarely however and need not necessarily lead to an increase of
the maximum velocities caused by the tidal motion.

§ 2. The reasons which led the technical commission of the
Congress of Paris in 1879 to expect currents with a velocity of
2—2,50 M. per second in an open canal, are twofold. In the: first
place, the commission gave some examples of currents with a velo-
city of 2—38,50 M. per second observed on the lower course of
rivers where similar differences exist between high and low-water
as on the Panama canal on the Side of the Pacific.*)

In the second place the commission published a memorandum of
Mr. Kurtz, one of its members, containing some summary calcula-
tions in regard to the velocities which must be expected in an open
canal.*)

It is evident that on the lower end of a river with a great
amplitude of the tide very considerable velocities of the current
may occur; but it does not follow that equal velocities w7// occur
in an open Panama-canal. This will be the case only if the remaining
circumstances which have a decisive influence on the velocity are
about the same in the two cases. Now it is evident that the velocity
of the current caused by the tidal motion of the water will be no
less dependent on the depth and in particular on the mean depth
for the whole of the width, than on the amplitude of the tide and
this irrespective of the question whether we have to do with a river or
with an open canal of relatively great length. In other words the
velocity of the current will depend as well on the proportion of
the amplitude to the mean depth as on each of these quantities
separately. Furthermore it is easily seen that in a river these velo-
cities will depend in a great measure not only on the discharge
but also on the changes of width and depth and on the inclination
of the bottom near its mouth. In fact, the examples communi-
cated by the commission show clearly that the velocity must be
dependent in a high measure on other causes besides the amplitude
of the tides. For among the examples of the commission we find
the Riviere de lOdet with an amplitude of the tides of 5 M. and
') See: Congrés international d’études du canal interocéanique. Compte rendu
des séances. Paris 1879 page 362.

*) Ib. p. 384 and Pl. IV fig. 6.

( 853 )

velocity of the high water flow of 3.50 M. and furthermore La
Charente with an amplitude of the tides of 6.85 M. and a velocity
of the high-water flow of 2 M.

As regards the calculation of Mr. Kuezrrz, it is as follows:

ATLANTIC oe oe PACIFIC,
B. High-tide level.
d= 385M*

Mean sea-level.

J= 277 M+

Sia Ye al

Low-tide level.

d= 195M4

a as ee aa ak RioCG th |)

| hte

According to the above figure the area of the wet section of the
canal on the side of the Pacific was adopted to be 385 M.? at high-
water and 195 M.? at low water.

The difference between the mass of water in the canal at high
and low tide is then taken for the volume of the prism ABC.

; : aod) ——= 895
therefore = 73000 « ae or M? = 6.935.000 M?.

As the interval between high and low tide is about six hours,
the change of the mass of water per second is found to be 321 M®
on an average. The mean wet section of the canal on the side of
the Pacific being */, (885 -+ 195) M’, that is 290 M?, Mr. Kuerrz
derives for the velocity of the inflow during the whole period of
high tide or for the outflow during the whole period of low tide

321
290

Furthermore, assuming that the most rapid change of the mass of
the water will occur about at the time at which the sea-level is

= 12TM.

( 854 )

equal to the mean level and besides, that this most rapid change is
equal to double the mean change, the maximum inflow is put at
2 < 321 = 642 M*.
As the wet section of the canal at the mean level is about 277 M?,
ae

we find = 2.32 M. per second for the maximum velocity.

272

It is easily seen that these calculations are valueless. For the fact
has been wholly overlooked that a certain time must elapse before
some rise or fall at the mouth of the canal on the Pacific will make
itself felt over the whole length of the canal. If therefore, shortly
after ebb, the level in the canal near its mouth begins to rise and,
shortly afterwards, the first inflow takes place, the level of the canal
further inland will still be fallmg and the water will there be
flowing out asa consequence. Similarly when shortly after the moment
of high tide on the sea, the level of the canal near its mouth begins
to fall and shortly afterwards outflow sets in, the level further
inland will still be rising and there the inflow will not yet have ceased.

Moreover the in- and outflow of the canal on the side of the
Atlantic has been left wholly out of consideration. They will certainly
not be small but will not take place at the same moments as the
in- and outflow on the side of the Pacific. We may see that the
difference in time, before mentioned, will not be insignificant but
will have a great importance, by considering that, on the Suez-canal,
ihe propagation of the high tide takes place with a velocity of about
10 M. p. second. Assuming the same velocity for the Panama-canal
the propagation of the tidal motion over the whole of the length
of the canal will require about 2 hours. As a consequence the
currents near the two terminals of the canal will have different
directions during a great part of the tide.

The incorrectness of the reasons for the conclusion of the congress
of 1879, according to which a lock is to be considered an absolute
necessity seems to have attracted little attention at that time, and
consequently the canal was originally executed with the intention
of building a sluice on the side of the Pacific.

FERDINAND DE Lesseps, who always considered it a great advantage
that the Suez-canal was executed without locks, probably never
favoured this lock in tbe project of the Panama-canal. This led
him in May 1886 to address himself to the French Academy of
Sciences, requesting it to institute an investigation about the influence
of the tidal motion of the Pacific and the Atlantic on the motion
of the water in an open Panama-canal.

The commission charged with this investigation reported on the

( 855 )

matter in the meeting of 51 May 1887. This commission consisted
of the members of the section of Geography and Navigation and
besides of the members Daupréz, Fave LALANNE, DE JONQUIERES and
BoussinesQ and the reporter Bouquet DE LA GRE.

This report, though short, contains the results of extensive com-

putations, which led the commission to the following highly remar-
kable and important conclusion.
“que, dans aucun cas, les courants dus a la dénivellation ne pour-
“ront depasser 24 noeud” (+1.29 M. par seconde, ‘et que cette vitesse,
“qui ne peut étre attemte tous les ans que pendant quelques heures,
“ne parait pas de nature a géner la navigation des bateaua a vapeur
“dans le canal que Con creuse actuellement « Panama’.

This conclusion was accepted by the Academy and the question
concerning the possibility of an open Panama-canal without Jocks
was placed in quite another light than that in which it appeared after
the congress of 1879.

Owing to particular circumstances, this conclusion of the French
Academy of Sciences has attracted comparatively little attention.
For in the same year that this conclusion was reached, the original
project of a sea-level canal with lock had to be given up and to
be replaced by a canal with several locks. It was the beginning of
the sufferings of the Panama-canal.

Since then the principal consideration has always been to limit
the excavations to the utmost. For this purpose the hilly country
required a canal at high level, consequently several locks.

§ 3. Therefore, if we wish to answer the question whether an
open Panama-canal without sluices is possible, we have to inquire
in the first place, whether the report of the French Academy of
Sciences, of 1887 is based on sound foundations.

What were these foundations ?

In accordance with observations at the tide-gauge at Panama the
differences between high and low water, in other words, the ampli-
tudes of the tides at the mouth of the Panama-canal were adopted
to amount to:

at neap tide, on an average 2.46 M.
2 spring 2 +) +) ” 5.06 M.
= » », Maximum in March or Sept. 6.76 M.

The commission now calculates the velocity of the current for this
maximum difference in height of the tides on the Pacific of 6.76 M.,

( 856 )

neglecting the usually small tidal oscillation in the Atlantic and
further starting from the following suppositions :

1. that experience shows that on a canal communicating on the
one side with a sea of variable level, on the other side with a sea
of constant level, the amplitude of the tidal curve diminishes uni-
formly from one sea to the other and further that the retardation
of the tide is proportional to the distance, that therefore :

if Y = half the amplitude of the tides of the Pacific,
/= length of the canal, 3
w = velocity of propagation of the tides,

the level y, with respect to the mean canal- or sea-level, at a
distance x from the Pacific, will be:

y= — v(a -+) con. (26 )
z l a)

2. that, in accordance with what has been observed on similar
canals, particularly on the Suez-canal between Suez and the Bitter-
Lakes, the velocity of propagation of the tide can be represented by
the well known formula :

ow ae 3
oes g @ aaa ) + Kv
where :
H= depth of the canal below mean sea-level,
v = velocity of the current,
kK = constant (0.4 at flood-time, 1.2 at ebb);

3. that, from the levels which have been derived by means of the
suppositions 1. and 2. for any moment and for two mutually not
too distant places, the velocity of the current for that moment may
be computed by applying the formula :

v = 56,86 YRi — 0.07.

By means of these suppositions the velocity of the currents have
been computed for places at 9, 27, 45 and 63 K.M. from the Pacific,
assuming a tide of the amplitude of 6.76 M. The results are as
follows *):

‘) The length of the canal which according to the project made at that time,
would amount to 72 K.M. has been put at 76 K.M. in the calculations to allow
for the curves. The bottomwidth was put at 21 M, the depth at 11.50 M. below
mean sea-level at Panama, and 9 M. at Colon, the slopes at 1 horizontal on
1 vertical.

Time elapsed | Distances from the Pacific.
since low tide ——— ——— a is ee
on the Pacific; 9 K.M. | 27 K.M. | 45 K.M. | 63 K.M.

Moon-hours. | Velocities of the current in M. per second.

0 eimestiescsn.00 | = 0.77 |-~0.60
3 —o.si| —0.90| —0.93| —0.79
1 — 0.60) — 0.8% | — 0.87) — 0.83
42 04s 2) | — 0.82: |, — 0.85
2 gusa =059| = 0.75 | =: 0.86
4 + 0.67) — 0.34) — 0.63 | — 0.81
3 + 0.84) + 0.35 | —0.42| —0.73
31 + 0.93} + 0.63] + 0.08| — 0.61
4 fe0r0S Weer 0s7e 8 0.43 | —.0.44
43 H402-- 420:-93 h -=0'80'| 0

5 4+.4.47|} +141.06| + 0.82 | + 0.51
54 + 4.16 | +41.41| + 0.86] + 0.66
6 Ef 09). 4.061 + 0.98:| +E 0.76
64 = 0,07. 4.01 | + 0.97 | + 0:85

current from the Pacific towards the Atlantic
5 P » Atlantic " » Pacific

ll ll

From these computations follows that the maximum velocity in
the canal on the side of the Pacific, due exclusively to the tidal
motion, will amount to 1.17 M. Supposing that there might be
some difference between the mean sea level of the Atlantic and the
Pacific and that this difference might amount to 0.50 M., the com-
mission concludes that the maximum velocity might then increase to
1.26 M. The commission thus finally arrives at the conclusion referred
to above.

§ 4. The two first suppositions on which the computations are
based will probably not seriously deviate from the truth. For they
are, at least partially, confirmed by what is observed on the Suez-canal.

The commission further points out, that the formula for the
velocity of propagation of the tidal wave, which has been derived
in the supposition that the amplitude of the tide is relatively small
as compared with the depth of the water, leads to results which, for
the Suez-canal, agree closely with the observations. For the formula

( 858 )

leads to a velocity of propagation of 10.06 M., whereas we find
9.54 M. by observation.

Matters stand somewhat differently for the third supposition. The
formula by which the velocities of the currents are computed is the
well known formula for permanent uniform motion. It is in the
nature of the thing that such a motion cannot occur in a canal where
a strong tidal motion takes place. But the question on which every
thing depends is not so much this, whether the use of this formula
leads to sufficiently correct velocities for any moment, as the following,
whether the computed maximum velocities are not too small.

In reference to this question we may remark that in general the
formula will lead to too small a value of the velocity during the
period that change in level is accompanied by decrease of inclination ;
to too great a value where the change is accompanied by an increase
of inclination.

If, taking this into consideration, we examine the parts of the
canal K.M. O—9 and K.M. 9—27, during the period of 4'/, to 6
hours after low tide on the Pacific, we get as follows:

Time elapsed Mean inclination

sincelowtide K.M. 0—9 | K.M. 9—27

4$ hours 0.000044 0.000040

ae 0.000048 | 0.000046
| me

5} =|: 0.000048 | 0.000047

6 , | 0.000044 0, 000045

From these data it appears that, during the half hour preceding
the epochs at which the velocities reach their maximum value at
k.M. 9 and 27 the mean inclination for the part O—9 as well as
for the part 9—27 has been little variable but increasing.

From this it follows that by the application of the formula at these
epochs we probably cannot have made any important error.

Meanwhile, in order to test the validity of the computations, we
have still to inquire whether the computed velocities, taken in con-
junction with the computed levels, satisfy the equation of continuity,

di _ dv dl

—— soe Se 4

Pe du 2
where / represents the area, v the mean velocity of the wet section
at the distance z from the Pacific, at the epoch ¢.

( $59 )

We can make out, approximately, in how far the computed levels
and the velocities satisfy this condition by availing ourselves of the
levels and velocities computed for each half hour and for the different
distances from the Pacific. We thus find as follows :

A. For the differences in the discharge at 9 and 27 K.M.
distance from the Pacijic.

Moon-hours | Area Velocity sae Per half hour in Excess of
fol
elapsed since I ee fo dD part 9—27 inflow over
low tide onthe — ; — -——)| outflow in
| | | hed .
Pacific. | 9 | 27 | 9 | 27) 9/27] m flow out flow __ half an hour
i
| M?.| M?.| M.| M. | es M°. M®. M®. M’.
ene
4A 450) 388/1 .02,0.93) 459) 361
° | Steal 914000 715000 +201000
5 | 475| 407/14 .47\4 .06| 556} 431
1.013000 807000 | = —-++206000
5s | 491) 4201.161.44] 570} 466
| 4.002000 | 828000 174000
6 | 498) 4281 .09/1.06) 543) 454

B. For the change of the mass of water contained in part 9—27.

Moon-hours Area Change of area Mean Change of mass
. during half
elapsed since | I an hour. change _ per half hour
low tide on the | for part for part
Pacific. 9 | 27 9 | 27 9-97, | 9—97,
| |
Me | M2 M2. M*. | M. | M*.
At 450 | 388 |
| +9} +19|/ +22 | + 396000
5 | 475 407
+16) 413) +14 | + 261000
5} | 491 | 420 | |
| +7) + 8); + 7} + 185000
6 4935 | 428
\

Comparing the last columns of the tables A and Bb we get the
following differences for part 9—27:
from 4'/, to5 hours + 195000 M°, or on an average per sec. + 108 M?.
mearelh Of 55 S5000 3 355 53 4 3 ap =e
Bonfy-,, 6 SET ROMNON LEE oe 3th 5 Jo. shh et eg

It appears from this comparison, that by the computed velocities,
taken in ‘conjunction with the computed levels, the condition of
continuity is not fully satisfied.

Therefore, assuming the levels to be correct, the velocities need
some correction.

( 860 )

Suppose these corrections for the consecutive half hours to be
for’ KeM vi 9 eee; cs; oy
for KM 27 = 630), 657 9,;
we find for the values of the corrections:

J, == + 0.1 pe: d,' = —.0,18 iM.
d= 440.12 5, ie eee
J, = — 0.04 ,, J, =+ 0,04 ,,
J,= — O08... J, 2 S0L <,,

Therefore, applying the corrections, for the velocities themselves:
at. KIM, -9 =. at KM. 27

at 4'/, hours 1,17 M. 0,80 M.
Age a 29. ie 0,965
5s tO) seme b> TAS 1 Ae
Bea i 108s. SMO mee

From these numbers it appears that we can satisfy the condition
of continuity at least for the part 9—27, during the period between
4'’, and 6 hours after low tide, by relatively speaking slight modi-
fications of the computed velocities.

It cannot be denied, however, that the circumstance of the condition
of continuity not being necessarily satisfied in applying this method
of computing the velocities, indicates that this method is uncertain
to some extent; though it appears that the uncertainty, at least as
regards the calculation of the maximum velocities, will be small.

Another reason of uncertainty in the computation of the velocities
lies in the value assumed for the coefficient of the formula for
uniform motion.

This value, 56,86, is not the result of a great number of obser-
vations made on rivers and canals of about the same inclination
and depth as the Panama-canal, but of observations for rivers of
considerably smaller depth.

We may of course test the validity of this coefficient, as well as,
more generally, the validity of the formula itself, by comparing the
velocities it yields for the Suez-canal with those really observed there.
Of the observations which have been made about the velocities in
that part of the canal which lies between the Bitter Lakes and
Port-Thewtik, those of 23 July, and 8 en 22 August and 6 September
1892 have been published ’).

These observations, however, are insufficient for a fair comparison.

1) See: The Suez-canal according to the posthumous papers of I. F. W. Conran
arranged by R. A. van Sanpick. Tijdschrift Kon. Instituut van Ingenieurs 1902—1903,
p. 89 and 90,

( 861 )

They have been made for two parts of the canal each 200 M. in
length and separated by only 4.9 K.M. One part was included in
that division of the canal which at that time had been widened to
a bottomwidth of 37 M. while the other, having a bottomwidth of
only 22 M., was situated a little beyond the point of transition to
the not yet widened canal. As a consequence the motion of the
water on the whole of this part of the canal, 4.9 K.M. in length,
cannot have been uniform *).

Moreover these observations are only relative to the velocities in
the middle of the current, observed by means of floats down to a
depth of 6 M. below the surface, whereas the velocity given by
the formula represents the average velocity for the whole of the
wet section. Meanwhile a comparison of these observations with the
results obtained by the formula might still give some idea about the
reliability of the formula.

The comparison of the observations referred to above with the
results yielded by the formula, putting the coefficient at 56.86, lead
to the following results:

OBSERVATIONS ON THE SUEZ-CANAL IN 1892.

| hein dif- |

ee a as : Computed
. Distance Observed velocities.
D BPS Direc- Ibetween erence of Averages during an | mean velo-
ay and hour o i Z aa ; ;

¥ tion of Si ‘tween the| hour in the city for the

1. ae 5—6 p.m. | ebb 4.9 —0.14/ —0.84|-—1.14 | — 0.58
8 Aug. 11—12am.| flood | 4.9 | + 0.09] + 0.69; + 0.87| + 0.47
a 5—6 p.m. ebb | 4.9 — 0.41 | — 0.80; — 0.93 | — 0.57
pee am | ebb. | 49 |- == 046) — 0.98 | — 1.05 | — 0168
he, t2—1 p.m.| flood | 4.9 + 0.07 | + 0.66) + 0.82 | + 0.46
6 Sept. 11—12a.m.| flood 4.9 + 0.07 | + 066 | + 089| + 0.47
—S 5—6 p.m. ebb | 4.9 — 0.10 | — 0.8 | — 9.98 | — 0.53

1) The first part was the widened part of the canal between K.M. 149 and
149.2; the other the not widened part between K.M. 144.1 and 144.3. The tran-
sition of the widened to the not widened part was situated at K.M. 144.4.

2) As the part of the canal from K.M. 149 to 144.4 had been widened the
observed difference of level is relative to the widened part.

( 862 )

From this table we derive for the proportion between the computed
average velocity for the whole of the wet section to the velocities
observed down to 6 M. in the middle of the widened part of the
canal, the following values:

at high water flow at ebb flow
(from the Red Sea) (towards the Red Sea)
1 isi WG 1.45
1.47 1.40
1.45 1.50
1.40 1.61
Mean 1.37 1.44

The true value of this proportion for the case in which observation
and computation agree, is unknown. But if we consider that the floats
went down to only 6M. below the surface, whereas the depth of the
water at flood tide was over 8.50 M. and at ebb time over 7.50 M.
and furthermore, that the canal had side slopes of 1 vertical on 2°/,
horizontal, we conclude that at all events the velocity in the middle
must have considerably exceeded the average velocity for the whole
of the section. As far as can be ascertained therefore, the formula
applied to the Suez-Canal leads to results which do not clash with
the observation.

More conclusive information cannot be derived from a comparison
of the computed velocities to the observed values. As long therefore
as complete observations, made for the widened Suez canal, concerning
the relation between the velocity of the current, the tidal motion and
the dimentions of the section, have not furnished us with more reliable
information about the value of the coefficient and about the question
whether the formula applies fully to the case, we cannot avoid a
relatively considerable uncertainty in the calculation of the maximum
velocity.

§ 5. <A closer examination is therefore required to decide in how
far the velocity of the current in an open canal may cause a hin-
drance to navigation and whether this hindrance cannot be overcome.

In discussing this question we must consider, on the one hand that
the computed velocities represent average velocities for the whole of
the wet section and that therefore the absolute velocities in the
middle of the canal will be more considerable; on the other hand,
however, that the computed velocities are relative to the greatest
possible differences in the height of the tide, The computed maximum

a a

( 863 )

velocities may occur therefore only on a couple of days every year.
And on these days only during a few hours.

In how far a relatively rare velocity of the current offers diffi-
culties to navigation is of course ascertained in the best way by a
comparison to canals on which under similar conditions similar
velocities occur. For such a comparison the Suez-canal offers the
best conditions. For this canal several observations about the velocity
of the current are known. Published observations, however, cannot
lay claim to completeness, at least not for the present purpose. In
the first place because they have not been frequent enough to justify
the belief that among them will have occurred these rare cases which
by an unfavourable coincidence of circumstances, must have given rise
to exceptionally great velocities. In the second place because the
Measurements are, aS a rule, relative to absolute velocities in the
middle of the canal and not to the average velocity for the whole
of the wet section.

Moreover, in comparing the Panama-canal to the Suez-canal we
have to consider that the dimensions of the former will be much
more considerable than those of the latter as originally executed.
Consequently such velocities as have caused no difficulties for the
Suez-canal will cause them still less for the Panama-canal.

For the Suez-canal between the Bitter Lakes and Suez originally
had a bottomwidth of 22 M. and a depth of 8 M. below mean
springtide low water, with which dimensions corresponds a cross
-section of 330 M?. On the other hand the sea level Panama-canal
would get a bottomwidth of about 45.7 M. (150 feet) and a depth
of about 12.2 M. (40 feet) corresponding with a cross section of
855 M?’.

Observations, made during the period 1871—1876, have brought to
light the following facts about the velocities of the current in the
-Suez-canal between the Bitter Lakes and Suez. *)

“The maximum velocity of the high water flow, running North-
“ward, amounts to 0.80 to 0.90 M. at the springtides of the months
“of May and November, to 1.45—1.35 M. p. s. in the months of
“January and February.

“The maximum velocity of the ebb flow running Southward amounts
“to 0.75—0.80 M. at the springtides of the months of May and Novem-
“ber, to 1.20—1.25 M. p. s. in the months of July and August.

“Along Port-Thewfik in the canal south of the main channel

1) Vide the paper of Mr. J. F. W. Conrap pp. 89 and 90.
59
Proceedings Royal Acad. Amsterdam. Vol. [X.

( 864 )

“towards Suez, bottomwidth 80 M., the velocity of the high water
“flow at springtide is 0.60 to 0.70 M., at neaptide 0.45 to 0.50 M.,
“in the winter-season with strong South wind 1.00 to 1.20 M. p.s.
“The velocity of the ebb flow at springtide is 0.55 to 0.60 M. In
“the summer with strong North wind 0.90 M. p. s.

“Outside the mouth of the canal at Port-Thewfik no velocity of
“the current has been observed.”

The observations of 23 July 1892 made under circumstances
which, as regards the flow, were certainly not unfavourable, led already
to velocities which, at flood tide, ranged from 0.95—1.03 M. and
were in the mean 0.97 M. at flood tide and’ 1.11 M. at ebb.

Mr. Davzats, chief engineer of the Suez-canal, speaking at the
meetings of the Technical subcommission of the International Con-
gress for the Panama canal in 1879, stated in regard to the sidings
of the Suez-canal, as follows °*):

“Dans les canaux ou le courant est faible, et la ot n’existe aucun
courant, il suffit de faire les gares d’un seul c6té; mais des que la
vitesse atteint 0.75 ou 1.50 M, il faut les établir des deux ecédtés et
en face Pune de l’autre’.

By this statement we are certainly justified in concluding that the
said engineer, founding his opinion on his experience of the Suez-
canal, deemed allowable velocities of the current of 1.50 M. The
small original bottomwidth of the Suez-canal of 22 M., however,
caused difficulties for the simultaneous navigation in both directions.

The following communications of Mr. E. QuELLENNEC, consulting.
engineer of the Suez-canal company, proves that these velocities of
the current offer no difficulties even for the big ships which at present
navigate the Suez-canal. These communications to the Board of Con-
sulting Engineers of 1905 are as follows :

“In the Suez section the velocity of the current very often exceeds
0.60 meter per second, and reaches at times 1.35 Meters per second.

“In the latter case the ships do not steer very well with the
“current running in; however the navigation is never interrupted
“on account of the current. In the Port Said section ships can
“moor with a current running in either direction; in the Suez
“section they always moor with the current running out’. ?)

The canal between the Bitter Lakes and the Red Sea has at
present a width of about 37 M., but a widening of the cross section

') See: Congrés international ete 1879, p. 361.
*) See: Report of the Board of Consulting Engineers for the Panama-canal, Was-
hington 1906, p. 176.

( 865 )

to 45 M. width and 10.5 M. depth is being executed. After this
widening, navigation will certainly experience still less difficulty
than at present. Meanwhile, and this point deserves attention, the
velocity of the current after the completion of the widening for the
whole of the canal between Suez and the bitter Lakes, will not be
lessened but increased. For, owing to the surface of the two Bitter
Lakes, which is about 23800 H.A., the widening will only cause
insignificant modifications in the level of these Lakes. Consequently
the fall of the water between the Red Sea and the Bitter Lakes
will be nearly unaltered after the widening both at high — and low
water. Under these circumstances the enlargement of the cross
section will necessarily cause increased velocity of the current.

The mere consideration of the maximum velocity which may
oceur during a few hours every year, and even then only on the
side of the Pacific, is evidently inadequate for reaching a true
estimate about the question whether the velocities of the current in an
open Panama-canal without lock will offer difficulties of any impor-
tance for navigation. We have to pay regard in the first place to
the velocities which will regularly occur on the whole length of the
canal at mean spring-tide and mean neaptide.

These velocities may be derived with some approximation from
those found by the French Academy for a maximum difference in
tide of 6.76 M.*), at least if we suppose that these velocities will
not considerably deviate from the truth.

We thus find for the maximum velocities
at K.M. 9 27 45 63

at mean neap tide: 0.70 M. 0.67 M. 0.59 M. 0.51 M.
eee epee) OL. = 8.96 . - 0.85 _,, 0.74 ,,

The following diagrams show the velocities of the current, for the
interval of from 9 to 63 K.M. distance to the Pacific, at mean spring
tide and mean neap tide, 0 to 6 Moon-hours after ebb on the Pacific.
They were derived from the calculations of the French Academy
of Sciences.

1) The approximation neglects the differences of the velocities of propagation of
the tide for different amplitudes. We thus obtain for the velocity 7, at an arbitrary
place, the amplitude being y’, the following value, which is expressed in terms
of the velocity v for an amplitude y:

fy! 4 0.07) = (v + 0.07) Va
y

or
e

( 866 )

SPRING TIDE. NEAP TIDE.

G S MM,
K.f1.65 45 Z2€ 2 Velocity: 3 4 ae 9K.M

+1.00M

+ o.75 Rh

: z
= =
8 +as5oMh =
iS 3S
= =
Ba +0.25M “En
2
1s)
+o0.25M
Z z
= = a sort =
=3'
~2)
= in
+o0.75it
- 100M

The figures inscribed in the diagram represent the hours elapsed since low tide in the Pacific.

”

§ 6. From the preceding considerations we may conclude that, as
far as we can judge by direct computation of the velocities, to be
expected in an open Panama canal, there is reason to think that
these velocities will indeed be somewhat, but not considerably greater
than those on the Suez-canal between the Bitter Lakes and the
Red Sea.

Meanwhile we ought not to forget, that both in these computations
and in our knowledge about the velocities which occur on the Suez-
canal there remains some or rather considerable uncertainty. This
uncertainty might only be diminished by more complete observations
than have been published as yet concerning the relation between
velocity of the current, tidal motion and dimensions of the eross-
section of the Suez-canal. .

We shall be enabled to get at a just estimate therefore about the
question whether an open Panama-canal without lock is possible, only
by following a way different from that of a comparison of the computed
velocities with those observed on the Suez-canal. This way may
consist in trying to get at a direct knowledge of the differences of
the velocities on the two canals by a comparison of the circum-

( 867 )

stances which will occur on the two. Afterwards the cireumstance
that, on the Suez-canal the velocity of the current offers no difficulty,
in conjunction with the probable value of the velocity of this canal,
will help us in deciding whether these differences are of such a
nature as to produce undoubted difficulties on the Panama-canal.

In making this comparison it will be permissible to assume that
the violent winds occurring in the Suez-canal, which cause velocities
of the current 0.30 to 0.50 M. in excess of those due to the tidal
motions, are not to be expected on the Panama-canal near the Pacific.

First, however, we have to inquire whether an open canal cannot
be executed in such a way that for that part where the current
will be greatest the difficulties caused by such great velocities can
be removed. It is evident that this would be possible only by giving
a very great width to the canal. This is practically impossible for
that part of the canal which intersects mountainous country, but it
is well feasible for that part of the canal which extends from the
Pacific to the Culebra mountain, that is to near Pedro-Miguel, a
part which for the greater part intersects low country.

If to this part of the canal, where just the greatest velocities will
occur, a bottomwidth is given of for instance 500 feet (about 150 M.)
instead of 150 feet (45.7 M.) no difficulties will be experienced from
any presumable velocity of the current.

Such a widening of the canal on the side of the Pacifie would
however increase the inclination and the velocity of the current in
the remaining part, at least if no particular measures are taken to
prevent such increase.

These measures would necessarily consist in making a reservoir
or lake in open communication with the widened part of the canal.
This reservoir or lake would have to be of such an area that it would
be capable of retaining the water which, during the rise of the level,
it would receive from the widened part in excess of what would
be discharged by the unwidened part. During the fall of the level
it would restore this surplus to the widened part.

From the nature of the thing this arrangement is theoretically
possible. Whether it be practically possible depends on the surface
which a determinate widening would entail.

A lake of somewhat over 800 H.A., such as is represented on
Plate I, is feasible in the low country bordering on the canal near
its mouth on the Pacific.

Starting from this area it is possible to determine the degree of
widening which may be given to the part near the Pacific in such
a way that, under given circumstances, for instance at spring tide,

( 868 )

no change will oceur in the gradient of the high and low water lines,
nor in the velocity of the current in the remaining part of the canal.

As soon as the amplitude of the tides exceeds that of springtide
the inclination and the velocity of the current will be somewhat in-
creased for the wider part, somewhat diminished for the remaining
part, as compared with what they would be without the widening
of the first part and without the addition of a lake. In the case of
a smaller amplitude of the tides the reverse will occur.

Owing to the situation of the ground the junction of the widened
canal with the lake must be made at a distance of about 12 K.M.
from the Pacific terminal of the canal. Not before 3 K.M. farther
however, that is not before 15 K.M. from ‘the sea, the surface of
the lake reaches a considerable breadth. Therefore if the inclination
of the high and low water lines remains nearly unchanged and if,
according to the most recent project, the length of the canal is
fixed at about 80 K.M., the amplitude of the tide in the lake may
reach (5.06 — 15 « 0.0632) M. = + 4.10 M.

With such an amplitude a mass of water may be received, in
the interval between high and low water, of 800 x 10.000 4.10 M*.
= 32,800,000 M?.

Assuming, as an approximation, that this mass is received within
a period of six hours, we find that on an average 1500 M?®. will be
received per second.

The surplus width of the part of the canal near the Pacific must
be determined in such a way, therefore, that on an average 1500 M*.
may be displaced — without increase of the velocity of the current —
in excess of what might be displaced if the width remained normal.

It is not well possible, without elaborate computation, to fix accu-
rately the surplus width necessary for the purpose. But it is easily
seen that this surplus width must be about 100 M. so that a bottom-
width of 150 M. might be given to the widened part extending
from the entrance of the canal to the junction with the lake. Corre-
sponding therewith the width at the spring tide level would be
about 250 M. At K.M. 64 this width might gradually be reduced
to the normal width.

It will be possible therefore to remove eventual difficulties offered
by considerable velocity of the current on the part of the canal
nearest the Pacific, by increasing the bottomwidth of this part.
(16 K.M. in length). |

Now let us consider how the case stands for the remaining part
of the canal, 64K. M. in length.

( 869 )

On this part the inclination of the high and low water lines
will amount to 3.16 cM. at mean springtide and to 1.52 eM. at
mean heaptide.

On the Suez-canal the inclination of the high and low water lines
between the Bitter Lakes and the Red Sea amounts to 2.52 eM.
per. K. M. at mean spring tide and to 1.48 cM. at mean neaptide.

Under the influence of the direction and force of the wind, the
height of the tides on the Suez-canal may be increased or diminished
by about 0.25—0,33 M.

As a consequence the inclination of the high and low water lines
may be increased by about 1 cM. per K. M.

As the distance of the Bitter Lakes to the Red Sea is about 28 K.M.,
this already enables us to conclude that the velocities of the current
in an open Panama-canal, for the first 28 k.M. on the side of the
Atlantic, cannot greatly differ from those which occur on the Suez-
canal (See Plate II).

If therefore — leaving out of consideration the absolute value of
the velocities — we may assume that the velocity of the current

will offer no difficulties on the Suez-canal even when it will have
been widened, then it follows that on an open Panama-canal, for
about the distance of 28 K.M. from the Atlantic, no difficulties will
be met with on account of the velocity of the current.

Finally as to the middle part of the canal extending for about
36 K.M. between K.M. 28 and K. M. 64 from the Atlantic.

For this part the dijerences between the velocities of the current,
occurring therein, with those occurring in the preceding 28 Kk. M.,
may be computed with sufficient accuracy by means of the equation
of continuity.

For, let ab be the canal’s surface for this part, at the epoch 7, a little
before low water, at the distance of 64 Kk. M. from the Atlantic.
Similarly let a’b’ be the canal’s surface a second later, then necessarily

B A 3
% a2 ee usd Yr

= lv — Iv,

from which :
36000 (B 4- By (Ay 4+ 4y,) [—1

1
Vv, —v)= See ga
( 1 ) 4 i. sy 3
Now the quantities /,/,, b, 6,, Ay and Ay,, are known for the
epoch ¢, at least if we admit that — as is the case on the Suez-

canal — the high- and lowwaterlines for the part 28—64 K.M. are

( 870)

Mean sea-level.

28 oO.
= be an
» H
;
= y
a
48

Area =
Width at surface — B. 6
Velocity = v.

—~
|
|
!

-4$ BUf

nearly straight lines, and further that the velocity of propagation of
the tides is known with sufficient accuracy, likewise owing to obser-
vations made on the Suez-canal. Therefore we will be able to
determine the difference of the velocities at 64 and 28 K.M. distance

tei) Rie eee ae
from the Atlantic, for the epochs at which —— isa small quantity.
f,
This will be the ease near the moment of low water.

For the difference of the velocities v, and v, during the half hour
preceding the moment of lowwater at K.M. 64, during which half
hour the velocity of the current will be maximum at that point, we
find as follows for spring-tide. We assume that between the distances
28 Wk.M. and 64 K.M. (from the Atlantic) there is a retardation of
the tides of just one hour :

at */, hour before lowwater: (v,—v) = 0.32 M. + 0.02 v
,, lowwater : (v,—v) = 0.12 M. + 0.015 ».

From these figures it appears, that during the half hour before
lowwater at K.M. 64 the digferences of the velocities of the current
are only to a small extent dependent on the value of the velocity v.
These differences, therefore, may be determined with sufficient pre-
cision, even if the velocity v is only approximatively known.

By observations made on the Suez-canal during the period 1871—

(2805s)

1876 the velocity-curve for a place near the Red Sea is known

both for springtide and for neaptide. It has been represented in

the following figures. ‘) .
Mean velocity-curves at the entrance of the Suez-canal.

Springtide. Velocity. Neaptide.
Loom

0.g0oMm

0.60M -

0.40M

0.20Nn

Highwater flow.
Highwater flow.

0.00 M
O.20M
O40oN

O.GoM

Ebb flow.
Ebb flow.

0.80M

UST Fs FR nO Ee

LooM

of 3 Gc : 5 (yas
ise Moon-hours after lowwater. tore S

The above velocity curves probably do not represent the mean
velocities but the velocities in the middle of the canal. They have
been derived from measurements made every hour partly by means
of floats partly by means of the current meter of WourTMann.

It deserves attention, however, that at the time of these observations
the Suez-canal had still only a depth of 8 M. below low water and
a bottomwidth of 22 M. The section of the canal is now being
increased to a bottomwidth of 45 M. and a depth of 10,5 M. below
low water. The velocities in the widened canal may perhaps exceed by
20 percent those observed on the canal during the period 1871—1876.°)

1) These curves are borrowed from the Etude du régime de la Marée dans le
canal du Suez par M. Bourpettes, in the Annales des Ponts et Chaussées of 1898.
They occur originally in a Note sur le régime des eaux dans le canal maritime
de Suez et & ses embouchures in 188! by Lemasson Chief Engineer of the canal-
works.

2) For the original cross section of 8 M. depth below low water, 22 M. bottom-
width and slopes of 1 vertical on 2 horizontal we have:

Area I= 304 M?; wet circumference O=57.9 M., consequently R= 5.25 M.
For the future cross section of 10.5 M. depth, 45 M. bottomwidth and slopes of
1 vertical on 21/, horizontal, we will have: J=749 M?., O=101.5 M. tberefore

‘ 7.37 von
RK =7.37 M. Now Dia rae

(5 )s

If, in consideration of this faet, we substitute in the second member
of the formula, for v the values observed in the period 1871—1876
increased by 20 percent, we finally find

‘/, hour before low water (v, — v) = 0.33 M.
at - » (v, —v)=013 M.

The differences 0.33 and 0.13 M. represent the differences of the
simultaneous velocities, not those of the maximum velocities at the
distances of 64 and 28 K.M. from the Atlantic.

At the moment that the velocity reaches its maximum at K.M. 64,
the velocity at K.M. 28, where the tides set in about an hour later,
will still be below the maximum at that place. According to the

observations on the Suez-canal we may assume that, at the epochs

mentioned, the velocities of the current at K.M. 28 will at least be
about 0.15 M. and 0.05 M. below the maximum of that place. Hence
we may conclude that the maximum velocities at K.M. 64 and 28
will certainly uot differ 0.18 M, and probably not much over 0.08 M.
We are sufficiently justified therefore in assuming that the velocity
at K.M. 64 may be about 0.145 M. in excess of that of K.M. 28.
As appears from what has been stated before the difference is
inferior to the increase of the velocity of the current on the Suez-
canal under the influence of the wind, which may amount to
0.380-——0.50 M. It cannot, therefore, cause any serious difficulty.

§ 7. For an open Panama-canal executed as follows:

From the Atlantic to K.M. 64 having the same normal cross section
as that of the project for the sea-level canal ;

from K.M. 64 to K.M. 68, which is the place where the canal
will be connected with a lake gradually widening ;

from K.M. 68 to the Pacific at K.M. 80 having bottomwidth
of 400 to 500 feet;

the following conclusions in regard to the velocities of the current
at springtide may be accepted :

On the first 28 K. M. of such an open canal, velocities of. the
current will oeeur at springtide which, on an average, will be about
equal to those, which will take place at spring tide and with a moderate
wind on the Suez-canal between the Bitter Lakes and the Red Sea
as soon as the widening of this canal will be complete.

On the subsequent 36 Kk. M. of such an open canal the maximum
velocities at springtide will exceed those on the preceding part by

——

\
ee ee

. om L
a
ae
i f i t
eS a
> .
¢-—*es¢ ‘
’ = =
2
“
J A
i
ij
4

(BY: }

about 0.15 M. They will not exceed however those on the Suez-
canal with a strong wind.

For the last 16 K. M. of such an open canal the maximum
velocities at springtide may be somewhat more considerable. On
account however of the great width, which may be given to this
part they will cause no serious difficulty.

Therefore, if we assume, as we have good reason to do, that even
at spring tide and with wind the velocities of the current on the
Suez-canal offer no serious difficulty to navigation we may conclude
that on a Panama-canal of the above description also navigation will
experience no difficulties on account of the velocities of the current.

Therefore, if we leave out of consideration the question whether
an open Panama-canal without tidal lock is to be preferred either
to a sea-level canal with such a lock, as proposed by the Board of
Consulting Engineers, or to a summit level canal with three locks,
as is now in course of execution, we may conclude, in the main
in conformity with the conclusion of the French Academy of Sciences
of 1887, but for different reasons:

That the velocities of the current due to tidal motion in an open
Panama-canal without tidal lock will be no obstruction to navigation.

Zoology. — “On the formation of red blood-corpuscles in the placenta
of the jlying maki (Galeopithecus). By Prot. A. A. W. Huprecat.

(Communicated in the meeting of March 30, 1907).

At the meeting of November 26, 1898, 1 made a communication
on the formation of blood in the placenta of Tarsius and other
mammals, which was later completed by a more extensive paper,
containing many illustrations (Ueber die Entwicklung der Placenta
von Tarsius und Tupaja, nebst Bemerkungen iiber deren Bedeutung
als hi&matopoietische Organe; Report 4 Intern. Congress of Zoology,
Cambridge 1898). The facts observed by me and the interpretation
founded on them, have not until now been generally accepted, and
in a recent very extensive discussion of the position of the problem
concerning the origin of the red blood-corpuscles in the 14 volume
of the “Ergebnisse der Anatomie und Entwicklungsgeschichte” (Wies-
baden 1905), by F. Werpenreicn, the author, when mentioning my
views, emits the supposition that 1 mixed up phagocytic and haemato-
poietic processes.

This conclusion was not based on a renewed and critical exami-
nation of the material, studied by me. I have regretted this, since
I have pointed out clearly and repeatedly that the numerous prepara-

( 874)
tions at Utrecht concerning this and other embryological problems are
always available for comparative and critical work, also for those
who do not share my views. Moreover it appears from the literature,
mentioned in WRIDENREICH’s paper, that the more extensive and illu-
strated article, quoted above, has remained unknown to him.

All this would not have induced me to return to this subject once
more, were it not for the fact that during the last months I have
become acquainted with the placentation-phenomena of a_ totally
different mammal in which these phenomena have never yet been
studied, namely Galeopithecus volans, which, like Tarsius, Nyeti-
cebus, Tupaja and Manis, was collected by me in the Indian Archipelago
in 1890—1891 as extensively as possible for embryological purposes.
During the first origin of the placenta of this rare and in many respects
primitive mammal’), phenomena are observed which elucidate the
process of blood-formation in the placenta in such ai uncommonly
clear manner that in this case it will be difficult to deny the evidence.

The formation of blood in the placenta of Galeopithecus may be
said to take place according to a much simpler plan than in Tarsius,
although the principal outlines remain the same and here also the
non-nucleate haemoglobine-carrying blood-corpuscles must be regarded
not as modified cells but as nuclear derivatives. Likewise the placenta
of Galeopithecus bears testimony that not only the maternal mucosa
but also the embryonic trophoblast takes part in the blood-formation,
while the thus formed blood-corpuscles — also those that are furnished
by embryonic tissue — circulate in the maternal blood-vessels only.

In Galeopithecus the process is simpler especially in this respect
that here no megalokaryocytes play a part in the formation of blood,
so that it is less easy — as WerpEnreicH did — to regard blood-
corpuscles that are set free (such as we notice it in Tarsius, when
the big lobed nuclei of these megalokaryocytes disintegrate) as being
on the contrary devoured in that moment by phagocytosis! *)

The haematopoiesis is started in Galeopithecus in the following
manner. At about the same time that the young germinal vesicle,
which has just gone through the two-layered gastrulation stage (gastru-
lation by delamination *)), has attached itself to the surface of the
strongly folded and swollen maternal mucous membrane, this mucous

1) W. Lecue is inclined (Ueber die Siugethiergattung Galeopithecus, Svenska
Akad. Handl. Bd. 21, N°. 11, 1886) to see in Galeopithecus a form which mast
be placed in the neighbourhood of the ancestral form of the bat.

*) Sectional series of Tarsius of a later date give a still clearer image than those
which served for my figures of 1898.

3) See on this point Anatomischer Anzeiger Bd. 26, 353.

membrane reacts in the manner, well-known in other mammals
(Tarsius, hedgehog, rabbit, bat,. ete.) by perceptible changes in
the uterine glands in the vicinity of this place of attachment and by
the formation of so-called trophospongia-tissue, consisting of a modi-
fication of the interglandular connective tissue, to which are added
proliferations of uterine and glandular epithelium.

As the final product of these preliminary phenomena we now see
that a part of the maternal mucosa where the germinal vesicle
has coalesced with the mucosa, presents a more compact proli-
feration, while nearer the periphery the uterine glands, by strong
dilatation of their lumen, differ clearly from the other uterine
glands, as this is also the case in Tarsius, Lepus and other mam-
mals during early pregnancy. The dilated glands may be followed
up to their mouth; this mouth, however, no longer connects the
glandular lumen with the uterine iumen, since in this place the
embryonic trophoblast has disturbed the connection and covers the
mouths of the glands.

This trophoblast now also shows unmistakable signs of cell-prolifer-
ation, although it does not at once attack and destroy the maternal
epithelium, as in the hedgehog, Tarsius, Tupaja, ete. but rather finds
itself facing this maternal epithelium in full proliferation, in the
manner stated by me also for Sorex'). Instead of being closely
adjacent, however, spaces are left open from the beginning between
trophoblast and trophospongia, which spaces are partly mutually
connected and partly are subdivided into smaller compartments by
trophoblastic villi, attaching themselves to the trophospongia-tissue.

In this manner the free surface of the trophoblast, facing the
embryo, obtains a knobbed appearance. *)

Already in early developmental stages, when there is as yet no
question of the folding off of the embryo and long before blood-
carrying allantoic villi have become interlocked with these tropho-
blastic villi for the further completion of the placenta, we find in
the spaces between trophoblast and trophospongia numerous blood-
corpuscles of which we can not say that they have been carried
thither by maternal vessels exclusively, although there can be no
doubt that a connection between these spaces and the maternal
vascular system is established at an early date. In the manner,
indicated above, these spaces communicate also with the uterine
glands which are here dilated. And in these glands as well as in

1) Quarterly Journal of Microscopical Science, vol. 35.
2) Certain modifications which I chserved when the germinal vesicle develops
in a uterus which is still in the puerperal stage, may be left out of account here,

( 876 )

the interglandular tissue and in the cells, lining the just mentioned
spaces, phenomena take place which force us to the conclusion that
a great number of these blood-corpuscles originate in loco. When we
follow these phenomena up to their earliest appearance, we find
that in the dilated glands in many places compact cell-heaps are
formed, which sometimes lie quite loose in the gland, but in other
eases are still found in direct connection with the cell-lining of the
gland. We must assume that this latter condition represents the
original one and that consequently we have here an epithelial proli-
feration by which new cell-material is carried into the region of
the future placenta.

The final product of these lumps of tissue, which in early stages
appear so distinctly as cell-heaps, is an agglomerate of non-nucleated
blood-corpuscles. The gradual transition of the nucleate cells into the
blood-dises may be followed step by step by successively comparing
preparations of the youngest and subsequent stages: often in one
preparation all transitions are found together. It then appears that
the conclusions I drew for Tarsius and Tupaja in 1898 are confirmed
here, viz. that the blood-discs are produced by gradual transitions
from the modified nuclei of the above-mentioned cell-heaps and that
in this process transitional stages are generally found, comparable to
what I called ‘“shaematogonia” in the above-quoted paper. They re-
semble polynuclear leucocytes from which they may be distinguished,
however (also according to Maximow and SIkGENBEEK VAN HEUKELOM ;
see report of the meeting of the Amsterdam Academy of Nov. 26,
1898), by certain characteristics. This phenomenon has been more
fully investigated by PornsaKxorr, who also regards the non-nucleate
corpuscles as nuclear derivatives and not as cells, deprived of, their
nuclei. In his paper‘) numerous illustrations are given of stages
corresponding to my haematogonia. It appears from the literature,
mentioned by Potsakorr that my paper of 1898, preceding his
publication, was unknown to him: the concordant results which we
have obtained at an earlier date, are confirmed in a striking manner
by the phenomena seen in Galeopithecus.

But blood-corpuscles are also produced by other sources. besides
these epithelial glandular proliferations. Between the dilated glands
we find in Galeopithecus in the trophospongia-tissue very conspicuous
eroups of large cells with a big, but circular nucleus. They show a
tendency to lie together in nests, which nests are more or less kept
together by elongated cells, forming a spurious wall which distantly
remind us of an endothelium.

1) Biologie der Zelle. In Arch. f. Anat. u. Phys. Abth. 1901, Pl. I and I.

( 877 )

These cells also are gradually dissolved into blood-corpuscles: as
the uterus grows and the trophospongia passes through its successive
developmental stages, they disappear: the blood-corpuseles which owe
their existence to them, fall into the above-mentioned spaces, from
whence they are taken up in the further circulation. The intermediate
stages that can be observed in this way of blood-formation, are in
fact an increase of nuclei by amitosis, as was also described by
Po.JAkorr and later a gradual formation from these nuclear derivates
of non-nucleated blood-dises.

To these two processes of blood-formation in the placenta of
Galeopithecus a third must be added in which not the mother is
the active agent, as in the two former cases, but the embryonic
trophoblast. Of this trophoblast we described above how it forms
the bottom of the cavities into which the newly-formed blood-corpus-
cles are discharged, and how it coalesces with the maternal trophos-
pongia to such an extent that for many cells, which here are closely
adjacent, it is impossible to determine whether they take their origin
in the mother or in the trophoblast of the germinal vesicle.

Yet in regard to the wall of the cavities, which separates them
from the lumen of the uterus, there can be no doubt that we have
here trophoblastic tissue only. About the active proliferation of this
trophoblast tissue there is no doubt, no more than about the question
whether the numerous parts of this trophoblast that project into the
cavities, partake in the haematopoiesis. As soon as these parts are
examined with strong powers it is quite evident that here the nuclei
of the trophoblast cells undergo similar modifications as were deseribed
above and that the final product of these modifications are again red
non-nucleated blood-corpuscles which are added to those already present
and originating from the mother. Now these corpuscles are, in the
same way as I observed ten years ago in Tarsius and Tupaja,
set free into the maternal circulation and carried along by it.

On the theoretical significance of the fact that the germinal vesicle
takes an active and important part in increasing the number of
units for the transport of oxygen in the maternal blood, I will not
expatiate here.

And for the histological details of the formation of the bloodplates,
resp. non-nucleated blood-corpuscles from an originally normal cell-
nucleus, I refer to the coloured figures of pl. I and II of PotsaKorr’s
paper in the 1901 volume of the Arch. f. Anat. u. Phys. (Anat.
Abth.). With his illustrations I can identify everything I have ob-
served in Galeopithecus. While in a very few cases there seems
to be a_ possibility that the blood-corpuscle owes its existence to a

( 878 )

change of the nucleus in its entirety, in the vast majority of cases a:
distinct amitotic disintegration is observed, the number of fragments
varying, but generally lying between three and five. As the already
modified nucleus dissolves into these fragments the eemparability
with polynuclear leucocytes seems more obvious, and the colour as
a rule approaches more and more to that which the blood-corpuseles
themselves assume in the artificially fixed preparation. The same fact
was stated by me also for Tarsius in 1898 and figured on Pl. 14
figs. 91—96.

Finally I point out, since my results and those of PoLJAkorr agree
in sO many respects, that also Rerrerer in the volume for 1901 of the
Journal de Anatomie et de la Physiologie (Structure, développement
et fonction des ganglions lymphatiques, p. 700) has obtained similar
results and is inclined to assume a still closer genetic relationship
between polynuclear leucocytes and haematogonia when he declares
that the leucocytes, liberated from lymphatic glands “‘finissent par
se convertir, dans la lymphe ou le sang, en hematies grace a la
transformation hémoglobique de leur noyau...”

Thus my observations on Galeopitheous form a link in the chain,
which begins with Heinrich MtLier in 1845 (Zeitsehrift fiir rationelle
Medicin vol. 3. p. 260) was then continued and upheld by Wuarton
Jones (Phil. Trans. 1846, p. 65 and 71) and Huxiry (Lessons in
Elementary Physiology, 1866, p. 63) and which, since in 1898
Tarsius added another link, has with increasing weight bound up
the question of the origin of the non-nucleated blood-corpnseles in
mammals to the conception that these elements in the mammalian
body are not equivalent with cells, but must be regarded as nuclear
derivatives.

(May 24, 1907).

CONTENTS.

ABSORPTION BANDS (Wave-lengths of formerly observed emission and) in the infra-
red spectrum. 706.
acip (On the nitration of phthalic acid and isophthalic). 286.
AcIDs (The six isomeric dinitrobenzoic). 280.
AIR (The preparation of liquid) by means of the cascade process. 177.
- AMBOCEPTORS (On the) of an anti-streptococcus serum. 336.
AMMONIA aud Amines (On the action of bases,) on s,trinitrophenyl-methylnitra-
mine. 704.
AMYRIN (On a- and £-) from bresk. 471.
Anatomy. A. J. P. van pDEN Broek: “On the relation of the genital ducts to the
genital gland in marsupials’’. 396.
— B. van Tricut: “On the influence of the fins upon the form of the trunk-
myotome”’. 814.
— L. J. J. Muskens: “Anatomical research about cerebellar connections” (3rd Com-
munication). 819.
ANILINEHYDROCHLORIDE (Three-phaselines in chloralaleoholate and). 99.
ANTHERS (On the influence of the nectaries and other sugar-containing tissues in the
flower on the opening of the). 390.
ANTHRACOsIS (On the origin of pulmonary). 673.
ANTIMONITE from Japan (On the influence which irradiation exerts on the electrical
conductivity of). 809.
ARRHENIUS (SvANTE) and H. J. HamBurcer. On the nature of precipitin-
reaction. 33.
Astronomy. J. Stein: “Observations of the total solar eclipse of August 30, 1905 at
Tortosa (Spain).’’ 40.
— <A. PannekoEK: “The luminosity of stars of different types of spectrum.” 134.
— A. PanneEKOEK: “The relation between the spectra and the colours of the
stars.” 292.
— J. A. C. OupEmans: ‘Mutual occultations and eclipses of the satellites of
Jupiter in 1908.” 304, 2nd part. 444.
— H. J. Zwiers: “Researches on the orbit of the periodic comet Holmes and on
the perturbations of its elliptic motion.” [V. 414.
60
Proceedings Royal Acad. Amsterdam. Vol. IX,

lr CONTENTS.

Astronomy. H. G. vaN pe SanpE Baxnuyzen: “On the astronomical refractions correspond-
ing to a distribution of the temperature in the atmosphere derived from balloon

ascents.” 578.

ATMOSPHERE (On the astronomical refractions corresponding to a distribution of the
temperature in the) derived from balloon ascents. 578.

BAKHUIS ROOZEBOOM (H. W.) presents a paper of Dr. A. Smits: “On the
introduction of the conception of the solubility of metal ions with electromotive
equilibrium.” 2.

— presents a paper of J. J. van Laar: “On the osmotic pressure of solutions of
non-electrolytes, in connection with the deviations from the laws of ideal gases.” 53.

— Three-phaselines in chloralalcoholate and anilinehydrochloride. 99,

— presents a paper of Dr. F. M. Jarerr: “On a substance which possesses nume-
rous different liquid phases of which three at least are stable in regard to the
isotropous liquid.” 359.

— The behaviour of the halogens towards each other. 363.

BAKHUYZEN (H. G. VAN DE SANDE). v. SANDE Bakuvuyzen (H. G. VAN DB).

BALLOON ASCENTS (On the astronomical refractions corresponding to a distribution of
the temperature in the atmosphere derived from). 578.
BaTavia (On magnetic disturbances as recorded at). 266.
satus (How to obtain) of constant and uniform temperature by means of liquid
hydrogen. 156.
BEMMELEN (w. VAN). On magnetic disturbances as recorded at Batavia. 266.
BENZENE-DERIVATIVES (On a new case of form-analogy and miscibility of position-
isomeric), and on the crystal forms of the six nitrodibromobenzenes. 26.
BINARY MIXTURE (On the shape of the three-phase line: solid-liquid—vapour for a). 689.
— (The shape of the empiric isotherm for the condensation of a). 750.
BINARY MIxTURES (The shape of the spinodal and plaitpoint curves for) of normal sub-
stances. 4th Communication. The longitudinal plait. 226.
— (On the gas phase sinking in the liquid phase for). 501.
— (On the conditions for the sinking and again rising of the gas phase in the
liquid phase for). 508. Continued. 660.
— (A remark on the theory of the y-surface for). 524.
— (Contribution to the theory of). 621. II, 727. ILL 826.
— (Isotherms of diatomic gases and their). VI. Isotherms of hydrogen between
— 104° C. and — 217° C. 754.
BLANKSMA (J. J.). Nitration of meta-substituted phenols. 278.
BLoop-corPuscLEs (On the formation of red) in the placenta of the flying maki
(Galeopithecus). 873.
BOEKE (J.). Gastrulation and the covering of the yolk in the teleostean egg. 800.
BO ESEKEN (J.). On catalytic reactions connected with the transformation of yellow
phosphorus into the red modification. 613.
BOLK (L.) presents a paper of Dr. A. J. P. van DEN Brogk: “On the relation of
the genital ducts to the genital gland in marsupials,” 396.

CONTENTS. ur

Botany. M. Nreuwenuuis- von UrexktLu-GtLpenbanp : “On the harmful consequences
of the secretion of sugar with some myrmecophilous plants”. 150,

— W. Burcx: “On the influence of the nectaries and other sugar-contatning
tissues in the flower on the opening of the anthers”. 390.

BRAAK (c.) and H. KameruincH Onnes. On the measurement of very low tem-
peratures. XIIL. Determinations with the hydrogen thermometer. 367. XLV.
Reduction of the readings of the hydrogen thermometer of constant volume to
the absolute scale. 775.

— Isotherms of diatomic gases and their binary mixtures. VI. [sotherms of hy-
drogen between — 104°C and — 217°C. 754.

BRESK (On g-and j-amyrin from). 471.

BROEK (A. J. P. VAN DEN). On the relation of the genital ducts to the genital
gland in marsupials. 396.

BROUWER(L. E. J.). Polydimensional vectordistributions. 66.

— The force field of the non-Kuclidean spaces with negative curvature. 116.
— The force field of the non-Kuclidean spaces with positive curvature. 250,

BURCK (w.). On the influence of the nectaries and other sugar-containing tissues
in the flower on the opening of the anthers. 390.

BuTyric EsTER of dihydrocholesterol, (On the anisotropous liquid phases of the) and
on the question as to the necessary presence of an ethylene double bond for
the occurrenc® of these phenomena. 701.

CARDIAC ACTION (An investigation on the quantitative relation between vagus stimu-
lation and). 590.

CASCADE PROcEsS (The preparation of liquid air by means of the). 177.

cases (A few remarks concerning the method of the true and false). 222.

CATALYTIC REACTIONS (On) connected with the transformation of yellow phosphorus
into the red modification. 613,

CEREBELLAR CONNECTIONS (Anatomical research about). 3rd Communication. 819.

Chemistry. A. Smrrs: “On the introduction of the conception of the solubility of
metal ions with electromotive equilibrium.” 2.

— J. Mout vaN Cuarante: “The formation of salicylic acid from sodium pheno-
late.” 20.

— F. M. Jagcer: “On the crystal-forms of the 2.4 dinitroaniline-derivatives, sub-
stituted in the NH,-group.” 23.

— J. J. van Laar: “On the osmotic pressure of solutions of non-electrolytes, in
connection with the deviations from the laws of ideal gases.” 53.

— H. W. Bakuuis Roozesoom: “Three-phaselines in chloralalcoholate and aniline-
hydrochloride.” 99.

— P. van Romsurcu: “Triformin (Glyceryltriformate).” 109.

— P. van Rompurcu and W. van Dorssen: “On some derivatives of 1-3-5-
hexatriene.” 111.

— J. J. van Laar: “The shape of the spinodal and plaitpoint curves for binary
mixtures of normal substances. 4th Communication. The longitudinal plait.” 226.

— J. J. BuanxsMa: “Nitration of meta-substituted phenols.” 278,

60*

IV C10 NATE NOTES.

Chemistry. A. F. Honteman and H. A. Sirks: “The six isomeric dinitrobenzoic acids.” 280.

— A. F. Horneman and J. Hursinea: “On the nitration of phthalic acid and
isophthalic acid.” 286.

— R. A. Weerman; “Action of potassium hypochlorite on cinnamide.” 303.

— F. M. Janeen: “On a substance which possesses numerous different liquid
phases of which three at least are stable in regard to the isotropous liquid.” 359.

— H. W. Bakuuis Roozesoom: “The behaviour of the halogens towards each
other.” 363.

— N. H. Conen: “On Lupeol.” 466.

— N. H. Conen: “On g- and B-amyrin from bresk.’’ 471.

— F£. M. Jazcer: “On substances which possess more than one stable liquid stat,
and on the phenomena observed in anisotropous liquids.” 472.

— F. M. Jagcer: “On irreversible phase-transitions in substances which may
exhibit more than one liquid condition.” 483.-

— A. F. Horueman and G, L. Vorrman: “z- and 6-thiophenic acid.” 514.

— A. P. N. Francurmont: “Contribution to the knowledge of the action of absolute
nitric acid on heterocyclic compounds.” 600.

— F. A. H. Scorememaxkers: “On a tetracomponent system with two liquid
phases.”” 607.

— J. Boiisexen: “On catalytic reactions connected with the transformation of yellow
phosphorus into the red modification.” 613.

— F. M. Jancer: “On the anisotropous liquid phases of the butyric ester of
dihydrocholesterol, and on the question as to the necessary presence of an ethylene
double bond for the occurrence of these phenomena.” 701.

— P. van Rompurcu and A. D. MavrenprecHer: “On the action of bases,
ammonia and amines on s.trinitrophenyl-methylnitramine.” 794.

CHLORALALCOHOLATE (Three-phaselines in) and anilinehydrochloride. 99.

CHOLESTEROL (On the fatty esters of) and Phytosterol, and on the anisotropous liquid
phases of the cholesterol-derivatives. 78.

CINNAMIDE (Action of potassium hypochlorite on). 303,

cuay (s.) and H. Kamertincuw Onnes. On the measurement of very low tempera -
tures. X. Coefficient of expansion of Jena glass and of platinum between + 16°
and — 182°, 199. XI. A comparison of the platinum resistance thermometer
with the hydrogen thermometer. 207. XII. Comparison of the platinum
resistance thermometer with the gold resistance thermometer. 213.

COHEN (N. H.). On Lupeol. 466.

— On g-and f-amyrin from bresk. 471.

COMET HOLMES (Researches on the orbit of the periodic) and on the perturbations o
its elliptic motion. [V. 414.

COMMON porNTs (On the locus of the pairs of) and the envelope of the common chords
of the curves of three pencils. Ist part. 424. 2ad@ purt. Application to pencils of
conics. 548,

— (Lhe locus of the pairs of) of four pencils of surfaces. 555.

CONTENTS. ¥

COMMON POINTS (The locus of the pairs of) ofz-+-1 pencils of (n — 1) dimensional
varieties in a space of » dimensions. 573.

COMPLEXEs of revolution (Quadratic). 217.

compounDs (Contribution to the knowledge of the action of absolute nitric acid on
heterocyclic). 600.

CONCENTRATION (On the course of the P, T-curves for constant) for the equilibrium
solid-fluid, 9.

CONDUCTIVITY POWER (Researches on the thermic and electric) of crystallised conduc-
tors. I. 89.

conpuctors (Researches on the thermie and electric conductivity power of erystal-
lised). 1. 89.

CROMMELIN (c. a.) and H. Kamertincn Onnks. On the measurement of very low
temperatures. 1X. Comparison of a thermo-element constantin-steel with the
hydrogen-thermometer. 180. Supplement. 403.

CRYOGENIC LABORATORY at Leiden (Methods and apparatus used in the). X. How to
obtain baths of constant and uniform temperature by means of liquid hydrogen. 156.
XL. The purification of hydrogen for the eycle. 171. XII. Cryostat especially for
temperatures from — 252° to — 259°. 173. XIII. The preparation of liquid air
by means of the cascade process. 177. XIV. Preparation of pure hydrogen through
distillation of less pure hydrogen. 179.

cryosTaT especially for temperatures from —- 252° to — 259°. 173.

cRYSTAL-FoRMS (On the) of the 2,4 Dinitroaniline-derivatives, substituted in the
NH,-group. 23.

— (On a new case of form-analogy and miscibility of position-isomeric benzene-
derivatives, and on the) of the six Nitrodibromobenzenes. 26.

Crystallography. F. M. Jarcer: “On a new case of form-analogy and miscibility of
position-isomeric benzene-derivatives, and on the erystal-forms of the six Nitrodi-
bromobenzenes.” 26. -

— F. M. Jarcer: “On the fatty esters of Cholesterol and Phytosterol, and on the
anisotropous liquid phases of the Cholesterol-derivatives.” 78.

cusics (The locus of the cusps of a threefold infinite linear system of plane) with
six basepoints. 534.

CURRENT (Velocities of the) in an open Panama-canal. 849.

CURRENT-MEASUREMENTS at various depths in the North-Sea. 1st Communication. 56.

curves (On the course of the P, 7-) for constaut concentration for the equilibrium
solid-fluid. 9.

— of three pencils (On the locus of the pairs of common points and the envelope
of the common chords of the). Ist part. 424. 2nd part. Application to pencils

- of conics, 548.

cusps (The locus of the) of a threefold infinite linear system of plane eubics with
six basepoints. 534.

cycLE (The purification of hydrogen for the). 171.

cYcLic POINT of a twisted curve (Second communication on the Plucker equivalents
of a), 364. a

v1 CONTENTS.

DALHUISEN (a. F. H.). v. Winn (C. H.).

DIFFERENTIAL EQUATIONS (On a special class of homogeneous linear) of the second
order. 406.
DIHYDROCHOLESTEROL (On the anisotropous liquid phases of the butyric ester of),
and on the question as to the necessary presence of an ethylene double bond
for the occurrence of these phenomena. 701.
DINITROANILINE-DERIVATIVES (On the crystalforms of the 2.4), substituted in the
NH,-group. 23.
DISPERSION BANDS (Arbitrary distribution of light in), and its bearing on spectros-
copy and astrophysics. 343.
pDORSSEN (Ww. VAN) and P. van RomsBurcn: “On some derivatives of 1-3-5-
hexatriene.” 111.
EcLIPsEsS (Mutual occultations and) of the satellites of Jupiter in 1908. 304. 2nd
part. 444.
EGG (Gastrulation and the covering of the yolk in the teleostean). 800.
ELECTRICAL coNDUCTIVITY (On the influence which irradiation exerts on the) of
Antimonite from Japan. 809.
ELLIPTIC MOTION (Researches on the orbit of the periodic comet Holmes and on the |
perturbations of its). 414.
EMISSION and absorption bands (Wavelengths of formerly observed) in the infra-red
spectrum. 706.
EQUILIBRIA solid-fluid (On metastable and unstable). 648.
EQUILIBRIUM (On the introduction of the conception of the solubility of metal ions
with electromotive). 2.
— solid-fluid (On the course of the P,7-curves for constant concentration for the). 9.
— of systems of forces and rotations in Sp,. 820.
ERRATUM. 148. 378. 511. 59S.
rsrprs (On the fatty) of Cholesterol and Phytosterol, and on the anisotropous liquid
phases of the Cholesterol-derivatives. 78.
FTHANE and nitrous oxide (Graphical deduction of the results of KUENEN’s experiments
on mixtures of). 664.
ETHYLENE DOUBLE BOND (On the anisotropous liquid phases of the butyric ester of
Dihydrocholesterol, and on the question as to the necessary presence of an) for
the occurrence of these phenomena. 701.
EUCLIDEAN spaces (‘The force field of the non-) with negative curvature. 116.
— (The force field of the non—) with positive curvature. 250,
EXPANSION (Coeflicient of) of Jena glass and of platinum between + 16° and — 182°.
199.
EYSBROEK (H.). On the Amboceptors of an anti-streptococcus serum. 336,
rixs (On the influence of the) upon the form of the trunk-myotome, 814.
PISH-FAUNA of New Guinea (On the fresh-water). 462.
FLowreR (On the influence of the nectaries and other sugar-containing tissues in the)

on the opening of the anthers. 390.
PORCE-FIELD (The) of the non-Kuclidean spaces with negative curvature, 116.

CO; N, T-Bone TS: VII

FORCE-FIELD of the non-Euclidean spaces with positive curvature. 250.
FoRCES (Equilibrium of systems of) and rotations in Sp,. 820.

FORM-ANALOGY (On a new case of) and miscibility of position-isomeric benzene-deriva-
tives, and on the crystalforms of the six Nitrodibromobenzenes. 26.

FORMULAE (Some) concerning the integers less than x and prime to n. 408.

FRANCHIMONT (A. P. N.) presents a paper of Dr. J. Moun van Cuarante:
“The formation of salicylic acid from sodium phenolate.” 20

— presents a paper of Dr. F. M. Jarcer: “On the fatty esters of Cholesterol and
Phytostero], and on the anisotropous liquid phases of the Cholesterolderivatives”. 78.

— presents a paper of Dr. F. M. Jancer: “On substances which possess more than
one stable liquid state, and on the phenomena observed in anisotropous liquids”. 472.

— presents a paper of Dr. F. M. Jagcer: “On irreversible phase-transitions in
substances which may exhibit more than one liquid condition.” 483.

— Contribution to the knowledge of the action of absolute nitric acid on hetero-
cyclic compounds. 600.

— presents a paper of Dr. F. M. Jarcer: “On the anisotropous liquid phases
of the butyric ester of Dihydrocholesterol, and on the question as to the necessary
presence of an ethylene double bond for the occurrence of these phenomena.” 701.

“GALEOPITHECUS (On the formation of the red blood-corpuscles in the placenta of the
flying maki). 873.
Gas (A) that sinks in a liquid. 459,

— (The case that one component is a) without cohesion with molecules that have
extension. 786.

Gasks (On the osmotic pressure of solutions of non-electrolytes, in connection with
the deviations from the laws of ideal). 53.

— (lsotherms of diatomic) and their binary mixtures. VI. Isotherms of hydrogen
between — 104°C. and — 217° C. 754.

— (Limited miscibility of two). 786.

GASPHASE (On the) sinking in the liquid phase for binary mixtures, 501.

—- (On the conditions for the sinking and again rising of the) in the liquid phase

for binary mixtures. 508. Continued. 660.

GASTRULATION and the covering of the yolk in the teleostean egg. 800.

GENITAL DUcTs (On the relation of the) to the genital gland in marsupials. 396.

Geophysics. C. H. Winn, A. F. H. Datnutisen and W. FE. Ringer: “Current-measu-
rements at various depths in the North-Sea”. 1st Communication. 566.

GLYCERYL TRIFORMATE 109.

HAGA (H.). On the polarisation of Réntgen rays. 104,

HALOGENS (The behaviour of the) towards each other. 363.

HAMBURGER (u. J.) and Svante ARRHENIUS. On the nature of precipitin-
reaction. 33.

HEXATRIENE (On some derivatives of 1-3-5), 111.

HOLLEMAN (a. FP.) presents a paper of Dr. I’. M. Jarcrr: “On a new case of
form-analogy and miscibility of position-isomeric benzene-derivatives, and on the
erystalforms of the six Nitrodibromobenzenes”’. 26,

Vill CONTENTS.

HOLLEMAN (A. F) presents a paper of Dr. J. J. Buanksma: “Nitration of meta-
substituted phenols.” 278.
— presents a paper of Dr. J. Boisrxen: “On catalytic reactions connected with _
the transformation of yellow phosphorus into the red modification”. 613.
— and J. Huistnca. On the nitration of phthalic acid and isophthalie acid. 286.
— and H. A. Srrxs. The six isomeric dinitrobenzoic acids. 280.
— and G. L. VorerMaN. z-and $- thiophenic acid. 614.
HOLMES (Researches on the orbit of the periodic comet) and on the perturbations of
its elliptic motion. 1V. 414.
HOOGEWERFF (s.) presents a paper of R, A. WEERMAN: “Action of potassium
hypochlorite on cinnamide.” 303.
HUBRECHT (A. A. W.) presents a paper of Dr. F. Mutver: “On the placentation
of Sciurus vulgaris.” 380.
— presents a paper of Dr. J. Borke: ‘Gastrulation and the covering of the yolk
in the teleostean egg’’. 800.
— On the formation of red blood-corpuscles in the placenta of the flying maki
(Galeopithecus). 873.
wmutsinGa (g.) and A. F. Hotteman: “On the nitration of phthalic acid and iso-
phthalic acid.” 286,
uyDROGEN (How to obtain baths of constant and uniform temperature by means of
liquid). 156.
— (The purification of) for the cycle. 171.
— (Preparation of pure) through distillation of less pure hydrogen. 179.
— (Isotherms of) between — 104° C, and — 217° C. 754.
ick (On the motion of a metal wire through a piece of). 713.
INTEGERS (Some formulae concerning the) less than x and prime to x, 408. :
ions (On the introduction of the conception of the solubility of metal) with electro-
motive equilibrium. 2. :
yRRADIATION (On the influence which) exerts on the electrical conductivity of Anti-
monite from Japan. 809.
tsorueRM (The shape of the empiric) for the condensation of a binary mixture. 750,
1sorHERMS of diatomic gases and their binary mixtures. VI. Isotherms of hydrogen
between — 104° C. and — 217° C, 754.
ISOTROPOUS LIQUID (On a substance which possesses numerous different liquid phases
of which three at least are stable in regard to the). 359.
JAEGER (fF. M.). On the crystal-forms of the 2.4 Dinitroaniline-derivatives, sub-
stituted in the NH,-group. 23.
— On a new case of form-analogy and miscibility of position-isomeric benzene-
derivatives, and on the crystalforms of the six nitrodibromobenzenes. 26,
— On the fatty esters of Cholesterol and Phytosterol, and on the anisotropous
liquid phases of the Cholesterol-derivatives. 78.
— Researches on the thermic and electric conductivity power of crystallised

conductors. I, 89.

CONTENTS, Ix

JAEGER (F. M.). On a substance which possesses numerous different liquid phases
of which three at least are stable in regard to the isotropous liquid. 359.

— On substances which possess more than one stable liquid state, and on the
phenomena observed in anisotropous liquids. 472.

— On irreversible phase-transitions in substances which may exhibit more than
one liquid condition. 483.

— On the anisotropous liquid phases of the butyric ester of dihydrocholesterol,
and on the question as to the necessary presence of an ethylene double bond for
the occurrence of these phenomena. 701.
— On the influence which irradiation exerts on the electrical conductivity of
Antimonite from Japan. 809.
JENA GLass (Coefficient of expansion of) and of platinum between + 16° and — 182°, 199,
JOLLES (Miss T. c.) and H. KaMeruincH Onnes. Contributions to the knowledge
of the y-surface of van Den Waats. XIV. Graphical deduction of the results of
KUENEN’s experiments on mixtures of ethane and nitrous oxide. 664.
JULIUs (w. H.). Arbitrary distribution of light in dispersion bands, and its bearing
on spectroscopy and astrophysics, 343.
— presents a paper of W. J. H. Mouu: “An investigation of some ultra-red metallic
spectra.” 544.
— Wave-lengths of formerly observed emission and absorption bands in the infra-
red spectrum. 706. sal
JUPITER (Mutual occultations and eclipses of the satellites of) in 1908. 304. 2nd part.-444.
KAMERLINGH ONNES (H.). Methods and apparatus used in the Cryogenic Labora-
tory at Leiden. X. How to obtain baths of constant and uniform temperature by
means of liquid hydrogen. 156. XI. The purification of hydrogen for the
cycle. 171. XII. Cryostat especially for temperatures from — 252° to — 2599.
173. XIII. The preparation of liquid air by means of the cascade process. 177.
XIV. Preparation of pure hydrogen through distillation of less pure hydrogen. 179.
— Contributions to the knowledge of the y-surfuce of van DER Waats. XI, A

gas that sinks in a liquid. 459.
KAMERLINGH ONNES (H.) and C. Braax. On the measurement of very low

emperatures. XIII. Determinations with the hydrogen thermometer. 367, XIV.
Reduction of the readings of the hydrogen thermometer of constant volume to
the absolute scale. 775.
— Isotherms of diatomic gases and their binary mixtures. VI. Isotherms of hydro-

gen between — 104° C and — 217° C. 754.

KAMERLINGH ONNES (H.) and J. Cray. On the- measurement of very low
temperatures. X. Coefficient of expansion of Jena glass and of platinum between
+ 16° and — 182°, 199. XI. A comparison of the platinum resistance thermometer
with the hydrogen thermometer. 207. XII. Comparison of the plutinum resistance
thermometer with the gold resistance thermometer. 213.

KAMERLINGH ONNES (H.) and CU. A. Crommenin. On’ the measurement of very
low temperatures. IX. Comparison of a thermo-element constantin-steel with the

hydrogen thermometer. 180. Supplement. 403.

xX CoO SN Tab oN SEs:

KAMERLINGH ONNES (H.) and Miss T. C. Joxies. Contributions to the know-
ledge of the -surface of vaN DER Waats. XIV. Graphical deduction of the
results of KUENEN’s experiments on mixtures of ethane and nitrous oxide. 664.

KAMERLINGH ONNEsS (H.) and W. H. Keersom. Contributions to the knowledge
of the y-surface of vAN DER Waats. XII. On the gas phase sinking in the liquid
phase for binary mixtures. 501. XV. The case that one component is a gas
without cohesion with molecules that have extension. Limited miscibility of
two gases. 786.

KAPTEYN (w.). On a special class of homogeneous linear differential equations of
the second order. 406. —

KEESOM (Ww. H.). Contribution to the knowledge of the y- surface of VAN DER
Waais. XIII. On the conditions fer the sinking and again rising of the gas
phase in the liquid phase for binary mixtures. 508. Continued. 660.

— and KaMerLINGH Onnzs (H.). Contributions to the knowledge of the y-surface
of vAN DER Waals. XII. On the gas phase sinking in the liquid phase for
binary mixtures. 501. XV. The case that one component is a gas without cohesion
with molecules that have extension. Limited miscibility of two gases. 786.

KLUYVER (J. c.). Some formulae concerning the integers less than m and prime to
n, 408.

KOHNSTAMM (PH.). On the shape of the three-phase line solid-liquid-vapour fora
binary mixture. 639.

— On metastable and unstable equilibria solid-fluid. 648.

KORTEWEG (D. J.) presents a paper of L. E. J, Brouwer: “Polydimensional Vec-
tordistributions.” 66,

— presents a paper of L. E. J. Brouwer: “ The force-field of the non-Euclidean
spaces with negative curvature”. 116.

— presents a paper of L. E. J. Brouwer: “The force-field of the non-Euclidean
spaces with positive curvature”. 250.

K UENEN’S experiments (Graphical deduction of the results of) on mixtures of ethane
and nitrous oxide. 664.

LAAR (J. J. VAN). On the osmotic pressure of solutions of non-electrolytes, in con-
nection with the deviations from the laws of ideal gases. 53.

— The shape of the spinodal and plaitpoint curves for binary mixtures of normal
substances. 4th Communication. The longitudinal plait. 226.

LELY (c.). Velocities of the current in an open Panama-canal. 849.

Ligut (Arbitrary distribution of) in dispersion bands, and its bearing on spectroscopy
and astrophysics. 343.

LINEAR syYSTEM (The locus of the cusps of a threefold infinite) of plane cubics with
six basepoints. 534.

LIQUID (A gas that sinks in a). 459. ?

LIQUID CONDITION (On irreversible phase-transitions in substances which may exhibit
more than one). 483.

LIQUID PHASE (On the gas phase sinking in the) for binary mixtures. 501.

— (On the conditions for the sinking and again rising of the gas phase in the) for
binary mixtures, 508. Continued. 660.

——

CONTENTS. XI

LIQUID PHAsES (On the fatty esters of Cholesterol and Phytosterol, and on the aniso-
tropous) of the Cholesterol-derivatives. 78.
— (On a substance which possesses numerous) of which three at least are stable in

regard to the isotropous liquid. 359.

— (On a tetracomponent system with two), 607.

— (On the anisotropous) of the butyric ester of dihydrocholesterol, and on the
question as to the necessary presence of an ethylene double bond for the occur-
rence of these phenomena. 701.

LIQUID sTATE (On substances which possess more than one stable), and on the pheno
mena observed in anisotropous liquids. 472.

LIQUIDS (On substances which possess more than one stable liquid state, and on the
phenomena observed in anisotropous). 472.

LONGITUDINAL PLAIT (The). 226.

LORENTZ (H. A.) presents a paper of Dr. F, M. JazGER: “Researches on the thermic
and electrie conductivity power of crystallised conductors.” I. 89.

— presents a paper of J. J. van Laar: “The shape of the spinodal and plaitpoint
curves for binary mixtures of normal substances, 4th Communication. The longi-
tudinal plait.” 226.

— presents a paper of Dr. O. Postma: “Some additional remarks on the quantity
H and Maxwetv’s distribution of velocities.” 492.

— presents a paper of Dr. J. H. MeErBurG: “On the motion of a metal wire
through a piece of ice.” 718.

LuMINosITy (The) of stars of different types of spectrum. 134.

LUPEOL (On). 466.

MAGNETIC DISTURBANCES (On) as recorded at Batavia. 266.

MAKI (On the formation of red blood-corpuscies in the placenta of the flying). 873.
MARSUPIALS (On the relation of the genital ducts to the genital gland in), 396.
Mathematics. L. E. J. Brouwer: “Polydimensional vectordistributions.’’ 66.

— L. E. J. Brouwer: “The force field of the non-Euclidean spaces with negative
curvature.” 116.

— Jan DE Vrizs: “Quadratic complexes of revolution”. 217.

— L. E. J. Brouwer: “The force-field of the non-Euclidean spaces with positive
curvature.” 250.

— W. A. Versiuys: “Second communication on the Plucker equivalents of a
cyclic point of a twisted curve’. 364.

— W. Kapteyn: “On a special class of homogeneous linear differential equations
of the second order.” 406.

— J. C. Kuivyver: “Some formulae concerning the integers less than m and prime
to n.” 408. é

— F. Scuun: “On the locus of the pairs of common points and the envelope of
{he common chords of the curves of three pencils.’’ 1st part. 424. 2nd part. 548,

— W. A. Wyruorr: “The rule of Neper in the fourdimensional space.” 529.

— P. H. Scuoure: “The locus of the cusps of a threefold infinite linear system
of plane cubics with six basepoints’”. 534.

XII CON TEN TS.

Mathematics. F. Scuun: “The locus of the pairs of common points of four pencils of
surfaces.” 555.
— F. Scuun: “The locus of the pairs of common points of m + 1 pencils of
(n—1) dimensional varieties in a space of x dimensions.” 573.
— S. L. van Oss: “Equilibrium of systems of forces and rotations in Sp,.” 820.

MAURENBRECHER (a. D.) and P. van RomBurew. On the action of bases,
Ammonia and Amines on s. trinitrophenyl-methylnitramine. 704.

MAXWELL’s distribution of velocities (Some additional remarks on the quantity H
and). 492.
MEERBURG (J. H.). On the motion of a metal wire through a piece of ice. 718.
METAL WIRE (On the motion of a) through a piece of ice. 718. TP
Meteorology. W. vaN BEMMELEN: “On magnetic disturbances as recorded at Batavia.” 266.
— J. P. van per Stox: “The treatment of wind-observations,” 684.

METHOD (A few remarks concerning the) of the true and false cases. 222.

METHODS and apparatus used in the Cryogenic Laboratory at Leiden. X. How to obtain
baths of constant. and uniform temperature by means of liquid hydrogen. 156.
Xf. The purification of hydrogen for the cycle. 171. XII. Cryostat especially for
temperatures from — 252° to — 259°. 173. XIII. The preparation of liquid
air by means of the cascade process. 177. XIV. Preparation of pure hydrogen
through distillation of less pure hydrogen. 179. =

MISCIBILITY (On a new case of form-analogy and) of position-isomeric benzene-deriva-
tives, and on the crystalforms of the six Nitrodibromobenzenes. 26. oo

MIXTURES of ethane and nitrous oxide (Graphical deduction of the results of KUENEN’s

experiments on). 664.
MOGENDORFF (£. £.). On a new empiric spectral formula. 434.

MOLL (J. W.) presents a paper of Mrs. M. NrEUWENHUIS-VON UEXKULL-GULDENBAND:
“On the harmful consequences of the secretion of sugar with some myrmecophilous
plants.” 150.

MOLL (WwW. J H.). An investigation of some ultra-red metallic spectra. 544.

MOLL VAN CHARANTE (J.). The formation of salicylic acid from sodium
phenolate. 20.

MULLER (F.). On the placentation of Sciurus vulgaris. 380.

MUSKENS (t, J. J.). Anatomical research about cerebellar connections. (3rd Commu:
nication). 819.

NECTaRIES (On the influence of the) and other sugar-containing tissues’ in the flower
on the opening of the anthers. 390,

NEPER (The rule of) in the four-dimensional space. 529.

NEW GuINEA (On the fresh-water fish-fauna of). 462.

NIEUWENHUIS=VON UEXKULL-GULDENBAND (m.). On the harmful
consequences of the secretion of sugar with some myrmecophilous plants. 150.

NIEUWENHUYSE (P.). On the origin of pulmonary anthracosis. 673.

NiTRATION (On the) of phthalic acid and isophthalic acid. 286.

— of meta-substituted phenols, 278.

——

¢ OuNo EEN: Ty. XIII

NITRIC AcID (Contribution to the knowledge of the action of absolute) on heterocyclic
compounds. 600.

NITRODIBROMOBENZENES (On a new case of form-analogy and miscibility of position-
isomeric benzene-derivatives, and on the crystalforms of the six). 26.

NITROUS OXIDE (Graphical deduction of the results of KuENEN’s experiments on
mixtures of ethane and). 664.

NON-ELECTROLYTES (On the osmotic pressure of solutions of), in connection with the
deviations from the laws of ideal gases. 53.

NORTH-SEA (Current-measurements at various depths in the). 1st Communication. 566.

OCccULTATIONS (Mutual) and eclipses of the satellites of Jupiter in 1908, 304, 2nd
part. 444.

OSMOTIC PRESSURE (On the) of solutions of non-electrolytes, in connection with the
deviations from the laws of ideal gases. 53.

oss (Ss. L. van). Equilibrium of systems of forces and rotations in Sp,. 820.

OUDEMANS (J. A. c.). Mutual occultations and eclipses of the satellites of Jupiter
in 1908. 304. 2nd part. 444.

PANAMA-CANAL (Velocities of the current in an open). 849.

PANNEKOEK (a.). The Juminosity of stars of different types of spectrum. 134.

— The relation between the spectra and the colours of the stars, 292,
Pathology. H. Eysprork: “On the Amboceptors of an anti-streptococcus serum.” 336.
— P, NieuwenuvyseE: “On the origin of pulmonary anthracosis.”’ 673.

PENCILS (On the locus of the pairs of common points and the envelope of the common
chords of the curves of three). lst part. 424. 2nd part. Application to pencils of
conics. 548. ;

— (The locus of the pairs of common points of ~+ 1) of (n—1) dimensional
varieties in a space of x dimensions. 573.

— of conics (Application to). 548.

— of surfaces (The locus of the pairs of common points of four). 555.

PERTURBATIONS (Researches on the orbit of the periodic comet Holmes and on the)
of its elliptic motion. IV. 414.

PHASE-LINE (On the shape of the three-) solid-liquid-vapour fora binary mixture. 639.

PHASE-LINES (Three) in chloral alcoholate and anilinehydrochloride. 99.

PHASE-TRANSITIONS (On irreversible) in substances which may exhibit more than one
liquid condition. 483.

PHENOLS (Nitration of meta-substituted). 278.

puospHorus (On catalytic reactions connected with the transformation of yellow) into
the red modification. 613.

Physics. A. Smits: ‘On the course of the P,7-curves for constant concentration for
the equilibrium solid-fluid.” 9.

— F. M. Jagcer: “Researches on the thermic and electric conductivity power of
erystallised conductors,” 1. 89.

— H. Haga: “On the polarisation of Rontgen-rays.” 104.

— H. KameriincH Oxnes: “Methods and apparatus used in the Cryogenic Labo-
ratory at Leiden. X. How to obtain baths of constant and uniform temperature

XIV CONTENTS.

by means of liquid hydrogen. 156. XI. The purification of hydrogen for the
cycle. 171. XII. Cryostat especially for temperatures from — 252° to — 259°.
173. X{IL The preparation of liquid air by means of the cascade process. 177.
XIV. Preparation of pare hydrogen through distillation of less pure hydrogen”. 179.

Physics. H. KAmERLINGH OnNes and C. A. CroMMELIN: “On the measurement of
very low temperatures. IX. Comparison of a thermo-element constantin-steel with
the hydrogen thermometer”. 180. Supplement. 403.

— H. KameruincH Onnes and J. Cray: “On the measurement of very low tem-
peratures. X. Coefficient of expansion of Jena glass and of platinum between
+ 16° and — 182°. 199. XI. A comparison of the platinum resistance thermo-
meter with the hydrogen thermometer. 207. XII. Comparison of the platinum
resistance thermometer with the gold resistance thermometer.” 213.

— W. H. Junius: “Arbitrary distribution of light in dispersion bands, and its
bearing on spectroscopy and astrophysics”. 343.

— H. KameruincH Onnes and C. Braak: “On the measurement of very low
temperatures. XIII. Determinations with the hydrogen thermometer’. 367.

— E. E. Mocenporrr: “On a new empiric spectral formula”. 434.

— HH. KameErLINGH Onnes: ‘Contributions to the knowledge of the y-surface of
vAN DER Waats. XI. A gas that sinks in a liquid.” 459,

— O. Postma: “Some additional remarks on the quantity H and Maxwell’s distri-
bution of velocities.” 492.

— H. KameriincH Onnes and W. H. Kresom: “Contributions to the knowledge
of the y-surface of van per Waats. XII. On the gas phase sinking in the
liquid phase for binary mixtures”. 591.

— W. H. kKegsom: “Contribution to the knowledge of the y-surface of VAN DER
Waats. XIII. On the conditions for the sinking and again rising of the gas
phase for binary mixtures.” 508. Continued. 660.

— J. D. van per Waals: “A remark on the theory of the p-surface for binary
mixtures.”’ 524.

— W. J. H. Motu: “An investigation of some ultra-red metallic spectra.” 544.

— J. D. van per Waats: “Contribution to the theory of binary mixtures.” 621.
Il. 727. Ill. 826.

— Pu. Konnstamm: “On the shape of the three-phase line solid-liquid—vapour
for a binary mixture”. 639.

— Pu. Konnstamm: “On metastable and unstable equilibria solid-fluid.” 648.

— H. Kamerirncn Onnes and Miss T. C. Jouies: “Contributions to the know-
ledge of the -surface of van DER Waazs, XLV. Graphical deduction of the
results of Kuenen’s experiments on mixtures of ethane and nitrous oxide.” 664,

— W. H. Junius: “Wave-lengths of formerly observed emission and absorption
bands in the infra-red spectrum.” 706.

— J. H. Mreerpura: ‘On the motion of a metal wire through a piece of ice.” 718.

— J. D. van per Waats: “The shape of the empiric isotherm for the condensation

of a binary mixture.” 750.

C.O NTE NTS. XV

Physics. H. Kamertincu Onnes and C, Braak: “Isotherms of diatomic gases and their
binary mixtures. VI. [sotherms of hydrogen between — 104° C. and — 217° ©,” 754.

— H. KamertincH Onyes and C. Braak: “On the measurement of very low
temperatures. XIV. Reduction of the readings of the hydrogen thermometer of

constant volume to the absolute scale.” 775.

— H. Kameriincu Onnes and W. H. krrsom: “Contributions to the knowledge
of the y-surface of VAN DER Waats. XV. The case that one component is a gas
without cohesion with molecules that have extension. Limited miscibility of two
gases.” 786.

— F. M. Jarecer: “On the influence which irradiation exerts on the electrical
conductivity of Antimonite from Japan.” 809.

Physiology. H. J. HamBurcer and Svante ARRHENIUS: “On the nature of precipitin-
reaction.” 33.

‘— J. K. A. Wertuem Satomonson: “A few remarks concerning the method of
the true and false cases.” 222.

— H. ZwaarDEMAKER: “An investigation on the quantitative relation between
vagus stimulation and cardiac action, an account of an experimental investigation
of Mr. P. Woxrerson.” 590.

PHYTOSTEROL (On the fatty esters of Cholesterol and), and on the anisotropous liquid
phases of the Cholesterol-derivatives. 78.

PLACENTA (On the formation of red blood-corpuscles in the) of the flying maki
(Galeopithecus). 873.

PLACENTATION (On the) of Sciurus vulgaris. 380.

PLAITPOINT CURVES (Ihe shape of the spinodal and) for binary mixtures of normal
substances. 4th Communication. The longitudinal plait. 226.

PLaNTs (On the harmful consequences of the secretion of sugar with some myrmeco-
philous). 150.

PLATINUM (Coefficient of expansion of Jena glass and of) between + 16° and — 182°. 199.

PLUCKER EQUIVALENTS (Second communication on the) of a cyclic point of a
twisted curve. 364.

POLARISATION (On the) of Réntgen rays. 104,

POsTMA (0.). Some additional remarks on the quantity H and Maxwell’s distribu-
tion of velocities. 492.

POTASSIUM HYPOCHLORITE (Action of) on cinnamide. 303.

PRECIPITIN-REACTION (On the nature of). 33,

quantity H (Some additional remarks on the) and Maxwett’s distribution of
velocities. 492.

RINGER (W. E.). V. WIND (C. H.).

ROMBURGH (P. VAN) presents a paper of Dr. I. M. Jagcer: “On the erystal-forms
of the 2.4 Dinitroaniline-derivatives, substituted in the NH,-group”’. 23.

— Triformin (Glyceryl triformate). 109.

— presents a paper of Dr. N. H. Conen: «@ “On Lupeol”. 466. 6 “On g-and 6
amyrin from bresk”. 471. ;

— and W. van Dorssen. On some derivatives of 1-3-5-hexatriene. 11).

xVI CONTENTS.

ROMBURGH (Pp. VAN) and A. D. Mavrenprecuer. On the action of bases,
ammonia and amines on s. trinitrophenyl-methylnitramine. 704.
RONTGEN RAYS (On the polarisation of). 104.
— (A hypothesis relating to the origin of). 714.

ROOZEBOOM (H. W. BAKHUTS). Vv. Baknurs RoozEBoom (H. W.).
ROTATIONS in Sp, (Equilibrium of systems of forces and). 820.
saLicyLic acip (The formation of) from sodium phenolate. 20.
SANDE BAKHUYZEN (H. G. VAN DE) presents a paper of Dr. J. Srem:
“Observations of the total solar eclipse of August 30, 1995 at Tortosa (Spain)”. 45.
— presents a paper of Dr. A. PanneKoek: “The luminosity of stars of different
types of spectrum”. 134, .
— presents a paper of Dr. A. PaNNEKOEK: “The relation between the spectra and
the colours of the stars”. 292.
— presents a paper of Dr. H. J. Zwrers: “Researches on the orbit of the periodia

comet Holmes and on the perturbations of its elliptic motion”. IV. 414.
— On the astronomical refractions corresponding to a distribution of the tempe-

rature in the atmosphere derived from balloon ascents”. 578.

SATELLITES of Jupiter in 1908 (Mutual occultations and eclipses of the). 304. Qnd
part. 444. ‘

SCHOUTE (P. H.) presents a paper of Dr. W. A. Verstuys: “ Second communica-
tion on the Pliicker equivalents of a cyclic point of « twisted curve.” 364.

— presents a paper of Dr. F. Scuun: “On the locus of the pairs of common points
and the envelope of the common chords of the curves of three pencils”. lst part. 424.

— 2nd part. Application to pencils of conics. 548.

— presents a paper of Dr. W. A. Wyrtnorr: “The rule of Neper in the four-
dimensional space.” 529,

— The locus of the cusps of a threefold infinite linear system of plane cubies

with six basepoints. 534.

— presents a paper of Dr. I. Scnun: “The locus of the pairs of common points

of four pencils of surfaces”. 555.
— presents a paper of Dr. F. Scnun: “The locus of the pairs of common points
of n + 1 pencils of (v—1) dimensional varieties in a space of x dimensions”. 573.
— presents a paper of Dr. S. L. van Oss: “Equilibrium of systems of forces and
rotatious in Sp,.” $20.

SCHREINEMAKERS (F. A. H.). On a tetracomponent system with two liquid
phases. 607.

SCHUH (FRED.). On the locus of the pairs of common points and the envelope of
the common chords of the curves of three pencils. lst part. 424. 2nd part. Ap-
plication to pencils of conics, 548.

— The locus of the pairs of common points of four pencils of surfaces. 555.
— The locus of the pairs of common points of x-+ 1 pencils of (x — 1) dimen-
sional varieties in a space of m dimensions. 573.
scluRUS vuLGARIS (On the placentation of). 280.
SERUM (Qn the amboceptors of an antistreptococcus). 336.

(ore

GO WN. TEN TS; XVII

SIRKS (H. a.) and A. #. Hotreman, ‘The six isomeric dinitrobenzoic acids, 280.
SMITS (a.). On the introduction of the conception of the solubility of metal ions
with electromotive equilibrium. 2.
— On the course of the P,7-curves for constant concentration for the equilibrium
solid-fluid. 9.
SODIUM PHENOLATE (The formation of salicylic acid from). 20.
SOLAR ECLIPSE (Observations of the total) of August 30, 1905 at Tortosa (Spain). 45.

SOLUBILITY of metal ions (On the introduction of the) with electromotive equilibrium, 2.
SOLUTIONS“ of non-electrolytes (On the osmotic pressure of), in connection with the
deviations from the laws of ideal gases. 53.
space (The rule of NEPER in the fourdimensional). 529.
spaces (The force field of the non-Euclidean) with negative curvature. 116.
— with positive curvature. 250.
sPEcTRA (The relation between the) and the colours of the stars. 292.
— (An investigation of some ultra-red metallic). 544.
SPECTRAL FORMULA (On a new empiric). 434.
SPECTRUM (The luminosity of stars of different types of). 134.

—- (Wave-lengths of formerly observed emission and absorption bands in the infra-
red). 706.

SPINODAL and plaitpoint curves (The shape of the) for binary mixtures of normal
substances. 4th Communication. The longitudinal plait. 226.

SPRONCK (c. H. H.) presents a paper of H. KysBroex: “On the Amboceptors of
an anti-streptococcus serum”. 336.

-—— presents a paper of P, NizuwENHUYsE: “Qn the origin of pulmonary Anthra-
cosis”. 673.

sTars (The luminosity of) of different types of spectrum. 134.

— (The relation between the spectra and the colours of the). 292.

STEIN (J.). Observations of the total solar eclipse of August 30, 1905 at Tortosa
(Spain). 45.

STOK (J. Pp. VAN DER). The treatment of wind-observations. 684.

suGAR (On the harmful consequences of the secretion of) with some myrmecophilous
plants. 150.

W-SURFACE for binary mixtures (A remark on the theory of the). 524.

— of van pER Waats (Contributions to the knowledge of the). XI. A gas that
sinks in a liqnid. 459. XII. On the gas phase sinking in the liquid phase for
binary mixtures. 501. XIJI. On the conditions for the sinking and again rising
of the gas phase in the liquid phase for binary mixtures. 508. Continued. 660.
XIV. Graphical deduction of the results of KUENEN’s experiments on mixtures
of ethane and nitrous oxide. 664. XV. The case that one component is a gas
without cohesion with molecules that have extension. Limited miscibility ofgases. 786.

TEMPERATURE (How to obtain baths of constant and uniform) by means of liquid
hydrogen. 156.

— in the atmosphere (On the astronomical refractions corresponding to a distribu-

tion of the) derived from balloon ascents. 578.

xV1II CONTENTS.

TEMPERATURES (On the measurement of very low). 1X. Comparison of a thermo-
element constantin steel with the hydrogen-thermometer. 180. Supplement.
403. X. Coefficient of expansion of Jena glass and of platinum between
+ 16° and — 182°. 199. XI. A comparison of the platinum resistance thermo-
meter with the hydrogen thermometer. 207. XII. Comparison of the platinum
resistance thermometer with the gold. resistance thermometer. 213. XIII. Deter-
minations with the hydrogen thermometer. 367. XIV. Reduction of the readings
of the hydrogen thermometer of constant volume to the absolute scale. 775.

TETRACOMPONENT SYSTEM (On a) with two liquid phases. 607.

THERMO-ELEMENT constantin-steel (Comparison of a) with the hydrogen thermometer.
150. Supplement. 403.

THERMOMETER (Comparison of a thermo-element constantin steel with the hydrogen).
180. Supplement. 403.

— (A comparison of the platinum-resistance thermometer with the hydrogen). 207.

— (Comparison of the platinum resistance thermometer with the gold resistance). 2138.

— (Determinations with the hydrogen). 367.

— (Reduction of the readings of the hydrogen) of constant volume to the absolute
scale. 775.

THIOPHENIC ACID (z-and B-). 514.

HREE-PHASELINES in Chloralaleoholate and Anilinehydrochloride. 99.

rortosa (Spain) (Observations of the total solar eclipse of August 30, 1905 at). 45.

TRICHT (B. VAN). On the influence of the fins upon the form of the trunk-myo-
tome. 814.

TRIFORMIN (Glyceryl triformate). 109.

TRINITROPHENYL-METHYLNITRAMINE (On the action of bases, ammonia and amines
on s.). 704.

TRUNK-MYOTOME (On the influence of the fins upon the form of the). 814,

eWIsteD cURVE (Second communication on the Plucker equivalents of a cyclic point
of a). 364.

VAGUS STIMULATION (An investigation on the quantitative relation between) and
cardiac action, 590.

VECTORDISTRIBUTIONS (Polydimensional). 66.

veLocities (Some additional remarks on the quantity und Maxwell’s distribution
of). 492.

VERSLUYsS (w. A.) Second Communication on the Plucker equivalents of a cyclic
point of a twisted curve. 364.

vOERMAN (6, L.) and A. F. Hotnemay. a-and @- thiophenic acid. 514.

vy OSMAER (G. ©. J.) presents a paper of B. van Tricur: “On the influence of the
fins upon the form of the trunk-myotome.” 814.

VRIES (JAN DPF). Quadratic complexes of revolution, 217.

WAALS (VAN DER) (Contributions to the knowledge of the y-surface of). XI. A
gas that sinks in a liquid. 459, XIL, On the gas phase sinking in the liquid
phase for binary mixtures. 501. XIII. On the conditions for the sinking and

again rising of the gas phase in the liquid phase for binary mixtures, 508.

CO: N-T°E N TS, xIX

Continued. 660, XIV. Graphical deduction of the results of KUENEN’s experiments
on mixtures of ethane and nitrous oxide. 664. XV. The case that one compo-
nent is a gas without cohesion with molecules that have extension. Limited
miscibility of gases. 786.

WAALS (J, D. VAN DER) presents a paper of Dr. A. Smits: “On the course of the
P,T-curves for constant concentration fer the equilibrium solid-fluid”. 9.

' — A remark on the theory of the Q-surface for binary mixtures. 524.

— Contribution to the theory of binary mixtures. 621. [L. 727. ILI. 826.

— presents a paper of Dr. Ph. Kouxstamm: “On the shape of the three-phase

line solid-liquid-vapour for a binary mixture’. 639. ;

— presents a paper of Dr. Ph. Kounstamm : “On metastable and unstable equili-

bria solid-fluid”, 648.

— The shape of the empiric isotherm for the condensation of a binary mixture. 750.
Waterstaat. C. Lety: ‘Velocities of the current in an open Panama-canal”. 849.
WAVE-LENGTHS of formerly observed emission and absorption bands in the infra-red

spectrum. 706. 2

WEBER (MAx). On the fresh-water fish-fauuna of New-Guinea. 462.

WEERMAN (kr. A.). Action of potassium hypochlorite on cinnamide. 303.

WENT (Ff. A. F. C.) presents a paper of Dr. W. Burck : “On the influence of the necta-
ries and other sugar-containing tissues in the flower on the opening of the
anthers”. 390.

WERTHEIM SALOMONSON (J. k. A.) A few remarks concerning the method
of the true and false cases. 222.

WIND-OBSERVATIONS (The treatment of). 654,

WIND/(c. H.). A hypothesis relating to the origin of Réntgen-rays. 714.

— A. F. H. Datuuisen and W. E. Rrincer. Current measurements at various

depths in the North-Sea. (lst Communication). 566,

WINKLER (C.) presents a paper of Prof. J. K. A. WertHEIM SaLomonson: “A few
remarks concerning the method of the true and false cases.’ 222.

— presents a paper of Dr. L. J. J. Muskens: “Anatomical research about cerebel-

lar connections”, 3rd Communication. 819.

WOLTER SON (P.). V. ZWAARDEMAKER (H.).

WYTHOFF (w. a.). The rule of Neper in the four-dimensional space. 529.

YOLK (Gastrulation and the covering of the) in the teleostean egg. 800

ZEEMAN (P.) presents a paper of Dr. E. E. Mocenporrr: “On a new empiric
spectrai formula”. 434.

— presents a paper of Dr. F. M. Jazcer: “On the influence which irradiation

exerts on the electrical conductivity of Antimonite from Japan’. 809.

Zoology. F. Mutuer: “On the placentation of Sciurus vulgaris’. 380.

— Max Weser: “On the fresh-water fish-fauna of New Guinea”. 462.

— J. Boeke: “Gastrulation and the covering cf the yolk in the teleostean

ego”. 800.
— A. A. W. Husrecut: “On the formation of red blood-corpuscles in the
placenta of the flying maki (Galeopithecus)”. 873.

“= “Ys = en ne eds al es ee '
C20-N TE IN ATBay Ri
% ’ e

ZWAARDEMAKER (H.). An investigation on the quantitative relation between

perturbations of its elliptic motion. : 414.

Me Mere ht. toe
vagus stimulation and cardiac action, an account of an Experimental investigation
of Mr. P. WoLrErson. 590.

ZWIERS (H. J.). Researches on the orbit of the periodic comet aoe and on “the '

Q Akademie van Wetenschappen,

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ALS de Wis— en Natuurkundige

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