FOR THE PEOPLE
FOR EDVCATION
FOR SCIENCE
LIBRARY
OF
THE AMERICAN MUSEUM
OF
NATURAL HISTORY
j Sound
A.M.N.
KONINKLIJKE AKADEMIE
VAN WETENSCHAPPEN
: TE AMSTERDAM :
PROCEEDINGS OF THE
SECTION OF SCIENCES
VOLUME IX
JOHANNES MULLER :— : AMSTERDAM
: : JULY 1907 :
(Translated from: Verslagen van de Gewone Vergaderingen der Wis en Natuurkundige
Afdeeling van 26 Mei 1906 tot 26 April 1907. Dl. XV.)
KONINKLIJKE AKADEMIE
VAN WETENSCHAPPEN
: TE AMSTERDAM :
PROCEEDINGS OF THE ^o^Osi
SECTION OF SCIENCES
VOLUME IX
( — 1ST py^Rp _ )
JOHANNES MULLER :: AMSTERDAM
: : DECEMBER 1906 1 •
(Translated from: Verslagen van de Gewone Vergaderingen der Wis en Natuurkundige
Afdeeling van 26 Mei 1906 tot 24 November 1906. Dl. XV.)
CONTENTS.
Page
Proceedings of the Meeting of May 26 1906 . . 1
» » June 30 » 65
» » September 20 » 149
» »
» » » » » October 27 » 249
» » ;> » » Xovember 24 » 379
KONINKLIJKE AKADEMIE VAN AVETENSCHAPPEN
TE AMSTERDAM.
PROCEEDINGS OF THE MEETING
of Saturday May 26, 1906.
(Translated from : Verslag van de gewone vergadering der Wis en Natuurkundige
Afdeeling van Zaterdag 26 Mei 1906, Dl. XV).
COlsTTEIiTTS.
A. Smits: "On the introduction of the conception of the solubility of metal ions %\ilh
electromotive equilibrium", (Communicated by Prof. II. "W. Bakhuis Roozeboom), p. 2.
A. Smits : "On the course of the P,7'curves for constant concentration fur the equilibrium
solidfluid". (Communicated by Prof. J. D. vax der Waals), p. 9.
J. Moll vax Charante: "The formation of salicylic acid from sodium phenolate'. (Commu
nicated by Prof. A. P. N. Fraxchimoxt), p. 20.
F. M. Jaeger; "On the crystalforms of the 2,4 Dinitroanilincderivatives, substituted in the
NH2group". (Communicated by Prof. P. van Romburgh), p. 23.
r. M. Jaeger: "On a new case of formanalogy and miscibility of positionisomeric benzene
derivatives, and on the crystalforms of the six Nitrodibromobenzenes". (Communicated by Prof.
A. F. Holleman), p. 26.
H. J. Hamburger and Svante Arrhexics: "On the nature of precipitinreaction", p. 33,
J. Stein: "Observations of the total solar eclipse of August 30, 1905 at Tortosa (Spain)".
(Communicated by Prof. H. G. van de S.ande Bakhcyzen), p. 45.
J. J. VAN Laar : "On the osmotic pressure of solutions of nonelectrolytes, in connectioji with
the deviations from the laws of ideal gases". (Communicated by Prof. H. W. Bakhos Rooze
boom), p. 53.
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 2 )
Chemistry. — ''On the introduction of the conception of the solu
hility of metal ions /rith electromotive equilibrium." By Dr.
A. Smits. (Communicated bv Prof. H. W. Bakhuis Roozeboom).
(Communicated in the meeting of April 27, 1906.) .
If a bar of NaCl is placed in pure water or in a dilute solution,
the NaClmolecules will pass into the surrounding liquid, till an
equilibrium has been established ; then the moleiadar thermodynamic
potential of the Na CI in the bar has become equal to that of the
Ka CI in the solution.
As known, this equilibrium of saturation, represented by the equation :
ftNaCI = ftNaCl
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 Na Clmolecules 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 solidliquid under consideration this is not of
direct importance.
If, however, we immerge a metal e. g. Zn into a solution of a
salt of this metal, e. g. ZnSO^, 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 ions with a positive charge.
If the particles emitted by the bar of zinc were electrically 7ieutral,
then the zinc would continue to be dissolved till the molecuhir
thermodynamic potential of. the zinc in the bar of zinc had become
equal to that of the zinc in the solution, in which case the equation :
^zn — f* zn
would hold.
This, however, not being the case, and the emitted Z/iparticles
being electropositive, an equilibrium is reached lorn/ before the
thermodynamic potential of the zincparticles with the positive electric
charge in the solution has become equal to that of the zinc in
the bar of zinc 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 zinc emitting' positive Ziiion^, the surrounding solution becomes
electropositive, and the zinc itself electronegative. As known, this
gives rise to the formation of a socalled electric doublelayer in
the boundinglayer between the metal and the electrolyte, consisting
of positive Znions on the side of the electrolyte and an equivalent
amount of negative electricity or electrons in the metal.
By the formation of this electric doublelayer 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 Znions.
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 :
If the equilibrium is a purely electrical e(uilil»riiiin 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 electromotiv^e equilibrium, then with virtual displace
ment of this equilibrium, the sum of the changes of the molecular
potential j the sum of the changes of the electric energy will
have to be =z 0.
+
If we represent the mol. potential of the Znions by ;/;„ in case
+
of electromotive equilibrium, we know that (i~,t is much smaller
than Hsn or the mol. potential of the zinc in the bar of zinc.
If we now suppose that a Znion emitted by the zinc 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 v = valency of the metal and e = 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*
( ^)
+
de
V6
which increase is negative, because fi„ > [i^n •
111 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 Ve and that of the
zinc V„t , we know that in the above case I"c ]> Vm and Ve — V,,, = ^
indicates the potential diiference 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 A de.
From the principle of virtual displacement follows that with electro
motive equilibrium
+
^'''~^''" deiAde = 0, (1)
V e
or
+
^ ^ _ i^lfl^J^ (2)
V 6
Now we know that the inol. thermodyn. potential of a substance
may be S[)lit up as follows :
(1= ft' f/i; Tin C
where in diluted states of matter ft' may be called a function of
the temperature alone.
In nondiluted states however, (i depends also somewhat on the
concentration.
If we now apply this splitting up also to equation (2), we get :
. _ (li'.n(Xzn){RT ln C
where C represents the concentration of the Ziiions in the electrolyte.
If we now put :
•4
ii.nll'.n ^^^^^
RT ^ ^
we may say of this K that for diluted states of matter it will only
depend on the temperature, and will therefore be a constant at
constant temperature.
From equation (3), (4) follows
RT K
ve C
Mr. VAN Laar already pointed out that this equation, already
derived by him in the same way is identical with that derived by
K T P r
Nernst a = In — , in which therefore — stands instead of
Ï' 8 p p
K
— . P represents the "elektrolytische Lösungstension" of the metal,
and p the "osmotic pressure" of the metalions in the solution.
Rejecting the osmotic phenomenon as basis for the derivation of
the dilferent physicochemical laws, we must, as an inevitable conse
quence of this, also abandon the osmotic idea "elektrolytische Lösungs
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, whe]i we
seek the physical meaning of the quantity K in equation (5), it can
be so simply and sharply detined, that when we take the theory of
the thermodynamic potential as foundation, we do not lose anything,
but gain in every respect.
In order to arrive at the physical meaning of the quantity K, we
put for a moment
C = K
from which follows
A=:0.
From this follows that there is a theoretical possibility to give
such a concentration to the metalions in a solution that when we
innnerge the corresponding metal in it, neither the rnetahior the solution
yets elec.tncally chanjed.
How we must imagine this condition is sliowji by equation (2).
Let us put there A = 0, then follows from this for an arbitrary metal
_ +
l^m ~~ f*m
or in words the molecular potential of the metal in the bar is equal
to that of the metalions in the solution.
So it appears that we have here to do with an equilibrium
which is perfectly comparable with that between the Na CI in the
bar Na 01, and the salt in the solution.
(6)
The only difference is (his that l!ie molecules of a salt in solution
are neutral!}^ electric, whereas the metal particles in solution are
charged with positive electricity, hence the physical meaning of
+
the equation ft^ = [im 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 metalions have reached their concentration oj
saturation, and that K therefore represents the solubility of the
metalions.
To prevent confusion, it will be necessary to point out that the
fact that the dissolved metalparticles 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
copperions is supersaturated with respect to copper. Yet such a
coppersulphatesolution is in a perfectly stable condition, because
the copperions constitute a part of the following homogeneous
equilibrium,
CuSO,:^Cu" ^so:
which is perfectly stable as long as the solution is unsaturate or is
just saturate with O^/SO^molecules.
If we now, however, insert a copper bar into the solution, the
condition changes, because the Cwions which were at tirst only in
equilibrium with the CuSO^mo\^ and with the aSO/'Ious, must now
also get into equilibrium with the copper bar, and, the concentration
of the C'Mions with respect to copper being strongly supersaturate,
tiic (/Mions 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
zincsalt the concentration of the zincions always remains below the
concentration of saturation, which a})pears immediately when we
immerge a zincbar into such a solution; the zinc emits zinc 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.
( 7)
We know the potential dilierence 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
55,5 r + 1
in which v 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:
RT K
^« = In
vs 1
55,5r + l
or
RT
A„= In A'(55,5r + 1)
If we use ordinary logarithms for the calculation, we get:
RT
log Kihh.hv \ 1)
~" V8 X 0,4343 ^ ^ ' ^ ^
If we now express R in electrical measure, then
0,000198
t,^ — _: Tlog ^(55,5 v f 1)
V
and for / = 18 or r=291°
0,0578
• A„=z^— %A'(55,5r + l)
V
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, (he
value of A„ decreases and with it the value of loy K.
For the metals down to Fe {Fe 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 K will always be larger than C for the metals mentioned,
and as K denotes the concentration of saturation of the metalions,
( « )
Values of log K at 18°.
metal
ion
Ao
log K
metal
ion
Ao
log K
A
(+ 2,92; ')
i+^^Krt )
Co
— 0,045
— 1,805 2
Na
(» 2,54)
( 42,19 )
rn
» 0.049
— 1,872X 2
Bcf
(» 2,r)4)
( 42.92X2)
Sn
< » 0,085
<  2,49 X 2
Sr
(» 2,49)
( 42,00X2)
Pb
» 0,13
 327X2
Ca'
(» 2,28)
( 3S,42X2)
H
» 0,28
— 6,6
Mg'
» 2,26
38,07 X 2
Cu
)) 0.61
— 11,58 X 2
At
» 1,00
16,56 X 3
Bi
< >^ 0,67
<— 12.33 X 3
Mu
» 0,80
12,81 X 2
llff,"
» 1,03
— 18,84 X 2
Zh
» 0,49
7,45 X 2
Ag'
)) 1,05
— 19,92
Cd
)) 0,14
1,39 X 2
Pd
.. 1,07
— 19.03 X 2
Fe
» 0,0G3
0,065X 2
Pt
» 1,14
— 20,62 X ^
n
» 0,045
— 0,245X 2
Au
» 1,36
— 26,27 X 3
the metalions will not jet have reached their concentration of
saturation even in the most concentrated solutions of the corresponding
metalsalts. Hence, w^hen tlie 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 K would always be smaller than
C, can of course not occur. If log 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
[)lace through zero. Wliether 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 2^. On account of this
very small value of K, C is greater than K in nearly all copper
saltsolutions, or in other words the concentration of the Owions is
greater tlian the concentration of saturation. Hence copperions 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 K may be, it will nearly always be possible to
^) The values of a o between parentheses have been calculated from the quan
Illy of heat.
( 9 )
make C smaller than A'. In a coppersal tsoliition e.g. this can very
easily be done, as is known, by addition of /vCxV, which in consequence
of the formation of the complexions \Cu^{CjSI)^' , causes copperions
to be extracted from the solution. The solution, which at tirst had a
]iegative charge compared with the metal copper, loses this charge
completely by the addition oï KCN, and receives then a positive charge.
In the above I think I have demonstrated the expediency of
replacing the vague idea "elektrolytische Lösungstension" by the
sharply detined idea solubility of metiil ions.
Amsterdam, April 190G. Anory. Chem. Lah. of the University.
Physics. — "(M the course of the P,Tcurv€s for constant concentra
tioii for the equilihrium solidfluid.'' By Dr. A. Smits. (Commu
nicated by Prof. J. D. van der Waai.s.)
(Communicated in the meeting of April 27, 190G).
In connection with my recent investigations it seemed desirable
to me to examine the hidden connection between the sublimation
and meltingpoint 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 der Waals. Though
his results will be published afterwards, Prof, van der Waals allowed
me, witli a view to the investigations wdiich 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 etheranthraquinone ^) van der Waals also
discussed the F, 7lines for constant ,/; for the equilibrium between
solidfluid ^), and more particularly those for concentrations in the
immediate neighbourhood of the points p and cj^, 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, J'line
as it would be in the usual case, was not discussed.
i) These Proc VI p. 171 and p, 484 Zeitschr. f. pliys. Chem. 51, 193 and 52,
587 (1905).
2) These Proc. VI p. 230 and p. 357.
(10)
If we start from the difïerential equation in p,;v and T derived
by VAN DER Waals (Cont. II, 112).
Vsrdp = {,v,,vf)l^] clvf+dT . . . (1)
we get from this for constant x that
T
Vsfdp — dT (2)
or
T ('^ — — ^ (3)
If we now multiply numerator and denominator by r — as will
prove necessary for simplifying the discussion, we get :
(4)
In order to derive the course of the P, Tlines 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, .ïüg. 1 the lines ah and cd denote the two connodal
lines at a definite temperature. The line PsQs whose x = .r^ the
concentration of the solid compound AB 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 isobav MQIWB'U'Q'N
of the pressure of Ps will cut the connodal lines in two points Q
and Q', which points indicate the fluid phases coexisting with the
solid substance AB, and therefore will represent a pair of nodes.
The points for which — = or — ^ = are situated where
the isobar has a vertical tangent, so in the points D and JJ as
VAN DER Waals ') showed already before. In B the isobar passes
through the minimum pressure of the mixture whose x^^^xj), and
so it has there an element in common with the isotherm of this
concentration. In B' however, the isobar passes through the maxi
1) These Proc. IV p. 455.
A. SMITS. "On the
(11)
ïnnm pressure of the mixture wliose .i = /r/)', and will therefore have"
an element in common with tiic isotherm of the concentration xjo'.
As for the sign of  — we may remark that it is positive outside
the points D and D and negative inside them.
The ordinary case being supposed in the diagram, viz. Vs <C Vj\
we may draw two tangents to the above mentioned isobar from the
point Ps with the points of contact R and E. These points of con
tact now, indicate the points where the quantity l^,y=0, as van
DER Waals ^) showed.
This quantity is represented by the equation :
and denotes the decrease of volume per molecular quantity when
an infinitely small quantity of the solid phase passes into the coex
isting fluid pliase at constant pressure and temperature.
For the case that the coexisting phase is a vapour phase, Vsf 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 F) 0> ^^s even twice.
To elucidate this Prof, van der Waals called attention to the
geometrical meaning of V^f.
Let us call the coordinates of the fluid phases Q coexisting with
Fs, Vf and Xf and let us draw a tangent to the isobar in Q.
Then Ps F will be equal to Vsf if P is the point where this
tangent cuts the line drawn parallel to the axis of v through Ps.
If the point P lies above Ps, Vsf is negative, and if P' lies under
Ps, then Vsf is positive. For the case that the tangent to the isobar
passes thi'ough Pg, which is the case for the points E and R', Vsf=^ 0.
In this way it is very easy to see that for the points outside those
for which Vsf = 0, the value of Vsf is negative, and for the points
within them, Vsf is positive, but this latter holds only till the points
D and D' have been reached, where Vsf=^o:> Between Z) audi)',
Vsf is again negative. The transition from positive to negative takes
therefore place through od .
As each of the lines of equal pressure furnishes points where
1) These Proc. VI, p. 234.
(12)
— ^ = O and Vsf = O, when connecting the corresponding points
we obtain loci of these points, indicated by lines.
As, however, we simplify the discussion, as van der Waals has
a> ^ .
shown, when we consider the quantity  — . V^f instead of the
• dv/
quantity Vs/, because this product can never become infinitely great
and is yet zero when F,/ = 0, the locus of the points where
d'lp ^^ ^ . . • /. ^
—  . Vsf=0 IS given in fig. 1.
dv/
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.
ö>
Further the locus of  — =: is indicated, and we see that these
two lines intersect at the point where they pass through the line of
the compound.
In his lectures van der Waals has lately proved in the following
way that this must necessarily be so : If we write for — . Vg/
we see that when this quantity = 0, and when at the same time
Xs = ay :
or
^ = 0.
dv/
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 threephase equilibrium also on the
right, and by drawing the corresponding isobar M^Q^BJiJi^'B^'Q.'N'.
It appears then that here the points R^ and R^' lie within the points
Z)i and i)/, which points to a reversed situation (compared with
the left half) of the loci ^ . Vsf = and —==0. Van dek Waals
dl'/ • dv/
has also given this graphical proof.
As for the sign of the quantity r — on the right of the line
Ov/
( 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 :
d'xb \d.rdvj ö'5
^ — ^^ —  = or —  = (J (7 1
p
holds, lies entirely outside the locus r — = 0. Van der Waals M
proved this in the following way :
On the spinodel curve and  — must both be positive, and so
/d'xpY ^ ö> dp
also r^— . As T— =: — V IS positive outside the line for which
\0:cdi:J ov^ ov
 — =z 0, the spinodal line will alwavs have to lie outside the curve
^ = 0.
Or'
That the spinodal curve which coming from the left, runs between
the lines — — . Vg/ =z Q and  — = 0, cuts the line for —
Vs/=0 on the left of the line of the compound in two points q^
and q^ which will be discussed afterwards, follows from this, that
d^xp d^xb
on the line of the compound  — . Vsf=^ coincides with =
and that the line  —  alwavs lies within the spinodal line, whereas
drf
on the riglit of the line of the compound  — . Vg/ = lies within
Ovf'
ö>
the line  — = 0.
dv/
When we start from the maximum temperature of sublimation,
we get now i',.i'lines which have been indicated by T^, 7\, 7\ and
T^ in fig, 1 for the equilibria between solidfluid according to the
equation ')
ö> aiL'"i
i^^sVf)^^\{.Vs.vf)^\
•^ ■ ■—. ... (8)
dv/_
dxf d'lp
•) loc cit.
2) These Proc. YI, p. 489.
spinodal curve and on the curve .. „ . Vgf = 0.
( 14 )
The z;,,?;curve denoted by T^, relating to \\\q maximum temperature
of subUïiiation, consists of two branches, which pass continuously
into each other. The points of intersection with the connodal line
a b 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,,i'line cut the locus
With 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,x\\ne denoted by 1\. This
point of intersection is the point q^, it lies therefore both on the
V
If we now proceed to higher temperatures, detachment takes
place, and the ?;,.i'figure consists of two separate branches, one
of which, viz. the vapour branch is closed. This case is represented
by the v,xïme 1\, for which it is also assumed, that this temperature is
the minimummelting jwint of the compound, which follows from the
fact that the liquid branch of the v,a;\me T^, simultaneously cuts
the connodal line cd and the line of the compound.
With rise of temperature the closed i\.i'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 a b and the liquid branch the
connodal curve c d. This is represented by the v,.i'figure T^, which
represents the condition at the maximumthreephasetemperature, 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 q^, where the upper branch of the
spinodal curve and the curve ^r ^sf = intersect.
ö>
If we now also indicate the locus of the points where ^— . Wgf =
( 15 )
the peculiarities of the course of the PTlines may easily be derived
by means of the foregoing.
For the determination of tlie last mentioned locus, we start from
the equation:
"V =
P + ( ^)y, J ^'^' + (^^/)" . . . , (9)
The factor of Vgf being naturally positive and. (es/)y being always
negative, Ws/ can only be equal to zero in a point x where Vsf is
positive, so between the loci where Vsf=0 and ^r — = 0.
Ovf
Further it is now easy to understand that at the same time
with Vsf the quantity W^f will become infinitely great, there where
ö>
^—  =i 0. In order to avoid this complication van der Waals has
multiplied the quantity Wg/ by ^— ^ as equation (4) shows ; the
obtained product never becomes in finitely great now.
If we multiply equation (9) by t —  , we get :
/. + il^
Vsf + T~{^sfh ■ . (10)
öiyyyvjör/ dvf'
Now we know that the locus for ;r — . Wsfz=zO will have to lie
between that for ^ — . V^f =: and for ^ — = , as drawn in
dvf' ^ dvf^
fig. 1, which compels us to make ^^ — . Wsf^=0 and  — . Vg/ :=
intersect on the line of the compound.
That this must really be so, is easily seen, when we bear in mind,
that on the line of the compound the locus where ^— ^ = coincides
with that where ^;^ — . F^/ = , from which in connection with
equation (10) it follows immediately that at the same point also
 — . Wsf =z 0. In this way we arrive at the conclusion, that the three loci
dv;" ^ ^
^;^ —  = ,  — . Vsf = and ^ — . Wgf =z will intersect on the Ime
( 16 )
of the compound, and that therefore the loci
dvf'
Vsf ■= and
. Wsf = will interchange places on the left and the right of
d^
the line of the compound.
By means of equation (10) we understand now easily that the
sign of the quantity r —  . IFs/ must be negative outside the locus
 — . W,f = 0, and positive within it.
dv/
As connecting link for the transition to the PjJ'lines we might
discuss the TsTHnes; for this purpose we should then have to make
use of the following equation (Cont. II, 106)
rvj' ovf . o.vf_
OV/ OVf . OWi
dvf . d.vf
Ö>1 dT
By taking x constant we derive from this
{dT
or
T
dv
dT
7
— ihf)"
V.
sf
I shall, however, not enter into a discussion of the F7lines because
it is to be seen even without this connecting link, what the course
of the P,7Mines must be.
If now for simplication we call
Vsf = X^ and
W^= X
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 :
left
Xj — Xj —
X, =
right
x, — o
X.  ^. + V _
( 17 )
If now led by equation
(4a)
we (liiiw tlic 1\ 7line for a conccjilralion on the left of the curve
of the compound, we obtain a curve as gi\'en by GF'FD in fig. 2.
As we have assumed in our diagram, that the vapourtension
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.
F
•
/ ' '
■3 /
llf^
h '•■'
^y
' /
W^
Fig. 2.
This intersectiou 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 meltingpoint 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, Tline, 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 ri(/ht of the line of
the compound, the F, Tline 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 A\ and A'^ preceding it, we obtain a curve as
indicated by G^ F^ F^ D^. The situation of the loci A'^ ^ and
Xj = being different on the fight from that on the left, this
P, Tline differs from that just discussed. When now, starting from
the point G■^ resp. D.^, we follow the P,7^line, we meet first with
a point, where it is vertical, so we have just the reverse of the
preceding case. About the situation of the points P/ and F^ we
may point out, that jP/ always lies at lower temperature than F^.
The loci Xj = and A^ ^ intersecting on the line of the
compound, the P, Tline for the concentration of the compound will
have to give to a certain extent the transitioncase 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 Aj =
and Aj = 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
Aj and X^ that when the loci A"j := and A'^ =; have coincided,
the signs of A^ and X^ reverse simultaneously, on account of which
P( — ) retains the same sign, viz. remains positive. Combining this
with what we know about the course of the P,Plines somewhat
to the right and the left of the curve of the compound we are led
to the conclusion, that the P, 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
meltingpoint line must toucii the threephase line, so that the P,Pline
1) Verslag 21_April 1897, 482.
( 19 )
for the conoentration of tlie compound assumes a shape, as given
by the line G^ l\' F^ D^ in fig. 2.
If it were possible to make the degree of association of the com
pound smaller and smaller, the points FJ and F^ would move to
lower pressure and higher temperalure. Moreover these two points
and the neighbouiing })oint of intersection of the meltingpoint and
sublimation branches would draw nearer and nearer to each other,
till with perfect absence of dissociation these three points would
have coincided.
Another peculiarity will present itself for the case that we have
a threephaseline as desci'ibed by me before, viz. with two maxima
and one minimum'), for then there is a point where Xfi=zxL^) on
Fig. 3.
this line, and tiien it is immediately to be seen tiiat in consequence of
the coinciding of the points F' and F, we get for this concentration
a PjJ'line, as represented in (ig. 3, which curve has the form of
a loop.
Amsterdam, April 1906. Anorg. Cliem. Lab. oj the University,
1) These Proc. VIII, p. 200.
2) lu this point the direction of the threephase line is given by 2'~jyr,=
dT vdvi
2*
( 20)
Chemistry. — ''The formation of salicylic acid from sodiwn
phenolate.'" By Dr. J. Moll van Charantk. (Coinmimicated
by Prof. A. P. N. Franchimont).
(Communicated in the meeting of April 27, 1906).
The communication from Lobry de Bruyn and Tijmstra read at
the meeting of 28 May 1904 and tlieir subsequent article in the
Recueil 23 385 induced me to make this research. Their theory,
and particuhirly the proofs fuiven 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 phenokite
yv'iih. sodium phenylcarbonate, or what amounts to the same an
additive product of two mols of sodium phenolate with one mol. of
carbondioxide C6H50C(ONa).^OC6H5 might be the substance which
undergoes the intramolecular transformation to the salicylic acid
OH derivative and then forms, dependent on the tem
y perature, sodium salicylate and sodium phenolate
\ j; or else phenol and basic sodium salicylate. This
\<C.^ view is supported by previous observations of
V. various chemists and has been partially accepted
0C« H, ^^Iso by Claisen ').
As Lobry de Bruyn and Tijmstra give no analytical figures in
their paper it did not seem to me impossible that the phenolsodium
öcarboxylic 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 phenylcarbonate in the usual /^
manner, from sodium phenolate and carbon dioxide, ^«"^4 UJNa
and heated this to 100° in a sealed tube for 100 t(r\y,i
hours. On opening the tube considerable pressure was \
observed. This pressure was always found when QQ H
the experiments were repeated. The gas liberated proved to consist
entirely of carbondioxide and amounted to 74 — Va of that present
in the sodium phenylcarbonate. H' we argue that the sodium phenyl
carbonate under these circumstances is [)artia!l3' 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 (CgHsO')^ C(0Na)2. In the first ease 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 although
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 Landsberger, 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
phenylcarbonate in the manner indicated was, however, very trilling.
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
phenylcarbonate, so that, therefore, the reaction was not completed,
and that the tube after being heated must still contain a mixture of
unchanged sodium phenylcarbonate, sodium phenolate, sodium sali
cylate and free phenol, besides the said additive product (CbHbO)^
OH
C(0Na)2 and the salicylic acid derivative possibly CgH^ ONa
formed from this. I now thought it of great \y
importance to first study the behaviour of acetone CONa
with these substances as far as they are known. /^n u
(Jug Hj
Sodium phenolate dissolves in boiling acetone, from which it
crystallises on cooling in soft, almost white needles, several cm. Ion",
which contain one mol. of acetone. They lose this acetone, in vacuo,
over suli)huric acid. At the ordinary temperature acetone dissolves
only 0,1 7„.
Sodium phenylcarhonate placed in caiefuUy dried acetone gives otf
( 22 )
carbon dioxide with a slight elevation of temperature. The quantity
amounts to about ^/g 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 niixtiue is extracted with ether,
also the ether, contains a quantity of phenol 'corresponding with the
total amount obtainable from the sodium phenylcarbonate. 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 + C(3, + NaHC03 + 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, witli 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 onehalf to
a full molecule. At iQ^ it dissolves in about 21 parts of acetone.
Disodiuin salicylate was prepared by adding an (957o) alcoholic
solution of salicylic acid to a concentrated solution of sodium
ethoxide in alcohol of the same 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 \y\{\\ these substances now being known,
the experiment of heating the sodium phenylcarbonate 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 phenylcarbonate. A quantity of carbon
dioxide was collected corresponding with an amount of unchanged
sodium phenylcarbonate representing 50 — 60 7o of the reaction
product. Another portion was extracted with ether and yielded about
20 7o 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 disodiumsalicylate.
It, however, contained )hcnol, j)i'obal)ly from sodium phonohite.
It seems strange there is such a large quantity of free phenol
( 23)
in the heated sodium phenylcarboiiate, and as no disodiumsalicylate
has been found it cannot liave been caused by the formation of
that compound.
I have not been able to hnd the lookcil 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 saHcylic acid from sodium phenylcarbonate is
not so simple as is generally imagined.
A more detailed account of research will appear in the "Recueil".
Chemistry. — ''On the cnj stal forms of the 1,^DinitroanUineden
vatlves, substituted in the ^R^(/roup\ By Dr. F. M. Jaegkr.
(Communicated by Prof. P. van Romburgh).
(Communicated in the meeting of April 27, 190G).
More than a year ago I made an investigation as to the form
relation of a series of positionisomeric Dinitroanilmederivaüves ^).
On that occasion 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 Nf^I.^ group.
Among the compounds then investigated, there were already a few
124: Dinitromiilhieoevwaiives kindly presented to me by Messrs.
VAN Romburgh and Franchimont. Through the agency of Prof.
VAN Romburgh and Dr. A. Mulder, I have now received a series of
other derivatives of 2,4DmitroaniUne 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 formsymbolic, that the morphotropous rela
tion of the great majority of these substances is clearly shown. They
all possess the Same familycharacter 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) Jaeger, Ueber morpholropisclie Beziehungen boi den in der AniinoGruppe
substituierten Nitfo Anilinen; Zeils. f. Krysl. (1905). 40. 113— VG.
Name of the compound
Survey of tl
M.p.
Mol.
weight.
Equiv. Vole.
(iu the solid
state.)
12Nitro Aniline.
14NitroAniline.
124DinitroAniline.'
•l_24CTrinitroAniline.
14NitroDiethylA.*)
i2iDinitroMethylA.
i24DinitroEthylA.
l24DinitroDime1hylA.
l_24GTrinitroüimethylA.
l_24DinitroMethylEthylA.
124DinitroDiethylA.
l_24, + 13 4üinitroDiethylA.
(Double rompound.)
124 GTrinitroDiethylA.
'l_24DinitroEihylnPropylA.
124GTriiiitro EthylIsopropylA.
124DiiiilroIsopropylA.
i24Dinitro DipropylA.
124GTrinitroDipropylA.
lQZ^DinitroIsol^utylA.
';_24GTrinitroIsobutylA
'124 niiiitror>iïsobutylA.
]_9_4_DinitroAllylA.
■12 4DinitroMethylPhonylA.
'124Dinitro EthylPhenylA.
'1 24 D nitroBonzylA.
12 4Diiiitro Methyl DenzylA.
i_24DinitroEthyl Benzyl A.
124DinitroPlieiiylBenzylA.
124 GTriiiitroEihylNiUMnilino.
124GTrinitrjIsopropylNitraniline.
1234GTetranitroMethyiNitraniline.
72°
14G°
182=
ltO°
78°
178°
1140
87°
154°
59°
80^
59°
1G4°
55°
109°
95°
40°
138°
80^
95°
112°
70°
IGG'^
95°
110°
144°
73°
108°
90°
108°
140°
138
138
183
228
194
197
211
211
250
225
239
478
2?4
253
298
225
207
312
2.39
284
295
223
273
287
273
287
301
349
301
315
332
95.70
90.03
113.30
129.39
102.07
125.24
145.41
142 95
165.05
157.45
173.94
i(3G1.02)
192.41
189.13
211.80
153.79
202.50
227.23
172 70
190.53
250.21
157 9:J
194.10
210,48
187.50
204.41
219 87
250.00
183.09
201.53
189.71
*) On the isomorphism and the complete miscibility of this compound with pNitrosodiethylanilinej
ystallograpliicallyinvestigalerl derivatives of 1— 2— 4— DinitroAniline.
AxialElements:
Topic Parameters;
a :
a :
.a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a :
a ;
b :
b :
b :
b :
b :
b :
b :
'J.
b :
b :
b :
'J. :
b :
b :
b :
b :
b :
b :
a
b :
b :
'J.
b :
b :
b :
I) :
a
b :
b
b :
h :
b :
b :
b
b :
b ;
c = 1.3GG7 :1: 1.1585.
c = 2 .0350 : 1 : 1 .4220 ; ;5 zz 88° 10'
c  1.982G :1 : 1 4088; r^=85° 1^'
c = 1 .G5G0 : i : '1 .5208 ; /? == 80°47i'
c = 1 . 0342 : 1 : . 9894 ; ,9 = 80°34'
c =: 1 .228G : 1 : 0.9707 ; ;? = 8.30^8'
c = 1 2251 :1 ; 0.9745
= 33^25^' 1^ = 83^22' y = 75^41 '.
c = 1.2154 :1 : 1.0803.
c = 1.293G :1 : 1.3831.
c = 1.1497
= 75°40' ;3 =
c = 1.2045
e = 1.3435
= 1.1750
c = 1.0535
c = 2 0102
= 75°0' li =
c = 1.1527
= 118043' ;9
c = 1.0191
c " 1 3327
=:11904()' ^s
c = 0.7104
:1 : 1.GG39.
= 92°33,' y r= 08^57'
:1: 1.1.513.
:1 : 1.3013; ,9 = 80039'
: 1 : 94G2 ; ;3 =z 8G"28'
;1 : 0.9227.
0.7325
1.0717
c = 1.0251
= 71010^' i?
c ~ 1.1518
99^17' y = 1
: 1 : 1 5790.
i=104°33' y^
:1 : 0.9240.
: 1 : 9055
=rz111°0' y
:1: 0.3591;
: 1 : :',47().
:1 : 0.9124;
:1 : 0.9G32
= 11100' y
:1 : 1,0908;
1 010.1'
85' 1:^1'
 102O35'
;3 = 85034i'
,3 = G305i'
IIG04O'
,5 = 80^1'
: c : 0.4933 :1 : 0.G5SG; ,3 = 78°6i'
c = 1.0385
— 107^57' /3
c = 1.508G
0=1. 7258
c =: 1.1373
= 1 .4187
c = 1..3924
c = 1.0G8G
1 : 8.580
131J047' y
:1 : 1.327G;
:1 : 1.3087;
:1 : 1 3G45;
:1 .
:1 : 0.9JG8;
:1: 1.4712;
= 78023'
li = 7I04O'
,3 = 84°5'
;3 = G4024'
;3 = 8G023i'
P = 78°33'
,3^76°37'
X : '^ :
5.3G35 :
3 9245 :
4..54G5
6 .5400 :
3 2141 :
4 5704
G.820G :
3.4403 :
4.8407
5^9313 :
3.8119 :
5.7975
5.G210 :
5.4351 :
5.3775
5.8090 :
4.7281 :
4.5897
G.2321
5.0871 :
4.9880
5.8035 ■
4.77.50 :
5.1.583
5.8455
4.5184 :
G 2493
5.1900 :
4.5140 :
7.510G
G.0294
5 0058 :
5. 7031
6.30G4
4 G940 :
G 1083
G 5523
: 5.. 5702
5.2703
G.10'.5
5.7940
5 34GG
5.3440
4.G.3G0
7 3201
G 1042
: 5.9890
G..5382
8.1.550
G 1197
5.5414
G,1S70
8.7002 :
3 2145
G.7231
: 9.1782
3.1849
7.0080
: G.59G0
().0181
5.9480
: 5.8024
5.5890
5.3389
: 4.G34I
7.80.50
4.2994
: 8.71.5G
5.7401
7.2441
: G 9757
5.9S91
7.1735
: 4.7551
0.3129
G.9531
: 4.G08i
G.0310
6.4059
: 5. 7040
: 7.G85G
■I • 0) r^
7.3G7G : 5.2913 : 5.2743
7.1730 : 4.2987 : G.3243
lese Proceedings (1905) p. 658.
(26)
Crystallography. — "On a new case of formanalogy and misci
bility of positionisomeric benzenederivatives, and on the crystal
forms of the six Nitrodibromohenzenes.'" Bj Dr. F. M. Jaeger.
(Communicated by Prof. A. F. Holleman.)
(Communicated in the meeting of April 27, 1906).
§ 1. The following contains the investigation of the crystalforms
exhibited by the six positionisomeric Nitrodibromobenzenes, which
may be expected from the usual structurerepresentations of benzene.
It has been shown that, in this fully investigated series, there
again exists a miscibility and a formanalogy between two of the
six terms.
The above compounds were kindly presented to me by Prof.
Holleman, 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. Nitro23Dibromobenzene.
Structure: C^Hj . (NO,) .Br .Br ; meltingpoint: 53^ C.
(I) (2) (3)
Tlie compound, which is very soluble in most organic solvents,
Fig. 1.
crystallises best from ligrofn  ether in small, flat, pale sherry 
coloured needles which generally possess very rudimentary terminating
planes.
Triclinopinacoidal.
«:/>:C = 1,4778:1:1, 9513.
A= 90°30'
J5= 110°37'
C— 90°16V;
«= 90°45V;
/? = iio°367;
y = 89°59\/;
( 27 )
The ci'jstals, therefore, sliow a decided approach to the mono
clinic system ; on account, liowever, of their optical orientation, they
can only be credited with a triclinic symmetry.
The forms observed aie : a = jlOOj, strongly )redominant and very
lustrous; /; = 01ü}, smaller but yielding good ieflexes ; c = 00J,
narrower than a, bnt very lustrous ; o = jlil}, well developed and
very lustrous ; to :=: ill, smaller but very distinct ; .v = jillj, very
narrow but readily measurable.
The habit is elongated towards the /^axis with tlattening towards jJOOj.
Pleasured : Calculated :
a:h = (100) : (OlO) =* 90^67; —
a:c = (100) : (001) =* 69 23 —
a\o = (100) : (ill) =* 65 11
c:o = (OOl) : (111) =* 75 477^ —
b:o = (OlO) : (iTf) =* 36 6 —
a: w = (100): (lfl)=r 50 52 50^49'
c : oj = (001) : ( ill) r= 56 52 56 43
h : o) = \(dlO) . {Ill) = 46 28 46 35
: (o = (1ÏÏ) : (111) = 47 13 47 2973
a:s = (TOO) : (liT) = 49 59 50 4973
h : s = (OTO) : (Iff) =: 45 48 45 527,
C : s = (OOf) : (111) = — 56 4
: s = (111) : (Til) = 63 39 63 5973
Readily cleavable, parallel 100.
The extinction on 100 amounts to about 2672 in regard to the
óaxis ; in convergent light a hyperbole is visible occupying an eccen
tric position.
The sp. gr. of the crystals is 2,305 at 87 the equivalent volume 121.47.
B. Nitro25DibroinoBenzene.
Structure: C„H3 . (NOjr,) . B^o) • Br,,); m.p. : 84°,5.
This compound has been )reviously studied crystallographically by
G. Fels, (Zeits. f. Kryst. 32, 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 } ligTOïn have the form of
small plates flattened towards {00J (tigs. 2 and 3). They are pale
yellow and very lustrous.
(28)
Fig. 2.
1 : 2,0214.
a= 90°57V;
^=z 113°2173'
y=r 90' 2'.
Fig. 3.
Triclinopinacoidal .
a :/';:c = 1,4909
A= 9r 37;
B=z 113°21V;
C= 90°27'
Forms observed: c = jOOlj, strongly predominant and retlecting
ideally; a = jlOOj, and ?*=:101, nsnally developed equally broad
and also yielding sharp reflexes ; b ^= \010\, smaller, readily measur
able ; m = \liO], large and lustrous; ^; = 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 ease.
Measured
a.b = (100) : (010) =*89°33'
b:c = (010) : (001) =*88 567,
; a = (001) : (100) =66 387,
; m = (010) : (110) =*35 597,
: r = (100) : (101) =*43 45
m == (001) : (110) = 75 46
; m = (100) : (110) = 53 33
?• =: (OOi) : (lOl) = 69 37
7?i=(10i):(110)= 65 20
7?ï=rr(113):(110)= 60 59
: b = (lOi) : (010) =r 89 55
: p = (101) : (113) = 50 53
Calculated:
c
b
a.
c
a
c
r :
p:
r
r
75^387;
53 337,
69 367,
65 11
60 447,
89 22
( 29 )
ReadiJj cleavable, parallel in.
The optical orientation is that of Fhi,s, in which his forms 010,
{001 and llT assume, respecti\cly, in inv project the symbols
001, 110 and jOlOj. It may be remarked that Fels has incor
rectly stated the structure and also the melting point. Moreover, his
angles (11Ï) : (100) and (111) : (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 uiita\ourable than with me. I have
never observed forms 552 and 15 . 15 . 4
The sp.gr. ar 8° is 2,368; the equiv. volume: 118,06.
Topical axes : / . ip : to = 5,2190 : 3,5005 : 7,0758.
On comparing the said positionisomeric derivatives, one notices at once
not the great similarity between the two compounds, which, although
constituting a case of directisomorphism, still y^ry closely resembles it.
Nitro2oDlbromobenzene.
Triclinopinacoidal.
a:h:c= 1,4909 : 1 : 2,0214.
.4=91°3V; i?==113°21V; ^=90°27'
«=90°57\/; ^=113^217,' y=90°2'
i:\p:a)z= 5,2190 : 3,5005 : 7,0758.
However :
Forms : jlOOj, jOlOj, jOOlj, T01,
110, 113.
Cleavable parallel 110.
Habit tabular towards 001.
N itro2^Dlhromobenzene .
Triclinopinacoidal .
a:h:c= 1,4778 : 1 : 1,9513
^=::90°30'i?=110°37' (7=90"a6V;
«=90°45V3'/?=110°367;y=89°59V;
X : ip : a> = 5,2565 : 3,5571 : 6,9409.
However :
Forms : 100, jOlOj, jOOlj, jlTfl,
jlllj and llITi
Cleavable parallel 100.
Habit tabular towards 100.
We, .therefore, still notice such a difTerence 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 123derivative (53°) is depressed by
addition of the 125derivative. The melting point line has also
1) The binary raellingcurve possesses, — as proved by means of more a exact
determination, — a eutedic point of 52° G. at 2% of the higher melting com
ponent ; therefore here the already published meltingdiagram is eliminated. There
is a hiatus in the series of mixedcrystals, from ± i'^/o to circa 48"'o of the 123deri
vative. I shall, however point out, that the possibility of such a hiatus thermody
naraically 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 tribromotohienes
(Dissertation, Lejden 1903) a continuous form; the difference is
caused by the lesser degree of formanalogy which these substances
possess in proportion to that of the two said tribromotohienes.
The third example of miscibility, — although partially — , and of
formanalogy of positionisomeric benzenederivatives^) is particularly
interesting.
Mixed crystals were obtained by me from solutions of both com
ponents in acetone f ether.
They possess the form of fig. 1 and often exhibit the structure of
a sand timeglass or they are formed of layers. With a larger quantity
of the lowermelting deri\'ative, long delicate needles were obtained
which are not readily measurable. The melting points lie between
± 75" and 84^°; I will determine again more exactly the mixing limits.
C. Nitro24Dibromobenzeiie.
Structure : CeH, . (N0J(,) . Br(2) . Br(4) ; ra. p. 61°. 6.
Recrystallised from alcohol, the compound forms large crystals
flattened towards a and elongated towards the caxis. They are of
a sulphur colour.
Triclinopinacoidal.
a:b:c = \,\Z01
A— 97°13V;
B= 113°30\/;
Cz=z 90°38V;
1,1698.
a= 97°36'
/?= 113°37'
y= 87°33'
Fig. 4.
Forms observed : a = 100j predominant and
very lustrous; b = 010 and c = 001}, equally
broad, both strongly lustrous; p== 110, narrow
but readily measurable; ö^jlllj, large and
yielding good reflexes.
The compound has been measured previously,
by Groth and Bodewig (Berl. Berichte, 7, 1563).
My results agree in the main with theirs; in
the symbols adopted here, their a and />axes
have changed places and the agreement with
the other derivatives of the series is more
conspicuous.
1) The examples now known are 1235, and l'iiQTribwmobenzene ;
1^3bTribromoiSDinUro and l^iQTribromo3bDmitrotohiene\ and 125,
and 1^3Nitrodibromobenzene, partially miscible.
( 31 )
Measured : Calculated :
a: h = (100) : (010) =* 89^217; —
a. c = (100) : (001) = m 29\/; —
h: c = (010) : (001) =* 82 46\/; —
p.a = (1Ï0) : (100) =*■ 46 36 —
c:o = (001) : (ill) =* 48 42 —
o:p = (111) : (110) = 51 43 (circa) 52" 1'
c.p^ (001) : (IlO) = 100 29 (circa) 100 43'
Cleavable towards jOlOj ; Groth and lioDEwur did iiol find a
distinct plane of cleavage.
Spec. Gr. of the crystals = 2,356, at 8° C, the equiv. vol. = 119,27.
Topic Axes: / : tf' : to = 5,2365 : 4,6304 : 5,4166.
Although the analogy of this isomer with the two other triclino
isomers is plainly visible, the value of a : h is here quite different.
In accordance with this, the derivative melting at 847/ loivers the
melting point of this substance. A mixture of 877o 124 and 137^
120 Xitrodihromobenzene melted at 56*^. There seems, however, to
be no question of an isomorphotropous mixing.
D. Nitro26Dibromobenzene.
Structure: CgHj (NOj)(i) , Br(2) . Br(6);
m.p. 82°.
Recrystallised from alcohol the compound
generally forms elongated, brittle needles
which are often flattened towards two
parallel planes.
Monoclinoprlsinatic.
a:h:c = 0,5678 : 1 : 0,6257.
J = 83°24'.
Forms observed: ^=:010, strongly pre
dominant; (i = \0\l\ and o = lll about
equally strongly developed. The crystals
are mostly flattened towards b with incli
nation towards the aaxis.
( 32 )
Measured : Calculated :
q:q= (Oil) : (Oil) =* 63°43V; —
0:0 =(111): (111) =* 47 52 —
0:q={lll): (Oil) =* 74 207, —
0:q =:: (111) : (01 1) = 45 427, 45°42'
^:^* =(0.11): (010)= 58 87, 58 87,
b:o =(0J0):(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 eqniv. vol.: 127,09.
Topic parameters : x : V' : to = 4,0397 : 7,1147 : 4,4516.
E. Nitro35Dibroniobenzene.
Structure: C^Hj (NOj(i) . Br^s; . Br(5); m.p. : 104°,5. The compound
has already been measured by Bodkwig (Zeitschr. f. Kryst. 1. 590);
my measurements quite agree with his.
Monodinoprismatic.
BoDEWiG finds «:/^:c = 0,5795:l: 0,2839, with ,^=56°12'. Forms:
110, llOOj, 001 and jOJlj.
I take ^ := 85°26' and after exchanging the a, and caxis
a : 6 : c = 0,5678 : 1 : 0,4831,
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: / : if. : to = 4,3018 : 7,5761 : 3,660J.
The great analogy in the relation a:b of this and of the previous
substance is remarkable ; also that of the value of angle /?.
P. Nitro34DibromoBeiizene.
Structure CgHg (NO,) i; . Br 3) . Br^4: ; mi). 58^ C. Has been measured
by Groth and Bodewig (Berl. Ber. 7.1563). MonocUnoprismatic.
rt: 6 = 0,5773:1 with i?=78°31'. Forms 001, T10 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 meltingpointline of the already desciibed type
in the monoclinic derivatives in which some degree of formanalogy
is noticeable. However, in none of the three binary mixtures this
was the case ; the lower melting point was lowered on addition of
the component melting at the higher temperature, without formation
of mixed crystals. For instance :
A mixture of 82,37o J23 and 17, 7 y,l35'Nitrodibromobenzene
melted at 487,° C.
A mixture of 76,57o 126 and 23,57„ lZDNitrodibro7nobenzene
at 687,° C.
A mixture of 90,57o 134 and 9,57o i2GMtrodibromobenzene
at 54° C.
Moreover, no mixed crystals could be obtained from mixed solutions.
The slight formanalogy with the N itrodichlorobenzenes ^) investi
gated by me some time ago is rather remarkable.
Nitro%'^DichloroBenzene (62° C. rhombic) and Nitro1QDichloro
Benzenc (71° C. monoclinic) exhibit practically no formanalogy with
the two Dlbivmocompoumh. There is also nothing in the Dichloro
derivatives corresponding with the isomorphotropous mixture of the
23 and 25Z)/6römöproduct. The sole derivatives of both series
which might lead to the idea of a direct isomorphous substitution
of two CI by two i>ratoms are the Nitro'6^DihalogenBenzenes
(65° C. and 104°, 5 C); the melting point of the Z)/c/i/o?'öderivative
is indeed elevated by an addition of the DibromoéQv'waiivQ.
As a rule, the differences in the crystalforms 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
NitroDibrornoBenzenes than in the corresponding C%/(>?'oderivatives.
Zaandam, April 1906.
Physiology. — ''On the nature of precipitinreaction." By Prof.
H. .1. Hamburger and Prof. Svante Arrhenius (Stockholm).
(Communicated in the meeting of April 27, 1906).
One of tiie 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. GG<S.
3
Proceedings Royal Acad. Amsterdam. Vol. IX.
(34)
toxin into the bloodvessels, the result is, that this is bound and free
antitoxin proceeds. Ehrlich 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 Weigert has stated the
biological law, that when anywhere in the body tissue is destroyed,
the gap usually is tilled up with overcompensation. So, it may be
assumed, that wiien 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 case of this general phenomenon the forming of
precipitin is to be considered.
When a calf is repeatedly injected with horseserum, which can be
regarded as a toxic liquid for the calf, then after some time it
appears that in the bloodsei'um 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 antitoxin 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 oimi 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
(Uhi.enhuth, W^assermann 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 iabbit 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 rehited species of animals by a quantitati\'e way, and
in connection with this a method ^) was proposed to determine accu
1) H. J. Hamburger, Eine Matliode zur Differenzirung von Eiweiss biologisch
verwandier Tliierspecies. Deutsche Med. Wochenschr. 11)05, S. 212.
") H. J. Hamburger, Zur Untersuchung der quantitativen Verhaltnisse bei der
Pracipiüureacliüii. Folia haemalologica. 11 Jahrg. N". 8.
( 35 )
ratelj the quantity of precipitate which is formed by the precipitin
reaction. This method also permitted tr» investigate quite generally
the conditions ^vhich nilo the formation of precipitate from the two
components.
Immediately two fticts 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 (pi'ecipitinogen
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 Arrhenius and ^Iadsen '').
This conclusion has become also the starting point of the now
following researches of iv/iich the purpose mas to investigate by
quantitative iixuj the principal conditions by which precipitin reaction
is ruled.
Methods of investigation.
To a fixed quantity of calfserum') (precipitin =i antitoxin) increas
ing quantities of diluted horseserum (precipithiogen subslance =
1) Eisenberg. Beitrage zur Kenntniss der specifischen Pracipitalionsvorgange
Bulletin de I'Acad. d. Sciences de Cracovie. Glass, d. Sciences Mathem. et nat.
p. 289.
) AscoLi. Zur Kenntnis dcr l^racipitinewirkung. Miinchener Med. Wochenschr.
XLIX Jahrg. S. 398.
3) They used sera of other animals.
*) Arrhenius und Madsen. Physical chemistry to toxins and antitoxins. Fest
skrift ved indvielsen of Statens Serum Institul. Kjobenhavn 1902; Zeitschr. f.
physik. Chemie 44, 1903, S. 7.
In many treatises the authors have continued these investigations ; compare e.g.
still :
Arrhenius. Die Anwendung der physikalischen Chemie auf die Serumtherapie.
Vortrag gehalten im Kaiserl. Gesundheitsamt zu Berlin am !22 Sept. 1903. Arbeiten
aus dem Kaiserl. Gesundheitsamt 20, 1903.
Arrhenius. Die Anwend. der physik. Chemie auf die Scrumtherapeutischen Fragen.
Festschrift f. Boltzmann 1904. Leipzig, J. A. Barth.
5) To make it easy for the reader, we speak here only of calfserum and horse
serum. Compare the third note on this page.
3*
( 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.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 presei'ved 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 NaClsolution of 17o
Each time two parallel proofs were taken as a control. The
capillary portion of the funnel shaped tubes used for this experiment
had a calibrated content of 0.04 cc. Each division of the tubes thus
corresponded to 0.0004 cc.
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 tirst 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 1. c. for further particulars of the method.
2) Fuller details of other proofs taken on other days with calfhorseserum, also
of experiments with serum obtained by injecting rabbits with pig, oxen, sheep
and goatserum will be communicated elsewhere.
( 37 )
TABLE I.
1 cc of
the
mixture of \
cc.
Volume of tiie
precip
itate,
after centri
The quanlity
cal/serum
(precipitin
or
fugating lor :
of precipitate
found in Ice.
serum
containing anti
of the mi,xtures
calculated for
to.\in) j
. . . CO. Iiorse
the total quan
tity of the
serum
V.
(precipitino
mixed compo
nents according
gen or
toxin containing
to the last
observation.
serum.
i h.  i h.  h li
. ih
"
èh.
 20m
.  15ra.
0.04
3 ''•
horseserum
Va»
1 — Vj — not to be measured accurately
04
3 '
))
»
1 — Vi — »
» »
»
»
0.08
3 "
»
»
3 _ 3 — 3
— 3
—
3
— 3
— 3
3.08
0.08
3 "
»
»
3 — 3 — 3
— 3
—
3
— 3
— 3
3.08
0.1
2 "
»
))
i2 — 11 — 10
— 10
—
10
— 10
— 10
10.5
0.1
2 '
»
»
^'2 _ 11 _ 10
— 10
—
10
— 10
— 10
10.5
0.16
2 »
»
»
20 — 23 — 20
— 18
17
— 17
— 17
18.4
0.16
2 '
»
»
20 — 23 — 20
— 18
—
17
— 17
— 17
18.4
0.2
2 '
»
»
32 — 20 — 24
 22
—
21
— 21
— 21
23.1
0.2
2 "
»
»
33 — 26 — 24
— 22
—
21
— 21
— 21
23.1
0.13 »
))
»
48 — 43 — 39
— 34
_
32
— 32
— 32
36.2
0.13 »
f>
»
48 — 43 — 39
— 34
—
32
— 32
— 32
36.2
0.15 »
»
»
52 — 45 — 40
— 36
34
— 34
— 34
39.1
0.15 »
»
»
50 — 45 — 40
— 36
—
34
— 34
— 34
39.1
0.18 »
»
1»
65 — 61 — 54
— 48
42
— 43
— 43
50.7
0.18 »
»
»
65 — 61 — 54
— 48
—
42
— 43
— 43
50.7
0.2 »
»
»
65 — 62 — 55
— 49
45
 45
— 45
54
0.2 »
»
))
05 — 62 — 55
— 49
—
45
— 45
— 45
54
0.25 »
»
»
78 — 73 — 65
— 58
55
— 53
— 53
66.3
0.25 »
»
»
78 — 73 — 65
— 58
—
54
— 53
 53
66.3
0.3 J)
»
»
85 — 80 — 70
 62
58
— 57
— 57
74.1
0.3 ))
»
»
84 — 80 — 70
— 62
58
— 57
— 57
74.1
t ^8 )
So e. g. the quantity of precipitate when 0.3 cc. horse serum Is added to
1 cc. 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 II.
1 CC of the
Volume
of the precipitate, after centrifu
The quantity of
mixture of 0.5 cc
gating for
precipitate found
in Ice of the
horseserum
mixtures calculated
for the total
/50
 . . . cc
quantity of the
mixed ompnuents,
calfserum.
ih.  ih.
 h.  ih.  ^h. 20m. 15m.
according to the
last observation.
U.4 cc
calfserum.
1  i
— not to be measured accurately
0.1 »
»
1  i
— » » » » »
0.3 »
»
2—2
_ 2— 2— 2— 2—2
1.6
0.3 »
»
2—2
— 2— 2— 2— 2 2
1.6
0.5 »
»
6—5
_ 5_ 5_ 5_5_5
5
0.5 »
»
7—5
_5_5_5— 5— 5
5
0.7 »
»
48  30
_ 32 — 28 — 25 — 25 — 25
30
0.7 »
»
50 — 38
— 33 — 29 — 25 — 25  25
30
0.9 »
»
84 — 65
_ 57 _ 50 — 43 — 43  43
51.6
0.9 »
»
81 — 63
_ 55 _ 49 _ 43 _ 43  43
51.6
1.4 »
»
95 — 81
— 67 — 58  52 — 50 — 50
80
1.1 »
»
94 — 81
_ 08 — 56 ^ 52 — 50 — 60
80
1.3 »
»
92 — 79
— 66 — 59 — 59 — 55 — 55
99
1.3 »
»
97 — 80
_ 09 — 60 — 59 — 55 — 55
99
1.5 »
»
96 — 84
_ 74 _ 65 — 62 — 59 — 59
118
1.5 »
»
95 — 84
_ 73 — 64 — 62 — .59  59
118
1.9 »
))
90  75
_ 05 — 55 — 53 — 51 — 51
122.4
1.9 »
»
89 — 75
_ 05 — 55 — 53 — 51 — 51
122.4
f39)
of 51. G. If Instead of 0.9 calfserum 1 cc. was used llie (luanlity of liorseserum
would necessarily liave amounted to 0,5 X r, a — ^>^^ ^^ ^^ '^^ appears that by llie
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 i).
This decrease must be attributed partly to the solubility of the
precipitum in NaClsolulion, a solubility wliich is felt the more strongly
as a greater quantity of diluted horsescruui is added. (Compare also
Fol. Haematol 1. c).
So we see that the clear liquid above the precipitate contains,
besides free precipitin and free precipitinogen substance, as has already
been slated, also dissolved precipitate.
These three substances must form a variable ecpiilibrium, which
according to the rule of (xUi.dbkkg and Waacü; is to be expressed
by the foUowiiig relatioii.
Concentration of the free precipitinogen subst. X Concentr.
of the precipitin = ki X Concentr. of the dissolved precipitate .... (1)
in this /(j is the constant of reaction.
Meanwhile it appears from the experiment, that a greater quantity
of precipitate must be dissoh^ed than corresponds with this equation,
or to express it more clearly, than corresponds with the concej)tion
that the solubility of the precipitate in NaCl solution is the only fact
by which the quantity of sediment decreases.
To take away the difüculty, the hypotiiesis 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._,Oj with COjj. As is
known CaH^O^ is precipitated by CO^, but by addition of more CO.^
the sediment of CaCOj decreases again, while COj with CaCOj 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 aflirmcd.
Let us now try, reckoning both with the solubility of the precipi
tate in NaOlsolution 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 liorse
serum may still appear from the following series of experiments taken on anotlier
day (Table 111). This series has not been used for the following calculation.
(40)
TABLE III.
1 CC of the mixture
of 1 cc calf serum
 . . . cc horse 
serum i/bo
Volume of the precipitate after centrifu
gating for:
ih.
h. — ih. — è h. — 20 m. — 15m. — lOm
The quantity of
precipitate found
in Ï cc. of the
mixtures calculated
for the total
quantity of the
mixed components
according to the
last observation.
0.1 cc
horseser.
/so
38 —
38 _ 28 — 24 — 23 — 23 — 23
25.3
0.1 »
»
»
40 
32 _ 29 — 24 — 23 — 23 — 23
25.3
2 »
D
»
66 —
54 _ 48 — 44 — 42 — 42 — 42
50.4
0.2 »
»
»
59,—
50 _ 45 _ 43 _ 41 _ 41 _ 41
50.4
0.3 »
»
»
88 —
09 — 65 — 56 — 55 — 55 — 55
71.5
0.3 »
»
»
87 —
08 — 65 — 56 — 55 — 55 — 55
71.5
0.4 »
»
»
98 —
76 — 70 — 62 — 58 — 57 — 57
79.8
0.4 »
»
»
89 —
73 _ 08 — 62 — 58 — 57 — 57
79.8
0.6 »
»
»
84 —
02 — 57 — 49 — 44 — 44 — 44
70.4
0.6 )■>
»
»
71 —
57 _ 53 _ 47 _ 43 _ 43 — 43
68.8 .
0.7 »
»
»
65 —
49 _ 45 — 39 — 37 — 37 — 37
62.9
0.7 »
»
»
66 —
49 _ 45 _ 39 — 37 — 37 — 37
62.9
8 »
»
»
61 —
45 _ 40  38 — 33 — 31 — 31
55.8
0.8 »
»
»
62 —
45 _ 40 _ 38 — 33 — 31 — 31
55.8
0.9 »
»
»
41 —
30 — 26 — 25 — 22 — 21 — 21
39.9
0.9 »
»
»
41 
30 — 26 — 25 — 22 — 21 — 21
39.9
1 »
»
»
24 —
17 _ i5 _ 15 _ 13 — 13 — 13
26
1 »
»
»
25 —
10 — 15 — 15 — 13 — 13  13
26
1.2 »
))
»
2 —
2—2—2—2—2—2
4
1.2 »
))
»
2 —
2 2 — 2 2 2 2
4
1.4 »
»
»
not 1
to be measured
1.4 »
»
»
»
) » »
(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.
Firstly the quantity of free prcliniinogen substance. Let A be the
total quantity of that substance used for an experintient. To determine
how much of this is still present in the li([uid in free stale, it is to
be determined how much is bound. Bound is :
1. a certain quantum to form the j)recipitate which is present in
solid condition. If we set down as a rule that 1 mol. pi'ecipitum
proceeds from 1 mol. precipitinogen substance And 1 uiul. precipitin,
then the wanted precipitinogen substance will be expressed by P, if
the molecular quantity precipitate also amounts to P.
2. a quantity /; F 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. precii)itinogen substance
and then that y of this compound is present, then together 2y 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 =z
A—PpV—2y
f ' (2)
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:
l^f . a quantity P for the same reason as is given at the calcula
tion of the free precipitinogen substance (see above).
2"<^. a quantity pV, likewise as explained there.
3'*^. a quantity necessary to form the compound precipitatepreci
pitinogen substance. While in this compound but 1 mol. precipitin
is present, only ly is to be charged. So that the quantity of pre
cipitin which remains in free state, amounts to B — P — pV — y.
While the volume of the liquid amounts to V, the concenti'atioii
of the free precipitin is =
( 42 )
B—P—pV—y
V
(B)
As for the concentration of dissolved iirecipitate in the third place,
this must be expressed bj
(4)
So the equation (1) becomes :
A—P—pV—2.y B—P—pV~y_ pV
V V ' V
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
= ko concentr. compound precipitinogen subst. — precipitate.
APpV2y pj[_ y_
or
(A— P— pV— 2y) p = k, y (6)
By putting shortly PjpF=i^' and by substituting the value of
y of equation (6) into equation (5) we obtain
(aP^2p — ^ (bP'p ^^^ = K ?V"
(7)
In this equation are known :
1st. j\^ the total quantity of precipitinogen substance (diluted
horseserum added) ;
2"'^. B, tlie total quantity of precipitin (calfserum) used ;
3'd. V, the volume of the liquid resulting from the mixing of
the two components ;
4*^. P, the quantity of solid precipitate directly observed.
unknown are :
1st. 2?, the quantity in percentages of precipitate which is dissolved
(so 2^ represents the sohibility of the precipitate) ;
2''^. k^, the constant for reaction of the formation of precipitate;
3'd, k^, the constant for reaction of the formation of the com
pound precipitateprecipitinogen substance ;
(43)
4'^. P' , this is however P\p V and therefore known as soon
as /; 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 ({uanlities correspond in very satisfactory way wilji those
which are observed.
Let us observe that to avoid superfluous zeros J cc calfserum
(/i) is taken = 100.
While as appears from the experimenis in the case in question
1l cc calfserum is equivalent to nearly Vs fx* horseseruui 1 : 50, i cc
horseserum 1 : 50, that is A, obtains a value of 300.
• . . 0,04
bo, where m (he lirst experiment —— cc. horse serum was used
Ö
0,04
A obtains a value of —  X 300 = 4.
o
In the experiment, where on 1 cc. calfserum 0,3 cc. horseserum
was used, with a value B = 100, A becomes 0,3 X 300 = 90.
Let us now combine the two tables to one by calculating for the
second table how much Veo horseserum is used on 1 calfserum..
We .see that the comforuilty between the determined and calcu
lated precipitate [col. HI and I V) is very satisfactory. The average
oj 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 toxinantitoxin 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 toxinantitoxin 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 injectingexperiments in animals.
So there is good reason to expect that a further study of the
precipitinreaction will facilitate too the insight in other toxinanti
toxin reactions.
(44)
I II
ice. cal f serum, B = 100.
TABLE IV.
Ill
IV
Used quantity
of
horseserum
Vso ("« '•'''••
cal f serum).
Used quantity
of
horseserum
expressed
in the just
accepted units
A.
Determined
volumes of the
precipitate
in 1 cc. of the
mixtures.
Calculated
volumes of the
precipitate
in 1 cc. of the
mixtures.
Difference
between
III and IV.
0.013 cc.
0.027 »
4
8
not to be measu
red.
3
2
3.9
f 0.9
0.05 »
15
10
10.3
+ 0.3
0.08 »
24
17
17.8
+ 0.8
0.1 »
30
21
23 6
4 2.G
0.13 »
39
32
29.7
— 2.3
0.15 »
45
34
34
0.18 »
54
43
40.1
— 2.9
0.2 »
GO
45
43 9
— 1.1
0.25 »
75 ^^
'•^ 51
62
52.1
— 0.1
0.2GG »
79
51
53.6
f 2.6
0.294 »
88.3
55
57.1
+ 2.1
0.3 »
90
57
57.5
f 0.5
0.33 »
100
59
58.9
— 0.1
0.385 »
115.4
55
57.4
+ 2.4
0.457 »
137
50
51.3
+ 1.3
557 »
1G7
43
41.3
— 1.7
0.713 »
214
35
26.8
+ 1.8
1 »
300
5
5.5
+ 0.5
1.G7 »
500
2
— 2
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 Guldberg and Waage. 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 Clsolution.
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 Guldberg and Waage. The case is to be
compared with the precipitation of Ca (OH)^ by COj. By excess of
CO, a part of the resulting Ca CO3 is transformed in a soluble
bicarbonate. So Ca H^ 0, takes the function of the precipitin and
CO3 that of the precipitinogen substance.
Astronomy. — ''Observations of the total solar eclipse of August ZO,
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 Tortosa 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 coordinates 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:
<p = 40°49' 13".43 ; X=lm 58s 18 east of Greenwich.
In these results the spheroidal shape of the earth is accurately
taken into account. Later measurements made by Mr. J. Ubagh
gave the same results. Electric timesignals, 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: r/ =r 40°49' 14".8. The height above the sealevel is 55 meters.
The instrument at my disposal for the eclipse observation was a
new e(iuatorial of Mailhat (Paris), 2"^. 40 focal length and 16 cm.
aperture, provided with an eyepiece with a double micrometer.
I have determined the screw value of one of the two screws from
18 transits of circumpolar stars near the meridian. 1 found for it:
i?j z= 60".3534 ±0".0117;
the value of the other screw was determined by measuring the
intervals by means of the first:
i2^ =: 1.00010 . 72,.
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—30 August, Mr. B. Berloty, a clever
observer had observed 20 clockstars, so that the accuracy of the
determination of the clockerror left nothing to be desired.
During the phase observations the objectglass was reduced to
25 mm. by means of a screen of pasteboard. The eyepiece 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 eyepiece, which was more than 20' in diameter. At my
signal "top" the moments of the observations were noted by Mr. Belda,
who was seated in front of a mean time standard clock, which
before, during, and after, the observations was compared with the
sidereal clock; 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 dei'ivation 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 R, — 15'59".63 (Auwers)
,, ,, moon ro = 15'32".83 (KuESTNEKandBATTERMANN)
Parallax of the moon nr„ = 57' 2". 68
( 47 )
OBSERVATIONS.
F 1 r s t c o n t a c t : 1 l^r^39' . 1 (mean time of Greenwich.)
Length of the chords (corrected for refraction)
li 56 28 .2
294.93
57 12 .1
390.24
57 35 .2
437 22
58 ?0 .0
507.74
59 8 .2
566.98
59 38 .9
608.94
12 9 .2
642.58
1 25 .0
721.69
2 49 .9
798.82
' 4 18 .3
876.43
4 57 .0
906.12
5 44 3
935.04
6 15 .9
959.75
6 53 .2
983.94
7 18 .9
1004.93
8 1 .2
1030 37
8 43 .3
1052.50
9 23 .3
1078.17
9 49 .1
1096.89
10 16 .1
1106.16
10 42 .2
1124.37
11 9 .3
1138.90
11 26 .1
1144.49
11 56 .3
1160.37
12 24 .3
1178 82
Second contact:
h m s
1 16 13 .2
Third contact:
1 19 7 .2
Length of the chords
h m s
2 15 53 .0
1297.92
(48 )
Fourth contact
Lfn
gth of the chords
h ra s
2 17 17 .3
1256 94
18 4 .5
1232.27
18 25 .3
1219.81
18 42 .5
1209.51
19 13 .3
1193.25
19 38 .2
1181.49
20 45 .0
1157.42
21 5 .3
1129.77
21 28 .3
1117.78
22 1 .0
1095.75
22 35 .3
1073.82
23 4 .1
1054.40
23 21 .3
1041.52
23 54 .3
1020.90
24 30 .0
993.28
25 2 .2
973.01
25 35 .3
9.50.47
26 2 .3
120.28
26 29 .3
903.24
26 52 .3
880 81
27 13 .3
863.90
27 36 .2
845.41
28 7 .6
819 14
28 43 .3
779.01
29 5 .3
762.98
29 38 .6
726.38
30 2 .3
697.40
30 22 .3
677.17
30 52 .3
637.13
31 14 .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
h m s
2 34 44 .7.
(49)
Right ascension of the sun, Aug. 30, 12^ M.T. Gr. «© = 158°10'44".24
Declination „ „ „ „ „ „ öq = 9° 9'33".19
Right ascension of the moon ,, ,, ,, «( =: 157°42'47".95
(HansenNewcomb) .
Declination „ „ „ „ „ „ rf( = 9°53' 3".48
(HansenNewcomb\
Each observation gives an equation of condition for the determin
ation of the corrections A of the elements of tiie sun and moon.
Let these corrections be successively
AR, Ar, A«o, A«(, Arf©, A(f(, Ajt,
then we obtain by comparing the observed distances and chords
with those computed the following equations : (the coefiicients have
been rounded otf to two decimals).
EQUATIONS OF THE CHORDS.
I. Observations after the first contact. Obs. — Comp.
/' II
+7.98 A^ +7.97 Ar +7.14a« — 3.20 a^ +1 .07 At = +50.71 —10.30
(& (0
+5.50 „ +5.49 „ +4.88 „ —2.19 „ +1.12 „ +30.23 —5.57
+4.88 „ +4.87 „ +4.30 „ —1.93 „ +1.01 „ +37.59 +0.70
+4.10 „ +4.09 „ +3.50 „ —1.00 „ +0.81 „ +32.40 +1.78
+3.58 „ +3.57 „ +3.10 „ —1.39 „ +0.09 „ +22.49 —2.13
+3.34 „ +3.32 „ +2.87 „ —1.29 „ +0.04 „ +24.90 +0.27
+3.15 „ +3.13 „ +2.09 „ —1,20 „ +0.59 „ +22.45 —0.07
+2.78 „ +2.70 „ +2.34 „ —1.05 „ +0.51 „ +19.00 —0.49
+2.50 „ +2.48 „ +2.00 „ —0.92 „ +0.43 „ +10.00 —1.12
+2.29 „ +2.20 ., +1.85 „ —0.82 „ +0.38 „ +20.03 +4.00
+2.21 „ +2.18 „ +1.70 „ 0.79 „ +0.30 „ +19.00 +4.28
+2.13 „ +2.10,, +1.08 „ 0.75 „ +0.34,, +13.42 1.19
+2.08 „ +2.05,, +1.03 „ 0.73 „ +0.33 „ +15.00 +1.43
+2.02 „ +1.99,, +1.57 ,. 0.70 „ +0.31 „ +14.57 +0.47
+1.99 „ +1.95 „ +1.53 „ —0.09 „ +0.30 .., +17.85 +4.40
+1.94 „ +1.90 „ +1.48 „ —0.00 „ +0.29,, +15.73 +2.82
+1.89 „ +1.85 „ +1.43 „ 0.04 „ +0.27 „ +11.49 0.97
+1.85 „ +1.81 „ +1.38 „ 0.02 „ +0.20 „ +12.97 +0.87
+1.82 „ +1.78,, +1.35 „ 0.60 „ +0.25,, +10.44 +4.08
+1.80 „ +1.70 „ +1.32 „ 0.59 „ +0.25 „ +10.08 —1.50
+1.77 „ +1.73 „ +1.30 „ 0 58 „ +0.24 „ +13.51 +1.19
+1.75 „ +1.71 „ +1.27 „ 0.57 „ +0.24 „ +12.97 +1.77
+1.74 „ +1.09 „ +1.20 „ 0.50 „ +0.23 „ +9.37 1.70
+1.71 „ +1.07 „ +1.23 „ 0.55 „ +0.22 „ +10.37 —0.50
+1.09 „ +1.05,, +1.21 ., 0.54 „ +0.22 „ +12.71 +2.00
4
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 50 )
II. Observations before the last contact. Obs.— Comp.
fl.52
AS
+1.46
Ar
0.99 A
(
+0.49 a5
■o (
+0.16^
3
kTT =
= — 5.39
+ 0.51
+1.57
>j
+1.51
>5
—1.05
»
+0.52
»
+0.18
»
— 5.30
+ 0.97
+1.60
„
+1.55
,.
—1.09
»
+0.53
«»
+0.19
)>
— 5.84
+ 0.67
+1.61
n
+1.56
»>
—1.10
»>
\0M
»
+0.19
»1
— 7.39
— 0.76
+1.62
„
+1.57
»
—1.12
V
+0.55
>»
+0.19
»»
— 8.59
 1.88
+1.64
I»
+1.59
»>
—1 14
>»
+0.56
>i
+0.20
1»
— 8.19
— 1.31
+1.66
.,
+1.62
»?
—1.16
}>
+0.57
)>
+0.20
»
— 6.20
+ 0.81
+1.71
>»
+1.67
»
—1.23
)>
+0.59
.■
+0.22
»»
[+ 7.95]
—
+1.74
M
+1.70
„
1.25
»>
+0.61
»
+0.22
>>
— 2.58
+ 4.99
+1.75
»
+1.71
»
—1.27
)»
+0.02
>»
+0.23
»>
— 5.97
+ 1.72
+1.78
>>
+1.74
>•
—1.30
,,
+0.64
»
+0.24
»
— 7.71
+ 0.22
+1.82
»
+1.78
»■
—1.34
It
+0.66
»
+0.25
>»
— 7.88
+ 0.29
+1.85
»>
+1.81
>»
—1.38
»
+0.67
ij
+0.25
>J
— 8.47
— 0.07
+1.87
J»
+1.83
»
—1.40
)>
+0.68
j>
+0.26
»)
— 9.75
— 1.22
+1.91
>>
+1.87
»
—1.44
>>
+0.70
»»
+0.27
>j
— 8.05
+ 0.78
+1.96
«I
+1.93
>ï
—1.50
)i
'+0.73
>)
+0.28
»
— 6.19
+ 3.02
+2.00
1>
+1.97
J'
—1.54
'>
+0.75
»>
+0.29
»»
— 7.13
+ 2.15
+2.05
,.
+2.02
»
1.60
»j
+0.78
»
+0.31
i>
— 4.88
+ 4.94
+2.10
'J
+2 07
,•
—1.64
„
+0.80
M
+0.32
»
—13.04
— 2.92
+2.14
»
+2.12
)>
—1.69
>>
+0.82
IJ
+0.33
»>
— 9.47
+ 0.96
+2.19
»
+2.16
»
—1.74
„
+0.85
»
+0.34
ft
—12.50
— 1.80
+2.24
it
+2.21
„
—1.79
>j
+0.87
)»
+0.35
l>
—11.04
+ 0.04
+2 29
>>
+2.26
>)
—1.84
»
+0.89
»
+0.36
Ji
—10.50
+ 0.87
+2.36
)»
+2.34
„
—1.91
>»
+0.93
1»
+0.38
»'
— 8.40
+ 3.46
12.46
»>
+2.44
»>
—2.01
j«
+0.98
>J
+0.40
>>
—14.81
 2.31
+2.53
f'
+2.51
>)
—2.08
.,
+1.01
»
+0.42
ft
— 9.40
+ 3.53
+2.65
»
+2.63
.,
—2.20
)>
+1.06
»>
+0.44
!>
11.28
+ 2.38
+2.74
><
+2.72
„
—2.29
„
+1.11
»
+0.40
'J
—14.62
— 0.37
+2.83
>■
+281
»
—2.37
>>
+1.15
>>
+0.48
• >
—12.33
+ 2.47
+2.98
>»
+2.96
„
—2.52
»5
+1.22
•J
+0.52
>J
—16.88
— 1.15
+3.12
>1
+3.10
• >
2.65
)>
+1.28
1>
+0.55
»'
—15.37
+ 1.38
+3.29
;»
+3.27
J»
—2.81
)>
+1.36
)5
+0.58
>
—18.89
— 1.29
+3.49
>,
+3.48
n
—3.00
?»
+1.45
,.
+0.63
..
—19.50
— 1.52
+3.90
,)
+3.89
)>
—3.39
)l
+1.64
»
+0.71
)1
—18.51
+ 2.74
+4.20
»
+4.19
)»
—3.67
1)
+1.77
5»
+0.78
5>
—26.71
— 3.80
+4.37
>i
+4.34
,,
—3.82
?>
+1.85
>»
+0.81
)»
—38.87
10.28
(31 )
Equations of the contacts
I LR^ Lr^ 0.903 Zi«(_o — 0.405 Lö^q = + 3".78
ri LR — Lr — 0.9668 Zi«( ^ — 0.2007 A(f(_o +
+ 0.0004 Z'«(_c — O.OO3I) LaL(f f 0.0091 AM(_c = — 6".52 ^)
/// LR — Lr^ 0.3085 A«(_o  0.9489 Arf(_o +
+ 0.0104 ^''«(G + 0.0068 LaLó + 0.0012 Z.M(_o = + 4".02
/F LR^ Lr — 0.889 A«(q + 0.435 LÖ(q = — 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 tlie proportionality of the coefficients
we may use one single equation instead of the first 25 equations
after the 1^* 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(Ai2+Ar)f 56.2Zia25.2Ad= + 489".46  0.2>o{LRLr)  12. 9A^
— 81.6(A72+Ar)f65.1A«31.6Ad=+397".87 + 0.24(A7?A/)fl2.8A.T
whence :
LR + A? = + 1".05 — 0.015 Ad — 0.003 [LR — Lr) — 0.16 A.t.
A«= 4 7".428 + 0.465 Ad — 0.001 (LR — Lr)  0.02 Arr.
Neglecting the last terms, we find for the result from the equations
derived from the length of the chords:
Ai? + Ar = 4 1".05 — 0.015 Ad(_o
A«(_Q znz { 7".428 + 0.465 Ad(_o.
From the equations of the 2^^ and 3"^ contact we derive:
A«(_0 = + 7".793 + 0.464 Ad(_0.
A«(_o = + 7".13 + 0.667 (LR — Lr)
Ad(_o = — r'.43 + 1.437 [LR — Li).
And lastly the equations of the 1^^ and 4^^ contact yield :
A«(_o = f 8".35 + 0".468 Ad(o
[LR f Ar = — 3".781
The latter result for LR \ Ar, w^hich differs entirely from that
found above is little reliable. We can entirely account for it bj'
assuming that the first contact has been observed too late and the
last contact too earlv. It can hardlv be doubted that the 1*' contact
1) It is not allowed (as it is generally done) to neglect the quadratic terms
in the equations of the 2°'^ and 3"^^ contact, because the corrections A^and ^}, as
compared with the distance between the centre of the sun and that of the moon,
(in this case 40') are too large.
4*
( 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 = 10 cm.) 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. Th. Wulf
and J. D. Lucas kindly put at my disposal the results of their highly
interesting observations of the 2"^^ and the S'"^' contact, made at Tortosa
by means of sensitive selenium elements. (See for this Astron. Nachr.
N». 4071). They found:
beginning of totality l''16"il58,6
end „ „ 1 19 6,9,
which yield the following equations :
AR— Ar — 0.9G50 A«(_o — 0.2117 Acf(_G + 0.0004 AV<(o 
— 0.0039 A« A(f f 0.0092 AM(_e = — 5".73
AR — Ar I 0.3063 A«(_o — 0.9493 Arf(_o + 0.0105 AV(_o f
+ 0.0069 A« Ad + 0.0012 AM(_0 = + 4". 10
whence
A«(_0 =. + 6".42 + 0.653 {AR — Ar)
Aff(Q = — 1".76 + 1.404 {AR — Ar).
When we subtract the two equations A from each other we get :
A«(_o = + 7".238 + 0.465 A(f(_0 ,
which agrees exceedingly well with the result of the chord equations
A« = + 7". 428 f 0.465 Arf; but it also appears that it is impossible
to determine A«, Aö and AR — Ar separately from the combination
of the contact and chord equations.
In the derivation of the tinal result we have accorded the same
weight = 1 to the results of the chord measurements and to those
of the contact determinations made by Wulf — Lucas, and the weight
h to my observations of the 2"*^ and 3''^ contact. Thus we tind, leaving
out of account the tirst and the fourth contact :
Ai2 + Ar = + 1".07 — 0.02 {AR—Ar)
A«(_0 == + 6".66 + 0.66 {AR—Ar)
A(f(_o =  1".65 + 1.42 {ARAr).
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 iirst 25 obser
vations (excluding the tirst) amounts to ± 2. "53 ; that of the last 35
(excluding the last) is ± 2. "21,
( 53 )
Chemistry. —  ''On the osmotic pressure of solutions o f nonelectro
lytes, in connection ivith the deviations from the laws of ideal
gases." By J. J. van Laar. (Coniiininicated by Prof. H. W.
Bakhuis Roozeboom.)
Communicated in the meeting of Ai)ril 27, 1906).
1. Fiy 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 I normal, and as c is then about V34 [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 complex molecules) are present], the difference between the
exact expression — log (1 — x) and the approximate value x [formula
(2)] is not yet appreciable. It is however not so with the difference
between the molecular volume of the solution v := (1 — x) t\ \ xt\
{v supposed to be a linear function oïx, about which more presently)
and the molecular volume of the solvent i\, when i\ (the molecular
volume of the dissolved sugar) cannot be put equal to i^j. We shall see
that this difference for 1normal solutiojis amounts to 19%, so that by
means of the experiments we can very well ascertain, if we have to
make use of v or oi' i\. And these have really taught us, that the osmotic
pressures measured agree (and even with vevy great accuracy) Avitli
the calculated values, o?ilg when r^^ 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 tiie molecular
volume therefore of the whole solution, but the molecular volume
of the solvent in the solution. And this deprives those of their last
support, who in spite of all evidence (for not the dissolved substance,
but the solvent 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 whole volume v
and not in the volume Vj, which is perfectly fictitious with regard
1) Amer. Ghem.  Jouin. 34, 1905, p. 1—99. See also t.be extensive abstract
N' 274 in the Phys. Chem. Gentralblatt III (1906).
2) See inter alia my previous paper on this subject in These Proceedings, May
27, 1905, p. 49. (Some remarks on Dr. Ph. Kohnstamm's last papers).
( 54 )
to the solution, which would be equal to v only when i\ happened
to be equal to v^.
2. In order to compare the results, found by Morse and Frazer,
more closely with those for the osmotic pressure alread}'^ 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) [tlie 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 ^j :
f*l ('^' P) = f^i (Ö' Po) •
Now in general :
dZ
li,= — =C,(9,^RTlogc,,
when C, = k,T {log T  I) ^ {{e,\ ~ T{s,),), c, = ^ and
do
6^ = ^ — ; being given by
:= I fdv
pv — RT 2 Wj . log 2 n^ .
For binary mixtures of normal substances we may now introduce
the variable x and we obtain (^?z, is now := 1, so that the term
with log 2n^ vanishes), as may be supposed as known :
li, = C,Li.x^\p(vw~^^RTlog{lx), . (1)
when to is written for  pdv by way of abbreviation.
This expression is perfectly accurate for the above mentioned
mixtures. For the further calculation we now introduce the idea
"ideal" mixtures. They are such as for which the influence of the
two components inter se may be neglected. Then — — =: 0, and to
becomes a linear function of x. But also r — = 0, so that v becomes
Ï) 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
0. V ,, ,, ,, ,, J, J,
a
C.
d. the heat of mixing is = 0,
o
so that we may say : ideal mixtures are such for wliich the heat
of mixing is practically = 0, or ivith which no appreciable contraction
of volume takes place, when 1 — x Gr.mol. of one component is
mixed with x Gr.mol. of the second.
The conditions a, b, c and d are sinmltaneously fulfilled, when
the critical pressures of the two components are by approximation
of the same value.
3. For to — X rr we may now write oy^, as a =z{l — x) io^ 
ox
ö'o> r _ , dto
j tC (Oj, when  — = 0. Otherwise evidently to — x—z=zv)y^ —
 ^''A^J,  ''' "' \M)r ' ' J ^'' '^" '"""' '"""^ " " " èx='^'
and we get :
f*i ('^i/^) = C'l  «>i + p^'i + ^^' % (1  '^O j
f^i (ö»i>o)= <^i — ">i +Po^i '
always when v^ and lo^ are supposed to be independent of the
pressure. For else tOj and z'l would have another value at the
pressure p than at the pressure p^. We must therefore also suppose
that our liquids are incompressible. 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). Onlj' when x draws near to 1, and so the
osmotic pressure would approach to go, v^ (and so also coj must no
longer be supposed to be independent of j>.
By equating these two last equations, we get :
pv, + RT % (I — x) = p,t\,
hence
( 56 )
:;rr=pJ.„=^(%(l.^•)) , .... (2)
the expression already derived by me in 1894. ^)
1) Gf. Z. f. Ph. Gh. 15, 1894; Arch. Teyler 1898; Lehrbiich der matli. Ghemie,
1901; Arch. Teyler 1903; Ghem. Weekbl. 1905, N". 9; These Proceedings, June
21, 1905.
hi the original Dutch paper another note followed, which Mr. van Laar has
replaced by the following in the English translation.
A conversation with Dr. Kohnstamm suggested the following observations lo me.
db
Dr. Kohnstamm finds (These Proceedings, May 27, 1905) the quantity v — x ——
aw
in the denominator of the expression for ?r. 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 nonlinear variability of v.
Then we should viz. have:
IriilatinfT t.\ o'
'P
where, when calculating (o — x ^— by means of van der Waals' equation of state,
ó{v — b) . .,.,/' db\
also a term — px — ^^ appears, m consequence of which p [v — x^] occurs
ox \ ax J
H^^) :„ „„„ „„„ „.„..;„. „, A. . db^,
in the first member.
... öü db
Now it IS of no importance whether v is diminished by x ^^~ or by x j , as
o.v dx
d(v—b)
2)  — 5 approaches to both for small and for very large values of ^9. I therefore
obtained a correction term in the denominator, in connection with the size of the
molecules of exactly the same value as Dr. Kohn.stamm. That this did not always clearly
appear in my previous papers, is due to the fact that I then always introduced the
dv
approximation v — ^ v — ^'i, which was perfectly justifiable for my purpose.
dv /'d^v\
For as v — x^ = Vi — V2 ^^[ ^~^ ) ~ etc., this is sufficiently accurate for prac
tical purposes, (for ideal mixtures, where v is a Hnear function of x, it is of
course quite accurate).
Yet in a so early paper as the one cited by K. of 1894 (Z. f. Ph. Gh. 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
dv
used for the liquid phase) = ^ — . But this is in the a;notation nothing but
OUa'
dv .
V — x^— , tke physical meaning of which is : the molecular volume of the water
Ox
in the solution with the concentration x.
The phrase occurring on page 466 : "und niemals etwa Vn' — b im Sinne etc."
refers there to the wellknown attempts of Ewan and others. The same is the
case with the phrase in the paper on nondiluted solutions in the Gh. Weekblad
of June 7th 1905: "Ook heeft men getracht, etc." (j). 5).
( 57 )
We repeat once more: tliis expression holds from .v:=0 to x =
near 1, wlien the following conditions are satisfied :
a. the solution is an ideal hinanj mixture of ?/c>rma/ components;
b. the solution is practically incoDipressible.
Then (2) represents the additional pressure on the solution, in order
to repel the penetrating water (the socalled ''osmotic" pressure).
As however in all the experiments made up to now loater 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 x^ (just as the influence of the two components inter
se), so that in the above experiments, wheje .i'" may undoubtedly
be neglected (cf. § 1), formula (2) may certainly be used.
Let us, liowever, first reduce it to a form more practical for use.
4. Let us write (2) for this purpose :
RT ^ , RT
JT^^ (., + 7^ ,,^ ^ _ .) ^ __ .,. (1 _^ i _,)^ ^ _ _ (2«)
which is more than sufticient for solutions up to 1normal. Let us further
assume that c Gr. mol. are dissolved in 1000 Gr. H^O (called by
Morse and Frazer "weightnormal solutions"), then :
c c
when we put V34 <^ = c' (34 = 55,6 : 1,65 is the number of Gr. mol.
H,0 in 1000 Gr. at 18° C; cf. § J).
We find then :
RT c' ,
1 + 0' V '14 c'/
or when we restrict ourselves to terms of the second degree with
respect to c' :
RT RT c
., = _,(i_V,,) = _(>v„.).
In this 7^=3:82,13 (c.c.M., Atm.), and i', = 1001,4 : 34 cM' at
RT
18°. For we therefore find at 18° C. :
RT _ 82,13 X 291,04 _
34;r  l^^4:  ^^'^^'
hence
.Ti8° = 23.87 c (1 — 0,015 0) Atm . ... . (26)
We see from the calculation, as w^e 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 Vm <^ instead of V'es <^ = 0,015 c. (4 c' would then be = VaX Vss.o ^)
Let US now consider what the last expression would have become
for JtjgO, 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,0, then the
total volume will be (at 18°) 1001,4  190 c 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 f c Gr. mol., hence the molecular
volume of the solution will be:
_ 1001,4 + 190 c _ 1001,4 1 4 0,19 c
^' "~ 34 4 c ~ "~3 4~ 1 + 0,03 c *
For i\ we found however above :
_ 1001,4
""' ~ 34 '
so that the value of n^g" with v in the denominator instead of v,
w^ould have become :
1 + 0,03 c .
jr,,o = 23,87 c (10,015 c) X , T '^^ ^^°^"
i. e.
1 + 0,015 c .
jro = 23,87 c ^ '^ Atm.
'' 1 + 0,19 c
For (weight)normal solutions (c = 1) we should therefore have
found instead of ji,,° = 23,87 (1 — 0,015) _ 23,51 Atm., :r,,°
1,015
= 23,87 X ^j^ = 20,36 Atm.
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 v^.
5. But there is more. We shall viz. derive the expression for
the pressure which u)ould be exerted by the dissolved molecules,
1) With 0,5normal the two values would have been 11,85 and 10,98 Aim.,
whereas 12,08 Aim. has been found experimentally.
( •^>^> )
when they, according to the inaccurate interpretation of the osmotic
pressnre, could move free and undisturbed throughout the space of
the solution.
VAN DER Waals' equation of state, viz.
ET a
gives for the rarefied gasstate:
_RT
V a : RT
V — b V
RTf b a:RT
V \ V V
when we again content ourselves with terms of the degree 
Let us nov^ write:
then
a
Rf~^~'^'
V \ V J
where v now represents the volume, in which 1 Gr. mol. of the
dissolved substance moves. This volume is however evidently (cf.
also ^4):
1001,4 + 190 c
or
1001,4
^(1 + 0,19 c),
so that we get :
1001,4 (1 + 0,1 9 c) V 1001,4 (1 + 0,19 c)) '
RT y
or as 77^7:7 = 23,87 is (cf. § 4), and with y' = ^
1001,4 ' V • • ^ /'  f ~ 1001,4'
, = 23,87 c^^^^^..., ...... (3)
and this is an altogether ditferent expression from (2'^). Not oidy is
v^ replaced by v (which gives rise to the factor 1 j 0,19 c), but
we also find 1 — y' c instead of 1 — 0,01 5 c. In this y' is different
for every dissolved substance, dependent on the values of a and b,
whereas the coefTicient 0,015 has the same value for all substances
dissolved in water, independent of the nature of the dissolved substance
(cf. ^ 4). Also the coefficient 0,19 depends on the dissolved substance
on its molecular volume). Moreover y' depends also on I'on account
( 60)
oï a: RT. Except with H^, where y is negative at the ordinary tempe
rature, y is everywhere positive. But at liigher temperatures its vahie
is reversed, and becomes negative.
So, when comparing (2'') and (3), we see clearly, that it is ont
of the question that the socalled osmotic pressure should follow the
gas laws. Only with c = this would be the case, but for all other
values of c 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
RT RT
7t:^~{ log (1.0) = — X (1 + V, X' + . . .)
holds; for the gas pressure on the other hand:
RT
P =
V \ V J'
so that the deviations from the gas laws (at the ordinarj^ tempe
ratures) are even in Ojyposite se'use from the deviations of the osmotic
pressure for nondiluted solutions.
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 <jas pressure. The molecules of the dissolved
substance have nothing to do with the osmotic pressure except in
so far as they reduce the uiater in the solutions to another state of
concentration (less concentrated), wiiich causes the pure water (concen
tration 1) to move towards the water in the solution (concentration
1 — x) in consequence of the impulse of diffusion. On account of
Rl'
this a current, of which the equivalent o/ pressure =^ — {log{\x)),
arises in the transition layer near the semipermeable membrane,
Avhich current can only be checked by a counter pressure on the
solution of equal value : the socalled osmotic pressure.
This is in my opinion the onli/ 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 6\ (cf § 2) are
no longer the same in the two members of
/*i ('^', ^) = Ih (o, T,),
( 61 )
whereas the terms pv are now the same. In tliis case 7^ would have
to be <[ 7\, because the temperature exerts an opposite influence
on the change of ft from the pressure.
In consequence of the lei'm RT loij (1 — ai), ftj {x) will be <^ [i^ {o). ^^
must therefore be increased. Now— ^ =: v^ , hence positive , whereas
op
—  I — I = — (é?i \ pi\), so negative. So the value of /.tj (.f), which is
too small in consequence of x, can again be made equal to that
of fij (o), either by Increase of pressure ("osmotic" pressure), or hy
lowering of the tem'perature ("osmotic" temperature).
It would, however, be advisable to banish the idea "osmotic
pressure" altogethei' from theoretical chemistry, and oidy speak of
it, when such dilFerences of pressure arc actually met with in case
of semipermeable walls (cellwalls, and such like).
6. Appendix. Proof of some properties, mentioned in § 2.
a. In a previous paper in these Proceedings (April 1905) I
èv
derived for — the perfectly accurate expression [equation (4), p. 651) :
dh
1 {v—hyda
dv
dx
RT v' dx
ex
1
RT v^
dh da
With ^ = /? and — = 2« ya, in which {i = b.^ — l>i and « = \/a.. — i/a, ,
dx dx
this becomes :
2a)/a {v—by
dv _ ~ RT^ ^"^~
RT v'
, .. . ^^
And now we see at once, that this passes mto /? or — , when
dx
/? [/a i=z ctv ^
a\/a
For then in the numerator becomes equal to "/f hi the de
dv dh d\' d^'b
nominator. But when ^— =:— , then also t — =0, as — ^=0, andv
Ox dx ox^ dx*
is a linear function of x.
(62 )
[We above derived tlie condition ^\/a=zav from the general
dv dv db
expression for —  . If we knew^ this condition beforehand, ^ ^ — 
o,v Ox ax
would immediately follow from this by differentiation, and then it
dv
would not be necessary to start from the general expression for ^
Ox
b. On p. 651 [equation (5)] of the paper cited the perfectly
general expression :
ö''0_ 2 {av—^[/ay
bx^~^ 2ö/„ {v—by
was derived for r— , which becomes therefore = 0, when again
ox
Jo"© d^v
pdv — pv=za) — pv. And as r^ and t— are
Ox' o.r"
both = when av = ^[/a, also r — will be = 0, in other words lo
ox^
is a linear function of x.
c. The heat of dilutmi. It is given by the formula
L — — 7'^ — [^h!^ _ ^h^
"" ~ or L ^ T
This is viz. the socalled differential heat of dilution per Gr. mol.
/ m \
of the solvent when an Gr. mol. solvent ( x = — ■ — I are added to
\^ m \ nj
a solution consisting of m Gr. mol. dissolved substance and n Gr.
mol. solvent.
This becomes [see equation (1)] :
L. = T
d ri I / do)\ ( dv\
If ::— = 0, then vi — x c — = <o,; and v — x :— will be = i\, when
ox^ Ox a.^■
Lf " '
t—  z=: 0. But then L^ =0. q. e. d.
And hence also the total heat of mixing will be = 0, when x Gr.
mol. of the 2"^ component are mixed with 1 — x Gr. mol. of the
1^^ component.
d. The peculiarities mentioned in \ 2 under a, h and d, which
(63)
characterize the socalled ideal mixtures, are therefore all satisfied
when
^ [/a =1 a V.
This yields :
when it is permissible — for liquids far from the critical temperature
— to replace v by b. Hence we get :
or
{b, — b,) [/a, = {y/a, — [/a,] 6,,
or also
6, [/a^ =: b^ l/a,,
hence
b, ~ b,'
from which w^ see, that the case of ideal mixtures occurs, when
the critical pressures of the components have the same value.
e. Finally
b' fa\2{b, \/a,  b, \/a,y
be
)a' \b )
a
SO we see that also — will be a linear function of x, when
b
6, j/ttj r= 6j /a, or Pi = p^. In this way also c of § 2 has been proved.
(June 21, 1906).
KONINKLIJKE AKADEMIE VAN WEÏENSCIIAITEN
TE AMSTERDAM.
PROCEEDINGS OF THE MEETING
of Saturday June 30, 1906.
(Translated from : Verslag van de gewone vergadering der Wis en Natuurkundige
Afdeeling van Zaterdag 30 Juni 1906, Dl. XV).
co2srT'E3sra?s.
L. E. J. Brouwer: "Polydimeusional Vectordistributions". (Comnuinioatcd by Prof. D. J.
Korteweg), p. 66.
F. M. Jaeger: "On the fatty esters of Cholesterol and l'hyfcosterol, and on the auisotropous
li(]uid phases of the Cholesterolderivatives" (Communicated by Prof. A. P. N. Franchimont),
p. 78.
F.M.Jaeger: "Researches on the thermic and electric conductivity power of crystallised
conductors" I. (Communicated by Pruf. II. A. Lokentz), p. 8'J.
II. W. Bakhuis Roozeboom: "Threephaselines in chloralalcoholate and anilinchydrochloridc'",
p. 99.
II. IIaoa: On the polarisation of Röntgen rays", p. 104.
P. VAN RoMBURGii : "Triformiu (^Glyceryl triformatc)", p. 1Ü9.
P. VAX lloMBURGii and W. van Doussk.n: "On some derivatives of lSóhexatriene", p. 111.
L. E. J. Brouwer: "The force field of the nonEuclidean spaces with negative curvature".
(Communicated by Prof. D. J. Kokteweg), p. 116.
A. Pannekoek.: "The luminosity of stars of ditlcrent types of spectrum". (Communicated by
Prof. H. G. VAN DE Sande Bakiiuyzen), p. 134.
Errata, p. 148.
Proceedings Royal Acad. Anislerdani. Vol. IX.
( 66 )
Mathematics. — '* Polydimensional Vectordistributions" . ^) By L. E. J.
Brouwer. (Communicated by Prof. D. J. Korteweg.)
Let us call the plane space in which to operate >S„ ; we suppose
in it a rectangular system of coordinates in which a 6^ represents
a coordinatespace of p dimensions. Let a /^^distribution be given
in Sn', ie. let in each point of Sn a /9dimensional system of vectors
be given. By Xa, «.^....a we understand the vector component parallel
to C^t 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 1. The integral of f'X in S» over an arbitrary curved
bilateral closed S^ is equal to the integral of ^'+^ F over an arbitrary
curved *S^+i, enclosed by S^ as a boundary, in which P + ^l^is
determined by
n.
XX.
^
ÖX.
^2 78
V+1
d.v.
where for each of tlie terms of the second member the indicatrix
{ctfj^aq^. . . aq Oq ) has the same sense as {a^ «, . . . «^i). We call
the vector Y the first derivative of )'X.
Proof. We suppose the limited space Sp\.\ to be provided with
curvilinear coordinates ii.^ . . . ?^y,}i determined as intersection of curved
C)/s, i. e. curved coordinatespaces of ;>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 oï f^^Y becomes when expressed in differen
tial quotients of rX :
ÖX.
Y.
^,+x
dx.
Yai ... a
"/>ll
dxo
^+^
du^^
d.v.
du.
du
Z'+l
6?Mj
du
/;+!•
1) The Dutch original contains a few errors (see Erratum at the end of Ver
slagen 31 Juni 1906), wliich have been rectified in this translation.
(67)
We now unite all terms containing one of the components of
pX, e.g. X\23...p' We then find:
dXi23..
+
dX
UZ...p
d^^_l_2
d.r/,_.,
diCj
ÖMj
dwj
d.?^/,+i
dcTj
^Up\\
du^fi
d.r^,_.2
Ö^j
dw,
öm,
d?v+i
do;,
c?Mj . . . rf?fy^fi f
Öm,
di»
■/>+2
dic.
du^ . . . (fw^+i 4
f . . . (n — 7? terms).
If we add to these the following terms with the value :
dx^
d^i
ÖMj
d..,
dXi23...;)
•
d^Fj
•
•
d.,
d.rj
dwp+i
ÖM/.+1
d^,
dj7,
dwj
d„,
d.z
'M/)+i
+
dX
123...P
dic.
d^r,
d^
d?/,
d.r/,
 • • • (jo terms),
du.
• <^ï</;4l 4
fZ;/,
(\+\ +
the ?2terms can be summed up as:
dXi23...p , d.Pj
r au^  — aUj
dXi23...p
öm
C?M
Z'+l
V^+1
du
.+1
p+\
~ du,
<^"/.ll
5*
( 68 )
Tjet US suppose this determinant to be developed according to tlie
first column, let us then integrate partially each of the terms of the
development according to the differential quotient of JTios... .y, , appearing
in it; there will remain under the (/>[ l)fbld integration sign
P ip ~\~ 1) terms neutralizing each other two hv two. Thus for instance:
du.
diijj^i
X
and dn.
<^"y;+l
X
d\v.
123.../)
\23...p
dii^dujj^i
d\v
dxn
du.
dup
bdp
blip
d,.,
bxp
bu^
b.vp
blip
b\t
bu^bup^i
as they transform themselves into one another by interchangement
of two rows of the matrixdeteiminant.
So the y;fold integral iemains only, giving under the integration sign
X
I 23 ... ;)
bxj^
1 ^dii,
Oil,
± 1
b.v
bup\.[
'dupj^i .
b.vp
— dll^
bxp
biip^
 ^'//)li
to be integiated over the boiindaiy, whilst in a definite point of that
boundary the A"^^ term of the first column gets the sign f when
for the coordinate ?ip the point lies on the positive side of the boundary.
Let us now liud the integral of X123...;) over the boundary and
let us for the moment suppose ourselves on the part of it lying for all
(69)
u's oil the positive side. The iiidicatrix is in the sense u^ ?/,... W/.fi and
if we integrate A'123.../; successively over tiic components of tlic
elements of I)oundaiT according to the curved 6//s we find:
S/'
 — dua
Ouu.
JU:,
ÖX,
Öj^aj
""^1
bxi,
1'
dUu:
1^
where («/,( i «, • • • «/.' = (1 2 3 . . .yj !'^;lj j ; so that we can write
as well
rf
da
du
du.
— dn^
Oh,
dx
d>i,/—
— du,j—i
dx
b'((j\
duy—i
Xn
]2^
^=l,2...(/ + l;
0.7;,
du
du
7+1
dx^
7+1
d»^,+l
or
ƒ■>
123... /y
1 ^r— du
du.
dx^
dxp
dui.^^
d.r:
J' 7
f/''y,+l
du.
'' du.
da
du
— du^jj^i
dx,
■p+\
du
du
7^+1
y.+l
If we now move to other parts of the boundary we shall conti
nually see, where we pass a limit of projection with respect to one
of the coordinates u, the })rqjection of the indicatrix on the relative
curved Cp change in sense.
So in an arl)itrary point of the boundary the integral is found in the
same way as pn the entirely positive side; we shall tlnd oidy, that
for each coordinate u^j for which we are on the Jiegative side, the
corresponding term under the sign ^ will have to be taken nega
tively, by which we shall have shown the equality of the //fold
( ^0 )
integral of pX over the boundary and tlie (/> f 1) fold integral of
/'"^' F over the bounded Spj^\.
We can also imagine the scalar values of I'X set off along the
normalAS,j_;/s. As such the integral over an ai'bitrary curved bilateral
closed Sn—p can be reduced to an {n — ^j f l)diraensional vector
over a curved Sn—p+\, bounded by Sn—p If again we set off the
scalar values of that vector along its normalAS^i, the vector p—'^Z
appears, which we shall call the second derivative of pX. For the
component vectors of i'~'^Z we find:
^ The particularity may appear that one of the derivatives becomes 0.
If the first derivative of an "'A'' is zero we shall speak ofan,„_iA'',
if the second is zero of an ,„+iA'.
Theorem 2. The first derivative of a /'A is a ^ pX, the second a
'' pX ; 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 rX over a closed aS'^,^^ i,
then we can substitute for the addition given by an Sfj4.\ element
the integral of pX along the bounding Sp of that element. Along
the entire >S/, + i each element of those Sj, boundaries is counted twice
with op[»osite indicatrix, so that the integral must vanish.
The analogous jtroperty for the second derivative is apparent, when
we evaluate the integral of the normal vector over a closed /S„_^,^i.
By total derivative we shall understand the sum of the first and
second derivatives and we shall represent the operation of total
derivation by v
h=n
Theorem 3. v' = ~^^ ^r~~
A=l
Proof. In the first )lace it is clear from theorem 2 that the
vector v" is again a pX. Let us find its component A'i2....p
The first derivative supplies the following terms
(71 )
2\ =
^
öF,
q \....p
q=p^l
where
u—p
dWg
u—\
VF ^ a.,„
U—\
+ sign for 0.^12.. (ul) {u + l) ... p)={q I ... p))
dXl2...;,
So
T — V* V ^'^912...(m1XmH). ..p
f — sign for (j/^ 12 ...((/ — !)(?< 41) ..;>) = (^ 1 ...jo) J
q—u
■^^ 0''Xl2...;,
The second derivative supplies the terms
^'="2* d^.
«=1
f+ sign for (u 12 ...(wl)(u f 1) ...;>)=: (12 .. p)
r for (qn I. ..{u\){uil)...p)^(q 12. ..p)\
or
, r, 'V^ ÖA^12.. («!)(«+ 0...;,
where ^i2...(h— i)(m+i).../) = T. a ^
9=/'+ 1
dXl2...;,
9=/'^ 1
^^' ( sign for (n 12 ... (?^l) (« + 1) ... ;)) = (12 ...p)] .
u=p q=H
g^ j7 _ V^ '^ Ö'X^12..(ul)(«+l)..;t,
u = 1 9 = ^)  1
d^'ud^'^
( — sign for {qui . . {u — l){u{'l)..p) r=: {ql2 ..p)\
u = p
'^ 0'Xi2..
tt=l
( 72 )
The terms under the sign ^ JS" of Tj are annulled hv those of
1\, so (hat only
h = n
i:
h — \
is left.
Corollary. If a veetordistribntion v V is given, then the vector
r Vdv
distribution I — „, integrated over the entire space, has tor
J /;„(» — 2)r"
second derivative T^. (if /,„/•''"' expresses the surface of the
"'sphere in ;S;,).
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 Sn ^) 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 yM", from which it is derived by the operation
dv
J
X„(/.2)r"2
Theorem 5. A vectordistribution is determined unifoi'mly by its
total derivative of the first order.
For, from the first total derivative follows the second, from which
according to the preceding theorem the vector itself.
We shall say that a vectordistribution has the pelil property, if
the scalar \alues 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 S,r 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
1) Generally the condition is put : the function must become infinitesimal of
order ?i— 2 willi respect to the reciprocal value of the distance from the origin.
We can, however, prove, lliat the being infinitesimal only is sulUcienl.
( 73 )
(]eri\ative. in otlier words eacli vectordistribiitioji has a potential
and thai i)<iifMili;il i uiiifnrinlv determined liv it.
Proof. Let T" be the fuiven distribution, then
is its potential. For \7^P=x7F, or y(^P) = vF, or \jP=V.
Farther follows out of the field property of V, that P is uniformly
determined as \7~ of v I'^, so as V of V. So P has clearly the
potential property: it holm I, however, not have the field property.
X.B. A distrihiition not to be regarded here, because it has not
the field profierty, thougii it has the potential property, is e. g. the
tictit ions force field of a single agens point in aS,. 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 "'' T the first derivative of /' F and TyFthe second;
we can then break up /' V into
and
From the preceding follows immediately :
p II— \ p
Theorem 7. Each /,_i T has as potential a j, V. Each p__i V has
as potential a p I .
V
We can indicate of the ^;_^i I' the elementary di^>trilultion, i. e. that
p
particular ^j+i V of which the arbitrary Sn integral must be taken to
i'
obtain the most general /;ri F.
For, the general p^\V is "^ of the general ''"*" F, so it is the
general >S„ integral of the ^ of an isolated {p f l)dimensionai
vector, which, as is easily seen geometrically, consists of equal /'vectors
in the surface of a /'s[)here with infinitesimal radius described round
the point of the given isolated vector in the /i'^,_i of the vector.
V
111 like manner the general ^.i F is the ^y of the general P^ I ,
( 74 )
so it is the general aS„ integral of the v^' of an isolated P> 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 R,ip+u normal to that vector.
From this follows:
Theorem 8. The general p V is an arbitrary integral of elemen
tary fields E, and E^, where :
pi
ƒ» Zdv P^
, where » Z consists of the y"* vectors in the
surface of an infinitesinal P ' sphere aS/;., (1)
p+i
JYdv P'^^
, where ^ Y consists of the p+^ vectors normal
to the surface of an infmiiesinal "P^sphere Sp^^. ... (2)
For the rest the fields E^ and E^ must be of a perfectly identical
structure at finite distance from their origin; for two fields ^i and ^,
with the same origin must be able to be summed up to an isolated
^vector in that point.
We can call the spheres >§/>>// and Spz vvitli their indicatrices the
elementary vortex si/stems Voy and Fo. 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 m S^.
The field E^. The elementary sphere Sp. becomes here two points
lying quite close to each other, the vortex system Vo^, passes into
two equal and opposite scalar values placed in tiiose two points. It
cos If)
furnishes a scalar potential —  in which <ƒ denotes the angle of the
radiusvector with the *S\ of Yo~, i. c. 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 scalar value) here again becomes
COS iD
; so the field itself is obtained by allowing all the vectors of
( 75 )
a field E^ to rotate 90''. As on the other hand it has to be of an
identical structure  to A\ outside the origin we may call the Held
E^ resp. E, "dual to itself".
In our space the field E^ can be realized as that of a plane,
infinitely long and narrow magnetic band with poles along the edges ;
the field E^ as that of two infinitely long parallel straight electric
currents, close together and directed oppositely.
The planivector {vortejc) field in S^.
The field E^. The elementary sphere /S/>, is a circlet, the elementary
vortex system Vo~ a current along it. It furnishes a linevector
siti (f
potential =i — — directed along the circles whicii ])roject themselves
on the plane of Vo^ as circles concentric to Voz, and where </ is
the angle of the radiusvector with the normal plane of Vo,. The
field is the first derivative (rotation) of this potential.
Jlie field E,. The elementary sphere Sjyy is again a circlet, tiie
elementary vortex system Vo^j assumes in the points of that circlet
equal 'vectors normal to it. The ' Fpotential consists of the ' I^'s
normal to the potential vectors of a field E^ ; the field E^ is thus
obtained by taking tlie normal i)lanes of ail planivectors of a field ^i.
As on the other hand E^ and E^ are of the same identical structure
outside the origin, we can say here again, that the field E^ resp. E,
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 ecpial structure, but one field is perfectlv
normal to the otiier.
So if of a field the two generating systems of currents are
identical, it consists of isosceles doublevortices.
The force field in S,.
The field E^. Vo, gives a double point, causing a scalar
cos (p
potential — , where </ is the angle of the radiusvector with the
r
axis of the double point; the derivative (gradient) gives the wellknown
field of an elementary magnet.
The field E^. Voy consists of equal planivectors normal to a
small circular current. If we represent the [»lanivector potential by the
linevector normal to it, we shall find for that linevector ^^ directed
( 76 )
along the circles, which project themselves on the plane of Vo,
as circles concentric to Vo, and where <p is the aiigle of the radius
vector with the normal on the circular current. The field E^ is the
second derivative of the plan i vector potential, i.e. the rotation of
the normal linevector.
According to what was derived before the field E^ of a small
circular current is outside the origin equal to the field E^ 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
/l
scalar potential out of that of single agens points I — X l^ie second
derivative of the field), and the vector potential out of that of rectilinear
1
elements of current (perpendicular to — X the first derivative of the
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 Schering (GiHtinger Nachr.
1870, 1873; compare also Fresdorf Diss. Gottingen 1873; Opitz
Diss. Gottingen 1881) and Killing (Orelle's Journ. 1885) construct
the potential of elliptic s[)ace, starting from the suj^position tliat
as unity of fiehl must be possible the field of a single agens point,
leads to absurd consequences, to which Klein (Vorlesungen iiber
NichtEuldidische Geometrie) has referred, without, however, proposing
an improvement. To construct the potential of the elliptic and
si)hcrical spaces nolhing but the field of a double point must be
assumed as unity of field, which would lead us too far in this
( 77 )
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 lime. It is an integral of vortex fields as
thej 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 Sn •
The field E^. Voz again gives a double point, which furnishes a
cos (p
scalar potential , where (p is the angle between radiusvector and
^ ^,7i — 1
axis of the double point; its gradient gi\es what we might call the
field of an elementary magnet in S^
lite field E^. Voy consists of equal plani vectors normal to a
small " — sphere Spjj. To find the plani vector potential in a point
P, we call the perpendicular to the aS„ — i in which Sp,i is lying
OL, and the plane LOP the "meridian plane" of P ; w^e call
ip the angle LOP and OQ the perpendicular to OL drawn in
the meridian plane. We then see that all planivectors of Voy have
in common with that meridian plane the direction OL, 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 n — 2"'^ power of their
distance to P, and placed in P, neutralize each other two by two;
and the former consist of pairs of ecpial 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
sin (p
a iilanivector potential lying in the meridian plane = c . The
field L\ is of this potential the v = V^, inid 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 1"^. by magnets,
2'"'. by vortex systems consisting of the plane vortices erected normal
to a snuill "—sphere. We can also take as the cause the spheres
themselves with their iiidicatrices and say that the field is formed
by magnets and vortex spheres of n. — 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 voitex 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 plant vector) are given, then it is the V of a potential :
r div. dv r rot. dv , . ^ , , i /. , i
I 1 I : this formula takes the held as an
integral of fictitious fields of agens points and of single vortices.
Crystallography. — "On the fatty esters of Cholesterol and
Pliytosterol, and on the anisotropous liquid phases of the
Cholesterolderivatives." By Dr. F. M. Jaeger. (Communicated
by Prof. A. P. N. Franchimont.)
(Communicated in tlie meeting of May 26, 1906).
§ 1. Several years ago I observed that pliytosterol obtained from
rapeseedoil suffers an elevation of the melting point by a srAall
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 Bomer ') on the meltingpointelevations
of phytoterol by cholesterol and also of cholesterolacetate by phyto
sterolacetate. Apart from the fact that the crystallographic data
from 0. Mügge led me to the conclusion, that there existed here an
uninterrupted miscibility between heterosymmetric components, a
further investigation of the binary meltingpointline 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 Bomer 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 acidresidues into the molecule of cholesterol would modify the
behaviour of the estei'S in regard to the phenomenon of the optically
anisotropous liquid phases, first noticed with the acetate, propionate
and berizoate, with an increasing carboncontent of the acids. Finally
I wished to ascertain whether there was question of a similar
meltingpointelevation as with the acetates in the other terms of the
series too.
1) BöMER, Zeit. Nahr. u. Genussm. (1898), 21, 81; (1901). 865, J070; the last
paper s,with Winteu) contains a complete literature reference to which 1 refer.
( 79 )
§ 2. In the lirst place the esters of cholesterol and phytosterol
had to be prepared.
The cholesterol nsed, after being repeatedly recrjstallised from
absolute alcohol + ether, melted sharply at 149°. 2. The phyto
sterol was prepared by Merck, by Hesse'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 Mügge,
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 objectglass 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.
Fig. 1.
Cholesterol,
fused and then solidified.
Fig. 2.
Phytosterol,
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
'j Hesse, Annal. der Chemie, 192. 175.
( 80 )
traversed by a dark cross, so that the whole conveys tlie 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 crj'stals is optically negative.
Cliolesterol, however, presents a quite diiferent image. When melted
on an objectglass, the substance
contracts and forms small droplets,
wdiich are scattered sporadically and,
on soliditication, look like little nug
gets with scaly edges, wliich mostly
exhibit the white of the higher order.
That the microscopical distinction
in this manner is much safer than
by Mügge's method, may be seen from
fig. 3 where phytosterol and choleste
rol are represented as seen under the
l'ig 3 microscope, after being crystallised
Pliytosterol and Gholeslerol fiom from alcohol. ^4 is cholesterol, B phy
950/0 Alcoliol. tosterol.
§ 3. Of the fatty esters, I have prepared the acetates, propionates,
hutyrates and isobiity rates by heating the two alcohols with the pure
acidanhydride 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 sodinm hydrocarbonate, and
then recrystallised from alcohol { ether, afterwards from ethyl
acetate f ligroin, or a mixture of acetone and ligroin, until the nielt
ingpoint was constant. Generally, I nsed equal parts by weight of
the alcohol and the acidan hydride.
The fonniates, valerates, isovalerates, capronates, capnjlates and
caprinates were prepared \)\ means of the pure anhydrous acids.
These (valeric, caprylic and capric acids) were prepared synthetically
by Kahlbau:\[ ; the isovaleric acid and also the anhydrous formic
acid were sold commercially as pure acids "Kahlhau.m". Generally,
a six hours heating of the alcohol witii a Httle 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 nnich greater than with the above described method of preparing.
Both series of esters crystallise well. The phytosterolesters in soft,
flexible, glittering scales ; the formiate and the valerates present some
difiiculties in the crystallisation, as they obstinately retain a trace of
(81 )
an adhesive byproduct which it is (lil'ticiilt to remove. The choles
terolesters gi\e much nicer crystals; the formiate, acetate and hen
zoate have been measured macroscopically ; the other derivatives
crystallise 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 '). Iji the case of the caprylate, liic 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, plateshaped crystals, from boiling ligroïn ^).
The phytosterolesters retain their white colour on exposure to the
light; the cholesterolesters gradually turn yellowish but may be
bleached again by recrystallisation.
The determination of the melting points, and in the case of the
cholesterolesters, also that of the transitiontemperatures: solid — ^
anisotropousliquid, was always executed in such manner, that the
thermometer was placed in the substance, whick entirely surrounded
the mercuryreservoir. Not having at my disposal a thermostat, I
have not used the graphic construction of the coolingcurve, in
the determinations, but simply determined the temperature at wiiich
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 phytoslerol are still doubtful, and where the
molecules contain from 28 to 37 carbonatoms. 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
case of cholesterolesters, botli temperatures, remaijied constant on
further recrvstallisation .
1) I have even succeeded lately in obtaining the formiate in large transparent
crystals from a mixture of ligroïn, ethyl acetate and a little alcohol.
2) The crystals of the cajjrinate are long, flat needles. They form monoclinic
individuals, which are elongated parallel to tlie öaxis, and (latlcned towards Ü01
The angle 3 is 88° a 89°; there are also the forms: jlOUJ and jToij; measured:
(1U0):(101) = ±20.°. The optic axial plane is 01ü; inclined dispersion: p>v
round the first bissectria. Negative double refraction. On jOOlj there is one optical
axis visible about the limits of the held. The crystals are cnrvc(lil;iii( .
Proceedings Royal Acad. Amsterdam. Vol. IX.
(82 )
§ 4. I give in the following tables the temperatures observed
etc.^) Next to my data are placed those of Bomer as far as he has
published them. The temperatures in [J will be discussed more in
detail later on.
I. FATTY ESTERS OF CHOLESTEROL.
ti
k
^3
Bömer's data:
ChoL Formiate
—
[± 90° J
960.5
—
90°.
» Acetate
~
[80 a 90°]')
il2°.S
—
1130.5
» Propionate
93°.0
107O.2
—
96°
111°
» «Butyrate
9G°.4
i07°.3
—
96°
108°
» Isobutyrate
—
—
126°. 5
—
)) «Valerate
91°. 8
99°.2
—
—
—
» Iso valerate
—
[+ 109°]
llOo.G
—
—
» Capronate
910.2
iOO°.4
—
—
—
» Caprylate
—
[± tOloj
106O.4
—
—
» Caprinte
82°.2
90O.6
—
—
—
» Benzoate
145°.5
i78°.5
—
146°
1780.5
» Phtalate )
—
—
—
—
1820.5
» Stearinate ^)
—
—
—
6
)°
Benzoates and phthalates alihough not being fatty esters, have nevertheless
been included.
1) According to Schönbeck, Diss. Marburg. (1900).
2) According to Bomer loco cit.
3) According to Berthelot. 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 —^ anisotropousliquid ; 2. transition: aniso
tropousliquid — > isotropousliquid ; 3. transition : solid — > isotropousliquid.
This distinction has been retained, particularly on account of the cases of labile,
liquid crystals discovered here.
( 83 )
did colourphenomena observed during the ('oolin;^; of the clear,
isotropou'ï, fused mass to its temperature of soliditication, and also
during the heating in the reverse way. Tiiese colour phenomena are
caused by interference of the incident light, every time the turbid
anisotropous liquidphase 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 Reinitzer, until the
temperature t.^ 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
sliown by the butyrate and normal \alerate, also \ery well by the
capronate and caprinate.
The temperatures in \_] t^ answer to anisotropous liquid phases
which are labile in regard to the isotropous liquid, and irliich double
refracting liquids are, therefore, only realisable in undercooledfused
material. Of this case, which is comparable with the monotropism, as
distinguished by LehmaN!* from the case of enantiotropous transfor
mations, the acetate is the only known example up to the present. Xoav
the number of cases is increased by three, namely the formiat>>, the
caprylate and without any doubt also the isovalerate, to w^hich I will
refer presently. Cholesterolformiate and caprylate melt therefore,
perfectly sharply to a clear liquid at, respectively 96^',° and 106. °2.
If, however, the clear liquid is suddeidy cooled in cold water,
one notices the appearance of the turbid, anisotropous, morelabile
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 doublerefracting 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 cholesterolcaprinate may probably
exhibit two anisotropous liquid phases. Although, personally, I never
noticed more than one single phase, and Prof. Lehmann'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 tiro phases implies that they would not be miscible
in all proportions with each other.
6*
. ( 84 )
§ 6. Tlie beliaviour of cholesterolisohutyrate is a very remarkable
one. Microscopic and macroscopic investigation shows absolutely nothing
of an anisotropons 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 differentlybehaving 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 acidresidue, which contrary to that of
the other esters, is branched.
All this induced mo, to prepare the analogous ester of isovaleric
acid; perhaps it might be shown also here that the branching of the
carbonchain of the acid destroys the phenomenon of the anisotro
pons liquid phase. At first T thought this w\as indeed the case, but
a more accurate observation showed that in the rapid cooling there
occurs, if only for an indivisible moment, a hhile anisotropons
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 capry late! 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 others, that the occur
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 c. s.).
§ 7. We now give the melting points of the analogous phytosterol
esters which, with one exception, do not exhibit the phenomenon of
the doublerefracting liquids. As the phytosterols from different vege
tal)le fats seem to differ from each other, and as B()Mer does not
mention any phytosterol esters from Calabarfat in particular, 1 have
indicated in the second cohuim only the limits 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 acid
residues of increasing carboncontent, takes place much more rapidhj
than with cholesterol. On the other hand, the succession of the melting
points of the acetate, propionate, butyrate and /i valerate is more
regular than with the cholesterolderivates.
All phytosterolesters share with phytosterol itself the great {qr
(85)
II. FATTY ESTERS OF PHYTOSTEROL.
! Limits arcoiding to
j Bömkh:
1
PhytosterolFormiato
110°
103°— 1130
PhytosterolAcetdte
12904
123°— 135°
Phytos<erolPropionate
105°.5
104^—116°
Phy tosterül Bu t yrate
91°.2
85°^ 90°
PhytosterulI^obutyrato
117".
—
Pliylosterclnonn. Valerate
f, = C7° ; /', = 30°
—
PliytosterolI^^ovalerat"^
1003.1
—
dency to crystallise from the melted mass in .^tphaeroh'tes ; with an
increasing carboncontent of the fatty acidresidue, these seem gene
rally' to become smaller in cii'cumference.
The formiate crystallises particularly beautifully; this substance
possesses, moreover, two solid modifications, as has been also stated
by Prof. Lehmann, who is of opinion that these two correspond with
the two solid phases of the cholesterolderivative. In the phytosterol
ester the sphaeroliteform is the morelabile one.
On the other hand, when recrystallised from monobromonaphthalene
or almondoil, they form under the microscope wellformed 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 difticulty 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 TrUaurln and Triinyristin by Schey. ^)
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 doublerefrac
ting layer on the object glass, whicii 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) Schey. Dissertatie, Leiden (1899) p. 51, 54.
( 86 )
Accordinp, to Prof. Lehmann, normal phjtosterolvalerate forms
very beautiful liquid crystals, which are analogous to tliose of chole
sterololeate ; 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 Schey with his higher tryglicerides also owe
their origin to the occurrence of labile, doublerefracting 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 I'egard to each other.
It has been sufiiciently proved by Bomer that the meltingpoint
line of cholesterol and of phytosterol is a rising line. In connection
with Mügge's and my own crystal determinations w^e 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 iniscibility of this new kind of
crystal with both comjionents, or whether an eventual transformation
in the solid mixing phases proceeds so slowly that a transition
poijit in the meltingpointline escapes observation, cannot be decided
at present.
The matter is of more interest with the esters of both substances.
According to Bomer ^) the formiates give a meltingpointline with a
eutectic point ; the acetates, however, a continuously rising melting
pointline.
The method of experimenting and the theoretical interpretation is,
however, rather ambiguous, as Bomer prepares mixed solutions of
the components, allows these to crystallise and determines the melt
ingpoint of the solid phase tirst deposited. By his statement of the
proportion of the components in the solution used, he also gives an
incompletfc and confusing idea of the connection between the melting
point and the concentration.
Although a rising of the binary meltingpointline may, of course,
be ascertained in this manner quite as well as by otiier means
— and Bömer's merit certainly lies in the discovery of the fact
1) Compare Bomer, Z. f. Nalir. u. Gen. M. (1901) 546.
2) BöMER, Z. f. Nahr. u. Gen. Mitt. (1901) 1070. In connection with tlie dimor
phism of the formiates, a mixing series witli a blank is however very probable in
this case.
(87)
itself — the determination of I he binary nieltingpointline must be
reckoned fiinh} as soon as it is to render quantitative services, whicli
is of importance for the analysis of butter ; for if the meltingpoint
curve is accurately known, the cpiantity 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\
a:.''
/Ji7
too
fOO
90
60 TO. 3/0
bO
JO
20
fP
50 ^Z^
Fig. 4.
Cholesterol, and PhytoslerolAcetate.
Altliongh 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 7o cholesterolacetate to
7o is nearly horizontal, it follows that the composition of mixtures
can be veritied by the meltingpoint, when the admixture of phytosterol
in the animal fat does not exceed 60 "/o. The results are the most
accurate when the quantity of phytosterolester^) amounts to 27o — 'AOVo
In practice, this method is therefore applicable in most cases. The
cholesterolacetate used in these experiments melted at 112. °8; the
phytosterolacetate at 129.°2.
A mixture of 90 «/^ Choi. Acet. f 10 7„ Phyt. Acet. melts at 117°
»
B
80 »
>
»
»
73.3 »
»
»
»
60 »
»
»
»
42.4 »
))
»
»
20 »
»
»
))
10 »
))
+ 20 .)
+ 26.7 »
j 40 »
+ 57.6 ))
4 80 »
+ 90 »
120.°5
122. °5
125°
128°
129.°1
129.°2
1) It should be observed that although Bomer, 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 (juantitative 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 cholesterolacetate is
0,5 — 1"/,,, the meltingpoint is practically not altered ; when it is 2"/,, however,
the amount is easy to determine.
( 88 )
Probably, a case of isomorphotropoiis relation occurs here with
the acetates ; both esters are, probably, inonocliiiic, although this is
not quite certain for the cholesterolester. This is pseudotetragonal and
according to Von Zepharovich : monoclinic, with /? =r= 73°38' ;
according to Obermayek: tricUnic, with /^=106^17', «=90°20', y=90°6',
while the axial relations are 1,85 : 1 : 1,75.
The phytosterolester has been approximately measured microsco
pically by Beykirch and seems to possess a monoclinic or at least a
triclinic symmetry with monoclinic limitvalue. In my opinion both
compounds are certainly not isomorphous. In any case it might be
possible that even though a direct isomorphism does not exist in
the two esterseries, there are other terms which exhibit isomor
photropous miscibility in an analogous manner, as found for the ace
tates by BöMER. 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 :
31.87o cholesterolbutyrate f ()8,27o phytosterolbutyrate indicate
for t, 8i; and for t^ 83° etc. etc.
For the formiates, the lo^vering had been already observed by Bömer;
other esters, also those of the isoacids behave in an analogous
manner : at both sides of the meltingdiagram occurs a lowering of
the initial melting points. It is, however, highly probable that in
some, perhaps in all cases, there exists an isor/zmorpholropous mixing
with a blank in the series of the mixed crystals.
The anisotropons liquid phase of cholesterolesters gives rise in this
case to anisotropons liquid mixed crystals. I just wish to observe that
for some of the lowermelting esters, such as the bntyrale, capronate,
caprinate, normal valerate, etc, the temperature t^ 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. Bakhuis Roozeboom called my attention, that in those substances
where t^ answers to the morelabile condition, the at tirst more labile
liquid mixed crystals, 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 lowmelting derivatives Or else a
similar study of the lowmelting liquid mixed crystals by means of
the ultramicroscope might yield something of importance.
Zadudam, May 1906.
( 89 )
Physics. —  ''Researches on llw thermic and electric conductivity
power of crystallised conductors." I. By Dr. F. M. Jaeger.
(Communicated by Prof. H. A. Lorentz).
(Communicated in llie niecllng of May 20, 19ÜG).
1. Of late years, it lias been attempted from various sides to
find, by theoretical means, a connection between tiie piienomena of
tlie thermic 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. Drude ^), J. J.
Thomson ^) and E. Riecke ") 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 tJiat the quotient of the electric
and thermic conductivity power of all metals, independent of their
particular chemical nature, is a cimstant, directly proportional to the
absolute temperature.
When w^e assume that the electrons in such a metal can move
freely with a \elocity depending on the teniperatuie, 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 ). and the electric conductivity
power a may be indeed represented by a constant, proportional to
the absolute temperatuie T.
The theories of Drude and Lorentz only differ as to the ab
;. 4 fa\
solute value of the quotient; according to Drude =   T;
Ö 3 \e J
;. 8 fct\\^
according to Lorentz  =  7'. In these expressions )., o and
r) 9 \ej
T have the above cited meaning, whilst a is a constant and e
represents the electric charge of the electron.
By means of a method originated by Kohlrausch, Jaeger and
Diesselhorst have determined experimentally the values for  with
ij P. DauDE, Ann. Phys. (1900). 1. 566; 3. 369.
2, J. J. Thomson, Piappoit du Congres de physique Paris (1900). 3. 138.
S) E. Riecke, Ann. Phys. Cliem. (1898). 66. 353, 545, 1199; Ann. Phys. (1900).
2. 835.
^) H. A. Lorentz, Proc. 1905, Vol. Vil, p. 438, 585, 084.
( 90)
various metals ^). The agreeuient 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 18°, the value 47X^0^ may be deduced in
aT
C.G.S. units, for the expression — . (Compare Lorentz, loco cit.
e
p. 505); according to Drude'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
crystals, the propounded theories render it fairly probable for all
such conducting crystals that:
— z= — r= — , and therefore also: ^ : Xy\ ),, = a^ •■ Oi, : Gg.
Ox Gy Gz
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
conducti\'ities should be numerically e€[ual to that of the thermic
mainconductivities.
Up to the present but little is known of such data. The best
investigated case is that of a slightly titaniferous Haemitate of
M W. Jaeger und Diesselhorst, Bed. Silz. Ber. (1899). 719 etc. Gomp. Reinganum,
Ann. Phys. (1900) 2, 398.
2) With Al, Cu, Ag, Ni, Zn, the value of  at 18' varies between 636x10**
and 699X10«; with Gd, Pb, Sn, Pt, Pd between 706X10« and 754x10^; with
Fe between 802 and 882 X 10^ therefore already more. With bismuth  at
18° = 962 X 10* Whilst in the case of the other metals mentioned the values of
 at 100" and at 18^ are in the average proportion of 1,3:1, with humxith the
<l
proportion is only 1.12. In their experiments, Jaeger and Diesselhorst employed
hltle 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 dilfer very considerably.
( 91 )
Swedish origin whicli lias been investigated l)y H. Backstrom and
K. Angstrom ') as to its lliennic and electric condnctivity power.
Iji tliis ditrigonal mineral, they found for the qnotient of the thermic
conductivity power in the direction of the chief axis (c) and in tiiat
perpendicular to it (^/) at 50" :
^1.12.
For the qnotient of the electric resistances w at the same tempe
rature they fonnd :
— =1.78, and, therefore:  = 1.78.
lOa Or.
From this it follows that in the case of the said conductor, the
theory agrees with tlie obserxations as to the relation between the
conducti\ ity powers only qualitatively, but not quantitatively, and
— contrary to the usually occurring deviations — the proportion of
the quantities I is smalle v than that of the quantities o.
Jannettaz's empirical rnle, according to which the conductivity for
heat in crystals is greatest j)ai'allel to the directions of the more
complete phmes of cleavage, ap[tlies 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 (Miller), 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 })lan was conceived to investigate in a series of determinations
the thermic and electric conductivitypower of some higher and also
of some lowersymmetrical crystalline conductors, and, if possible,
of metals also. For the moment, I intend to determine the quotient
of the conductivities in the ditFerent main directions, and afterwards
perhaps to measure those conducti\ ities themselves in an absolute
degree.
I. On the thermic and electric conductivities in crystallised Bismuth
and in Haematite.
Measurements of the thermic and electric conducti\ity of bismuth
are already known.
Matteucci ^) determined the thermic conductivity, by the well
^) H. Backstrom aii<l K. Angstuöm, Ofvers. K. Vetensk. Akad. Füili. (1888).
No. 8, 583; Backstrom ibid. (1894), No. 10, 545.
•2) Matteucci, Ann. Gliim. et Phys. (3). 43. 4G7. (1855).
( 92 )
wellknown method of Ingenhousz, by measurement of the length
of the melted off waxy layer which was pnt on the surface of
cylindrical rods of bismuth, cut // and J_ 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.
Jannk.ttaz's rule applies in this case, because the complete cleavability
of ditrigonal bismuth takes place along 111 (Miller), therefore,
perpendicularly to the main axis. Jannkttaz ^) 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, Lownds^) has again applied the Sénarmont method
to bismuth. He finds for the quotient of the demiellipsoidal axes
1.19 and, therefore for the quotient of the conductivities 1.42.
The last research is from Perrot "). By the Senarmont method
he finds as the axial quotient of the ellipses about J. 17 and conse
quently for the quotient of the conductivities J_ and // axis 1,368,
which agrees foirly well with the figure found by Lownds. Secondly,
Perrot determined the said quotient by a method j)roposed 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 0^ and d^ placed
at a given distance at ditferent sides of a block of the substance
under examination begin to melt. As indices were used; ctNapJdyla
mine (i^' = 50° C), oNitroanUhie (<*> = 66^C.), and NaphtJialene
(^ = 79° C).
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 Sp;narmont's method.
He, however, rightly observes that this concordance between the
two results is quite an nccldental o/z^^ and that the method of Thoulet
and SoRET must not be considered to hold in all cases. The proof
thereof has been given by Cailler in a theoretical paper ; '') the
agreement is caused here by the accidental small value of a quotient
hi
— , in which / represents the thickness ot the little plate of bismuth
k
^) Jannettaz, Ann. cle chim. pliys. 29. 39. (1873).
2) L. Lownds, Phil. Magaz. V. 152. (1903).
'^) L. Perrot, Archiv. d. Science phys. et nat. Geneve (1904 . (4). 18. 445.
4) Thoulet, Ann. de Chim. Phys. (5). 26. 261. (1882)
s) G. Gailler, Archiv. de Scienc. phys. et nat. Geneve (1904). (4). 18. 457.
( 93 )
and h and k the coeflicients of external and internal conductivity.
§ 4. 1 have endeavoured to determine the (juotient of the chief
conductivities by the method proposed by W. Voigt.
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 X^ and ).^ are the two chief conductivities of a plate of bismuth
cut parallel to the crystallographic main axis, and if the angle which
tlie two main directions form with the line of demarcation equals
45°, then according to a former formulae:
§ 5. The bismuth used was kindly funushed to me by Dr. F. L.
Perrot, to whom I again wish to express my hearty thanks.
The prism investigated by me is the one which Dr. Perrot in
his publications'') indicates with M, and for which, according to
Senarmont's method, he found for — the value 1,390. The prism
given to Dr. van Everdingen yielded in the same manner for
 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 <p = 45°.
It soon appeared that in this case the Voigt method ^) was attended
by special difficulties which, as Prof. Voigt 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 conducti\ity of the metals, in connection with
the peculiarity just mentioned. On the advice of Prof. Voigt 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 j>lau did not yield xevy good results.
1) These Proceedings. (190G). March p. 797.
2) p. 4, note 10.
3) Voigt, Göltinger Nachr. (189Ü). Hcit 3, p. 1 — 16; ibid. (1897j. Heft 2. 1—5
( 94)
However, at last, I succeeded in getting a satisfactory coating of
the surface by substituting for white wax the ordinary, yellow
beeswax. This contains an adhesive substance probably derived
from the honey, and, when mixed in the proper proportion with
elaidic acid it yields the desired surface coating.
I have also coated ^) the bottoms of the plate and the sides, except
those which stand J_ 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 generall}' become curved.
It is further essential to heat rapldh/ 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 f.
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
f = 22^12' and therefore:
 = 1,489.
Ac
§ 6. The value now found is somewhat greater than that found
by Perrot. 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 Senarmont,
Janettaz and Roentgen, 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 diiferences were still
greater than those which are communicated here. Although a more
extended research, including some plates kindly lent to me by Prof.
VoiGT, showed that these ditferences are not so serious as I suspected,
at first the deviation exists ahimys in the same direction.
For instance, I measured the angle « of a plate of an Apatite
cry stsd from Stillup in Tyrol and found this to be 17°. From the
') RicHARz's method of experimenting (Nalurw. Rundschau, 17, 478 (1902)) did
not give sufficiently sharply defined isotherms and was therefore not applied.
(95 )
position of the isotherms it also follows that P.r > h so that — ^1,35.
In a quartzplate obtaine<l from Prof. Voigt I found s = 30^°,
therefore — z=l,/5. In a plate of Antimonite from Ski kok u in Japan
'■a
cut parallel to the plane 010}, — was found to be even much larger
than 1,74, which value is deduced from the experiments of Senar.mont
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
TucHSCHMiDT determined the heatconductivity of the latter mineral
according to Weber's method in absolute degree. His experiments
h
give the value 1,646 tor the quotient — .
The deviations are always such that if P, ^i*., the values of the
quotient — turn out to be larger when Voigt's method is employed
instead that of de Senarmoxt. 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 caxis was found
not to be homogeneous and to contain gasbubbles. I repeatedly
measured the angles £ of a beautifully polished preparation of Prof.
Voigt, and found fairly constantly 10è°, whilst the position of the
isotherms showed that ).a was again larger than ).c.
For the Haematite we thus obtain the value: — = 1,202. The
;..
\alue found by Backstköm and ANCiSTUoM for their mineral with the
aid of Christiansen'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
yn : X, in both crvstal phases, if bv this is meant ( ) : (   the
values :
Wwh Bismuth:  = 1,128.
With Hunnatite:  = 1,480.
yc
In this mv measurements of — are combined with the best value
( 96 )
found by van E verdingen ') with Perrot's prism, namely — ^1,68,
and witli the vahie found bv 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. Tlie said values
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 prol)able by the electron theory.
In the first place it will be observed that the agreement is much
better with bismuth than with haematite. However, this may be
expected if we consider that the theory has been proposed, in the
first instance, for metallic conductoj'S. Tiie influence of the peculiar
nature of the o,vide when compared with the true vietal 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
the chemical nature on one side, and the given values of — on the
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 crystallise ditrigonally and have an analogous axial quotient;
for bismuth: a : c = 1:1,3035 (G. Rose); for haematite a : c =
1 : 1,3654 (Melczer). 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 pseudocubic
construction is very distinct; the planes of complete cleavage which
answer the forms 111 and 111 approach by their combination the
regular octahedron in a high degree. Although haematite does not
1) VAN EvERDiNGEN, Avclilves Néerlaiid. (1901) 371 ; Vers). Akacl. v. Wet. (1895—
1900); Gomm. Ptiys. Lab. Leiden, 19, 26, 37, 40 and 61. See Arcliiv. Nêerl.
p 452 ; rods No. 1 and No. 5.
( Ö7 )
possess a perfect plane of cleavage, it mny be cleaved in any case
along 111 with testaceons plane of separation. It adniits of no doubt
that the elementary parallelepipeds of the two cr3^stal structures are
in both phases pseudocubic rhombohedral contigurations and the
question then rises in what proportion are the molecular dimensions
of those cells in both crystals?
If, in all crystalphases, we imagine the whole space divided
into volumeunits 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
J\I
the size of those volume elements is proportionate to — , in which
I\f represents tlie 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
31
this equivalentvolume — , the dimensions of those cells will be reduced
for all crystals to a same length unit, namely all to the length
of a cubicside belonging to the volumeelement of a crystal phase,
whose density is expressed by the same number as its molecular
M , ,
weight; then in that particular case V=z— = 1. If we now calculate
the dimensions of such an elementary parallelopiped of a Bravais
structure whose content equals the quotient — and whose sides are
d
in proportion to the crystal parameters a : ö . c, the dimensions
X, t^ and o) thus found will be the socalled topic parameters oïWiq
phase which, after having been introduced by Becke and Muthmann
independently of each other, have already rendered great services
in the mutual comparison of chemicallydifferent crystalphases. In
the particular case, that the elementary cells of the crystalstructure
possess a rhombohedral form, as is the case with ditrigonal crystals,
the parameters x> ^ and ^ become equal to each other (= (>). The
relations applying in this case are
<? =
sin a . sm
a
sin
V \^ A 2
with nin
A I 2 sin a
If now these calculations are executed with the values holding
here: 5i = 207,5; Fe,0, = 159,G4; fZij; = 9,851 (Pekrot); t//,>,Os = 4,98,
then
7
Proceedings Royal Acad. Amsterdam. Vol. IX.
(98)
VBi=2l,0QA and Fi>'e,03= 32,06,
and with the aid of the given rehitions and the values for « and A
we find for each phase : ^ )
Qbj ^ 2,7641
Q ~ 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
following relation:
Xr,
©
=:q' : q' = 1,32.
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.
In our case the quotient — may therefore be written for both
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
Qi' ' Qi^i perhaps q^^ sin «j : q^^ dn «^ = 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 ciuotient — in the two
Xc
phases should then be directly proportional to the reticular density
of the main netplanes of Bravais's structures.
A choice between this and the above conception cannot yet be
made, because «^ and «^ differ too little from 90''. Moreover, a further
investigation of other crystals will show whether we have to do
here with something more than a mere accidental agreement. Similar
investigations also with lowersymmetric conductors are at this
moment in process and will, I hope, be shortly the subject of further
communications.
Zaandam, May 1906.
1) For bismuth a = 87°:34' and J. = 87o40': for haematite a==85°42' and
A = 86°0', The angle A is the supplement of the right angle on Ihe polar axes
of the rhorabohedral cells and x is the Hat angle enclosed between the polar axes.
( 99 )
Chemistry. — '^ llireepliaseUnes in chloralalcokolate and aidluie
hydroddoride". By Prof. H. W. Bakhuis Roozeboom.
It is now 20 years since tlie study of the dissociation piieno
niena of various solid compounds of water and gases enabled nie
to find experimentally the peculiar form of tliat threephaseline 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 der Waals and the frequency of its
occurrence was proved afterwards by the study of immj other
systems.
That this threephaseline 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 ^^, Mines, iiave
been again studied by van der Waals, Smits and myself either
qualitatively or qualitativequantitatively.
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 threephaseline of a
compound, recently deduced by Smits for the case in which a
7*
( 100 )
minimum occurs in the pressure of the liquid mixtures of its
components.
Mr. Leopold has now succeeded in giving experimental contributions
in regard to both questions, bj means of a series of \'ery accurately
conducted researches where chloralalcoholate and anilinehjdrochloride
occur as solid compounds.
Solid compounds which jield two perceptibly volatile components
(such as PCI3, NH,.H,S, PH3.HCI, C0,.2 NH, etc.) have been investi
gated previously, hut 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 threephase 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
f alcohol) was much smaller than in the second (aniline f 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.
( 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 threephase 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 threephase line, T and T,
are the respective maxima for the vapour pressure of solutions with
excess of either alcohol or chloral and saturated with chloralalco
holate; the minimum is situated very close to the melting point F.
In the second system (anilinehydrochloride Fig. 2) the first maxi
mum, in presence of excess of HCl is situated at such an elevated
■^00
CM
^^,
\
^
^I
\
^^
^
^
2.50
B,
200
150
^
/
100
—
■
'
H
li
B
^1
<
H
1
50
A
___^
Ta
"sTi
F^
'A,
^
D
L.
F,
Fig. 2.
( 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
Tj is situated at the same side and is removed further from the
melting point than in Fig. 1.
1\ minimum 7^ melting point
p 16 cM. 22.5 cM.
t 197° 199°2
The determination of these lines and also that of the equilibria
lines foi compound f 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 fdl 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 LG oï aniline hydro
chloride was thus determined. On this line then follows the piece
GF of the threephase 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 threephase line G may be traced to very near
the melting point F, where one passes on to the line FA, for the
equilibrium of the fused compound with its vapour.
We have here, therefore, the first experimental confirmation of
the normal succession of the ^?, Mines when those are determined
with a pure compound which dissociates more or less.
Theoretically, the minimum T^ in the threephaseline must be
situated at the left of the terminal point G of the sublimationline.
The difference here, although small, is yet perfectly distinct:
T, G
p 16 cM. 16.5 cM.
/ 197° 198°
In the case of chloralalcoholate the points 7\ and G both coincide
so nearly with F that this point is practically undistinguishable from
ihe triple point of a nondissociating compound, both LF and FA,
or their mctastable pi'olongation FA' appear to intersect in F. Moreover,
the investigation of the melting point line proved tliat chloralalco
holate in a melted comlilion is but little dissociated.
( 103 )
In both compounds the />,^hnes 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 threephaseline.
In the case of a slight excess of chloral (Fig. 1) this was followed
from D over J\ to F^ just a little below the melting point, and from
there one passed on to the liquidvapour line F^A^, which was
situated a little above FA.
In the case of a slight excess of aniline the piece DT^TfiF^
could be similarly followed (Fig. 2). In this occurred the minimum T^,
whilst the piece GF^ coincided entirely with the corresponding part
of GF, which had alreadj' been determined in the experiment with
the pure compound. Just below F the compound had disappeared
entirely and one passed on to the liquidvapour line F^A^, which,
unlike that in Fig. 1, was situated below FA.
If the excess of the component is verj' trifling, liquid is formed
only at higher temperatures of the threephaseline, 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 chloralalcoholate a similar line BE (Fig 1) was determined,
situated decidedly above LF. At E, liquid occurred and a portion of
the threephaseline EF was followed up to a point situated so
closely to F that the liquidvapourline, which was then followed, was
situated scarcely above FA.
The excess of chloral was, therefore, exceedingly small, but in
spite of this, BE was situated distinctly above EF. The position of
BE depends, in a large measure, on the gasvolume 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 chloralalcoholate in sucii a state of jnirity that
it exhibited the lowest imaginable sublimationline LF, which meets
tiie threephaseline 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 CTF of the threephaseline. Such hap
pens, for instance, always when we use crystals of tiic compound
which have been crystallised from excess of alcohol. They then
contain sufficient motherliquor.
( 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 CTF, whilst the superfused liquid
gives the vapour pressureline FA^ which is situated much lower.
Ramsay has found this previously without being able to give an
explanation, as the situation of the threephaseline was unknown
at that period.
In the case of anilinehjdrochloride, 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 BE and B^E^. From ^^ the threephaseline was
followed over the piece E^H^ afterwards the liquidvapourline i:/i/i.
From E also successively EH and HI. With a still smaller excess
of hydrogen chloride we should have stopped even nearer to F on
the threephaseline.
In the case of chloralalcoholate 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 threephaseline where the
maximum and minimum have disappeared in the lower branch of
the threephase 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 lo better understand the general behaviour
of such substances.
Physics. — " On the polarisation of Röntg en 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 Röntgen bulb are partially
polarised, in agreement with a prediction of Blondlot 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 Ihe pi'iniary 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 iiad tried to examine the same question by a somewhat different
metiiod. A pencil of Röntgex rays passed through a tube in the
direction of its axis, without touching the wall of the tube. A photo
graphic fdm, bent cylindrically, covered the inner wall of the tube
in order to investigate whether the secondary rays emitted by the
air particles showed a greater action in one direction than in another.
I obtained a negative result and communicated this fact to Barkla,
who advised me to take carbon as a \'ery strong radiator for secon
dary rays. I then made the following arrangement.
s.
>
s.
s.
A.
Ph
R
A 'JlS
n
p
c
LA
^F
F'ii
Let /Si (fig. 1) be the front side of a thickwalled leaden box,
in which the Röxtgen bulb is placed; S.^ and S^ brass plates
10 X 10 cm. large and 4 m.m. thick. Their distance is 15 cm. 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 R^ and R^ slides into it ^).
An ebonite disk E in which a carbon bar is fastened fits in
tube B. This bar is 6 cm. long and has a diameter of 14 m.m. At
one end it has been turned off conically over a length of 2 cm.
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 Röntgen rays, which passed through the apertures
in Si, 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 Röntgen rays.
If we accept Barkla'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 Röntgen
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 B or not. If namely
the inner surface of cover D was coated by a jjliotographic 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 Röntgen 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 raj^s 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 Röntgen rays ^).
When in this way the symmetrical passage of the Röntgen 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 inductioncoil of 30 cm. striking distance was used
with a turbine interruptor. A storage battery of 65 volts was used ;
ij Take for lliis experiment the above described arrangement, but a carbon bar
of 1 cm. diameter and 4 cm. long.
( 107 )
the current streiigtli amounted to 7 amperes; the RöntgEiN 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
RöNTGEN rays are polarised at the utmost only to a very slight
amount, and 2"^ 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 Barkla's experiments
on the polarisation of secondary rays, wdiich he has shown also by
means of electroscopes and described Proc. Roy. Soc. Series A vol.
77, p. 247, 1906.
S
G.
5:
A.
R
4
U
n
JL
n
p
'F
Fi^."^
u
Let the arrow (fig. 2) indicate the direction of incidence of the
RöNTGEN rays on the carbon plate K large 8 X ^ cm. 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 inductioncoil was used with a
( 108 )
Wehnelt interruptor; the voltage of the battery amounted to 65 Volts
and the current to 7 Ampères. 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 tertiarj^ 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 sidetube F was
adjusted to the outside of cylinder A, and this tube F was placed
in an horizontal position during the experiment. A metal tube with
a narrow axial hole fitted in tube F, so that in the dark room,
after taking away a small caoutchouc stopper which closed F, 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 Barkla 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 Röntgen 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 cm.,
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 ring was perfectly concentric: the arrangement proved therefore to be
exactly symmetrical.
( 109)
This agreement makes it already very probable tiiat the Röntgen
rays also consist in transversal vibrations; these experiments however
yield a firmer proof for this thesis. Tf namely we accept the suppo
sition of Barkla 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 /o/i^iVz^c/ma/ vibration of the primary Röntgen
rays would, in the experiment discussed here, lead to a maximum
action of the tertiary rays in a vertical plane and not in an hori
zontal plane, as was the case.
Groningen, Physical Laboratory of the University.
Chemistry. — "Triformin {Gli/ceri/l irifonnate)". By Prof. P. van
ROMBURGH.
Many years ago I was engaged 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°/o, 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 Lorin'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 dicompound.
I will only mention a few series of experiments which I
made at Buitenzorg, first with Dr. Nanninga and afterwards w4th
Dr. Long. In the first, a product was obtained which had a sp.gr. 1.309
at 25", and gave on titration 76.67o of formic acid, whilst pure
triformin requires 78.47o The deficiency points to the presence of
fully 107o of diformin in the product obtained.
1) Gompt. Rend. 93 (1881) 847.
2) Gompt. Rend. 100 (1885) 282.
( 110 )
In the other, the diformin, was treated daily, during a month,
with a large quantitj 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. 1 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 107o of diformin.
Afterwards, when 1007o 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 liqueiied 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 0^ 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.
/if, = 1.4412.
lo
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 it 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
( 111 )
with diformin, however, cannot be distilled under those circum
stances, but is decomposed with evolution of carbon monoxide and
dioxide and formation of allyl foimate.
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 allyl
formate, some formic acid, and further, small quantities of allyl
alcohol. In the flask a little glycerol is left ').
Triformin is but slowly saponiiied 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 ot
the higher fatty acids.
Chemistry. — "On some derivatives of l^^hexattiene". By
Prof. P. VAN RoMBUKGii and Mr. W. van Dorssen.
In the meeting of Dec. 30 1905 it was communicated that, by
heating the diformatc of sdivinylglycol we had succeeded, in pre
paring a hydrocarbon of the composition CgHg to which we gave
the formula :
CH, =: CH — CH rr: 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 Ladenburg flask in an atmosphere
of carbon dioxide.
The main portion now boiled between 77° — 78°. 5 (corr.; pressure
764.4 mm.).
Sp. gr.i.3.5 0.749
n/),3.5 1.4884
Again, a small quantity of a product with a higher sp. gr. and a
larger index of refraction (;ould be isolated.
1) This decomposition of triformin has induced me to study the bchavioui" of
the formates of different glycols and polyhydric alcohols on heating, hivesligations
have been in progress for some time in my laboratory.
2) Meeting 30 Sept. 1905.
( 112 )
In the first place the action of bromine on the hydrocarbon was
studied.
If to the hydrocarbon previously diluted with chloroform we add
drop l)y drop, while agitating vigorously with a Witt 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 snbstance. 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 Liebig gave 66.847o, CgHgBr,
requiring 66. 65°/o.
A second bromine additive product, namely, a tetrabromide was
obtained by the action of bromine in chloroform solution at 0" 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 lies
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 1 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 Griner 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.
( 113 )
product melt ing 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 raeltingpoint 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 tirst 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,
w^e must attribute the formula
CH, == CH  CHBr — CHBr — CH =r 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 Thiele would
not apply in this case of two conjugated systems.
Experiments to regenei'ate the gl^^col 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 Sab.\tier and Senderens, hexatriene
may be readily made to combine with 6 atoms of hydrogen. If its
>) 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. IX.
( 11^ )
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 remoxal 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.,,o = 0,6907 nij,,^ = 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 Brühl and by Eykman ^).
Therefore, the product obtained from hexatriene was shaken for
some time with fresh portions of fuming sulphuric acid until tliis
was no longer coloured. After this treatment were obtained
one fraction of
B. p. 69°70°, Sp.gr.,, 0.6718 njj,, 1.38250.
and another of
B. p. 69°.7— 70°5, Sp.gr.,^ 0.6720, ud,, 1.38239.
An nhexane prepared in the laboratory, according to Beühl ^) by
Mr. ScHERiNGA gave the following values
B.p. 69°, Sp.gr.,, 0.664 nn,, 1.3792
whilst an ?zhexane prepared, from diallyl according to Sabatier and
Senderens, by Mr. Sinnige gave
B.p. 68.5°— 70,° Sp.gr.,, 0.6716, 71d,, 1.38211.
It is, therefore, evident that the hexane obtained b}' Sabatier's, and
Senderens' process still contains very small traces of impurities.
There cannot, however, exist any doubt that 135hexatriene
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 135hexatriene, two
conjugated systems are found and we might therefore expect the
occurrence of a 24hexadiene :
CH,— CH=CH— CH=:CH— CH,
1) Brühl (B. B. 27, (1894) 1066) finds Sp.gr..t^  0.6603, nD.n = 1.3734.;
Eykman (R. 14, (1881) 187) Sp.gr.^^  0.6652 nOu  1.37725.
2) Ann. 200. 183.
( 115 )
or, of a 25hexadiene :
CH3— CH=CH— CH,— CH=CH,.
The first, still having a conjugated system can again absorb two
atoms of hydrogen and then yield hexene 3.
CHg — CHj — CH=CH — CHj — CHj
whilst the other one cannot be hj'drogenated any further ').
The results obtained seem to point out that both reactions have
indeed taken place simultaneously, and that the final product of the
liydrogenation is a mixture of hexadiene with hexene.
10 grams of 135hexatriene 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.yr.,, 0.7326 jin,, = 1.4532
II. „ 78° — 79°.5. „ _ ^, = 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.(jr.,, 0.7146 nj)^, 1.4205
The fraction 75°— 80 gave nn,, 1.4351.
An elementary analysis of the fraction boiling at 72°. 5 — 74° gave
the following result :
Found Calculated for Cfi^„ Calculated for C„Hj^
C 87.06 87.7 85.6
H 13.32 12.3 14.4
The fraction investigated consists, therefore, probably of a mixture
of CgHj„ and C,,Hjj. 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 GHo^GHCHaCHaGH^GHa should be formed, this will not readily
absorb more hydrogen either.
8*
( 116 )
Mathematics. — "The force field of the nonEuclidean spaces
with negative curvature''\ Bj Mr. L. E. J. Brouwer. (Commu
nicated by Prof'. D. J. Korteweg).
A. The hyperbolic Sp^.
I. Let us suppose a rectangular system of coordinates to be placed
thus that ds = \/ A^ du^ \ B^ dv^ f C^ dw^, and let us assume a line
vector distribution X with components Xu, Xo, X^o, then the integral
of X along a closed curve is equal to that of the plani vector F over
an arbitrary surface bounded by it; here the components of Y are
determined by :
_ 1 \ d{x,B) a (x^ c) \
For, if we assume on the bounded surface curvilinear coordinates
è and tj, with respect to which the boundary is convex, the surface
integral is
r,^ /a. die dv dto\ f d {x„ B) a (x„, c)\
J 2^\M ' d^~a^ • dïJV~a^;; aT";'^*'^'^
Joining in this relation the terms containing X^ C and adding and
a {x,o c) dio aw
subtracting — ^^ . ^;^ . v we obtain :
ƒ
^^^ \h{X,oC) bw b{X^C) bw
di dy]
a?j a§ a§ a?/
Integrating this partially, the first term with respect to % the second
to %, we shall get iXwCdw along the boundary, giving with the
integrals j Xv B dv and  Xu A du 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. QQ — 78) ^) we call the planivector Y the
first derivative of X.
1) The method given there derived from tlic indicatrix of a convex boundary
that for the bounded space by frontposition of a point of tlie interior ; and the method
understood by the vector Xpqr... a vector with indicatrix opqr.... We can however
determine the indicatrix of the bounded space also by postposition of a point of the
interior with respect to the indicatrix of the boundary; and moreover assign to
tlie vector Xyqr... the indicatrix pqr...o. We then find :
( 117 )
Analogously we find quite simply as second derivative the scalar :
~ ABC 2^ ÖW
According to the usual way of expressing, the first derivative is
the rotation vector and the second tiie divergency.
II. If X is to be a ^_X, i. e. a second derivative of a pLanivector
i, we must have:
_ 1 iö(r.i?) ö(r,.Q)
BC \ öif ÖÜ (
and it is easy to see that for tljis is necessary and suflieient
III. If X is to be a o^, i e. a first derivative (gradient) of a
scalar distribution y, we must have :
X — — — X = — — X — ^^
.4ÖU ' ' Bdv Cdw
and it is easy to see, that to this end it will be necessary and
sufficient that
Y=0.
IV. It is easy to indicate 'comp. Schering, Göttinger Nachrichten,
1870) the (jX, of wjjich tlie divergency is an isolated scalai' value in
the origin.
It is directed according to the radius vector and is equal to :
1
sinh'^r
when we put the space constant = 1 ').
Ö...
V,+ 1='^'^P+1
^p+1
7 — 'V P~^ ^
q
g p n
These last definilions 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 nonEuclidean spaces.
) For another space constant we have but to substitute in the following formulae
r
R '°' '■•
( 118 )
It is the tirst derivative of a scalar distribution :
— 1 \ coth r,
and has in the origin an isolated divergency of 4:jr.
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
than  and in the direction perpendicular to the radius vector of
r
lower order than e"^ .
For a 0^ 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 Green holds for two scalar distributions (corap.
Fresdorf, diss. Göttingen, 1873):
r oil? r C ^^ r. C
tip — dO — I ^ y " ip . cZt =: \y^ — dO — I \p ^^(p . dx
j = I aS \grad. <p, grad. i^j dt. j .
If now (f and if? both vanish at infinity whilst at the same time
Urn. (p\l^ e'' ^ 0, then the surface integrals disappear, when we apply
the tiieorem of Green to a sphere with infinite radius and
i (f . s^"" \p . dr z=z j If? . y" ^ . cZt,
integrated over the whole space, is left.
Let us now take an arbitrary potential function for (p and
— \ { coth r for tf% where 7' represents the distance to a point P
taken arbitrarily, then these functions will satisfy the conditions of
vanishing at infinity and /m. (p if? e'' = 0, so that we find :
Ajt . (f p := I ( — 1  cof/i r) \y'^(p . dr.
So, if we put — 1 + (^oth r ^ Fiir), we have :
1.. _r\27ox
F,(r)dT (/)
VI. We now see that there is no vector distribution with the field
property, which has in finite nowhere rotation and nov/here diver
gency. For, such a vector distribution would have to have a potential,
having nowiierc rotation, but that potential would have to be every
where 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 spaceintegral of such
fields.
For such elementary fields we find thus analogously as in a Euclidean
space (1. c. p. 74 seq.) :
1. a field Ey^, of which the second derivative consists of two
equal and opposite scalar values, close to each other.
2. a field E^, 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 tlie fields E^ and E^ are here
again of the same identical structure.
VIII. To indicate the field E^ we take a system of spherical
coordinates and the double point in the origin along the axis of the
system. Then the field E.^ is the derivative of a potential:
cos (p
sinh^r
It can be regarded as the sum of two fictitious "fields of a
single agenspoint", formed as a derivative of a potential — 1  coth7\
which have however in reality still complementary agens at infinity.
IX. The field E^ of a small circular current lying in the equator
plane in the origin is outside the origin identical to the above
field E^. Every line of force however, is now a closed vector
circuit with a line integral of 4.t along itself. We shall find of this
field E^ a planivector potential, hing in the meridian plane and
independent of the azimuth.
In order to find this in a point P with a radius vector r and
spherical polar distance </? we have but to divide the total current
between the meridian plane of P and a following meridian plane
with difference of azimuth dO, 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 f, \ï dli
is the element of the parallel circle, 2 the above mentioned current
and H the vector potential under consideration, we find :
d^ ■=z dh . Xds sin r,
( 120 )
whilst the condition for H is:
d (Ildh) = dh ds X sin f.
So we have but to take — for H.
'dh
To find 2" we integrate the current of force within the meridian zone
through the spherical surface through y*. The force component perpen
coshr
dicular to that spherical surface is 2 cos tp —_ — — , therefore :
sink r
f
/coshr
2 cos (p — ; . siiihr d'p . sinhr sin (f dO = dd coth r . sin^ (p.
sinker
So:
2 ^ coshr
H ■=: — :=. — ; : = — sm (f.
dh sinhr sin (fdd 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 = , the volume product of these three vectors, i. e. the
sinh'^r
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 \\i{S^,S^) is a symmetric function of the two
vectors \m\ij of which we know that with integration oï 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 aS, 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 «j. So we can regard V^{S^,S^) as a force in the direction
of S^ by an element of current unity in the direction of S^.
With this we have found for the force of an element of current
with intensity unity in the origin in tlio direction of the axis of the
system of coordinates :
( 121 )
coshr
— ; — sin y,
smh^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 r only.
Let us call that scalar value U, and let us regard a small elemen
tar}^ 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 :
Ö , . d
— ^ I Usin (p sinh r dw] dr — —iUcos (p dr] dw.
or d(fi r ^ r
This must be equal to the current of force through the small
rectangle :
c^sli r
— 7— sin <p . sinh r d(p. dr,
sinh r
from which we derive the following differential equation of Ü
with respect to ?' :
d
* U — ^ \U sinli r\ z=z coth r,
or
the solution of which is:
U =z cosech r — \r sech^ ^ r { c . sech^ ^ r.
Let us take c = 0, we shall then find as vector potential V of
an element of current unity E:
cosech r — ^ r sech'' ^ r ^ F^ (?),
directed parallel to E.
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
E, or likewise the vector potential in the direction of E caused by
an elementary magnet with intensity unity in the direction of G.
. So, if we call of two vectors unity is and 7"^ the potential x (£"» i'),
the symmetric function /% {r, cos y, where r represents the distance
of the points of application of the two vectors and (p their angle
after parallel transference to a selfsame point of tlieir connecting line,
we know that this function x gives, by integration of e.g. E over
( 122 )
a closed curve e not onlj the negative energj of a magnetic scale
with intensity unity bounded by e in the üeld 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 yector V of an element of current,
that v^hen 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 2X i e. of a field of currents
is obtained as an integral of the vectors T^of the elements of current.
XII. We can now write that in an arbitrary point:
IX=^/ j^:il^F,{r)dx, {II)
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 :
The extinguishment of the scalar potential is greater than that of the
vector potential ; for, the former becomes at great distances of orderc^^'",
the latter of order re~''. Farther the latter proves not to decrease
continuously from go to 0, but at the outset it passes quickly
through to negative, it then reaches a negative maximum and
then according to an extinguishment re~^' it tends as a negative (i. c.
directed oppositely to the generating element of current) vector to zero.
XIII. The particularity found in Euclidean spaces, that
Fj (r) z=. F^ {r) =T  — , is founded upon this, that in Euclidean spaces
r
the operation of twice total derivation is found to be alike for scalar
distributions and vector distributions of any dimensions (I.e. p. 70).
Not so in nonEuclidean spaces; e. g. in the hyperbolic Sp^ we
find for the V of a scalar distribution u in an arbitrary point
( 123 )
when choosing that point as centre of' a system of Rie^iann normal
coordinates
(
i. e. a system such that as = —
^'" = (0^ + 5? + a
but as V^ of a vector distribution with components A^, Y and Z,
we tlnd for the .I'component X^i:
/ d'X d'X d'X^
^ V ÖA^ ^'f ^^''
The hyperbolic Sp^.
I. As first derivative Y of a vector distribution X we find a
planivector determined bj a scalar value:
1 j a(X,^) a(A,^)j
As second deri\ative Z we find the scalar :
1 j d(X,^) ^{X,A) ]
II. If X is to be a ^X, i. e. a second derivative of a planivector
with scalar value ip we must have :
Bdv AÖU
to which end is necessary and sufficient : ^= 0.
If
have
If A' is to be a qX, i.e. a first derivative of a scalar ^ we must
A^u ' '' Bbv
to which end is necessary and sufficient: F=3 0.
III. The oA', 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 dislribntion
I colli A r.
( 124 )
The divergency in the origin of this field is 2jr.
The scalar distribution lcoth\r 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 n 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 :
1 r \2/ qX
qX ■=. \i/ I AL I coth \r dr,
^ J 2:7r
lx=^^/J
\V oX
\L^F,{r)dr (J)
V. For the field E^ of an agens double point we find the gradient
of the potential:
cos ip
sink r
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 E^ 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 E^. For that field E^ we
find as scalar value of the planivector potential in a point P the total
current of force between P and the axis of the agens double point,
that is :
sin if) 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 coth r, the volume
product ^ 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 V.
Of tp 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 magnetic strip S^ S^ with intensity unity, in
( 125 )
other words the force in the direction of V l>y a couple of rotation
^^^,5^. So we can regard tf^ as the foi'ce 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^.
YII. Let us now find the scalar value U, function of /', which we
must assign to a planivector potential, tliat the "field of an element
of rotation unity" be its second derivative. We must have:
dU _
dr
U =z I cosech r.
And we find for an arbitrary 2^'
—^ I cosech r dr,
1^
^^FMdx (//)
And an arbitrary vector field X is the total derivative of the potential
VIII. We may now^ wonder that here in iSp, we do not find
F^ 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 ^S^^,. 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 >$/>, 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 S^n forces of order g— '", this furnishes in a spherical layer with
thickness dr and infinite radius described round the origin as centre an
energy of order e— 2' X «^"~"^^'' dr = e("— s)» dr ; which for n ]> 3 would
( 126 )
give when inteo;rated with respect to r an infinite energy at infinity of
Spn So for ?i ]> 3 are excklded by the field property only vector distri
butions which cannot have physical meaning.
For 7Ï = 2 however the postulation lacks its right of existence ;
more sense has the condition (equivalent for ?z > 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 V'
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)
m the Euclidean Si\ under given divergency distribution a minimum
energy ?
According to the theorem of Green we have for this :
/fbu\ r du ddu r d.du r
62 I ^— (Zr = I JS"  . :^ . dt =z I w . — . dO — I u w ^ du , dr,
\d.vj J dx dx J dv J
so that, as \7^öit is 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 ^i,
cos (fi
derived from a potential — ; — .
sink r
The lines of force of this field E^ have tlie equation,
sin (f coth r = c.
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 0potential.
( 127 )
The same holds for the arbitrary o^; of (he lines of force one
part goes to infinity ; the potential lines however are closed.
X. If we now have to lind the field with rotation only, giving
for given rotation distribution a niinininni enei'gy, 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
general 2^ is composed of fields E„, derived from a planivector
SZTl (£)
potential —_ (whilst we found under the postulation of the field
sink r
property sin (f coth r).
In contrast to higher hyperbolical spaces and to any Euclidean
and elliptic spaces the fields E^ and E^ cannot be summed up here
to a single isolated vector.
For this field E^ and likewise for the arbitrary 9 A' the lines of
force (at the same time planivector potential lines) are closed curves.
XI. We have now found
Jx = xv/
2jr
1
I coth I r dx,
1 r \ï7 2X
2X = \2/ I Az I coth \ r dr.
And from this ensues that also the general vector distribution X
having under given rotation and divergency a minimum energy is
equal to :
rwA . rwA
Xdiv. + Xrot. = \V I ~— I coth ^ r dx + \y I ^ I coth \ r dx.
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 Xdiv.', neither
(as according to § IX all lines of force of Xrot. are closed curves
and a flux of exclusively closed vector tubes has no reciprocal
energy with a gradient distribution) with Xrot. ; so that the energy
of Xdiv. + Xrot. + F is larger than that of Xdiv. + Xrot. •
So finally we have for the general vector distribution of minimum
energy X:
'J
Z^A
A = V I . I coth I r dx.
2n
( 128 )
C The hyperbolic Spn
I. Let us suppose a system of rectangular coordinates, so that
ds =1 \/A^U^^ \ . . . . AnUn^,
and let us suppose a linevector distribution X with components
Xi,,. Xn, 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 :
Y is the fi}\bi derivative or rotation of JT.
Further the starting vector current of X over a closed curved
Spn—\ is equal to the integral of the scalar Z over the bounded
volume of that Spn—i ; here
z — — — >
A^ An ^^ ^X^j.^
Z is the second derivative or divergency of X
II. If X is to be a oA", i.e. a second derivative of a planivector
S", we must have:
^=<i = I r~ 2^
x. =^ .  . 7. '^ ^
?1
The necessary and sufficient condition for this is:
If X is to be a o A, i. e. a first derivative of a scalar ff, we must
have:
X — ^^
The necessary and sufficient condition for this is:
r=o.
III. Tlie oX, which has as divergency an isolated scalar value in
tlie origin (comp. Opitz., Diss. Göttingen, 1881), is directed along
the radius vector, and if we put the space constant equal to 1 is
equal to
1
It is the first derivative of a scalar distribution
/;
( 129 )
dr
sinh^^~^ r
z= 10 u {')•),
and it has in the origin an isolated divergency of kn (if k^ /'"~' ex
presses the spherical surface of the Euclidean space Spn)
IV. For two scalar distributions <p and ip the theorem of Gheen
holds (conip. Opitz., I.e.):
I <p— . dOnl — I <ƒ W' • (^'^n = I tp— . dOn—l — jtp V^» • dt»
(=ƒ
S{\7 (f, V tf') . dXn
If at infinity <p and tp both become whilst at the same time
Urn (/)if>e("i)' = 0,
then for an "—'sphere with infinite radius the surface integrals dis
appear and we have left
i<p . V'tp . dtu = 1 1
integrated over the whole space.
If here we take an arbitrary potential function for <p and iVn (r)
for if?, where r represents the distance to an arbitrarily chosen
point P — these functions satisfying together the conditions of the
formula — we have :
^ri <Pp = I W'„ (r) . ^7" (fi . dtn.
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 iOn{ir)^F^{r), for an arbitrary o^:
IX^ST/ j)lL^F,{r)dr: (/)
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 E^, of which the second derivative consists of two
equal and opposite scalar values close to each other.
2. Fields E^, of which the first derivative consists of planivectors
distributed regularly in the points of a small "sphere and perpen
dicular to that "—^spijgpe.
9
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 130 )
At finite distance from their origin the fields E^ and E^ are of
identical structure.
VI. In order to indicate the field E^ we assume a spherical
system of coordinates ^) and the double point in the origin along
the first axis of the system. Then the field E^ is the derivative of
a potential :
cos (f
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 of a potential ?y„(7") to which, however,
must be assigned still complementary agens at infinity.
VII. The field E^ of a small vortex"— ^spjjgj.g according to the
space perpendicular to the axis of the double point just considered is
identical outside the origin to the field E^. Each line of force is now
however a closed vector tube with a line integral kn along itself.
We shall find for this field E^ a planivector potential //, lying in
the meridian plane and dependent only on r and (p. It appears then
simply that this H is sx iX.
Let s be an (n — 2)dimensional element in the ?z — 2 coordinates
existing besides r and (p, then it defines for each r and (p an element
on the surface of an "—^.^pj^gig Qf ^ gj^e dh = C8 sink "~ r sin ^~~g),
and for the entire SjJn what may be called a "meridian zone".
We then obtain for the current of force S, passing inside a
meridian zone between the axis of the system and a point P with
coordinates r and (p, if ds represents an arbitrary line element
through P in the meridian plane under an angle f with the direction
of force :
d^ z=: dh . X ds sin f,
whilst we can easily find as necessary and sufficient condition for H:
d (Hdh) = dh . ds . X sin f ;
SO we have but to take — for H.
dh
1) By this we understand in S'p» a system which with the aid of a rectangular
system of numbered axes determines a point by 1. r, its distance to the origin,
2. 0, the angle of the radius vector with Xi, 3. the angle of the projection of
the radius vector on the coordinate space X^ . . . X„ with X^, 4. the angle of
the projection of the last projection on the coordinate space X^ . . . X„ with X^ ;
etc. The plane through the Xjdirection and the radius vector we call the meri
dian plane.
( J31 )
To ÜJid 2" we integrate the current of force inside the meridian
zone passing through the "—^spherical surface through P between the
cosh r
axis of the system and 1\ As we ha\c (n — l)cos<p— for the
force component perpendicular to that spherical surface we find:
?
ƒ* cosh r ,
[ti — 1) cos <p —7 . sinh r cUp . c8 sink 't^r siti ^—^(p ■=. ce sm ^*^<fi coth r.
sink "r
2" cosh r
II z= ~— =z — si7i <p.
dh sinh "—V
VIII. If thus are given in different points a line vector L
unity and an "— ^^ector W unity and if we put along their con
cosh r
nectine: line a line vectoi' , then the volume product il? of Z, IF
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 L.
We know of if7(L, IF) that with integration of W along a closed
curved Si)n—2 Q it represents the current of force of a magnet unity
in the direction of L through Q, in other words the negative reci
procal energy of a magnet unity in the direction of L and a
magnetic "—'scale with intensity unity, bounded by Q, hi other words
the force in the direction of Z by a magnetic "~'scale bounded
by Q, in othei words the force in the direction of Z by a vortex
system, regularly distributed over Q and perpendicular to Q. So we
can regard xp {L, W) as the force in the direction of L by a \'ortex
nnity, perpendicular to W. With this we have found for the force
of a plane vortex with intensity unity in the origin :
cosh r
— : sin (f,
sinh "— 'r
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 (p is here tlie angle of
the radiusvector with the aS/>„— 2 perpendicular to the vortex elemejit.
IX. For the fictitious field of a vortex element in the origin intro
duced in this way (which meanwhile lias vorticity everywhere in space)
we shall find a planivector potential, directed everywhere "parallel"
to the vortex element and of which the scalar value Ü is a function
of r only.
Let us suppose a point to be determined by its azimuth parallel
9*
( 132 )
to the vortex element and then farther in the Sp^*~^ of constant azi
muth by a sj'stem 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 (p here the angle of the radius vector with the Spn—2,
perpendicular to the vortex element ; let further s be an {n — 3)dimen
sional element in the n — 3 last coordinates, then this defines for
each r and (p an element on the surface of an "— 3gp}^gj.g^ of a size
dk := ce sink n—^r cos ^^—^(p.
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 Spn—i 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 {n — 2)fold integral along the boundary
to a {n — l)fold integral over the volume according to the definition
of second derivative, we find :
d
{U cos (f . dr . CE sink "— ^r cos ^—^(p\ dtp —
d
dr
\U sin (p . sink r d(p . ce si7ih n—3^ (.Qg n— 3^ d^, ^^
cosh r
=z ce sink »— ^r cos "~3«) . sink r dip . dr . — 7— — sm (p.
sinh "— 'r
d U cosh r
(11 — 2) U sinh 1 in — 2) U cosh r =: —r ^ •
^ ' dr si?ih''^r
dU cosh r
h (n—2) tanh 1 r . £/ == — — .
dr smh^^r
The solution of this equation is:
I r 1
U = . cosh 2(n2)ir . coth «3i»' .dir^ — — 
2»— 3 J {n—2)sinh »^r
So we find as planivector potential V of a plane vortex:
1
^ I coth n3Lr .dhr^ F, (r),
{n—2) sinh "— 2^ 2«— ^ cosh
directed parallel to that plane vortex.
Let us now call E the "— 2yector, perpendicular to the plane vortex,
the field of which we have examined, and let us also set off" the
vector potential V as an "2vector ; 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 aS/;„_2 in a
Spn—\ perpendicular to G, it indicates the force in the direction of
G caused by the current element E, or also the vector potential
in the direction of E, caused bj an elementary magnet with
intensity unity in the direction of G.
Let us now call the potential x {E, F) of two "'vectors unity
E, F the symmetric function F^{r)cos(p, where r represents the
distance of the points of application of both vectors and (p 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. E is
integrated over a closed curved Spn—2 which we shall call 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 F of the vector potential caused by a
system of vortices about e with intensity unity. ,
From this ensues again for the vector potential F of a vortex
element, that when the vortex element is integrated to a system of
vortices about a closed curved Spni it becomes the vector potential
determined according to § VII of that vortex Spn2; so that the
vector potential of an arbitrary 2A" is obtained as integral of the
vectors V of its vortex elements, in other words :
2X=^2/j^^FAr)dr, (//)
where for each point the vector elements of the integral are first
brought over to that {)oint 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:
f^r^ ^1 (r) dx + f^^ F, (r) dr.
The extinguishment of the scalar potential is the stronger, as it is
at great distances of order ^t"')'", the vector potential only of
order re'»^^".
( 134)
Astronomy. — ''The luminosity of stars of different types oj
spectrum." Bj Dr. A. Pannekoek. (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. Vogel'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 Pickering
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
VoGEi, that die 2"^^ type is subdivided. The natural groups that can
be distinguished are: class A: the great majority of the white stars
(Sirius type), Vogel's la; class B: the smaller number of those stars
distinguished by the lines of helium, called Orion stars, Vogel's lb.
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 (Procyoii) ; 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^^ fjp^' which ap
pi'oach to the 3^^ type, such as Arcturus (Pickering reckons among
them the H and I as indistinct representatives). The 3^ 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 2^ type, XIII— XIV the 2^ 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 III a group of Hues is gradually falling out, namely the hydrogen
lines of the other series, which are characteristic of the WolfRayet
( 135 )
stars or the socalled fifth type stars (Vogel II/>), it is obvious that
we must place these stars at the head of the series, as it has also
been done by Miss CannoiN in her investigation of the southern
spectra (H. C. 0. Ann. Bd. 28) ').
Some of these stars show a relative intensity of the metallic
lines ditFerent from that of the ordinary stellar spectra ; Vogel 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 IIIc — XIII c, in contradistinction to
which the great majority are called a stars.
According to the most widely spread oi)inion 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 maximurh 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 ;
wheth^r 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 Campbell's results (Astronomy and Astrophysics XIII, p. 448),
the characteristic lines of the WolfRayet 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/3' 5414, Hy' 4542,
Uy .4201, principal line 4G8G) ; it is that group wliich in Maury's classes I— III
occurs as dark lines and vanishes and which in the classes towards the other
side (class Oe — Ob Gannon) 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, 5G92, 5594, 5470, 4654,
4443. The brightest hand is 4654; its relative intensity as compared willi the
H Hne 4689 gradually increases in the series: 4, 47, 5, 48, 42 (Campbell's
star numbers).
( 136 )
observed. The hypothesis maj' 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 Bradleystars 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 1^"^ 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 lightgiving 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
Ejnar Hertzsprung : Zur Strahlung der Sterne^), where Maury's
classification of the spectra has been folio w^ed. 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 pi'()})cr molion
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, Ihougli this star properly belongs to the sun and the G stars.
•5) Zcitsclirift fiir wissenscliaftliclic Pbotograpbie Bd. III. S. 429.
( 137 )
Spectrum
Magn. for
P. M. for
Maury
Draper C.
P. M. 0"01
Magn. 4.0
IIIV
B
4.37
0.012
VVI
B— A
7.25
0.045
VII— VIII
A
8.05
0.005
IXXI
F
9.0G
0.103
XIIXIII
F— G
11.23
0.279
xiiixiy')
G
7 93
O.OGi
XV
K
9.38
0.119
J^— XVI
K— M
7,77
0.057
XVIIXVIII
M
8.28
0.072
XIII that form the transition from F to G ; for the hiter 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 W 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 auxiliary
quantities for all the Bradleystars. Let t and v be the components of
the proper motion at right angles with and in the direction of the
antapex, X the spherical distance of the starapex, then
2 V sin X
2 sm* X
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
1
of the other component — ^ t is, at a random distribution of the
71
directions, equal to half the mean linear velocitj^ 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
^) 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
p= 1002 ('«4)
we have
2 pv sin X S pt
q4.o = ^ ■ , , and T4.0 = •
^ sm / n
In this computation we have used Maury'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 (/? y de S) we have taken
only one star (e). In the following table are combined the results
of the two computations.
Spectrum
Typical
mean
mean
Maury
Dr Cat.
star
?i
j«
T
7
^4.0
^4.0
II
II
II
II
I— III
B
£ Orionis
33
3,57
0.007
0.018
0.007
0.013*
IV— V
BA
y 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
O.OGl
IX— XII
F
Procynn
94
4.14
0.089
0.1.53
0.095
0.130
XIII— XIV
G
Capella
09
4.08
0.141
0.1.57
O.IGO
0.199
XV
K
Arcturus
101
3.90
0.123
0.119
120
0.09G
XVI— XX
M
BeteJgeuze
01
3.85
0.049
0.0G8
0.050
O.OGl
In both the series of results the phenomenon found by Monck and
Hertzsprung manifests itself clearly. I have not, however, used
these numbers T4.0 and ^40, but have modified them first, because it
was not until the computation was completed that I became ac
quainted with Hertzsprung's remark that the above mentioned c stars
show a very special behaviour ; their proper motions and parallaxes
arc 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
^ I In his list of parallaxes Hertzsprung puts the question whether perhaps the
bright southern star a Carinae (Canopus) belongs to the c 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 Gannon has paid no regard
( 139 )
Cluss
n
""4.0
^4.0
2t/^
I
5
0.009
0.022
0.8
II
13
005
009
1.1
III
'14
000
015
0.8
IV
18
014
023
1.2
ir
10
010
044
0.7
V
11
00!)
042
0.4
VI
IG
030
008
0.9
VII
30
040
080
9
VIII
41
043
055
1.0
IX
25
050
004
1.0
X
10
070
171
0.8
XI
22
103
001
3.3
XII
23
170
282
1.2
XIII
18
297
340
1.7
XIV
21
192
305
1.3
xiy
20
077
025
0.2
XVyl
20
234
148
3.2
~ XV li
35
105
070
3.0
XY C
40
059
087
1 4
XVI
19
049
071
14
XVII
19
049
032
3.1
XVIII
10
050
075
1.3
XIXXX
7
057
078
1.5
to the difference between tlie a and tlie c stars. Yet all llic same this question
may be answered in the aftirnialive; on both spectrograms of tins star occur
ring in her work, we see very distinctly the line 4053.8, which in Capella and
Sirius is absent and wliich is a typical line for the c stars. Hence follows that
« Carinae is indeed a c 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 Maury
separately ; class XV is divided into three subdivisions : XV A are
those whose spectra agree with that of a Boötis, XV C are those which
agree with the redder « Cassiopeiae, while XV ^ embraces all those
that cannot with certainty be classed among one of the other two
groups.
The values for T4.0 and g'4.0 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 ^''s divided by 4.2
yield the mean parallax of stars of different spectral classes for the
magnitude 4.0 (.T0.4). Reversely, we derive from the q'^ 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 at a distance for which g = 0". 10,'
hence with the parallax 0".024. Finally the last column 2t/5' contains
the relation between the mean linear velocities of tlie group of stars
and our solar system.
In the following table we have combined these values in the same
way as before.
Spectrum  Typical
Maury I Dr. Cat. star
4.0
74.0
4.0
L for
7=0". 10
Irjg
I— III
IV— V
VI— VIII
IX XII
XIII— XIV
XV
XVI— XX
B
BA
A
F
G
K
£ Orionis
32
V Orionis
45
Sirius
87
Procyon
80
Capi Ua
59
A returns
101
Betelgeuze
01
0.005'
0.013
0.040
0.101
182
0.120
0.050
014
0.03G
0.0G3
0.141
. 224
09G
061
0.0033
51
0.0080
7.7
0,015
2.5
0.034
0.50
0.053
0.20
023
1.1
0.015
2.7
8
0.7
1.3
1.4
l.G
2.5
l.G
^ 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 readied.
(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 Campbei.l found 19.9 kilometres for the
velocity of the solar motion, and 34 kilometres for the mean velocity
of all the stars, Frost and Adams dcMivcd 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 2xlq. Hence the Orion stars are the particularly slow ones and
the Arcturian stars (class XV) are those which move witii the greatest
speed.
§ 5. When we look at the values of g'4.0 or those of jtj.o or
//o.io, derived from ttiem, 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 q was larger for
the 2'^ type as a wiiole than for the tirst (the Orion stars included)
has long been known ; some time ago Kapteyn derived from
the entire Bradley Draper material that on an average the 2"^ type
stars (F G K) are 2,7 times as near and hence 7 times as faint as
the l^t type stars (A and B). This result perfectly agrees with the
ordinary theory of evolution according to which the 2'^' type arises
from the 1*^ 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 WolfRayet stars follow next
to class I, an investigation of their proper motion, promised by
Kapteyn, will be of special interest.
Past the G stars, the solar type of the series, the brightness again
increases. The values obtained here for q confirm in this respect the
results of MoNCK and Hertzsprung.
1) Publications Yerkes Observatory. Vol. II. p. 105.
(142 )
Against the evidence of the q^ 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 q 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 .'T4.0':
Derived from q jtq.o
Observed
^ ^0.0
II—]
IV
0"
.0255
(6)
IV—
VI
.106
(5)
VII
VIII
.153
(10)
IX—
XI
.226^:
1 (6)
XII
XIII
.442
(2)
XIV
.567
(5)
XV
.151
(8)
XVI
.171
(3)
XVII— XVIII
.115
(3)
I— III
0".021
IV— V
.054
VI— VIII
.094
IX— XII
.21
XIII— XIV .33
XV .14
XVI— XX .096
In general Hertzsprung'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
also 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. Monck 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
( lt3 )
ing light is less, the cm^Iolu' of the I'adiated light will be redder.
Therefore it is bejond doubt that a redder colour corresponds at
any rate with a less degree of railiance per unit of surface.
Then only one explanation remains: tke K and M stars {the redder
2"^ type stars like Arcturus and the d''^ iyp^i) 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 yellowwhite 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 tliat the K and M stars
have much larger masses than the F's and (rs. This result is the
more remarkable in connection with the coiuiusion derived above
about their greater mean velocity. If the stars of our stellar system
form a group in the sene that theii' velocities withiti the group
depend on their mutual attraetiou, \ve may expect that on an
average the velocities will be the greater as the masses are smaller.
No diftleulty from this arios for the Orion stars with small speed,
because the same circumsttinces 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 \elocity are characterized by this
thesis as belonging to a separate group, wliicii through whatever
reason must originally have been endowed \vith 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 in\ estigate separately the systematic motions of the K stars
which hitherto have been classed without distinction with the F and
G stars as 2"'^ 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 is 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 tlie 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 Gr — 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 Maunder has drawn
attention to several circumstances, which indicate that the spectral
type rather marks a difference in constitution than difference in the
stage 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 c stars, which, according to Hertzsprung,
have also a much greater luminosity, hence either less density or
greater mass than the similar a stars ; and the more so as these c stars
reach no further than class XIII. Yet to us this seems improbable;
the K stars are numerous, they constitute 20 7o of all the stars,
while the c stars 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 a stars 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 WolfRayet stars and also with the
4^'' type of SECcm (Vogel's 1116), 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 ns anything about the masses of
the stars themselves as will appear from the following consideration
(also occurring in "The Stars" by Newcomb). Let us suppose that a
binary system is 7^ 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 7i' to 1, the major axis of the orbit a in the proportion
of n to 1 and hence the time of revolution remains the same ;
the luminosity becomes ii^ 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 a be the angular semimajor axis, M the mass, P
the time of revolution, d' the density, ). the radiating power, ti the
parallax and q the radius of the spherical volume of the star, then
we shall have: rrM/^ — : the mass M is a constant value X o'd,
the apparent brightness H is sx constant X ^^Q^^» Eliminating from
this the parallax and the radius, we find
a' d^
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
. ■;. \U a'
cited before Maunder gives values for the density tf = c
HJ P*
in the supposition of equal values of ). ; he found for the Sirius stars
(l«t type) 0,0211, for the solar stars (all of the 2"^ 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 X^I(P is 200 times as large for the Sirius stars
as for the solar stars.
In a different form the same calculation has been made by
Hertzsprung by means of Aitken's list of binary system elements ^).
By means of — 2,5 log II—= 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 II { 4 log P — 6 log a = const. } 3 log X — 2 log ó
m — ^"/j log P  ^ log a z=z m^
then we have iiir = const. — 2,5 log X { Ys log ö.
If we arrange the values of vir after the spectra according to the
Draper Catalogue (for the Southern stars taking CannoxN; according
to the brightest component a 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 „ — 3.6J „ +0.14)
„ GandE — 0.49(11 „ —1.60 ,,+1.28)
The 3 stars of the type K (with H) give — 4.88 (yLeonis),
— 1.05 and +0,87, hence diifer so widely that no valuable result is
to be derived from them. To the extraordinarily high value for
X'^/ö^ given by y Leonis attention has repeatedly been drawn.
While for a great number of stars of the other classes the extreme
values of iiir differ by 3.5 magnitudes we fmd 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 timq^ as small as for these other stars. For the classes
A and F we fmd that X^/ö^ 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 asi?ii and F directly yield M sin^i; 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 ahiH 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 /? 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 '* ,r /^' . «.
P» (1+/?)'
It is not perfectly certain, of course, that on an average /:? is the
same for all classes of spectrum; if this is not the case the M'^
a' sin ^i
may behave somewhat different from the values of — =^^ — computed
here.
( 1^" )
Unfortunately, of tlie great number of spectroscopic double stars
discovered as yet (in Lick Observatory Bulletin N'. 79 a number
of M7 is given) the orbit elements of only very few are known.
They give, arranged according to their spectra:
Group II
IV (B)
Group VI
VIII (A)
Orion
type
Sirius
type
o Persei
0.61
^ Aurigae
0.56
7j Orionis
2.51
5 Ursae
(3.41)^)
Ö Orionis
0.60
Algol
0.72
/? Lyrae
7.85
a Androni.
0.36')
« Virgin is
0.33
«2 Gemin.
0.002
V Puppis
34.2
Group XII— XIV a 'F— G)
Solar type
« Aurigae 0.185
/Draconis 0.120
(W Sagittarii 0.005)
(X Sagittarii 0.001)
I Pegasi 0.117
^l Pegasi 0.234
Group XII— XIV ac
«Ursae min. 0.00001
SGeminorum 0.0023
1] Aquilae 0.0029
(f Cephei 0.0031
Group XV (K)
i5 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 solai stars. Very striking, however,
is the small mass of the c stars approaching towards a. Hence the
c stars combine a very great luminosity icitli a very small mass, and
consequently their density must he excessively small. If it should be
not merely accidental that the three regularly variable stars of short
period, occurring in Maury, 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.
•) Assumed period 100 days, velocity in orbit 32..Ü kilometres.
( 1^8 )
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 c 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 u Ursa e 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 3'^ 23^ l^™ makes me think that it has not been wholly an illusion.
ERRATA.
In the Proceedings of the Meeting of June, 1905, p. 81 :
line 7 from top, read: "cooled by conduction of heat",
16 „ „ for: ''Exh' PI. IV" read: ''Exit' PI. VI".
In Plate V belonging to Communication W. 83 from the physical
laboratory at Leiden, Proceedings of the Meeting of February 1903,
p. 502, the vacuum glass B\ has been drawn 18 cm. too long.
(August 21, 1906).
KONINKLIJKE AKADEMIE VAN WETENSCIlArPEN
TE AMSTERDAM.
PROCEEDINGS OF THE MEETING
of Saturday September 29, 1906.
(Translated from : Verslag van de gewone vergadering der Wis en Natuurkundige
Afdeeling van Zaterdag 29 September 1906, Dl. XV).
oo2vra:E3vrT's.
M. NiEuwENiiuisvox Uexküll Güldenbaxü: "Oil the harmful consequences of the secretion
of sugar with some myrmecophilous plants". (Communicated by Frof. J. W. Moll), p. 150.
II. Kameklingh Oxkes: "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. "Cryoetat especially for temperatures from —252° to —259°", p. 17?. (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. Kamerlixgh Onnes and C. A. Crommelin. "On the measurement of very low temperatures
IX. Comparison of a tliermo element constantinsteel with the hydrogen thermometer", p. 180.
H. Kamerlingii Onnes and J. Clay : "On the measurement of very low temperatures X.
Coefficient of expansion of Jena glass and of platinum between 1G° 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 Vries : "Quadratic complexes of revolution", p 217.
J. K. A. Wertheim Salomoxson: "A few remarks concerning the method of the true and
false cases". (Communicated by Prof. C. Winkler), 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).
11
Proceedings Royal Acad. Amsterdam. Vol. IX.
(150)
Botany. — "On the harmful consequences of the secretion of sugar
lüith some myrmecophilous plants." By Mrs. M. Nieuwenhuis
voN ÜEXKÜLLGüLDENBAND. Ph. D. (Comiiiunicated by Prof.
J. W. Moll).
(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 sugarsecreting myrmecophilous plants. The results
of these observations, extending over some 70 plants, are inconsistent
with the opinion expressed by Delpino, 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,
( i51 )
thick and darker eolourcMi points. On the inflorescences two kinds
of ants always abound, one large and one small species. Even when
the flowerbuds 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 belljar; after a few hours by means of Feeling'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 Delpino 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 s[)ecies remains on the flowers and is
content with the sugar there secreted, the big species also descends
to the basalleaves 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 louLier arnl the leaves, wdiich were produced in
this period, remained consequently ujiinjured. 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
uimamed tree, a Malvacea from IndoChina. This not only has
nectaries on the leaves and calyx, but also offers the ants a very
suitable dwellingplace in the stipules, which occur in pairs and are
bent towards each otlier. 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.
11*
( 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 L. 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 b}^ 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 grandijiora Roxb., Gmelina asiatica L., and Gmelina
bracteata, Nycticalos macrosj/phon 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 socalled
"foodbodies" (BuRCK'sche Körperchen) on the calyx of Thunbergia
grandijiora, it appeared to me that these are no "foodbodies" at
all, but ordinary sugarsecreting deformed hairs which I also found
on the bracts, leaves and leafstalks of this plant.
Further 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 Burck. It is much more dependent
on external factors, as e. g. the more or less free situation of the
plants, the weather etc.
As an example the creeper Bignonia Cliamberlaynii may be men
tioned. Of this plant on many days only 1,6 "/o of the fallen flowers
appeared not to have been bored by Xylocopa coeridea, 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 caljx
and the leaves. With Faradaya iKipuana Scheff., which stands in
the Botanical Garden at Buitenzorg surrounded by many other richly
flowering plants, the flowers are often perforated by a boring wasp ;
of the fallen flowers only 1 7o was undamaged. This was different
with another still unnamed species of the same genus which, as far
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 7o 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 Ipomoea carnea Jacq., a shrub having
nectaries as well on the leaves as on the calyx, the latter being
bored by Vespa analls and two Xylocopas. Collected in the morning
without regard to the weather of the preceding day 907o of the
fallen flowers were bored; after rainy days 57 7o of the flowers
were damaged and after sunny days even 99,1 7o 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 mutual 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 answei'S were obtained :
1. On what parts of plants is extrafloral secretion of sugar found ?
In the cases examined by me I 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 leafsheaths and calyx together, b. on the leafblade only
c. on the leafstalks, 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 Gmellna asiatka 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 cui)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
Beau v.).
With many species of Smilax 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 oï Nycticalos inacrosyplion, Spathodea
serridata and others. Attracting ants to the entrance of the corolla,
which is the very place where the animals causing crossfertilisation
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 Ule ^) has drawn attention to this as a result of
his investigation of American plants.
3. Is sugar secreted in all 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 belljar; this
w^as the case e.g. with the leaves of Gmelina aslatica. Consequently
they are not frequented by ants, although these insects always occur
on thé 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 pi'oducts 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, without a single animal visiting
them. (So with some species of Fassijlora).
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 Lujj'a species one may
see the ants at the nectaries peacefully busy by the side of a 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 socalled "ant
plants" of the Indian archipelago, seem to belong to the harmless
ones; the dangerous species with powerful mouthapparatus, e.g.
those which are called sennit ranggrang in West Java and according
to Dr. VoRDERMAN 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) Englers Bot. Jahrbücher. Heft III, 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. Hoio to obtain baths of constant and
uniform temperature by means of liquid hydrogen." By Prof.
H. Kamerlingh Onnes. Communicalion 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 tiien 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 liquid 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 iiad
therefore to avail ourselves of cooling by adiabatic expansion.
Tn Comm. N°. 23 of Jan. '96 1 made some remarks on what could
be derived from van der Waals' 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 farther frigorific agents. My efforts,
however, to obtain an apparatus for isentropic cooling by combining
to a regenerator the outlet and inflowtubes of a small expansion
motor, fed with compressed gas, had failed. Therefore I directed
my attention towards the then newly published (1896) application
of the JouleKelvin process (Linde's apparatus for liquefying air
and Dewar's jet of hydrogen to solidify oxygen).
Though the process of Linde was the most promising, because he
had succeeded with his apparatus to obtain licpiid air statically, yet it
was evident that only the principle of this method could be foliow^ed.
(157)
The cooling of an apparatus of dimensions like the first of Linde
(weight J300 kilogrammes) by means of li(ui(i 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 mentioned 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, Olszewski 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.
JMoreover 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 ciyostat (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 higiier than
they had formerly been for the bath of liquid oxygen. These require
ments could by no means be fulfilled before I 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 comjiressors 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 T 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. Flim, 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 expansioncock, where the pressures
are regulated according to the temperatures. Compare the theory of cooling with
the JouleKelvin process and the liquefying by means of the Linde process given
by van der Waals in the meeting of Jan. 1900.
( 159 )
The latter is represented scliemaiically by fig. 1 on PL I and
hardly requires fuitlier explanation. The compressed hydrogen goes
successively tlirongh the regenerator coils D^, D^, D^, D.^, C, B, A.
B is immersed partially in a bath of liquid air which, being admitted
through P, evaporates at a very low pressure; D^, D^, 6' and A
are surrounded by hydrogen expanding at the cock M, and D^ and D^
by the \'apours from the airbath in F. As, however, we can dispose
of more liquid air than we want for a suiHicient cooling of the admitted
hydrogen, and the \acuum j)ump (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^, wq have for simplicity
not yet added the double forecooling regenerator jD, 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 F with cooling spiral B and the principal regenerator
A in the vacuum glass E with a collecting vessel L, placed in
the case V, which forms one complete whole with the case U.
b. The principal regenerator, PI. 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 j\I.). As in the ethylene regenerator (Comm. N". 14, Dec. '94, and
description of Mathias ^), fig. IF) and in the methyl chloride regenerator
(Comm. N°. 87, ^larch '04, PL I) the windings are w^ound from the
centre of the cylinder to the circumference and again from the circum
ference to the centre round the cockcarrying tube j\J^, and are enve
loped together in fiannel and fit the vacuum glass F^ (the inner
and outer walls are marked with F^^ and F^J. Thence the liquid
hydrogen flows at F^^ into the collecting vessel Lq. At M^^ the
four coils are united to one channel which (comp. cock T in fig. 3
of Mathias' description I.e.) is shut by the pivot point J/jj moved
by the handle j\l^^. The packing 31^ hermetically closes the tube
M^ at the top, where it is not exposed to cooling (comp. Mathias'
description I.e.). The hydrogen escapes at the side exactly as at
the ethylene cock L, fig. 2 in Mathias' description I.e., through 6
openings 31^^ and is prevented from rising or circulating by the
screens M^^ and J/^g.
c. The newsilver refrigerator case F^ is suspended in the new
1) When using oxygen we might avail ourselves of cooling clown to a lower
temperature, which then must be carried out in two steps (comp. § 46).
) Le laboratoire cryogène de Leyde, Rev. Gen. d. Sc. Avril 1896.
( 160 )
silver case U^, from which it is insulated by flannel t/g^. 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 newsilver one P^.
The evaporated air is drawn off through a stout copper tube P,
(comp. § 4è). The 2 outlet tubes B,^ and P,,, of the spiral Pjj and P,,,
(each 23 windings, internal diameler 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 P^ and P,„ are soldered in the newsilver cover,
on which the glass tube F^ covering the index F^^ of the cork
float Pgi are fastened with sealing wax (comp. for nitrogen Comm.
N". 83 IV, March '03, PI. VII).
d. The forecooling regenerator spiral Cj, C^, C^, and C^ is
wound in 4 windings like A, wrapped in flannel and enclosed in the
cylinder of the newsilver case U^. The four windings (internal diam.
of the tubing 2.4 m.m., external diam, 3.8 m.m., number of laj'ers 81,
length of each tube 20 M.) branch oif 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 thinwalled newsilver tube Cj
shut at the top.
The hydrogen blown off is expelled through the tube U^.
e. The liquid hydrogen is collected in a newsilver reservoir L^,
fitting the vacuum glass L^, which by means of a little wooden block
Fy rests on the woodcovered bottom of the insulated case F^, which
is coated internally with paper F^^ and capoc Fja Thanks to L^
the danger of bursting for the vacuum glass is less than when the
hydrogen should flow directly from Pj into the glass Po a 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 Pjoo)
which by means of a silk cord L^.^, slung over the pulleys P^,
and Pjj is balanced by an iron weight P^^, moving in a glass
tube Fji, which can also be pulled up and down with a magnet
from outside. The float is a box L^^ of very thin newsilver, the
hook Pjoj 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 Fj,, which is connected with the case by means of a thin
walled newsilver tube Fj^.
The hydrogen is drawn off through the newsilver siphon tube JVji,
which is continued as the doublewalled tube N^^ ^boi» leading
( 161 )
towards the delivery cock JSf^i. Here, as at the ethylene cock
(description of Mathias 1. c. fig. 2), the packing iVj and the screw
thread are in the portion that is not cooled. The pin ^'^j, made of a new
silver tube, passes through the cockcarrying tube iV^. Both the outlet
tube iVo and the delivery cock jV^ 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 iYgj^ and along Kha
{Kd on PL II). The outer wall jVjoi, ^^503 of the doublewalled tube is
insulated from the side tube T^ji at the case F30 by means of wool.
The glass L is covered with a felt cover Zg, fitted at the bottom
with a sheet of nickelpaper to prevent radiation towards the liquid
hydrogen. This cover fits tightly on the lower end E^ of E and
rests on the tube ,^501 ^"d the pulleycase L^^.
f. We still have to describe the various safety arrangements to
prevent the apparatus from bursting when the cock M 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
TFi, 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 TF30 with several
screens W^^ all of varnished cardboard to collect the mercury and
to reconduct it into the glass IF, (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 thinwalled india rubber tube V^^, which is
drawn over the perforated brass cylinder wall V^^ (separated from
it by a thin sheet of tissuepaper), breaks. The safety apparatus is
connected with the case V^ by a wide newsilver tube F^^.
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 F431 and V^^^ after being exhausted is filled through
the cock V^^ with hydrogen under excess of pressure; during the
exhaust the india rubber cylinder F^,, is pressed against the india
rubber wall F^,.
An arrangement of an entirely identical construction protects the
case C/j, 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 suflicient to protect the refri
( 162 )
gerator space F by means of the tube Y opening below mercmy.
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 boiliiig 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 F, 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 L 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 doublewalled
siphon tube N^^, N^^^ and the double walled cock N^, iV^soi
The newsilver case V, from which the vacuum glass L is insulated
by laj^ers of paper V^^ and the refrigerator vessel F by a layer of
flannel, and in the same way the newsilver case f7, are further pro
tected from conduction of heat from outside by separate wrappings
of capoc T^gi, packed within a cardboard cover Y^^ 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 b).
The airtight connection between the case U and the case Y is
effected by the India rubber ring Ua, which fits on the glass and
on the strengthened rims JJ^^ and Fso of the newsilver cases. India
rubber of somewhat larger dimensions can only be used for tightening
purposes when it is not cooled. In this case the conduction along the
newsilver wall, which is insulated from the vacuum glass by layers
of paper, is so slight that the ringshaped 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
Uc, which fits tightly on U^ and Y^ and which is filled with hydrogen
under excess of pressure tiirough the cock Ud. Thanks to the small
conduction of heat of newsilver no cooling is to be feared for the
connections of Y^^ and U^^ no more than for the packings of the
cocks il/3 and N^.
h. The cases Y and U are joined and form one firm whole by
the three rods Uh with 1 he screwfastenings f/g( and F^^. The vacuum
glass E^, held by the India rubber ring Ua, rests with a wooden
ring E^ and a newsilver cylinder f/,i against the refrigerator vessel F.
( 163 )
The whole construction can stand exhaustion, which is necessary
to fill theapparatus with pure h J drogen. After the case Z7, of which the
parts Z7i and U^ are connected together by beams, and the case V
are mounted separately, the vacuum glass E 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 F^ for air and the
outlet tube ^5 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 Si q. The compressed hydro
gen is admitted through Kc, the hydrogen blown off is let out
through Khd or Khc.
i. Before the apparatus is set w^orking it is filled with pure
hydrogen (the cock M being open) by means of exhaustion and
admission of pure hydrogen along Kc. In the drying tubes 2)^ 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 foi'
operations with pure gas (comp. Comm. N". 14 of Dec. 'Gl, § 10,
and N". 51, Sept. '99, § 3), I have used for the hydrogen circulation
slowiy running compressors (see PI. II Q) at 110 and p 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 Hows 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 ' 1 ich are required
wiien 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 (© PI. II, displacing 20 M' per hour) the
gas is raised in the first cylinder (doubleacting with slide) from
1 to 5 and in the second cylinder (plunger and valves) from 5 to 25
atmospheres; in the second compressor Sp (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 tw^o first cooling spirals (those of © PL II) an oil
separator is connected.
Safetyvalves lead from each reservoir back to the delivery;
moreover the packings are shut ofl* with oilholders (Comm. N". 14
'94 and N". 83, PL VIII). The hydrogen that might escape from
the packing at Sp is collected.
b. The high pressure compressor forces the hydrogen through two
steel drying tubes ^a and T}b filled with pieces of sodium hydroxide
(comp. § 2, i, 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. c) 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 sack 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 tlie compression. Therefore we have used
for this purpose two zinced gasometers, Gaz a and Gaz b, PL II, with
tinned welds (holding each 1 M.') floating upon oil ^), which formerly
(comp. Comm. N". 14, Dec. '94) have been arranged for collecting
ethylene ^).
The cock Kpa {Kph) 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.
~) 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 Kph [Kpi)
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 oilgasholders 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. II at "^Iha. When we wish to
lique'fy hydrogen, this is blown off into the gasometer through /{^ (/iTA^,
Kpe and Kpb for instance to Gaz b), 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 Sp is repumped along Kpf and Kpc through Ka and Kf
into the reservoirs ^Xha.
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 liquid air.
a. The liquid air is sucked into the refrigerator vessel F (PI. I),
which by A"^ (PI. II) is coupled to the vacuumpump 5, along the
tube Pb connected with the siphon of a vacuum bottle 21a 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
k (fig. 1) with 3 tubes; one of these d is designed to raise the
pressure in the bottle with a small handpump, the other c is connected
to a small mercury manometer, and the third b 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 c 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. Amslerdam. Vol. IX.
( 166 )
FiR. 1
to conduct liquid air from a larger stock into the bottle. With the
cap a closed glass tube h 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 Aj, as shown in fig. 2, with the three tubes and with
a double wall li^ of very thin newsilver 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 h^ and the
whole piece rests on the neck of the bottle by means of a wooden
block i. After it is placed on the bottle the cap is wrapped round
with wool.
With a view to ihe transport the vacuum glass is placed in a
cardboard box with fibre packing.
When the siphon is not used it is closed witii 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 Pb (PI. I)
is connected with the siphontube h (fig. 2) with a piece of india
rubber tubing. To prevent breaking of the india rubber, which through
the cold has become brittle, the newsilver tubes are arranged so
that thej 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, PI. 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
mercnry manometer tube Y.
b. The air is caused to evaporate at a pressure of 15 mm., which
is possible because a BuRCKHARDTWEisspump % 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 S 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 ?5, displacing 360 M' per
hour, is exhausted by a small vacuumpump, displacing 20 M' per hour^)
(indicated by ^ on PL II).
§ 5. How the Viquefactor is set ivorking.
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 Q) and ^
PL II along Kc, is caused to stream through during some time
with wide open cock il/, PL I, for the forecooling of the whole
apparatus. Then the cock M 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 Kha PL I fig. 2 [Kd PL II)
for the forecooling of the siphon N^^ PL I and the cock N,
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 evaporates under low pressure, for instance towards j?.
2) V. D. Waals 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 L rime is seen on the tube
jV,„„ PL I, fig. 2 near the cock N.
b. The gaseous hydrogen escapes along Klid (PI. II) 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 Kpa
and Kijb. New pure hydrogen is then admitted from ^ha, PI. II,
along Kg.
c. The float (Xjoo P^ I) does not begin to indicate until a fairly
large quantity of liquid hydrogen is collected.
^ 6. The sip/wnmg of liquid hydrogen and the demonstration of
liquid and solid hydrogen.
a. When the float L^^^, PI. I, 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 b
etc., PI. 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 one
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. PI. 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 topelevation.
The evaporated hydrogen escapes along d\ and d'\ and further along
Ko (see PI. II) to the gasholder. The letters of the figures have the
same meaning as in fig. 2 ; for the explanation I refer to the de
scription of that figure in § 4.
The conduction of heat in the thin newsilver is so little that
the newsilver tubes can be soldered in the caps h^ and that they
are sufficiently protected by a double wall /i^ of newsilver with
a layer of capoc between, which is again thickly enveloped in
wool.
It has occurred that the India rubber ring k' 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 hj' drogen flowing from
( 169 )
the cock jSF, PI. I, we connect
eu.
Fi?. 3
with the tube A^o and the India
rubber tube J,, instead of the
silvered tlasks of PL II and
PI. Ill, a transparent vacuum
cylinder fig. 3a, closed by an
India rubber ring with a new
silver cap witii inlet tube. After
the cock is opened the India
rubber outflow tube c/j 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. 36, a glass cover
is placed on the top, the glass
may be left standing in the
open air without the air con
densing into it, which would hasten the evaporation. In the same
manner I have sometimes filled nonsilvered vacuum flasks holding
1 liter, where the liquid hydrogen boils vividly just as in the glass
mentioned before. The evaporation is of course much less and the
rising of the bubbles stops when the vacuum glass or the vacuum
flask is placed in liquid air.
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. 4). 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 HjO 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. Tlirough 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 bo taken
away from the pouring off cap.
( 170 )
a. 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 ViJ, there they collect to a
white pulver which, when the hj'drogen is shaken, behaves as heavy
sand would behave in water. When the 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 airjiump. A starchlike
white cake is soon formed, which can be moved np and down
with the wire.
e. To fill a vacuum flask as shown on PI. Ill we first cool it
by washing it out with liquid air. The connection at jV„, PI. I fig. 2
and PL III, is brought about simply by drawing a piece of india
rubber tubing N^^ over the newsilver tubes N^ and Co fitting into
each other, round which tlannel is swaddled. This again is enveloped
in loose wool. When some bottles are connected they are filled with
pure hydrogen through the tube h^ 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 L^^^ (PI. 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. Ill for
Hydr. c. The disconnection at N\, is simply effected by taking off
the flannel band C^, heating the piece of india rubber tubing N^^
(unvolcanized) 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 N,.
§ 7. 7\ansport to the cryostat, closure of the cycle.
a. The vacnum glasses filled with liquid hydrogen (see Hydr. d
on PL II) are transported to the room where the cryostat ^r 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 thinwalled 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 tlae bottom.
( 171 )
into which the hydrogen is siphoned. To this end tlie tube b'\ of Pi. Ill
is connected (again by a piece of india rubber tubing, enveloped in
flannel and wool) to the inflow tube a^ of the eryostat and the
tube d^ to an inflow tube of pure hydi'ogen under pressure, which is
admitted from üvAc, PI. II, along Kwa. 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 ürst the apparatus after having been exhausted should prelimi
narily be filled with pure hydrogen. The liquid hydrogen is not
admitted into the eryostat ^r until the latter has been cooled —
coupled in another way (see the dotted line on PI. II) — by means
of pure hydrogen which has been led from d\hc through a cooling
tube immersed in liquid air. This refrigerator is of a similar construc
tion as the nitrogen condenser PI. VII of Comm. N". 83 (March '03).
Instead of Nliq should be read H^ and instead of Ox liq, Aër liq,
which is siphoned from the vacuum flask 21c. (comp, § 6).
During the siphoning of the liquid hydrogen into ^r the rapidity
of the influx is regulated after a mercury manometer, which is con
nected with the tube c on the cap h, PI, III (comp. fig. 2 of ^ 4).
b. From the eryostat the evaporated hydrogen escapes along 3%^
into the compressor (^), PI. II, which can also ser^•e as vacuumpump
and which precautiously through .p and Kfixi the dotted connection Kf
stores the gas, which might contain minute impurities, in the separate
reservoir ^ilid; or it escapes along Y^^ and Kpe or /i^x/ 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. W. 94(/ 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 PI. IV and is also to be found on PL II at 3
The impure hydrogen from :)\hb is admitted through Kn and along
a drying tube into a regenerator tube (see PI. IV) consisting of two
tubes enclosing each other concentrically, of which the outer a serves
for the inflow, the inner b for the outlet. Outside the apparatus
a and b are separated as a^ and b^, within the apparatus from the
point c downwards a is continued as «i and subsequently as the spiral
( 172 )
a, to terminate at the top of the separating cylinder d, from which
the gas escapes through h^, and the impurities separated from the
hydrogen as liquid escape along e and Km (comp. PI. II). The liquid
air, with which the cooling tube and the separating cylinder are cooled,
is admitted along / and the cock ?« (and drawn from the vacuum glass
215, PI. II) ; a float dr indicates the level of the liquid air. The eva
porating air is drawn oiï by the vacuumpump S (PI. II) along Kt.
The refrigerator vessel p is protected against heat from outside by
a double wall q of newsilver with capoc v packed between, of
which the lower end is immersed in a vacuum glass i\ while the
whole is surrounded with a layer of capoc enclosed in a varnished
cover of cardboard 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 evaporation
takes place.
The cock Km is turned so that some more bottles of known
capacity are collected of the blown oiF 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 V20 Vo ^^ i'' ^^d
along Kl to the gasholders, and compressed by v^ and S^ in dJid.
h. 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 (PL I)
closed, after which 2)^ and 2)^ (PI. II) are blown off to the gas
holders along Ka and Kg, and Kc 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 M and Kc closed, we return to the ordinary temperature and
analyze the gas, which in 2) 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 worked,
we go on admitting a quantity of preliminarily purified hydrogen of
Vjo Vo ^nd take care that the impurities are removed, we gradually
obtain and maintain without trouble a sufficient quantity of pure
hydrogen.
H. KAMERLINGrH ONNES. 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. Fl. I.
Uiii
3Cê<]
H. KAMEKLINGH ONNES. Methods and apparatus used in the cryogenic lahora
torv at Leiden. X. How to obtain baths of constant and uniform temperaturi
means of liquid hydrogen. Fl 1.
Fi,, 2.
Proceedings Royal Aca^l. Amateidain. Vol.
H. KAMERL
tempe
PI. II.
o    j ■
KAMERLINOH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden. X. How to obtain baths of constant and uniforn
temperature by means of liquid hydrogen.
T.I. 1. 1. 1. 1 .1.1. 1. 1. 1
Proceedings Royal Acad. Amslcrdam. Vol. IX.
H. KAMERLINGH OXNES. 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,
PL m.
Proceedings Royal Acad. Amsterdam. Vol IX.
H. KAMERLINGH ONNES. Methods and apparatus used in the cryogenic
laboratory at Leiden. XI. The purification of hydrogen for the cycle.
PL IV.
Xi
Proceedings Royal Acad. Amsterdam. Vol. IX.
ilNGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden.
Cryostat especially for temperatures from  252'^ to — 259".
PL \
\ /
( 173 )
XII. Cryostat especially for temperatures from
— 252= to —259^
§ 1. The principle. In X § 1 I liave said that we succeeded in
pouring into the crjostat of Comm. N". ^i'J VIII a bath of liquid
hydrogen, maintaining it there and making measurements in it, but
then the vacuum glass cracked. By mere chance it hiippened
that the measuring apparatus which contained the work of several
series of measurements came fortli 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 PI. 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 newsilver case, which forms, as it were,
a lining (see X, L PI. I). P'urther 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 protitably
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 cryostat 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 olher 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 here. PI. II 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 PI. VI
of Comm. N". 83, March '03. Instead of Bu Vac on the latter plate,
the compressor © serves as vacuumpump here (see PI. II of the
present paper).
b. The measuring apparatus (as on the plate of Comm. N°. 94''
VIII I have represented here the comparison of a thermoelement
with a resistance theimometer) are placed \vithin the protecting
cylinder §„ of the stirring apparatus. This is held in its place by 4
glass tubes l^^ fitted with caps of copper tubing ^^^ and %^^ at the
ends of the rods.
The beaker Ba, containing the bath of liquid hydrogen, is supported
by a newsilver cylinder Ba^, in the cylindrical rim Ba^ of which
the glass fits exactly ; the beaker is held in its place by 4 flat, thin,
newsilver suspension bands running downwards from Ba^ and
uniting below the bottom of Ba. 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
case U. On Ba^ rests the cover ^Vm in which a stopper is placed
carrying the measuring apparatus. The india rubber band effects
the closure (comp. also Comm. Nos. 83, 94'" and 94<0.
c. In the case U the vacuumglass B\, of which the inner wall
B\^ is protected by the thin newsilver cup Bb, is suspended by
bands L\ and supported by the wooden block L\. The cardboard
cover B\ forces the evaporated hydrogen, which escapes between
the interstices of the supporting ridges, over the pasteboard screen
B\^^ with notches B\^^ along the way indicated by arrows, to escape
at 2\^. The case is lined with felt, covered with nickel paper (comp.
Comm. W. 14, Dec. '94, and Comm. W. 51, Sept. '99).
d. The keeping of liquid hydrogen within an enclosed space, oi
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 Avhen I first poured off a bath of liquid oxygen within
a closed apparatus (comp. Comm. N". 14, Dec. '94).
The bottom of the case U is made a safety valve of very
large dimensions; as cover W^ of pei'forated copper with strengthened
ridges it fits into the cylindrical case Ub, which is strengthened
with the rim W. Over the external side of this co^•er (as in the
safety tubes for the hydrogen liquefactor) a thin india rubber sheet
IFj — 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 W^ 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 indiarubber
with the air must be prevented, it is surrounded with hydrogen;
this is done by filling the india rubber cylinder \Va, drawn over
the supporting ring Ub^ and the auxiliary cover Wb, with hydrogen
along Wc.
( 176 )
The cords Wd serve to press the auxiliary cover Wb 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 Ub^ and the principal body of
the case is made of newsilver to prevent the cooling of the lower
rim. The entire lower part is stutfed with layers of felt and w^ool
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 newsilver 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 § 4 a). The newsilver tube is put into the
newsilver side piece Ud, which is soldered on the case and, being
stuffed with capoc 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 tlie beaker Ba, 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. Remai'ks on the measurements ivith 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. d= 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 PI. Ill in Comm. N°. 83 (Febr. and March '03).
If the pressure is reduced down to 54 m.m. the tapping noise of
the valves of the stirring apparatus becomes duller. This is a warning
that solid hydrogen begins to deposit.
( 177 )
XIII. The preparation of liquid air hy means of the cascade process.
§ 1. Efficiency of the regenerative cascade method. In none of
the communications there was as yet occasion to treat more in
detail of the preparation of liquid 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. X". 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 (wiiile in general for the cryostats
these two cycles were sufficient, cf. the end of XII § 1). Larger
quantities of oxygen could be used in consequence, for which (as
mentioned in '94) a Brotherhood compressor was employed (comp.
the description 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 crvogenic
laboratory by H.H. Francis Hyndman in "Engineering" 4 Mrch '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 iji 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 through 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 airliquef actor . 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 PI. 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 RgRf, PI VI tig. 1. This spiral branches
off from the tube Rq^^ in the soldered piece Rg^^ and carries four
branches Rg^, Rg^, Rg^ and Rg^. 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 newsilver tube p^, round which it is drawn in the new silver case p.
The four windings are united below to one soldered piece to the
spiral Rf, 8 M. long, w^hich is immersed in a bath of liquid oxygen
and whence the liquid air flows through Rf\ 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 ?•„, 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 (o spend
the time at our disposal on other problems, as enough liquid air is
AMERLINGH ONNES. Methods and apparatus used in the cryogenic laboratory at Leiden.
XIII. The preparation of liquid air by means of the cascade process.
PI. VI.
Fig. 1.
^ceedino■s Boyal Acad. Amsterdam. Vol, IX
( 179 )
produced by the regenerative cascade. Enough but not too much,
because for operations with liquid hj^drogen (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 jmre 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 PI. I
and III at N^) or with a storage bottle, exhausted and filled with
liquid hydrogen as indicated in X §> 7. Then C (exhausted beforehand)
in the vacuum glass B is filled several times out of A, and the
vacuum glass B is connected with B^^ to the liquefactor and exhausted
like A and also filled with liquid hydrogen and connected with the
ordinary airpump at B^ so that the hydrogen boils in B atGOm.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 HoO and SO4H2.
How completely the water vapour can be freed in this manner appears from a
calculation of Dr. W. H. Keesom, for which he m.ade use of the formula of Scheel
(Verb. D. phys. Ges. 7, p. 391, 1905) and from which follows for the pressure of
water vapour (above ice) at — 180°G. 10— i® 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 (SOg vapours, greasevapours 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, when 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 Hters (Morlev, Amer. Journ.
of So. (3) 34 p. 149, 1887). This quantity of 1 m.gr. is probably only for a
small part water (Morley, 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 sulïicient contact with the phosphorous pentoxide were
ensured. But in this way the uncertainty, which remains on this point, is removed.
2) This application follows obviously from what has been suggested by Dewar,
Proc. Chem. Soc. 15, p. 71, 1899,
( 180 )
Fig. 5.
shut Cj 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 B 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. — "■(Jn the measurement of very loio temperatures. IX.
Comparison of a theimioelement constantiiisteel with the hydrogen
thermometer y By Prof. H. Kamerlingh Onnes and C. A.
Crommelin. Communication N" Sö*^ 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 thermoelement^), a gold and a platinumresistance thermometer'')
1) Gomp. comms. N". 27 and 89. (Proc. Roy. Ac. May 189G, June 1896, and
Feb. 1904).
3) Gomp. comms. N». 77 and 93. (Idem Febr. 1902 and Oct. 1904).
( 181 )
each iiulividually with two gas thermometers and also with each
other, wliile the deviation of the gas thermometer would be determined
by means of a differential thermometer '). Nitrogen liad originally
been chosen by the side of hydrogen, afterwards nitrogen Las 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 platinumresistance
thermometer ^), and these, for which others will be substituted in
Comm. X^ 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 abovementioned thermoelement 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 thermoelement.
§ 2. Comparisons made by other observers.
a. Constantiniron elements have been compared with a hydrogen
thermometer only by Holborn and Wien ^) and Ladexburg and
Krügel ^). 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 — 189M). They hold that the temperature can be
represented by the formula
t = aE \ bE'
and record that at an obserxation for testing purpose in boiling
oxygen ( — 183°. 2 at 7GÜ m.m. mercury pressure) a good harmony
was obtained.
Ladexburg and Krügel deem Holborn and Wien's formula unsatis
factory and propose
t = aE I bE' + cE\
They compare the thermoelement 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°. 94''. (Idem June 1905).
2) Gorap. comm. N'\ 93. (Idem Oct. 1904).
3) Silz.ber. Ac. Berlin. Bd. 30, p. G73, 189G, 'and Wied. Ann. Bd. 59, p. 213. 1896.
*) Ghem. Ber. Bd. 32, p. 1S18. 1899.
13
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 182 )
RoTHE^) could only arrive at an indirect comparison with the hydrogen
thermometer. He compared his thermoelements constan tiniron at
• — 79° with the alcohol thermometer which Wiebp: and Böttcher *)
had connected with the gas thermometer and at — lOl'' with a
platinumresistance thermometer which at about the same tempera
ture had been compared with the hydrogen thermometer in the
Phys. Techn. Reichsanstalt by Holborn and Dittenberger ').
The thermostat left much to be desired ; temperature deviations
from 0^.4 to O'^.l occurred within ten minutes (comp. for this § 7).
As RoTHE confined himself to two points, he had to rest content with
a quadratic formula and he computed the same formula as Holborn
and WiEN.
From the values communicated for other temperatures we can
only derive that the mutual differences between the deviations of the
different thermoelements constantiniron and constantincopper 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.
b. Among the thermoelements of other composition we mention
that of Wroblewski "), who compared his newsilvercopper 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 t.
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 0°.l. As, however, Wroblewski 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.
Dewar's ^) investigation of the element platinumsilver 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. fur Instrumentenk. Bd. 22 p. 14 and 33. 1902.
2) , , „ Bd. 10 p. 16. 1890.
5) Drude's Ann. Bd. 6 p. 242. 1901.
*) Sitzungsber. Ac. Wien Vol. 91. p. 667. 1885.
6) Proc. R. S. Vol. 76, p. 317. 1905.
( 183 )
how far thermoelements are suitable for the accurate determination
of low temperatures (for instance to within V,/ 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 thermoelements and auxiliary apparatus.
We shall consider some modifications and improvements which
have not been described in § i of Comm. N". 89. The first two
{a and b) 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 thermoelement 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 I of Comm.
N°. 89. The wires a and b are soldered on the bottoms
of small holes c, bored in the protecting block and are
insulated each by a thinwalled glass tube. If the con
struction of PI. 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 consequenlly 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 thermoelement is immersed in
li(jnid 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
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 ca block of greater
thickness is required than for that of Comm. N". 89, PI. I) has not
been applied to the element used, we need not fear uncertainties on
this point thanks to the very careful construction of tlie latter.
b. When temperatures below — 253" liave to be determined we
might fdl 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 PI. IV, fig. 4, Jc) but in
parafifm, so as to obtain perfect insulation, which, as experience has
taught, is not guaranteed by the glass wall. Tlie tubes are continued
beyond the sealing places of the platinum wires c^ c^ Cjand c^, (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 Westonelements have been ainal
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. W. 89, thermoelectromotive 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. Jaeger, Die Normalelemente, p. 57.
( 185 )
juncture are in contact with each other, but we only require that
a definite ele;ctromotive force for a definite temperature of the bath
in which the element is immersed should be accurately indicated,
(for the rest comp. § 5).
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 cm. above the copper rim of the copper
block of a copper tube, 5 cm. 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 measureme?its of the electromotive forces.
a. The apparatus and connections which have been described in
§ 3 of Comm. K". 89 have been mounted entirely on paraffin, with
w^hich also the enveloping portions of the apparatus are insulated.
Only the wares running between the different rooms stretched on
porcelain insulators, of which the high insulationresistance has
repeatedly been tested, have no paraffininsulation. The icepots are
hanging on porcelain insulators. As a matter of coui'se, all parts of
the installation have been carefully examined as to their insulation
before they are used.
h. The necessity of continually packing together the ice in the
icepots has been argued before in Comm. N°. 89.
c. The plugcommutators are of copper. All contacts between different
metals in the connection have been carefully protected from variations
of temperature by packing of wool or cottonwool, from which they
are insulated by paraffin in cardboard 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 Westonelements 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 shortcir
cuiting in the copper commutators in the conductions, leading from
the thermoelements and the Westonbattery to the connections, whether
all electromotive forces in the connections are so small and constant
(not more than some microvolts), that elimination through the re^•ersal
( 186 )
of the several commutators may be considered ag perfectly certain.
§ 5. The control of the thermoelements.
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 constantinsteel,
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 electromoti\e forces which are raised
in the element outside the places of contact, are exceedingly small.
§ 6. Corrections and calculations of the determinations of the
electromotive forces.
a. In the following sections R^ Be R' have the meaning which has
been explained in Comm. N°. 89 § 3. E^, Ec and E' signify the
electromotive forces of the observationelement, the comparisonelement
and the Westonbattery respectively. If we have obtained 11,0 Re and
R' it follows that :
Rw _ _ Rw
Ew '==
As a test we use :
E,o = — Ec or E,, — — E'.
Rp R'
E'
En =^ Rr '
R'
b. In order to find R^ we read on the stops of the resistance box
R'w (in the branch of small resistance),' and R'w (in the branch of
great resistance) which are switched in parallel to form /?^..
«. To none of the resistance boxes temperature corrections had
to be applied (nor to those given by Re and R' either).
/?. To R'w we sometimes had to add the connecting resistance of
the stops.
y. To Rw is added the correction to international ohms according
to the calibration table of the Phys. Techn. Reichsanstalt.
Ö. To R'w is added the amount required to render the compen
sation complete, which amount is derived from the deflections on
( 187 )
■H
"«il
ca co
ca
O o
&.§
O H
o
•it Tl
ü ö
^^
o ,^
O CO
o Oi
ft S
P4 ^
r— I O
c
O
Ü
u
o M
rH
t/i
O
IS
CD
1 +
o 2
't: rt
ü,
o
o
00
II
f^
tó
O
bC
O
5^
C5
o
00
O
00
o
o
00
■r
II
II
II
x»
•d**
^
+
o
00
co
fi^
+
o
o
o
co
ft^ 0:5
^
r^
bC
b.
Ci
00
co
ry
O
iO
(—1
l~
r^
_o
<v
^
11
u
c*
ffi
CD
0)
eS
a
bC
CC
o
u
03
c8
■u
O
X
3!
u
O
o
Ol
tsi
tó
.4J
<*,
<v
""
o
o
O *^ O
in lO CD
c^ o
t 00
§ § g
co o
CD CD
^
co
s
s
co
s
co
§
lO co lO
— 1 (M co
^ s
co CD CD
co co CD
CD CD
CD CD
+
Cl
+
o
o
o
Ol
S^ ft^
+
+
o l—
II II
tó
bC
O
C^ O)
co <j3
bo
m
co
co
co
65.63
65.70
65.84
66.00
( 188 )
the scale of the galvanometer at two values of R'\o (see tables I
and IV).
, c. In order to find Ec, Re and E'c, which with regard to Re have
a similar meaning as R\o and E\o with regard to Rw , are treated
ilike R'w and R\ concerning the corrections a, /?, y and d. The
I thence derived result R"'c holds for the temperature at which the
j water boils in the boiling apparatus at the barometric height B
I existing there during the observation.
I' 8. R"'c is corrected to the value which it would have at a pres
Isure of 760 m.m. mercury at the sealevel in a northern latitude of 45°.
• d. To find R' the corrections mentioned sub y and ö are applied
; to the invariable resistance R'.
I' e. E', referring to the temperature i of the Westonbattery, is
derived from Jaeger's table ^).
§ 7. Survey of a measurement. Table I contains all the readings
: which serve for a measurement of the electromotive force namely
! foi' 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 comparisonelement 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 — 253° the
sensitiveness of the element constantinsteel is considerably less
than 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) Jaeger, Die Normalelemente 1902. p. 118.
'i) Comp. Comm. N". 83, § 5 and PI. III.
3) Together with the readings we have also recorded the temperature of the
room [tk) and of the galvanometer {Uj) ; 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. For the notation of
the combination P3 + Qo of die comparisonelements we refer to Comm. N'. 89 § 2.
( 189 )
From table I directly follows
TABLE II.
Corrections and results.
Obsprvationelement. Comp.nrisonelement.
Westoneleraents.
corr. ^ R'„, = + . 00 1 n ' corr. ^ i^'^ = j . 00 1 P.
corr. yi2',„= f 00080 p c^rr. '/ R'^ = — 0.00015 si
coTr.SR'\^= + 179 n corr. I R\ = ^ 149 n
corr. 7 ^', = — 2.4n
corr. $ Zt'i = + 0.654 p.

R'"^ =50.3163n
baron).hght.45°N.B.=:76.21cM.
corr. e /ï'^'^= — 0.0373X1
Final results.
1
i?j^ = 53 . 6404 a Re = 50 . 2787 a
i?' = 7998.3A
/' = i8o.8
^'=1.0187 volt.
Z'„= 6.8312 milliv.
4u3'
^^.=6.4037 milliv.
TABLE III
£w
Ec
6.8312
6.8308
6.8310
6.4037
6.4039
6.4038
Mean
6.8310
6.4038
§ 8. The temperatures.
a. The thermoelement 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 )
00 1
X.
CM
.
bü
(M
00
^
k.
vj
(T^
C3
in CO 00
— ^
— ^
CM
1
"TH
CO
(1
O)
lO co t^
co co co
CO ^~, 03
co >*
TH C5 TH
^H
to
vt •«* «^
•H
tó
ö:i
TH
d
.2
ü
«*^
%
j3
S
;z;
lO
CM
s
c
c
03
O)
O)
s
0)
a
a
CU
H
% S:
O
O
5
'E
00
H
ij
co
02
•«* lO
1
03
03
h
rH co
CO CO co"
t> co (M
00
'/.
CS
«^ >* «*
000
0)
_;
II
^_j
00 05 05
TH TH TH
+j
5i
tó
OJ
CM
II
1 II
fl
*
^ .^ >.^
•iH
p
>•
ü
'o?
s
cö
i— 1
■4.3
:3
co
4J
+*
HH.
bc
S 8
G^
'C^
6%
lO 1 1 lO
II II
'E
>^ co
co
03
co rH l?ï lO
a
CO
Uj
2
co 00 TH 04
Cm
^
^^'
TH
6
.2
0,
T^ T^ (?i CN cïi
>* ve< «* «* N^
:z
a
Ih
03
O
Oh
CO
a +
o; co
13
lo
«* 00
o.
Hl
bt
d
3^
&s S^
ffi
c
ij
00 TH
c
r—i
ft
f*
P4
s
T3
B
03
y2
HHI
s
c
>
co ^
E'S
ü
HH>
03
Cu
O.
100
d+
"'000
II II
H>
«=*
00 00
oi co'
co er:
♦f
OJ
00 <M m CJ5 co
iS
4>
^5
te
ca
tH
o;
CM co ►* co
^ vt >^ 5 >^
a
t
00
T<
II
c^ t
1— <
^ w
ii
00 co
03
O
fi^ 0:5
<o
00 (M
n
fcX)
00 «^
ft
+0
ÖC
CD 00
TH TH
OJ CS
rf
++
b.
co co
+3
lO in
(M 00 1(0 lO
fl>
+08
tH
co co * co CT>
5
co
C
4)
g
1 ^
II II
5^ 5^
06
CT)
CO
10
TH
d
t/'
s
N
d d d d d
«Sj< Vt !t ^ *
Cm
,2
co
00 10
d T^
^
ir
.2
"3
•rH
33
bc
00 I^
d d
;h
(D
1
L.
(h
co co
CA2
+
'S
<D
s
Oi '^ï in i> (M
•H
lO
os
3
(h
00 oj TH co
CiJ
>
o
Ci a> Q
+
cö
fi
co co "^ ^ »*
fO
O
oj
>
lO lO
rH
1
+
~
HH>
'f>
03
0)
T< 00
d d
co co
tn
(1
fj
>
03
1— 1 HH
1H >— 1 1— 1 1— 1
^
II II
a
a
bC
10
a
, , HM
1— 1 >— 1  1— [
o
<u
l
( 191 )
TABLE V.
Corrections and results.
Observationelement.
Comparisonelement.
Westonelements.
corr. /3.f?'^y = +0.001 a
corr. y.R'^ + 0.00531 SI
corr.<J ii"^j, = + 20xi.
corr.iS.R'^ = + 0.001 n
corr. y.i^'^ = + 0.0084 n
corr.o\/i"'c = — 209x1
corr. y. R\^ — 2 . 4 xi
corr. 0. J?', = + 0.8xi
R'"^ = 50.4133 XI
Bar.hght. 45° N.B. =70.82 cM.
corr.eR'"^ = — 0.1459f2

Final results.
/?j„=55.9981fi
i?c = 50.2014 fi
i?' = 7998.4 Ü
^' = 18°. 5
^' = 1.0187 volt.
E^ = 1A3'21 milliv.
2ii24'
^^ = 6.4075 milliv.
bath a tube is mounted symmetricallj with the thermoelement, 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/.
b. 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 — E^ in millivolts, column V the number of
observations, column VI the greatest deviations in the different deter
minations of Eio of which the appertaining E^u is the mean, column VII
the same reduced to degrees.
TABLE VI.
CALIBRATION OF THE THERMOELEMENT
CONSTANTINSTEEL.
I
II
III
IV
V
VI
VII
20
27 Oct. 05
— 58° 753
2.3995
3
0.0006
0?016
21
30 Oct. 05
— 88.140
3.4825
3
29
81
17
8 July 05
— 103.833
4.0229
3
56
168
46
7 July 05
— 139.851
5.1469
3
6
21
18
2G Oct. 05
— 139.873
5.1469
4
12
41
19
20 Oct. 05
— 158.831
5.6645
3
15
59
11
27 June 05
— [182.692]
6.2297
3
10
46
28
2 Mrch. OG
— 195.178
6.4717
4
28
150
12
29 June 05
— [204.535]
6.6382
3
31
186
27
2 Mrcli. 06
— 204.694
6.6361
4
26
156
14
30 June 05
— [212.832]
6.7683
3
8
56
13
6 July 05
— 212.868
6.7668
3
15
106
29
3 Mrch. 06
— 217.411
6 8221
3
14
112
15
6 July 05
 217,416
6.8310
3
4
32
'iO
5 May 06 .
— 252.93
7.1315
4
17
39
31
5 May 06
— 259.24
7.1585
1
—
—
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. W. 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 Ec for the ditferent days from the mean
vahie, and also the mean largest deviation of the vahies o[ Ec found
on one day amounts to 3 microvolts, which amount shows that in
the observation of the comparisonelement 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
observationelement yields for this mean only 1,8 microvolt.
§ 10. Indirect determinations.
In order to arrive at the most suitable representation of Ei^ as a
function of t, 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 thermoelement and a platinumresistance 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 THERMOELEMENT
CONSTANTINSTEEL.
I
II
III
IV
V
VI
VII
22
13 Dec. 05
— 29° 825
1.2523
3
0.0005
0°012
24
14 Dec. 05
— 58.748
2.3980
4
6
16
23
13 Dec. 05
— 88.161
3.4802
3
6
17
1
23 Jan. 05
— 103.576
4.0100
5
9
27
3
30 Jan. 05
1— 182.604]
6.2270
4
32
147
5
16 Mrch. 05
[ 182.828]
6.2340
3
13
60
4
2 Febr. 05
— 195.135
6.4730
3
20
107
17 Mich. 05
— 195.261
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
233
8
3 April 05
— 212.940
6 . 7686
4
15
106
25
25 Jan. 06
— 217.832
6.8;i76
4
29
232
( 194 )
§ 11. Representation of the observations by a formula.
a. It was obvious that the formula of Avenarius:
t f t
100 ^ V1Ö0
can give a siifücient agreement for a very limited range only. If,
for instance, the parabola is drawn through 0°, — 140° and — 253°,
we find:
a= + 4.7448
6= + 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 —J 83°, we find:
a— i 4.4501
ft = + 0.57008,
while at — 140° the deviation still amounts to 1°.3.
Such a representation is therefore entirely unsatisfactory.
b. With a cubic formula of the form
100 ^ \iooJ ^ vioo
we can naturally attain a better agreement. If, for instance, we
draw this cubic parabola through 0°, —88°, —159° and —253",
we find :
a=z + 4.2069
6= + 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 i, expressed in E (comp. § 2), gives much larger
deviations. ^)
c. A formula, proposed by Stansfield ^) for temperatures above
0°, of the form
1) As we are going to press we become acquainted with the observations of
Hunter (Journ. of phys. chem. Vol. 10, p. 319, 1906) wlio supposes that, by
means of a quadratic formula determined by the points — 79^ and — 183°, he can
determine temperatures at — 122° to within C^.l. How this result can be' made
to agree with ours remains as yet unexplained.
2) After the publication of tbe original Dutch paper we have taken to hand
the calculation after the method exposed in § 12 of a formula of the following form:
E = a^ 4 h I — I + c ( ^ 1 4 e ( ~
100 ^ viooy ^ Vi^v ^ vi^o
We hope to give the results at the next meeting.
3) Phil. Mag. Ser. 5, Vol. 46, p. 73, 1898.
( 195 )
^=aT f hlog T+ c,
where 1 represents the absolute temperature, [)roved absolutely
useless.
cl. 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 /. To this end we have tried two forms :
'^='w, + \m) ^Inoj +\m) +\w,) ■ • • <^)
and
^="100 + ^1100; +IÏÓ0J +lïööj +\m) ■ ■ <^^
First the constants of the two equations are determined so that
the equations satisfy the temperatures — 59°, — 140°, — 159°, — 183°
and — 213°. (^1) indicated at — 253° a deviation of 113.1 micro
volts, (B) a deviation of 91.8 microvolts. We have preferred the
equation (B) and then have sought au equation [BIY) which would
represent as well as possible the temperature I'ange from 0° to —217°,
two equations (^1 and ^III) which would moreover show a not too
large deviation at — 253°, for one of which (Bill) a large deviation
was allowed at — 217°, while for the other (Bl) the deviations are
distributed more equally over all temperatures, and lastly an equation
(i^II) which, besides —253°, would also include —259°.
§ 12. Calculation of the coefficients in the formula of five
terms. 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 de Sande Bakhuyzen in which
instead of the 5 unknown coefficients 5 other unknown values are
introduced which depend linearly on the former ^). For these are
chosen the exact values of E for the five observations used originally,
or rather the differences between these values and their values found
to the first approximation.
Five auxiliaiy 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
') 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 R^ The deviations of the observed values
from those derived from this first formula are given in column III
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 {B) (comp. § 11):
a^— ^4.32044 e, = + 0,011197 .
b, = ] 0,388466 /, = — 0,00446381 .... (BIV)
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 ^3 = + 0,053261
^3 r= + 0.436676 ƒ3 =+ 0,003898) . . . {BUI)
C3 = + 0,048091
and
a, — \ 4.35603 e, — \ 0,103459
6j= + 0,531588/i = f 0,0118632j {BI)
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 {B)
a^—4_ 4.35905 «, = + 0,111619 
6, = + 0,542848 ƒ, = + 0,01321301 . . . {BII)
C3 = + 0,172014 ]
The deviations from the observations shown by these diff'erent equa
tions are found under {W—R,) {W—R,) ( IF— A\) and ( TF— A\) 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 interpolationcoefficients
of Lagrange.
( n)7 )
TABLE VIII.
DEVIATIONS OF THE CALIBRATIONFORMULAE FOR THE
THERMOELEMENT CONSTANTINSTEEL.
"l
II
III
IV '
V \I
VII
VIII
No.
t
WR,
(ÏV/i„)
(WR,) {WR,)
OVR,)
oyiu)
22
24
2Ü
21
23
I
17
'16
18
19
3
11
5
4
28
G
12
27
7
26
14
13
8
29
15
25
30
31
— 23.825
— 58.748
— 58.753
— 88.140
— 88.101
—103.576
—103.833
—139.851
—139.873
—158.831
[182.604]
[—182.7501
[—182.828]
—195.135
—195.178
195.201
[—204 535]
—204.694
—204 895
—212.765
[—212.832]
—212.868
—212.940
—217.411
217.416
—217.832
—252.93
—259.24
0,0080
—
13
+
44
+
14
+
43
+
5
—
2
+
41
+
36
+
63
+
15
—
6
+
76
+
34
—
17
—
11
+
21
+
58
+
38
+
45
—
36
+
52
—
36
+
87
—0.0080
—
—0 0030
+
29
+
14
+
1
+ 47
+ 28
' + 5
f 40
+ 87
1
+
26
+
31
+
4
—
20
—
20
+
31
+
13
—
10
+
25
—
i3
—
20
+
68
0 0032
+ 26
+ 32
+ 4
— 20
20
+ 33
+ 17
+ 29
— 19
— 40
+ 37
—0.0013
+ 16
+ 8
— 16
— 18
— 10
4 38
+ 12
— 20
+
— 45
0.0011
+ 11
+ V
+
11
— o
+ 31
+ 4
20
+
— 18^
— 20 + 280
+ 90 I + 490
14
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 198 )
To observ^ation 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) ').
§ 13. 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, db 3.2
R^ dtz 2.6 (2.1 when leaving also out of account — 217°j
R^ ±1.8
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 thermoelement constantinsteel between 0° and — 217° by the
fiveterms 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
constantinsteel is hardly suitable (comp. § 7).
In conclusion we wish to express hearty thanks to Miss T. C.
Jolles and Messrs. C. Braak and J. Clay 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 witli the true vakies
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 wouUl be reduced to four.
( 199 )
Physics. — On the measurement of very loio temperatures. X.
Coefficient of expansion of Jena (jlass and of platinum hetio een
j 16^ and —182°." By Prof. H. Kamerlingh Onnes and
J. Clay. Coniiniinication N°. 95^ from the Physical Laboratory
at Leiden.
(Communicated in the meeting of June 30, 1906).
§ 1. Introduction.
The difference between the coefficients a and h in the expansion
r i ^ f ^ \) ^
formula for the linear expansion ^ = ^o ^ + k' tttt; + ^^ I tttt: ) P^^
and k^ and h^ in the formula for the cubic expansion
^1 ' 100 ^ ' Viooy i
between O"" and — 182° found by Kamerlingh Onnes and Heuse
(comp. Comm. N". 85, June '03, see Proceedings of April '05) and
those found by Wiere and Bottcher and Thiesen and Scheel for
temperatures above 0° made it desirable that the strong increase of
b at low temperatures should be rendered indubitable 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 b which
better represent the results of the measurements than those given in
Comm. N^ 85.
By means of the formula
Wt = W, (1 + 0,00390972 t — 0,039801 t'),
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 — 18r,42 „ „ — 182°,99
in table V read  86°,98 „ „ — 87°,71
and — 18r,22 „ „ — 182°,79
1) That the coefficient of expansion becomes smaller at lower temperatures
is shown by J. Zakrzewski 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 b. Our investigation refers to the question whether
b itself will increase with lower temperatures.
14*
( 200 )
Thence follows
Jena glass 16"« a — 781 b = 90
ki = 2343 k, = 272.
Thüringer glass (n". 50) a = 920 b = 120
k, = 2761 k,= 362.
1903.
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 h 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 h, when this is calculated for tlie 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 ScheeTj (Zeitschr. f. Instr.
April 1906 p. 119) published his resuU that the expansion of pla
tinum from — 190° to 0° is smaller than follows from the quadratic
formula for the expansion above 100'^. For the expansion from { 16°
to — 190^ Scheel finds — 1641 u per meter, while — 1687 f* 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 vahie of b below 0*^.
The necessity of adopting a cubic formula with a negative coeffi
cient of f may be considered as being in harmony with the
negative expansion of amorphous quartz found by Scheel (1. c.)
between — 190° and 16° when we consider the values of a and h
in a quadratic formula for the expansion of this substance between
0" and + 250°.
( 2()i )
A more detailed investigation of these questions ought to be made
of course witii more accurate means. It lies at hand to use the
method of Fizp]au. Many years ago one of us (K. 0.), during a
visit at Jena, discussed with Prof. Pulfrich the possibility of placing
a dilatometer of Abbe 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 Jaquerod^) give for the coefficient of expan
sion of a Jiot further determined kind of glass between O'' 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 Thüringer glass.
The mean coefficient of expansion between 0° and — 190° for
Thüringer glass found at Leiden in 1903 is 0,00002074.
§ 2. Measurement of the coefficient of expansion of Jena glass and
of platinum between (f 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 cm. 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 thermoelement constantinsteel (comp. Comm. N°. 95^,
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 Scheel, Zcilschr. f. Instrk. April 1905, p. 104 and April 1906,
p. 118. Randall, Phys. Revie v 20, p. 10, 1905 has construcLed a shnilar apparatus.
3) Travers, Senter and Jaquerod, Phil. Trans, A 200.
( 202 )
conducting pieces of glass, which partij projected out of the batii.
The scale (comp. Comm. N°. 85) was wrapped round with a
thick layer of wool enclosed in cardboard 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 w^ere read on three thermometers at the bottom, in the
middle and at the top.
The scale and the points of the glass rods were illuminated by
mirrors reflecting daylight or arclight, wiiich 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 Avas economized. For the
measurement with liquid oxygen we required with the
first tube V/^ liter per hour and 74 liter with the
second. Of N,0 we used with the first only ^^ liter
per IY4 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. The}' 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 Avhich 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
oxide such variations in the distribution of the tem
perature can be entirely neglected.
With some measurements we have observed that the
length of the rods, when they had regained their
ordinary temperature after cooling, first exceeded the
original length, but aftei' 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) witii a tine capillary tube was filled
with mercury. First the level of the mercury was compared with
an accurate thermometer at the temperature of the room in a water
bath in a vacuum glass. Then the apparatus was turned upside down
so that the mercury passed into the reservoir B, which is a little
greater than A. Subsequently.! 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 B no mercury could flow back
TABLE I. — JENA GLASS 16 1".
Date Time
Temp,
scale
h
•^160
^f\
1^0
s
A
46 Dec.
2A.35
15.7
1026.285
1026.280
40.620
15.9
1904
3A 50
16
.286
.279
40.786
17.0
4A.22
10.3
.292
.290
40.845
17.4
20 Dec
U.50
15.3
1025.571
1025.559
s. 3.503
5.021
40.6
2A 10
15.4
.560
.550
7n.25.029
38.28
86.78
2A.30
15.4
.571
.561
i. 6.300
7.191
\=22.1
21 Dec.
3A.15
14.6
1026.308
1026.291
»i.40.523
15.1
3A.45
14.7
.299
.284
15.1
4A.15
14.7
.308
.289
»4 40.583
15.6
22 Dec.
lOA 50
15.0
1025.108
1025.091
s. 2.105
5.021
A,=30.8
12A.15
15.0
.112
.095
m. 9.880
38.28
—181.48
12A.50
15.0
.115
.098
i 5.005
7.191
>,=18.0
23 Dec.
12A.30
15.8
1026.341
1026.341
«.40,606
15.6
3/5.
15.6
.339
.339
15.2
3A.30
15.6
.335
.336
«/.40.537
15.2
11 Jan.
3A.40
15.4
1026.288
1026.278
40.634
15.9
1905
4^.30
15.5
.291
.280
40.703
16.4
( 204 )
to A. When A had regained the temperature of the room the
niercurj was passed again from B into A and the apparatus
replaced into the same waterbath as before. Tiie de\'iation of the
level of the mercury was of the same order as the reading error of
the thermometer, about 003\ A perceptible thermical hysteresis
therefore we do not find.
TABLE II. — PLATINUM.
Date
Time
Teiiii
ir,
/^„
16 D.T.
5A5D
16.5
1027.460
1027.401
17.0
1904
16.4
1027.461
1027.459
17,0
17 Doc')
dh 45
16 6
16 3
1026,620
1026,618
1020.630
622
10/^5
16.3
613
617
19 Dec.
Sk
14 S
1027.459
1027.442
15.5
8A30
14.8
457
1027.440
15.5
20 'ec.
?,k
15.5
1026.627
1026,630
5 3. 47 5
4.993
"40.2
■
S/i 30
15.5
630
633
m
80.32
3h 55
15.4
031
635
i 7 575
8.653
'':ïï.5
21 Ore.
4/^40
14 7
1027.460
1027.441
15.5
5A1Ö
14 9
459
444
15 5
6/i
14.8
459
442
15,5
22 Dec
10/i40
15.3
1025,963
1025.951
^2.140
4.993
*'28.9
11/MO
15.3
1025.973
9u1
m
182.6
'18.5
1A45
14.9
1025.964
947
f 5.649
8.653
23 Dec.
11// 25
15.7
15 6
15.7
1027.434
440
440
1027". 436
441
442
15.0
15
15.2
3 Febr.
2/*
15.4
15.4
1027.403
459
1027.459
455
15.2
15.2
1) Journ. Chem. Soc. 63. p. 135. 1893.
( 205 )
In table II (p. 204:) the temperatures are used wliicli are found
with the thermoelement. A controlmeasurement with the thermo
element placed in the same vacuum tui)e without rod gave for the
temperature in nitrous oxide — 87^,3 instead of — 86^,32.
The mean value of the two deterininations is used for the calculation.
Another reason for the measurement of the temperature of the
bath with a thermoelement 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 Hunter *) has given
— 86^.2 for that temperature.
§ 3. Results.
Jena glass 16 III a 835 b 117
k, 2505 k, 353
Platinum a 905,3 b 49,4
k, 2716 k, 148,4.
' 1905
As regards platinum:
Bexoit finds from 0^ to 80° a 890,1 b 12,1
Scheel from 20' to 100' a 880,6 h 19,5
HoLBORN and Day from 0' to 1000^ a 886,8 b 13,24
As to the differences between the values obtained now and those
of Comm. K". 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 b and the necessity of a cubic formula.
There is every reason to try to combine our determinations on
Jena glass above and below 0"" 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, Se)t. 1900, § 20) we
find in the formula for the linear expansion below 0^ and in the
corresponding one for the cubic expansion
ii = K
1 + !«' ^ + ^' (Aj^^ ^' r^LY! 100
100 ' viooy viooy
Jena glass 16 III a 789,4 k'^ 2368,1
b' 39,5 k', 120,2
c' — 28,8 k'. 86,2
1) With this measurement in N2O we have not obtained a temperature deter
mination with the thermoelement. 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.
2) Journ. Phys. Chem. May 1906, p. 356.
( '206 )
§ 4. Controlexperiment.
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 III. — JENA GLASS ENDS.
Date
Temp,
scale
Lt
/^,6 =
^ft
K,
&
A
42 April 1905
lOAio
15.4
227.684
227 683
15.4
\\h
.686
.685
15.4
\\h 43
15.4
227.684
227.682
15.5
15.4
.681
.679
15.5
N^O
_
3A50
15.4
227.533
227 536
s 3.473
5.021
42.3
4A24
15.4
.543
.541
^i =
4A52
15.4
.550
.548
i 5.490
7.191
32.3
13 April
17.4
227.677
227.681
17.1
14 April
16.2
227.675
227.676
15.9
\0h 10
■
0,
2A50
18.4
227.474
227.482
s 1.941
5.021
35.5
4A22
18.9
.482
.494
i 4.683
7.191
8.9
15 April
16.6
227.725
227 727
15.7
\\h\
16.6
.724
.726
16.0
4A20
16.4
227.706
.708
15.8
4A46
16.4
.711
.713
16.0
IG April
14.1
227.706
227.702
13.6
17 April
14.2
227.682
.685
227.678
.681
14.0
I 207 )
manner as those of the rods in the vacnimi glass. Now we have
taken only a double glass filled with wool, enveloped in a cardboard
funnel and tube for letting out the cold vapours.
The measurements are given in table III.
The ).'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 b found in § 2 yields :
Ly^o = 227,547 while we have found Ly^o = 227,544
Lo, =227.487 „ „ „ „ Lo, =227,488.
In conclusion we wish to express hearty thanks to Miss T. C.
Jolles and Miss A. Sillevis for their assistance in this investigation.
Physics. — ■ "On the measurement of very loio temperatures. XI. A
comijarlson of the platinum resistance thermometer loitli the
hydrogen thermometer" By Prof. H. Kamerijngh Onnes and
J. Clay. Communication N°. 95^ from the Physical Laboratory
at Leiden.
(Communicated in the meeting of June 30, 1906).
§ 1. Introduciion. The following investigation has been started
in Comms. N°. 77 and N". 93 YII of B. Meilixk 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 cojitain pure hydrogen, but that it was contaminated by
air. The modifications which are consequently required in tables
V and YI of Comm. N°. 93 and which particularly relate to the very
lowest temperatures, will be dealt wiili i]i 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 pro\'e the existence of the point of
inflection which may be expected in the cur\'e (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 e\en becomes infinite at the absolute temperature (comp.
( 208 )
Snppl. K°. 9, Febr. '04). And this has been done especially becanse
temperature measurements with 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 tlie conclusion has been drawn that between
O' and — 180° 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 hydjogen 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 wiiether the thermometer wdien
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 w^ere furnished at different times
by Heraeus, 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. Ghem. LII, 4, 1905. The wire of which
the calibration is given by Olszewski, 1905, Drude's Ann. Bd. 17, p. 990, is appa.
rently according to himself no platinum wire. (Gomp. also § 6, note 1).
( 209 )
have availed ourselves alread}' in the regulation of the temperatures
in the investigation mentioned in Comm. N". 94^^.
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°, —J 59°, 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°. 95^^ § 10 and are not used in the
derivation and the adjustment of the formidae. For the meaning of
W — Raj 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 Gallendar, used at low temperatures by Travers and
GwYER, Z. f. Phys. Ghem. 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. Dev^ar Proc. R. Soc. 64, p. 227, 1898.
The cahbration 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. 9G8, '04).
( 210 )
TABLE I.
COMPARISON BETWEEN THE PLATINUM RESISTANCE
THERMOMETER AND THE HYDROGEN THERMOMETER.
Date
Temperature
bydrogt^ntherm.
Resistance
measured
Remarks
0°
0°
137 884 a
1 mean (f 5 measurement.*^.
27 Oct.
'05
5 h.
2 h. 50
— 29.80
— 58 75
121.587
105.640
30 Oct.
'05
3 h. 50
— 88.14
89.277
8 July
'05
10 h. 12
— 103.83
80.448
26 Oct.
'05
5 h. 20
— 139.87
59.914
7 Julj
'05
4 h. 25
— 139.85
59 920
26 Oct.
'05
3 h. 16
— 158.83
48.929
27 June
'05
1 b. 40
[— 182.69]
34.861
}r—Rjj = — O.OG]
30 June
'00
11 h.
— 182.75
34.858
27 June
'05
3 b. 50
[ 195.30]
27.598
jr—Rjj=]0.0S2 .
2 March
'06
3 h. 35
— 195.18
27.595
29 June
'05
11 h. 6
[— 204.53]
22.016
}F—Rjj — — OAiO
2 March
'06
1 h. 30
— 204.69
22.018
30 June
'05
3 h.
[— 212.83]
17.255
jfr—Ejj~ — 0.0S'2
5 July
'05
5 h. 53
— 212.87
17 290
5 July
'05
3 h. 20
— 217.41
14.763
3 March
'05
10 h.
— 217.41
14.770
5 May
'06
3 h.
— 252.93
1.963
5 May
'06
5 h. 7
— 259.24
1.444
( 211 )
W, = T^e j 1 + 0,39097 (^^^  0,009862 (^^
For instance cat —139° it gives W—E: +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 inflection is certain
(comp. sub d). Therefore it is evident that a quadratic formula will
not be sufficient for lower temperatures.
h. 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 :
''( ^ +0,0,583861 *
Wt= W, j 1+0,393008 — _0,0J3677(^— J +0,0368386^^^^
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 t, 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 :
lF: = ^+^ïöó + 'tï^J+\ïööJ+lT273;Ö9j • (^)
Wt _ t f ^ Y / t y no^ lo^ \
1f7~ "^""ÏÖÖ"^ [JÖö) ^yïööj "^ '\"7^"~ 273,09 J "^
"10^ 10^
+ «
T» (273,09)
Wt . t /" t y / t Y /lo» 10
.]■
(^)
l + a \b{ +c ] +d{ +
,Ö9J
W^ ' 100 ' \100J ' \100J ' \T 273
/10« 10
"^ ' V T^ ~ (273,09)7 ' ^^^
We shall also try a formula with a term — instead of — .
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 de Sande Bakhcyzen (comp. Comm. N°. 95a) in
two different ways. First we have obtained a set of constants Ai
with which a satisfactory accurate agreement was reached down to
— 217°, a rather large deviation at — 252° and a moderate deviation
at — 259°. Column W—Rai of table II contains the deviations.
Secondly we have obtained a set of constants which yielded a fairly
These values deviate slightly from those comraunicated in the original.
( 212 )
accurate agreement including — 252", but a large deviation at — 259°,
These are given in table II under the heading W — Rati
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 IF — Rb , and a
solution of the form C which agrees only to — 252° and to which
W — Re relates.
The constants of the formulae under consideration are :
^I
^11
B
C
a
^ 0,399625
+ 0.400966
+ 0.412793
+0.40082
L
— 0.0002575
4 0.001159
+ 0.013812
+0.001557
c
+ 0.0049412
+ 0.0062417
4 0.012683
+0.00557
d
+ 0.019380
+ 0.026458
+ 0.056221
+0.01975
e
— 0.0033963
—0.16501
TABLE II.
COMPARISON BETWEEN THE PLATINUM RESISTANCE
THERMOMETER AND THE HYDROGEN THERMOMETER.
Temperature
observed with
the hydrogeri
Number
of obser
vations
with the
Resistance
observed
WRj^l
^^I^All
1FBjj
JF~Rc
thermometer.
hydrogen
therm.
in n
0°
137.884
— 29.80
3
121.587
+ 0.025
+ 0.066
+ 0.210
+ 0.063
— 58.75
3
105.040
+ 0.011
— 0.011
+ 0.153
+ 0.0.^8
— 88.14
4
89.277
— 012
— 0.050
 0.001
+ 0.008
— 103.83
3
80.448
— 0.023
— 0.061
— 0.075
— 0.015
— 139.87
3*
59.911
+ 0.004
— 0.005
— 0.082
— 0.005
— 158.83
3
48.929
+ 0.023
+ 0.044
+ 0.008
— 182.75
2
34.858
— 0.029
+ 0.027
+ 0.083
— 035
— 195.18
2
27.595
+ 0.009
+ 0.061
+ 0.148
+ 007
— 204.69
1
22.018
 0.014
+ 0.012
+ 100
— 0.014
— 212.87
3
17.290
— 0.024
— 0.065
— 0.001
— 0.031
— 217.41
4*
14.706 •
+ 0.028
— 0.048
+ 0.270
+ 0.007
— 252.93
2
1.9G3
+ 2.422
+ 0.057
— 0.001
— 259.24
1
1.444
+ 0.199
— 4.201
(213)
In those cases where the W — R have been derived f^om two deter
minations the values in the 2""^^ 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 Aj, this mean error is expressed in resistance
± 0,025 i2, 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 i2, 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 iialf 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.
Jolles and Mr. C. Braak for their assistance in this investigation.
Physics. — "On the measurement of very loio temperatures. XII.
Comparison of the platinum i^esistance thermometer ivith the
gold resistance thermometer. By Prof. H. Kamerlingh Onnes
and J. Clay. Communication N°. 95'^ from the Physical labora
tory at Leiden.
(Communicated in the meeting of June 30, 1906).
§ 1. Introduction. From the investigation of Comm. N°. 93, Oct.
'04, VIÏI 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 soems 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 Meilink'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.
2) Owing to e being negative [B] gives no minimum; a term like that with e
does not contradict, however, tiie supposition ?<;  go at T — (§ 1) as the foimula
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
temperatures — acquired a special xaXne through this peculiarity
of gold.
As will appear from what follows, the 2^omt 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". 93.
The pure gold was furnished through the friendly care of Dr.
C. HoiTSEMA. It has been drawn to a wire of 0,1 mm. in diameter
by Heraeus.
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 tempera
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.
W. 95^).
§ 3. The Results, obtained after the direct and the indirect method
are given in column 3 of table III and indicated by d and i
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. W. 95^ belonging to the resistance of the
platinum thermometer.
( 215 )
TABLE III.
CALIBRATION OF THE GOLD RESISTANCE THERMOMETER.
Datfi.
Temperature
resistance.
Observed
gnld resistance.
W
R
A
W
R
BI
^^^//
1900
51.915«?
1 Febr. 5 b. 57
— 28°96
46 137 e
—
0.002
—
0.018
4 0.029
» 3 h. 40
— 58.58
40 326 1
+
22
+
12
+ 46
» 12 b. 25
 87.43
34 640 t
27
—
3
 2
12 June 2 b. 20
 103.82
31. 432 (^
—
24
+
15
+ 4
» 11 b.
— 139.86
24.284^
—
28
+
16
+ 10
17 Jan. 3 b. 20
— 159.11
20.394 2
—
28
—
1
f 14
1 June 11 h. 50
— 182.75
15.559 rf
+
6
+
4
+ 39
» 5 b. 8
 195.18
12.980 <;
+
30
+
19
+ 40
» 4 h.
 204.69
10.966 (/
+
24
+
18
+ 1
» 3 b.
— 212.87
9.203^/
—
2
— 76
12 Jan. 11 b.
— 210.25
8.400?
—
30
—
17
— 128
18 May 4 h. 10
 252.88
2.364^
+
1.059
+
2
— 6
» 6 h.
— 259.18
2.047 rf
+
1.334

3.277
6
111 order to agree with Dewaii, we ought to have found for the
resistance of the gold wire at the boiling point of hydrogen 1.70812
instead of 2.364 52. Also the further decrease of the resistance found
by Dewar ^) in hydrogen evaporating under a pressure of SOniM.is
greater than that was found by us. We may reinarlt that this latter
decrease of the resistance according to him would belong to a decrease of
4 degrees on the gas thermometer, and that we in accordance with
Travers, Senter and Jaquerod ^) 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 resifitance by a
formula. As to this we refer to what has been said in Comm.
1) Dewar, Proc. Roy. Soc. Vol. 68 p. 360. 190L
2) Travers, Senter and Jaquerod, Phil. Transact. A. 200.
Proc. Roy. Soc. Vol. 68, p. 361, 1901.
15*
( 216 )
N°. 95<^, XII. § 6. The resistance of the gold wire ran be represented
fairly well as far as — 217° as a function of the temperature by a
formula of the form A.
—1=1+ 0,39070 h 0.017936 +
Wo 100 ^ V looy ^
+ 0,0085684 (^—j + 0.0080999 (^— ^^^j . (A)
This formula A is not fit to include the hydrogen temperatures.
For the deviations W — Ra comp. table III.
We have therefore made use of a formula B, and
t \ / «
_i = 1 + 0,382404 (^J + 0,0102335 (^^J +
/^ t Y /lOO 100 \ 1 .„ ..
+ 0,0035218(J 0,0268911 (^^^+ [ (B I)
I /100\' / 100 \^
4 0,0052211 U — „ ^^
I V T J V273,09y/
is in good harmony down to — 253°, while
^;== 1 + 0,394548 (4) + 0,0200118(4)'+
/ t A» /lOO 100 \ 1
+ 0,0102889 (j + 0,0229106 (^ " ^„^ j \(B U)
'loo^' / 100
 0,00094614 11
T J V273,09
)0 n
gives a fair harmony also at — 259^
The deviations are given under the headings W—Rbi and W — Rbu
in columns 5 and 6 of table III. The mean error of an observation
with respect to the comparison with formula B I is ±0,017 52 in
resistance and ± 0°,09 in temperature. Formula B I gives for the
point of inflection of the gold resistance — 220°.
Mathematics. — "■Quadratic complexes of revolution'' By Prof.
Jan de Vries.
§ 1. 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 i2
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 )
aiid it can be distijigui.hcd as a .ri/imnetric complex of revolution.
Til is is the case with the complexes of tangents of surfaces of
revol lit lull.
We determine the general equation of the quadratic complexes of
revolution with axis OZ in the coordinates of ra^s
Ih = ''^' — ■^*' ' Vi — V — V ' 2>3 = ^ — '^'.
i\ — y~ — ^y' ' P5 = •^■' — •^' ' Pt — '^i/' — 3/'^'
Bj substitution of
Pi = «Pi — i^^ ' Pa = /?i^ + «i^ » Ps = ^3'
1\ = «P. — t^/'s ' Ps = /5/>4 + «i>6 ' Pe = Pa'
(where «'  Z^'' = 1) in tlie general quadratic equation we easily
find that the equatioji of an 12 can contain terms only with
ipi' ^P.'h (pi'iPs'h P3' Pe^ ipiPs—P.Pi) and (p^p^ \p,P,)
As the latter combination can be replaced by — />3 ^^^ in consequence
of a wellknown identity we liiid tVjr i2 the equation
^(Pi'+P.=)fi?P3^ + 2Q>3Pa+A^a^+^iP.^+P3^) + 2^T/>iPaP,y^)=0. (1)
If C=0, equation (1) does not change when x is replaced by
— X ; so it represents a symmetrical complex.
The coordinates of rays
q^:=u — u' , q^ = v — v' , g^ =r k — w ,
q^ ■=! vw' — u'v' , ^5 zzz icic — riiv' , q^ z= uv' — vu' ,
where u, v and lu represent the coordinates of planes are connected
with the coordinates p by tlie wellknown relations
Pi'i4=P2'9ó=Pz'q,= Pa qi^p.q^— p, ■ q»
So i2 can also be represented by
^q.'^q^) + ^q^'+^Cq,q,+Bq,^^A{q,^\q,^)^2F{q,q,q,q,) = 0. . (2)
This equation is found out of (1) by exchanging p/c and qf;, and
of A, B, a D, E, F and E, I), C, B, A, — F.
§ 2. The cone of the complex of the point {x\y\z') has as
equation :
A{xw'yJrA{yy'y\B{zz'y^2C{y'xx'y){zz')JrD{y'x~x'yy^
\E{z'yy'zy^E{z'xxzy^2F{xx){x'zz'x)^2F{y—y'){y'zz'y)=0.{^)
In order to find the equation of the singular surface we regard
the cones of the complex whose vertices lie 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 z^
which is to be rejected and substitution of .ï' } ^' = r* for x"", we
find the equation
( 218 )
D{AE  F') r' + {AE J^ BD — C' — F') r' {Ez' — 2Fz^A)\
{ B {Ez'' — 2Fz i Ay = O (4)
As this can be decomposed into two factors of the form
Lr^ { M {Ez^ — 2Fz f ^), th'i singular surface S consists of two
quadratic surfaces of revolution.
These touch each other in the cyclic points /j and I^ of the plane
XOY and in the points B^ and B^ on OZ determined by
Ez^ — 2Fz\^A — {).
The two surfaces cut each other according to the four isotropic
right lines indicated by the equations
A^ ^ y"" = Q and Ez"" — 2Fz \ Az=:Q . . . . (5)
If 52 is symmetric (C=0) the two parts of the singular surface
have as equations
{AE — F') (.r' 4 if) + B {Ez^ — 2Fz\A) = 0, . . (6)
D i^' + y') + Ez' — 2Fz ^ A = (7)
If we find ^ = and D = 0, then S breaks up into the four
planes (5) and i2 is a particular tetraedal complex.
Out of (3) it is easy to find that the cones of the complex of the
points B^, B^, /j and I^ break up into pencils of rays to be counted
double.
These points shall be called hisingular.
§ 3. The rays of the complex resting on a straight line / touch
a surface which is the locus of the vertices of the cones of the
complex touched by /. 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, z') of an arbitrary cone of the complex with OZ are
indicated by the equation
iE{x^ + y^) + Bl^ z'^  2 [F (.^'^ + y^) + Bz] z' + [A {.V^ + y^) ^ Bz^] = 0.
This has two equal roots if
{{AE  F^) (.^.^ ^y') + B {Ez  2 Fz ^ A)] {x^ \y') = ^ . (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 i2 is symmetrical, it forms part
of the singidar surface as is proved out of (6).
Also the axial surface of the right line /<» lying at infinity in
XOY breaks up into two planes, and a quadratic surface. Its
1) Sturm, Liniengeometrie III, p. 3 and 6.
( 219 )
equation is found most easily by regarding the rays of the complex
normal to XOZ. From x = x\ z = z' ensues p^ = 0, i)^ = 0,
2)^ = zp^, p,^ = 0, j^g = — ivp^. By substitution in (i) we lind
{A \ Da' { Ez' — 2 Fz)p^' = 0,
and from this for the indicated surface
J) (.^.^ j^ ,f) ^ Ez' ~ 2 Fz \ A = . . . . (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 l<x,. We can show this
by determining the equation of the axial surface of the right line
z' ^0, y' =zb, and by putting in it b = co. We then find
{Ez' 2Fz } A) {I) {w' J^ tf)\ Ez' — 2Fz ^ A\ = . (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^I^BJ^ as basis.
If in the equation of the cone of the complex the sum of the
coefficients of x"^, y^ and z'^ is equal to zero, then the edges form x^
triplets of mutually perpendicular rays. The vertices of the triortho
gonal (equilateral) cones of the complex belonging to ^ form the
surface of revolution
{D + E) {x^ + 2/') + '^Ez' — 4i^^ + (2^ + ^) = . . (11)
It 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 4 from a right line to OZ is determined by
l^^^^l— (12)
Pi + P,
the angle X between a ray and XOY by
tg^X= — — (13)
So the condition l^ tang X = a furnishes the complex
P8P9 = «(i>l' +Pï') (14)
Here we have a simple example of a symmetrical complex of
revolution.
The equation
V = «(Pa^+P,') (15)
( 220 )
determines a complex £1 whose rajs form with the axis a constant
angle, so they cut a circle lying at infinity.
The equation
<^W=p:+P^ (16)
furnishes a complex Si, whose rays cut the circle .r' }?/' = a'.
For XOY cuts each cone of the complex according to this circle.
If I represents the distance from a ray then
I, _ P,' +P,' +Pe' .^rj.
~ P^" + P2' ^ Pz'
If XOY is displaced along a distance c in its normal direction,
p, and ps pass into (p^ — ci\) and (ps \ cp^). So for the distance
/, from a ray to the point (0, 0, c) we have
, , ip^ + Ps' + Pe') i 2c {p,p,  p,p,) + 0^ (Pi" + Pu') ,, «,
l^ — _ . (ly)
Pr hP, +Pz
If in this equation we substitute — c for c we shall find a relation
for the distance 4 from the ray to point (0,0, — c).
The equation
a, I,' + «, I,' = /?
furnishes a complex £2 with the equation
}^{a,a,)ö{p,p,p,p,) = (19)
This symmetrical complex is very extensively and elementarily
treated by J. Neuberg {Wiskundige Opgaven, IX, p. 334 — 341, and
Annaes da Academia Polytechnica do Porto, I, p. 137 — 150). The
special case «1 ^1 + «2 ^2 = ^ "^^^^ treated by F. Corin {Mathesis, IV,
p. i77_i79, 241—243).
For ^1 = 4 we find simply
P.P.  P.P. = ^ (20)
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 as centre ^).
§ 5. When there is a displacement in the direction of OZ the
coordinates of rays p^, p„ p^ and p„ do not change whilst we obtain
P, = /" + hp, and p, = p, — hp„
so
Pi Pa + Pa Ps = Pi Pi + P. Ps'
The forms {p,^ + Psl and (Pi Pi — P. P.) are now not invariant.
1) This complex is tetraedral. See Sturm, Liniengeometrie, I, p. 364.
( 221 )
When in equation (1) of tlie complex 52 the coefficients E and F
are zero, the complex ii. 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^){x^ \ y"")^ AB^O; (21)
so it consists of a cylinder of revolution and the double laid plane
at infinity.
§ 6. By homographic transformation the complex i2 can be changed
into a quadratic complex with four real bisimjular points.
If we take these as \'ertices of a tetrahedron of coordinates
O^OJJJJ^, it is not diflicult to show that the equation of such a complex
has the form
Ap\, + Bp\^ + 2 Cp,,p,, + 2Dp,,p,, ^2Ep,,p,, = 0. (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{DE)y,^y,'^^2\AB{CD){CE)]y,y,y,y, + B{DE)y,^y,:^ = . (23)
So this consists of two quadratic surfaces, which have the four
right lines 0^0^, 0^0^, OJJi and <J.JJ^ in common.
For A = 0, B = the complex proves to be tetraedral.
For D = E the equation is reducible to
^P\. + Bp\, + 2{C D)p,,p,, = 0,
and indicates t\vo linear complexes.
For the axial surfaces of the edges 0^0^ and O^U^ we lind
,x,x,\2Ax,x,\{DE).v,x,\ = () .... (24)
and
' x,x,\2Bx,x,^{DE)x,x,\^0 .... (25)
For a point (0, t/^, 0, yj of the edge OJJ^ the cone of the complex
is represented by
Ay.^x,^ ^r2{CE)y,y,x,x,\By,^x^ = ^,. . (26)
so it consists of two planes through 0^0 ^.
This proves that the edges 0^0^, 0^0^, OJJ^, OJJ^ are double
rays of the complex ').
1) See Sturm, Liniengeometrie III, pp. 416 and 417.
( 222 )
Physiology". — • "A feiv reniarhs concerning the method of the true
and false cases.'' By Prof. J. K. A. Wertheim Salomonson.
(Communicated by Prof. 0. 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 tlie measure of precision
(Pracisionsmasz) wlien observing differencethresliolds, afterwards to
determine tliese differencethresholds.
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
cases 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. Fechner 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?
Fechner 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 to true cases, v false cases and t dubious cases, he
calculated his measure of precision as if there had been iv {• \t
true cases and \t \ 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 t cases amongst
the true and the false cases.
Fechner 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. Muller,
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 wellknown law of errors. From
the figures for the true false and dubious cases the thresholdvalue
may afterwards be calculated.
I need not mention some other methods, e.g. that of Foucault,
( 223 )
that of Jastrow, because the nielliod of Fouoautt is certainly in
correct (as has been demonstrated among otliers by G. E. Muller),
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 Jie 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 iv, of
false cases v and of dubious cases t.
Generally it is admitted that the reagent has indeed perceived
correctly iv times, that he has been mistaken v times, that he
was in doubt t times. If this premiss were correct, Fechner's or
G. E. Müller'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 plijsical cause is not perceived, in
which liowever 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 ?ü—z; cases that
we may suppose the physical cause to have been really and correctly
perceived ; in all other cases, in 2t> + i 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 perceived and nonperceived cases, the number of which
I will indicate by § and /. The supposition tliat 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 Fechner to exist between the
numbers of true and false cases.
As is well known, Fechner added to the number of true cases,
obtained by the experiment, one half of the dubious cases: he
used therefore in his calculation a rectified number of true cases
10^ =z w \ \ t. In the same manner he corrected the number of false
cases by adding to them likewise one half of the dubious cases :
v'=:v\\L
In calculating the number of my perceived cases, I get § = iv — v,
whilst the number of nonperceived cases is represented by x = ^ + 2v.
Evidently I may also express the number of perceived cases by
As Fechner has given for the relative value of the corrected
number of true cases the expression :
Dh
and for the corrected relative number of false cases the expression
Dh
Dh
( 225 )
we obtain from these immediately for  and x the two relations :
Dh
2 r
and
Dh
= 'vJ'
6 ' «' dt.
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. Müller's
formulae give the same result, saving only the wellknown dif
ference in the integrallimits: these latter being and {Su^D) hu
I need scarcely add that my remarks do not touch in the least
the question about "threshold value" between Fechner and G. E.
Muller.
It is evident, that the result of the calculation of a sufiiciently
extensive series of experiments according to the principles, given in
my remarks should give numbers, closely related to those either of
Fechner or of G. E. Muller — depending on the limits of inte
gration. Still I wish to draw special attention to the fact that the
formulae of G. E. Muller 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 Cattell 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 / are both = 50 V^. 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 jjlait.) 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 :
2
RT =  [w (1— .v) {av^l/aY + a{v—by] . . . (1 )
was derived for the spinodal lines for mixtures of ?2örma/ substances,
on the supposition that a and b are independent of v and T, and
that 01,=! 1/^1^2, while
{€w^]/ay [(1— 2.tO v—^x (1— A')/?] 4
a{v—h){v — 36)
^av—^\/a){av—2^\/a)^
^Vaivby
= (2)
.v{l — ,v)
was found for the ?;,A'projection of the p)^<^'dpoint line, when
a—{/a^ — \/a^ and ^ = b^—bi.
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 ^^0,i.e. b^:=b^, was
assumed, so that then the proportion ^ of the critical temperatures of
the two components is equal to the proportion Jt of the two critical
[/a, b T
pressures. If we then put =(p, =:to, — = t (where 7 „ is the
"third" critical temperature, i. e. the plaitpoint temperature for
V = b), the two preceding equations become :
T = 4a> [41;iO + {(f + .xy (l^ri . . . . (la)
(rp +.7)»(lto)='(l3a>) ^ ^ ^
(12.) + 3{cp 4 ..) {lo>y + ^'^ ^ ; ^ ^ = 0. (2«)
It now appeared that the plaitpoint curve has a double point,
when cf = 1,43, i.e. = ji = 2,89. If <9 > 2,89, the (abnormal) case
of fig. 1 (loc. cit.) presents itself (construed for (p = l, ^ = {l{ 7^)^ = 4);
if on the other hand 6 < 2,89, we find the (normal) case of fig. 2
(loc. cit.) (construed for (f := 2, <9 = 2^/^).
At the same time the possibility was pointed out of the appearance
of a third case (fig. 3, loc. cit.), in which the branch of the plaitpoint
( 227 )
line running from C^ to Cj was ttince touched by a spinodal line.
Here also the hrancli C^A is touched hy a spinodal line [in the tirst
two cases this took place oidy once, either (in fig. 1 , loc. cit.) on the
branch C^A {A is the point x =zO, v^ b), or (in fig. 2 loc. cit.) on
the branch C^A (C„ is the beforementioned third critical point)].
So it appeared that all the abnormal cases found bj Kuenen may
already appear for mixtures of 2)erfectly normal substances.
It is certainly of importance for the theory of the critical phenomena
that the existence of two different brandies 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 fdT\ , 1
■^1 V«'^Vo ^
^i/^fv.V,i/^)i) ..(3)
^ V ^
was derived for the molecular increase of the lower critical temperature
for the quite general case a^^a,, b, ^b„ which equation is reduced
to the very simple expression
A = ^i^i) (3„)
for the case Jt = 1 {p^ = p^).
This formula was confirmed by some observations of Centnekszwer
and BÜCHNER.
d. The fourth paper appeared in the Archives Teyler of Nov. 1905.
Now the restricting supposition /? =: (see b) was relinquished for the
determination of the double point of the plaitpoint line, and the quite
general case a,^a,, b^ < b, was considered. This gave rise to very
intricate calculations, but finally expressions were derived from which
for every value of ^ = ^ the corresponding value ot' jr = — and
^1 Pi '
also the values of x and v in tiic double point can be calculated.
Besides the special case & = jt (see b) also the case .t = 1 was
examined, and it was found that then the double point exists for
<9 ^ 9,90. This point lies then on the line v =z b.
^) The three papers mentioned have together been published in the Arch. Néerl. of
Nov. 1905. * .
( 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 threephasepressure is greater than
the vapour pressures of the components). For the first condition
was found:
for the second
4 Jt [/ Ji
'<W^i' <''
which conditions, therefore, do not always include each other '').
After this the connodal relations for the three pi'incipal types were
discussed in connection with what had already been written before
by KoKTEWEG (Arch. Néerl. 189J) and later by van der Waals (These
Proceedings, March 25, J 905). The successive transformations of main
and branch plait were now thrown into relief in connection ivith the
shape of the jilcLitjwint line, and its splitting up into tivo branches as
examiried by me.
ƒ. In the sixth paper (Arch. Teyler of May 1906) the connodal
relations mentioned were first treated somewhat more fully, in which
also the p, .^'diagrams were given. There it was proved, that the
points i?i, R^ and R\, where the spinodal lines touch the plaitpoint
line, are cusps in the jj>,Tdiagram.
Then a graphical representation was plotted of the corresponding
values of 6 and Jt 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, namelj^ characterize all possible pairs of substances by
the values of 6 and üt, 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
{:t, 6) one type passes into another. As to the transition of type I
to II (III), it is exactly those combinations for which the plaitpoint
line has a double point. In fig. 1 (see the plate) every point of the
t) Inserted in the Arch. Néerl. of May 1906.
2; These results were afterwards confirmed by Verschaffelt (These Proceedings
March 31, 1906; cf. also the footnote on p. 749 of the English translation).
( 229 )
plane denotes a combination {6. rr), to which every time a certain
pair of substances will answer.
T
n :
Pi
X
V6
1,00
7,50
en 0,13
0,96
en
0,040
2,57
en 2,57
1,19
7,21
» 0,13
0,94
9
0,036
2,49
» 2,60
1,71
6,^2G
Ü 0,13
0,84
s
0,025
2,26
5 2,68
1,88
5,76
Ö 0,13
0,78
»
0,021
2,18
» 2,71
2,04
5,42
» 0,12
0,72
i
0,018
2,11
» 2,74
.2,22
4,94
Ö 0,12
0,63
»
0,014
2,02
B 2,79
2,89
2,89
B 0,12
0,24
»
0,003
1,73
B 2,87
9,90
1,00
)) 0,11
O,0i
B
0,001
1,00
» 2,95
00
—
» 0,11
—
»
0,000
—
B 3,00
In the said figure the line C'APB denotes the corresponding
values of 6 and rr from <9 = to 6» = 9,9. For C' <9 = 0, .t = 9,
for .4 6* = 1, rr = 7,5 ; with 6 — 2,22 corresponds n = 4,94. (Case
71 = 19' or a, = aj; for P rr = 6» = 2,89 (Case rr = ^ or b^ = b,);
forB <9 = 9,9, rr = 1. For values of ^ > 9,9 the double point would
lie on the side of the line v = b, where v <^b. It appears from the
figs. 23, 24 and 25 of the said paper, that then the line BB (t ^ 1)
forms the line of demarcation between type I and II (III). For
starting from a point, where t <^ 1 (however little) and ^ is com
paratively low% wiiere therefore we are undoubtedly in region II (III),
we see clearly that we cannot leave this region, when with this
value of T that of 6> is made to increase. For we can never pass
to type I, when not for realizable values of v (so <^b) a. double
point is reached, and now a simple consideration (see the paper
cited) teaches, that for n <^1 a double point would always answer
to a value of v <^b.
Now it is clear that ^ = 0, .t = 9 is the same as ^ = oo, rr := ^Z,;
that 8 = :t = 2,89 is identical with 6 = n = 1/2,89 = 0,35 ; etc., etc.
(the two components have simply been interchanged), so that the
line CA' will correspond with the line C' A, wiiile A' B' corresponds
with AB. If we now consider only values of 6 which are ]> 1, if
in other words we always assume T, ^ T^, we may say that the
16
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 ABB we have the
abnormal type I (C,H, + CH.OH, ether +H,0); on the left of^'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 C\Co
and C\B (see figs. 23 — 25 loc. cit.) in the second region. It is namely
easy to show, (loc. cit.), that for jr > 1 the branches of the plait
point line are either C\C\ and C^A (type II and III), or C,A and
CC (type I), while for jr <[ 1 these branches are C^C^ and C^B
(type II and III) or C\B and C,C\ (type I). The line jt = 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 ^>1) But in practice it will most likely never happen,
that with ^ > 1 a value of j1 corresponds which is much smaller
than 1, for a higher critical pressure goes generally together with a
higher critical temperature. We may therefore say that with a given
value of Jt 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 II (or III) appears when 8 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 R^ and /?/). This investigation 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^^a^ , b^^b^.
Only the special cases ^=0{b^=:b, or üt = 8) and ji = 1 admitted
of calculation, though even then the latter was still pretty complicated.
Then it appeared, that for ^ = the region of type III is exactly
= 0, that it simultaneously appears and disappears in the double
point P, where jr = ^ = 2,89. But in the case ^i = 1 the region
lies between 8 = 4,44 and 8 = 9,9 (the double point). This is
therefore QB in fig, 1 ; i. e. for values of <9 > 1 and < 4,44 we
find type II (see fig. 2«); for <9 = 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 III (fig. 2^) with two points R, and R/,
where the spinodal lines touches the plaitpoint line. This type
disappears in the double point P, where (9 = 9,9 and R, and R^'
coincide in P (fig. 2^), and passes for values of 6» > 9,9 into type I
(fig. 2^). We point out, that the figs. 2«— 2« represent an intermediate
case (i.e. between Jt = 8 and ji~l, see fig. 1), for in the case of
( 231 )
TT =z 1 the branch AR^Cq would coincide with AB (;ü = b). There
fore the special value 4,44 has been replaced by 6*^ (the value of
6 in Q) and the value 9,9 by 6*^, (the value of & in P).
Of the curve which separates type II from type III we know as
jet only the points F and (2 (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. Teylerhave
proved, that this very abnormal type III is possible for mixtures of
normal substances. If the critical pressures of the two components
are the same {n == 1), then we meet with this type when 6 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 numeric results
of our investigation will naturally be modified, when b is not assumed
to be independent of v and T, or when one of the two components
should be associating substances. This will cause the types III and I
to make their appearance earlier than has been derived above (i. e.
with lower values of S with for the rest equal values of tc), but
that qualitatively everything will remain unchanged. This appears
already from the fact that the substitution of the quite general assumption
^1 \. ^a fo^' tlie simplified assumption b^ = b^ (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 6
and jr, and that also the calculations for the limits of type III
(in the second paper in the Arch. Teyler) may be carried out for
the quite general case b^^b^. So the phenomena remain qualitatively
the same for very different pairs of values of b^ and /;,, and will
therefore not change essentially either, when one delinite 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 T 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 T, or
when that substance forms complex molecules.
The longitudinal Plait.
2. In former papers it has been demonstrated that in the neigh
bourhood of Co a minimum plaitpoint temperature makes its appearance
( 232 )
both with type I in the line CoG, and with type II in the line Co^,
and that tiierefore with decrease of temperature a separate plait
begins to detach itself starting from C^ at a definite temperature
Tq (the plaitpoint temperature in Co), 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 wellknown way
a closed connodal curve begins to form within the connodal line
proper, wiiich 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. 3'' and 3^*). In many
cases this branch plait has already appeared before the plait which
starts from Co, begins to develop at somewhat lower temperature.
Later on the two branches coincide (at the minimum temperature
in D), and then form again a continued branch plait (fig. 3'). ^).
Now for the special case h^ = ^^ the point D lies always very
near Co (see the paper in these Proceedings referred to under b.
in § 1). If then e.g. Tj7\ = T/„ then r,„/r„ = 0,96, when Tm
represents the temperature in the minimum at D. The real longi
tudinal plait round Co exists then only at very high pressures
(fig. 3^), while the ojien 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 calculation proves that in the quite general case b^ ^ b, the
point D may get much nearer in the neighbourhood of R^, and also
that the temperature in the plaitpoint Cq may be comparatively high,
so that in opposition to what has been represented in fig. 3" the
longitudinal plait has already long existed round Co 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 tig. 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 .I'l and a', approach to limiting values at ^^ ^ x, without
the longitudinal plait ever closing again — as was formerly considered
possible [cf. inter alia van der Waals, Coiit. II, p. 190 (1900)].
For in consequence of the minimum at D the longitudinal plait
always encloses the point C^. Only at temperatures higher than T^y
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 M has broken through
the connodal curve proper of the transverse plait ; or by fig. 3*,
before a longitudinal plait has developed round Cq.
Of course we may also meet with the case, that the plait round
Co coincides with the branch plait at the moment that the latter
with its plaitpoint just leaves tlie transverse plait, as shown in fig. 5^,
but this involves necessarily a relation between S and t, 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 /Vö»i
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 longitudinal plait round C^. The
latter penetrates then further into the transverse plait, till its meets
the isolated closed connodal curve in D (fig. 6''), after which the
confluence with it takes place in the unrealizable region (fig. 6^).
This plait is then the longitudinal pjlait 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 6 of the critical temperatures of the two components
is in the neighbourhood of 1, and the proportion n of the critical
pressures is at the same time pretty large.
A clear representation of these different relations is also given by
the two ih Tdiagrams of fig. 7 and fig. 7«. (The temperature of C^
is there assumed to be lower than that of R^, but it may just as
well be higher). The plaitpoints 7/ on the part R^A below the cusp
are the unrealizable plaitpoints (see also figs. 3— 6); the plaitpoints
p on the part RJ^I before M also (then the isolated closed connodal
curve has not yet got outside the main plait); the plaitpoints F
beyond M 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 B lies near R^, the critical
mixing point M 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 can 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 /?,, D and M, will be published elsewhere
(in the Arch. Teyler). It is, however, selfevident 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 x^ of the vapour phase is neither in fig. 4^^, nor in
fig. 5« or 6«, the same as the concentration of the two coinciding
liquid phases .^1,2, 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 der Waals, Cont. II, p. 181 (1900)]. Now we
know namely, that when x^ lies between x^ and x^ at lower tem
peratures, this need not continue to be so till x^ and x^ have coincided.
The latter would be quite accidental; in general one of the maxima,
e. g. in the j9,A^line, which lie in the unstable region between x^ and
x^, will get outside the plait before x^ 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 footnote on p. 665.
'aal plait],
^J>
J. J TAN LAAP .The sbspe of the spinodnl and plaj
lal subetacccb ■ iFourth com
: The lüngitudinal plaitj
^i .:(;■■•"'
.
f
1 .*. ,
?
J]p'l '
yf '
;\j. ..>'■'■'
xt" ;
..:..
: i i i>».
■ 1 1 ,,••/.■
.•';/'■
1 1 ; ;
t_
7
( 235 )
Already in a previous paper (These Proceedings Jnne 27 1903} I
had elaborately demonstrated this, and somewhat later (These Proceed
ings 31 Oct. J 903) KuENEN arrived at the same opinion independently
of me. ') And in 1900 Schreinemakers (Z. f. Ph. Ch. 35,p. 462—470)
had experimentally demonstrated that one maximum leaves the
longitudinal plait for exactly the same mixture (phenol and v^^ater),
for which van der Lee thought he could theoretically prove, that
Finally I shall just point out that in the peculiar shape of the
p, Tdiagram of the plaitpoint line (tig. 7) in the neighbourhood of
the point D, and in the fact that the two critical moments represented
by figs. 4« and 4'^ (as B and Al 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 Guthrie as by Rothmund [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 lO"" above this temperature.
1) G. f. also KuENEN : Theorie der Verdampfung und Verfliissigung von Gemischen.
Leipzig 1906, p. 170, note.
••2) For the rest the assumption .'C3=.Xi,2 at the point M leads, as the calcula
tions teach, not only to strange, but to highly absurd conclusions.
3) G. f. also Friedlander, Ueber merkwiirdige Erscbeinungen in der Umgebung
des kritiscben Punktes. Z. f. Ph. Gh. 38, p. 385 (1901).
(October 25, 1906).
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.
KONINKLIJKE AKADEmE VAN WETENSCHAPPEN
TE AMSTERDAM.
PROCEEDINGS OF THE MEETING
of Saturday October 27, 1906.
(Translated from: Verslag van de gewone vergadering der Wis en Natuurkundige
Afdeeling van Zaterdag 27 October 1906, Dl. XV).
OOlsTTEIvTTS.
L. E. J. Brouwer: "The forcefield of the nonEuclidean spaces with positive curvature"»
(Communicated by Prof. D. J. Korteweg), p. 250.
W. VAN Bemmelen : "On magnetic disturbances as recorded at Batavia", p. 266.
J. J. Blanksma: "Nitration of metasubstituted phenols". (Communicated by Prof. A. F.
Holleman), p. 278.
A. F. Holleman and H. A. Sirks: "The six isomeric dinitrobenzoic acids", p. 280.
A. F. Holleman and J. Huisinga: "On the nitration of phthalic acid and isophthalic acid",
p. 286.
A. Pannekoek: "The relation between the spectra and the colours of the stars". (Commu
nicated by Prof. H. G. van dk Sande Bakhutzen), p. 292.
R. A. Weerman: "Action of potassium hypochloiite on cinnamide". (Communicated by Prof.
S. A. Hoooewerff), p. 303.
J. A. C. OuDEMANS : "Mutual occultations and eclipses of the satellites of Jupiter in 1908",
p. 304. (With one plate).
H. Eysbroek: "On the Amboceptors of an antistreptococcus serum". (Communicated by
Prof C. H. H. Sphonck), p. 336.
W. H. Julius : "Arbitrary distribution of light in dispersion bands, and its bearing on spec
troscopy and astrophysics", p. 343. (With 2 plates).
F. M. Jaeger: "On a substance which possesses numerous different liquid phases of which three
at least are stable in regard to the isotropous liijuid". (Communicated by Prof H. W. Bakhuis
Roozeboom), p. 359.
H. W. Bakhuis Roozeboom: "The behaviour of the halogens towards each other", p. 363.
W. A. Versluys: "Second communication on the Pliicker equivalents of a cyclic point of a
twisted curve". (Communicated by Prof. P. H. Scuoute), p. 364.
H. Kamerlingh 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.
17
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 250 )
Mathematics. — ''The force field of the nonEuclidean spaces
with positive curvature" by Mr. L. E. J. Brouwer. (Commu
nicated by Prof. D. J. Korteweg).
(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 *S/>„'s. But on account of the fmiteness 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 Spn that the total sum of the divergency is 0; for the
outgoing vectorcurrents out of the different spaceelements destroy
each other. This proves already that as elementary qX we can but
take the field of a double point.
Schering (Göttinger ^Nachrichten 1873), and Killing (Crelle's Journal,
1885) give as elementary gradient field the derivative of the potential
; — = Vn (r). ')
stn"~^ r
r
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.
II. If we apply for a spherical Spn the theorem of Green to the
whole space (i. e. to the two halves, in which it is divided by an
arbitrary closed Spn\, together), doing this particularly for a scalar
function <f which we presuppose to have nowhere divergency and
a scalar function having only in t^^'0 arbitrary points P^ and P^
equal and opposite divergencies and nowhere else (such functions
we shall deduce in the following), we then find
i. 0. w. <p is a constant, the points P^ and P^ being taken arbitrarily.
1) A, B and G refer to : "The force field of the nonEuclidean spaces with nega
tive curvature". (See these Proceedings, June 30, 1906).
2) We put the space constant = 1, just as we did in the hyperbolic spaces.
( 251 )
So there is no oX possible with nowhere divergency, thus no ^X
having nowhere rotation and nowiiere divergency, and from this
ensues :
^ A linevector distribution in a spherical Spn is determined uniformly
by its rotation and its divergency.
III. The general vector distribution in a spherical Spn must thus
be obtainable again as an arbitrary integral of :
1. fields E^, whose second derivative consists of two equal and
opposite scalar values close to each other.
2. fields E^, 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 E^ and E^ have
an identical structure.
IV. For the spherical Sp^ there exists a simple way to find
the field E^ 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
doublepointaxis — in the plane q and (p, on the sphere r and <p —
we have :
^ Q ^= tan 4 r.
cos (f
The potential in the plane: — becomes on the sphere:
Q
\cos (p cot \ r.
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 E^. (If we place in the opposite point of
the double point an other double point in such a way that tiie
unequal poles correspond as opposite points, we find as potential
cos <p
^ COS (f, {cot \ r]rtan { r) = — — , which is the Schering potential of a
sm r ^
double point).
V. Here too we can meanwhile break up the field of a double
point into two fictitious "fields of a single eigens point"; for this
■K
we have but to take \\cot^rdrz=:~lsin^^r^F{r)', so that for an
r
17*
( 252 )
arbitrary gradient distribution holds
lX = \T/ f ^^—F{r)dt (i)
The "field of a single agens point" has however divergencies every
where on the sphere.
VI. Out of the field E^ we deduce in an analogous way as under
B § VI the field E^ of a rotation double point normal to the agens
doublepoint of the field E^. As scalar value of the plani vector potential
we find there:
^ sin (p cot I r,
as we had to expect, completely dual to the scalar potential of the
field E,.
As fictitious force field of a unityrotationelement we deduce out
of this (in the manner of B § VI) .
è cot ^ r,
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 rotationelement:
^cot h rdr:z^F (r),
SO that for an arbitrary 2X:
. r w 2A
And an arbitrary vector field is the \7 of a potential
VX
F (r) dr.
s
2jr
E. The spherical Sp^.
I. The purpose is in the first place to find E^; 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 B with poles P^ and P^, and
on ^ a principal circle C with poles Q, and Q, determining on B
meridian circles M cutting C in points H.
( 253 )
We can construct in the first place out of the Schering potential
the potential of two double points, in P^ and P,, the positive
poles of which are both directed toAvards Q^ (so that in opposite
points unequal poles correspond). Let us determine a point S of the
hypersphere by the distance PS = r and /^ QPS = (p (where for P
and Q the index 1 or 2 must be taken according to S lying with
Pi or with Pj on the same side of P), then this potential (a) becomes
cos w
sin^ r
where the sign + ( — ) must be taken for the half hyperspheres
between P^ (Pj and B.
This field has no other divergency but that of the double points
P, and P,.
If we now reverse the sign of the potential in the half hyper
sphere on the side of P,, we obtain the potential (^):
cos<p
sin' r
The divergency of this consists in the first place of two double
points, one directed in P^ towards Q^ and 0)ie 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 B varying in intensity according to cos y.
II. 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 cos (f. Now a field of a magnetic scale
in B 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=: ^ .t — ^ QHS
(P and Q to be provided with indices in the way indicated above
according to the place of S) = tan~^ \cos (p 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(pf{r), wiiere
ƒ (ƒ•) = i cos<p . tan^ [cos (f tan r\ chxt , {dm representing the element
I r
cos (f 2
( 254 )
of the solid angle about P), whilst this integral field has as only
divergency a magnetic scale in B with intensity proportional to
cos (p.
Effecting the integration we obtain :
■n
/(r) =z 2jr j sin <p cos (p ta7i~^ [cos (p tan r] dip.
Ir ]
— cot r 4 ^—  ,
sin r )
and for the corresponding potential function (y) we find :
Ir )
— cot r \ — : I .
sin^ r]
III. If we take the difference of the field (^) multiplied by ^ and
the field (y) multiplied by — the magnetic scale in B disappears
and we have left the field (d) :
— r
. \ cot r\ ,
Jt \ sin ^r )
which field has as only divergency two double points in P^ and P,
of which in the opposite points equal poles correspond.
The sum of this field (rf) and the field («) multiplied by h must
now give a field having as divergency a single double point with
unitymoment in Pj, i. o. w. the field ^j.
We therefore find on the half hypersphere between Pj and B:
1 ijt — r )
— cos (p I — ; 1 cotr\
jr ( sill ^r )
and on the half hypersphere between P^ and B :
1 \—r )
— cos (p I ; \cotr\ ,
n { sin ^r )
or if we define on both halves the coordinates 7^ and <p according to
Pi and Pj Qi we obtain the following expression holding for both halves:
1 ijt — r I
~ cos (p I — : \ cotr) :ir '^^ ( v ^^^ ^ •
3t ( sin ^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  ip (?•) c?r = F^ (r).
( 255 )
Then for an arbitrary gradient distribution holds :
Jx = v/J
 ^ ^/ ° FAr)dr. . . , . . . (/)
4jr
V. The field E^ 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
4:ji along itself.
According to the method of A § IX we find of this field E^ the
planivector potential H in the meridian plane and independent of the
azimuth.
We find when writing n — r = ^ :
So
2 =  sin ^(p(l \ ^ cot r) dd:
jt
1 . I \ ^cotr
H =. — sin<p ; ,
n sin r
vanishing along all principal circles in the opposite point.
From which we deduce for the force of an element of current
with unityintensity in the origin directed according to the axis of
the spherical system of coordinates :
1 , 1 \ ^cotr
— sin (p ; ,
n smr
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 :
d I , , 1 d I )
—\U sin w sin r dcp] dr — ^ \U cos (pdr \ d(p z=z
Or \ ) 0(p { )
1 , \ \ ^cotr
= — sin<p ; . dr . sm r d<p.
n sin r
Or:
d ) 1
U — — Usinr\——i\{^cotr),
Or 1 ^
of which the solution is
c 1 I i3'
cos^ \r n \ cos* \r sinr\
( 256 )
We choose c:=0, and we find a,s vector potential V of a unity
element of current :
^> .... lJiL_A=^,H.
jt ^cos' 4 r smr]
directed parallel to the element of current. The function i^^ (r) vanishes
in the opposite point.
. For an arbitrary flux now holds:
I '.„'
lX=sr/J)l^^FAr)dr (//)
And finally the arbitrary vector field X is the V of the potential:
F. The spherical Spn
I. To find the field E^ we set to work in an analogous way as
for the spherical Sp^. The principal sphere B becomes here a.
"^sphere B; the principal circle C of the points H a principal
«— ^sphere C of the points H.
For the potential («) is found:
cos w
for the potential (/?) :
cos ip
  s^w"~"' r
this field (j3) has in the sphere B a magnetic "— ^scale.
The potential (y) is integrated out of fields tan~ ^ \ cos ip tan r \
according to cos (p, the first zonal "—^spherical harmonic on B. This
integration furnishes when dw represents the element of the wdimen
sional solid angle about P:
cos<pf{r),
where :
ït
f(r) =. I cos<ptan~^ I cos <ptanr\ dw =■ ^/j— 1 1 sin^~^(p cos(ptan~^  cos(ptanr\ dcp =
n
k,i—i f , tan r d<p
I sm "O) 
ij ^1
n — \J \\ tan ^r cos ^(p
[hn defined as under C § III).
( 257 )
Putting under the sign of the integnil a factor si?i "^ ta7i %• outside
1
the brackets and, by regarding that factor as  — {l\'cos^(f>tafi^r),
cos r
writing the integral as sum of two integrals to the former of
which the same division in two is applied, etc., we find, if we write
I sin^ r dr =z Sh :
(nl)/(r) .
n— 1^ _ — g^f^ 7j— 2y. QQs r Sn—2 — siil »— 4/> cos r aS„_4 . .
Kl
.... — sin ^r cos r S^ \ ji {1 — cos r)
(for n even)
=. — sin «— 2r cos r Sn—2 — sin n—4j. (.qs r /S„_4 ....
.... — sin r cos r S^ \ 2 r
(for n odd)
(nl)(n3)....
(n— 2)(/j — 4)
— I sin n—ir dr = (w — 1) Sn—2 i sin »— V c?r,
(for n even)
^^ (nl)(n3)....
— I sin "— V dr =. (n — 1) S,i—2  sin ^—^r dr.
(71—2) (/I— 4).
(for n odd)
If we write ê„ for 2 . jr . 2 . jr . 2 . . . ., to n factors, we have
kn = » and =z On—}.
" (n2) (n4) . . . .' K
Therefore :
r
ƒ (?') sin n— V = ^„ I sin "— ir c?/* ,
and the potential (y) becomes :
r
COS y /* ,
kn— isin"~^r dr.
sin ""~'r^
II. We find the field {(f) by taking difference of field (^) multi
1 1
plied by è and field (y) by 7^ — = 7 — , i. e.
( 258 )
sin «—V dr \ sin «— V
cos (p rj
''— V dr
cos (p '' " ^ '
sin "—Ir Sn—l sin "—V Sn—\
This field has as only divergency two double points, in Pj and
P3, of which equal poles correspond in the opposite points. The field
E^ is then obtained by adding to it the field («) multiplied by è.
We find on the half "sphere between P^ and B:
n
1 COS ^ . .
'r dr.
)s <p r ,
— — I sm "■
 ' ■ Sn 1 sin
On the half "sphere between P, and B:
r
1 costp r ,
." ' . — I S171 n^^r dr.
Sn—i sin^~^rj
Or, if we define on both halves the coordinates r and y according to
P, and PjQi, we arrive at the expression holding for both halves:
1 cos<p r ,
. . I sm "— 'r dr ■^;:^ ip„ \r .) cos (f.
Sn—\ sin "— ^r J
r
III. For the potential of the fictitious "field of a single agens
point" we find :
ƒ•
ipn {r) dr — P, (r).
And for the arbitrary gradient distribution holds
lx^y/J\^FAr)dr (/)
Of the divergency distribution of F^{r) in points of a general posi
tion we know that, taken for two completely arbitrary centra
(fictitious agens points) with opposite sign and then summed up,
it furnishes : 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 F^ (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 Pi and out of this the field P^.
The differential equation becomes :
( 259 )
'sin "— V . 1 :=: o sin «— 'r (fl)
dr I dr \
sm «— 'r . =c t sin "—V dr.
dr
= c \ sin ""
ƒ■■"■■
If the field E^ is to be composed out of the function F^ (r) then
the opposite point of the centre may not have a finite outgoing
vector current; we therefore put I sin "— V dr =i 0, so that we get
 =. ; I sin «— ir dr,
sm n— V ^
dF,
dr
which corresponds to the above result.
IV. The field E^ of a small vortex «— ^sphere according to Spn~i,
perpendicular to the axis of the just considered double point, is iden
tical to that field Ei outside the origin; but now each force line is
closed and has a line integral k,i along itself.
According to the method of C ^ VII we shall find of this field
E^ the planivector potential H, lying in the meridian plane and depen
dent only on r and cp ;so that it is a iX We find :
dh = ce sin ^~r sin ^'~'^<p.
Force in rdirection :
I sin "
~V dr
cot V
(n — 1) cos (fi \ — [ —— .  — ——, \ = {n—l) cos (p . «>„ (r.)
9
2 z=. I (n — 1) cos <p ojn {r) . cs sin «—2^ ^j^ "—2^ . sin r dtp ■=.
z=. oi^^r . cs sin " — V sin " — ^(f.
■ ^ . . _
H = — =2 On (r) sin r sin g) zn'An (^) sin (p.
dh
From this ensues for the force of a plane vortex element with
unityintensity in the origin :
Xn (r) sin (p,
( 260 )
directed parallel to the acting vortex element and projecting itself on
that plane according to the tangent to a concentric circle; whilst tp
is the angle of the radiusvector with the Spn—2 perpendicular to
the vortex element.
V. In the same way as in C § IX we deduce from this the
plant vector potential F of a vortex element directed everywhere
parallel to the vortex element and of which the scalar value is a
function of r only. That scalar value U of that vector potential is here
determined by the differential equation :
\U COS (fi . dr . cs sin ^—^r cos ^~^(p \ d(p —
0^ ( )
\U sin<p . sin r d(p . ce sin "— 3r cos »— 3y  dr z=z
br I )
r= /„ [r) sin <p . sin r dip . dr . ce sin f—^r cos "~^g}.
dU
(w— 2) ü sinr — {n — 2>) U cosr ■=. Xn {r) sm r.
dr
dU
_ {n2) Utghr=yin W
dr
^ = orn 2U • r^^' '^""'^ 2 ^ • Xn (r) dr ,
cos 2(.n— 2;i r J
r
a function vanishing in the opposite point, which we put ^ F^ (r).
We then find for an arbitrary flux :
lX:^\f/J^^FAr)dT (//)
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 :
X = V j ƒ ^ F, (r) dr + ƒ ^ ^. W ^^1
G. The Elliptic Sp»^
Also for the elliptic Sp^ the derivative of an arbitrary linevector
distribution is an integral of elementary vortex systems Voy and
VOz, which are respectively the first and the second derivative of
( 261 )
an isolated line vector. For elementary o^ ^^'e shall thus have to put the
field of a divergency double point.
r dr
The Schering elementary potential I — ^ t'„ (r) is here a plu
J sin " — 'r
r
rivalent function (comp. Klein, Vorlesungen über NichtEuklidische
Geometrie II, p. 208, 209) ; it must thus be regarded as senseless.
II. The unilateral elliptic Sp^ is enclosed by a plane Spn—\,
regarded twice with opposite normal direction, as a bilateral singly
connected *S/9„segment by a bilateral closed Spn—\ If we apply to
the Spn enclosed in this way the theorem of Green for a scalar
function (p 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 :
i. 0. w. ^ is a constant, the points P^ and P, being arbitrarily chosen.
So no u^ is possible ha\ing nowhere divergency, so no X having
nowhere rotation and nowhere divergency; and from this ensues:
A linevector distribution in an elliptical Spn is uniformly deter
mined by its rotation and its divergency.
III. So we consider :
1. the field E^, with as second derivative two equal and opposite
scalar values quite close together.
2 . the field E^ 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 E^ and E^ are of
identical structure.
IV. To find the potential of the field E^ we shall represent it
unibivalently'' on the spherical Spn; the representation will have as
divergency two doublepoints in opposite points, where equal poles
correspond "as^opposite points ; it will thus be the field (rf), deduced
under F § II, multiplied by 2 :
( 262 )
VïTT
COS <p
I'.
ƒ■
Xn (r) cos (p.
sin "— V 4 Sn—l
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 Sj),!—! 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 Spn—], have the equation :
I r . )
sin "— 'yi Jsin "—V  [n — 1) cot r I sin "— 'r dr\ = dr 1.
' %J )
r
The Sjin—i of zero potential consists of the pole Spn—\ and the
equator Sp,.—i 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
j A„ (r) dr. It is rational to let it become in the pole S}^—] ; so
we find :
Xn{r)dr = F,{r),
r
and for the arbitrary gradient distribution holds :
72
IX = ^ C^^F,{r)dT (/)
We could also have found F^ {r) out of the differential equation
{H) of F § III, which it must satisfy on the same grounds as have
been asseited there. For the elliptic aS/>« also we find:
dF,
c .
I sin " — V dr
dr sin " — V
But here in the pole Spn—x, Ijhig symmetrically with respect to
the centre of the field, the force, thus I sin^^—'^rdr must be 0; so
that we find :
dr
( 263 )
1/2 TT
= r— — — I sin n— V dr.
sm ^~^rj
VI. In the usual way we deduce the iJT, which is planivector
potential of the field E^.
dh = c£ sm n— S/* sin "~2 ^^
Force in /direction :
^ = I (n — 1) cos ^ . ii)i (r) . CE sin '^—r sin '^—^(p . sin r d(p =:
■= fi„ (r) . c£ sm «— ly sm "~'y.
/if = — = f*n (^') SÏW r sin (p = x„ (r) sin <p.
dh
From which ensues for the force of a plane vortex element with
unity intensity in the origin :
Hji (r) sin y,
directed parallel to the acting vortex element and projecting itself
on its plane according to the tangent to a concentric circle; <p is
here the angle of the radiusvector w^ith the 82)71—2 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 elHptic
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 SjJn (in
their ^sphere the two indicatrices are then oppositely directed).
The vector potential in a point of the elliptic Spn 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 C/i normal to the radius vector, according to the
formula :
U
cos
i=— i—  fcos^(n2)irXn{r)dr{
(p cos ^(n— i; ^r ^
+ • .(n 2) 1 r^' '^""'^ ^ ^' • ^» (^) ^^'^
TT — r
2. a component f/, through the radius vector, according to the
formula :
u 1 r
sm (f cos ^vn—'i) 1 r j
r
 . J ,, , fcosKn2)ir.Xn{r)dr.
sm ^K^^—^J \r ^
■K — r
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 (r,, then
2X=\y I '^ ^^ ' ^^ dx {II)
^^ J k.
holds for an arbitrary flux in the elliptic 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:
='ƒ
kji
VIII. In particular for the elliptic Sp^ the results are:
Potential of an agens double point:
cos cp J. 2 COS w i(è^— ^) , I
\ cotr) ^
shi^ r \ S^ rr I sin V
or if we put \:x — rz=y .
2cosip \ y \
7t sm r
Equation of the boundary lines of force:
( 265 )
sin Vp (I f y cot r) =z ziz l.
Potential of a single agens point :
2
— . y . cot r.
rr
Vector potential of an elementary circular current;
2 . I ^ y cotr
— sin (f . ; .
ct sin r
So also force of an element of current :
2 . \ \ y cotr
sin If . ; .
ct sin r
Linevector potential of an element of current :
, . cos ifi \ ^^ ct { r'
according? to tiie racliusvector : 1 —
Ct \ cos^ A r sin r si?i^ ^ r
2 ' oc/t ƒ ot/t Q
sinq\ i/3' 2r—ct \r
normal to the racliusvector: 1
TT— 1
Ct ( cos^ I r sin r sin^ ^ r '
IX. For the elliptic plane we find :
Potential of an agens double point :
cos ff cot r.
Equation of the boundary lines of force :
sin (fi z=z dz sin r, or ff =.
Potential of a single agens point :
— I sin r.
Scalar value of the plani vector potential of a double point of rotation:
sin (p
sin r
Thus also force of a rotation element :
sin cp
siïi r
Planivector potential of a rotation element :
I cot ^ r.
We notice that the duality of both potentials and both derivatives
existing for the spherical Sj),, has disappeared again in these results.
The reason of this is that for tiie 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
1
great ; — .
sm r . 
Postscript. Ill the formula for vector fields in hyperbolic spaces:
Pot. X = Pl^ F, {r) dx + {^%— F, (r) dx
nothing for the moment results from the deduction but that to xy Y
and \y 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 \^ X and \y X in finite.
For the y/ at infinity pro surfaceunity of the infinitely great
sphere is <^ order e~^'\ the potentialeffect of this in finite becomes
<^ order re («2)' X c^' z=r v^ ("!)'•; so the forceeffect <^ order
g— (n— i)r. gQ the forceeffect of the entire infinitely great spherical
surface is infinitesimal.
And the \y at infinity pro surfaceunity is <[ order — ; it fur
1
nishes a potentialeffect in finite < order e^"^'''. — , thus a force
effect <i order e— («— D'. — ; so the forceeffect, caused by the infi
1
nite, remains <i order —  .
The reasoning does not hold for the force field of the hyperbolical
>S/^2 in the second interpretation (see under B § VIII), 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 Bemmelen.
(Communicated in the meeiing of September 29, 1906).
Some months ago Mr. Maunder of the GreenwichObservatory
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 infiuence of the synodic rotation of the sun to
the occurrence of disturbances.
Mr. Maunder concludes from an inspection of the disturbances
( 267 )
recorded at Greenwich (and also at Toj'onto) that they show a
tendency to recur after a synodic rotation of the 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 sunspots."
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 anotlier 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 1 called the postturbation '), the
w^ell 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, loith this dijj'erence, 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 a.vis of the earth.
') Gf. Meteorologische Zeitschrift 1895, p. 321. Terrestrisch Magnetisme I p. 95,
11 115, V 123, VIII 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 above
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
casu the horizontal intensity, which in Batavia is most liable
to disturbance);
4. for a period with nearly constant scalevalue of the curves
(1 mm. = ± 0,00005 C.G.S.);
For each storm has been noted:
1. the hour of commencement;
2. „ „ „ expiration;
3. „ „ „ maximum;
4. the intensity.
Mr. Maunder calls a storm with a sudden start an Sstorm;
analogously I will call one with a gradual beginning a Gstorm.
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 Gstorm 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 Gstorras
(^69 )
commencing at the odd hours for one half to the preceding and for
the other half to the following hour.
Because a storm as a rule expires gradually, it is often'irapossible
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 meaji Hforce reaches its lowest
level is a better timemeasure for the stormmaximum, 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 distrihution of the heginning of storms.
It is a known fact, that the starting impulse is felt simultaneously
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 7^12™
„19 „ 7
„28 „76
Mean 7^ 7^^153
True difference l^l^l^K
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 ,Simpulse 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 Bataviairapulses show indeed an unequal
hourly distribution. We find them more frequent at 6'^ and 10'' a. m.
and 7^ p. m.
( 270 O
Hourly distribution of ASimpulses.
Hour
Number
ino/o
Hour
Number
ino/o
a. rn.
4.1
noon
12
4.7
i
2.5
13
5.0
2
3.0
14
3.3
3
2_2
15
3.9
4
4.1
16
4.4
5
3.9
17
3.6
6
6.3
18
4.1
7
4.5
19
5.5
8
5.1
20
3.6
9
5.8
21
3.9
10
6.1
22
3.6
11
5.0
23
3.3
This same distribution we lind again in the case of the 6^storms,
but much more pronounced; a principal maximum at about S^^ a. m.,
and a secondary one at 6^' p. m.
Accordingly the hour of commencement of the (rdisturbances 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 (rdisturbance also favour the development of an >Sim pulse.
Hourly distribution of the commencement of Gstorms (in 7o)
Hour
Intensity: 1
» 2
» 3 and 4
All
noon
2 4 6 8 10 12 14 16 18 20 22
6.0 6.0 5.1 6.7 18^ 17.9 7.4 4£ 6.0 8_£ 6.5 7.1
4.5 4.9 4.2 5.7 20^ 16.4 7.3 5.6 5^ 9^ 8.4 7.6
7.1 3.5 4.3 7.5 18?5 13.4 3.9 5.5 5.1 9.1 11.8 10.2
5.4 5.1 4.6 6.3 19.7 16.5 6.8 5.2 5.6 8.7 8.3 7.8
( 271 )
o
Hourly distribrution of the maximum (in 7o)
Hour '2 4 6
noon
10 I'i U 1Ü 18 20 22
Intensity
'11.5 6.7 5.51.84.712.812.65.3 6.1 6.110.916.2
2 I 10.9 9.1 5.91.21.4 7.3 7.63.9 7.610.813.515.2
» 3 and 4
All
Intensity
» 3 and 4
11.2 4.0 2.41.60.8 4.0 7.26.812.013.617.219.2
14.1 7.4 4.9 1^2.4 8.6 9_2 4^ 7.8 9.7 13.3 16.3
12.3 1G_7 10.9 5.8 4_3 13^ 5.8 2.9 2_^ 5.110.1 9.4
11.3 T? 3.6 5.6 3_3 8.5 11_^ 8.5 6.1 5.2 14.1 14.9
12.2 9.3 5.8 3.2 3.5 7.7 9.0 6.4 8.0 7.7 10.3 17^
All
11.9 10.2 6.04.63.6 9.2 9.26.5 6.3 6.311.614.8
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 Gstorms.
The period has a principal maximum at lO"! p. m. and a secondary
one at noon; and being compared to the diurnal periodicity of the
commencement of Gstorms, it is evident, that: On the hours when
the chance for a maxiinumagitation begins to increase, ive may expect
most storms to. take a start.
Hence we may come to the following supposition.
The susceptibility of the earth's magnetic field to magnetic agitation
is liable to a diurnal and semidiurnal ijeriodicity. Whatever may be
the origin of the increase 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 S'^ 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 I he early moniing, is reached in
the afternoon, and a second rise follows till a maximum is attained,
shortly before midnight.
( 272 )
The daywaves however are smaller and shorter, the nightwaves
larger and longer and also more regular in shape. These regular
nightwaves 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
a. m.
163
12
66
2
172
14
60
4
204
16
43
—
~~
6
140
18
50
8
60
20
46
10
36
22
49
Quite in agreement with the above mentioned conclusions, the
curve representing the diurnal periodicity of the finalhour is nearly
the reverse of that for the maximum.
Evidently the hour (the end of the day) has been strongly
favoured.
Resuming we may according to the Batavia disturbancerecord
draw the following conclusions :
1^*. the origin of Sstorms is cosmical ;
2"*^. the origin of (jistorms may he also cosmical, hut the com
mencement is dependent on the local hour;
3^'^. the development of all storms, concerning the agitation, is hi
the same way depeiident on the local hour.
Storms and sunspots.
lu the following table the year has been reckoned from April 1^
till April l«t 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 •/„
Suuspot
Intensity
Year
i
c
)
3ar
id 4
All
number
S
G
S
G
S
G
S
G
1880/81
37.5
2.7
4.2
1.5
6.6
1.2
2.3
1.6
5.1
81/82
56.9
1.4
6.7
0.0
5.9
1.2
2.3
0.8
5.6
82
70.8
1.4
2.8
5.3
6.6
13.0
6.1
7.9
5.2
83/84
68.8
6.8
7.8
7.5
5.4
6.9
5.3
7.1
6.2
84/85
59.5
2.7
4.6
9.8
6.4
9.3
6.1
8.2
5.7
85/86
45.7
4.1
1.4
9.8
4.1
10.6
6.9
9.0
3.6
86/87
19.6
0.0
1.1
2.3
5.9
1.9
7.6
1.6
4.5
87/88
11.0
2.7
7.8
3.0
4.3
3.7
4.6
3.3
5.6
88/89
6.4
5.4
5.7
4.5
4.1
2.5
3.1
3.8
4.5
89/90
5^
10.8
4.9
5.3
3.3
1.2
4.6
4.6
4.1
90/91
13.0
13.5
8.1
2.3
3.8
0.6
3.1
3.8
5.2
91/92
47.4
6.8
4.6
5.3
5.1
6.9
8.4
4.7
5.5
92/93
74.5
9.5
6.4
8.3
3.3
12.4
4.6
10.3
4.6
93/94
85.2
9.5
6.0
9.0
3.8
11.8
5.3
10.3
4.8
94/95
74.2
5.4
4.6
8.3
7.7
3.7
6.9
5.7
6.5
95/96
57.4
6.8
5.3
5.3
6.9
5.0
8.4
5.4
6.6
96/97
38.7
4.1
4.2
5.3
5.4
2.5
3.1
3.8
4.6
97/98
26.5
4.1
5.3
3.0
5.4
3.7
7.6
3.5
5.7
98/99
22.9
2.7
8.5
4.5
5.9
1.9
3.8
3.0
6.5
From these numbers it appears that those for the Gstorms show
no correspondence with the sunspotnurabers, also that those for the
Sstorms show a correspondence which is emphasised according as
the intensity increases, and finally that the Sstorms show a maximum
when the Gstorms have a minimum and the reverse.
This latter fact is apparently caused by the circumstance of the
storms hiding each other, the Gstorms being eclipsed by the Sstorms
in a higher degree during greater activity of the sun, than the S
storms by the Gstorms. Indeed a simple inspection of the diagrams
( 274 )
shows that the agilation of Gstorms is greater during a sunspot
maximum, than in minimumjears. Also in maximumyears the S
storms of intensity 1, are hidden by their stronger brothers to such
an extent, that the elevenyearly periodicity is nearly the reverse
for them.
Annual distribution of S and Gstorms.
(Only the uninterrupted period April 1, 1883— April 1, 1899
has been considered).
Numbers
Month
s.
G.
January
31
54
Februairy
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 Sstorms can be
noticed. The Gstorms have no annual periodicity as to their frequency,
whereas the Sstorms 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 Simpulse during the day and year.
Comparison ivith Greenwichstorms.
Maunder 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 recordingsheets 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
1'' 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 Batavialist, 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 diifer no less than 60
degrees in latitude. But the Greenwich dates, quoted from a complete
magnetic calendar, prepared by Mr. Ellis and extending from 1848
to 1902 give no separation of G and Sstorms. Thus it is not
possible to decide whether at Greenwich the Gstorms lack an annual
periodicity in their frequency.
TJie 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, tvhich might teach us much about the
manlier the Sstorms 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 GreenwichObservations,
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 Sl cases and have found on an average :
HAD AH AZ
25 V W + 77 r f 39 y (1 7 = 0.00001 C.G.S.).
Batavia. The preceding impulse is missing for H and Z, only for
D it is often present.
Summarizing them, we have:
Station
H
D
Potsdam
+
W
Greenwich
+
W
ZiKaWei
1
E
Batavia
+
W
Cape Hoorn
+
W
( 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 ,, ,, 3.5 min. ; 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 disturbancecurves 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.
Z
Consequently with one exception for D and one for Z we find that :
the commencing impulse of the Sstorms is the i^everse of the vector
oj postturhation, 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 Maundbr's results. From the statistics
based on the record of disturbances at Batavia it might be concluded
that it is chiefly the Sstorms that find their origin in the sudden
encounter of the earth with such a stream.
And as the earth is first struck at its sunsetarc, it is not impos
sible that the Gstorms, 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
Gstorms find their origin by electrified particles being propagated
by the lightpressure 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. Arrhenius 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 Bezold, 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 Simpulse will be the consequence.
The sudden charge of the extreme layers of the atmosphere with
negative electricity, Avill 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 Sstorms; 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 latitudeparallels
and accordingly the vectors of postturbation to point to the true
south and not to the soutiierly end of the earth's mean magnetic axis.
( 278 )
Perhaps we may find an explanation for tliis fact in the influence
no doubt exerted by the earth's mean magnetic held 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 Kelvin, of
allowing an expenditure of the sun's energy causing magnetic disturb
ances, much too great to be admitted.
Chree (Terr. Magnet. X, p. 9) points to the fact, that also Maunder's
deflned streams require far too great an expenditure of energy.
According to my opinion we have only to deal with (he 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 rotationenergy 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. — "titration of metasubstituted phenols" . By Dr. J. J.
Blanksma. (Communicated by Prof. Holleman).
(Communicated in the meeting of September 30, 1906).
Some years ago') I pointed out that by nitration of metanilro
phenol and of 35dinitrophenol tetra and pentanitrophenol are formed.
This showed that the NO^groups in the mposition do not prevent
the further substitution of the Hatoms in the o and pposition by
other groups. I have now endeavoured to increase these two cases
by a few more and have therefore examined the behaviour of some
msubstituted phenols which contain, besides a NO^group in the
mposition, a second group in the mposition, namely of
C,R, . OH . NO, . (CH3,OH,OCH3,OC,H3,Cl,Br) 1.3.5.
Of these phenols the 5nitromcresoP) and the monomethylether
of 5nitroresorcinoP) were known. The still unknown phenols were
made as follows:
The 5nitroresorcinol (m.p. 158°) from its above cited monomethyl
ether by heating for five hours at 160° with (307J HCl, or by
reduction of 35dinitrophenol with ammonium sulphide to 5nitro
1) These Proc. Febr. 22, 1902. Rec. 21. 241.
2) Nevile en Winther Ber. 15. 2986.
3) H. Vermeulen Rec. 25. 26.
( 279 )
3aminophenol (m.p. 165°) and substitution of the NHjgroup in this
substance by OH.
The monoethylether of 5nitroresoi'cinol (m.p. 80*^) was prepared
(quite analogous to the methylether) from 5nitro3aminophenetol;
the 3Cl (Br) 5nitrophenoi was obtained by substituting the NH, group
in the 5nitro3aminoanisol by CI (Br) according to Sandmeyer and
then heating the 3Cl (Br)5nitroanisol so obtained m.p. (101° ^) and
88^); with HCl as directed. We tiien obtain, in addition to CHjCl,
the desired product 3Ci (Br) 5nitrophenol (m.p. 147° and 145°).
The 35substituted 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
Hj SO4. 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 NOjgroups ^).
on OH OH OH
/\ /\ /\ /\
^"OjX /CH3 NOaX /OH NOaX /OCHjCOCsHs) No\ /Cl(Br)
\/ \/ \^ 80° \/ 145°
I' I' y 1
OH OH OH OH
NOü/XnOs N0s/\n02 NOs/^XnOs NOj/XnOj
U75°J 1 152° 1 I 1150 I  ■ I 147° I
XOA /CH3 NOjV /OH NO^X /OCHsfOCjHs) NOjV/ClCBr)
NO, NO2 NO, ^^^° NO;,'^'"
withlH^O withlHsO withlNHj withlNHaCgHs
OH OH OH OH
NO3/XNO2 NOa/X^^O, NOjy^X^Os NOs/X^O,
H0\ /CH3 H0\ /OH NHoX /NHj CbHjHNV /NHCsHs
NO» NO2 NOo NOo
In this scheme are given only the melting points of the as yet
unknown compounds.
Tetranitrom. cresol yields on boiling with water trinitroorcinol ;
1) 91° according to de Kock Rec. 20, 113.
) A comparative research as to these properties in the different compounds
has not vet been instituted.
( 280 )
in tlie same manner, tetranitroresorcinol ^) yields trinltroplilorogliicinol ;
tetranitrochloro and bromophenol also yield trinitrophloroglucinol on
boiling with water or, more readily, with Na, CO, solution. By the
action of NHg or NH, Cj Hj etc. in alcoholic solution various other
products are obtained, such as those substances included in (he 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 tetrachloride may be used.
If, in the above cited 35substituted phenols the OHgroup is sub
stituted by OCH3 it is not possible to introduce three nitrogroups.
For instance the dimethylether of 5nitroresorcinol yields two iso
meric trinitroresorcinoldimethylethers (principally those with the
melting point 195°, just as in the nitration of 5nitromxylene) ') ;
similarly, the methylether of 5nitromcresol (m.p. 70°) yields the
methylethers of three isomeric trinitromcresols, principally the
compound with m.p. 139°. The constitution of these substances is
not yet determined.
Amsterdam, September 1906.
Chemistry. — Prof. Holleman presents a communication from him
self and Dr. H. A. Sirks: ''The six isomeric dinitrohenzoic acids."
(Communicated in the meeting of September 29, 1906).
Complete sets of isomeric benzene derivatives Cg ^,^,6 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 dinitrohenzoic acids which Dr. Sirks has executed under
my directions.
The considerations which guided me in the choice of this series
') According to Henriques (Ann. Ghem. 215, 335), tetranitroresorcinol (m.p. 166°)
is formed by the nitration of 25dinitropheno]. In Beilstein'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.
3) Rec. 25, 165.
( 281 )
of isomers were the follo\>inf^. Firstlj^ all six isomers were known,
altiiough the mode of preparation of some of them left much to be
desii*ed. Secondh', this series gave an opportunity to test V. Meyer's
"ester rule" with a much more extensive material than hitherto and
.to stud)^ what influence is exercised by the presence of two groups
present in the different positions in the core, on the esterilication
■velocity^ and to compare this with that velocity in the monosub
stituted benzoic acids. Thirdly, tlie dissociation constants of these acids
could be subjected to a comparative research and their values con
nected with tliose 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 7?znitroben
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); (1.2,4) and (1,2,6) are com
mercial articles whilst (1,3,5) was not wanted because the orre
sponding acid, as already stated, w^as readily accessible by direct
nitration of benzoic acid. As will be seen the three dinitrotoluenes
which had to be prepared are all derivatives of mnitrotoluene and
it was, tlierefore, tried which of those might be obtained by a further
nitration of the same.
?/iNitrotoluene, which may now be obtained from de Haen 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
nitrationproduct a considerable amount of 1,3,4 dinitrotoluene crys
tallised out, which could be still 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 ?7initrotoluene,
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 tiie other isomers, whose formation in the
nitration of y//nitrololueno 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
17o 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 1,3,4dinitrotoluene 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 7?initroto
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, 4dinitro
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 Beilstein'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.
3.4
3.5
2.3
2.5
2.6
2.4
Dinitrotoluenes
58.3
92.6
59.3
50.2
65.2
70.1
Dinitrobenzoic acids
163.3
206.8
204.1
179.0
206.4
180.9
Ethyl esters
74.0
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 )
CO,H(
orCH,
on 1 ' toluenes
e.<ters
3.4
3.5
2.3
2.5
2.4
2.6
1.2594
•1.2772
1.2625
1.2820
1.2860
1.28:3
1.2791
1.2935
1.2825
1.2859
1.2858
1.2923
Water at 4° as unity. Corrected for upward atmospheric pressure
and for expansion of glass.
Conductivity jwwer. This was determined in the usual manner
with a Wheatstonebridgë and telephone at 25° and at 40°. As the
acids are soluble in water witli diflieulty v =: 100 or 200 was taken
as initial concentration; the end concentration was ?; = 800 or 1600.
In the subjoined table the dissociation constants are shown.
Dinitrobenzoic acid;
3.4
3.5
2.3
2.5
2.4
2.6
at 25^^
K=:100k
at 40°
0.163
0.171
0.163
0.177
1.44
1.38
2.64
2.10
3.85
3.20
8.15
7.57
On comparing these figures it is at once evident that the acids
with orthoplaced nitrogroup 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 orthoplaced 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 pliiced ortho : in N/mo solution 26dinitro
benzoic acid is ionised already to the extent of 90 °/g. Again, a NO^group
in the paraposition increases the dissociation constant more than one
in the metaposition ; and for the two acids 2,3 and 2,5 which both
have the second group in the metaposition, K is considerably larger
for 2,5, therefore for the nonvicinal acid than for the vicinal one, so
that here an influence is exercised, not only by the position of the
19*
( 284 )
groups in itself, but also by their position in regard to each other.
It also follows that Ostavald'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
COgH on 1
acid
K calculated
K
found
3.4
0.23
0.16
3.5
0.20
0.16
2.4
4.1
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. 7o alcohol to a N./,oo 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 acids
3.4
3.5
2.3
2.5
2.6
2.4
/*j(^0 i" alcoh. sol.
"aOo ^" aqueous sol.
1.1
161.5
1.15
162.5
1.75
293
2.25
321
2.7
355.5
2.9
335.5
from which it appears that also in alcoholic solution the acids with
an orthoplaced nitrogroup are more ionised than the others.
Esterification velocitij. 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 Goldschmidt 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 7o by volume.
In order to be able to compare not only the esterificationconstants
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. NOo »
0.0071
—
—
0. » »
0.0010
—
—
3.4 dinitro »
0.0086
0.033
0.077
3.5» B
0.0053
0.028
0.060
2.3 » »
0.0005
0.0025
0.0071
2.5 » »
0.0003
0.0027
0.0076
2.4 » »
0.0002
0.0017
0.0056
2.6 » »
unm
easurably s
mall
As will be seen, E is by far the largest for benzoic acid and each
subsequent substitution decreases its value.
On peinising this table it is at once evident that in the dinitroben
zoic acids two groups can be distinguished. Those with an ortho
placed nitrogroup have a much smaller constant than the other two.
Whilst therefore the dissociation constant for acids with an ortho
placed nitrogroup 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 esterilication constant and vice versa.
( 286 )
Dinitrobenzoic acids
diss, const, at 40°
esterif. const at 40°
3.4
0.171
0.033
3.5
0.477
Ü.028
2.3
'1.38
0.0025
2.5
2.10
0.0027
2.4
3.20
0.0017
2.6
7.G
< 0.0001
On perusing the literature we have found that this regularity
does not exist in this series of dinitrobenzoic acids ouly, 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 
Groningen '
Sept. 1906, Laboratory of the University.
Chemistry. — Prof. Holleman presents a communication from
himself and Dr. J. Huisinga. ''On the nitration of phthalic
acid and iso phthalic 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 Miller (A. 208,
233). Isophthalic acid can yield three isomeric mononitroacids. 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 mononitroacid
whose structure has remained doubtful.
The investigation of the nitration of phthalic and isophthalic 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 (b}^ Miller) was known,
whilst in the case of the mononitroisophthaUc acids it had yet to
be ascertained which isomers are formed there from.
We commenced by preparing the five mononitroacids derived
from phtaUc acid and isophthalic acid in a perfectly pure condition.
In the case of the a and /jnitrophthalic acids no difficulties were
encountered, as the directions of Miller, 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 nitroisopkthalic acid was prepared by nitration
of isophthalic acid. It crystallises with 1 mol. of H,0 and melts at
255 — 256^ whilst it is stated in the literature that it crystallises
with JVs ï^ol. of HjO 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 /?ixylene (from Kahlbaum) an
isophthalic acid was obtained which still contained terephthalic acid
which could be removed by preparing the barium salts.
The motheiliquors of the symmetric nitroisophthalic acid appeared,
however, to contain such a small quantity of the byproducts that
the preparation of the nitroacids (1, 3, 2) and (1, 3, 4) was out of
the question. These were therefore, prepared as follows :
Preparation of asymmetric nitroisophthalic acid (1, 3, 4). On
cautious nitration of mxylene at 0° with nitric acid of sp. gr. 1.48
a mixture is formed of mono and dinitroxylene which still contains
unchanged 7?ixylene. 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 xylenegave about 85 gr. of mononitroxylol (1, 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 nitroisophthalic 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 7o ^t 25°. Unlike the symmetric acid, it crystal
lises without water of crystallisation in small, fairly thick, platelike
crystals. It is very readily soluble in hot water, alcohol and ether,
Preparation of the vicinal nitroisophthaUc acid {1, 3, 2). Grevingk has
observed that in the nitration of mxylene with nitric and sulphuric acid
CH,
N0,/\
there is formed, besides the symmetric dinitromxjlene  
\/CIl3
NO2
as main product, also the vicinal isomer   . On reduction
NO2
w^ith hydrogen sulphide both dinitroxylenes pass into nitroxylidenes
CHg
which are comparatively easy to separate. The nitroxylidene  
\/CH3
yields by elimination of the NH^group vicinal nitromxylene. Whilst
however, Grevingk states that he obtained a yield of 257o of vicinal
nitroxylidene we have never obtained more than a few per cent of
the same so that the preparation of vicinal nitromxylene in this
manner is a very tedious one, at least when large quantities are
required. When it appeared tlmt 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
nitroisophthalic 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
nitroacid 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 tive
nitrophthalic acids, in parts per 100.
«nitrophtalic acid 9nitrophthalic 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 mononitroacid ; 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 «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 ,jacid was calculated by making
use of a table which had been constructed previously and in which
was indicated whrch j?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 7o "" and SO5 °/o <?nitroplithalic acid.
The quantitative nitration of isophthalic 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 7o 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 7o 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 7o of symmetric and 3.1 °/o 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 7o ortho, 76,5 7o meta
and 1.27o paranitrobenzoic acid the following is noticed.
COJI
/iXcOjH
As in phthalic acid IJ ^1 the positions 3 and 6 are meta in
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 «acid (the vicinal) is the main product and
the /3acid the byproduct, because in the latter the nitrogroup
must be directed by one of the carboxyles towards para and
because ^9nitrobenzoic acid is formed only in very small quantity
in the nitration of benzoic acid. As regards the isophthalic acid
COaH
''i\ it might be expected that the chief product will be sym
\l l\ metric acid but that there will also be byproducts. (1, 3, 2)
\V^ ^ and (1, 3, 4) the first in the largest quantity, although it
should be remembered that a nitrogroup 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
vicinal 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. Huisinga has now endeavoured to calculate,
( 291 )
more accurately than before, from the relative proportion in which
the isomers C„H,.AC and CbH^BC are formed by the introduction of
C in CeHjA or CgHsB, in Avhat proportion the isomers CgHjABC
are formed by the introduction of in CgH^AB. He observes first
of all that in a substance C^H^A there are two ortho and twometa
positions against one para position so that if the relation of the
isomers is as CgH^AC p : q : r (ortho, meta, para) this relation for
each of the ortho and meta positions and for the [)ara position will
be V, p : V, q : r
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 Hatom should be represented by a sum.
But only the j)roportion of those directing forces are known and
not their absolute value; the force which, in the nitration of
nitrobenzene, pushes the NO^gro.up towards the ?/iposition may be
of quite a different order than the force which in the nitration
of benzoic acid directs the same group towards the mposition.
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 37.67o ortho, 62.17o
para and 0.37o of meta nitrobromobenzene is formed, the substitution
in the different positions of the benzene core takes place in the
Br
proportion   ; for the proportion in which the isomers are
Ü.15\ /0.15
62.1
formed in the nitration of odibromobenzene the calculation gives
l8.8 + 0.15/^\Br
I I or 62.25:18.95; or 76.7 "/„ asymmetric nitro
C2.I f 0.15\ >^ 18.8 40.15
G2.1 f 0.1.5
odibromobenzene and 23.3 7o vicinal whilst the experiment gave
81,3 "/'o 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
( 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 substitnents present are unequal, when
Huisinga's method of calculation cannot be applied. The proof there
of is laid down in the subjoined table which gives the figures of
pi'oportion 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.
Cl:ClorthoCl:Cl meta
BnBrortho
Br:Br meta
CojH:COjH
ortho
COg H : CO2 H
meta
found
7:93
4:96
18.3:81.7
4.6:95.4 49.5:50.5
3.1:96.9
product
18:82
9:91
23.3:76.7
13:87
82 :18
*10.6:89 4
■ sum.
18:82
15:85
23.3:76.7
19:81
55.6:44.4
*38 : 62
*totalquant
ity byproduct.
C0„ H : CI ortho COj H : CI meta
COo H : Br ortho
CO2 H : Br meta
found 16.0:84.0 8.7:91.3 19.7:80.3 11.4:88.6
product 17.7:82.3 17.7:82.3 23.3:76.7 23.3:76.7
A fuller account of this investigation will appear in the Recueil.
Amsterdam, org. lab. Univ. 1906.
Astronomy. — ''The relation hetioeen the spectra and the colours
of the stars." By Dr. A. Pannekoek. (Communicated by
Prof. H. G. VAN DE 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 l^S 2^ and 3^ types are usually
called the white, the yellow and the red stars, although accurately
spoken the colour of the socalled yellow stars is a very whitish
unsaturated yellow colour and that of the socalled 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 X 587 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 X 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 selfluminous celestial bodies will in general he 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
Zöllner's colorimeter, which produces quite different colours, has
given so few satisfactory results, and on the other hand why the scale
of Schmidt, who designates the colours by one series of figures,
where 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. Osthoff at
Cologne, in the A. N. Bd. 153 (Nr. 3657—58). This list in which
the colours of all stars to the S''^ 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 spectralphotometric 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 Maury'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 OsTHOFF 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 Osthoff'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 ]
[II VI
Classe VII— VIII
Classe
i IX— XII
Mg.
Col.
Mg.
Col.
Mg.
Col.
1.78
1.46 (5) .
0.1
1.2 (3)
1.0
2.7 (2)
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
3.06 (8)
3.70
2.86 (7)
3.55
2.57 (6)
3.65
3.73 (10)
4.00
2.47 (8)
3.82
2.95 (6)
3.85
3.40 (8)
4.15
2.91 (7)
4.00
2.86 (5)
4.10
3.69 (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 (4)
4.96
2.66 (5)
Classe XIII— XIV
Classe
XV
Classe XVI— XVIII
Mg.
Col.
Mg. Col.
Mg.
Col.
0.2
3.4 (1)
0.7 4.5 (2)
0.95
6.45 (2)
3.07
4.71 (7)
2.12 5.1
SO (6)
2 50
6.40 (6)
a.54
4.61 (7)
2.92 5.66 (9)
3.22
6.65 (6)
3.98
4.72 (9)
3.37 5.74 (9)
3.72
6.65 (4)
4 24
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
7.22 (8)
4.14 5.85 (11)
4.45 6.08 (6)
4.87 6.43 (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 :
CI. Ill— VI c = 2.15 + 0.35 (771 — 3)
„ VII— VIII 2.27 + 0.36
„ IX— XII 3.17 + 0.39
„ XIII— XIV 4.45 + 0.42
„ XV 5.47 + 0.39
„ XVI— XVIII 6.60 I 0.20
Thus we find about the same coefficient in all groups except in
the last. The value of the coetTicients is chiefly determined by the
difference between the observed colours of the very bright stars of
the 1^* magnitude and of the greater number of those of the 3^ and
4^*^ 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 a 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 for m = 3), according to brightness and deriving
thence mean values we find :
—0.91
—0.47
0.02
fO.27
+0.39
+0.60
A linear relation c = c, \ 0,34 (m — 3) yields the computed values
given under C, and the differences obs.comp. — C,. These are
distributed systematically and show the existence of a nonlinear
relation. A curve, which represents as well as possible the mean
values, gives the computed values C^ and the diiferences, obs.comp.
— 6\. Fora 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
m
C — Cj
c\
0.3
—1.03
—1.10
1.6
—0.63
—0.54
2.91
+0.02
+0.04
3.73
+0.32
+0.31
4.12
+0.48
+0.40
4.73
+0.50
+0.52
0C,
0C,
+.07
—.12
— 09
— 16
— 02
+ 04
+ 01
+ 05
+ 08
+ 09
— 02
— JO
( 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. Avhitish)
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 b^^^ 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.
Helmholtz in his Physiologisclie Optik has given a theory called
"Theorie der kürzesten 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 "yellowwhite" 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 not agree with what we have found here, in the supposition
that Helmiioltz's '•'yellowwiiite" is also yellowwhite in our scale, i. e.
is also represented by a positive number in Schmidt's scale. We
also tind 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. Xow the expression "yellowwhite" is vague, but
if we consider that what is called white in the scale of Schmidt is
whiter, that is to say bluer tlian 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 Schmidt
would be called 3 a 4 (Capella 3, 4), then the principal colour,
if Helmholtz's theory is true, instead of being yellowwhite 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 ijifluGiice 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 obseiver is sure to have taken care that most
of the stars were observed at a proper altitude (for in.stance between
SO"* 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 Osthoff 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 classmeans 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
O^O
fO.56
5
+ 0.05
— g'O
+ 0.14
+ 0.26
— 1 7
fO.35
4
+ 06 '
— 10.2
■ +0.35
4
+ 32
—3.3
— 0.17
6
+ 09
13.2
+ 0.33
6
+ 57
— 5.0
f 0.50
5
+ 12 i
— 15.0
+ 1.17
6
+ 79
 6.6
+ 0.22
4
+ 17 !
— 18.2
+ 0.93
6
+ 1.32
— 8.0
—0.05
+ 22
20
Proceediugs 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. 0,1 0,2 0,3 0,4 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 a Bootis (A)
or with a 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 B's lie between the two.
Class
Colour
Number
Class
Colour
Number
I
2.47
6
XII
3.68
17
11
2.36
10
XIII
4.12
13
III
2.30
9
XIV
4.45
12
IV
1.94
14
xrv
5.09
9
IV
1.62
10
XVA
5.18
18
V
2.11
9
XVB
5.35
26
VI
2.16
10
xvc
5 55
31
VII
2.27
23
XV
6.34
5
VIII
2.37
34
XVI
6.47
17
IX
2.64
20
XVII
6.80
15
X
3.11
14
XVLII
6.74
15
XI
3.40
9
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 colournumber, [/ 0,20 = 0,45 ; the real accuracy will be
greater, 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 delinite class, and also the real deviations
of the single stars from their classmeans.
( 299 )
With 9 stars (out of 355) the deviation exceeds a unit of colour ;
the reduced colours are here :
/? Can. maj. Ill 1,2 8 Hydrae XIII 5,2 7j Persei XV^ 6,8
o^ Cygni 1X1,4 ft Persei XIV 5,5 llUrs.min. XV5 6,6
d Delphini IX 3,8 o, Cygni XfV 6,5 5 Ononis XVII 7,9
In this investigation we have, as it was said before, excluded the
c and ««cstars, the L (bright lines), the P (peculiar spectra) and
the C (composed spectra). It is important to examine the c and the
«cstars among them more cl.osely in order to see whether thej show
a distinct difference in colour from the «stars of tlie same class
number. In the mean 11 «cstars give a deviation of j 0,1 (from
f 0,5 to — 0,3), and 12 cstars 0,7; so these last ones are a little
redder than the dtstars. Here, however, the great individual deviations
are very striking ; the extreme vahies are :
Q Cassiop XIII + 2,5; /, Orionis III + 1,8 ; 47?Camelop F/+ 2,0;
3FCamelop F7^+ 1,5; n I^eonis VII — 0,3 ; /? Orionis VI — 1,2,
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 colournumbers 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 {n Aurigae, n Hydrae, u Herculis) have the whitest colours ;
both the earlier and the later stages of evohition 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 colournumbers for each of the groups formed before are:
CI. I III
2.35
IV— V
1.87
VI— vin
2.30
IX— XII
3.20
XIII— XIV
4.58
XV
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 selfluminous 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. KöNiG on the relative quantities of the elemental colours red,
green and blue as functions of the wavelength in white sunlight. Ii
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
can calculate the quantities of the red, green and blue in this second
source of light. If we call the numbers of König R (A), G {X), B {X),
which are chosen so that
r72(A)fa= 1000 I G{X)dX=l()0^ i B(k)dX—lOO(i
and a f{X) represents the brightness of another source of light, then
^f{X) R{X) dX Cf{X) G{X) dX and Cf{X) B{X) dX
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 :
c
X e dX
where T is the absolute temperature and « and c constants. For two
sources of light of different temperatures the relation of the inten
sities is :
c/J_ 1_\ 6 6'
//n ~^^^ ^"^ ^ 1ft ^
f{X) = e zz:<? = 10 ,
if h =z c( I and b' = 0.43 b. As unit for X we adopt 0,00 J mm;
jf'o is supposed to be given, then b' is a function of the variable
tem})erature T only and may be called the degree of glowing with
regard to tlie glowing of a body at a temperature 7\. If we adopt
( 301 )
for h' ciifFerent values (c = 15000 about) ^), we can calculate iov
each of Ihem the brightness and colour of the light, as well as the
temperature 2\ We then find for the degrees of glowing {1,0
and —1
6'= 4 1 69200 R + 68100 G + 175800 B
1000 R + 1000 ö.f 1000 B
— 1 17,7 E + 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
h' — \l Col. = 221 i^ [ 218 Ö + 562 i? Br. = f 4,6 Mg.
// = — 1 Col. = 445 R + 39Q G i 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 R and 344 B, hence corresponding
in tint to ;. 466; the second colour is a mixture of 480 white and
521 of a yellow consisting of 285 R and 236 G, hence corresponding
to the wavelength / 587. A degree of glowing b' =^ — 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 MtJLLER's spectralphoto
metric measurements appeared to correspond to 1,05 atmosphere,
we have calculated ƒ(/) and thence found for the remaming quantity
of light, the initial quantity beijig 1000 R f 1000 G + 1000 B :
783 R\ 771 Ö + 571^,
or reduced to 1000 as the sum,
368 72 4 363 G f 269^;
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 72 + 248 + 184 5
hence when reduced to a sum of 1000
372 72 + 3610+267 5
i; 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^ 750G^ 5000 \ 3750\ 3000' G 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 fi'om 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
spectralphotometric 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 A with regard
to ;i500 we find
A = 650 600 550 500 450 400
with abs. 1.05 atm. +0.114 +0.083 + 0.051 0.000—0 084—0.231
with glowing— Vj + 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 wavelengths.
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 temperatui'e and at the same time a greater atmospheric
absorption than the other. An increase of the degree of glowing of
+ \/j 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 K 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 bandabsorption, which begins at the /iTstars and which is
characteristic for the M stars (the 3'<^ fype) indicates a lower tem
perature.
( 303 )
Chemistry. — "Action of Potassium hypochlorite on Cinnamide" .
By Dr. R. A. Weerman (Communicated by Prof. Hoogewerff).
(Communicated in the meeting of September 29, 1906).
From the experiments of Baucke ^) on propiolamide and those of
Freundler ^), VAN LiNGE ') and Jeffreys '') on cinnamide it appears
that in the case of these unsaturated acids, the Hofmann 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: lirst 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 CgH.CH z=CHNH,,
suffers decomposition under the said circumstances. ®).
The first, however, is 7iot the case as from cinnamide may be
prepared the urea derivative :
C,H,Cii3=:Cn— N"
C,HAi=ChCoNh
where consequently onehalf 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 Hoogewerff 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 foi"med and not the urethane, this may be
explained by the experiments of Stieglitz and Earle '), which show
that isocyanates react very readily with halogenamides ®).
1) Rec. 15, 123.
2) Bull [3] 17, 420.
5) Dissertation van linge, Bazel 1896.
^) Am. Ghem. Journ. 22, 43.
°) On account of the great analogy existing between the Lossen transformation
of hydroxamic acids and the Hofmann reaction, this first supposition was not very
probable, as Thiele had prepared from the acylated cinnamohydroxamic acid the
urethane GgHjCil —C^l — N'^— G'^oGsH. A second indication, though less conclusive,
in the more distant analogy between the Beckmann rearrangement and the
Hofmann reaction was the formation of isochinolin from the oxime of cinnamaldehyde.
(Ber. 27, 1954).
6) Thiele, Ann. 309. 197.
') Am. Ghem. Journ. 30, 412. G 1904, I, 239.
^) This is the reason why, in the preparation of urethanes according to Jeffreys,
the sodium elhoxide 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
Avith 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,0 and 0,4682 grm. CO,
0,1654 „ „ 0,1863 „ „ „ 0,4467 „ „
0,1654 „ „ 13,9 CC.N at 19^° and 765 m.M.
Found 73,68 5,78
pCt. C pCt. H 9,70 pCt. N
73,66 5,85
Theory C,3H,,N,0,: 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. Oudemans.
(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 n and
f indicate whether the satellite is tiear 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 conjunclion. Eastern elongation, denoted by e . e, has an
amplitude of S', western elongation, iv.e, one of 9^ .
Not to interrupt the text unnecessarily, all particulars have found a
place at. the end of the paper.
FIRST PART. OCCULT ATIONS.
In the numbers 3846 and 3857 of the Astronomische N^achrichten
we find two communications relative to observations of the occul
tation of one satellite of Jupiter by another. Tlie first (1) is b}^
Mr. Ph. Faüth at Landstuhl, dated 8 December 1902, with post
scripts of 29 December 1902 and 14 January 1903. The other (2)
by. Mr. A. A. Nijland at Utrecht, dated 27 February 1903.
Fauth notes in addition that Houzeau, in his' Vademecum, p. 666
mentions a couple of similar observations (3), and further that Stanley
Williams, on the 271'^ March 1885 at 12'' 20'", saw the third satellite
pass the first in such a way that the two satellites combined had a
pearshaped appearance. (4)
The satellites of Jupitei' move in orbits but little inclined to the
plane of Jupiter's equator. Laplace assumed a fixed plane for each
satellite; the plane of the satellite's orbit has 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
Earth, 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 haAe
been observed in 1902, we must expect that these phenomena will
be again visible in 1908 (5).
( 306 )
To facilitate these observations I tliought 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 Bessel at Koningsberg
in 1834—39, and by Schur at Göttingen 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 Gill and Finlay 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 ditference 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, I9^',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 2''10 " M. T., its declination
being 16°48'5 North, whereas the Sun's declination is 22°3ü' North.
From these data I find for the 8^'' of Julj, for Uirecht, duly making
allowance for refraction :
Setting of the upper limb of the sun at 8''20'^i mean time,
„ Jupiter „ 9 44 5 „ „ .
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 5'' 5'" mean time,
Setting of Jupiter ,, 7 23 ,, ,, .
We thus find that on July 8, 1908, at Utrecht, the setting of the
sun precedes that of Jupiter by l''24'^5; at the Cape by 2''20^.
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 etc.
was neglected.
The scale of this drawing gave 4" to 1 mm. The radii, of the
orbits therefore were: for I 27 9 mm.; for II 4445 mm.; for III
709 mm. and for IV 1247 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,
llh2m.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 :
21 8 :
25 :
35 :
66 25:
71 0 :
YVf occulted by Illn,
IV
/
III/
Iln,
II„,
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 édiptiques
of Damoiseau, 2nd part. (8). In the case that the elongations did
not perfectly agree, a slight computation led to a more accurate
resulf for the time of conjunction (9).
In the case that the two satellites moved apparently in opposite
directions, (which happens if the one is in the further part of its
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, i.e.
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
upper limb
of the Sum
Setting
of
Jupiter
Difference
908 June 1
ShlOm
llh54m
3h44m
„ li
8 20 5
11 19
2 58 5
» 21
8 24
10 44
2 20
July 1
8 24
10 9
1 45
„ 11
8 18
9 34
1 16
„ 21
8 75
8 59
51 5
( 309 )
For the CVape of Good Hope :
1908 June 1
41,590,
9>48'^
4"19
. 11
457 5
8 46 5
3 49
„ 21
4 58
8 16
3 18
July 1
5 2
7 46
2 44
, 11
5 65
7 16
2 9
„ 21
5 13
6 47
1 34
The circumstances are tlius 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 Xos. 8, 9, 12, 13, 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 Ko. 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), June 13
9''28"i M. T.
Grw.,
. II eclipsed,
(X°.
31), „ 20
12 51
'J
II
(N°.
51), July 4
18 15 7 „
'5
II ,. ,
(N°.
65), „ 13
12 3
J>
IV „
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; foi* according to the iV. Almanac, the reappearance of
IV from the shadow of the planet takes place at 12'^l"'^h^ M.
T, Greenwich and the predicted eclipses of this satellite are occasionally
a few minutes in error. A few minutes later, according to the N.
Almanac at 12''16™, II enters the disc 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 27^,
IV. Occultation Reappearance 15 13,
IV. Eclipse Disappearance 18 5 6 ,
IV. Eclipse Reappearance 22 52 2 .
( 310 )
iN O T E S.
(1) The article of Fauth, abridged, runs thus :
— — — Ausser den in Hoüzeau, Vademecum p. 666 aufgeführten
Beobachtungen, (vid. below Note 3), kenne ich aus neuerer Zeit nur
einen Fall : Stanley Williams sah am 27 Marz 1885 an einem 7 cm.
Rohre mit 102facher Vergrösserung um 12'> 20'" den III Trabanten
vor dem I, wobei beide ein birnformiges Objekt bildeten.
— — — In fünf Wochen konnte ich drei Bedeckungen verfolgen,
wobei auzunehmen ist, dass inir durch schlechte Witterung etwa 10
andere Gelegenheiten entgangen sein mogen, nnter denen sicher
einige Bedeckungen vorkommen. Nach meiner Erfahrung können Kon
junctionen der Jupitermonde unter sich weit genauer beobachtet
werden als Bedeckungen durch Jupiter oder Vorübergange vor ihm.
Somit mochten die hier angegebenen Beispiele Anlass bieten, in den
spateren Oppositionen Jupiters den durchaiis nicht seltenen Bedeck
ungen oder wenigstens Berühriingen nnd sehr nahen Konjunktionen
der Trabanten unter sich mehr Aufmerksamkeit zii schenken, zumal
schon kleine Instrumente zur Wahrnehmung der Phasen einer event.
Bedeckung genügen. Die Beobachtungen der letzten Zeit sind :
1. Oct. 7; II bedeckt I; die S. Rander berühren 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 nördlich am
N. Rand von III vorbeigeht; Konjunktion um 8^^ 7'" 3%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'" 20\
Instrument: 178 mm., Vergrösserung 178 fach.
Landstuhl, 1902 Dez. 8.
P.S. vom 29 Dezember. Am Abend des 24 Dezember gelang
noch mals die Beobachtung einer Bedeckung, bei welcher I über IV
hinwegzog. Aus je fünf vor und nachher notierten Zeitmomenten
folgen als Mittelwerte 6'^ 24'",25, 24'",625, 24'",50, 24"\625 und
24"!, 50. Die Konjunktion fand also statt &' 24™ 30\
Der ührstand war um 3^^ mit dem Zeitsignal verglichen worden.
IV Stand ein wenig südlicher 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 vorüberging,
P.S. vom 14 Januar (1903). Heute Abend, am 14 Januar, bewegte
1) i. e. Mittlere Europaische Zeit, l^' later than Greenwichtime.
( 311 )
sich der Trabant III über II hinweg. Die sehr schlechte Liift liess
nur den ersten Kontakt auf etwa &' 2'" feststellen. üm 6'' 18"^ mochten
sich beide Koniponenten so weit getrennt haben, dass dies in einem
weniger schlechten Augenblick bemerkt wurde; um 6^^ 32"^, dem
nachsten blickweisen Auftauchen der beiden Lichtpnnkte, waren diese
um etvva einen Durchmesser von einander entfernt. Die Bedeckung
war fast genau central. Ph. F.
(2) Mr. NiJLAND writes in N". 3857 of the Astronomische Xachiichten:
,Am 15 Juli 1902 fand eine Konjunktion der Trabanten
II nnd III statt, welche ich bei guter Luft am Refraktor (Brennweite
319 cm., OefFnnng 26 cm.) mit Yergr. 248 beobachten konnte. Es
wurde III nahezu central von II bedeckt. Einige Minuten lang blieb
eine feine schwarze Linie zwischen den beiden Sclieibchen sichtbar,
welche um 14i'10"lls M.Z. Utrecht verschwand and um 14'i20"^31s
wieder erschien ; die Konjunktion musz also um 14'^15"^21'' statt
gefunden haben. Dass diese Trennungslinie vor und nach der Kon
junktion immer dieselbe Richtung hatte, und zwar scheinbar senk
recht auf der Balmebene der Trabanten stand, mag als Beweis dafür
gelten, dass der Yorübergang wirklich nahezu central gewesen ist.
Dann lasst sich aber aus dieser centralen Passage die Surnme der
Durchmesser der Monde II und III mit erheblicher Genauigkeit
bestim men.
Nehme ich für die mittlere Entfernung l^ — O die Halbmesser
der Bahnen gleich 177'',8 und 283'',6, so finde ich für die relative
Bewegung von II und III zur Beobachtungszeit 13 ",86 pro Stunde.
Aus der beobachteten Zeitdauer von 10'"20^^ = 0''172 folgt dann für
die Summe der beiden Durchmesser, 2", 38. Wird i^siehe die Angaben
von Douglass, Astr. Nachr. 3500) für das Verhaltniss der Durch
messer von II und III "/n angenommen, so finde ich, in vorzüglicher
Uebereinstiramung mit den a. a. O. genannten Werten, für den Durch
messer von II 0",87 und von III 1^,51 (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. Nijland, this gentleman allowed me to consult
his reduction of tlie 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''*78tj and the sum of the dia
meters 2 '"STi. Moreover their proportion was, evidently erroneously and
( 312 )
against the real intentiop. put at 4 to 11 instead of at 4 to 7. We
thus get for the diameters 0"'8G3 and 1"'511, which is still in good
agreement with the result of Mr. Nijland. 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 Houzeau, Vademecum (Bruxelles, 1882), p. 666 :
On rapporte line occultation du satellite II par le satellite III,
observée a Sommerfeld, prés de Leipzig, par C. Arnoldt, le l^^"
novembre 1693, (Whiston, The longitude discovered by the eclipses,
8°, London, 1738), et iine autre du satellite IV, également par le
III'"*', vue par Luthmer k Hanovre, le 30 octobre 1822 {Nature, 4**,
London; vol. XVII, 1877, p. 148).
1st Remark. The little book of Whiston 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 G. Arnoldt. 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 Derham ^), 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 2).
But I do not find the obseivation 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 observation
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 Luthmer
was communicated by him to Bode who inserted it in the (Berliner)
Astronomisches Jahrbuch fur 1826, p. 224 :
"Am 30 Oct. Ab. 6" 55' Bedeckung des vierten ^ Trabanten vom
dritten."
1) Poggendorff's Biograpkisches Wörterbuch, (article W. Derham) 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' = 38^48» East of Greenwich for the longitude
of Hannover, this is = 6'» 16^ 12^ 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, XVII (Nov. 1877April 1878) p. 149 (not J 48) we fmd
in "Our Astronomical Column'":
"Jupiter's S.^tellites. — 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. ^) The following cases may be men
tioned: — On the night of November 1, 1693, Christoph Arnoldt, of
Sommerfeld, near Leipzig, observed an occultation of the second satellite
by the third at lOb 47m apparent time. On October 30, 1822, Luthmer,
of Hannover, witnessed an occultation of the fourth sateUite by the
third at 6^ oSm mean time.
It thus appears that the editor of Nature 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 Stanley Williams in any of tlie 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 liis 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 ''Sirius".
He had moreover the courtesy of commuiiicatiug to me the original
account of the observation in question. From this account the
following passages may be quoted :
Occultation of sateUite I by satellite III.
1885 March 27, ... . 2'/, incii refractor. Power 102.
11''55"^ (Greenwich mean time). They are now only J?^.9^ free from
contact. ^1 , like an elongated star witii little more tlian a
black line between the components.
12'00'" to 12''04m. After steady gazing I cannot see any certain
separation between the satellites, and thciefore 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.
1) 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)
12'40'". They now appear as one elongated satellite. At times a
trace of flie notches is apparent.
12^'20"\ 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 sraallness 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^ey Williams mentions the remarkable fact
that he too observed on 15 July 1902 the same conjunction which
has been described by Nijland. His instrument was a reflector of
GYa inch, with a power of 225. The following are the particulars
as communicated :
1902 July 15, 13H5m2. Satellites II and III are in contact. The
one will occult the other. See diagram uiQ,., •
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 disc is considerably elongated now.
14''02'"2. Satellites II and III in contact as in diagram adjoining
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 13'i52"^ was not real.
The above times are Greenwich mean times. Satellite 111 was on
the farther side of its orbit moving east, 11 on the near side moving
west. As the disc of 111 is larger than that of II, the phenomenon
should be described as a transit of 11 over or across 111, rather than
an occultation of one satellite by the other.
The arithmetical mean of 13'45"i2 and 1412^2 is 13i'53"i7, which
16 1"1 earlier than Nuland's observation.
(5) For tlie numbers which follow we refer to K.mskk's "Stemm
hemeT\ 4th Edition, \). 707 and following.
In the 4'''> Vol. of his J/tr(«n«</zo<^ 6Wd.v^é^, p. 62, Tisser and, following
SouiLLAUT, adopts inclinations for the orbits of III and IV, which
( 315 )
respectively exceed those given in the "Sterrenhemel by  4" and — 8".
According to Leverrier we iiave, for the orbit of Jupiter in 1908,0 :
Ascending Node = 99'31'56",
Inclination = i 18 29 .
The fixed plane of tlie first satellite coincides witii the plane of
Jupiter's equator: the longitude of tiie ascending node on the plane
of Jupiter's orbit, for the beginning of 1908 is tiierefore 315°33'35",
the inclination 3° 4' 9".
Furthermore ^ve have for the four fixed planes relative to the
plane of Jupiter's orbit :
Long. asc. node
Inclination
I
315°33'35"
30 ^ 3„
II
III
315 33 35
315 33 35
3 3 4
Epoch 1908.0.
2 5911 I
IV
315 33 35
2 39 57
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 ha\'e for the respective fixed planes in 1908, according
to TiSSERAND :
Change in
long, asc, node
1000 days
Inclination
II
122°293
— 33°031
0°28' 9"
II
26 173
— 6 955
10 44
V
238 982
— 1 856
13 51
The effect of these inclinations, however, is but trifiing. At the
distance of 90^ from the node they produce only deviations
for II of l"46,
,, III „ 0"89,
„ IV „ 2 01.
The determination of tiie position of the fixed planes, as also
that of the planes of (lie orbits of the satellites relative to these,
will be much improved by the measurements which De 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 )
Tn the meeting of our section of last March a provisional account
of these measures by de Sitter was communicated bv Messrs J. C.
Kapteyn and E. F. van de Sande Bakhuyzen ^).
Our computations were then already too far ad\'anced 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 occultations 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" =: i2,
Inclination 2 717 =1,
Now, if Rqr, Tj^ and ^ represent the radius vector, the longitude
and the latitude of Jupiter; R., L^ the radius vector and the longi
o o
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 :
R^. cos 8 si?i (Lr,r — i2) — i?«, sin ^ cot J ^= R+ sin {L — i2) ,
T r 'T o o
which is satisfied July 8, 1908 at 19^38'n3. For at that moment
log R^ = 0728527 % R^ = 0007179
L,r = 141^23' 9"0 L^ = 286°40' 3"5
r o
[3 =\ 52 26 73 i^ =: 336 48 52
so that our equation becomes
1423706 — 2204190 = — 0780484
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 sel^uel to the thesis of
Mr. DE Srn'Eu. This thesis, mainlained by him al Groningen on 17 May 1901, bears the
title: Discussion of Heliometerobservalions of Jupüefs satellites made hysir David
Gill K.C.B. and W. H. Finlay 3Ï.A. Further particulars will be given in the
Annals of the Roycd Observatory at the Cape of Good Hope.
( 317 )
(6) In 1833 — 39 Bessel, 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 Fralxhofer; the objective had an aperture of
702 Par. lines and a focal distance of 11314 Par. lines = 7 feet
10 inches 34 lines, Paris measure, (1584 and 25522 c.M.). The
mean error of a single observation of distance (which properly was
the mean of eight pointings) appeared to be
for I + 0"26, for the mean distance resulting from all the measures, + 0"055
, II ±024, „ „ „ „ „ r, , . r, ±0067
, Hi ±031, „ „ , r, , » , , „ ±0042
, IV ±043, „ „ „ „ , n n . n ±0045
Mean. ±031, „ , , „ , » , , , ±0052
ScHUR, at Göttingen, 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
w^as 34 Par. lines, something less than half that of the heliometer
of Koningsberg; the focal distance was 3 feet (1137 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 Bessel's instrument, Schur, instead of taking the
mean of eight pointings, was content with the mean of four pointings,
which also make a complete measuremeiit.
The mean errors of each observation obtained by Schur for a
complete set of four measures was :
for I ± 0"34,
„ II ± 44,
„ III ± 37,
„ IV ± 42,
Mean: ± 0"39,
a result, which, taking into account the shorter focal distance, may
be considered fairly good. Bessel as well as Schur aimed not so
much at the determination of the positioJi of the orbits of the
satellites as at that of the mass of Jupiter.
( 318 )
ScHUR improved in different respects the reduction of the observations
of the measures made by Bessel. In consequence, the mean errors
of the single determinations of Bessel were considerably lessened.
The numbers quoted just now, became :
for I ± 0"21,
„ II d= 10,
„ III ± 26,
„ IV db 30,
Mean: ± 0"24.
As has been mentioned already. Gill and Finlay, acting on a
suggestion formerly made by Otto Struve^), did not measure the
distances and the angles of position of tiie satellites relative to the
centre of the planet, but relative to each other. (The instrument
at their disposal, a heliometer of Repsold, aperture 7^ inch = 1905
cm., 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 Sitter have not been deterred by this conside
ration. They found ± 0"087, a number considerably less than that
of Bessel, for the probable error of the measurement of a single
distance. Mr. de Sitter even finds that the probable error of the
mean distances (the real unknown quantities) does not exceed
± 0"020 or d= 0"021.
(7) It may be remarked that Mr. de Sitter 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 Struve, in the first supplementary vol. of
the Pulkowa observations, 1st page at the bottom.
( 319 )
apsides. He further introduced two unknown quantities, viz. tiie
constant errors which might vitiate the observations of the two
observers Gill and Finlay. He thus also obtained a total of 29
unknown quantities. It need not be said that the sohition 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, 1 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. de Sitter has done good work, vide the
communication already mentioned, presented in the meeting of last
March by Messrs Kapteyn and E. F. van de Sande Bakhuizen.
(8) The same volume, which contains the ecliptic tables of Damoiseau,
contains also in a second part (not mentioned on the title) tables
''pour 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, Marth, Gill,
Finlay, 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 x and x\
of the two satellites, expressed in radii of Jupiter, but also their
difference in time.
( 320 )
Observer
Date
Occul
tat'on
of
by
Error
Relative
hourly
motion
Corr.
of Table
y'—y
Fauth
1902 Oct 7
"/
In
0,025
1,278
m
r
+ 0,04
»
» » 23
III^
n.
0,02
1,130
+ 1,1
+ 0,08
»
». Nov. 10
V
Ill,,
0,00
0,883
0,0
+ 0.13
»
» Dec. 24
lYy
In
0,10
1.089
— 5.5
+ 0,005
»
1903 Jan. 14
"«
IIT„
0,11
0,314
—19,2
— 0,05
Nijiand
1902 July 15
III^
"n
0,08
0,751
+ 6,4
— 0,01
Stanley Williams
» » »
))
»
0,07
0,751
+ 5,3
— 0,01
» »
1885 March 27
I,.
"I«
0,00
0,292
0,0
+ 0,01
The observation of Luthmer in Hannover, of Octob. 30, 1822 is
not contained in this table. Its calculation yields the result :
Jovic. Long.
Amplitude
X
yQmJ
III
10s26"77
8%22°25
— i5'21
+ 018
IV
9 674
7 222
— 14 40
— 074
Difference + ^81 + 0'92
So there is a difference in the amplitudes, of 0'81, = 081 X
18"37 = 14"9, in the latitudes, of 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
Damoiseau for 1822. The difference in sign of the latitudes y and y'
is explained by the fact that the longitude of the ascending node of
the fixed plane was 1044°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. 0'280. It would thus require
nearly three hours to annul the difference of 0'81.
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
( 321 )
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. O"? earlier
inferior conjunction (mean
of ingress and egress) 16 July 5"40m0 ,, „ O^^S
superior conjunction 17 July 23''54"i4 ,, „ 0^7
all three less than a minute.
Now, as the conjunctions in the Kautical Almanac have been
calculated by the aid of Damoiseau's tables écUptiques {mokm^ixWow
ance for some slight corrections indicated by Adams) the difïerences
must be solely due to the fact that in Damoiseau's second part the
main terms only of the equations and perturbations bave been taken
into account.
The same tables represent as accurately the superior conjunction
of I on January 1, 1908, i4"4"i2 M.T. Grw. = January 2, 2"13™55
civil time of Paris; the error amounts to 0^07 or O^'Ol linear
measure only, aii arc traversed by the satellite in 0'"5.
(On the terms taken into account in the second part of the tables
of Damoiseau vide 3''^^ appendix below).
In his letter Mr. Stanley Williams mentions another rare obser
vation, made as well by himself as by the Spanish observer J. Comas
of Tails, (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 lor this computation.
The unit, the radius of Jupiter, is 18"37. Soüillart states that he
found mentioned in the papers of Damoiseau that this number was
borrowed from Arago. According to Houzeau, Arago must have
made the determination by means of the double image micrometer
(an invention made nearly simultaneously by himself and Pearson;
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
Houzeau, p. 647— 650; See, Astron. Xachr. N". 3670 (15 Aug. 1900).
( 322 )
Hourly
chaji
ge
of the elongation x as a
function
of
the
am
plitude.
f
I
II
III
IV
r
r
r
r
Os
0^
Os(12)
Oc
895
4
0,708
3
0,560
2
0,420
2
Cs
0°
6s
Om
5
11
25
0.891
10
0,705
8
0,558
7
0.418
4
5
5
25
10
11
20
0,881
17
0,697
14
0,551
10
0,414
8
10
5
20
15
11
15
0.864
23
0,683
18
0,541
15
0,406
11
15
5
15
20
11
10
0,841
30
0,665
23
0,526
18
0,3:5
14
20
5
10
25
11
5
0,811
36
0,642
29
0,508
23
0,381
17
25
5
5
1
11
0,775
42
0,613
33
0,485
26
0,364
20
7
5
5
10
25
0,733
48
0,580
37
0,459
30
0.344
22
5
4
25
'10
10
20
0,685
53
0,543
42
0,429
33
0,322
25
10
4
20
15
10
15
0,632
57
0,501
46
0,396
36
0,297
27
15
4
15
20
10
10
0,575
62
0,455
49
0,360
39
0,270
29
20
4
10
25
10
5
0,513
66
0,406
52
0,321
41
0,241
31
25
4
5
o
10
0,447
69
0,354
55
0,280
43
0.210
32
8
4
5
9
25
0,378
72
0,299
57
0,237
45
0,178
34
5
3
25
10
9
20
0,306
75
0,242
59
0,192
47
0,144
35
10
3
20
15
9
15
0,231
76
0.183
60
0,145
48
0,109
36
15
3
15
20
9
10
0,155
77
0,123
61
0.097
48
0,073
36
20
3
10
25
9
5
0.078
78
0,062
62
0,049
49
0,037
37
25
3
5
3
9
0,000
0,000
0,000
0,000
9
3
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.
l^t Appendix. What is the maximum duration oj the several occuh
tations of one satellite by another ?
We have seen above that it took 19'"2 to annul the small
difference of the elongations of O'll (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 u and u' ,
x' X
are absolutely equal, the denominator of the fraction ; is zero.
( 323 )
The case then corresponds to that of the "Station of Venus" and
it is a very ancient problem to compute its epochs.
Let be 7' and r' the radii vectores of two satellites ; 6 and 6' the
corresponding amplitudes, then for the occultation :
r sin ^ = r' sin 6' .
The condition of an equal change of longitude leads to :
^dS , ^, d&
r cos 6 — = r cos 6 — .
dt dt
Now, if T and T' represent the sidereal periods, we have, neglecting
the apparent movement of Jupiter :
dO dd' _l 1 _ 1 ^ 1
It ' ~dt~'T ' Y'~?l^ ' /V^'
consequently :
r~V2 cos 6 = r'~V2 cos 6\
from which :
r
cos^ ^ = — cos^
r T
& — — sin^ 0' .
r'
r r
Adding
.
sin" &
=: — sin' d\
r'
3 get
1;+ (
   sin' e\
r' \
y, J r
Therefore,
r'
putting — = ^t,
11
^l 1 1
n'
f^'  1 f*' + f^ + 1
A
fi
u
sin' e 
fi'
ii' + ii^i
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 circum
( 324 )
stances in question. Let the amplitudes be between 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 11, but the
motion of I is quicker than that of II. 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 II, 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 II, the distance I — 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 See 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 See 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 Damoiseau.
Diameter Radius
I l"07 =z 0' 058 0^029
II 95 052 026
III 1 56 085 0425
IV 1 41 076 038
^) 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 W. 3764, (21 Jan, 1902).
Therefore
)
I
+
II
I
+
III
I
+
IV
II
+
III
II
+
IV
II]
[ +
IV
( 325 )
Sum of the diameters Sum of the radii
. . O'llO 0'055
. . 143 0715
. . 134 067
. . 137 0685
. . 128 0064
. . 161 0805
For the mean radii vectores we will take two figures more than
did Damoiseau in his tables, and we will adopt for the purpose the values
found by Souillart in Damoiseau's papers, (Souillart, second paper,
Mémoires présentés par divers savants a I'Academie des Sciences,
Tome XXX, 2™« Série, 1889 ; p. 10) ').
I 60491,
II 9 6245,
III 153524,
IV 270027.
The result of our computation is, that the time between the first
contact and the central occultation is :
for I and II I and III I and IV II and III II and IV III and IV
1^324, l'i245, 111103, 2i>263, li'774, 3^725;
between the central occultation and the second contact:
1^204:, J'>161, li>059, 21^190, 1^767, 31^725,
therefore in all
2''528, 2i'406, 2''162, 4''453, 3i'541, 7''450,
or
2h32m^) 2''24'", 2^10'", 4i>27"S 3^32", 7^27"^
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 + r'), i. e. that between two central conjunctions there
1) According to Souillart, Damoiseau derived these numbers in the follo^Ying
way: He adopted the mean distance of IV, in accordance with Pound's determi
nation = 496"0, and took 18''37 for Jupiter's semidiameter, so that, by division
riv = 2700102834. 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 produced in the radii vectores by
the perturbing force.
I beg leave to remark that 496"0 : 18''37 is not 2700102834 but 27000544366.
Happily the 4'!', 5"\ 6i", 7"' and S'h f,gure have no appreciable influence on our
computations, nor probably on those of Souillart. For the rest the 2"'i appendix,
further below, may be consulted on such numbers of many decimals.
2) 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 y/2. It thus becomes
for I and II, I and III, I and IV, II and III, II and IV, III and IV
3^574, 311402, 3''057, Q^2m,. 5i>006, 10ii43,
or:
3i>34'^, 31^24'^, 31» 3'", GHS'^i, 5'' 0"\ 10^126'^.
These numbers hold only for those very rare occasions in which 1^^
the occultation is central and 2"*^. 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"^. Appendix. Investigation of the uncertainty, existing in the
determination of the synodic jjeriods of the satellites.
In his introduction to the Tables Ecliptiques, p. XIX, Delambre says :
"Nous n'avons aucune observation d'éclipse antérieure èt 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 III for IV
0^00062, 0s00188, 0^0050, 0^0176.
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 Damoiseau, 20 years after Delambre, published new eclipse
tables^) for the satellites of Jupiter, he adopted the period of I un
1) The tables of Delambre and Damoiseau 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 Delamrre, 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 Bessel in his Astrono
mische Untersuchiingen and by Marth in the Monthly Notices of the Royal
Astronomical Society, Vol. LI, (1891).
( 327 )
changed, but applied the following corrections to the remaining ones :
II I 0^005 127 374,
III + 029 084 25,
IV — 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
176, our estimated uncertainties are only diminished by about — 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
,, „ II, III and IV to 2 decimals ,, ,, „
Th'e 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 !
3'^ Appendix. Meaning of the equations taken into account in
the 2°'^ part of the tables of Damoiseau.
On p. 321 we have referred to the 3"^ appendix for information
as to the equations which have been taken into account for each
satellite in the second part of the tables of Damoiseau. We will now
supply this information; we will denote by U, Uo, u\, z^ii,W]iiand
Miv the mean longitudes of the sun, of Jupiter and of the four
satellites; by :to the longitude of the perihelium of Jupiter, by rt' that
of the Earth, by :i'iii and ttiy the perijovia of III and IV ; by //the
longitude of the ascending node of Jupiter's equator on its orbit ;
finally by An, Am and Aiy 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 follo^ving below we have
taken the daily motion of the argument of each equation from the
tables in the second part of Damoiseau. 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. (Ill), (V), (VII) and (VIII) the letter i (the number of
synodic periods) is multiplied.
These daily motions are so nearly equal for several of the equations
of II, III 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 synodic period in days, we
get 360° \ a fraction. The 360° are of no account ; the fraction
is the factor of /; we thus recognise which is the equation w^e have
to deal with. In the preface of the second part of Damoiseau we
look in vain for any information on the subject.
I. For this satellite five terms have been taken into account.
W. 1 with an amplitude of 1°16, is the equation of the velocity
of light ; its argument is U — u^.
N°. 2, (amplitude 0°'29), is the equation caused by the ellipticity
of Jupiter's orbit ; the argument is the mean anomaly of Jupiter z^^ — n^^.
N°. 3 is 180° + the mean anomaly of the Earth, f7— rr'; by
its aid and that of W. 1 /. e. 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.
W. 4 with an amplitude of 0^45, shows the perturbation caused
by II in the motion of t. The argument is ui — uu.
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 ui — Ai.
II. Seven terms. N". 1, 2 and 3 have the same arguments as the
analogous terms for I ; the amplitudes of N°. 1 and 2 are half those
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 l°06), shows the perturbation caused by III in
the motion of II. The argument is uu — um.
N". 5, 6 and 7 serve for the latitude.
N" 5, (amplitude 3°"05), has the argument uu — tiv ;
N». 6, ( „ 47), „ „ „ uii—Au ;
N«. 7. ( „ 03), „ „ „ wii— ^m.
III. Nine terms. Nos. 1, 2 and 3 are the same as for I and II;
the amplitudes of W. 1 and N". 2 are 0°29 and 0°07.
( 329 )
N°. 4, (amplitude 0'^07), has the same argument as N°. 4 for II,
but it now shows the perturbation caused by II.
N". 5, (amplitude 0°15), is the equation of the centre; argument
Win — ^iii
N". 6, (amplitude 0''04), has the argument i^m— ^iv, 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 um — Jt, um — Am
and tiiii — Aiy.
IV. Seven terms.
Nos i, 2 and 3 are similar to those of the preceding satellites.
N" 4, (amplitude 0°"83), is the equation of the centre, argument
UlY — ^Tiv
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 Ua — Jr„.
N" 6, (amplitude 0''24), depends on the argument of the latitude
of the satellite itself; argument miv— ^iv
NV 7, (amplitude 0°04), is a minute perturbation, caused by III;
its argument is uiy — Am .
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 Damoiseau 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^'i columm contain the numbers of the occulted
and the occulting satellite. The appended letters ƒ 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 3% near if it is between
3^ and 9^ Furthermore ee denotes an eastern elongation, for which
the amplitude is about 3^ and iv e a western elongation, for which
the amplitude differs little from 9\
22
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 330 )
At tlie conjunction the elongations, counted along the orbit of
Jupiter, are equal ; they are to be found in the next cohnnn. 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 conjunction were central. In a few cases (Nos. 20, 23, 30, 48,
53 and 64), we find ij = y, consequently ij — y = 0. If the tables
were correct these conjunctions would be central. But in testing the
tables by the conjunctions observed by Messrs Fauth, Kijland and
Stanley Williams the difference of the ?/'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.
i. C. OUDEMANS. .Mutual Occultations and Eclipses of the Satellites of
Jupiter in 1908."
Starting point: tlip oeocentric superior conjunction of I on July 12, 1908,
at llh 2m.3 M. T. (ireenwich =^ 11'' 11'". Gó M. T. Paris = 23'' ll"i.t35 civil time Varh
30S6 4 a
Scale
1
30 168 ÜOO 000
; Imm. = 8" heliocentric.
100
200 300 ^00 500
Unit: the radius ot the earth's equator
600
700
oceedin^s Roval Acad. Anislurdaui. Vol. IX.
( 331 )
RESULTS.
Geocentric conjunctions of two satellites in June and July 1908.
Civil time
at Paris
a
Mean time
t Greenwich
n =
f =
near
far
1
Occulted
satellite y
Occulting
satellite y'
y' ~v
c
o
ol
§s
«
o
No.
6l
.1^
o "
X = x'
visible at
1
1 Jun
e öhoöin
31
Ma
y 17h46m
I
n
II
n
+2rl8
Or 16
— 0rl25
+0,03*
45m
Mt. Hamilton
2
1 »
18 25
1
June 6 16
I
we
II
n
—6,03
+0,31
+0,29
—0,02
11
Cape
3
2 »
1 55
1
))
13 46
'/
IV
n
—3,15
+0,18
—0,12
—0,30
7
Washington
4
2 »
13 16
2
s
1 7
"V
"/
—8,51
+0,51
+0,38
0,13
60
Madras, H. Kong
5
2 »
14
2
»
1 51
III
IV
n
—8,16
+0,51
+0,13
—0,38
11
Madras, H. Kong
6
2 »
14 4,5
2
»
1 55
"/
IV
n
—8,20
+0,37
+0,13
0,24
10
Madras, H. Kong
7
2 »
21 10
2
D
9 1
"V
"/
—4.54
+0,31
+0,20
0,11
41
Utrecht
8
9
3 »
3 »
3 15
3 50
2
2
»
»
15 6
15 41
"/
"V
I
?!
I
n
—0, 335
0,87
0,00
40.15
0,03
0,00
—0,03
—0,15
4
6
At the 1
same time c/lf
10
4 »
4 »
4 »
14 46
16
17 15
4
4
4
»
»
2 37
3 51
5 6
I
n
»
»
II
n
a:=+5,015
a;'=45,07
4,34
x=z+3,54
x'=i3,48»
—0,29
0,23
—0, 25*
0, 19*
+0, 03^
+0,035
\ Two contacts at\
Jthe same side. II just jg
breaches central cun}^
jj unction but then S
/retraces its steps. /'*
The satellites
will b
e visible as one body during nearly 2^ hours.
11
5 »
7 51
4
»
19 42
I
we
"/
—6,05
+0,33
+0,28
—0,05
12
Sydney
12
13
14
6 »
6 »
9 »
16 24
18 11,5
5 6
6
6
8
1)
»
4 15
6 2
16 57
I
n
III
n
0,48
+0,82
9,19
0,00
—0,06
+0.59
— 0,01
—0,17
+0,43
—0.01
—0,11
0,16
4
6
13
At the
same time d %
andJII eclipsed
Mt. Hamilton
?
15
16
10 »
10 »
4 26
5 23
9
9
»
16 17
17 14
"/
J
n
1
n
+0,29
0,58
+0, 14
+0,02
0,05'
— o,ai
0, 19*
—0,03
6
5
At the
same time c/o/;
17
10 1.
6 28
9
»
18 .18,5
'"/
I
n
—1,54
+0,15
+0,05
—0, 10
6
Mt. Hamilton
18
19
11 »
12 ö
3
10 34
10
11
»
»
14 51
22 25
I
we
II
n
+9,01
6,03
—0,32
+0,32
—0,52
+0,28
0,20
0,04
183
11
Washington
Wellington
20
13 »
18 46,5
13
»
6 37
'V
I
n
0,82
+0,01*
+0,0P
0,00
4
(ƒ ¥
21
22
23
13 »
15 »
17 »
21 37
23 55
7 53
1
13
15
16
»
9 28
11 46
19 4i
I
we
III
n
II
n
I
n
+i,31
—5,99
—0,92
0,07
+0,32
+0,02
1
—0,16
+0,29
+0,02
—0,09
—0,03
0,00 ■
6,5
10
4
II eclipsed
(Utrecht)
(Atl. Ocean)
22*
( 332 )
Hong Kong
Madras, H.Ko
Madras
Madras, H. Kt
Cape
(At). Ocean)
Madras
Mt. Hamiltoi
Hong Kong
1 Central Ash
Cape
Berlin etc.
Washington
II eclipsed
ivil lime Paris.
II
No.' *
ks 9=0
) 2,9
i
54i
O 10,5 9 8,9
U 18.8 ;i\ 24,8
15,3
24,8
6.2
0.9
55i
561
57
58)
59[)^7,2
601
I
61 !
62
63
5
lOs 2°8
5 18,7
6
OslSoe
5 0.2
3 20,8
11 9,2
2 11,8
1,9
5 19,7
2 22, 5
11 9, 7
2 11,8
1.9
4 ir>,9
7
lis 28^1
9 10,
1 2,
11 20,
oceultatioii of lUf by Uf,
K)8, 12''i)"'.3 eivil lime at
po
10 1,,G
11 0,1
2 11,
III
Mean
Longitf
Os 17045
1 20,32
O 25,16
0,32
Os 707
1 20,3
25,2
0,3
1,7
11 24j 4
64
05
66
67
68
69
70
71
72
5 + 0°53
6 — O, 32
7 0,00
IX + O, 05
+ 0,26
x' = — 2v 00 y' = + 0r,04
de
t
s,
enwich.
Limber of bis
that between
January) 1880
\vy) 1858 and
ir by treating
vliich contains
). B^or on the
on of the four
5, one headed
iles".
be used for
3
3,25
—
0,13
—
0,04
+
0,04
+
0,15
+
0,01
3
3,28
4
9,25
10 24,03
2 23,5
6
2^2403
1 20,3
25,2
0.3
7
2^ 2^0
1 20.3
25, 2
3
5 10,1 4 17,8
8s G03
1 20,3
25,2
0,3
10 22.1
9
6s20°4
1 20,3
25, 2
0,3
9 12,2
7 4 2O0i
8 +0,11
9 +0,03
IX — 0, 11
+ 2,04
ar' = — 9r^0 1 f = + 0r,055
+ 0,453 A t
pAt
83 =: + 59'"
tion: 12 9 ,3,
,tion : 13 ST^.
value, we still do not get
le of II becomes 9*2756,
id:
t5 A/
0"',9
8 ,3
Civil time at Paris,
Mean „ „ „ ,
' ,, ,, ,, Greenwich.
( 334 )
i.lv 15, 111
III
0.50
'.
0.27
+1.20
5.12
1.
a. 12
".
2,2!
'".
■H."
"1.
*•"■
'
"
.858
8.5^.
7.S75
1. 08 , 0. 14 ' 0.18=1
10.200
8.17.00
3.90
10. 2»8 1 0.18"0
11.28^
lIs»
0.20
11 S2.0
3 8.0 jl. 28.0,0 4.2
10 20,7
2 2.03
2 2,.
I 'Z ' °>l
1 2.3
Id)
5 13.4
0.5.0 5 28.4^4 2.7
3 215
11 .8,83
8 1.3
11 18.8 ill 24.8
11 20.1
Julj 10O8
7 13.23
10 2i7
4 3.
3 20.2
5 0.0
8 27,7
2 12,39
1. 15,3
3 20.8
9 22,5
10 1.5
E
10 28.85
4 21.14
3.82
°':::
1.1.8
0.'
11 19.0
':z
.iso
1.24,8
1 0.2
0.0
""'■:
11 0,7
11 0.3
1.5,0
4 4.3 IS.7
7 4.7
24,4
.5,35
0.7,2
5 19,7
4 1r..9
,12,,5
2
0.10
2
0,08
5 + 053
4
+ 0.43
6 + .•27
'
+ 0,00
_ 0.32
7 0.»
4 5,35
SG
4 17,3!
 ix + o.or.
S0
4 .7,37
 IX + 0.05
AmpUlude
21.90
+ 1.32 A
.plli.d.
11 17,08
+ 0,28
■=2
20 ,
. + 0r,.4
J^
2.0O
>■ = +
Or,0*
PL'ond example we will inke llie ruciillatjoii
I our «Irnwing gavo June 2. I9l)8. 12i'9'"'
1 first approximation.
■■
„1
1908 J„.,
u!J°i.
.L
loo
4.1L
S.1»! 10.^,=..
j „,^3
4.*»
lyingil.
«■'t:
2.21=3
.loi/o.
■■■
9.3
1 20
37
0.0
0.5
0.1
0.5
:0 23.5..20.7
0.3: 0.0
3 11.4
0.0
12H7
1 20.32
35.10
0.3S
1 20.S
25,2
0,3
120,3
95.2
180.3
25.9
120.3
25.2
0.3
ia3
IK.2
0.3
2 3
84
9 27.0
4 0.7
5 ..1
.. 0.5 1 3 18.3
2.S.0
9 2*7
3 3.25
223,5
5.0,.
4 17.S
.0 21.
0.9.2
<
0
28
1
0.13
^
0

:
3 +2^80
*
0.04
+ 0.04
+ 0.15
+ 0.01
7 +«>0.
8 +0...
S0
4 9
58
Amplitude
23
33
= 8
« , + o..t
Anplitudi'
wZ^
+ 2.04
■■
= 0,0
r
= + 0..05
Rcsull lo lal\ M 17»16"i7 Cml 1
= 5 le 7 Moan
= 6 7 35
— 8,84 + 0,280 i / = — 9,01 + 0,453 A I
+ 0,17 = + 0,173 i I
i I = + 09S3 = + 5»"
Finit approximalioii : 12 9 ,3,
Second nppioximatioo : 13 6 ,3.
If we repeat the rompiilation will, this value, ive still lio not gel
equaiilj' of the elonsalions. Tlie amplitiitle ol II liecoiiies 9275IÏ,
Ihiit for 111 10.26 08. Furllicrnoni ive llticl :
 S',535 + 0',327 A / = — 8',56 + 0',4645 i (
+ ,025 = + 0,1865 A (
A(= + OM82 = + 10",9
\niieet (oiiiiliuiies tlie oEliei Iieadeil Aiinees. bis!
The hflli hue of llii^ eonipiilalinii (JiiU 111181 i
all the colli inctton^ of ihii month
13»
19
•2 Civil lime at
1
19
9
2 Mean „ „
86 „ „ „
( 336 )
Pathology. — ''On the Amboceptors of an Antistreptococcus .<ierum.''
By H. Eysbroek. (From the Pathological Institute of Utrecht).
(Coinmuiiicated 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 immunesera, is thermostatic and is named differently by
different investigators, according to the idea which they make of its
influence (Amboceptor of Ehrlich, Substance sensibilisatrice of Bordet,
Fixateur of Metchnikoff). 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 Fodor ^) and Flügge ^) in
normal bloodserum, to which is given the name of alexin by Buchner.
Next to this name at present the denominations complement (Ehrlich)
and cytase (Metchnikoff) are used.
Had Metchnikoff in 1889 already pointed out the analogy between
hemolytic and bacteriolytic processes, later investigations have com
pletely comfirmed this supposition.
In 1901 Bordet and Gengou ^) published a method to demonstrate
the presence of a "substance sensibilisatrice" in the serum of an
animal, which was immunized against a certain microorganism, 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 microorganism belonging to it from
others, by means of a combination with the complement.
Using the above mentioned method of Bordet — Gengou, Besredka"*)
succeeded in pointing out an amboceptor also in an antistreptococcus
') Deutsche Med. Wochenschrift, 1887, W. 3i, S. 745.
2) Zeitschrift fur Hygiene, Bd. IV, S. 208.
S) Annates de I'Inst. Pasteur, T. 15, 1901, p. 289.
*3 Annates de I'Inst. Pasteur, T. 18, 1904, p. 363.
( 337 )
serum prepared bj himself. This serum was obtained from a horse,
which foi 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 antistreptococcus 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 diflterent processes of disease, must be regarded as
representatives of one and 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 subdivision in the large group of the pathogenic strepto
cocci, would be of great importance for the bloodserumtherapy.
Some years ago Schottmüller ^) tried to give a new division, based
on biological grounds instead of the older morphological division in
streptococcus longus and streptococcus brevis (von Lingelsheim "),
Behring ^). 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 Schottmüller divides the pathogenic
streptococci as follows:
1. Streptococcus pyogenes s. erysipelatos.
2. Streptococcus mitior s. viridans.
3. Streptococcus mucosus.
Several other investigators (Eug. Fraenkel *), Silberstrom *),
1) Munch. Med. Wochenschrift, 1903, No. 20, S. 849; No. 21, S. 909.
2) Zeitschrift fur Hygiene, Bd. X, S. 331.
8) Gentralblatt. fur Bakteriologie, Bd. 12, S. 192.
4) Munch. Med. Wochenschrift, 1905, N'\ 12, S. 548; m 39, S. 1869.
5) Gentralblatt fur Bakt., le Abth., Orig., Bd. 41, S. 409.
( 338 )
Baümann ^) have latterly come to the same result in an almost
similar waj'.
Besredka ') 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 {JB, C), from which Besredka 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 antistreptococcus serum, which I used, 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 asciticbouillon of the
different streptococci and of a bouillonculture 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 antistreptococcus serum, whilst
animals used for controll died shortly after.
The method, followed by me, is that of Bordet — Gengou '). For
^) Munch. Med. Wochenschrift, 1906, N". 24, S. 1193.
2) 1. c.
3) 1. c.
( 339 )
each experiment six tubes were used, which contained consecutively ^) :
N°. 1 : ' 'i„ c.C. complement, \'^ c.C. emulsion of streptococci,
7j c.C. antistreptococciis serum.
N". 2 : '/lo c.C. compl., \\ c.C. emulsion of str., 7, c.C. normal
horseserum.
N". 3: 7,0 c.C. compl., 7, c.C. physiological NaCl, 7, c.C.
antistreptococcus serum.
N". 4: 7io C.C. compl., '/, c.C. physiol. NaCI, 7, c.C. normal
horseserum.
N". 5 : \/,, c.C. physiol. NaCl, '/, c.C. emulsion of str., \', c.C.
antistreptococcus serum.
N°. 6: 7j„ c.C. physiol. NaCl, \/, c.C. emulsion of str., 7, cC.
normal horseserum.
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 7io c.C.
of a mixture, composed of 2 c.C. of hemolytic serum and 1 c.C.
corpuscles of a rabbit, which were buspended 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 nonexistence
of the hemolyse in the first tube.
It is necessary to repeat all these controllexperiments 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 V^„ c.C. of the complement was added. After three
hours 7io c.C. of a mixture, composed of 2 c.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 bloodserum of a guineapig was used. The strep
tococci, which were to be examined, were cultivated on Loeffler's coagulated
bloodserum and after 24 hours suspended in physiological NaGl to a homogeneous
emulsion. The antistreptococcus serum was heated in advance for one hour to
56° C, as well as the fresh normal horseserum, used for controll, and the hemoljtic
serum originating from guineapigs, which were treated 3 or 4 times with 5 c.C.
of defibrinated blood of rabbits. The physiological NaCI used, was always a solution
f 0,90/0.
( 340 )
follows: no heniolyse in the first (least diluted) tube, a little hemo
Ijse in the 2"^ 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
BoRDET — 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 controlltubes, 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
streptococcusserum, 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 antistreptococcus 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 antisti'eptococcus serum.
The investigation however did not confirm this supposition. All
streptococci, no matter what their origin, shoioed a strong combination
with the complement under the influence of the antistreptococcus serum.
Keeping to the specific of the amboceptors, the conclusion of
Besredka ^) 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 guineapig,
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
1) 1. c.
( 341 )
deviating growth on the iisiuil eiilturemedia, shows very distinct
differences from the other pathogenic streptococci, whether from man
or from animals, tlie conchision is at liand, tiiat 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 antistreptococcus
serum is also active against microorganisms, which do not belong to
the streptococci viz, pneumococci and meningococci.
By the above is fully shown, that 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 Bordet — Gengou, 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 antistrepto
coccus serum originates, is a most favourable testanimal as regards
the forming of antibodies, then the above mentioned facts would
remain the same.
DoPTER^) has recently found, that the amboceptor, present in the
serum of a horse which has been treated with dysenteria bacilli
(type Shiga) during 18 months, next to the action on these bacilli,
1) Annales de I'lnst. Pasteur, T. 19. 1905, p. 753
( 342 )
also presented the selfsame effect against the socalled pseudo or
paradysenteria bacilli (type Flexner, Kruse). Asserting the specific
activity of the amboceptor, he decides on "l'unité specifiqne" of
the djsenteria 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 antistreptococ
cus serum w^ill have a favourable effect on patients suffering from
pneumonia, typhus, anthrax etc. 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 guineapigs of
nearly the same weight received partly a small quantity of anti
streptococcus serum (2 — 3 c.C), which contained some anthrax bacilli
(one eye of a deluted twelve hours, old culture on bouillonagar),
partly normal horseserum (2 — 3 c.C.) with an equal dose of anthrax
bacilli. A favourable effect of the antistreptococcus serum compared
to normal serum was never perceptible. The animals died generally
about the same time, within 48 hours.
Yet Predtetschensky ^), who has made such investigations with
rabbits, is of opinion that a favourable effect can be perceived from
antidiphtheria as well as from antistreptococcus 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 horseserum.
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 KoLiiE and Wassermann") do with regard to a
meningococcus serum prepared by them. In the meningococcus serum
of JocHMANN (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 specific activity in
relation to a prolonged administering of antigens is known for other
substances in immunesera 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 Nuttall ')
was able to get a precipitation with the bloodserum of all kinds
1) Gentralblatt fur Bakt., le Ablh., Ref., Bd. 38, S. 395.
2) Deutsche Med. Wochenschrift, 1906, n» 16, S. 609.
3) Blood immunity and blood relationship, Cambridge, 1904, p. 74, 135, 409.
C 34 )
of mammals even with a very strong precipi tinserum, which was
obtained with and against an arbitrary mammiferalbumen ("mamma
lian reaction"). Hauser ^) comes to a similar result; only quantitative
differences remain.
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 in 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 of 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) Munch. Med. Wochenschrift, 1904, n" 7, S. 289.
2) After Wood, Lummer and Pringsheim, Ebert, especially Pucgianti has inves
tigated the anomalous dispersion of various metallic vapours. In Nuovo Cimento.
Serie V,' Vol. IX, p. 303 (1905) Pugcianti describes over a hundred hnes, showing
the phenomenon.
^) VV. H. Julius, "Dispersion bands in absorption spectra." Proc. Roy. Acad.
Amsl. VII, p. 134140 (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 tiie other rays of the
spectrum ; absorption and emission changed relatively little. The
result was, that the distribution of the light in the neighbourhood of
Dj 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 efl"ect 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. Wood^) 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.
NN' (see fig. 1) is a nickel tube of 60 centimetres length, 5.5 cms.
diameter and 0,07 cm. thickness. Its middle part, having a length
of 30 cms., is placed inside an electrical furnace of Heraeus (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. Wood, Phil. Mag. [6], 3, p. 128; G, p. 362.
( 345 )
shut airtight 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 M' , 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 b {b' and a'), placed diametrically
)=•
Fig. 1,
opposite to each other and pro\ided 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 Le
Chatei.ier pyrometer is fitted airtight,
while on d a glass cock with mercury
lock is cemented, leading to a mano
meter and a Geryk airpump. 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
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 tw^o nickel tubes A and B of 0,5 cm. diameter,
leading from a to a' and from b to b' 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 cms. In the four openings of
the caps, ^1 and B are ftistened airtight 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
Fig. 2.
( 346 )
p"
not alter their shape through tension. At the same time the rubber
insulates A and B electrically from JSfN'. 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
aircurrent 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.
Fig 3 gives a sketch of the whole arrangement.
The light of the positive carbon L is concentrated by
•* the lens E on a screen Q, having a slitshaped aper
ture of adjustable breadth. The lens F 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.
If now the opening in the diaphragm F has the
shape of a vertical narrow slit and if its image falls
exactly on the slit of the spectograph, then in this latter
the continuous spectrum of the arclight appears with
great brightness. If the tube JVN' is not heated, D^
and 7^3 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
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
by a simple combination of lenses and mirrors without
passing the electric furnace. So on each photograph that
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 cms.) with 14436 lines to the inch, and two sil
vered mirrors of Zeiss; the collimator mirror has a
Fig. 3. focal distance of 150 cms., the other of 250 cms. Most
of the work was done in the second spectrum.
( 347)
Wlien heating the sodiuiii for the tirst time a pretty large quan
tity of gas escaped from it (according to Wood hydrogen), whicii 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, nexev went beyond 450°. The
inner wall of A^X' 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 ou 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 Jewett ^). He gives the following table.
temperature
density
368°
0.00000009
373
0.00000020
37G
0.0000003.5
380
0.00000043
385
0.00000103
387
0.00000135
3'JO
0.00000160
;95
0.00000270
400
0.00000350
40G
0.00000480
408
0.000C0543
412
0.00000590
418
0.00000714
420
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
^) F. B. Jewett, A new Method of determining the VapourDensity of Metallic
Vapours, and an Experimental Application to the Cases of Sodium and Mercury.
Phil. Mag. [6J, 4, p. 546. (1902).
23
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 348 )
saturated sodium vapour is about — — of that of tlie atmospheric air
at O"* and 76 cms.
Observations.
If we now regulate the intensity of the current in the furnace in
such a manner that the thermocouple 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 falls from the middle towards the ends, but since the
surfaces of equal temperature are practically perpendicular to the
beam of light, all rays pass nearly rectilinearly through the vapour.
Accordingly the spectrum is only little changed ; the two Z)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 aii' 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 va]:»our density has decreased
a little. The reason is that rays of light with very great refractive
indices are now bent towards cf (fig. 3), rays with very small indices
towards q; hence in the image of the slit P which is formed on >S,
rays belonging to regions on both sides of tlie Z>lines no longer
occur, while yet this image remains perfectly shai'p since the course
of all other rays of the spectrum has not been perceptibly altered.
If now at the same time the tube B is heated by a current of e.g.
20 Ampères, by which the density gradient^ in the space between
the tubes is increased, the breadth of the lines becomes distinctly
greater still. The heat geuerated in the tube by the current is about
1 caloiy per second ; it is, howevei', for the greater pari conducted
away to the cooled ends of the tube, so that tlie iisc of leiiiperalure
can only be small.
By switching a current key and a cock, A and 7> can be made
to suddenly exchange parts, so that A is heated, B cooled. The dark
bands then shrink, pass into sharp Z)lines and then expand again,
until, after a few minutes, they have recovered their original breadth.
P'ine 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, liowever, there certainly exist currents in the mass of vapour
which cause the distribution of density to be less regular. Also when
A and B 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 ascribed
either to motion of the lightemitting 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 Z)lines appear as in «, PI. I. If
now A is cooled below this temperature, B raised above it, the
dark Dlines only broaden in the direction of the shorter wave
lengths, while at the side of the longer wavelengths the intensity of
the light is even increased, since now also anomalously bent rays
from the radiation field j)' can reach the point S through the slit Q.
(see j5, PI. I). The spectrum /9 passes into y when the temperature
difference between .1 and B is made to change its sign or also when the
original temperature difference is maintained and the slit in P is made
much broader towards j) instead of towards //. A small shifting of
the whole diaphragm P (starting from the condition in which it was
when taking /?) so that S falls exactly in the shadow, causes the
spectrum ö 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' whicii 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 tinfoil 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 wavelengths)
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
of tig. 4, then the spectrum s is obtained. When the
density gradient is diminished the figure shrinks, ?; if
now the density gradient is made to change its sign
and to increase, the spectrum proceeds through the
stages a (gradient exactly zero) and ^j to 0.
Fig. 4. The relation between the shape of the opening
in the diaj^hragm and that of the bright spots in the spectrum might
easily liave been foretold from the shape of the
dispersion curve. Having, however, experimentally
found the relation " between the two figures for a
simple case as the one above, it is not difficult to
design for any desired distribution of light the shape
of the required opening in the diaphragm. The
Fig. 5. flower I and its inversion n required the diaphragm,
represented in fig. 5. By reversing the gradient the image i passes into x.
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 sunspots, faculae or prominences. On
Plate II a number of arbitrary distributions of light have been
collected. They w^ere all produced in sodium vapour of 390^ on the
average; «' is again the spectrum with equal temperatures of the
tubes A and B. In v on the dark dispersion band D^ a bright
double line is seen, reminding us of the spectrum of the calcium
flocculi of Hale. In the same negative D^ also shows a line double
line, which however is no longer visible in the reproduction. The
spectra <p, /, if' imitate the origin of a sun spot and prominence
spectrum ; tp namely represents the spectrum of the quiet solar limb
with radially placed slit ; in x ^ prominence appears and a spot with
phenomena of reversal ; if' shows all this in a stronger degree. If
now the density gradient is made to change sign, the image first
shrinks again lo (p after which it expands to w, in a certain sense
the inversion of if'. The remarkable aspect of these gradual changes,
admitting of perfect regulation, is only imperfectly rendered by the
photographs.
The 1'elation between the curvature of the rays a7id
the density gradient.
The (uestion arises whether it is prohahle thai circumstances as
were realised in our experiments are also met with in nature, or in
( 351 )
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 j)laces of the field reproduced by the lens F,
then the i>>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 9 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 ö 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 :
Q il z=. d : Ö.
Putting d=l cm., cZ=110 cms., /=27 cms. this gives: (►=3000 cms.
The average density L of the sodium vapour was in this case about
of that of the atmospheric air.
1000 ^
Let us see how" 9 changes with the density gradient.
We always have :
n
n ^ '
if n represents the local index of refraction of the medium for the
. dii
ray under consideration and n = — the change of this index per cm.
in the direction of the centre of curvature. Approximately we have,
for a given kind of light:
n—\
From this follows:
A
■ zrz.
: constant =
n 
^RL
+ 1
dn
^A
n
'd^^
R —
ds
9
RL
R
+ 1
dA '
ds
( 352 )
but since for rarefied gases n differs little from unity, even for the
anomalously dispersed rays which we consider, Rt\ may be neglected
with regard to 1 and we may write
o^liK ^">
R —
ds
For every kind of light q 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 B, 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 B would have been
pretty easy to determine thermoelectrically ; 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 B 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.
dL ^
The second method at once gives an average value of — for
ds
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 q has been
determined.
Now Wood (Phil. Mag. [6], 8, p. 319) gives a table for the values
of n for rays from the immediate vicinity of the Z)lines. These data,
however, refer to saturated sodium vapour of 644°; but we may
deduce from them the values of n 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 X = 5875) becomes — = 11 times greater, and at fur
50
ther heating from 508° to 644" again — ^ 12,5 times greater (now
found by interference measurement with light fiom the mercury line
A = 5461) ; hence from 390^ to 644° the refractive power increases
in ratio of 1 to 11 X 12,5 = 137.
Since now for rays, situated at 0,4 ANGSTRÖMunit from the D
( 353 )
lines ^) we have n — 1 = ± 0.36, (as the average of three values
taken from Wood's table on page 319), we ought to have with
sodium vapour at 390^ for the same kind of rays
0.36
n—\ = = 0.0026.
137
The density A at 390° is, according Jewett, 0.0000016, hence
n~\ 0.0026
R = — — = = 1600.
A 0.0000016
Then from formula (2) follows
dA 1 1
— = — = = 0,0000002.
ds Rq 1600 X 3000
Dispersion bands in 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 cur\ature 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 sufticient 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 inire 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 e in our plate shows that the extremities of the peaks corre
spond pretty well to light of this wavelength; for they approach the Dlines to
a distance which certainly is no more than V15 of the distance of the Dlines
which amounts to G Angstr. 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 arc at a distance of 1 cm. from
the crater, at Ü.0001, then we have already an average gradient 5000 times as
large as that used in our experiments.
( 354 )
When we bear this in mind, many nntil now mysterious phenomena
will find a ready explanation. So e.g. the fact that Liveing 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 evapoi'ated 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
RuNGE 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 RowiiANo'schen Atlas ganz scharf sind, und dann stets etwas
kleinere Wellenlange haben. So haben wir 4703,33, Rowland 4703,17 ;
wir 5528,75, Rowland 5528,62. Unscharfe nach Roth verleitet ja
leicht der Linie grösseie WQllenlange zuzuschreiben; so gross kann
aber der Fehler nicht sein, denn die RowLANo'sche Ablesung liegt
ganz ausserhalb des Randes unserer Linie. Wir wissen daher nicht,
woher diese Differenz riihrt." Kayser has later ^) given an explana
tion of this fact, based on a combination of reversal with asymme
trical widening; but a more probable solntion 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
arc, 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) Liveing and Dewar, On the reversal of the lines of metallic vapours, Proc.
Roy. Soc. 27, p. 132136: 28, p. 367—372 (1878—1879).
2) Kayser und Runge, Über die Spektren der Elementc, IV, S. 13.
^) Kavser. Hundbuch der Speklroskopie II, S. 366.
( 355 )
to see light emitted by the vapour, in which light different wave
lengths occur, all greater than the exact wavelength of the serial
lines. The observed displaced lines of the second secondary series
are consequently comparableto appaicnt emission lines of the spectrum
d of our plate I.
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 wavelength.
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 increases when we move away from the
positi\e 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 wavelengths
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 Stark's theory are formed
in the arc 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
deser\es 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 uutil now neglected principle of raycurving
has undoubtedly been at the root of the matter are found in Kayser's
handbook II, p. 292—298, 304, 306, 348351, 359—361, 366.
Dispersion hands 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 rajs 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 — — of
this amount, so that the vertical density gradient is
0.001293
= 16 X 1010.
1050 X 760
The horizontal gradients occurring in the vicinity of depressions
1
are much smaller; even during storms they are only about——
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 Schmidt'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. 134, 140 and 323.
2) Arrhenius. Lehrbuch der kosmischen Physik, S. 676.
( ■■^5'! )
to the radius of curvature of luminous rays whose patii is horizontal
at a point of its surface. Tliis radius of curvature is. consequently
Q = 7 X lO'o cms., a value which we may introduce into the expres
sion for the density gradient .
dA _ 1
ds Rq
The refractive equivalent R for rays that undergo no anomalous
dispersion varies with different substances, to be sure; hut in an
approximate calculation we may put R = 0,5. Then at the height of
the critical sphere we shall have :
dL 1
0.29 X lO'o,
ds 0.5 X 7 X lO'o
(this is 50 times less than the density gradient in our atmosphere).
All arguments supporting Schmidt'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 wavelength of which differs but very
little from that of one of the sodium lines, may seem to come from
points situated some arc 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
9' = 6 X iOio cm.
If we further assume that of the kind of light under consideration
the wavelength is (\4 Angstromunits greater than that of JJ^, then
for this kind of light R'=:^QOO, as may be derived from the obser
vations of Wool) and of Jewett ^) ; we thus find for the density
gradient of the sodium vapour
— = — — = 0.0001 X 1010,
ds R'q' 1600 X 6 X l<>'^
a quantity, 2900 times smaller than the density gradient of the
gaseous mixture.
Hence if only part of the caseous mixture consists of sodium
" 3000
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 ivhich these substances are
presetit 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 nonradially 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 Schmidt (Phys.
Zeilschr. 4, S. 232 and 341) according to which the chief constituent of the solar
atmosphere would be a very light, until now unknown gas.
W. H. Julius. ARBITRARY DISTRIBUTION OF THE LIGHT IN DISPERSION BANDS.
PL. 1.
a>
'
H
^:l
<^
Proceedings Royal Acad. Amslerdam. Vol. IX,
Hüiotypc, Van Leer, Amsterdam.
W. H, Julius. ARBITRARY DISTRIBUTION OF THE LIGHT IN DISPERSION BANDS.
PL. n.
Proceedings Royal Acad. Amslerdam. Vol. IX.
HtHUype, Van Leer. Awuterdam.
( 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 Fraunhofer's lines not
only absorption lines, as Kirchhof 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 ivhich possesses numerous ^) different
liquid phases of lohich three at least are stable in regard to
the isotropous liquid." By Dr. F. M. Jaeger. (Communicated
by Prof. H. W. Bakhuis Roozeboom).
§ 1. The compound which exhibits the highly remarkable phenomena
to be described, is cholesterylcinnamijlate: C^. H^j O^C .CH iCHCgHj.
I have prepared this substance by melting together equal quantities
of pure cholesterol and cinnamylchloride in a small flask, which
w^as 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 li(iuid stale and viceversa, which yet I cannot explain at this
moment.
( 360 )
The solidiüed mass is dissolved in boiling ether, and the brown
liquid is boiled with animal charcoal for an hour in a refluxappa
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 etheralcohol, 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, thinw^alled testtubes, whilst
surrounded of a cylindrical airbath, and w'hilst the thermometer
was placed in the liquid completely w4iich 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 orangered, 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 t [155°. 8 C] at which the substance sohdifies; the break in the
curve is distinct as the heat eflect 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 obsei've 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) BoNDZYNSKi and Humnicki (Zeitsehr. f. physiol. Chem. 22, 396, (1896), describe
a cinnamylale which as regards solubility etc. agrees with mine, but which melts
at 149°. This is evidently identical with my first temperature of transiliou.
( 361 )
On cooling, the following phenomena occur: At about 200° the
isotropous liquid becomes turbid, at 198° the doubljrefracting 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 undercooling by inoculation, solidifies at 155°. 8.
If the solid substance is melted under Lehmann'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 phasetraject 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
isotropousliquid 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 isotropousliquid. The whole shows much
resemblance to a gradual dissociation and association between more
or less complicated moleculecomplexes. It is quite possible that
the transitions solidliquid 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 cliole
( 362 )
sterjlesters usually display the phenomena of the doublyrefracting
liquidconditions. All this seems probable to the investigator, the more
so as it has been proved by Lehmann, that in my other cholesteryl
esters, even in the case of the caprinate, botli or one of the two
anisotropous liquidphases were always labile and only realizable on
undercooling; some of them, such as the «^(^butyrate, only exhibited
their labile anisotropous liquidphases, when containing some impurities
and not when in a pure condition. With the idea of a gradual
dissociation of compound moleculecomplexes 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 alwaj^s the
endphenomenon, 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 anisotropousliquid 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
interferencecolours, 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 doublyrefracting, 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 t^ solid. The view expressed by IjEHmann, 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 continuity of the aggregate
conditions, treated of here.
§ 3. I wish to observe, finally, that cholesterylcinnamy late 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
temperaturelimits.
Zaandam, 26 Oct. '06.
1; A still more dislinct case of this [jheiiomeiion has now been found by me
in ^.phytosterylpropionate, which 1 hope soon to discuss in another communication.
( 363 )
Chemistry. ''The behaviour of the halogens towards each other ' .
By Prof. H. W. Bakhuis Roozeboom.
If the phasedoctrine 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 ICI3 and
ICl. He also showed that it is probable that ICl, on melting, liquefies
to a very large extent without dissociation, whilst on the other
hand ICl, is almost entirely dissociated into ICl ( 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
ICl is also exceedingly small in the vapour, it being already known
that it is very large in the case of ICl,.
From the investigation of Meerum Terwogt ') it has been shown
that Br and I form only one compound BrI which in the solid state
forms mixed crystals both with 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 ICl, is a feeble and
ICl a strong compound. IBr is also a feeble compound and no com
pound exists between CI 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 4 CI.
From the researches of Moissan and others it follows that Fluorine
yields the compound IF^ which is stable even in the vapourcondition.
With Bromine, the compound BrFg 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 phasedoctrine, there does not exist as yet a
reasonable certainty as to their number or their stability.
Mathematics. — "Second communication on the Plücker equivalents
of a cyclic point of a twisted curve." By Dr. W. A. VERSiiUYS.
(Communicated by Prof. P. H. Schoute).
§ 1. If the origin of coordinates is a cyclic point [n, r, 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:
A* = a <",
y=.h^ «"+'• + h^ v^+r+\ f b^ ^«+'•+2 { etc.,
e = Co <«+'+"> f Cj «»+'+'»+i f Cj «"+'+»»+2 J\ etc.
Let gi be the greatest common divisor of n and r, let q^ be that
of r and m, q^ that of m and n\r and finally q^ that of n and
r j m.
If q^ = q^ =: q^ = q^ = 1 the PLtJCKER equivalents depend only
on n, T and m. In a preceding communication ^) I gave the Plücker
equivalents for this special case ").
§ 2. If the 4 G. C. Divisors q are not all unity, the PLtJCKER
equivalents of the cyclic point (?i, ?', ni) depend on the values of the
coefficients h and c, just as in general for a cyclic point of a plane
curve given by the developments:
X = V\
y =: t^+^ \ d^ i«+'«+i f fZj i«+"»+2 } etc.,
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.
") The deduction of these equivalents is to be found among others in my treatise :
"Points sing, des courbes gaudies données par les equations: x = t>', y — tn\r^
2 = ^1+'+'"," inserted in "Archives du Musee Teyler'\ série II, t. X, 1906.
3) A. Brill and M. Noether. Die Entwicklung der Theorie der algebraischen
Functionen, p. 400. Jahresbericht der Deutschen MathematikerVereinigung, 111,
1892—93.
( 365 )
If the coefficients c and h are not zero, if no special relations
exist between these coefficients and if besides n, r and m are greater
than one, the cyclic point {n, r, m) is equivalent to
n — 1 stationary points ^ and to
\{n — 1) {n 4 r — 3) + g,  1 : 2 nodes H.
The osculating plane of the curve C in the cyclic point {n, r, m)
is equivalent to
m — 1 stationary planes « and to
\{m — 1) (r f m — 3) + ^, — 1 : 2 double planes G.
The tangent of the curve C in the cyclic point {n, r, m) is equi
valent to
7' — 1 stationary tangents &, to
\[r — l){7i \ r — 3) + ^1 — 1 : 2 double tangents to and to
\[r — 1) {r \ m — 3) [ ?j — 1 : 2 double generatrices tu' of the
developable formed by the tangents of the curve C.
§ 3. The cyclic point (n, r, m) of the curve C is an tz j rfold
point of the developable of which C is the cuspidal curve.
The cyclic point {n, r, m) counts for
{n \ r — 2) [71 + r j ^^^)
points of intersection of the cuspidal curve C with the second polar
surface of for an arbitrary point.
Through the cyclic point (ji, r, m) of the cuspidal curve C pass
\n {n + 2r f m — 4) + ^3 — g'J : 2
branches of tiie nodal curve of the developable 0.
All these nodal branches touch in the cyclic point {n, r, m) the
tangent of the cuspidal curve C (the A'axis).
They have with this common tangent in the point of contact
(^ + r) [n + 2r + in — 4.) J^q^q^\:2
points in common.
The nodal branches passing through the cyclic point {n, r, m) all
have in this point as osculating plane the osculating plane ^ = of
the cuspidal curve C.
These nodal branches have with theii' osculating plane z = in
the cyclic point {n, r, m)
(,^ + ,, 4_ „,) [n + 2?' + 111  4) + (^,  q,\ : 2
points in common.
§ 4. The case of an ordinary stationary plane ft, the point of
contact of which is a cyclic 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 53 ]> 1. If
r ^ 1 the coefficients b and c must satisfy special conditions.
If r =zl then through the cyclic point {n, r, m) of the cuspidal
curve pass eilher 5', : 2, or {q^ — 1) : 2 of these nodal intersecting
branches. All intersecting nodal branches have a common tangent
in the plane ^ = if r = 1.
§ 5. The case of an ordinary stationary point ^ (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 ^'^^l. lï q^'^1 and m = l these particular nodal
branches are always present. \i q^'^1 and also m ]> 1 the coefficients
h and c must satisfy special conditions. These particular nodal
branches have in the cyclic point {n, v, in) a common osculating
plane (differing from the plane 2 = 0) if m = 1.
§ 6. The tangent to C in the cyclic point {n, r, m) is an rfold
generatrix y on the developable 0. The r sheets of the surface
passing through the generatrix g all touch the osculating plane z =
of C in the point {n, r, m).
The generatrix g moreover meets in q — {n { 2 r \ m) points R
a sheet of the surface 0, when is of order q.
In every point R the generatrix g meets r branches of the nodal
curve. These ?" branches form, when m'^ r a singularity {r, r, rn — r)
and the osculating plane of these nodal branches is the tangent
plane of along g.
If m <^ 7' these r nodal branches form a singularity (r, in, r — m)
and the osculating plane of these r nodal branches is the tangent
plane of along the generatrix intersecting g in R.
If r = m these r nodal branches form a singularity {r, r, 1).
§ 7. In general the singular generatrix g will meet only nodal
branches in the cyclic point {n, r, m) and in the points R. If ^'^ ^ 1
the generatrix g may meet moreover nodal branches arising from
the fact that some of the r sheets, which touch each other along g pene
trate each other. These nodal branches meet g in the same point Q.
If g'j > 1 and ?? = 1 there is always such a point of intersection Q.
\i q^'^l and n > 1 the coefficients b and c must satisfy some special
conditions if the sheets passing through g are to peneti'ate each other.
( 367 )
Physics. — ^'On the measurement of very low temperatures. XIII.
Determinations with the hydrogen thermometer" Bj 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
hydrogen 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. Nos 95« 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. W. 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
c and e (Plate II, Comm. N". 27) at the end of the capillary. The
dimensions of the thermometer reservoir of Comm. N°. 60 (80 c.M'.)
did not present any difficulty in our measurements, the bath in the
cryostats (see Comm. N«s. 83, 94^, 94^ and 94/ (May and June
1905 and June 1906)) offering sufïicient 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 thermoelenient '). 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 determinatioji 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 PI. VI belonging to Comm. N". 60.
By means of an indiarubber tube and a Tpiece /„ the thermometer
{a, h, c, d, e, h, k) is connected on one side with the manometer
/j, 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 {ii^, n^ (airtrap) n^. Besides from
the manometer and the barometer joined at iii^, the pressure can
now also immediately be read from the difference in level of the
mercury in n^ 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 Jaquerod, (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 (Gf. Comm. Nos. 95" and 95<^).
As soon as one of suitable dimensions will be ready, the experiment will be repeated.
^) 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.
^) We must be very careful that no narrowings occur.
6) A great deal of time must be given to exhausting the reservoir witli 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.
•) Travaux et Mémoires du Bureau International, Tome VI.
( 369 )
n. By means of hydrogen prepared in the apparatus of Comm.
N". 27 with the improvements described in Comm. N". 94« (June 1905)
§ 2. After liaving 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.
h. By means of hydi'ogen prepared as described in Comm,
N". 94/ 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 maj 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 Coram. N". 94 ƒ 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).
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 III Conini. 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 ')), b. 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 sufllicient, 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 b 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:
Vq the volume of the reservoir at 0°.
M, the volume of that part of the glass capillar}^ that has the same
temperature t 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.
M,' and ?/," the volumes of the parts of the glass capillary in the
cryostat outside the bath at temperatures t^ and t^'.
w, the volume of the part of the glass capillary outside the cryostat
(Wj'") and of the steel capillary at the temperature t^.
^) For determinalious on liquid hydrogen no float was used. The level of the
liquid in the bath was derived from the volume of the evaporated gas.
) The lowest part from f^ to ^12 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.
( 371 )
u^ the volume at the steel point of the volumenometer.
/?! and j?2 the variation of the volume V^ caused by the pressure
of the gas.
If Ht is the observed pressure, and //„ and u have the same
meaning as in Comm. N". 60, the temperature is found from the
formula :
^4
1 + cd,
(1)
^L !+«« '^l + atyi + at^"^ 1^at,
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 u.
That of the thermometer reservoir has been calculated by means of a
quadratic formula, of which the coefficients ^i and k^ have the fol
lowing values : k, = 23.43 X 10^ k, = 0.0272 X 10^ ').
Put
+ ^f V + ^^^ + r^— = ^
I { at\ 1 { at\ 1 + ««3 1 + ««4 l+««
H.
Vo+^. + ^^^ + n\ + u\ + ii'\ +
1 + 15«
then follows from the above for the temperature:
= A,
(2)
K + ^^ + u,^\:s ^ + V, k, t'
Ht \\at
t = — ,
^ ^ u ]
Ht l\at\ " '
(3)
If the term with t^ is omitted, we find an approximate value for
the temperature. Now t may be calculated again, while in the term
with f 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. 95« and corresponding to the
resistance measured in the observation given in Table I of Comm.
N°. 95^ (in the last case even almost simultaneous).
^) 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 — O'^.OU 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 K 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.
TABLE I.
DETERMINATION IN A BATH OF LIQUID HYDROGEN.
(ABOUT —253°). READINGS.
Mays, '06,3.103.30
A
B
C
D
E
F
G
H
A'
974
20.17
9.1
Point
14.75
7.9
975
17.86
8.1
/ lower top
23.00
9.0
297
22.02
9.3
15.5
15.5
15.4
fe \ meniscus riin
26.01
9.0
298
19.90
10.3
15.0
15.1
21.926
nome
15.5
14.3
;! i higher top
15.10
7.9
974
20.17
9.1
\ meniscus rim
18.43
7.9
975
17.86
8.1
15.5
89.294
/ lower top
21.03
8.7
297
22.02
9.3
15.5
^ \ meniscus rim
22.70
8.6
298
19.90
10.3
15.5
s (
o 1
u 1
(2 1 higher top
25.82
9.6
1058
28.07
11.3
15.7
103.279
\ meniscus rim
27.98
10.0
1059
25.43
11.0
15.7
/ lower top
22.98
9.3
297
22.02
9.3
15.4
15.5
15.4
« meniscus rim
25.99
9.3
298
19.90
10.3
15.4
15.2
15.3
14.3
g higher top
15.06
7.9
974
20.17
9.1
\ meniscus rim
18.42
8.0
975
974
17.86
20.17
8.1
9.1
15.5
Point
14.76
8.0
975
17.86
8.1
( 373 )
TABLE IT.
DETERMINATION IN A BATH OF LIQUID HYDROGEN
(ABOUT —253°). CORRECTED AND CALCULATED DATA OF
THE OBSERVATION.
B'
F
F'
lower meniscus
height
5 i higher menisf'us
height
lower meniscus
■£ I height
«
higher meniscus
height
296.53
4.39
976.21
1.46
297.46
0.77
1058.87
0.83
296.70
976.37
297.48
1058.90
14.8
14.9
14.9
14.8
14.9
13. S
81.53
14.8
15.0
0.14
The correction was applied for the difference in level of barometer
and manometer (cf. also Comm. N°. 60). In this way we find Bt,
the pressure of the gas in the thermometer.
TABLE III.
DETERMINATION IN A BATH OF LIQUID HYDROGEN.
(ABOUT —253°). DATA FOR THE CALCULATION,
a, =0.ai05 cm3
Wj' =0.0126 »
t',=
162°
aj" = 0.0140 »
i"2 =
0°
«3 =0.6990 »
h =
140.5
«4 =0.2320 »
t, =
14°. 9
«/'= 0.1141 »
Hj,= S\
53 m.m.
y^= 82.265 cm3
^, = —0.0041 » ,
h = ^
O.0021cm3
£^0 = 1091.88 mm
u = 0.7991 cm3
( 374 )
Prom the indication of the float the value of u^ is found, iij and
uj' are chosen such that the circumstances are as closely as possible
equal to those for which the distribution of temperature in the
crjostat is determined. ■ We get now the table III, in which H^ 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 :
«= — 252°.964
and after application of the correction for the quadratic term :
t = — 252°.964 + 0°.035 = — 252°.93.
§ 7. Accuracy of the determinatmis of the temjjerature.
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. 27^^^ '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. 95'^ Tab. VI„ observation N". 30, and
Comm. N". 95^ Tab. I) yielded :
3"20' — 252°.926
3"58' — 252°.929
the two others with another resistance :
2u35' _ 252°.875
3u 7' — 252°.866'')
Determinations of one and the same temperature on different days
1) From the values of « found by Ghappuis at different pressures and from
Berthelot's calculations follows by extrapolation from Ghappuis' value for
p = 1000 m M. a = 0.00366262 for _p = 1090 m.M., from Travers' value of a, for
700 m.M. with the same data ;e = 0.00366288 for p=^1090 m.M.
2) 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.
H. KAMERLINGH ONNES and C. BRAAK. "On the measurement of very
low temperatures. XIII. Determinations with the hydrogen ther
mometer.'"
Plate I.
Proceedings Boyo.l Acad. Amsterdam. Vol. IX.
( 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"S '05 — 139°.867
Oct. 26^1', '05 — 139°.873
July 6th, '05 — 217°.416
March 3'd, '06 — 217°.424
June 30'^, '06 — 182°.730
July Q^^, '06 — 182°.728
For the deviation of the 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 S'^, '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 6t\ '05 and March 3'^, '06 — 217°.420
June 30^^ '06 5"50' — 217°.327 i
6u 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 appeal's 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 Coram. N^. 95% though both refer to the same resistance. This diffe
rence is due to the fact that in Gomra. N^. 95'' the result of one reading has
been used, and here the mean has been given of more readings.
2) It gives namely for the probable error 0^.0029, so only a trifling difference
with the above.
( 376 )
1°
may without difficulty be effected accurate to — 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. W. 94/, 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 :
H5«= 53.82 m.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
1) For this Travers, Senter and Jaquerod (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. J 3th '05 i/„ = 1092.11 mm.
on Febr. 26'^ '06 ^„ = 1091.93 mm.
The determinations before and after every period of observation
give but slight diiferences when compared. A^ a rule the pressure
decreases slightly as in the second of the abovementioned 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"^ '05,
Hq = 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. Id'^ '02
Ho = 1056.04 mm.
was found, and the pressure on June 8^^ '04 was
Ho = 1055.43 mm.
while during further measurements up to July 7'^ '04 the pressure
retained a value which within the limits of the errors of observation
remained equal to this.
Chappuis^) 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 Kuenen
and RoBSON and by Keesom 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 eludes sur les Ihennomètres a gaz, Travaux et Mémoires du Bureau
International. T. XIII p. 32.
( 378 )
absorption which comforms slowlj 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.
ERRATA.
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 1. 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.
1. 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.
(Translated from: Verslag van de gewone vergadering der Wis en Natuurkundige
Afdeeling van Zaterdag 24 November 1906, Dl. XV).
COZvTTElsrTS.
F. Muller: "On the placentation of Scinrus vulgaris". (Communicated by Prof. A. A. W.
HrBRECHT), p. 380.
W. BiRCK: "On the influence of the nectaries and other sugarcontaining tissues in the
flower on the opening of the anthers". (Communicated by Prof. F. A. F. C. Went), p. 390.
A. J. P. VAN DEN Broek : "On the relation of the genital ducts to the genital gland in
marsupials". (Communicated by Prof. L. Bolk), p. 396.
H. Kamerlingh Onnes and C. A. Crommelin: "On the comparison of the thermoelement
constantinsteel with the hydrogen thermometer", p. 403.
W. Kaptetn: "On a special class of homogeneous linear differential equations of the second
order", p. 406.
J. C. Klctver : "Some formulae concerning the integers less than « and prime to »", 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 Bakhüyzen),
p. 414.
Fred. Schuh: "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. Sciioute), p. 424.
E. E. MoGENDORFF: "On a new empiric spectral formula". (Communicated by Prof. P. Zeeman),
p. 434.
J. A. C. OiTDEMANS : "Mutual occultations and eclipses of the satellites of Jupiter in 1908",
2nd part. Eclipses, p. 444. (With two plates).
H. Kamerlingh Onnes : "Contributions to the knowledge of the ^surface of van der Wa.ils.
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. Muller. (Communicated bj Prof. A. A. W. Hubrecht).
(Communicated in the meeting of September 29, 1906).
I. T h e very earliest 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 Tshaped 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.
II. P r ep 1 a c e n t a r y 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 antimesometral
(i.e. antiplacentary) 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 tlie 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 honi, left open between the fixations of the o\ules,
as also the mesometrally situated parts ; all these processes begin
subepithelially, gradually penetrating deeper and deeper. These
successive processes thus gradually give rise to dishshaped layers
of varying structure, surrounding the ovule at the antimesometral
side and the character of which is most sharply pronounced in the
points that are at the greatest distance from the mesometrium. ]iy
the extension of the antimesometral part of the long end of the
( 381 )
Tshaped slit, a broadening is brought about here, which, progressing
more and more in the mesometral direction, finally produces a space,
the crosssection of which presents a shape like that of a cone,
truncated mesometrally by the old transverse part of the T, and
bordered antimesometrally 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 subepithelial
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 dishshaped 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 cellelements, accompanying the proliferation of the
subepithelial multinuclear zone, the final result being a system of
cavities, separated by thin cellpartitions 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",
plasmalumps of different size, which assume a dark colour and
contain many giant nuclei witii 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 Rejsek. 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 MiNOT, ScHOENFELD and others, described for the rabbit, since these
elements are also found in Sciurus, only nuu*h hiter. 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 proUferation 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 (verj^ 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 preplacentary stages is characterised for
the ovule by the origin of the amnion etc. 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 mesometrally 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 cellheaps rise every
( 383 )
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 difïerentiation 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
pycnotic nuclei dissolve in the plasm and a mass is formed, epithelial
symplasm, in which finally greater and smaller vacuoles are e\enly
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 )
TIT. Placentary stages (After the beginning of the for
mation of the alluntoid placenta). In the om[)haloid 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
mesometrall}' situated formations. The entoderm, covering these monster
cells, is very narrow and smallcelled ; 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
tow^ards 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 crosssections.
The dilatation now affects very strongly as well the placentary
part of the foetal chambers as their mutual connecting pieces, so that
the omphaloid part becomes smaller and smaller, while the formerly
existing combshaped 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 svmplasm is formed, and, as was the case in the
hyj)erplastic process, not ever^wiiere simultaneously, so that small
partitions of svmplasm still surround more healthy groups. Outside
the placentai'y 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, onelayered
invaginations in the crypts, 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 lias 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 ürst 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 tirst time at the expense of maternal tissue),
which, when they become larger, bend the basal trophoblast layer
(cytotrophoblast) inwards and finally lill 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 priniordium 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 allantoisstrands with foetal vessels; these are surrounded by the
cytotrophobJast, 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, etc.) ; the deciduacells 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 supraplacentary 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 .deciduacells 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 tlie dilated connecting pieces, now entirely incorporated
in the foetal chambers and whose mucosa, attenuated by extension,
only possesses crypts still, that are squeezed fiat, and a rather tliick
epithelium which for a part turns into symplasm. Against all these
extraplacentary parts lies the extraplacentary 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 lia\e 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 i)art of the mucosa, perhaps
joins with the meanwhile jnoliferating glandular remains in the
depth : the umbilical vesicle is lifted oif 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 pi'ovided with a new
epithelium.
Comparative considerations. Among Rodents the in
vestigation of the times at wiiich various processes and organs of
the ovule (not of the foetus) are found, leads to the following series :
Sciurus — 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 low^er, 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
Sciurus, 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 preplacentary
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 antimes ometral wall of the uterus. This coniiectiou only
ends in Sciurus towards parturition, in Mus and Cavia already \'ery
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, etc. 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 latei' foetal ones, when the former have disappeared.
In Lepus Schoenfeld 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 Jenkinson at much earlier
stages, Kolster 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 prolifeiating "Gegenpolcellen" of v. Spee,
which perforate the zone at the vegetative pole; the mateinal 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. Spee, 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 morpho logy of the extra
( 389 )
placentary paris of the foctal chamber might perhaps slied liglit on
this subject.
The now following appearance of the allantoic! placenta is found
latest in Sciurus, earliest in Cavia. The tendency, increasing in the
wellknown 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 stromaepitheliuni, later de
generation, disintegration and resorption witli penetration of the
trophoblast into this mass, all temporarily clearly distinct and rela
tively slow, in Cavia we fuid 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 ètre 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 foelal blood. The allantois
remains passive, the foetal mass grows further and further round
the allantoisramifications, 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 supraplacentary (as J en kinson 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
preplacentary period will ha\e been more extensively carried out
(also in regaid 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 Fleischmann ; 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 "
Bj Dr. W. BuRCK. (Communicated bj Prof. F. k. F. C. Went.)
(Communicated in the meeting of September 29, 1906).
The consideration that the opening of the anthers is preceded bj
a very considerable loss of water ^) and that with very many plants,
e.^. Compositae, Papilionaceae, Lobeliaceae, Antirrldneae, 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 sugarcontaining 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 belljar,
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 Diervilla ( Weigelia) rosea or jloribunda,
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 üfth 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 FrUillaria imperialis to 90 % of the
weight of the anthers, with OrnUhogalum umbellatum to 86 %, with Diervilla
floribunda to 87%, with Aesculiis Hippocastanum to 88%, with Pi/rns japo
nica to 80 7^, with different cultivated tulips 59—68 %. etc. With plants whose
anthers burst 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 o/o, with Lathyrus latifolius to 24 7».
( 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 Fehling's solution shows that as well
the stamen as the whole corolla and even the corollar slips, show
the wellknown reaction, indicating glucose.
Of Digitalis purpui'ea 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 Fehling'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 LamarcJciana, the anthers of which burst already in
the bud, a flowerbud 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 Fehling's solution gave the same result
as was found above with Digitalis.
Similar experiments were made with the flowers of Antirrhimwi
majus L., Lamium album L., Glechoma hederacea L., Salvia argentea
L., Nicotiana affinis 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., Echiumvidgareli.,
Calceolaria pinnata, Hibiscus esculentus, Anoda lavateroides, Malva
vulgaris Tr., Torenia asiatica, Corydalis luteaDc, Colchicum autum
nale L., LysimacJiia vulgaris L., Atropa Belladona L. and lïhinanthus
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 Fehling's solution.
These experiments indicate that the loater is luithdrawn from the anthers
by an osmotic action, having its origin in the glucosecontaining 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
vv^ith Fehling'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 ha\'e coalesced.
There is rather question here of a quantitative difference than of a
special property, peculiar to these flowers.
2. With Stellar ia 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 Fehling'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 tiie
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 tlie 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 Pai)ilionaceae, of which I investigated Lupiniis luteus L.,
Lupinus grandifolius L., Lathy ras odoratns L., Lathy rus latifolius L.
and Vicia Faha L., the anthers are known to open already in the
closed flower. The petals precijntate cuprous oxide from Fehling's
solution, but exert no influence on the opening of tlic 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 Stellaria media and the men
tioned PapiUonaceae, behave with respect to the opening of the
anthers in a space, saturated with watervapour :
Stellaria Holostea L., St. graminea L., Cerastium Biehersteinii C.
arvense L., Cochlearia danica L., Sisymbrium Alliaria Scop., Crambe
hió'panica L., Bunias orientalis L., Capsella Bursa pastoris Much.,
Hesperis violacea L., H. matronalis L., Thlaspi arvense L., Alyssum
maritimum Lam., and further Lychnis diurna Sibth., Silene injlata
Sm. Galium MoUugo L., Asperula ciliata Rochl., Campanula media
L., C. lat if alia 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, ^vith Stellaria media, the anthers
are removed before they have discharged water to the nectaries, one
finds all the same the nectaries amply pro\ided 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 to 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 I 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 sugarcontaining tissues
of the flower, a( 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 watervapour. In so far as they possess nectaries,
these latter appeared to exert no influence on the bursting of the
anthers.
Ranunculus acris L., R. hulbosus 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. lanatum Michx, Aegopodium Podagraria Spr.,
Carum Ca7'vi L., Fimjnnella magna L., Valeriana officinalis L.,
Ligustrum vulgare L. Majantliemum bifolium Dc, 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 Spkengel published his view
of the matter. Also after Sprengel, in the flrst iialf 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 honeysecretion for the fertilisation of the flowers by
the intervention of insects, to which Sprengel had drawn attention,
the sugarcontaining tissues and the secreted liquid were still in
another respect useful to the plant.
After Darwin had in 1859 brought to the front again Sprengel'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 crossfertilisation for the maintenance of the vital energy
( 395 )
of the species, on the other hand witli 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 Bonnier ^) who,
in his extensive })aper on the nectaries, in which as well the ana
tomical as the physiological side of the problem were submitted to
a very extensive in\'estigation, proved that sugarcontaining 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 Avith 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 sugarcontaining tissues as in
the socalled "insect flowers". Even with anemophilous plants he
found "nectaires sans nectar", e. g. with Avena sativa, Triticum
satlvuiii and Hordeum murinum. A number of plants which under
ordinary conditions of life contain no nectar, he could induce to
nectarsecretion by placing them under conditions, favourable for
this purpose.
At the end of his paper he reminds us that an accumnlation of
reserve materials. Avherever a temporary stagnation in the develop
ment exists, may be considered a A'ery 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 accumulated nourishing substances in the endosperm or in the
cotyledons of the embryo. These reserxe materials, turned into assi
milable compounds, theji 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 fiower is a very common phenomenon ').
1) Gaston Bonnier. Le.s neclaires. Etude critique, anatomique et physiologique.
Annales des sciences naturelles. Tome VIII. 1878.
) On this point see also : Paul Knuth, Über den Nachweis von Nektarien auf
chemischern Wege. Bot. Centralbl. bXXVI. Band, 1898, p. 76 and Rob. Stager,
chemischer Nachweis von Nektarien bei Follenbluraen und Anemophilen. Beihefte
zum Bot. Centralbl. Band XII. 1901, p. 34.
26
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 396 )
Kut 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, independent of
the hygroscopic condition of the air.
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 crossfertilisation, 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 genital
gland in marsupials." By A. J. P. v. d. Broek. (Communicated
by Prof. L. Bolk).
(Communicated in the meeiing of October 27, 1906).
In the following communication the changes will be shortly described
which the cranial extremities of tlie 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 obsei'vations 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 k) is situated at the medial
1) Ch. Darwin. Origin of species. Sixth Edition. 1872. Chap. IV, p. 73 and
The effects of Gross and Seitferlilisation. Edition 1876, Chap. X, p. 402.
( 397 )
side of the mesonephros and is attaclied to it by a narrow band
(afterwards the mesorchiuin or uiesovarium) (Fig, lin). The genital
ducts are developed on their whole length. The Wolffian duct
{iv.y.) joins transversal niesonephridial tubules in the mesonephros
but has no connection whatever as yet with the genital gland. The
Miillerian duct (Figure 1 '»■;/■) couunences with an ostium abdomi
nale (o. a.) aiul runs as far as the region of the mesonephros is
concerned at the lateral side of the Wolftian duct.
relation of the genital gland and genital ducts
in an indifferent stadium.
k. genital gland.
o.a. Ostium abdominale tubae.
g.s. genital cord.
w.g. Wolftian duct.
m.y. Miillerian duct.
s.it.g. Sinus urogenitalis.
FiR. 1.
We tirstly 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 mesouephridial tul)ules 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. </.). 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.y.).
In how tar the remnant of I lie Wolffian duct lias ielation to the
2(j*
( 398 )
little tubules which I described and represented in the mesovariam
of a fullgrown Petrogale penicillata, remains out of discussion here^)'.
Fig. 2.
Relation of the J genital ducts
to the ovarium.
ov. Ovarium.
m. Mesovarium.
0. a. Ostium abdominale lubae.
m. g. MuUerian duct.
w g. Wolffian duct.
tr. c. Transversal combination of both
the genital cords,
s. u. g. Sinus urogenitalis.
Fig. 3.
Relation of the genital ducts
to the ovarium.
ov. Ovarium,
m. Mesovarium.
0. a. Ostium abdominale tubae.
t. Tuba Falloppie.
u. Uterus.
V. "Vagina.
w.g. Remnant of the Wolftian ducts.
w'.g'. „ „ „ , „
g. s. genital cord.
tr. c. Transversal combination of both
the genital cords,
s. u. g. Sinus urogenitalis.
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) V. D. Broek, Untersuchungen über die weiblichen Geschlechtsorgane der
Beuteltiere. Petrus Camper III.
(1399 )
the Wolffian duct grows forlli and takes its course archwise through
the mesorcliium in the testicle. (Fig. 4 id. //.). 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 53 m.m.) the Woftian 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.
Fig. L
Relation of the genital ducts
to the testicle.
t. Testicle.
m. Mesorchium.
m.g. Remnants of the Miillerian duct.
w.g. Woltïian duct (vas deferens).
^.6'. Genital cord.
s.u.g. Sinus urogenitalis.
v.a. Vas aberrans.
Relation of the genital duct
to the testicle.
t. Testicle.
7)1. Mesorchium.
m.g. Remnants of the Miillerian duct.
d.r/. Glandule part in the epididymis.
iv.g. Wolffian duct (vas deferens).
g.s. Genital cord.
s.u.g. Sinus urogenitalis.
Meanw^hile the Miillerian duct is for the greater part reduced. The
cranial extremity remains as a remnant of the duct either beginning
with an ostium abdoiuiualo or not, and oudiuu 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 (Didelpliys)
with the Wolffian duct gi'own 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 Müllerian 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 tiiem 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 tlie communication between
the two parts (Didelpliys, 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 Disselhorst the communication that it agrees with that of other
animals. As on this immediatel_y follows: "die Spermatogenese war
in vollem (lange", it seems to me that this communication relates
more to the structure of epithets of the tubules than to the nature
of the connection of testicle and epididymis.
A comparison witli what we tind 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 Wolflian
duct, which then becomes the tnbo)a!"ovarian tube, which was first
described by Roth ■^) and recognised by Mihalkovics '') as a {)art of
the Wolffian duct. Where however in Marsupials the Wolffian duct
penetrates into the genital gland, the tuboparovarian 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, whicli
have appeared afterwards, and must be considered as a second
generation of tubuli seminiferi (Coert)'') or as homologa of the
1) R. Disselhorst. Die mannliclien Geschlechtsorgane der iMonotremen und
einigei' Marsupialen.
Semon's Zoölogische Forschungsreisen in Auslraliën und den Malayischen Archipel.
1904. p. 121.
~) Quoted y^y Mihalkovics.
^) Mihalkovics, Unlersuchungcn fiber die Enlwickelungsgeschiclite dcr Urogenital
organe der Craniolen.
Internal. Zeitschrift fur Anatomie und Histologie. Bd. 2.
^) Coert, Over de ontwikkeling der geslaclitsklier bij de z ogdieren. Diss. Leiden
1898.
( 401 )
"Markstrange" of tho ovarium (Mimai.kovics), or as tubules of the
mesonephros grown into the tissue of the testicle (Kollmann) ^) is
not found in marsupials. If, during further developmeut a network
resembling the rete testis, arises in the marsupial testicle, it must
be considered as a i)art 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, tlie 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 ieduced to minimal
rests (vasa aberrantia). In the mass of tissue, which represents the
socalled 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 tlie epididymis of other
mammals.
To explain the ditTerences 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 witii the testis, which connection furnishes the later vasa
eiferentia testis, we read the following in the extensive investigations
of CoERT^): 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 epitlioHuni 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 destroj'ed, while on the
other hand many new cells are foimed (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 reestablished
in another place I have not been able to investigate.
1) Kollmann, Lehrbuch der Eutwickelungsgeschiclite des Menschen.
2) I.e. p. 96.
( ^02 )
My opinion is that these investigations siiow tliat tlie vasa efferentia
testis must not be considered as simple tubules of tlie mesonephros,
but newly formed tubules, which use quite or for the greater part
the way given to them by the tubules of tlie mesonephros. And that
they are able to use this way finds its cause in this, that, according
to Felix and BtJHLER^) there is most probably no idea of a functioning
of the mesonephros in monodelphic mammals, even not in the pig,
where it 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 ci'anially and brings about itself
the connection between the gland and its excretory duct.
At last the tubes, whicli occur secondary and independently oi
the tubules of the mesonephros in the tissue of the epididymis, might
be explained in the same way, i.e. as tnbules which have the same
signification as the coni vasculosi, but foi' 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 socalled 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 abo\'e, 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 surel}'' be the case a( a transformation of a lubule of the
mesonephros to a connecting duct.
1) Felix und Bühler, Die Entwickelung der Hani und Ciesclilechlsorgane in
Hertwig's Handbucli dcr vergleiclienden und experimentellen Ent wiek elungsgescbichte
der Wirbeltiere.
( 408 )
Physics. — "Siipplement io Commwiication .\\ 95'^ from the
Phi/Mical Lahonitorii of Leiden, on the comparison of the
thermoeJement constantinsteel unth thf hi/drogen thermometer \
By Piof. TI. Kamf,rt,ingh Onnks and C. A, Crommelin.
§ 14. Corrected representation of the oh.'^ervations by a f re 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 tlie 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 Tal)le VIII, are
used, the mean errors in microvolts become :
for formula {Bl) ± 3.0
(^11) ±3.4
{Bill) ± 2.8 (2.5 without —217^)
(^IV) ±2.1
instead of
(^I) ± 2.8
(^11) ± 3.2
[BIW) ± 2.6 (2.1 without  217°)
{B\\) ±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 {Bl), {B\\),
(Z>I1I^. It appeared possible for {B\\') 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^,h^,c^,e^R\\(\f^ (see §12) we get then
a\ = + 4.32.51 3 e\ = + 0.023276
b\ = f 0.4091 53 ƒ , = — 0.0025269
c\ = 10.0015.563
The deviations are given in Table IX under W — R\.
ij The correction amounted to 0°,0S1 in temperature or to 1.7 microvolt, in
electromotive force.
( 404 )
§ 15. Representation of tlw oh.^ii'rr itions hy means of a four
term formula.
We have now quite carried out the calculation of a formula of
the form
' ^ A / ? ^' / ^ \' r t
E—a
100
+ ^>
100
+ '•
too
+
100
{C)
announced in note 2 of § 11, by the method of E. F. v. d. Sande
Bakhuyzen, which proved to facilitate matters greatly again.
Four solutions (C) were found, viz. (CI), (CII), (CllI) representing
the observations down to — 253°, whereas in (CIA ) only agreement
down to — 217° has been sought for.
The coefficients in millivolts are the following :
\
2
3
4
a
\ 4.30192
+ 4.30571
+ 4.30398
+ 4.33031
b
+ 0.357902
+ 0.366351
+ 0.363681
+ 0.421271
c
— 0.0250934
— 0.0192565
— 0.020071
+ 0.018683
e
4 0.0257462
+ 0.0270158
+ 0.0270044
+ 0.035268
The residuals have been given in tenth parts of microvolts in
Table IX under W—Rci, W—Rcii, W—Rcjii, W—Rciv
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 (CI), (CII), (CIII), and only those down to —217" for
(CIV), for
(CI) ± 3.0
(CII) ± 2.9
(CIII) ± 3.0
(CIV) ± 2.3
If — 182° is excluded, they become:
(CI) ± 2.7
(CII) ± 2.6
(CIV) ± 1.8
The mean errors of (CI), (CII), (CIII) must be compared with
those of (^I) and (^III), ihose of (CIV) with those of {BIY).
( 405 )
This comparison teaches thai the four term formula for the represen
tation of tlie observations may be considered to be almost equivalent
to the live 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 obserAations are required, amounts
to four. That three are not sufficient was already proved in § IJ.
This appears also clearly, when the mean error is determined, which
rises to ±7.6 microvolts for the three term formula.
T A H L E IX.
DEVIATIONS OF THE CALIBRATIONFORMULAE FOR THE
THERM 0ELEMENT CONST ANTINSTEEL.
IH
II
III
IV
V
VI
VII
NO.
t
jr— /?;
W
■%
^f'RcM
^^Rcm
W—R(,^y
22
— 29.82
— 12
+
20
+ 1^
+ 18
— 19
24 and 20
— .58.75
+ 10
+
30
+ 20
+ 29
+ 4
21 ard 23
— 88.15
+ 14
+
1
+ 1
+ ^
+ 1
1 and 17
— 1'i3.70
—
—
29
— £8
— 30
— 20
10 and 18
— 139.80
+ •
—
20
— 2i
— 31
— 17
19
— 158.83
 10
—
10
— 10
— 18
— 10
3, 11 and r^
[ 182.73]
+ 20
+
40
+ 44
+ 35
+ 34
4, 28 aid
— 195.19
+ 2
+
23
+ 21
+ 1i
+ 11
12, 27 and 7
— 204.70
— 20
9
— 11
— 19
— 18
20,14, 13 and 8
— 212.85
+ 24
+
21
+ 21
+ i:^
+ 21
29, 15 and 2:.
— 217.55
 15
30
— 29
 ''"'
— 23
30
— 252.93
+ 280
+ 20
+ 20
+ 1.50
31
— 259.24
+ 485
+
115
+ 141
+ 143
+ 313
( 406 )
Mathematics, r "On a special chus of homogeneous linear dif
ferential equations of the second order" . By Prof. W. Kapteyn.
The differential equation of Legendkk
d^y dy
.i(l:^') ~  2.^ ;^ + n {n\l) y =
is satisfied by a' polynomiiim P„ {x) of the ?z*^ degree and by a
function Qn (x) which may be reduced to the form
1
dz
Qn (4 = j ^37
— 1
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 2/jr Fn {x).
In analogy to this we have examined the question: to determine
all homogeneous linear differential equations of the second order of
the form
_ d^ii dy
dx^ dx
where the coefficients are polynomia in ,x, which possess the property
that y^{x) being a first particular integral, the second integral may
be written
. . CyMdr
y^ (■^') = I
J .r, — z
'J.
where a and /? represent two real values, supposing moreover that
this integral has a meaning everywhere except on the line of dis
continuity.
Let
R{x)^kr^xP , S{x)^kH.,xP , T{x)=.kt^,xP
Ü
then we obtain firstly the conditions
n {x) = (..«) (.;/?) r {x) = (xa) (../?) V(),. xP
Ü
/— 2
.S (.7;) = H' {x) 4 (x—a) (x^) 2 h,, XP.
If now we put
G,'=JzPy:'{z)dz , G;=JzPy,'{z)dz , G,, =j zP y^ (z) dz
( 407 )
and ' '1
N={ai^)G:G:2G:
m = ~G:
n = {a^^)G:~G;G,
the further necessary conditions may be deduced from the equation
1 4 J =
where / ajid J represent the following polynomia of degree ;. — 1
/—I
1= :e{q,,n^ (jp_^ M) .fp
/—I
. . ... . . , + ^ [j^^/^ + 0^ + 1) Qp+\] 't + l^'/Ji + P9/>\ ^«] ■'^^
;— I
J= 2 (r^+i Gp' + .s^+i 6^^' 4 tpj^x Op)
p=0
;.— 2
{ X 2! {rp + 2 Gp" + .^^_2 Gp + <^__2 6?^;)
+ •...,
+ ,.—2 ^ (r^+;_i 6^;' 4 6>+;,_l G; + tp^;,^, Gp)
p=0 . :,,,.,. ..^;. ^,,,. .
+ .t— 1 2 {rp^. Gp" + .v+' ^p' + tp+y Gp).
p=0
From this we may easily deduce that if ). = '2, the most general
differential equation of the second order possessing the property in
question is ';
(^«) i^^ ■— + Vh ('^'«) {^^ + 2.r  «  I?] ^ + {t.^Jrh'^:)y=,
where a, /?, t^ and ^j are arbitrary constants.
When P. = 3 the most general equation may be ^vritten
a A' ax
J^{t,x'^t,x\t,)y^{)
Here however the ten constants must satisf} the following three
conditions
., + («+.i) ., + («•^+«p?+/J0 <, = iQ, + («4^) ^^
( 408 )
Mathematics. — ''Some formulae concerning the integers less than
n and prime to n." By Prof. J. C. Kluyver.
The number ^ {ii) of tlie integers v less than n and prime to n
can be expressed by means of the divisors d.
We have
(f (n) = 2" (1 (d) d\ [dd' = n)
din
if we denote b}' [i {q) the arithmetical function, whicli equals if q
be divisible by a square, and otherwise equals  1 or — 1, according
to q 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 v are expres
sible as a function of the divisors (/.
This general relation may be written as follows ^)
k=d'
2f{v) = 2ix{d)2f{kd).
din l=\
For the proof we have to observe that, supposing (/??,, n) '^ D, the
term ƒ (w) occurs at the righthand side as often as d in a divisor
of D. Hence the total coefficient of the term f{m) becomes
2 (x {d),
dID
that is zero if D be greater than unity, and J when m is equal to
one of the integers v.
We will consider some simple cases of Kronecker's equation.
First, let
The equation becomes
k=d' gxn 1
^ e^'= 2: (i{d)2 e^^'i = 2 li (d.) e^(^ —. ,
■J din i—\ ,11» e^^ — 1
or because of
If Ave write
din k=l djn
2 H{d) z= 0,
din
gxn I
din e^"^ — d
:E — 2 aid) — ,
V en  1 din e^'d  1
1) Kronecker, Vorlesungen fiber Zahlentheoiie. I, p. 251.
( 409 )
we maj introduce the Bernoui,lian functions fk{^), defined by tlie
equation
 — r^+ ^.^•VK6'j,
and lience show that
By equating tlie corresponding terms on the two sides we get
:Ej\,n f1 = ( 1)'"' ^, ^ m(^)^^''" + ^
\n J Im: din
as a first generalisation of the relation
din
Observing tliat we have
there follows for two integers n and yz', both having the same set
of prime factors,
^r f''^
\n
In the same way an expression for the sum of the k^^ powers
of the integers v may be obtained. Expanding both sides of the
equation
V djn e^<^ — 1
we find
^ :evI^ = 2 ii{d)dJ^Md!).
k' J din
Other relations of the same kind, containing trigonometrical functions
are deduced by changing x into 2jtii\
From
djn gSTTjarf — I
we find by separating the real and imaginary parts
2 cos 2jt,w z= i .s/n 2jr.vn S ft {d) cot Jt.vd^
dhi
^ sin 27Ï.XV ■= sm* Ji.vn 2 ft (d) cot jxa'd.
dju
( 410 )
In particular the first of these equations gives a simple result if
we put .^' = — + e, where e is a vanishing quantity. Asjhe factor
dn 2 Jt,xn tends to zero with « the whole righthand side is annulled
but for the term in which (/ = n.
So it follows that
2 cos =: (I («),
and we have fx (??), originally depending upon the prime factors of
n, expressed as a function of the integers prime to n.
1
Similarly we may put in the second equation cc=: — and write
sm nv ^ , ,. ^d
^ = ^ fi ((i) cot — .
n din 2n
Still another trigonometrical formula may be obtained by the sub
stitution x = — \ ^ Let D be the greatest common divisor of the
n
integers n and q, so that
n — n,D , q = q^D ;
then as e vanishes, we have to retain at the righthand side only
those terms in which qd is divisible by n, or what is the same the
terms for which the complementary divisor d' divides D.
Hence, Ave find
:E cos ?^ = ^ ,x f 4 V' = ^ ^ f* ("o^) T • (*^' = ^>
n d'lD \d J djD d
Instead of extending the summation over all divisors d of D, it
suffices to take into account only those divisors 6 of //, that are
prime to ?ï„. In this way we find
1 1
D:E ii {n,d) — = fx (n„) D 2 (i (d)  ,
dlD «
and as the second side is readily reduced to
<f{u) ( u ^ (f{n)
n
.A
we obtain for any integer q, for which we have {n,q)^D,
<i)'
( 411 )
Concerning the result
2;rr
^' cos = (X (n)
a slight remark may be made. To each integer v a second v' = n — v
is conjugated ; hence denoting by Qn an irreducible fraction <^ ^ with
the denominator n, we may write
2^ cos 2jrQn = (i{n),
and also
22 cos 2jrQn =■ 2 l^i^)
n _ 7 " — ."7
Now for large values of </ the fractions Qn will spread themselves
not homogeneously, but still with some regularity more or less all
over the interval — ^ and there is some reason to expect, that in
the main the positive and the negative terms of the sum 2 cos 2jtQ,t
uSg
will annul each other, hence the equation
22 cos 2jr()„ = 2 n{n)
is quite consistent with the supposition of voN Sterneck, that as ^ takes
larger and larger values the absolute value of 2 n{n) does not
exceed \/</.
Another set of formulae will be obtained by substituting in
Kronecker's equation
Ay) = log[e » — e
Thus we get
/
^ 2:iix 2T/v^
\ k=d' /2:rü 1.U
2log[
■J
] = 2 ix{cl)2 log \e^—e «
din i=rl ^
or
^ Inix •ln.a\
/ Inixd' \
2 log
g n _ g « 1
= 2ii{d)log[e " l)
<. /
din ^ ^
and after some
reductions
71 ^ StiV
2 log 2 sin — {v — x) = 2 (i{d) log 2 sin — .
n djn ' d
By repeated differentiations with respect to x we may derive from
this equation further analogies to the formula
(p {n)r=z 2 n (d) d'.
dhi
So for instance we obtain bij diHerentiatiiig two times
27
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 412 )
JtV
sm'
V
d\n
[X ((/) d!
and bj repeating the process
^2m
— :e
V L^^^
lm
log sin y
B,n 22'«
j,2«
.'/ = 
^m d/n
a result included in the still somewhat more aeneral relation
n^ 2 2
V ]c=\ ynh—vf din
which is self evident from.
Returning to the ec^uation
T€ . JttV
2 log 2 dn y — [v — x) z=z ^ n [d) log 2 sin —
V " n din d
we obtain as x tends to zero
2 log 2 sin — = — 2 [i {d) log d.
V '»' din
111 order to evaluate the righthand side, we observe that for
n = Pi"^ p/' • . . we have
d
S f* (d) log d =
dlii
dy
(1 — ell H Pi) (1 — eV^oöiJ'j) .
,V=rU
So it is seen that, putting
— 2 n (d) log d = Y (n),
din
the function y {n) is equal to zero for all integers n having distinct
prime factors, and that it takes the value loc/ p, when n is any powei
of the prime number p.
Hence we may write
JtV
112 sin — = ev W,
V n
a result in a different way deduced by Kronecker ^).
Again in the equation
n 2 sin — (r — .v) = /ƒ I 2 sin
din
d
we will make ,v tend to —
If n be odd, all divisors (/ and (/' are odd also and we have
at once
^) Kronecker, Yorlesungen über Zahlentheorie. I, p. 296,
( 413 )
d'—i , ,
77 2 CO. — = 77 ( 1) 2 ' = ( l)if(«).
If n =: '2m and //i be odd, we shall have r/ (//;) = 7; (?i). Half the
numbers y, prime to //i and less than m will be equal to some
integer v, the other half will be of the form vin.
Hence we have
2jri? 2jrx , , , .cry.
772 sin — = (— l)if(n) 77 2 sin = (— l)iK") 77 2 5<n — ,
V n y n X m
and therefore
77 2 Sin —
ii sin / n \
/72co. =( 1)W.;^ ^ = ( l)W»).'UJ''">
•J n _ Jiv
77 2 sm —
n
Ijastly, if n = 2m, and m be even, we shall have (f (ni) = è y ('O
Now each of the numbers y. prime to m and less than rii at the
same time will be equal to some integer v and to one of the dif
ferences V — ?M. Reasoning as before we have in this case
2nrr , . / 2jry.y , . { ^ , ny\
772 mi = (— IjiK"; 77 2 sin = (— l)^''("i 77 2 sm — .
n y \ 1^ J y. \ in J
= (_ l)ir(")e ^"^
From the foregoing we may conclude as follows. If we put
77 2 cos '— = (— l)i?(« é''^«) ,
the arithmetical function >.(??) is different from zero only when
n is double the power of any prime number [>, in which case we
have X (n) =z lo(j jj .
Again we introduce here the irreducible fractions Qn less than 
with the denominator ?z ; then denoting by ^J (q) the least common
multiple of all the integers not surpassing q we may write
2 2 log 2 sin nr()„ = 2! Y{n) =^ log M{g),
n < 7 " < .7
and therefore
f :iy.\
77 2 sin —
JIV
n2 cos — = ( —
n
JIV
77 2 sin —
V n
2 2 log 2 cos jtQn =: ^ ).(ti) =1 /o_a 1/ ( \
27*
(414 )
If we consider the quotient log M{g) . log g as an approximate
(but always too small) value of the number A{g) of prime numbers
less than g, to Kronecker's result
2
^ id) = 7 — 2" log 2 sin jiQn
log gn<q
we may add
:g\ 2
A\  ] = :S log 2 cos jtQu ■
^ 2
Astronomy. — "Researches on the orbit of the periodic comet Holmes
and on the perturbations of its elliptic motion. IV." By Dr.
H. J. ZwiERS. (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 :i^'of May till the Sl^t 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 Wolf 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
« = 61° 51' d= + 42°28'
for 13''52'"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. Wolf was so kind, to communicate immediately
to me the places as they had been obtained, after carefully measur
ing the plates. Although Wolf declared in a note to the observed
( 415 )
position of the 25^'^ of September^) tliat the briglitness had ijicreased
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 comnuiiiicato here the results as I had the pleasure to receiv^e
them from prof. Wolf.
1. "Den Kometen Holmes habe ich auf der Platte von 28 August
recht winklig an die 4 Sterne
A.G. Bonn 3456, 3462, 3472, 3493
angeschlossen, und die Messungen nach der Turner'schen Methode
reduziert. Ich finde fur 1906.0 :
« = 4h 7m 34^84 (ƒ= + 42°30'59"9
fur die Aufnahmezeit : 1906 Aug. 28, 13^ 52™1 Kgst. Das iiusserst
schwache zentrale Kernchen wurde dabei eingestellt. Die Messung
und Rechnung bezieht sich auf die mittleren Orte der 4 Sterne fur
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: 12^46^0 M.Z. Kgst.
«1906.0 =^ 32'n 1 0^02 (fi9u6.o = + 47° 34' 54"6
Ich habe auch den letzten Ort (viz. of Aug. 28) mit nur 3 Sternen
nochmals gerechnet fweil ein Stern sehr ungiinstig war) und fand
fur 1906 August 28: 13'' 52'"1 Kgst.:
«1906.0 = 4h 7ra 35^00 (fi906.o = + 42° 30' 58"3
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. Kopff hat gestern den Ort einer Aufnahme vom
10 Okt. 1906 des Kometen Holmes ausgemessen
1906 Okt. 10 : 9h IniO Kgst.
«1906.0 = 4h 34m 48S94 ^f^g^g^^ = _^ 490 54 592
Sterne : A.G. Bonn 3759, 3768, 3777 Der Komet war
1) Astron. Nachr., N". 4123, S. 302.
( 416 )
diesmal sclion 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
8"80, I find for the Heidelberg Observatory the following constants :
X= — 0ii84"i54^^8
tg(p' = 0.06404
^ = 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.
TABLE I.
NO.
Red. on app. pi. Parallax
Apparent geoc. place.
Ax
Ao
A«
Ao
'X
\
2
3
+ 1.888
+ 2.929
+ 3.593
II
—8.55
—8.57
—7.51
s
— 0.191
— 0.217
— 0.298
+1.24
+0.92
+2.35
h m s
4 7 36.697
4 32 12.732
4 34 .52.235
o 1 II
+42 3a 50.99
+47 3 i 46.95
+49 54 54.04
1
I used for comparison with the ephemeris my original computations,
which contained in « as well as in ö one decimal place more than
the published values. The computed places and their comparison with
the observed positions, are given in the following table.
TABLE IL
Local time
Aberration
time.
Comp. apparent place
Observ. — Comp.
a
a
ö"
Aug. 28.553602
Sept. 25.507699
Oct. 10.351449
d
0.013211
.012005
.011462
h m s
4 7 29.753
4 32 4.2.55
4 34 43.017
o 1 II
+42 30 24.28
+47 34 29.94
+49 54 43.02
s "
+6.94 +26.7
+8.48 +17.0
+9.22 +11.0
( 417 )
Togetlier with the ephemeris I coinnuiiiicated a table containing
the variations of the right ascension and the declination by a variation
of tlie periiieUon passage of + 4 or — 4 days. In comparing'the
above given values — C' witii 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 «. The deviations in ö cannot be used so well for that
purpose, as the variations of rf, resulting from a variation of T, 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 AT= — 4 days that the positive
errors in d will not entirely disappear by a variation of T.
By means of a rough interpolation I derived from the 3 differences
— C in right ascension the following corrections for the time of
perihelion passage .
Observ
of Aug. 28
,, Sept. 25
„ Oct. 10
AT =
0.0900 day
0.0916 „
0.0896 „
In the average AT= — 0.0904 day, which at the rate of a mean
daily motion of 51 7 "448 corresponds to an increase of the mean
anomalies of 46"8.
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: V by 40", 2° by 50". I interpolated the following
sun's coordinates (with reference to the mean equinox of 1906.0)
from the Naut. Almanac.
TABLE III.
1906 X
1
r
Z
Aug. 28.540391
Sept. 25.495694
Oct. 10.339987
— 0.9134887
— 1.0018399
— 0.9565810
+ 0.3947635
— 0.0318699
— 0.2616405
+ 0.1712510
— 0.0138250
— 0.1135029
For the reduction to the apparent places I added to the mean a
of the comet: f {■ g sin {G j «) tg ö, to its mean 6: g cos {G \ a).
The following table contains the computed apparent places in the
two suppositions.
( 418 )
TABLE IV.
\M =
+ 40"
Md =
+ 50"
NO.
a.
a.
S
1
h m s
4 7 35.758
o / II
+ 42 30 34.72
h m s
4 7 37.266
a 1 II
+ 42 30 37.38
2
4 32 11.451
+ 47 34 31. 40
4 32 13.248
+ 47 34 31 . 85
3
4 34 51.050
+ 49 5 i 42.20
4 34 53.060
+ 49 54 41.99
A sufficient control is obtained here by comparing the values for
A 31= 0" (epheraeris), A .¥= + 40" and A if = + 50".
In comparing with the observed apparent places w^e obtain the
following differences — C :
TABLE V.
AM = + 40"
1
A/l/ = + 50"
N».
Aa
A^
Aa
AJ
1
s
+ 0.939
II
+ 16. ^7
— 0.560
1 1
+ 13.61
2
+ 1.281 + 15.49
— 0.516
+ 15.10
3
+1.185 + 11.84
— 0.825
+ 12.05
By means of interpolation between the values of A « we find as
resulting value for A J/ ~ 46"412, leaving the following errors :
N».
Aa
A<r
1
2
3
S
— 0.03
+ 0.13
— 0.10
1'
+ 14.7
+ 15.2
+ 11.9
From this follows that by a variation of M alone, the differences
— C in a 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 Jt,
ip or n ; at the end I will add a few words on these elements. So
we must try to bring it about bj variations in the position of the plane
of the orbit, viz. of i and ^, and for this reason I determined the relation
between those elements and the computed places of the comet. As
from the two suppositions A M = \ 50" seems to be nearer to the
truth, I computed the appaient places of the comet : for A J/ =: f" ^^"
Ai={10" and AcTl = and also for AJ/= + 50" Ai =
AcH) = — 10". Probably a somewhat larger value of AcTl
had been more convenient. The following table gives the variations
of a and Ö in the two cases.
TABLE VI.
A2= + 10"
A 06 = _ 10"
N».
A«
Ao
Aa
A
1
— 0.149
II
+ 10.00
+ 0.040
II
+ 1.26
2
— 0.108
+ 11.95
+ 0.067
+ 0.83
3
— 0.111
+ 12.88
+ 0.080
4 0.56
The numbers from the tables V and VI give the following values
of the differential quotients of a and ö with respect to M, i Sind ^,
which will be used as coefficients in the equations of condition.
Aug. 28
Sept. 25
Oct. 10
da
dM
f 0.1508
+ 0.1797
} 0.2010
dM
+ 0.266
+ 0.039
— 0.021
da
di
— 0.0149
 0.0108
— 0.0111
dd
di
+ 1.000
+ 1.195
+ 1.288
da
dSl
— 0.0040
— 0.0067
— 0.0080
dd
dSl
— 0.126
— 0.083
— 056
( 420 )
For « 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 15 cos d, and instead of A cTl I introduced
as unknown quantity.
Equations of condition.
a. From the Right ascensions :
Acfl
0.22202 LM + 9.21681,, Li + 9.64568„ = 0.79873„
0.25966 „ + 9.03853,, „+ 9.83118;, „ =0.71776,,
0.28811 „ f 9.03023„ „ f 9.88800„ „ =0.90136,,
h. From the Declinations :
LSI
9.42488 LM + 0.00000 Li + 0.10037„ = 1.13386
8.59106 „ + 0.07737 „ + 9.91908,, „ = 1.17898
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 LM,
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 :
Acfl
4 9.9278 LM — 0.39596 Li — 3.8260 = — 31.495
^ 10
— 0.39596 „ + 4.1375 „ — 2.7434 „ = + 49.637
— 3.8260 „ — 2.7434 „ + 3.8423 „ = — 23.951
These equations are much simpler if we introduce besides LM,
only one of the two unknown quantities. If we try e.g. to represent
the observations only through variations of M and i we have not
only LSh=0 but the third equation falls out entirely.
1. Solution for LSI = 0.
The results are :
LM = — 2" 7042
At = + 11.74
and the remaining errors :
1. A«= + 0^014 A(f= + 2"59
2. = + 0.097 + 1.18
3. = — 0.151 —3.13
( 421 )
2. Solution for Ai ^0.
In this (!ase we find :
AM = — 9"0461
A^ =z — 2'32"41
and for the remaining errors :
1. A«=: + 0^185 Ad—  — 3"18
2. +0.089 +2.80
3. — 0.226 + 3. 32
3. Solution lüith 3 unknovm quantities:
Tlie results are :
AM = — 5"3045
Ai r= + 7.32
A^=: 1'2.90
and according to the equations of condition there remain the following
differences Obs. — Comp.
1. A«=r + 0^088 A(f= — 0"23
2. + 0.095 + 1.34
3 —0.181 —1.01
As we see the solution with AcTl = and that with A/=: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 Ö 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
jLt = 517"447665
log a =0.5574268
T = 1906 March 14.09401
(f =24°20'25"55
e =0.4121574
t = 20°49' 0"62
.T =346 2 31.63 1906.0
SI =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 tlie 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 006 in « and 1"6 in declination, or of 2"39
in arc 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 coordinates :
X — [9.993 7648.63] sin {v + 77°37'28"36)
y — [9.876 2140.59] dn {v — 20 58 46.82)
z z= [9.832 7020.56] sm {v — 1 46 46.76)
The following table contains the computed apparent places of the
comet and the differences Obs. — Comp.
TABLE VII.
NO.
a
Aot
At?
1
h m s
4 7 36.602
o 1 It
+ 42 30 51.32
s
f 0.095
— 0.33
2
4 32 12.633
+ 47 3445.69
+ 0.099
+ 1.20
3
4 34 52.412
+ 49 54 55.19
— 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 (i, n and tp.
The elements from which the ephemeris for 1 906 has been derived
are those given in "Système VII" p. 78 of my Deuxième Mémoire,
reduced to 1906 by applying the perturbations, arising from the
action of Jupiter. The mean error of the obtained value for yc 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 n. Taking the
obtained AM for the 25^1' of Sept. we get :
44" 6955
A a = H = 4 0" 016787
^ ^ 2662.50 ^
and thus the real error of ft should be 67 times the mean one.
Adopting this correction of fi, the mean anomalies for the 28''i of
August and the 10^^ Qf October would be only 0" 469 smaller and
0" 249 greater than the adopted ones.
It is more probable that the correction of AI arises from neglected
perturbations of tiiat element by Saturn. This perturbation is given
by the formula
t t
^^'^ =ƒ!?'''+ƒƒ
de
dt
to fo
Even if instead of the sum of the values each term was known
separately it would be equally impossible to conclude from the value
J^dii
— — dt, or the correction
(It
of II for 1906. Observations during a much longer period can only
decide in this case.
Something like this holds for :it and <f. During the short period of
the observations, we may even substitute for a part of the correction
hM corresponding variations of jr and (p. If we keep to the plane
of the oi'bit, the apparent place, except for small variations in the
radiusvector (of little influence near the opposition), depends wholly
on the longitude in the orbit, or on
I =: Jt \ V.
So we can apply small variations to the elements without varying
perceptibly the computed positions, if only
A l=z Ajr A Ü =
or
A jr =: — A y.
This relation provides us with the means to throw a pai't of the
correction found foi M on rr or on cp or on both together. In the
first case we have to satisfy the equation
dv
A .T = — —  A M.
dM
( 424 )
du
We can derive the values of ^ directly from the comparison of
the two former computations with A M= { 40" and A il/ =: ] 50".
And so I find for the three dates of the observations:
AM— — O.bOe Arc
— 0.549 Ajt
— 0.573 A;r
If we keep Jt constant and want to substitute apart of the correc
tion of M by a variation of <fi, we must satisfy the relation
Aï; =
or
\0(p Jv const.
fdM\
I derived the values of T — bv computing from the three values
VO^/y const.
of V, with a varied excentricity, the corresponding values of the
mean anomaly. Hence I got for the three observations:
AM = — 1.040 A(p
— 1.186 A^
— 1.260 Ar/)
Although the coefficients as well those of Ajt 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 M to allow a determination of
A3J, A(f) and Ajt separately from the three observations.
Leiden, November 1906,
Mathematics. "On the locus of the imirs of common points and
the envelope of the common chords of the curves of three
pencils" (1^* part). By Dr. F. Schuh. (Communicated by
Prof. P. H. Schoute).
1. Given three pencils (6V), (Cs), (C«) oj plane curves of degree
r, s, t. To find the locus L of the pairs of points throitgh ivhich
passes a curve of each of those pencils.
Let P and P' 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
( 423 )
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 Pand
P' to be a fixed point of intersection of two of the pencils.
Suppose the pencils (C\) and (C/) show « fixed points of inter
section and that this number amounts to j? for the pencils (G) and
{Cf) and to y for the pencils (C,) and (6s).
The degree n of L is determined from its points of intersection
with an arbitrai'y straight line /. On /we take an arbitrary point Q,.,. and
through Qrs we let a C, and a Cs pass, which cut each other besides
in the basepoints and in Qrs still in rs — y — 1 points. Through
each of these points we let a curve Ct pass. These rs — y — 1 curves
Ct cut / in t{rs — y — 1) points Qt, which we make to correspond
to the point Q,.s • To find reversely how many points Qrs correspond
to a given point Qt of / we take on / an arbitrary point Qr through
which we allow a 6'. to pass cutting the Ct through Qt in rt — /?
points differing from the basepoints. Through each of those points
we allow a Cs to pass, of which the points of intersection with /
shall be called Qs. To a point Qr now correspond 6^ {rt — /?) points
Qs and to a point Qs correspond r {st — a) points Qr. The Irst — ar — ^s
coincidences QrQs are the t points of intersection of / with the Ct passing
through Qt and the points Qrs corresponding to Qt, whose number
therefore amounts to 2 rst — «r — ^s — t.
So between the points Qrs and Qt of / we have a {rst — yt — t,
2 rst — (xr — ^s ■ — ^)correspondence. The 3 rst — ar — ^s — yt  — 2t
coincidences are the points of intersection of / with L and the points
of intersection of / with the curve of contact of the pencils (6r) and
(6s), i. e. the locus of the points of contact of the curves Cr and Cs
touching each other. If there are two systems of curves (fti, v^) and
(Ma' ^2) ^)j tli6 order of that curve of contact is
^i^v^ + ftjVj + n^ii^ ^).
1) A system of curves (^^4, v) is a simply infinite system of curves, of which
fz pass through an arbiu'arily given point and v touch an arbitrarily given straight line.
~) This order is found by counting the points of intersection with an arbitrary
hne J. To this end we consider the envelope of the tangents of the curves of the
system (,«1, v^) in its points of intersection with /; this envelope is of class juj+vi,
the tangents of that envelope passing through an arbitrary point Q of I being
the tangents in Q to the ^wj curves of the system through Q and the line I
counting v^ times. In like manner does the system (:/2> "2) gi^^ an envelope of class
fji2 + vo. The («1 + vi) (uo + V2) common tangents of both envelopes are the
line I counting vjvj times and ,mi,"2 4" pr'2 "^Ps''i other lines whose points of
intersection with I indicate the points of intersection of I with the curve of contact.
For a deduction with the aid of the symbolism of conditions see Schubert, "Kalkiil .
der abzahlenden Geometrie", p. 51 — 52.
( 426 )
If we take for the systems the two pencils (G) and (Cs) then
(n^ = jLt, =r 1 and (as ensues imnriediately from the principle of corre
spondence) Vi = 2 (r — 1), r, = 2 (s— 1). So the order of the curve of
contact is
2r\ 28—3.
For the number of points of intersection of / with L remains
3rstar^syt—2t—{2r+2s—3) = 3{rst + l)2{r\s\t){ar\^sir7t).
So we find:
The locus L of the pairs consisting of two movable points by which
a curve of each of the pencils is possible is of order
n = 3 {rst + 1) — 2 (r + s + — («^ + i^s + tO 5
here a is the number of fixed points of intersection of the pencils
[Cs) and (Ct), ^ that of the pencils (Ct) and (Cr) arid y that of
(Q and (Cs).
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 {€].), {Cs) and (G) have respectively
r% 6'' and f different basepoints, so that we can only allow the
basepoints of one pencil to coincide in part with those of an other
pencil. Then « is the number of common basepoints of the pencils
(Cs) and (G) (which can however also belong to (G)), etc. If the
pencils have no common basepoints (« = /? = y = 0), the order of
the locus becomes
3{rst + 1) — 2 (r + s 1 t).
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 (Cr) only we call Ar, a common base
point of the pencils (CI) and (CO which is not a basepoint of the pencil
(6r) we call Ast and a common basepoint of the three pencils we
call Arst If' <^ is the number of points Am tlien the number of
points Ast amounts to «' =: « — cf, that of the points A,t to ^' = ^ — d
and that of the points Ars to y' = y — rf, whilst the number of points
Ar is equal to /•' — ^' — y' — d, etc. By introduction of «', ^', y' and
d the order n of this locus proper becomes
( 427 )
w = 3 (r s < + 1) — 2 (;• h .v + t) — («' r + /i' .:j + y' /) — rf (r + s + i).
F'rom this we see fluit the order of the lociis proper is lowered by r
on account of u common basepoint .'1^^ . Tf there are no points
Arst {(f = 0) one can easily account for that lowering of order
bv noticing that from the total locus the ('. passing through
Ast separates itself, as not belonging to the locus proper. The point
Ast furnishes namely together with an arbitrary point of that C a pair of
points satisfying the question ; of which points however only the latter is
movable ^). Farthermore we see that a point Arst diminishes the order
of L by r\s{t, a fact one cannot account for by separation, the
total locus becoming indetinite ').
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 (Cj) only is an (6^ — « .— l)fold
point of L. In fact, the curves Cs and Q passing through Ar have,
A,, and the basepoints excepted, still dt — a^l points of intersection
each of which combined with .1,. furnishes a pair of points satisfying
the question. The tangents in .1,. to the curves CV passing through
the st — a — 1 mentioned points of intersection are the tangents of
L in the multiple point.
To determine the multiplicity of a point Agt we remark that to
obtain a pair of points satisfying the question and of which one of the
movable points coincides with Ast , it is necessary for C] to pass
through Ast (by which it is determined), whilst Cs and Q ^vhich
always pass through Ast niust present a movable point of intersection
in Ast , thus must touch each other in Ast . The question now rises :
How often do two curves Cs and Ct touching each other in Agt in
tersect each other again on the curve C, passing through Agt ? To
answer this question we introduce an arbitrary Cs intersecting the
above mentioned C, in r.i — y — 1 points differing from the basepoints.
Through each of these points we allow a Ci to pass which gives
rise to a correspondence between the curves 6s and Ct (so likewise
between its tangents in Ast) where rs — y — 1 curves C< correspond
to a Cs and rt — (■? — J curves Cs to a Ct. Thus for the curves Cs
and Ct touching each other in Ast it happens (/'.v } ^'^^ — /? — 7 — 2)
1) If Asi counts foi f lixed points of intersection of the curves C, and C/, the
Gr passing through Asi separates itself i times by whicli tlie degree of i is lowered
by it:
'^) If Arsi counts for . lixed points oi' intersection of Cs and Ci, for t iixed points
of intersection of Cr and Ct and for y. iixed points of intersection of 6V and G, then
Arti diminishes the order of L by e r } C s + y.t; this holds for a point ,1,, too,
but then we must ngard 2^ and y, as being zero.
28
Proceedings Royal Acad. Amsterdam. Vol. IX
( 428 )
times that Cs and Cr have besides yls< another movable point of inter
section, being at the same time movable point of intersection of G and
Cr Here is included the case in which this second point of inter
section coincides with Agt , thus where the cuives (1 and Ci touch
Cr in Ast ; then only one movable point of intersection of Cs and
Ct still coincides with Ast, vv^hilst there need be no other movable
point of intersection lying on Cr, so that in this way we get no
pair of points furnishing a branch of L passing through Agt . So
the point Agt is an (i^s { rt — /? — y — 3)fold point of L.
To determine the multiplicity of a point Arst we have to consider
how many times three curves Cr, Cs and Ct touching each other
in Arst pass once more through a same point. To this end we con
sider an arbitrary Cr and the Cs which touches this Cr in Arst
Through each of the rs — y — 1 points of intersection of these C, and
Cs, differing from the basepoints, we allow a Ct to pass. Then the
question arises how many times this 6^^ touches Cr and Cs in Arst
Let us call Irs the common tangent in Arst of Cr and Cs and It the
tangent of Ct in that point. To 4s correspond 7's — y — 1 lines
It. To find reversely how many lines Irs correspond to an arbi
trary line It we consider an arbitrary Cr intei'secting the C deter
mined by It in rt — ^ points differing from the basepoints. Through
each of those points of intersection we imagine a Cg. If 4 and I
are the tangents in .4,.,^ of Cr and Cs then rt — i? lines 4 corre
spond to 4 and ó'^ — « lines 4 to 4. The rt \ st — a — /? rays of
coincidence indicate the lines 4s corresponding to It; to those rays
of coincidence however belongs the line It itself, which must
not be counted, so that rt\st — a — j3 — 1 lines 4s corresponding to
It remain. So between the lines 4s and It exists an {rs — y — i,rt\
 st — « — ^ — l)correspondence.
The required lines 4s< are indicated by the st\tr\rs — («/?y) — 2
rays of coincidence of this correspondence of which however three
must not be counted. When namely the contact in Arst of Cr and
Cs becomes a contact of the second order one of tlie?'5 — y — 1 points
of intersection differing in general from the basepoints of Cr and Cs
coincides with Arst, namely in the direction of 4.< • The Ct passing
through that point of intersection will touch 4s in Arst in other
words It coincides with 4s • As however the curves C, and Cs, but
not the curves Cr and Ct , neither the curves Cs, Ct have in Arst 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 o curves with a common basepoint, between which a pro
jective correspondence has been in such a way arranged that the
( 429 )
curves must toucli 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 irst wanted.
From this ensues that tlie multiplicity of the point Aj^i amounts to
,t 4 tr + /.•  i<( + ^? + y) _ 5.
So we find :
A basepoint of tlm pencil (C,) only is a
{st — « — 1)
fold /tomt of the locus proper L. A common basepoint of the
pencils {Cs) and {Q) /vhich is not basepoint of (C,) is a
{rs J,rt^y~ 3)
fold point of L and a common basepoint of the three pencils is a
{^t \ tr \ rs — (t — ^ — 7 — 5)
fold point of L^).
4. With the help of the preceding the points of intersection of
L with an arbitrary cur\'e of one of the pencils, e.g. a CV, are
easy to indicate. These are :
1. The r^ — ^ — V h ^ points Ar counting together for
(,.'  ^ y + d) (..«  «  1)
points of intersection.
2. The i? — Ö points Art counting together for
{^ — Ö) {sr j St — « — 7 — 3)
points of intersection.
3. The 7 — d points Ars , giving
(y  6) {tr f ^s — a — i — 3)
points of intersection.
4. The Ö points Arsi, giving together
') If there are no points Arsi (5 = 0) and tlioielbre the total locus is not inde
finite, we can also ask after the multiplicities of the points Ar and Ast as
points of the total locus. Now the improper part of the locus consists of x curves
Cr , <3 curves Cs and y curves Ct . Of these pass tlnough a point yJr the « curves
Cr and through a point Ast the /3 curves C, , the y curves Ct and one of the
curves Cr . From this ensues :
A point Ar is an (st — 1), a point Au an {rs\rt  2) fold point of the
total locus.
So the multiplicity of Ar as a point of the total locus is not changed by the
coincidence of the basepoints, whilst the multiplicity of Asi is equal to the sum
of the multiplicities which this point would have if it were only basepoint of the
pencil (r>) or (mly basepoint of the pencil (C).
28*
( 430 )
ö{st \ tr \ rs — a — ^ — Y — h)
points of intersection.
5. The movable points of intersection of L vvitlj (',■ ; these are
those points of intersection which displace themselves when we
choose another Cr These are found as the pairs of common points
of the simpl}^ infinite linear systems of pointgi'oups intersect on Cr
by the pencils {Cs) and (G). The number of these are found from
the following theorem :
If there are on n carve of yenus p two .sinvply infinite linear
systems of pohityroups consisting of a and b points, the number of
common ijairs of pjoints of those systems is
(«  1) (6  1)  p.
In our case a = rs — y, b = rt — ? and (as Cr is an arbitrary curve
of the pencil (6';)) p ^ ^ {r — 1) (r — 2). For the number of pairs of
common points we therefore find
^rs _ y _ 1) (r« _ ^ _ 1) _ 1 (,. _ 1) (r 2),
and for the number of movable points of intersection of L and C, •
2{rs — y—l){rt — ^—l)~{r— 1) (r  2).
So the total number of points of intersection is:
r{drst [ 3 — 2r — 2s — 2« — ar — ^s — y«),
in accordance with the formula we have found foi' the order of L.
5. The pairs of points FF' through which a curve of each of
the pencils is possible determine on L an involutory (l,l)correspon
dence ; in the following we shall indicate F and F' as corresponding
points of L.
If F falls into a doublepoint of L differing from the base
points, then in general two different points Z'" and P" will correspond
to F according to our regarding F as point of the one or of the other
branch of L passing through F. The curves of the pencils passing
through F now have two more common points F' and F", so that
we get a triplet of points FF' F", through which a curve of each
of the pencils is possible.
It may however also happen that tiie points F' and F" coincide.
In that case correspond to the two branches througii F two branches
through F', so that F is likewise doublepoint of L. The curves of
the pencils passing through F have now but one other common
point P', but now the particularity arises that J* or P' can be
displaced in two ways such that the other common point is retained.
So FF' is then to be regarded as a double corresponding pair of
points.
( 431 )
If reversely we have a tri)U't of {)oiiils PP'P" lying on curves
of each of the pencils, then P is a doublepoint of L, 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 desci'ibes a branch passing
through P' and in such a way that a branch passing through P"
is described. The curve L has thus two branches PI and P2 passing
through P to which the branches PI and P'2 correspond. Through
the point P' (which is of course likewise doublepoint of Zr^as well
as P") a second branch P'3 passes and through F" a second branch
P"3, which branches correspond mutually. If a point Q describes
the branch PI the curves C,, Cs, Ct passing through Q have a
second common point describing the branch P'\, 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 P' appears and disappears.
Triplets of points PP'P", and therefore doublepoints of L
ditfering from the basepoints, there will be as a triplet of points
depends on 6 parameters and it is a 6fold condition that a curve of
each of the pencils must pass through it. So we have:
The curve L ha.s doublepoints, differing from the basepomts of the
pencils, belonging in triplets together and forming the triplets of points
through ivhich 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 doublepoint of such a pair correspoiid the branches through the
other doid)lepoint of the pair.
6. The number of coincidences of the correspondence between P
and P' can be determined as follows. The points P and P' coincide
if the cur\'es C,, Cs and Ci passing through P have in P the same
tangent. Then P must lie on the curve of contact R,.s of the pencils
(O and {Cs) as well as on the curve of contact Rri of (C,) and (C,).
The number of points of intersection of those curves of contact
which are of order '2r f 2s — 3 resj). 2r \ 2t  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^t , and so they must not be counted. The curve
A^•.s• namely passes once through a basepoint A,, or A^ and three
times through a common basepoint Ays or .4,5^^ in fact in a point of
( 432 )
Rrs two movable points of intersection of Cr and Cs, coincide so that
the point Ajs as a point of the curve of contact is found when Cr
and Cs show in Ars a contact of the second order which takes place
three times. Further Rrs passes through the doublepoints of the curves
Cr and Cs, of which the number for the pencil {Cr) amounts to
3(r — 1)^ and for the pencil {Cs) to 3(5 — 1)', which follows imme
diately from the order of the discriminant.
Each of the r* — ^' — y' ■ — ö points Ar is a simple point of inter
section of Rrs fïnd Rrt (simple, the tangents in Ar to Rrs and Rrt
being the tangents of the curves Cs and Ct passing through Ar,
diifering thus in general), but no point of Rst ■ Each of the «' points
Ast is a double point of intersection of Rrs and Rrt , as those curves
of contact in Ast have a simple point with the same tangent, namely
that of the Cr passing through Ast ; these points are also points of Rst ,
namely threefold ones. Each of the /?' points Art is threefold pomt
of inlersection of Rrs and Rrt (it being simple point of Rrs and
threefold point of Rrt ) and lies at the same time on Rst ; the same
holds for the y' points .4,* . Each of the ff points Arst which are common
basepoints of the three pencils is 9fold point of intersection of Rrs
and Rrt , being threefold point of each of those curves ; moreover it
is threefold point of Rst B'inally the 3(?" — 1)" doublepoints of the
pencil {Cr) are simple points of intersection of Rrs and Rrt , but not
points of Rst ; of the curves Cr, Cg and Ct passing through such a
doublepoint Cr has an improper contact with Cs and with Ct, without
however Cs and Ct touching each other.
From this we see that the curves of contact Rrs and Rrt have
,,» _ ^' _ y' _ (f ^ 3 (/• — 1)'' = 4r» — 6r + 3 — ^' — r' — d
points of intersection which are not points of Rsi , and so do not
furnish coinciding points ]\ P' . Moreover Rrs and Rrt have
2«' + 3ii' 4 3y' I 9J
points of intersection coinciding with the common basepoints, which
do fall on Rgt , but which do not give any coinciding points 1* and
P', as for this it is necessary that of three curves C,, C^^ and Ct
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 F and /*' remains :
^2r 4 2s — 8) (2r f 2«  8) — (4r' — 6/ + 3 — J' — y' — d) —
— (2«' f 3/3' + 3y' + 9(f) =
= 4(.sr tr h rs) — 6{r f s + f) + 6 — 2 («' + /J' f y' f 4d).
So we find :
It happens
4(,^ + tr f rs) — 6(/' + s \ t) ] Q — 2(« + /? + y + d)
( 4B3 )
times that the two points F and F' 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 >S. We find this number by regarding the correspondence
between the rays SP and aS'P', which we call I and /'. This is an
involutory [ii, /i)correspondence where n represents the order of the
locus L of the points P and P' ; for on an arbitrary ray / (or /')
lie n points P (or P'), to each of whicii one point P' (or P) cor
responds. So there are 2/i 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 aS' we find :
2 \^rst + 1) _ 2 (r f .. 4 ^) — {ar + /?« + yt)\ — 4(«« + <r + rs) —
— 6(r f « f + Ö — 2 (« } /? + y f ff)j = Qrst _ 4 (s« + «r + rs) +
+ 2 (r + s +  2« {r _ 1) _ 2/? (.s  1)  2y («  1) + 26.
These rays of coincidence however coincide in pairs. For if the
line connecting tlie corresponding points P^ and P/ passes through
S, then to PxP^ regarded as line / correspond n 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. Pj. Likewise to P^P^
regarded as line /' correspond n lines /, of which also two coincide
with P^Pi, 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 M'ith respect to the line y = X
on account of the correspondence being involutory) indicates by its points of
intersection with the line y = x the rays of coincidence. If B is the point of
representation of the rays I and /' coinciding in PiPi', the curve of representation
is cut in two coinciding points B by a line parallel to the ?/axis as well as by
a line parallel to the .caxis, on account of PjPi' regarded as / or /' corresponding
twice to itself regarded as /' resp. I. So B is doublepoint of the curve of repre
sentation, so that the lino y = .r furnishes two points of intersection coinciding
with B.
( 434 )
The envelope of the lines connectlnj paifs of points, through lohich
a curve of each of the pencils is possible, is of class
3 rst — 2 {st{tr\rs) + (r+s+0  a{rl) — /?(sl) — y (^1) f rf =
= 3 rst — 2 (st^trirs) + (r+s^^  «'(r1)  ^'{s~l)  r'(«l) 
S. If the pencils have no common basepoints then the class of
the envelope is 3 rst — 2 (.s^ + tr + rs) f (r + .s^ + t). By a common
basepoint Ast of the pencils {Cs) and (G) that class is lowered with
r — 1. IViis is because point Ast has separated itself from the
envelope r—1 times. Tn fact, the curve (■',. passing through Ast has
separated itself from the locus of the points P and P' . If we take
P arbitrarily on this C,, the corresponding point P' coincides with
Ast' So an arbitrary line passing through Ast is to be regarded
(^. — 1) times as a line connecting P and P' , as any of the r — 1
points of intersection with t; differing from Ast may be chosen for I\
If the three pencils have a common basepoint Arst 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 P'
coincides with a point .4,,.^, so that the line 7V^' passes through that
point Arst^ and therefore is not quite arbitrary. As the class of the
envelope proper is lowered by the point Arst with r \ s ]; 1 — 4 it
follows, that Arst separates itself (r \ s \ 1 — ^) 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 nem empiric spectra] formula." By E. E.
MoaKNDORFF. (Commuuicated by Prof. P. Zeeman).
By the fundamental investigations of Kaysek and Runge and those
of Rydberg the existence of spectral series was proved. The fornnilae
of these physicists, however, give in general too great de\'iations for
the first lines of a series. I have tried to improve the formula given
bv Rydberg:
11=1 A .
(m 1 ay
Particularly noteworthy in Rydberg's formula is tlie universal
constant 7V„. From Balmer's formula, which is included as a special
case in RvDBEiKi's formula, follows for hydrogou for the observation
corrected to vacuo X. = 109675.
(435)
Assuming tor a hioiikmiI lluil the A'^ was also variable for the
ditferent series, 1 lia\'(^ calculated the constants A, a and A^,, from three
of the best observed cuixes. For iY„ the following values were found :
Principal Series
1^' associated series
Second
Lithium
Natrium
Potassium
Rubidium
Caesium
Hydrogen
Helium
Natrium
Potassium
Silver
Magnesium
Zinc
Oxygen
Natrium
Magnesium
Calcium
Zinc
Aluminium
109996
107178
105638
104723
104665
109704
109703
110262
109081
107162
108695
107489
110660
107819
105247
103702
105399
105721
These values ha\'e been calculated from wa\e frequencies not
corrected to vacuo.
x\s appears fi'om these values JSf^ is not absolutely constant. As
Kayser ') found in another way, we see, however, that relatively
1) Kayser, Haiullxicli II. p. ."iS^.
( 436 )
iVfl 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 iV„ = 207620 calculating from the middle lines 7\^„ = 138032,
and from the lines with smaller X JST^ = 125048.
The first asssociated series of aluminium behaves therefore quite
abnormally.
In Rydberg'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
ti = A
>
in which 7i represents the wave frequency reduced to vacuo, A, a
and b are constants which are to be determined, m passes through
the series of the positive integers, starting with m = l. In most
cases 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 Rydberg — Schuster is satisfied
in those cases where besides associated series, also a principal series
is observed.
A spectral formula has also been proposed by Ritz ^).
In my thesis for the doctorate I ha^e 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 X^ , the limit of error of observation under F, the deviation
according to the formula proposed by me under A, the deviation
according to the formula of Kayser and Runge 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 b.
The constants are calculated from the wave frequencies reduced
to vacuo "'*).
1) The B in Kayser and Runge's formula varies within considerably wider limits
than the N„ of Rydberg's formula.
2) Ann. d. Phys. Bd. 12, 1903, p. 264. W. Ritz, Zur Theorie der Serienspectren.
2) Where it was possible, 1 have always taken these values from the "Index of
Spectra" from Marshall Watts.
( 437 )
Lithium.
Principal series: A = 43480,13 ■<( = { 0,95182 ; /> = + 0,00722
V' ass. series : A = 28581,8 ; (/ = + 1,998774; 6 = — 0,000822
2°'i „ : A = 28581,8 ; r/ = + 1,59872 ; /> = — 0,00321
S'd „ ; A = 28581,8 ; .7. = + 1,95085 ; A = f 0,00404
Tlie associated series converge here evidently to one limit.
The difterence of wave frequency between the limits of principal
and associated series is 43480,13^21581,8 = 14898,33. The wave
frequency of the 1^* line of the principal series is 14902,7. So the
formula satisfies the law of RydbergSchuster pretty well.
PRINCIPAL SERIES.
m
>w
F
K
A. K. R.
1
G708,2 *
0,20
+ 108
2
3232,77*
0,03
3
2741,39
0,03
— 0,06
4
2562,60*
0,03
5
2475,13
0,10
— 0,22
0,2
6
2425,55
0,10
— 0,18
— 0,01
7
2394,.54
0,20
— 0,13
+ 0,30
8
2373,9 L. D.
?
+ 0,02
+ 0,75
9
2359,4 L. D.
?
+ 0.17
41,18
FIRST ASSOCIATED SERIES.
m
/v>^
F
1
A
A.K.R.
\
6103,77'
0,03
(»
2
4602,37*
0,10
3
4132,44
0,20
— 0.11
4
3915,20*
0,20
— 0,20
5
3794,9
5,00
+ 0,09
— 0,35
6
3718.9
5,00
— 4,94
— 2,25
7
3670,6
5,00
— 1,06
 1,41
( 4B8 )
SECOND ASSOCIATKÜ SEUIES.
m
>w
F
A
A. K. R.
\
8127,0* S
0,30
— 65
2
4972,11
0,10
— 0,13
3
4273,44*
0,20
4
3985,94
0,20
+ 0,22
5
3838,30
3,00
+ 2,40
0,2
THIRD ASSOCIATED SERIES.
m
Aw F
A
A. K. R.
1
6240,3* S
0,40
—
2
4636,3* S
0,40
—
3
4148,2 S
1,00
+ 1,6
—
4
3921,8 E H
?
— 0,88
Tlie capitals after the wavelengths denote the observers: L. D.
LivEiNG and Dewar ; S. Saunders and E. H. Exner and Haschek.
Where no further indication is given, the observation has been made
by Kayser and Runge.
Natrium.
Principal series (the lines of the donblets with greatest X)
A = 41447,09 ; a = 1,147615 ; /> =  0,031484
Principal series (lines of the doublets with smallest X)
A = 41445,20 ; a = 1,148883 ; /; = — 0,031908.
For the calculation of the limit of the associated series Rydberg
Schuster'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 l«t ass. series A = 24491,1 ; a = 1,98259^ b = \ 0,00639
For the 2"'i ass. series A = 24491,1; ./ = 1,65160; /> = — 0,01056
( 439 )
PRINCll
'AL SERIES.
m
/w
1
' i
A
A. K. R.
1
589C,lü
—
+ 7S
1
0890,19
—
+ 86
2
3303,07*
0,03
2
3302,47*
0,03
3
2852,91
0,05
— 0,14
3
2852,91
0,05
— 0,06
4
£680,46*
0,10
4
2680,46*
0,10
5
2593,98
0,10
+ 0,03
\ 0,03
5
2593,98
0,10
— 0,02
+ 0,09
6
2543,85 L. I).
0,10
— 0,06
+ 0,10
6
2543,85 L. D.
0,10
— 0,14
+ 0,24
7
2512,23 L. D.
0,20
+ 0,03
+ 0,50
7
2512,23 L. D.
0,20
— 0,10
+ 0.60
FIRST ASSOCIATED SERIES.
m
>w
F
A
A. K. R.
1
8184,33*
L.
0,2
2
5682,90
0,15
0,01
3
4979,30*
0,20
4
4665,20
0,50
— 0,13
+ 0,52
5
4494,30
1,00
— 0,28
+ 0,50
6
4390,70 L.
D.
7
+ 0,28
f 1,30
7
4325,70 L.
D.
?
+ 4,00
+ 1,76
SECOND ASSOCIATED SERIES.
11404
6154,62*
5149,19*
4748,36
4542,75
4420,20 L. D.
4343,70
0,10
0,10
0,15
0,20
?
7
+ 1,00
O
o
+ 0,12
+ 0,65
+ 0,02
+ 2,00
+ 100,
o
o
+ 1.39
+ l,>r.
— 1,36
( 440 )
Zinc.
For this element I liaxe oalculated the formulae of the 1^*^ and
2"*^ associated series for tlie 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
good agreement.
The formula gives as 1®' line of the i^^ associated series of Zinc the line
8024,05, which has not been observed. The 8''Mine of the first associated
series 2409,22 has not been observed either. As 9^^ ijp^^ Qf ji^jg ggrigg
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 1^^ associated series is :
= 42876,25
109675
0,007085\'
m + 0,909103 —
in J
and for the 2°** associated series
n — 42876,70 
109675
m + 1,286822
0,058916
FIRST ASSOCIATED SERIES.
m
>w
F
A
A. K. R.
1
—
—
—
2
3345,13*
0,03
— 0,08
3
2801,00*
0,03
+ 0,03
4
2608,65*
0,05
+ 0,06
5
2.510,00
0,20
+ 0,01
— 0,11
6
2463,47
0,20
— 0,14
— 0,39
7
8
9
2430,74
0,30
+ 0,22
+ 9,00
2393,88
0,05
— 0,05
~
( 441 )
SECOND ASSOCIATED SERIES.
m
/W
F
A
A. K. R.
1
4810,71 *
j 0,03
4 58
2
3072.19*
■ 0,05
i 0,00
3
2712,60*
'■ 0,05
+ 0,02
4
2567,99*
i 0,10
+
0,11
o,ai
2493,67
! 0,15
+
0,12
— 0,04
6
2U9,76
0,25
—
0,11
0,20
Thallium.
The forrnula for the 1^* associated series is
109675
71 = 41466,4 —
m f 1.90141 —
109675
for the satellites :
n = 41466,4 — 
(yn 4 1,88956
and for the second associated series :
71 = 41466,4
0,00366\
m J
0.00085
)■
109675
(
m + 1,26516
0,07108
The limit has been"^calculated from three lines of the 1^^ associated
series ; only two more lines were required of the satellites and of the 2"*^
associated series. So in this spectrum all the constantshavebeen calculated
from 7 lines and 31 lines are verj well represented by the formula.
FIRST ASSOCIATED SERIES.
m
Aw
F
A
A. K. R.
1
3519,39'
0,03
—
2
2918,43
0,03 !
— 0,04
—
3
2709,33*
0,03
—
4
2609,08
0,03 1
+ 0,04
—
5
2552,62
0,10
—
6
2517,50
0,10 1
— 0,06
— 0,34
7
2494,00
0,10
— 0,03
— 0,19
8
2477,58
0,10 1
— 0,09
+ 0,06
9
2465,54
0,20
— 0,17
+ 0,24
10
2456,53
0,20 '
— 0.15
+ 0,47
\i
2449,57
0,30
«.,17
+ 0,68
12
2444,00
0,30
 0,28
f 0,79
13
2439,58
0,30
0,24
+ 0,95
( 442 )
SATELLITES.
m
>w
F
A
A. K. R,
1
3529,r>8*
0,03
+ 0,02
2
2921,63
0,03
+ 0,00
— 0,07
3
2710.77*
0,03
+ 0,13
A
2609,80
0,03
— 0,03
— 0,02
5
2553,07
0,10
— 0,05
— 0,12
SECOND ASSOCIATED SERIES
m
w
F ! A
A. K. R.
1
5350,65*
0,03
— 168
2
3229,88*
0,03
— 21,7
3
2826,27
0,05
— 0,05
— 3,65
4
2665,67
0,05
— 1,32
 1,69
5
2585,68
0,05
— 0,16
+ 0,01
6
2538,27
0,10
— 0,17
+ 0,04
7
2508,03
0,15
— 0,14
 0,01
8
2487,57
0,20
— 0,06
+ 0,08
9
2427,65
0,20
— 0,34
— 0,21
10
2462,01
0,30
— 0,20
— 0,03
11
2453,87
0,30
— 0,17
+ 0,07
12
2447,59
0,30
— 0,05
+ 0,22
13
2442,24
0,30
— 0,37
— 0,01
I shall just add a few words on the spectrum of Aluminium.
None of the foiniulae 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 N^ were calculated from three of the
1^^ Unes of the series.
The formula runs :
109675
w = 48287,9 —
/ 1,038060\'
( m + 0,89436 + j
( 443 )
The constants have been calculated from the lines 4, 5 and 6.
ALUMINIUM. FIRST ASSOCIATED SERIES.
m
/yy
F
A
A. K. R.
1
3< 82,27
0,f'3
—
268,82
f 384,8
2
2568,08
0,03
+
3,46
+ 53,5
3
2367,i6
0,03
+
2,52
+ 6,1
4
2263,83*
0,10
f 0,03
5
2204,73*
0,10
+ 0,17
6
2168,87*
0,10
— 0,13
7
2145,48
0,20
+
0,06
— 0,31
8
2129,52
0,20
+
0,11
— 0,21
9
2118,58
0,20
—
0,^8
+ 0,44
The agreement with the first lines (1, 2 and 3), leaves much to be
desired. The value of the constant h is here 1,03806, greater than
the value of a in that formula; this does nof, occur with any of the
other series.
With 4 constants, so with :
n =z A
109675
b c
m \~ a \ [ —
m m
a better result is most likely reached. When the constants h and here
probabl}' 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 in. 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.
29
1'iccceding.s lloyal Acad. Amslei'dam. Vol. IX.
( 444 )
Astronomy. — ''Mutual occultations and eclipses of the satellites
of Jupiter in 1908." By Prof. J. A. C. Oudemans.
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 tlie eclipses of one satellite hj another
we have but one, incomplete account given in a private letter of
Mr. Stanley Williams 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'h August 1891 and was observed by Mr. J. Comas at
"Valls in Spain and by the writer at Hove. Mr. Comas' observation
"was published in die French 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."
" "1891 Aug. 14. 67, inch reflector, power 225. Definition good,
" "but interruptions from cloud. Satellite I. transitted on the S. Equa
" "torial belt, (N. component). Immediately on its entering the disc
" "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 shadow^s of satellites I. and II. were confounded together at
" "this time, there seeming to be one very large, slightly oval, black
" "spot. At ll'i59''^ the two shadows were seen neatly separated,
""thus, @® . The preceding shadow must^be that of II., the follow
" "ing and )iiuch smaller one that of I.. At 12'ilO™ 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 disc near the limb. Definition good, but much thin cloud
" "about." "
"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 II., so that it
"became lost to view immediately, instead of shining, as usual, for
J. A. C. OUDEMANS. "Mutual occultations and eclipses of the satellites of Jupiter
in 1908." Second part: Eclipses.
(/5
1
Scale
30 168 000 000
. On this scale the sun's diameter is 0.24 meter and its
distance 25.783 m.
Proceedings Royal Acad. Amsterdam. Vol. IX.
( ^^5 )
''some time as a brillaiil disc. Also that the minute dark spot seen
"at ll^'49f" 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 'Jl''59^ was due to a part only of the shadow of II. being
"projected on the disc of Jupiter, the other part of this shadow
"having been intercepted by satellite I. 3)
"But combining Mv. 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."
"P.S. The above times are Greenwich mean times. The Xautical
"Almanac time for the trïinsit ingress of satellite I. is llhSS""." 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: l^t whether the shadows of the
foremost satellite reaches that of the more distant one in even/ helio
centric conjunction and 2"^ whether tlie occurrence of total eclipses
is possible in any cae. 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 <j, e, c, a, h, d, f
and //. For clearness, sake the figure is given below (Plate I).
Now suppose that I is placed either at a or at h. In both cases
the other satellites will be in\'ohed in its shadow cone as soon as
they come: 11/ at d, III /at _/' and IV; at It.
The points of intersection with the orbit of II are c and d If
II„ is at c then I,j may be eclipsed in a but also If in b; III, at f
and Wf at h.
But if Wf is in d then only IIL and IV, can be eclipsed, the former
at ƒ and the latter at h.
The points of intersection with the orbit of III are e an<l f If
III is at e there is the possibility of an eclipse for II„ at c, I„ at o,
\f at h, Wf at d and IV, at It. If on the other hand it is in ƒ there
is siu'h a possibility oidy for IV, at li.
It is evident ihat IV can only cause tiie eclipse of another satel
29*
( 446 )
lite if it is at the position g, one of the other three sateUites being
then at one of the points of intersection already mentioned.
Each of the satellites mi,^ht thns 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 Milues 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. Ill,; may cause a total eclipse of 1I„ and In ; If
may nearly produce such an eclipse of 11/; 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/ .
„ II„ „ „ „ III/ and IV/,
mlV/
„ IV„ „ ,, ,, 11/ and III/.
In the tifteen 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 radii. In computing
however the occultations observed by Messrs Fauth and Nijland it
appeared that this difference in latitude, according to the tables of
Damoiseau, is sometimes slightly greater. The latitudes found by these
tables are therefore not entirely trustwoi'thy. 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, viz the drawing of the orbits
of the satellites is the same as for the computation of the geocentric
conjunctions (see 1^"^ part). First however the epochs of the helio
centric superior conjunctions must be derived from the epochs of
the geocentric 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, for the computation of thejovicentiic
( 447 )
latitude of the Earth, wliich iiee<l not l>e knriwu in the present case.
The number of coluniii in (Mir tatile 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 eases an
eclipse is possible.
(1) The account of Mr. José Comas is as follows :
Ombres de deux satellites de Jupiter et eclipse. — Dans la nnit
(\\\ 14 aoiit, j'ai observe iiii phénoraène bien rare: la coincidence
partielle, sur Jupiter, des ombres de ses deux premiers satellites, et
par suite l'éclipse de Soleil pour le satellite I prod uit par le satellite II.
A 11'^ (temps de Barcelone) '\ I'ombre du satellite II est entree
sur la jtlanete. Pres du hr>rd. elle n'était pas noire, mais d'un gris
rougeatre. Comme I'image était fort agitée, j'ai cessé d'observer,
mais je suis retourné a I'observation vers ll^^l^ pour observer
I'immersion du premier satellite, qui a eu lieu a 11^42"! (grossis
sement 100 fois; lunette de 4 pouces . J'ai été surpris de voir
disparaitre lo ') a son entree sur le disque, ne se dëtachant pas
en blanc, quoiqu'il se projetat iir la liande foncée equatoriale
australe.
A 11*'52'", avec des images plus tranquilles et un grossissement
de 160, je remarquai que I'ombre complètement noire que Ton
voyait était allongée dans une direction un peu inclinée vers la
droite, relativement a Taxe de Jupiter. La phase maxima de l'éclipse
du satellite I était déja passée de quelques minutes. A 11^56™ je
pris le petit dessin que j'ai riioiiiieur de vous adresser; les deux
ombres se touchaient encore ';. Aussitót elles se séparèrent et,
quoique je n'aie pas pu noter l'instant du dernier contact, je crois
être assez pres de la vérité, en disant qu'il s'est effectué vers ll^'SS"".
L'empiètement dune ombre sur l'autre pourrait être de la troisième
1) Barcelone is 2^10' East of Greenwich; mean time at Barcelone is therefore
8ni40s later than of Greenwich.
2) Since a few years the Nautical Almanac mentions the names of the Satel
lites of Jupiter proposed by Simon Marics: Io, Europa, Ganymedes and Gallisto.
3) Tliis 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 dn diamètre. Dans cette supposition la distance minima des
centres des deux onibres a du avoir lieu vers 11''47'" et Ie premier
contact vers 11''37'". Le premier satellite pénétra dans Ie disque de
la planète a 11^42"^ comme j'ai dit plus haut, done l'éclipse a com
mence quand le satellite se prqjetait encore dans l'espace, cinq
minutes avant l'immersion.
L'invisibilité de l'ombre d'Europe sur lo peut s'expliquer par la
mauvaise qualité des images. Toutefois, la pénombre et l'ombre du
II satellite ont été suffisantes pour diminuer notablement l'éclat du
premier.
(2) The meaning evidently is that, as seen in an inverting 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 vSouth band of the South
belt. It is well known that the socalled Red ^ipot is there situated.
(3) 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 Damoiseau, second part, the time of
the heliocentric conjunction of the two satellites is 23"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'^i M. T. Grw.
I Transit ,, 11 33 ,, ,, ,,
II „ „ 11 58 „ „ „
I Shadow. Egress 13 18 ,, ,, ,,
11
)5
13 45
I Transit
51
13 51
11
J>
14 49
If from the 1^' , 2"*^, 5'^^^ 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
motion in five minutes, consequently also that of their shadows, was
very minute.
Mr. Stanley Williams seems not to have perceived a shadow
before II '49" M. T. Greenwich; Mr. Comas already saw an oblong
shadow at ll''43^20' M. T. Greenwich. For the rest Mr. Stanley
I 449 )
Williams makes the shadow of II larger than that of T whereas
in the estimation of Mr. Comas they were e(ual. It seems hardly
doiihtful but tlie English observer must be right.
(5) In 1901 See 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. iVstron. Naclir. N". 3764,
21 Jan. 1902. The communication of See 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 l)y 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 held and the satellites were coloured greenish
yellow by a screen tilled with protochloride of copper and picric acid.
The results for the diameters turned out to be smaller in every case
than those formerly found. The difference w^as attributed to irradiation.
The results, reduced to the mean distance of Jupiter to the sun
(5,2028), are as follows.
Satellite
At nisht
In daytime
Ditïerence, attributed
to irradiation
I
4 ",077 + 0"ai8
0"834 ± 0",006
0"243 + 0"019
II
,976 ± ,043
,747 + ,007
,229 ± ,0435
III
4 ,604 + ,033
1 ,265. + ,009
,339 + ,039
IV
1 ,441 + 0,018
1 ,169 + 0,006
0,372 ± 0,019
It is remarkable that the brightest satellite, III, shows also the
strongest irradiation. If howe\'er we consider the difference insufTi
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 exceedinglv small.
( 450 )
It seems worth while to call atteiilioii to the differences between
the diameters found i)y the same observer in ']9()() and 1901.
I
II
III
IV
1900
0"672 ± 0"098
0,624 db 0,078
1,361 d= 0,103
1,277 ± 0,083
1901
0",834 ± 0",()06
,747 ± ,007
1 ,265 ± ,009
1 ,169 ± ,006
1901—1900
1 0"162
f 0,121
— 0,096
— 0,108
Stone, at Oxford, once tohl me that Airy, in a conversation on
the determination of declinations at the meridian circle, remarked to
him: "I assure von. 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. In how far are the tables of Damoiseau stlU reliable ?
In the first part of this paper, pages 319 and 321, we explained
why we felt ourselves justified in using the tables of Damoiseau 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 oi' the Astroiio
inische Nachrichten and of the Monthhj 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 wiih the data of the Nautical Almanac.
In order to simplify, we requested him to note only the observation
of the last light seen at disap})earance and the first light at reajipear
ance^). We intended to extend our investigation from 1894 to 1905
^) Delambre in the introduclion to his tables, does not stale explicitly the
precise instant to which his tables refei but from some passages we may conclude
thai he also means the instant as here defined. So for instance on page LIII
where he says: "Les demidureées ont été iin pen diminuées, pour les raiyprochcr
des observations qiCon a faites deiniis la découverte des lunettes achromatiques" .
That Laplace also takes it for granted that such is his real meaning, appears
from Gh. Vfll, 8lh book of the Mécanique Celeste.
( 451 )
or 190(1 Imt after liaving eom»lete(l som^ four years tliere seemed
reason lo iliink ihal lliere was hardly need for fiiitlier information.
Tlie geiieral result arrived at was, that the tables were still sulKi
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 w^e will not suppress its results
though it cannot at all claim to be complete. It never was om
intention to make it so, and the journals appearing in France, in
America etc. have not been searched.
The following observatories have contributed to our investigation.
Aperture of the telescopes
in ni.ni.
Greenwich 102, 170, 254, 714.
Utrecht 260
Uccle 150
Jena (Winkler) 162
Halifax (Gledhill) 237
Pola 162
Christiania 74, 190
Kasan 66, 81, 84, 96, 244
Göttingen 161
Windsor (Tebbutt) near Adelaide 203
Lyon (a single observation) 2
At Greenwich, Cliristiania 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 wdiich still would be
pei'feclly justified, even if there had been a regularly 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
sition.
•1894/95
1895/96
1897
1898
9418
1894/95
1895/96
1897
1898
1894
1895
1895/96
1897
1898
1899
1895
Mean
Num
Mean
Corr. N.A.
Disapp.
ber.
error.
+ 37s
f 30
— 19*
+ 11
— 78
I 52
+ 73
— 72
— 36
f151s
flOl
f 87
+181
+266
+361
+ 21ni45s
Mean
Corr. N.A.
Reapp.
Num
Mean
h{D+R)
ber.
error.
Mean
3
+14
9
8
2
18
15
6
2
±32
4
225
6
18
3
26
5
20
3
+22s
4
19
9
13
4
i9
4
19
3
22
I.
— 18^
— 5
+ 7
II.
— 42
— 4
+ 11
— 15
III.
—2428
—127
— 50
+ 37
+ 10
—126
IV.
— 17m 9s
25
+ 4s
+ 9S.5
32
4
+ 15
12
6
— 12
13
6
+ G
7
+lls
— 39s
15
75
+ 5
19
6*
+ 34
10
9
— 30
9
95
— 26
3
+38s
— 45s
4
33
— 13
9
22
+ 19
9
22
+109
1
60
+138
4
33
+118
+138S
+ 7s
45
9*
4
±17s
12
10
14
11
±253
ir
34
20
1895/96
1897
+ 3 49
— 02
10
2
±25
457
— 3 17
+ 1 16
±22s
60
+ 16 ±17s
+ 37 I —41
(453)
Average mean error of a single observation.
Disappearance Reappearance Mean  Dolambre *) Introd. p. LIV
I
II
III
IV
± 258
45
37
80
± 20s
29
(iG
00
± 22s 5
37
51,5
70
1785
1 88, 5
'72,5 rejecting the observations
deviating more than 3 mi
nutes).
According to these nnmbers the comphiints abont the i'lcreased
inaccnraey of the tables of Damoiseau seem rather exaggerated, at least
for the (irst and secoiul satellites.
Taking into account the mean errors contained in the last column
we get tlie most probable correction at the epoch 1894 — 98
for I f S^0 with a mean error of ± 2^6
similarly „II — 3 ,8 „ „ „ „ „ ± 5 ,4.
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 foi' 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 'm the duration of the eclipse. The observations
of Mr, Winkler 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'"48« 21'"58^ + 20'^53s
at reappearance — 19 36 — 18 33 — 19 4,5
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 IV4
in order to set mean errors.
( 454 )
The explanation of these extravagant differences mnst rather he
songht, either in a correction needed l\y the longitude of the node
of the satellite's orbit or in the adopted tlattening of Jupiter. It is
also possible that for suchlike eclipses the diminution of light is
very slow.
For tlie rest, according to the Nautical Almanac, this eclij^se
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 Jan>'. D. 1'' 36^16^ M. T. Gr., R. 2^ 8'"17^ duration 32^1 1«
2 Feb>. „ 19 26 12 „ „ „ „ 20 36 58 , „ 1'' 10 46
19 „ „ 13 24 6 „ „ „ „ 14 59 3 , „ 1 34 57
8 March „ 7 24 14 „ „ „ „ 9 18 28, „ 1 54 J4.
Only, according to ScottHansen, who, on the NorthPolar expedition
of Nansen, was in charge of the astronomical observations, the
satellite has not been eclipsed at all on the 17^'^ of January ^).
On the 2 "f^ 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
Sheephanks 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
Xh44mi4s _ 39m57.s 5 _ li.i4mie%5,
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 Damoiseau's value r— r exceeds already
that found by direct measurement by most observers. Taking into
account however the results obtained by De Sitter, 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 18941896. Scientific Results, edited
by Fridtjof Nansen. VI. Astronomical Observations, arranged and reduced under
the supervision of H. Geelmuyden, p. XXIV.
( 455 )
the fixed plane is t'ouiid tlie value := 0^,2504:= 15' 2"4, which exceeds
Damoiseau's inclination only bv somewhat less than a minute.
The remaining eclipses of l\ in 1895 and the two following years
do not show any extraordinary divergencies.
Now, as in 1908 the eclipses of the sateUites 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 xllmanac, which is based
on the tables of Damoiseau (taking into account only a few necessary
corrections), may be considered sufficient for preparing ourseh'es for
the coming observations. The only exception would be for an early
eclipse of IV after a period in wdiich it is not eclipsed at all.
Utrecht, 23 November 1906.
( 456 )
RESULTS.
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 ;
To. = Tokio ; We. = Wellington ; Wi. = Windsor.
n^=
near
f =
ar
Mean time
"^2
S3 ^
No.
at Greenwich
Ö 01
.S .
c «
X^=X^
1^ CD
Ü <>
*^ CO
~ "53
*^ to
y—u'
Visible at
4
1 April 4h 8m
1/
II«
+5r70
—Or 30
—Or 25'^
— 0r04^'
Kas., Taschk., Madras, HK., Perth
2
2 »
18 3
1/
III»
+3,21
—0,16
—0,20
+0,04
Lick, Fl., Tac, AA., Harvard.
3
3 »
4 15
11/
III«
—2,49
+ 0,08
+0,10
—0,02
Kas., Taschk., Madr., HK., Perth, To
4
3 »
9 51
11/
\n
+1,50
—0, 105
—0, 09
—0,01*
Grw.,Pulk.,Kas.,Taschk.,LaPl.,Rio
5
3 »
11 10
IV/
III«
6, 19^
+0,40
+0,32
+0,08
Grw., Pulk., Kasan, La PI., Rio.
6
3 »
16 26
IV/
I»
—4,03
+0,30
+0, 19
+0,11
Lick, Fl., Tac, AA., Harv., La PI.
7
4 »
16 52
IV/
II.
+6,03
—0.20
—0,27
+0,07
Lick, FL, Tac, AA., Harvard.
8
4 »
17 21
!ƒ
II»
+5, 75
—0,31
—0,25
—0,06
Lick, FL, Tac, AA., Harvard.
9
5 »
19 56
Ill/
II»
—9, 24
+0,54
+0,30
+0,24
Wi., We., Lick, FL, Tac, AA.
•10
6 »
20 12
111/
In
+3,61
—0, 18^
0, 21
+0,02'^
Wi., We., Lick, FL, Tac, AA.
11
Ö »
22 58
11/
I«
+1,37
—0, 10
—0,09
—0,01
Perth, Tokio, Wi., We.
12
8 »
6 31
he.
II«
+5,82
0, 31
0, 25
—0,06
BresL, Pulk., Kas., Taschk., Madras.
13
9 »
20 52
1/
II I«
+3, 85^
0,18
—0,24
+0,06
WL, We., Lick, Fl.
14
10 »
7 28
11/
III«
—2, 05^
+0,06
+0, 09^
—0,03*
Grw., Pulk., Kas., Taschk., Madras
15
10 »
12 4
11/
I»
+1,24
—0, 09
—0,08
—0,01
Grw., Pulk., Kas., llarv., La PL, Rio.
10
11 » ,
( II +7, 87 )
15 42
II
IV
1 J
—0,36
0,48
+0,12
Lick,FL,Tac.,AA.,lIarv.,LaPl.,Rio.
11 »'
smallest
distance
(lV+7,96)
17
11 »
19 43
h.e.
\\n
+5,88
—0,31 5
—0, 26
—0,05*
Wi., We., Lick, FL, Tac, AA.
18
11 »
20 24
le e.
IV„
+5,99
—0,32
—0, 38
+0,06
Wi., We., Lick, Fl, J
19
12 »
23 33
Ill/
\U.e.
—9,41
+0,54
+0,45
+0,09
Perth, HK., Tokio, Wi., We. ■
20
13 »
3 57
III/
1V„
—7,35
+0.42
+0,27
+0,15
Kasan, Taschk., Madr., HK.
21
13 »
23 22
III/
In
+2,64
—0,16
—0,19
+0,03
HK., Perth, Tokio, Wi., We.
22
14 »
1 11
11/
1.
+ 1,11
—0,09
—0,06
—0,03
HK., Perth, Tokio, Wi.
23
15 »
8 57
\c.e.
\u
+ 3,93
0,32
—0, 26
—0,06
Grw., Pulk., Kas., Taschk., Rio.
24
16 »
23 44
1/
Illn
+4,45'>
—0, 23
—0. 26
+0,03
HK., Pe., To., Wi., Wc
25
17 »
10 41
11/
III,,
—1,61
+0,03
+0,08
—0,05
Grw., Pulk., Kasan, La PL, Rio,
26
17 »
14 17
11/
L.
+0,ï>8
0,09
—0,06
—0,03
Grw.,Fl.,Tac.,AA.,Harv.,LaPl.,Rio.
27
18 »
22 11
I..*.
1I«
+5,97
—0,32
0,26
—0,06
Perth, To., Wi., We.
( 457 )
Mean time
at Greenwich
f =
near
far
x=x'
«3
«3 *±
o'JZ
y—y'
No.
Eclipsing
satellite
Visible at
28
19 April 5l»15m
IV/
III«,...
— 15r28
+ 0r80
+ 0r85
— 0r05
Kasan, Taschk., Madras.
29
19 »
22 57
IV/
II»
— 8,69
+ 0,48
+ 0,43
+ 0,05
HK., Perth, Tokio, Wi., We.
30
20 »
3 14
III/
II«
— 9,54
+ 0,55
+ 0,45
+ 0,10
Taschk., Madras, HK., Perth, To
31
21 »
1 45
III/
In
+12,27
 0,12
— 0,13
+ 0,01
Madras, HK., P., Tokio.
32
21 »
1 46
IV/
In
+ 2,26
— 0,05
 0,12
+ 0,07
Madras, HK., P., Tokio.
33
21 »
1 52
IV/
III/
+ 2,33
— 0,05
— 0,12
+ 0,07
Madras, HK., P., Tokio.
34
21 »
3 23
11/
I»
+ 0,845
0,088
— 0,044
— 0,044
Taschk., Madr., HK., P., Tokio.
22 »]
12 19
IV ƒ
"ƒ
+ 6,62
— 0,39
 0,44
+ 0,05
Grw., Pulk., Harv., La PI., Rio.
35
15 13
id.gr.
dist.
8,W and 7.84
— 0.32
— 0,40
+ 0,08
Lick, FI., Tac, A.A., Harv., La PI
I
17 56
»
»
+ 8,88
— 0,38
— 0,44
+ 0,06
We., Lick., Fl., Tac, A. A., Harv
36
22 »
9 41
IV ƒ
III(o.e.)
+ 14,84
— 0,66
— 0,83
+ 0,17
Grw., Pulk , Kasan, La PI., Rio.
37
22 »
11 27
I(« e.)
II»
+ 0,01
— 0,32
 0,26
— 0,06
Grw., Pulk., (Kasan), La PI., Rio.
38
24 >
2 41
If
III»
+ 5,02
— 0,255
— 0,28
+ 0,025
Taschk., Madr., HK., P., To.
39
26 »
41
h.e.
II»
+ 6,03
— 0,33
— 0,25
— 0,08
HK., P., To., Wi.
40
27 »
7 5
Ill/
11/
— 9,62
+ 0,54
+ 0,45
+ 0,09
Bresl., Pulk., Kasan, Taschk.
41
28 »
4 28*
1/
III»
+ 1,57
— 0,08
— O,0j
+ 0,01
Kasan, Taschk., Madr., HK.
42
28 »
5 36
11/
I»
+ 0,58
— 0,03
— 0,065
+ 0,035
Kasan, Taschk., Madras.
43
28 »
13 29
IV ƒ
III»
+ 6,41
— 0,36
— 0,36
0,00
Grw., Tac, AA., Harv., La PL, Rio.
43*
28 »
16 18
II
III
11+7,31
III + 7,80
— 0,38
— 0,49
+ 0,06
.i—x' diminishes gradually in absolute value, reaches its minimum' 0,49 at the time assigned and then
increases again. So there is no eclipse.
Taschk., Madr., HK.
Tac, AA., Harv., La PI., Rio.
Lick, Fl., Tac, AA., Harv., La PI.
Lick, FL, Tac, AA.
Taschk., Madr., HK., P., Tokio.
Grw., Pulkowa.
Grw., Pulk., Kasan, Taschk.
Grw., Pulk., Kasan, (Taschk ).
Tac, A. A., Harv., La PL, Rio.
We., Lick, FL, Tac, AA.
To., Wi., We., Lick.
44
29
»
3 45
45
29
»
13 57
46
1 May 17 5
47
1
»
18 43
48
3
D
3 16
49
4
»
11 7
50
5
1)
7 48
51
5
»
8 17
5
»
14 2
52
5
»
17 57
5
»
21 39
1/
IV»
he.
II»
"ƒ
III»
11/
I»
I«...
II»
Ill/
II...».
11/
I»
III/
In
III/
11/
greatest
dist.
III/
11/
+ 0,27
— 0,02
— 0,08
+ 0,08
+ 6.05
 0,33
 0,26
— 07
— 0,74
 0,W
— 0,62
+ 0,61
+ 0,44
 0,07
— 0.02
 0.05
+ 6,06
 0,32*
— 0,24
 0,085
— 9.61
+ 0,54
+ 0,45
+ 0,09
+ 0,31
 0,07
— 0.Ü2
— 0,05
 0,10
 0,09
— 0,01
— 0,08
+ 4,63
 0,26
— 0,26
0,00
+6"64+6^85
— 038
— 0,36
+ 0,02
+ 8,43
— 0,42
 0,49
+ 0,07
( 458 )
No.
Mean time
at Greenwich
n = near
/•=far
o <u
bO^
.r" 0)
3^
" C3
o cs
W U3
W »
Visible at
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
9
10
10
11
11
12
12
12
13
13
14
14
14
15
15
15
16
17
17
17
18
80 18
81 120
» 16b38ni
» 6 27 .
Mei 6 37 5
» 7 4
» 20 5 't
» 23 25
» 5 29
» 6 O
» 16 41
» O 13
» 15 26
» 9 54
» 10 O
» 10 41
» 3 38
» 19 31
» 4 29
» 8 57
» 13 46
» 9 48
» 22 59
)) 23 7
H 11 59
. 9 4
M 1] 22
.) 16 9
)) 3 12
^) 20 36
» 8 8
!«.«.■
II«
IV„
I"
IVr
II«
IV ƒ
III«
11/
l„
I„
HI„
ïw.e.
III„
I«
Iln
(e.e.)
h
II«
In
II.
111/
Uw.e.
111/
ïn
H/
In
111/
"ƒ
111/
iiy
U
II«
In
II«
UU
IV„
(e.e.)
ï(w.e.)
II«
!ƒ
IV„
"ƒ
IV/
H/
I»
1/
IV„
I«
II»
III»
IV„
iw.e.)
u
II«
k«>.e.)
II«
111/
11/
111/
11/
+ 6'03
— 3,>7
— 3,18
+ 6,90
+ 0/18
— 2,04
— 5,36
+ 5,99
— 0,97
— 5,80
— 9,46
+ 0,14
+ 0,04
+ 0,58
+ 9,32
+ 5,89
+ 0,10
4 1517
— 5,93
+ 559
— 018
— 0,09
 5,69
+ 5,75
— 14,82
+ 1,22
— 6,00
— 8,935
+ 9,57
— 0>215
+ 0,20
+ 0,19
— 0,38
— 0,06
+ 0,12
+ 0,32
— 32
+ 0,05
+ 0,32
4 0,25
— 0,03
— O 38
— 0,10
— 0,55
— 0,33
— 0,05
— 0,87
+ 0,32
— 0,32
— 0,06
— 0,06
+ 0,30
— 0,31
+ 0,80
— 0,08^
+ 0,32
+ 0,51
— 0,56
— 0'24
+ 0r025
+ 0,18
+ 0,02
+ 0,20
— 0,01
— 0,29
— 0,09
 0,00
— 0,06
+ 0,13
— 0,01
+ 0,34
 0,02
— 0,38
+ 0,06
+ 0,10
— 0,05
+ 0,32
0,00
+ 0,43
 0,18
+ 0,01
— 0,04
+ 0,12
— 0,50
— 0,13
+ 0,03
— 0,45
— 0,10
— 0,25
 0,08
+ 0,02
— 0,07
 0,75
— 012
+ 0,33
— 01
 ü,29
— 0,03
— 0,02
— 0,U4
+ 0,02
— 0,08
+ 0,24
— 0,54
— 0.23
— 0,08
f 0,70
+ 0,10
 0,02
— 0,06^
+ 0,32
0,00
+ 0,41
+ 0,10
 0,45
— 0,11
L ck., Fl., Tac, AA., Harvard.
Bresl. Pulk.,Kasan,Taschk.,Maf1r.
Bresl. Pulk., KasanTaschk ,Madr.
Bresl., Pulk., Kasan,Taschk.,Madr.
Wi., Wellington.
Perth, Tokio, Windsor.
Kasan, Taschk., Madras.
Kasan, Taschk., Madras.
Lick., Fl., Tac, AA., Harvard.
IIK., P., Tokio.
Lick., FL, Tac, AA., Harvard.
Grw., Pulk., Ka^an, La PI., Rio.
Grw., Pulk., Kasan, La PI., Rio.
Grw., Pulk., La PI., Rio.
Taschk.. Madr., IIK.
We., Lick., Fl.
Taschk., Madras.
Grw., Pulk. Rio.
Tac, AA , Harv., La PI, Rio.
Grw., Pulk., Kasan, La PI , Rio.
Perth, To., Wi.
Perth, To, Wi.
Grw., La PI., Rio.
Grw., Pulk., Kasan, La PI., Rio.
Grw., Pulk., La PI., Rio.
Lick, Fl , Tac, AA., Harvard.
Taschk., Madras, UK.
Wi., Wellington.
Grw., Pulk., Kasan.
Plate II.
J. A. C. OÜDEMANS. 'Mutual occultations and eclipses of the satellites of Jupiter
in 1908." Second part: eclipses.
N. B. The continuous circles show the contour of the satellites, the dotted circles represent
the penumbra.
Scale 1 : 314 250 000.
In in a.
Uf in d. Ill/ in ƒ. IV/ in h.
12mm z^ 1" heliocentric.
ly in b.
Uf in (/ 111/ in /. IVy in //.
lly in d.
111/ in /. IV/ in h.
Totaal. Totaal.
Illn in e.
Un in c. 1/1 in a. 1/ in b. Uf in d.
IVy in h.
jUy^in^
IV/ in h.
U\n in ('. Un in c. In in a.
IV» in g
If in b. 11/ in d.
111/ in /.
Proceedings Royal Acad. Amsterdam. Vol. IX.
( 459 )
Physics. — ''Contribution to the knowledge of the xpsurface of
VAN DER Waals. X[. A gas that sinks in a liquid." By Prof.
H. Kamerlingh Onnes. (^omniuiiicatlon N". 96 from the
Physical Laboratory ot Leiden.
If we have an ideal gas and an incompressible liqnid withont
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. Everj^ experiment in which a gas is compressed
above a liquid, is practically an application of the theory of binary
mixtures of van der Waals. Li 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 gasphase
becomes comparable with that of the liquid phase.
If the theory of van der Waals 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 it
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 nol
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 Jzsurface construed with the unity of weight
the projection of the nodal chord on the zfplane runs parallel to the line v = 0.
30
Proceedings Royal Acad. Amsterdam. Vol. VIII.
( 460 )
In order to ascertain myself of this I compressed by 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 if'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 h of helium
must be small, from which follows again that a 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 a has really a positive value, whether it is
zero, or whether (what is also conceivable) a is negative, will have
to be decided by the determination of the isotherms of helium.
(December 21, 1906).
KONINKLIJKE AKADEMIE
VAN WETENSCHAPPEN
: TE AMSTERDAM :
PROCEEDINGS OF THE
SECTION OF SCIENCES
VOLUME IX
( — 1ST PART  )
JOHANNES MULLER :— : AMSTERDAM
! • nFCFMRFR 1Qn8: :
(Translated from: Verslagen van de Gewone Vergaderingen der Wis en Natuurkundige
Afdeeling van 26 Mei 1906 tot 24 November 1906. Dl. XV.)
Printed by
DE ROEVER KRÖBER & BAKELS
AMNH LIBRARY
100139142